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A Review on Turbine Trailing Edge Flow

A Review on Turbine Trailing Edge Flow International Journal of Turbomachinery Propulsion and Power Review 1 2 , Claus Sieverding and Marcello Manna * Turbomachinery and Propulsion Department, von Kàrmàn Institute for Fluid Dynamics, Chaussée de Waterloo 72, 1640 Rhode-St-Genèse, Belgium; sieverding@vki.ac.be Dipartimento di Ingegneria Industriale, Università degli Studi di Napoli Federico II, via Claudio 21, 80125 Napoli, Italy * Correspondence: marcello.manna@unina.it; Tel.: +39-081-768-3287 Received: 15 February 2020; Accepted: 11 May 2020; Published: 20 May 2020 Abstract: The paper presents a state-of-the-art review of turbine trailing edge flows, both from an experimental and numerical point of view. With the help of old and recent high-resolution time resolved data, the main advances in the understanding of the essential features of the unsteady wake flow are collected and homogenized. Attention is paid to the energy separation phenomenon occurring in turbine wakes, as well as to the e ects of the aerodynamic parameters chiefly influencing the features of the vortex shedding. Achievements in terms of unsteady numerical simulations of turbine wake flow characterized by vigorous vortex shedding are also reviewed. Whenever possible the outcome of a detailed code-to-code and code-to-experiments validation process is presented and discussed, on account of the adopted numerical method and turbulence closure. Keywords: turbine wake flow; vortex shedding; base pressure correlation; energy separation; numerical simulation 1. Introduction The first time the lead author came in touch with the problematic of turbine trailing edge flows was in 1965 when, as part of his diploma thesis, which consisted mainly in the measurement of the boundary layer development around a very large scale HP steam turbine nozzle blade, he measured with a very thin pitot probe a static pressure at the trailing edge significantly below the downstream static pressure. This negative pressure di erence explained the discrepancy between the losses obtained from downstream wake traverses and the sum of the losses based on the momentum thickness of the blade boundary layers and the losses induced by the sudden expansion at the trailing edge. Pursuing his curriculum at the von Kármán Institute the author was soon in charge of building a small transonic turbine cascade tunnel with a test section of 150 50 mm, the C2 facility, which was intensively used for cascade testing for industry and in-house designed transonic bladings for gas and steam turbine application. These tests allowed systematic measurements of the base pressure as part of the blade pressure distribution for a large number of cascades which were first presented at the occasion of a Lecture Series held at the von Kàrmàn Institute (VKI) in 1976 and led to the publication of the well-known VKI base pressure correlation published in 1980. This correlation has served ever since for comparison with new base pressure data obtained in other research labs. Among these let us already mention in particular the investigations carried out on several turbine blades at the University of Cambridge, published in 1988, at the University of Carlton, published between 2001 and 2004, and at the Moscow Power Institute, published between 2014 and 2018. In parallel to these steady state measurements, the arrival of short duration flow visualizations and the development of fast measurement techniques in the 1970’s allowed to put into evidence the existence of the von Kármán vortex streets in the wakes of turbine blades. Pioneering work was performed at the DLR Göttingen in the mid-1970’s, with systematic flow visualizations revealing the Int. J. Turbomach. Propuls. Power 2020, 5, 10; doi:10.3390/ijtpp5020010 www.mdpi.com/journal/ijtpp Int. J. Turbomach. Propuls. Power 2020, 5, 10 2 of 55 existence of von Kármán vortices on a large number of turbine cascades in the mid-seventies. This was the beginning of an intense research on the e ect of vortex shedding on the trailing edge base pressure. A major breakthrough was achieved in the frame of two European research projects. The first one, initiated in 1992, Experimental and Numerical Investigation of Time Varying Wakes Behind Turbine Blades (BRITE/EURAM CT92-0048, 1992–1996) included very large-scale cascade tests in a new VKI cascade facility with a much larger test section allowing the testing of a 280 mm chord blade in a three bladed cascade at a moderate subsonic Mach number, M = 0.4, with emphasis on flow visualizations and 2,is detailed unsteady trailing edge pressure measurements. The VKI tests were completed by low speed tests at the University of Genoa on the same large-scale profile for unsteady wake measurements using LDV. In the follow-up project Turbulence Modelling of Unsteady Flows on Flat Plate and Turbine Cascades in 1996 (BRITE/EURAM CT96-0143, 1996-1999) VKI extended the blade pressure measurements on a 50% reduced four bladed cascade model to a high subsonic Mach number, M = 0.79. Both programs not 2,is only contributed to an improved understanding of unsteady trailing edge wake flow characteristics, of their e ect on the rear blade surface and on the trailing edge pressure distribution, but also o ered unique test cases for the validation of unsteady Navier-Stokes flow solvers. A special and unexpected result of the research on unsteady turbine blade wakes was the discovery of energy separation in the wake leading to non-negligible total temperature variations within the wake. This e ect was known from steady state tests on cylindrical bodies since the early 1940’s, but its first discovery in a turbine cascade was made at the NRAC, National Research Aeronautical Laboratory of Canada, in the mid-1990s within the framework of tests on the performance of a nozzle vane cascade at transonic outlet Mach numbers. The experimental results of the total temperature distribution in the wake of cascade at supersonic outlet Mach number served many researchers, in particular from the University of Leicester, for elaborating on the e ect of energy separation. The paper starts with the evaluation of the VKI base pressure correlation (Section 2) in view of new experiments. This is followed with a review of the advances in the understanding of unsteady trailing edge wake flows (Section 3), the observation and explanation of energy separation in turbine blade wakes (Section 4), the e ect of vortex shedding on the blade pressure distribution (Section 5) and the e ect of Mach number and boundary layer state on the vortex shedding frequency (Section 6). This experimental part is complemented with a review of the numerical methods and modelling concepts as applied to the simulation of unsteady turbine wake characteristics using advanced Navier-Stokes solvers. Available numerical data documenting significant vortex shedding a ecting the turbine performance even in a time averaged sense, are collected and compared on a code-to-code and code-to-experiments basis in Section 7. 2. Turbine Trailing Edge Base Pressure Traupel [1], was probably the first to present in his book Thermische Turbomaschinen, a detailed analysis of the profile loss mechanism for turbine blades at subsonic flows conditions. The total losses comprised three terms: the boundary losses including the downstream mixing losses for infinitely thin trailing edges, the loss due to the sudden expansion at the trailing edge (Carnot shock) for a blade with finite trailing edge thickness d taking into account the trailing edge blockage e ect and a third term te which did take into account that the static pressure at the trailing edge di ered from the average static pressure between the pressure side (PS) and the suction side (SS) trailing edges across one pitch. Thus, the profile loss coecient  reads: 0 1 B C B te C B C = 2 Q + B C sin ( ) + k d (1) p 2 te @ A 1 d te where: Q + Q ss ps Q = ( ) g sin 2 Int. J. Turbomach. Propuls. Power 2020, 5, 10 3 of 55 is the dimensionless average momentum thickness, and: te d = te g sin( ) the dimensionless thickness of the trailing edge. The constant k appearing at the right-hand-side of Equation (1) depends on the ratio: te d = te Q + Q ss ps that is, k = 0.1 for d = 2.5 and k = 0.2 for d = 7, while a linear variation of k is used for 2.5 < d < 7. te te te Terms containing squares and products of Q + Q /d were considered to be negligible. ss ps te Most researchers are, however, more familiar with a similar analysis of the loss mechanism by Denton [2], who introduced in the loss coecient expression  , the term cp d quantifying the trailing p b te edge base pressure contribution, with: p p 2 b cp = (2) 1/2V re f For commodity may be taken as the isentropic downstream velocity . However, there For commodity V may be taken as the isentropic downstream velocity V , . However, there re f 2,is was a big uncertainty as regards the magnitude of this term, although it appeared that it could was a big uncertainty as regards the magnitude of this term, although it appeared that it could become become very important in the transonic range and explain the presence of a strong local loss very important in the transonic range and explain the presence of a strong local loss maximum as maximum as demonstrated in Figure 1, which presents a few examples of early transonic cascades demonstrated in Figure 1, which presents a few examples of early transonic cascades measurements measurements performed at VKI and the DLR. (performance of VKI blades B and C are unpublished). performed at VKI and the DLR. Pioneering experimental research concerning the evolution of the turbine trailing edge base Pioneering experimental research concerning the evolution of the turbine trailing edge base pressure from subsonic to supersonic outlet flow conditions was carried out at the von Kármán pressure from subsonic to supersonic outlet flow conditions was carried out at the von Kármán Institute. Institute. In 1976, at the occasion of the VKI Lecture Series Transonic Flows in Axial Turbines, In 1976, at the occasion of the VKI Lecture Series Transonic Flows in Axial Turbines, Sieverding presented Sieverding presented base pressure data for eight different cascades for gas and steam turbine blade base pressure data for eight di erent cascades for gas and steam turbine blade profiles over a wide profiles over a wide range of Mach numbers [3] and in 1980 Sieverding et al. [4] published a base range of Mach numbers [3] and in 1980 Sieverding et al. [4] published a base pressure correlation (also pressure correlation (also referred to as BPC) based on a total of 16 blade profiles. referred to as BPC) based on a total of 16 blade profiles. Blade g/c Ref A 30° 22° 0.75 [5] B 60° 25° 0.75 VKI C 66° 18° 0.70 VKI D 156° 19.5° 0.85 [6] Figure 1. Blade profile losses versus isentropic outlet Mach number for four transonic turbines. Blade A Figure 1. Blade profile losses versus isentropic outlet Mach number for four transonic turbines. Blade data from [5], blade B and C unpublished data from VKI, blade D data from [6]. A data from [5], blade B and C unpublished data from VKI, blade D data from [6]. Int. J. Turbomach. Propuls. Power 2020, 5, 10 4 of 55 All tests were performed with cascades containing typically 8 blades and care was taken to ensure in all cases, and over the whole Mach range, a good periodicity. The latter was quantified to be 3%, in the supersonic range, in terms of the maximum di erence between the pitch-wise averaged Mach number (based on 10 wall pressure tappings per pitch) of each of the three central passages and the mean value computed over the same three passages. The correlation covered blades with a wide range of cascade parameters, as outlined in Table 1: Table 1. Parameters range for Sieverding’s correlation. Parameter Symbol Value Pitch to Chord Ratio g/c 0.32–0.84 Trailing edge thickness to throat ratio d /o 0.04–0.16 te Inlet flow angle 45 –156 Outlet flow angle 18 –34 Trailing edge wedge angle  2 –16 te Rear suction side turning angle " 0 –18 Of all cascade parameters only the rear suction side turning angle " and the trailing edge wedge angle  appeared to correlate convincingly the available data, although the latter were insucient to te di erentiate their respective influence. In fact, in many blade designs both parameters are closely linked to each other and, for two thirds of all convergent blades with convex rear suction side, both " and te were of the same order of magnitude. For this reason, it was decided to use the mean value (" +  )/2 te as parameter. The relation p /p = f(p /p ), is graphically presented in Figure 2. The curves cover b 01 s2 01 a range from M  0.6 to M  1.5, but flow conditions characterized by a suction side shock 2,is 2,is interference with the trailing edge wake region are not considered. Comparing the experiments with the correlation (results not shown herein), it turned out that 80% of all data fall within a bandwidth Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 5 of 65 5% and 96% within10%. Figure 2. Sieverding’s base pressure correlation; solid lines (resp. dashed lines) denote convergent 129 Figure 2. Sieverding’s base pressure correlation; solid lines (resp. dashed lines) denote convergent blades (resp. convergent-divergent blades) [4]. 130 blades (resp. convergent-divergent blades) [4]. An explanation for the significance of " for the trailing edge base pressure is seen in Figure 3, 131 An explanation for the significance of for the trailing edge base pressure is seen in Figure 3, presenting the blade velocity distribution for two convergent blades with di erent rear suction side 132 presenting the blade velocity distribution for two convergent blades with different rear suction side turning angles of " = 20 and 4.5 , blade A and B, together with a convergent/divergent blade with an 133 turning angles of = 20° and 4.5°, blade A and B, together with a convergent/divergent blade with internal passage area increase of A/A = 1.05, blade C. The curves end at x/c = 0.95 because beyond, 134 an internal passage area increase of / =1.05, blade C. The curves end at / = 0.95 because the pressure distribution is influenced by the acceleration around the trailing edge. 135 beyond, the pressure distribution is influenced by the acceleration around the trailing edge. 136 The rear suction side turning angle ε has a remarkable effect on the pressure difference across 137 the blade near the trailing edge. For blade A one observes a strong difference between the SS and PS 138 isentropic Mach numbers, respectively pressures, while the difference is very small for blade B. On 139 the contrary, for blade C the pressure side curve crosses the SS curve well ahead of the trailing edge 140 and the PS isentropic Mach number near the trailing edge exceeds considerably that of the SS. The 141 base pressure is function of the blade pressure difference upstream of the trailing edge. Blade A Convergent blade ε = 20° Blade B Convergent blade ε = 4.5° Blade C Convergent divergent blade (A/A*) = 1.05 142 Figure 3. Surface isentropic Mach number distribution for two convergent and one 143 convergent/divergent blades at =0.9, based on data from [3]. 144 It is also worthwhile mentioning that also plays an important role for the optimum blade 145 design in function of the outlet Mach number. Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 5 of 65 129 Figure 2. Sieverding’s base pressure correlation; solid lines (resp. dashed lines) denote convergent 130 blades (resp. convergent-divergent blades) [4]. 131 An explanation for the significance of for the trailing edge base pressure is seen in Figure 3, 132 presenting the blade velocity distribution for two convergent blades with different rear suction side 133 turning angles of = 20° and 4.5°, blade A and B, together with a convergent/divergent blade with Int. J. Turbomach. Propuls. Power 2020, 5, 10 5 of 55 134 an internal passage area increase of / =1.05, blade C. The curves end at / = 0.95 because 135 beyond, the pressure distribution is influenced by the acceleration around the trailing edge. 136 The rear suction side turning angle ε has a remarkable effect on the pressure difference across The rear suction side turning angle " has a remarkable e ect on the pressure di erence across 137 the blade near the trailing edge. For blade A one observes a strong difference between the SS and PS the blade near the trailing edge. For blade A one observes a strong di erence between the SS and PS 138 isentropic Mach numbers, respectively pressures, while the difference is very small for blade B. On isentropic Mach numbers, respectively pressures, while the di erence is very small for blade B. On the 139 the contrary, for blade C the pressure side curve crosses the SS curve well ahead of the trailing edge contrary, for blade C the pressure side curve crosses the SS curve well ahead of the trailing edge and 140 and the PS isentropic Mach number near the trailing edge exceeds considerably that of the SS. The the PS isentropic Mach number near the trailing edge exceeds considerably that of the SS. The base 141 base pressure is function of the blade pressure difference upstream of the trailing edge. pressure is function of the blade pressure di erence upstream of the trailing edge. Blade A Convergent blade ε = 20° Blade B Convergent blade ε = 4.5° Blade C Convergent divergent blade (A/A*) = 1.05 Figure 3. Surface isentropic Mach number distribution for two convergent and one convergent/divergent 142 Figure 3. Surface isentropic Mach number distribution for two convergent and one blades at M = 0.9, based on data from [3]. 143 convergent/divergent blades at =0.9, based on data from [3]. 2,is It is also worthwhile mentioning that " also plays an important role for the optimum blade design 144 It is also worthwhile mentioning that also plays an important role for the optimum blade in function of the outlet Mach number. Figure 4 presents design recommendations for the rear suction 145 design in function of the outlet Mach number. side curvature with increasing Mach number from subsonic to low supersonic Mach numbers as successfully used at VKI. (a) (b) (c) (d) Figure 4. Recommended values of " (a) and l/L (b) for the design of the blade rear suction side for increasing outlet Mach numbers. Full (c) and close-up (d) view of the parameters characterizing the turbine geometry. Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 6 of 58 (a) (b) Int. J. Turbomach. Propuls. Power 2020, 5, 10 6 of 55 The rear suction turning angle " for convergent blades should decrease with increasing Mach number reaching a minimum of 4 at M  1.3 (maximum Mach number for convergent blades). 2,is Note that similar trends can be derived from the loss correlation by Craig and Cox [7]. They showed that in order to minimize the blade profile losses the rear suction side curvature, expressed by the ratio g/e, where g represents the pitch(c and) (d) e the radius of a circular arc appr oximating the rear suction side curvature, should decrease with increasing Mach number. 154 Figure 4. Recommended values of (a) and / (b) for the design of the blade rear suction side for Commented [M17]: Please add explanation for subgraph and 155 increasing outlet Mach numbers. Full (c) and close-up (d) view of the parameters characterizing the For a given rear suction side angle " the designer is free as regards the evolution of the surface add “a,b,c,d” in the figure. Please unify the all figures which 156 turbine geometry. angle from the throat to the trailing edge. It appears to be a good design practice to subdivide the rear have subgraph like figure 51. 157 For a given rear suction side angle ε the designer is free as regards the evolution of the surface suction side length L into two parts, a first part along which the blade angle asymptotically decreases to Commented [MM18R17]: Done. 158 angle from the throat to the trailing edge. It appears to be a good design practice to subdivide the the value of the trailing edge angle, followed by a second entirely straight part of length l, see Figure 4. 159 rear suction side length into two parts, a first part along which the blade angle asymptotically With increasing outlet Mach number, the length of the straight part, that is the ratio l/L increases, but it 160 decreases to the value of the trailing edge angle, followed by a second entirely straight part of length 161 , see Figure 4. With increasing outlet Mach number, the length of the straight part, that is the ratio does never extend up to the throat. 162 / increases, but it does never extend up to the throat. For calculating the trailing edge losses induced by the di erence between the base pressure and 163 For calculating the trailing edge losses induced by the difference between the base pressure and the downstream pressure, Fabry & Sieverding [8], presented the data for the convergent blades in 164 the downstream pressure, Fabry & Sieverding [8], presented the data for the convergent blades in 165 Figure 2 in terms of the base pressure coefficient , defined by Equation (2), see Figure 5. Figure 2 in terms of the base pressure coecient cp , defined by Equation (2), see Figure 5. 167 Figure 5. Base Figupr re 5. essur Base e prcoe essu re c cients oefficiecorr nts coesponding rresponding toto the the base base pressu pr re c essur urves of Fi e curves gure 2 of [8]. Figure 2 [8]. Since the base pressure losses are proportional to the base pressure coecient cp , the curves give immediately an idea of the strong variation of the profile losses in the transonic range. As regards the low Mach number range, the contribution of the base pressure loss is implicitly taken into account by all loss correlations. Therefore the base pressure loss is not to be added straight away to the profile losses as predicted for example with the methods by Traupel [1] and Craig and Cox [7] but rather as a di erence with respect to the profile losses at M = 0.7: 2,is te = cp cp bp b b,M =0.7 2,is g sin( ) Martelli and Boretti [9], used the VKI base pressure correlation for verifying a simple procedure to compute losses in transonic turbine cascades. The surface static pressure distribution for a given downstream Mach number is obtained from an inviscid time marching flow calculation. An integral boundary layer calculation is used to calculate the momentum thickness at the trailing edge before separation. The trailing edge shocks are calculated using the base pressure correlation. Two examples are shown in Figure 6. Calculation of eight blades showed that 80% of the predicted losses were within the range of the experimental uncertainty. Besides the data reported by Sieverding et al. in [4,6], the only authors who published recently a systematic investigation of the e ect of the rear suction side curvature on the base pressure were Granovskij et al. Of the Moscow Power Institute [10]. The authors investigated 4 moderately loaded rotor blades (g/c = 0.73, d /o = 0.12,  85,  22) with di erent unguided turning angles (" = 2 te 1 2 to 16 ) in the frame of the optimization of cooled gas turbine blades. A direct comparison with the VKI base pressure correlation is dicult because the authors omitted to indicate the trailing edge wedge Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 7 of 58 168 Since the base pressure losses are proportional to the base pressure coefficient , the curves Int. J. Turbomach. Propuls. Power 2020, 5, 10 7 of 55 169 give immediately an idea of the strong variation of the profile losses in the transonic range. As regards 170 the low Mach number range, the contribution of the base pressure loss is implicitly taken into account 171 by all loss correlations. Therefore the base pressure loss is not to be added straight away to the profile 172 losses as predicted for example with the methods by Traupel [1] and Craig and Cox [7] but rather as angle  . Nevertheless, a comparison appeared to be useful. Figure 7 presents the comparison, after te 173 a difference with respect to the profile losses at =0.7: conversion, of the base pressure coecient: Commented [M19]: Please confirm and relayout the = − , . ( ) sin p p b 2 number of equation in order.. 174 Martelli and Boretti [9], used the VKcp I base pres = sure correlation for verifying a simple procedure p p 02 2 175 to compute losses in transonic turbine cascades. The surface static pressure distribution for a given Commented [MM20R19]: No need to add an eq. 176 downstream Mach number is obtained from an inviscid time marching flow calculation. number here, and wherever it was not inserted in the used by Granovskij et al. [10], to the base pressure coecient (2) based on V , used by Fabry and 177 An integral boundary layer calculation is used to calculate the momentum thi2, cknes is s at the original manuscript Sieverding at VKI [8]. The data of Granovskij et al. [10] (dashed lines) confirm globally the overall 178 trailing edge before separation. The trailing edge shocks are calculated using the base pressure 179 correlation. Two examples are shown in Figure 6. Calculation of eight blades showed that 80% of the trends of the VKI base pressure correlation (solid lines). However, the peaks in the transonic range are 180 predicted losses were within the range of the experimental uncertainty. much more pronounced. (a) (b) 182 Figure 6. Example of profile loss prediction for transonic turbine cascade, adapted from [9]; (a) low Commented [M21]: Please add explanation for subgraph Figure 6. Example of profile loss prediction for transonic turbine cascade, adapted from [9]; (a) low 183 pressure steam turbine tip section, (b) high pressure gas turbine guide vane. Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 8 of 58 pressure steam turbine tip section, (b) high pressure gas turbine guide vane. Commented [MM22R21]: Done 184 Besides the data reported by Sieverding et al. in [6] and [4], the only authors who published 185 recently a systematic investigation of the effect of the rear suction side curvature on the base pressure 186 were Granovskij et al. of the Moscow Power Institute [10]. The authors investigated 4 moderately 187 loaded rotor blades ( =0.73, / = 0.12 , ≈85°, ≈22°) with different unguided turning 188 angles (= 2° to 16°) in the frame of the optimization of cooled gas turbine blades. A direct 189 comparison with the VKI base pressure correlation is difficult because the authors omitted to indicate 190 the trailing edge wedge angle . Nevertheless, a comparison appeared to be useful. Figure 7 191 presents the comparison, after conversion, of the base pressure coefficient: 192 used by Granovskij et al. [10], to the base pressure coefficient (2) based on , used by Fabry and 193 Sieverding at VKI [8]. The data of Granovskij et al. [10] (dashed lines) confirm globally the overall 194 trends of the VKI base pressure correlation (solid lines). However, the peaks in the transonic range 195 are much more pronounced. 196 Figure 7. Comparison of Granovskij’s base pressure data (dashed lines) with the VKI base pressure Figure 7. Comparison of Granovskij’s base pressure data (dashed lines) with the VKI base pressure 197 correlation (solid lines). correlation (solid lines). 198 Also, cascade data reported by Dvorak et al. in 1978 [11] on a low pressure steam turbine rotor Also, cascade data reported by Dvorak et al. In 1978 [11] on a low pressure steam turbine rotor 199 tip section, and by Jouini et al. in 2001 [12] for a relatively high turning rotor blade (∆ = 110° , and a 200 smaller pitch to chord ratio / = 0.73 ), are in fair agreement with the VKI base pressure correlation, tip section, and by Jouini et al. In 2001 [12] for a relatively high turning rotor blade (D = 110, and a 201 although the latter authors state that below / =0.45, their data drop below those of the BPC. smaller pitch to chord ratio g/c = 0.73), are in fair agreement with the VKI base pressure correlation, 202 However, some other cascade measurements deviate very significantly from the VKI curves. although the latter authors state that below p /p = 0.45, their data drop below those of the BPC. 203 Deckers and Denton [13], for a low turning blade model and Gostelow et al. [14] for a high turning 2 01 204 nozzle guide vane, report base pressure data far below those of Sieverding’s BPC, while Xu and However, some other cascade measurements deviate very significantly from the VKI curves. 205 Denton [15], for a very highly loaded HP gas turbine rotor blade (∆ = 124° and / = 0.84 ) report Deckers and Denton [13], for a low turning blade model and Gostelow et al. [14] for a high turning 206 base pressure data far above those of the BPC. nozzle guide vane, report base pressure data far below those of Sieverding’s BPC, while Xu and 207 The simplicity of Sieverding’s base pressure correlation was often criticized because it was felt 208 that aspects as important as the state of the boundary layer, the ratio of boundary layer momentum Denton [15], for a very highly loaded HP gas turbine rotor blade (D = 124 and g/c = 0.84) report 209 to trailing edge thickness and the trailing edge blockage effects (trailing edge thickness to throat base pressure data far above those of the BPC. 210 opening) should play an important role. The simplicity of Sieverding’s base pressure correlation was often criticized because it was felt 211 As regards the state of the boundary layer and its thickness, tests on a flat plate model at 212 moderate subsonic Mach numbers in a strongly convergent channel by Sieverding and Heinemann that aspects as important as the state of the boundary layer, the ratio of boundary layer momentum to 213 [16], showed that the difference of the base pressure for laminar and turbulent flow conditions was trailing edge thickness and the trailing edge blockage e ects (trailing edge thickness to throat opening) 214 only of the order of 1.5–2% of the dynamic head of the flow before separation from the trailing edge. should215 play an For t important he case of sup role. ersonic trailing edge flows, Carriere [17], demonstrated, that for turbulent 216 boundary layers the base pressure would increase with increasing momentum thickness. On the 217 contrary, supersonic flat plate model tests simulating the overhang section of convergent turbine 218 cascades with straight rear suction sides showed that for fully expanded flow along the suction side 219 (limit loading condition) an increase of the ratio of the boundary layer momentum to the trailing edge 220 thickness by a factor of two, obtained roughening the blade surface, did not affect the base pressure, 221 Sieverding et al. [18]. Note, that for both the smooth and rough surface the boundary layer was 222 turbulent. Similarly, roughening the blade surface in case of shock boundary layer interactions on the 223 blade suction side did not affect the base pressure as compared to the smooth blade, Sieverding and 224 Heinemann [16]. However, a comparison of the base pressure for the same Mach numbers before 225 separation at the trailing edge for a fully expanding flow and a flow with shock boundary layer 226 interaction on the suction side before the TE showed an increase of the base pressure by 10–25 % in 227 case of shock interaction before the TE. Since it was shown before that an increase of the momentum 228 thickness did not affect the base pressure, the difference may be attributed to (a) different total 229 pressures due to shock losses for the shock interference curve, (b) differences in the boundary layer 230 shape factor and (c) differences in pressure gradients in stream-wise direction in the near wake 231 region. 232 A systematic investigation of possible effects of changes in shape factor and boundary layer 233 momentum thickness on the base pressure in cascades is difficult. Hence, the investigations are Int. J. Turbomach. Propuls. Power 2020, 5, 10 8 of 55 As regards the state of the boundary layer and its thickness, tests on a flat plate model at moderate subsonic Mach numbers in a strongly convergent channel by Sieverding and Heinemann [16], showed that the di erence of the base pressure for laminar and turbulent flow conditions was only of the order of 1.5–2% of the dynamic head of the flow before separation from the trailing edge. For the case of supersonic trailing edge flows, Carriere [17], demonstrated, that for turbulent boundary layers the base pressure would increase with increasing momentum thickness. On the contrary, supersonic flat plate model tests simulating the overhang section of convergent turbine cascades with straight rear suction sides showed that for fully expanded flow along the suction side (limit loading condition) an increase of the ratio of the boundary layer momentum to the trailing edge thickness by a factor of two, obtained roughening the blade surface, did not a ect the base pressure, Sieverding et al. [18]. Note, that for both the smooth and rough surface the boundary layer was turbulent. Similarly, roughening the blade surface in case of shock boundary layer interactions on the blade suction side did not a ect the base pressure as compared to the smooth blade, Sieverding and Heinemann [16]. However, a comparison of the base pressure for the same Mach numbers before separation at the trailing edge for a fully expanding flow and a flow with shock boundary layer interaction on the suction side before the TE showed an increase of the base pressure by 10–25 % in case of shock interaction before the TE. Since it was shown before that an increase of the momentum thickness did not a ect the base pressure, the di erence may be attributed to (a) di erent total pressures due to shock losses for the shock interference curve, (b) di erences in the boundary layer shape factor and (c) di erences in pressure gradients in stream-wise direction in the near wake region. A systematic investigation of possible e ects of changes in shape factor and boundary layer momentum thickness on the base pressure in cascades is dicult. Hence, the investigations are mostly confined to variations of the incidence angle which, via a modification of the blade velocity distribution, should have an impact on both the shape factor and the boundary layer momentum thickness. Based on linear transonic cascade tests on two high turning rotor blades Jouini et al. [19] at Carlton University, (blade HS1A: g/c = 0.73, d /o = 0.082, = 39.5, = 31,  = 6, " = 11.5; blade HS1B is similar to te 1 2 te HS1A, but with less loading on the front side and = 29) concluded that discrepancies in the base region did not appear to be strongly related to changes of the inlet angle by14.5, however in broad terms the weakest base pressure drop in the transonic range were obtained for high positive incidence. Similarly, experiments at VKI on a high turning rotor blade (g/c = 0.49, d /o = 0.082, = 45, = 28, te 1 2 = 10, " = 10) did not show any e ect on the base pressure for incidence angle changes of10 [3]. te In conclusion it appears that for conventional blade designs, changes in the boundary layer thickness alone, as induced by incidence variations, do not a ect significantly the base pressure. Therefore, we need to look for possible other influence factors. Figure 3 showed that the e ect of the blade rear suction side blade turning angle " on the base pressure was in fact function of the pressure di erence across the blade near the trailing edge. Inversely, one should be able to deduct from the rear blade loading the tendency of the base pressure. The higher the blade loading at the trailing edge, the higher the base pressure. Corollary, a low or even negative blade loading near the trailing edge causes increasingly lower base pressures. This might help in explaining the large di erences with respect to the BPC as found by Xu and Denton [15] on one side and Deckers et al. [13] and Gostelow et al. [14], mentioned before, on the other side. To illustrate this, Figure 8 presents the base pressure data of Xu and Denton [15] for three of a family of four very highly loaded gas turbine rotor blades with a blade turning angle of D = 124 and a pitch-to-chord g/c = 0.84, tested with three di erent trailing edge thicknesses. The blades are referred to as blade RD, for the datum blade, and blades DN and DK for changes of 0.5 and 1.5 times the trailing edge thickness with respect to the datum case. The base pressures are overall much higher than those of the BPC which are indicated in the figure by the dashed line for a mean value of (" +  )/2 = 9. te A possible explanation for the large di erences is given by comparing the blade Mach number distribution of the datum blade with that of a VKI blade with a (" +  )/2 = 16 taken from [6], te Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 9 of 58 234 mostly confined to variations of the incidence angle which, via a modification of the blade velocity 235 distribution, should have an impact on both the shape factor and the boundary layer momentum 236 thickness. Based on linear transonic cascade tests on two high turning rotor blades Jouini et al. [19] 237 at Carlton University, (blade HS1A: / =0.73, / = 0.082, = 39.5° , = 31° , =6°, = 238 11.5°; blade HS1B is similar to HS1A, but with less loading on the front side and = 29°) concluded 239 that discrepancies in the base region did not appear to be strongly related to changes of the inlet angle 240 by 14.5° , however in broad terms the weakest base pressure drop in the transonic range were 241 obtained for high positive incidence. Similarly, experiments at VKI on a high turning rotor blade 242 (/ =0.49, / = 0.082, =45°, = 28°, = 10°, = 10°) did not show any effect on the base 243 pressure for incidence angle changes of 10° [3]. 244 In conclusion it appears that for conventional blade designs, changes in the boundary layer 245 thickness alone, as induced by incidence variations, do not affect significantly the base pressure. 246 Therefore, we need to look for possible other influence factors. 247 Figure 3 showed that the effect of the blade rear suction side blade turning angle ε on the base 248 pressure was in fact function of the pressure difference across the blade near the trailing edge. 249 Inversely, one should be able to deduct from the rear blade loading the tendency of the base pressure. 250 The higher the blade loading at the trailing edge, the higher the base pressure. Corollary, a low or 251 even negative blade loading near the trailing edge causes increasingly lower base pressures. This 252 might help in explaining the large differences with respect to the BPC as found by Xu and Denton 253 [15] on one side and Deckers et al. [13] and Gostelow et al. [14], mentioned before, on the other side. 254 To illustrate this, Figure 8 presents the base pressure data of Xu and Denton [15] for three of a 255 family of four very highly loaded gas turbine rotor blades with a blade turning angle of ∆ = 124° 256 and a pitch-to-chord / = 0.84 , tested with three different trailing edge thicknesses. The blades are 257 referred to as blade RD, for the datum blade, and blades DN and DK for changes of 0.5 and 1.5 times 258 the trailing edge thickness with respect to the datum case. Int. J. Turbomach. Propuls. Power 2020, 5, 10 9 of 55 259 The base pressures are overall much higher than those of the BPC which are indicated in the 260 figure by the dashed line for a mean value of ( + )/2 = 9°. 261 A possible explanation for the large differences is given by comparing the blade Mach number 262 distribution of the datum blade with that of a VKI blade with a ( + )/2 = 16° taken from [6], see see Figure 9. To enable the comparison, the blade Mach number distribution of Xu & Denton (solid line) 263 Figure 9. To enable the comparison, the blade Mach number distribution of Xu & Denton (solid line) presented originally in function of the axial chord x/c , had to be replotted in function of x/c. ax 264 presented originally in function of the axial chord / , had to be replotted in function of / . The The comparison is done for an isentropic outlet Mach number M = 0.8. 265 comparison is done for an isentropic outlet Mach number =0.8. , 2,is Commented [M23]: Please add explanation for subgraph Figure 267 8. Base Figurpr e 8. essur Base pre e vssu ariation re variatifor on for bl blades ades of of Xu & Xu & Den Denton; ton; blade R blade D datum RD cas datum e, blade D case, K thicblade k DK thick 268 trailing edge, blade DN thin trailing edge. Adapted from [15]. trailing edge, blade DN thin trailing edge. Adapted from [15]. Commented [MM24R23]: This is not a subgraph, and the geometries are well explained by the caption. Letter 269 Note that the geometric throat for the Xu & Denton blade is situated at / ≈ 0.34 , while for the Note that the geometric throat for the Xu & Denton blade is situated at x/c  0.34, while for the 270 VKI blade at / = 0.5 . At the trailing edge, the Mach number difference between pressure and referencing is inappropriate. VKI blade at x/c = 0.5. At the trailing edge, the Mach number di erence between pressure and suction Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 10 of 58 side for both blades are exactly the same, but contrary to the nearly constant Mach number for the VKI blade downstream of the throat, the blade of Xu and Denton is characterized by a very strong adverse 271 suction side for both blades are exactly the same, but contrary to the nearly constant Mach number 272 for the VKI blade downstream of the throat, the blade of Xu and Denton is characterized by a very pressure gradient in this region. As pointed out by the authors, this causes the suction side boundary 273 strong adverse pressure gradient in this region. As pointed out by the authors, this causes the suction layer to be either separated or close to separation up-stream of the trailing edge. Clearly, Sieverding’s 274 side boundary layer to be either separated or close to separation up-stream of the trailing edge. correlation cannot deal with blade designs characterized by very strong adverse pressure gradients on 275 Clearly, Sieverding’s correlation cannot deal with blade designs characterized by very strong adverse 276 pressure gradients on the rear suction side causing boundary layer separation before the trailing the rear suction side causing boundary layer separation before the trailing edge. 277 edge. Figure 9. Comparison of blade Mach number distribution for blade RD of Xu and Denton [15], 278 Figure 9. Comparison of blade Mach number distribution for blade RD of Xu and Denton [15], (solid 5 6 279 curve, =0.8, =8 10 ) with VKI blade (dashed curve, =0.8, =10 ). (solid curve, M = , 0.8, Re = 8 10 ) with VKI blade (dashed , curve, M = 0.8, Re = 10 ). Commented [M25]: Please add explanation for subgraph. 2,is 2 2,is 2 280 The possible effect of boundary layer separation resulting from high rear suction side diffusion Commented [MM26R25]: This is not a subgraph, and the The possible e ect of boundary layer separation resulting from high rear suction side di usion 281 resulting in high base pressures was also mentioned by Corriveau and Sjolander in 2004 [20], geometries are well explained by the caption. Letter resulting in high base pressures was also mentioned by Corriveau and Sjolander in 2004 [20], comparing 282 comparing their nominal mid-loaded rotor blade HS1A, mentioned already before, with an aft- referencing is inappropriate. 283 loaded blade HS1C with an increase of the suction side unguided turning angle from 11.5° to 14.5°. their nominal mid-loaded rotor blade HS1A, mentioned already before, with an aft- loaded blade HS1C 284 It appears that the increased turning angle could cause, in the transonic range, shock induced with an increase of the suction side unguided turning angle from 11.5 to 14.5 . It appears that the 285 boundary layer transition near the trailing edge with, as consequence, a sharp increase of the base increased 286 turning pressure, i.e angle . a sudde could n cause, drop in tin he b the ase p transonic ressure coeff range, icient as seen shock in induced Figure 10. N boundary ote that the layer transition 287 reported in the figure has been converted to − of the original data. near the trailing edge with, as consequence, a sharp increase of the base pressure, i.e., a sudden drop in 288 As regards the base pressure data by Deckers and Denton [13] for a low turning blade model the base pressure coecient as seen in Figure 10. Note that the cp reported in the figure has been 289 and Gostelow et al. [14] for a high turning nozzle guide vane, who report base pressure data far below converted 290 to those o cp of f Si the everdi original ng’s BPC data. , their blade pressure distribution resembles that of the convergent/ 291 divergent blade C in Figure 3 with a negative blade loading near the trailing edge which would As regards the base pressure data by Deckers and Denton [13] for a low turning blade model and 292 explain the very low base pressures. In addition, the blade of Deckers and Denton has a blunt trailing Gostelow et al. [14] for a high turning nozzle guide vane, who report base pressure data far below 293 edge, and there is experimental evidence that, compared to a circular trailing edge, the base pressure those of 294 Siever for ding’s blades with BPC, blunt their trablade iling edpr ge mi essur ght be co e distribution nsiderably lo rwer. esembles Sieverding an that of d Hei thenem conver ann [16] gent /divergent 295 report for flat plate tests at moderate subsonic Mach numbers a drop of the base pressure coefficient blade C in Figure 3 with a negative blade loading near the trailing edge which would explain the very 296 by 11% for a plate with squared trailing edge compared to that with a circular trailing edge. low base pressures. In addition, the blade of Deckers and Denton has a blunt trailing edge, and there is This image cannot currently be displayed. Commented [M27]: Please add explanation for subgraph. Commented [MM28R27]: This is not a subgraph, and the 297 Figure 10. Base pressure coefficient for mid-loaded (solid line) and aft-loaded (dashed line) rotor geometries are well explained by the caption. Letter 298 blade. Symbols: HS1A geometry, HS1C geometry. Adapted from [20]. referencing is inappropriate Int. J. Turbo Int. J. Turbo m m ach. ach. Propuls. Power Propuls. Power 2018 2018 , , 33 , x FOR PE , x FOR PE ER RE ER RE VI VI EE W W 10 of 10 of 58 58 271 271 suction suction side side for both blade for both blade ss are ex are ex actly actly t t hh e same, e same, bb uu t t contrary to th contrary to th e nearly constant Mach e nearly constant Mach number number 272 for the VKI blade downstream of the throat, the blade of Xu and Denton is characterized by a very 272 for the VKI blade downstream of the throat, the blade of Xu and Denton is characterized by a very 273 273 strong adver strong adver ss e pressure gr e pressure gr adient in this adient in this region. region. As As poi poi nn ted out by the a ted out by the a u u thors, thi thors, thi ss causes the sucti causes the sucti oo n n 274 side boundary layer to be either separated or close to separation up-stream of the trailing edge. 274 side boundary layer to be either separated or close to separation up-stream of the trailing edge. 275 275 C C lear lear ly, ly, Si Si ev ev ee rr ding’ ding’ ss cor cor rr el el at at ion cann ion cann ot ot dea dea l wit l wit hh b b la la d d e des e des igns ch igns ch ar ar act act ee ri ri zed b zed b yy v v ee ry st ry st rong rong adv adv ee rs rs e e 276 pressure gradients on the rear suction side causing boundary layer separation before the trailing 276 pressure gradients on the rear suction side causing boundary layer separation before the trailing 277 277 edge. edge. 278 278 Figure 9. Figure 9. Comparison of bla Comparison of bla dd e Mach number distri e Mach number distri bution f bution f oo r blade RD of r blade RD of Xu and Denton Xu and Denton [ [ 15], ( 15], ( ss olid olid 279 279 cu cu rve, rve, =0 =0.8.8, , =8 =8 1 100) with V ) with V K K I bla I bla dd e e (dash (dash ee d curve d curve , , =0 =0.8.8, , =1 =100). ). , , , , 280 280 The possib The possib le le effect o effect o f boun f boun dary dary la la yer yer se se paration paration res res uu lting lting from h from h igh re igh re ar s ar s uu ctio ctio n sid n sid ee d d iffus iffus ion ion 281 281 resul resultitn ing i g inn hi high b gh baase pressures was se pressures was aalso ment lso mentioioned by Corri ned by Corrivveeaauu aannd d Sjola Sjolannder i der inn 2004 2004 [20] [20], , 282 282 com compparing aring t thheir nom eir nomin inaal m l mid- id-lo loaded aded rot rotoor b r blad lade e HS HS1A, m 1A, meent ntio ioned a ned alre lready ady b beefore, fore, wit withh an an aft aft- - 283 283 load load ed ed bl bl ade ade HS HS 1C w 1C w itih th a a nn inc inc rr ea ea se se of of t h th e s e s uu ct ct ion ion si si de de ung ung uu ide ide dd t u tu rning rning angl angl ee fr fr om om 11 11 .5 .5 °° t t oo 1 1 44 .5 .5 °° . . 284 284 ItIt ap apppeear ars t s thhatat t thhe e incre increaased t sed tuurning rning angl angle co e coul uldd cause, in cause, in the transon the transonic ic range, range, shock shock induced induced 285 285 bounda bounda ry ry la la yer tra yer tra nn si si titon ion nea nea rr the tra the tra iliilng edge wi ing edge wi th th , , as conse as conse qq ue ue nce, nce, a sh a sh arp arp increase o increase o f the base f the base 286 286 pressu pressu re, re, i.e i.e . .a s a s uu dden dden dro dro pp in the b in the b aa se se pressu pressu re coe re coe ffic ffic ient as s ient as s ee ee nn in in Fig Fig uu re re 10. 10. Note that t Note that t hh e e 287 287 reported in reported in the figur the figur ee h h aa s b s b een converte een converte d to d to − − of of t t hh e o e o rr igin igin aa l d l d aa ta ta . . 288 288 As As reg reg aa rds th rds th e base pr e base pr essure d essure d aa ta by ta by De De ckers an ckers an d De d De nton [13] nton [13] for for a a low low turnin turnin g blade g blade mod mod ee l l 289 289 and G and G oo st st ee low low et et al al . [ . [ 11 4] 4] for for a hi a hi gh t gh t u u rn rn in in g noz g noz zz le le guide va guide va ne, who report ba ne, who report ba se pressure da se pressure da ta ta far below far below 290 290 those of those of Siev Sieverding’s BPC erding’s BPC, their blad , their bladee pressure pressure ddistribution r istribution reesembles th sembles that of the conv at of the convergent/ ergent/ 291 divergent blade C in Figure 3 with a negative blade loading near the trailing edge which would 291 divergent blade C in Figure 3 with a negative blade loading near the trailing edge which would 292 292 explain the ve explain the ve ry lo ry lo w base p w base p rr essures. essures. In In addi addi ti ti on, the b on, the b la la d d e of e of Deckers Deckers aa nn d Denton has d Denton has aa blunt blunt tra tra ili ili ng ng 293 edge, and there is experimental evidence that, compared to a circular trailing edge, the base pressure 293 edge, and there is experimental evidence that, compared to a circular trailing edge, the base pressure Int. J. Turbomach. Propuls. Power 2020, 5, 10 10 of 55 294 294 fo fo r b r b lades with bl lades with blunt tra unt trailiilng edge m ing edge m ight be co ight be co ns ns ide ide rr ably ably lower. lower. Siever Siever ding ding a a nn d d He He inem inem aa nn nn [1 [1 6] 6] 295 report for flat plate tests at moderate subsonic Mach numbers a drop of the base pressure coefficient 295 report for flat plate tests at moderate subsonic Mach numbers a drop of the base pressure coefficient experimental evidence that, compared to a circular trailing edge, the base pressure for blades with 296 296 bb yy 1 1 11 % % for for a a pp lat lat e w e w itih th s s qq uar uar ee d t d t rr ai ai ling ling edge edge com com pp ar ar ed t ed t oo t t hh at at wit wit hh a c a c irc irc ul ul ar t ar t rr ai ai ling ling edg edg ee . . blunt trailing edge might be considerably lower. Sieverding and Heinemann [16] report for flat plate tests at moderate subsonic Mach numbers a drop of the base pressure coecient by 11% for a plate with squared trailing edge compared to that with a circular trailing edge. Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 13 of 65 298 Figure 10. Base pressure coefficient for mid-loaded (solid line) and aft-loaded (dashed line) rotor 297 297 Figure 10. Figure 10. Ba Ba se se pressu pressu re coe re coe fficient f fficient f oo r m r m id-loade id-loade dd ( ( ss oli oli dd line) an line) an d aft d aft -loaded -loaded (da (da ss hed line) rot hed line) rot oo r r Figure 10. Base pressure coecient for mid-loaded (solid line) and aft-loaded (dashed line) rotor blade. 299 blade. Symbols: HS1A geometry, HS1C geometry. Adapted from [20]. 298 298 bl bl ad ad e. Symbol e. Symbol Symbols: s:s: HS1A geom HS1A HS1A geom geometry etry, etry, , HS1C HS1C ge HS1C ge geometry ometry. Adapt ometry. Adapt . Adapted ee d from [20] d from [20] from [20]. . . 300 It is important to remember that the measurement of the base pressure carried out with a single It is important to remember that the measurement of the base pressure carried out with a single 301 pressure tapping in the trailing edge base region implies the assumption of an isobaric trailing edge 302 pressure distribution. However, in 2003 Sieverding et al. [21] demonstrated that at high subsonic pressure tapping in the trailing edge base region implies the assumption of an isobaric trailing edge 303 Mach numbers the pressure distribution could be highly non-uniform with a marked pressure pressure distribution. However, in 2003 Sieverding et al. [21] demonstrated that at high subsonic Mach 304 minimum at the center of the trailing edge base, as will be shown later in Section 5. Under these numbers the pressure distribution could be highly non-uniform with a marked pressure minimum at 305 conditions it is likely that the base pressure measured with a single pressure hole does not reflect the 306 true mean pressure. In addition, the measured pressure would depend on the ratio of the pressure the center of the trailing edge base, as will be shown later in Section 5. Under these conditions it is likely 307 hole to trailing edge diameter / , which is typically in the range / = 0.15 − 0.50 . This fact was that the base pressure measured with a single pressure hole does not reflect the true mean pressure. 308 also recognized by Jouini et al. [12], who mentioned the difficulties for obtaining representative In addition, the measured pressure would depend on the ratio of the pressure hole to trailing edge 309 trailing edge base pressures measurements: “It should also be noted that at high Mach numbers the base 310 pressure varies considerably with location on the trailing edge and the single tap gives a somewhat limited diameter d/D, which is typically in the range d/D = 0.15–0.50. This fact was also recognized by Jouini 311 picture of the base pressure behavior”. It is probably correct to say that differences between experimental et al. [12], who mentioned the diculties for obtaining representative trailing edge base pressures 312 base pressure data and the base pressure correlation may at least partially be attributed to the use of measurements: “It should also be noted that at high Mach numbers the base pressure varies considerably with 313 different pressure hole to trailing edge diameters / by the various researchers. 314 Finally, it is important to mention that the trailing edge pressure is sensitive to the trailing edge location on the trailing edge and the single tap gives a somewhat limited picture of the base pressure behavior”. 315 shape as demonstrated by El Gendi et al. [22] who showed with the help of high fidelity simulation It is probably correct to say that di erences between experimental base pressure data and the base 316 that the base pressure for blades with elliptic trailing edges was higher than for blades with circular pressure correlation may at least partially be attributed to the use of di erent pressure hole to trailing 317 trailing edges. Melzer and Pullan [23] proved experimentally that designing blades with elliptical 318 trailing edges improved the blade performance. The reason is that an elliptic trailing edge reduces edge diameters d/D by the various researchers. 319 not only the wake width but causes also an increase of the base pressure compared to that of blades Finally, it is important to mention that the trailing edge pressure is sensitive to the trailing edge 320 with a circular trailing edge. This suggests that inaccuracies in the machining of blades with thin shape as demonstrated by El Gendi et al. [22] who showed with the help of high fidelity simulation 321 trailing edges could easily lead to deviations from the designed circular trailing edge shape and thus 322 contribute to the differences in the base pressure. that the base pressure for blades with elliptic trailing edges was higher than for blades with circular trailing edges. Melzer and Pullan [23] proved experimentally that designing blades with elliptical 323 3. Unsteady Trailing Edge Wake Flow trailing edges improved the blade performance. The reason is that an elliptic trailing edge reduces 324 The mixing process of the wake behind turbine blades has been viewed for a long time as a not only the wake width but causes also an increase of the base pressure compared to that of blades 325 steady state process although it was well known that the separation of the boundary layers at the 326 trailing edge is a highly unsteady phenomenon which leads to the formation of large coherent with a circular trailing edge. This suggests that inaccuracies in the machining of blades with thin 327 structures, known as the von Kármán vortex street. The unsteady character of turbine blade wakes is trailing edges could easily lead to deviations from the designed circular trailing edge shape and thus 328 best illustrated by flow visualizations. contribute to the di erences in the base pressure. 329 Lawaczeck and Heinemann [24], and Heinemann and Bütefisch [25], were probably the first to 330 perform some systematic schlieren visualizations on transonic flat plate and cascades with different 331 trailing edge thicknesses using a flash light of 20 nano-seconds only, and deriving from the photos 3. Unsteady Trailing Edge Wake Flow 332 the vortex shedding frequencies and Strouhal numbers. The schlieren picture in Figure 11 shows 333 impressively that the shedding of each vortex from the trailing edge generates a pressure wave which The mixing process of the wake behind turbine blades has been viewed for a long time as a steady 334 travels upstream. state process although it was well known that the separation of the boundary layers at the trailing edge is a highly unsteady phenomenon which leads to the formation of large coherent structures, known as the von Kármán vortex street. The unsteady character of turbine blade wakes is best illustrated by flow visualizations. Lawaczeck and Heinemann [24], and Heinemann and Bütefisch [25], were probably the first to perform some systematic schlieren visualizations on transonic flat plate and cascades with di erent trailing edge thicknesses using a flash light of 20 nano-seconds only, and deriving from the photos the vortex shedding frequencies and Strouhal numbers. The schlieren picture in Figure 11 shows impressively that the shedding of each vortex from the trailing edge generates a pressure wave which travels upstream. Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 11 of 58 299 It is important to remember that the measurement of the base pressure carried out with a single 300 pressure tapping in the trailing edge base region implies the assumption of an isobaric trailing edge 301 pressure distribution. However, in 2003 Sieverding et al. [21] demonstrated that at high subsonic 302 Mach numbers the pressure distribution could be highly non-uniform with a marked pressure 303 minimum at the center of the trailing edge base, as will be shown later in Section 5. Under these 304 conditions it is likely that the base pressure measured with a single pressure hole does not reflect the 305 true mean pressure. In addition, the measured pressure would depend on the ratio of the pressure 306 hole to trailing edge diameter / , which is typically in the range / = 0.15 − 0.50 . This fact was 307 also recognized by Jouini et al. [12], who mentioned the difficulties for obtaining representative 308 trailing edge base pressures measurements: “It should also be noted that at high Mach numbers the base 309 pressure varies considerably with location on the trailing edge and the single tap gives a somewhat limited 310 picture of the base pressure behavior”. It is probably correct to say that differences between experimental 311 base pressure data and the base pressure correlation may at least partially be attributed to the use of 312 different pressure hole to trailing edge diameters / by the various researchers. 313 Finally, it is important to mention that the trailing edge pressure is sensitive to the trailing edge 314 shape as demonstrated by El Gendi et al. [22] who showed with the help of high fidelity simulation 315 that the base pressure for blades with elliptic trailing edges was higher than for blades with circular 316 trailing edges. Melzer and Pullan [23] proved experimentally that designing blades with elliptical 317 trailing edges improved the blade performance. The reason is that an elliptic trailing edge reduces 318 not only the wake width but causes also an increase of the base pressure compared to that of blades 319 with a circular trailing edge. This suggests that inaccuracies in the machining of blades with thin 320 trailing edges could easily lead to deviations from the designed circular trailing edge shape and thus 321 contribute to the differences in the base pressure. 322 3. Unsteady Trailing Edge Wake Flow 323 The mixing process of the wake behind turbine blades has been viewed for a long time as a 324 steady state process although it was well known that the separation of the boundary layers at the 325 trailing edge is a highly unsteady phenomenon which leads to the formation of large coherent 326 structures, known as the von Kármán vortex street. The unsteady character of turbine blade wakes is 327 best illustrated by flow visualizations. 328 Lawaczeck and Heinemann [24], and Heinemann and Bütefisch [25], were probably the first to 329 perform some systematic schlieren visualizations on transonic flat plate and cascades with different 330 trailing edge thicknesses using a flash light of 20 nano-seconds only, and deriving from the photos 331 the vortex shedding frequencies and Strouhal numbers. The schlieren picture in Figure 11 shows 332 impressively that the shedding of each vortex from the trailing edge generates a pressure wave which Int. J. Turbomach. Propuls. Power 2020, 5, 10 11 of 55 333 travels upstream. Figure 11. Schlieren picture of turbine rotor blade wake at M = 0.8 [24]. 334 Figure 11. Schlieren picture of turbine rotor blade wake at 2,is =0.8 [24]. Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 12 of 58 In 1982 Han and Cox [26], performed smoke visualizations on a very large-scale nozzle blade at 335 In 1982 Han and Cox [26], performed smoke visualizations on a very large-scale noz zle blade at low speed (Figure 12). The authors found much sharper and well-defined contours of the vortices 336 low speed (Figure 12). The authors found much sharper and well-defined contours of the vortices 337 from the pressure side and concluded that this implied stronger vortex shedding from this side and Figure 40. Blade Mach number distributions for front and rear-loaded blades; adapted from [16]. from the pressure side and concluded that this implied stronger vortex shedding from this side and 338 attributed this to the circulation around the blade. attributed this to the circulation around the blade. Figure 12. Smoke visualization of the vortex shedding from a low speed nozzle blade [26]. 339 Figure 12. Smoke visualization of the vortex shedding from a low speed nozzle blade [26]. Beretta-Piccoli [27] (reported by Bölcs and Sari [28]) was possibly the first to use interferometry to 340 Beretta-Piccoli [27] (reported by Bölcs and Sari [28]) was possibly the first to use interferometry visualize the vortex formation at the blunt trailing edge of a blade at transonic flow conditions. 341 to visualize the vortex formation at the blunt trailing edge of a blade at transonic flow conditions. Besides the problem of time resolution for measuring high frequency phenomena, there was also 342 Besides the problem of time resolution for measuring high frequency phenomena, there was also the problem of spatial resolution for resolving the vortex structures behind the usually rather thin Blade 𝜶 𝜶 Ref g/c 343 the problem of spatial resolution for resolving the vortex structures behind the usually rather thin turbine blade trailing edges. First tests on a large scale flat late model simulating the overhang section A 30° 22° 0.75 [5] 344 turbine blade trailing edges. First tests on a large scale flat late model simulating the overhang section of a cascade allowed to visualize impressively details of the vortex shedding at transonic outlet Mach B 60° 25° 0.75 VKI 345 of a cascade allowed to visualize impressively details of the vortex shedding at transonic outlet Mach number (Figure 13a,b). C 66° 18° 0.70 VKI 346 number (Figure 13a,b). Following Hussain and Hayakawa [29], the wake vortex structures can be described by a set of D 156° 19.5° 0.85 [6] centers which characterize the location of a peak of coherent span-wise vortices and saddles located between the coherent vorticity structures and defined by a minimum of coherent span-wise vorticity. Figure 1. Blade profile losses versus isentropic outlet Mach number for four transonic turbines. The successive span-wise vortices are connected by ribs, that are longitudinal smaller scale vortices of alternating signs. (c) topology of wake vortex structure (a) turbine blade flat plate model. (b) schlieren photograph. behind a cylinder [29]. 348 Figure 13. Vortex shedding at transonic exit flow conditions [30]. 349 Following Hussain and Hayakawa [29], the wake vortex structur (c) topo es can logy be of w descr ake vor ibed by tex struct a set ure of (a) turbine blade flat plate model. (b) schlieren photograph. 350 centers which characterize the location of a peak of coherent span-wise vort behind ice a s cyli and nde sr a [dd 29]les . located 351 between the coherent vorticity structures and defined by a minimum of coherent span-wise vorticity. Figure 13. Vortex shedding at transonic exit flow conditions [30]. Figure 13. Vortex shedding at transonic exit flow conditions [30]. 352 The successive span-wise vortices are connected by ribs, that are longitudinal smaller scale vortices 353 of alternating signs. 354 A significant progress was made in the 1990’s in the frame of two European Research Projects, 355 i.e. Experimental and Numerical Investigations of Time Varying Wakes behind Turbine Blades 356 (BRITE/EURAM CT-92-0048) and Turbulence Modeling for Unsteady Flows in Axial Turbines 357 (BRITE/EURAM CT-96-0143) in which the von Kármán Institute, the University of Genoa and the 358 ONERA of Lille used short duration flow visualizations and fast response instrumentation in 359 combination with large scale blade models to improve the understanding of the formation of the 360 vortical structures at the turbine blade trailing edges and their impact on the unsteady wake flow 361 characteristics. Results of these research projects are reported by Ubaldi et al. [31], Cicatelli and 362 Sieverding [32], Desse [33], Sieverding et al. [34], Ubaldi and Zunino [35] and Sieverding et al. [21,36]. 363 The large-scale turbine guide vane used in these experiments was designed at VKI (Table 2) and 364 released in 1994 [37]. The blade design features a front-loaded blade with an overall low suction side 365 turning in the overhang section and, in particular, a straight rear suction side from halfway 366 downstream of the throat, Figure 14. Due to mass flow restrictions in the VKI blow down facility, the 367 three-bladed cascade with a chord length = 280 mm was limited to investigations at a relatively Int. J. Turbomach. Propuls. Power 2020, 5, 10 12 of 55 A significant progress was made in the 1990’s in the frame of two European Research Projects, i.e., Experimental and Numerical Investigations of Time Varying Wakes behind Turbine Blades (BRITE/EURAM CT-92-0048) and Turbulence Modeling for Unsteady Flows in Axial Turbines (BRITE/EURAM CT-96-0143) in which the von Kármán Institute, the University of Genoa and the ONERA of Lille used short duration flow visualizations and fast response instrumentation in combination with large scale blade models to improve the understanding of the formation of the vortical structures at the turbine blade trailing edges and their impact on the unsteady wake flow characteristics. Results of these research projects are reported by Ubaldi et al. [31], Cicatelli and Sieverding [32], Desse [33], Sieverding et al. [34], Ubaldi and Zunino [35] and Sieverding et al. [21,36]. The large-scale turbine guide vane used in these experiments was designed at VKI (Table 2) and released in 1994 [37]. The blade design features a front-loaded blade with an overall low suction side turning in the overhang section and, in particular, a straight rear suction side from halfway downstream of the throat, Figure 14. Due to mass flow restrictions in the VKI blow down facility, the three-bladed cascade with a chord length c = 280 mm was limited to investigations at a relatively low subsonic outlet Mach number of M = 0.4. The suction side boundary layer undergoes natural transition at 2,is Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 13 of 58 x/c  0.6. On the pressure side the boundary layer was tripped at x/c  0.61. The boundary ax ax 368 three-bladed cascade with a chord length = 280 mm was limited to investigations at a relatively layers at the trailing edge with shape factors H of 1.64 and 1.41 for the pressure and suction sides 369 low subsonic outlet Mach number of =0.4. The suction side boundary layer undergoes natural respectively, were clearly turbulent. The schlieren photographs in Figure 14 were taken with a Nanolite 370 transition at / ~0.6. On the pressure side the boundary layer was tripped at / ~0.61. The spark source, with Dt = 20 10 s. The dominant vortex shedding frequency was 2.65 kHz and the 371 boundary layers at the trailing edge with shape factors of 1.64 and 1.41 for the pressure and 372 suction sides respectively, were clearly turbulent. The schlieren photographs in Figure 14 were taken corresponding Strouhal number, defined as: 373 with a Nanolite spark source, with ∆ = 20 10 . The dominant vortex shedding frequency was 374 2.65 kHz and the corresponding Strouhal number, defined as: f d vs te St = (3) = (3) 2,is 375 was = 0.27 . was St = 0.27. (a) (b) (c) 377 Figure 14. Very large-scale turbine nozzle guide vane (VKI LS-94); =0.4, =2 x 10 case. (a) 6 Figure 14. Very large-scale turbine nozzle guide vane (VKI LS-94); M = 0.4, Re = 2 10 case. 2,is 2 378 test section, (b) surface isentropic Mach number distribution, (c) schlieren photographs. Adapted (a) test section, (b) surface isentropic Mach number distribution, (c) schlieren photographs. Adapted 379 from [32]. from [32]. 380 Figure 14c presents two instances in time of the vortex shedding process. The left flow 381 visualization shows the enrolment of the pressure side shear layer into a vortex, the right one the Figure 14c presents two instances in time of the vortex shedding process. The left flow visualization 382 formation of the suction side vortex. Note that the pressure side vortex appears to be much stronger shows the enrolment of the pressure side shear layer into a vortex, the right one the formation of the 383 than the suction side one, which confirms the observations made by Han and Cox [26]. suction side vortex. Note that the pressure side vortex appears to be much stronger than the suction 384 Table 2. VKI LS-94 large scale nozzle blade geometric characteristics [32]. side one, which confirms the observations made by Han and Cox [26]. Parameter Symbol Value Gerrard [38], describes the vortex formation for the flow behind a cylinder as follows, Figure 15. Chord c 280 mm The growing vortex (A) is fed by the circulation existing in the upstream shear layer until the vortex is Pitch to chord ratio / 0.73 Blade aspect ratio ℎ/ 0.7 strong enough to entrain fluid from the opposite shear layer bearing vorticity of the opposite circulation. Stagger angle −49.83° When the quantity of entrained fluid is sucient to cut o the supply of circulation to the growing Trailing edge thickness to throat ratio 0.053 Trailing edge wedge angle 7.5° Gauging angle (arcsin( /) ) 19.1° 385 Gerrard [38], describes the vortex formation for the flow behind a cylinder as follows, Figure 15. 386 The growing vortex (A) is fed by the circulation existing in the upstream shear layer until the vortex 387 is strong enough to entrain fluid from the opposite shear layer bearing vorticity of the opposite Int. J. Turbomach. Propuls. Power 2020, 5, 10 13 of 55 vortex—the opposite vorticity of the fluid in both shear layers cancel each other—then the vortex is shed o . Table 2. VKI LS-94 large scale nozzle blade geometric characteristics [32]. Parameter Symbol Value Chord c 280 mm Pitch to chord ratio g/c 0.73 Blade aspect ratio h/c 0.7 Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 14 of 58 Stagger angle 49.83 Trailing edge thickness to throat ratio d /o 0.053 te Trailing edge wedge angle  7.5 388 the growing vortex—the opposite vorticity of the fluid in both shear layer te s cancel each other—then Gauging angle (arcsin(o/g)) 19.1 389 the vortex is shed off. 390 Figure 15. Vortex formation mechanism; adapted from [38]. Figure 15. Vortex formation mechanism; adapted from [38]. 391 Contrary Contrary to t to the he bl blow ow down down tunnel a tunnel at t VKI, the Is VKI, the Istituto tituto di M di Macchine acchine e e Sistemi Sistemi Ener Energetici (I getici (ISME) SME) at the University of Genoa used a continuous running low speed wind tunnel. Miniature cross-wire 392 at the University of Genoa used a continuous running low speed wind tunnel. Miniature cross-wire 393 hot-wir hot-wire pro e probe be and a and a four four-beam -beam laser laser Doppler Doppler velo velocimeter cimeter are are used used for the for the measur measurement ements s of the of the unsteady wake. An example of the instantaneous patterns of the ensemble averaged periodic wake 394 unsteady wake. An example of the instantaneous patterns of the ensemble averaged periodic wake 395 characteristics characteristics is is present presented ed in F in Figur igure e 16 16.. A det A detailed ailed d description escription is is given by given by Uba Ubaldi ldi and Z and Zunino unino [3 [35 5]]. . The streamwise periodic component of the velocity, U U in Figure 16 (upper left), shows asymmetric 396 The streamwise periodic component of the velocity, − in Figure 16 (upper left), shows s s periodic patterns of alternating positive and negative velocity components issued from the pressure to 397 asymmetric periodic patterns of alternating positive and negative velocity components issued from the suction side. As already shown schematically in Figure 13, saddle points separating groups of 398 the pressure to the suction side. As already shown schematically in Figure 13, saddle points four cores, are located along the wake center line. On the contrary, the periodic parts of the transverse 399 separating groups of four cores, are located along the wake center line. On the contrary, the periodic component U U (upper right) appear as cores of positive and negative values, approximately 400 parts of the trnansver n se component − (upper right) appear as cores of positive and negative centered in the wake which alternate, enlarging in streamwise direction. The combination of the two 401 values, approximately centered in the wake which alternate, enlarging in streamwise direction. The velocity components give rise to the rolling up of the periodic flow into a row of vortices rotating in 402 combination of the two velocity components give rise to the rolling up of the periodic flow into a row opposite direction as shown by the velocity vector plots (lower left). 403 of vortices rotating in opposite direction as shown by the velocity vector plots (lower left). As illustrated by Gerrard [38] (see Figure 15), the vortex formation is driven by the vorticity 404 As illustrated by Gerrard [38] (see Figure 15), the vortex formation is driven by the vorticity in in the suction and pressure side boundary layers. The vorticity terms ! e and ! in the wake have 405 the suction and pressure side boundary layers. The vorticity terms and in the wake have been been determined taking respectively the curl of the phase averaged and time averaged velocity field: 406 determined taking respectively the curl of the phase averaged and time averaged velocity field: = e e @U @U @U @U s n s n ! e = and ! = , Figure 16 (lower right). The local maxima and minima and 407 − and = − , Figure 16 (lower right). The local maxima and minima and saddle @n @s @n @s saddle regions (the points where the vorticity changes its sign) define the location, extension, rotation 408 regions (the points where the vorticity changes its sign) define the location, extension, rotation and and intensity of the vortices. 409 intensity of the vortices. With increasing downstream Mach number, the vortices become much more intense as demonstrated in Figure 17 on a half scale model of the blade already presented in Figure 14 and Table 2, at an outlet Mach number M = 0.79 in a four bladed cascade, Sieverding et al. [36]. Contrary to 2,is schlieren photographs which visualize density changes, the smoke visualizations in Figure 17 show the instantaneous flow patterns and are therefore particular well suited to visualize the enrolment of the vortices. A close look at the vortex structures reveals that the distances between successive vortices change. In fact, the distance between a pressure side vortex and a suction side vortex is always smaller than the distance between two successive pressure side vortices. A possible reason is that the pressure side vortex plays a dominant role and exerts an attraction on the suction side vortex as already found by Han and Cox [26]. Int. J. Turbomach. Propuls. Power 2020, 5, 10 14 of 55 Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 15 of 58 (a) (b) (c) (d) 411 Figure Figure 16. 16. Inst Instantaneous antaneous realization of the realization of the ensemble ensemble aver averag aged streamwise velocity ( ed streamwise velocity (a a), ), transversal transversal velocity (b), velocity vector (c) and vorticity patterns (d) of the periodic flow [35]. 412 velocity (b), velocity vector (c) and vorticity patterns (d) of the periodic flow [35]. 413 With increasing downstream Mach number, the vortices become much more intense as 414 demonstrated in Figure 17 on a half scale model of the blade already presented in Figure 14 and Table 415 2, at an outlet Mach number =0.79 in a four bladed cascade, Sieverding et al. [36]. Contrary to 416 schlieren photographs which visualize density changes, the smoke visualizations in Figure 17 show 417 the instantaneous flow patterns and are therefore particular well suited to visualize the enrolment of 418 the vortices. A close look at the vortex structures reveals that the distances between successive 419 vortices change. In fact, the distance between a pressure side vortex and a suction side vortex is 420 always smaller than the distance between two successive pressure side vortices. A possible reason is 421 that the pressure side vortex plays a dominant role and exerts an attraction on the suction side vortex 422 as already found by Han and Cox [26]. (a) (b) (c) Figure 17. VKI LS94 turbine blade, M = 0.79, Re = 2.8 10 case. (a) four blades cascade, (b) surface 2,is isentropic Mach number distribution and (c) smoke visualizations. Adapted from [36]. The vortex formation and subsequent shedding is accompanied by large angle fluctuations of the separating shear layers which does not only lead to large pressure fluctuations in the zone of separations but also induces strong acoustic waves. The latter travel upstream on both the pressure and suction side as shown in the corresponding schlieren photographs obtained this time with a continuous light source, a high speed rotating drum and rotating prism camera from ONERA with a maximum frame rate of 35,000 frames per second (see Figure 18), as reported by Sieverding et al. [21]. (a) (b) (c) Int. J. Turbomach. Propuls. Power 2020, 5, 10 15 of 55 In image 1 of Figure 18 the suction side shear layer has reached its farthest inward position and Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 16 of 58 the local pressure just upstream of the separation point has reached its minimum value. Conversely, on the pressure side the separating shear layer has reached its most outward position. A pressure 423 Figure 17. VKI LS94 turbine blade, = 0.79, = 2.8 x 10 case. (a) four blades cascade, (b) wave denoted Pi originates from the point where the boundary layer separates from the trailing edge. 424 surface isentropic Mach number distribution and (c) smoke visualizations. Adapted from [36]. Upstream of Pi is the pressure wave from the previous cycle. It interferes with the suction side of 425 the neighboring The vortex fo blade rmation fromand wher subse e it is quent shedding reflected. In image is ac4companied b of Figure 18,yt large he suction angleside fluctuations shear layer of 426 is that e separ its most ating outwar shear dlaposition. yers which A pr does essur not e only wavelea originates d to large pr at the essu point re flof uct separation, uations in t denoted he zone o Si.f 427 The separ pr aessur tions but also in e wave further duce upstr s strong eamaco is due ustic w to the aves pr . eviou The la stter travel cycle. On upstrea the pressur m on both the pressure e side the pressure 428 wave and suction Pi extends side now as shown to the in the suction corre sidesponding of the neighboring schlieren photogra blade. The phs wave obtaiinterfer ned this ence time point with ofa 429 the continuous light source, a previous cycle has moved high sp up-str eed ro eam. tating dr It can ther um and rotating efore be expected prism ca thatmera the suction from ONERA side pressur with a e 430 distribution maximum frnear ame ra the te of 35 throat,0 r00 f egion ram ises per second ( highly unsteady see . Figure 18), as reported by Sieverding et al. [21]. Figure 18. Schlieren photographs of vortex shedding at two instances in time; M = 0.79, Re = 431 Figure 18. Schlieren photographs of vortex shedding at two instances in time; 2,is = 0.79, 2 = 2.8 10 [21]. 432 2.8 x 10 [21]. Holographic interferometric density measurements, performed at VKI at M = 0.79 by Sieverding 2,is 433 In image 1 of Figure 18 the suction side shear layer has reached its farthest inward position and Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 17 of 58 et al. [36], give further information about the formation and the shedding process of the von Kármán 434 the local pressure just upstream of the separation point has reached its minimum value. Conversely, vortices. The reference density is evaluated from pressure measurements with a fast response needle 451 The interferogram in Figure 19 shows the suction side vortex (upper blade surface) in its out 435 on the pressure side the separating shear layer has reached its most outward position. A pressure 452 most outward position i.e. at the start of the shedding phase. On the pressure side the density patterns static pressure probe positioned just outside of the wake assuming the total temperature to be constant 436 wave denoted Pi originates from the point where the boundary layer separates from the trailing edge. 453 point to the start of the formation of a new pressure side vortex. The pressure side vortex of the outside the wake. 437 Upstream of Pi is the pressure wave from the previous cycle. It interferes with the suction side of the 454 previous cycle is situated at a trailing edge distance of ≈2. This vortex is defined by ten The interferogram in Figure 19 shows the suction side vortex (upper blade surface) in its out most 455 fringes. With a relative density change between two successive fringes of ∆( ) = 0.0184 the total 438 neighboring blade from where it is reflected. In image 4 of Figure 18, the suction side shear layer is 456 relative density change from the outside to the vortex center is ∆( ) = 0.184. The minimum in outward position i.e., at the start of the shedding phase. On the pressur e side the density patterns point 439 at its most outward position. A pressure wave originates at the point of separation, denoted Si. The 457 the vortex center is = 0.552 compared to an isentropic downstream static to total density ratio to the start of the formation of a new pressure side vortex. The pressure side vortex of the previous 440 pressure wave further upstream is due to the previous cycle. On the pressure side the pressure wave 458 of ⁄ =0.745. cycle is situated at a trailing edge distance of x/d  2. This vortex is defined by ten fringes. With a te 459 Based on a large number of tests with holographic interferometry and white light interferometry, 441 Pi extends now to the suction side of the neighboring blade. The wave interference point of the 460 see Desse [33], Figure 19 shows the variation of the vortex density minima non-dimensionalized by relative density change between two successive fringes of D(/ ) = 0.0184 the total relative density 442 previous cycle has moved up-stream. It can therefore be expected that the suction side pressure 461 the upstream total density / , in function of the trailing edge distance / . There are two distinct change from the outside to the vortex center is D(/ ) = 0.184. The minimum in the vortex center is 443 distribution near the throat region is highly unsteady. 462 regions for the evolution of the vortex minima: a rapid linear density rise-up to distance / = 1.7 / = 0.552 compared to an isentropic downstream static to total density ratio of  / = 0.745. 0 463 followed by a much slower rise further downstream. 2 01 444 Holographic interferometric density measurements, performed at VKI at =0.79 by 445 Sieverding et al. [36], give further information about the formation and the shedding process of the 446 von Kármán vortices. The reference density is evaluated from pressure measurements with a fast 447 response needle static pressure probe positioned just outside of the wake assuming the total 448 temperature to be constant outside the wake. 449 The interferogram in Figure 19 shows the suction side vortex (upper blade surface) in its out 450 most outward position i.e. at the start of the shedding phase. On the pressure side the density patterns 451 point to the start of the formation of a new pressure side vortex. The pressure side vortex of the 452 previous cycle is situated at a trailing edge distance of ⁄ ≈2. This vortex is defined by ten 453 fringes. With a relative density change between two successive fringes of ∆( ⁄ ) = 0.0184 the total (a) (b) 454 relative density change from the outside to the vortex center is ∆( ⁄ ) = 0.184. The minimum in 465 Figure 19. Instantaneous density distribution (a) and variation of density minima with trailing edge 455 the vortex ce Figure 19. nter is Instantaneous ⁄ = density 0.552 comp distribu ared to a tion (a)n and isentropi variation c downstrea of density minim m stata ic with to tota trailing l densi edge ty Commente ratio d [M39]: Can it be in terms of (a)? 466 distance (b) at = 0.79, =2.8 x 10 ; adapted from [36]. Please check all figure captions format. distance (b) at M = 0.79, Re = 2.8 10 ; adapted from [36]. 456 of ⁄ = 0.745. 2 2,is 467 Comparing the vortex formation at =0.4 and 0.79 shows that with increasing Mach 457 Based on a large number of tests with holographi , c interferometry and white light interferometry, Commented [MM40R39]: Done 468 number the vortices form much closer to the trailing edge. This tendency goes crescendo with further 458 see Desse [33], Figure 19 shows the variation of the vortex density minima non-dimensionalized by 469 increase of the downstream Mach number as already shown in Figure 13 where normal shocks 459 the upstream total density / , in function of the trailing edge distance / . There are two distinct 470 oscillate close to the trailing edge forward and backward with the alternating shedding of the 471 vortices. A further increase of the outlet flow leads gradually to the formation of an oblique shock 472 system at the convergence of the separating shear layers at short distance behind the trailing edge, 473 causing a delay of the vortex formation to this region as demonstrated by Carscallen and Gostelow 474 [39], in the high speed cascade facility of the NRC Canada. The high speed schlieren pictures revealed 475 some very unusual types of wake vortex patterns as shown in Figure 20. Besides the regular von 476 Kármán vortex street (left), the authors visualized other vortex patterns, such as e.g. couples or 477 doublets, on the right. In other moments in time they observed what they called hybrid or random 478 or no patterns. The schlieren photos in Figure 20 show the existence of an unexpected shock 479 emanating from the trailing edge pressure side at the beginning of the trailing edge circle. 480 Questioning Bill Carscallen [40] recently about the origin of this shock it appeared that the shock was 481 simply due to an inaccuracy in the blade manufacturing of the trailing edge circle. Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 17 of 58 460 regions for the evolution of the vortex minima: a rapid linear density rise-up to distance / = 1.7 461 followed by a much slower rise further downstream. Int. J. Turbomach. Propuls. Power 2020, 5, 10 16 of 55 Based on a large number of tests with holographic interferometry and white light interferometry, (a) (b) see Desse [33], Figure 19 shows the variation of the vortex density minima non-dimensionalized by the upstream total density / , in function of the trailing edge distance x/D. There are two distinct 462 Figure 19. Instantaneous density distribution (a) and variation of density minima with trailing edge regions for the evolution of the vortex minima: a rapid linear density rise-up to distance x/D = 1.7 463 distance (b) at =0.79, = 2.8 x 10 ; adapted from [36]. followed by a much slower rise further downstream. Comparing the vortex formation at M = 0.4 and 0.79 shows that with increasing Mach number 464 Comparing the vortex formation at =0.4 and 0.79 shows that with increasing Mach 2,is the vortices form much closer to the trailing edge. This tendency goes crescendo with further increase of 465 number the vortices form much closer to the trailing edge. This tendency goes crescendo with further the downstream Mach number as already shown in Figure 13 where normal shocks oscillate close to the 466 increase of the downstream Mach number as already shown in Figure 13 where normal shocks trailing edge forward and backward with the alternating shedding of the vortices. A further increase 467 oscillate close to the trailing edge forward and backward with the alternating shedding of the of the outlet flow leads gradually to the formation of an oblique shock system at the convergence of 468 vortices. A further increase of the outlet flow leads gradually to the formation of an oblique shock the separating shear layers at short distance behind the trailing edge, causing a delay of the vortex 469 system at the convergence of the separating shear layers at short distance behind the trailing edge, formation to this region as demonstrated by Carscallen and Gostelow [39], in the high speed cascade 470 causing a delay of the vortex formation to this region as demonstrated by Carscallen and Gostelow facility of the NRC Canada. The high speed schlieren pictures revealed some very unusual types 471 [39], in the high speed cascade facility of the NRC Canada. The high speed schlieren pictures revealed of wake vortex patterns as shown in Figure 20. Besides the regular von Kármán vortex street (left), 472 some very unusual types of wake vortex patterns as shown in Figure 20. Besides the regular von the authors visualized other vortex patterns, such as e.g. couples or doublets, on the right. In other 473 Kármán vortex street (left), the authors visualized other vortex patterns, such as e.g. couples or moments in time they observed what they called hybrid or random or no patterns. The schlieren 474 doublets, on the right. In other moments in time they observed what they called hybrid or random 475 photos or no p inatterns. The sch Figure 20 showlieren photos the existencein Figure of an unexpected 20 show the exi shock emanating stence of fran u om the nexpe trailing cted sedge hock pressure side at the beginning of the trailing edge circle. Questioning Bill Carscallen [40] recently 476 emanating from the trailing edge pressure side at the beginning of the trailing edge circle. 477 about Questi the oning origin Billof Ca this rsca shock llen [4it 0]appear recented ly about that the the shock origin was of tsimply his shock it due appe to an ar inaccuracy ed that the in sh the ock w blade as manufacturing of the trailing edge circle. 478 simply due to an inaccuracy in the blade manufacturing of the trailing edge circle. (a) (b) (c) 479 Figure Figure 20. 20. Occurrence of different vortex patter Occurrence of di erent vortex patters s in in wak wake e of of t transonic ransonic bla blade de at at M = =1 1.07 .07.. ( (a a) ) re regular gular 2,is 480 vortex vortex street street,, ( (b b)) cou couples, ples, ( (cc)) doublets doublets [[39]. 39]. The question whether in distinction of the conventional von Kármán vortex street, a double 481 The question whether in distinction of the conventional von Kármán vortex street, a double row row vortex street of unequal vortex strength may exist, was treated by Sun in 1983 [41]. Figure 21 482 vortex street of unequal vortex strength may exist, was treated by Sun in 1983 [41]. Figure 21 presents presents an example of a double row vortex street with unequal vortex strength and vortex distances. 483 an example of a double row vortex street with unequal vortex strength and vortex distances. The The Int. J.author Turbomach demonstrated . Propuls. Power 2018 that, such 3, x FOR PE configurations ER REVIEW ar e basically unstable. 18 of 58 484 author demonstrated that such configurations are basically unstable. Figure 21. An example of vortex street with unequal vortex strengths and unequal vortex spacings; 485 Figure 21. An example of vortex street with unequal vortex strengths and unequal vortex spacings; adapted from [41]. 486 adapted from [41]. 487 4. Energy Separation in the Turbine Blade Wakes 488 In the course of a joint research program between the National Research Council of Canada and 489 Pratt & Whitney Canada of the flow through an annular transonic nozzle guide vane in the 1980s 490 certain experiments revealed a non-uniform total temperature distribution downstream of the 491 uncooled blades, Carscallen and Oosthuizen [42]. Considering the importance of the existence of a 492 non-uniform total temperature distribution at the exit of uncooled stator blade row for the 493 aerothermal aspects of the downstream rotor, the Gasdynamics Laboratory of the National Research 494 Council, Canada decided to build a continuously running suction type large scale planar cascade 495 tunnel (chord length 175.3 mm, turning angle 76°, trailing edge diameter 6.35 mm) and launched an 496 extensive research program aiming at the understanding of the mechanism causing the occurrence 497 of total temperature variations downstream of a fixed blade row, determine their magnitude and 498 evaluate their significance for the design of the downstream rotor. Downstream traverses with copper 499 constantan thermocouples reported by Carscallen et al. [43] in 1996 showed that the total temperature 500 contours correlated perfectly with the total pressure wake profiles, Figure 22. (b) Total pressure coefficient (c) Total temperature difference (a) Test section 502 Figure 22. Total pressure coefficient and temperature contours downstream of a nozzle guide vane at 503 a pressure ratio PR = 1.9; adapted from [42]. 504 In the wake center the total temperature dropped significantly below the inlet total temperature 505 while higher values were recorded near the border of the wake. The differences increased with Mach 506 number and reached a maximum at sonic outlet conditions. The question was then to elucidate the 507 reasons for these temperature variations. The research on flows across cylinders was already more Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 18 of 58 485 Figure 21. An example of vortex street with unequal vortex strengths and unequal vortex spacings; 486 adapted from [41]. Int. J. Turbomach. Propuls. Power 2020, 5, 10 17 of 55 487 4. Energy Separation in the Turbine Blade Wakes 4. Energy Separation in the Turbine Blade Wakes 488 In the course of a joint research program between the National Research Council of Canada and In the course of a joint research program between the National Research Council of Canada and 489 Pratt & Whitney Canada of the flow through an annular transonic nozzle guide vane in the 1980s Pratt & Whitney Canada of the flow through an annular transonic nozzle guide vane in the 1980s 490 certain experiments revealed a non-uniform total temperature distribution downstream of the certain experiments revealed a non-uniform total temperature distribution downstream of the uncooled 491 uncooled blades, Carscallen and Oosthuizen [42]. Considering the importance of the existence of a blades, Carscallen and Oosthuizen [42]. Considering the importance of the existence of a non-uniform 492 non-uniform total temperature distribution at the exit of uncooled stator blade row for the total temperature distribution at the exit of uncooled stator blade row for the aerothermal aspects 493 aerothermal aspects of the downstream rotor, the Gasdynamics Laboratory of the National Research of the downstream rotor, the Gasdynamics Laboratory of the National Research Council, Canada 494 Council, Canada decided to build a continuously running suction type large scale planar cascade decided to build a continuously running suction type large scale planar cascade tunnel (chord length 495 tunnel (chord length 175.3 mm, turning angle 76°, trailing edge diameter 6.35 mm) and launched an 175.3 mm, turning angle 76 , trailing edge diameter 6.35 mm) and launched an extensive research 496 extensive research program aiming at the understanding of the mechanism causing the occurrence program aiming at the understanding of the mechanism causing the occurrence of total temperature 497 of total temperature variations downstream of a fixed blade row, determine their magnitude and variations downstream of a fixed blade row, determine their magnitude and evaluate their significance 498 evaluate their significance for the design of the downstream rotor. Downstream traverses with copper for the design of the downstream rotor. Downstream traverses with copper constantan thermocouples 499 constantan thermocouples reported by Carscallen et al. [43] in 1996 showed that the total temperature reported by Carscallen et al. [43] in 1996 showed that the total temperature contours correlated perfectly 500 contours correlated perfectly with the total pressure wake profiles, Figure 22. with the total pressure wake profiles, Figure 22. (b) Total pressure coefficient (c) Total temperature difference (a) Test section Figure 22. Total pressure coecient and temperature contours downstream of a nozzle guide vane at a 502 Figure 22. Total pressure coefficient and temperature contours downstream of a nozzle guide vane at pressure ratio PR = 1.9; adapted from [42]. 503 a pressure ratio PR = 1.9; adapted from [42]. In the wake center the total temperature dropped significantly below the inlet total temperature 504 In the wake center the total temperature dropped significantly below the inlet total temperature while higher values were recorded near the border of the wake. The di erences increased with Mach 505 while higher values were recorded near the border of the wake. The differences increased with Mach number and reached a maximum at sonic outlet conditions. The question was then to elucidate the 506 number and reached a maximum at sonic outlet conditions. The question was then to elucidate the reasons for these temperature variations. The research on flows across cylinders was already more 507 reasons for these temperature variations. The research on flows across cylinders was already more advanced in this respect. Measurements of the temperature distribution around a cylinder for a flow normal to the axis of the cylinder, performed at the Aeronautical Institute of Braunschweig in the late 1930’s and reported by Eckert and Weise [44] in 1943, showed that the recovery temperature at the base of the cylinder reduced below the true (static) temperature of the incoming flow, so that the recovery factor: T T r = T T attained negative values in the base region (see Figure 23). The authors suspected that the low values were possibly due to the intermittent separation of vortices from the cylinder. These results were confirmed by Ryan [45] in 1951 at Ackeret’s Institute in Zürich who clearly related this low temperature to the periodic vortex shedding behind the cylinder as Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 19 of 58 Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 19 of 58 (b) Total pressure coefficient 508 advanced in this respect. Measurements of the temperature distribution around a cylinder for a flow 509 normal to the axis of the cylinder, performed at the Aeronautical Institute of Braunschweig in the late 510 1930’s and reported by Eckert and Weise [44] in 1943, showed that the recovery temperature at the 511 base of the cylinder reduced below the true (static) temperature of the incoming flow, so that the 512 recovery factor: (c) Total temperature difference 513 attained negative values in the base region (see Figure 23). (a) Test section 506 Figure 22. Total pressure coefficient and temperature contours downstream of a nozzle guide vane at Commented [M43]: Please add explanation for subgraph. 507 a pressure ratio PR = 1.9; adapted from [42]. Commented [MM44R43]: Done. Int. J. Turbomach. 508 Propuls. In th Power e wak 2020 e ce,nt5e , r 10 the total temperature dropped significantly below the inlet total temperature 18 of 55 509 while higher values were recorded near the border of the wake. The differences increased with Mach 510 number and reached a maximum at sonic outlet conditions. The question was then to elucidate the 511 reasons for these temperature variations. The research on flows across cylinders was already more cause for the energy separation in the fluctuating wake. He also noticed that the energy separation 512 advanced in this respect. Measurements of the temperature distribution around a cylinder for a flow was particularly large when a strong sound was generated by the flow. 513 normal to the axis of the cylinder, performed at the Aeronautical Institute of Braunschweig in the late 514 1930’s and reported by Eckert and Weise [44] in 1943, showed that the recovery temperature at the The existence of a low temperature field at the base of a cylinder was also observed by Sieverding 515 base of the cylinder reduced below the true (static) temperature of the incoming flow, so that the in 1985, who used an infrared camera to visualize through a germanium window in the side wall of a 516 recovery factor: blow down wind tunnel the wall temperature field around a 15 mm diameter cylinder at M = 0.4, 514 Figure 23. Evolution of the recovery factor in the azimuthal direction from the stagnation point see Figure 24. Unfortunately, due to a lack of time it was not possible to determine the absolute 515 (= 0 °) to the rear side of the cylinder ( = 180° ); adapted from [44]. temperature values. 517 attained negative values in the base region (see Figure 23). 516 The authors suspected that the low values were possibly due to the intermittent separation of 517 vortices from the cylinder. These results were confirmed by Ryan [45] in 1951 at Ackeret’s Institute 518 in Zürich who clearly related this low temperature to the periodic vortex shedding behind the 519 cylinder as cause for the energy separation in the fluctuating wake. He also noticed that the energy 520 separation was particularly large when a strong sound was generated by the flow. 521 The existence of a low temperature field at the base of a cylinder was also observed by Sieverding in 522 1985, who used an infrared camera to visualize through a germanium window in the side wall of a 523 blow down wind tunnel the wall temperature field around a 15 mm diameter cylinder at = 0.4 , 524 see Figure 24. Unfortunately, due to a lack of time it was not possible to determine the absolute 518 Figure 23. Evolution of the recovery factor in the azimuthal direction from the stagnation point Figure 23. Evolution of the recovery factor r in the azimuthal direction from the stagnation point 525 temperature values. 519 ( =0°) to the rear side of the cylinder ( = 180° ); adapted from [44]. Commented [M45]: Please add explanation for subgraph. ( = 0 ) to the rear side of the cylinder ( = 180); adapted from [44]. 520 The authors suspected that the low values were possibly due to the intermittent separation of Commented [MM46R45]: These two pictures have to be 521 vortices from the cylinder. These results were confirmed by Ryan [45] in 1951 at Ackeret’s Institute considered as a single entity. 522 in Zürich who clearly related this low temperature to the periodic vortex shedding behind the Figure 24. Side wall temperature field around a cylinder recorded by an infrared camera. 526 Figure 24. Side wall temperature field around a cylinder recorded by an infrared camera. Eckert [46] explained the mechanism of energy separation along a flow path with the help of the 527 Eckert [46] explained the mechanism of energy separation along a flow path with the help of the unsteady energy equation: 528 unsteady energy equation: @p DT @ @T @ c = + k +   (4) p i ij Dt @t @x @x @x i i j |{z} | {z } | {z } (a) (b) (c) The change of the total temperature with time depends on: (a) the partial derivative of the pressure with time, (b) on the energy transport due to heat conduction between regions of di erent temperatures and (c) on the work due to viscous stresses between regions of di erent velocities. As regards the flow behind blu bodies the two latter terms are considered small compared the pressure gradient term and Equation (4) then reduces to: DT @p c = Dt @t The occurrence of total temperature variations in the vortex streets behind cylinders was e.g. extensively described by Kurosaka et al. [47], Ng et al. [48] and Sunduram et al. [49]. The progress in the understanding of the mechanism was boosted with the arrival of fast temperature probes as for example the dual sensor thin film platinum resistance thermometer probe developed by Buttsworth and Jones [50] in 1996 at Oxford. Using their technique Carscallen et al. [51,52] were the first to measure the time varying total pressure and temperature in the wake of their turbine Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 20 of 58 = + + (4) () () () 529 The change of the total temperature with time depends on: (a) the partial derivative of the pressure 530 with time, (b) on the energy transport due to heat conduction between regions of different 531 temperatures and (c) on the work due to viscous stresses between regions of different velocities. As 532 regards the flow behind bluff bodies the two latter terms are considered small compared the pressure 533 gradient term and Equation (4) then reduces to: 534 The occurrence of total temperature variations in the vortex streets behind cylinders was e.g. 535 extensively described by Kurosaka et al. [47], Ng et al. [48] and Sunduram et al. [49]. 536 The progress in the understanding of the mechanism was boosted with the arrival of fast 537 temperature probes as for example the dual sensor thin film platinum resistance thermometer probe 538 developed by Buttsworth and Jones [50] in 1996 at Oxford. Using their technique Carscallen et al. Int. J. Turbomach. Propuls. Power 2020, 5, 10 19 of 55 539 [51,52] were the first to measure the time varying total pressure and temperature in the wake of their 540 turbine vane. Figure 25 presents the results for an isentropic outlet Mach number =0.95 and a 541 vortex shedding frequency of the order of 10 kHz. The probe traverse plane was normal to the wake vane. Figure 25 presents the results for an isentropic outlet Mach number M = 0.95 and a vortex 2,is 542 at a distance of 5.76 trailing edge diameters from the vane trailing edge. In a later paper concerning shedding frequency of the order of 10 kHz. The probe traverse plane was normal to the wake at a 543 the same cascade, Gostelow and Rona [53], published also the corresponding entropy variations from distance of 5.76 trailing edge diameters from the vane trailing edge. In a later paper concerning the 544 the Gibb’s relation: same cascade, Gostelow and Rona [53], published also the corresponding entropy variations from the Gibb’s relation: T 02 − = 02− s s = c ln R ln 2 1 p T p 01 01 545 The results are presented in Figure 26. The results are presented in Figure 26. (a) (b) Figure 25. Contour plots of phase averaged total pressure (a) and total temperature (b) behind turbine 546 Figure 25. Contour plots of phase averaged total pressure (a) and total temperature (b) behind turbine Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 21 of 58 vane; adapted from [51]. 547 vane; adapted from [51]. Figure 26. Time resolved measurements of entropy increase [53]. 548 Figure 26. Time resolved measurements of entropy increase [53]. The variation of the maxima and minima of the total temperature in the center of the wake vary 549 The variation of the maxima and minima of the total temperature in the center of the wake vary between a minimum of 15 to a maximum of 4 with respect to the inlet ambient temperature, 550 between a minimum of −15° to a maximum of −4° with respect to the inlet ambient temperature, Figure 27. At the border of the wake the temperature raises considerably above the inlet temperature, 551 Figure 27. At the border of the wake the temperature raises considerably above the inlet temperature, while the time averaged temperature in the wake center is about10 . 552 while the time averaged temperature in the wake center is about −10°. 553 Figure 27. Variation of total temperature maxima and minima through the wake; adapted from [51]. 554 In 2004, Sieverding et al. [36] published very similar results for the turbine vane shown in Figure 555 17. The wake traverse was performed at a trailing edge distance of only 2.5 in direction of the 556 tangent to the blade camber line, which forms an angle of 66° with the axial direction. The traverse is 557 made normal to this tangent. 558 The steady state total pressure and total temperature measurements are presented in Figure 28. 559 Similar to the results obtained at the NRC Canada, the wake center is characterized by a pronounced 560 total temperature drop of 3% of the inlet value of 290 K which corresponds to about −9°, a variation 561 which is of the same order as that reported in Figure 27. On the borders of the wake, total temperature 562 peaks in excess of the inlet temperature are also recorded. The mass integrated total temperature 563 value across the wake (denoted with a <∗ ) should be such that < /< = 1 , but lack of 564 information on the local velocity did not allow to perform this integration. (a) Steady total pressure (b) Steady total temperature Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 21 of 58 548 Figure 26. Time resolved measurements of entropy increase [53]. 549 The variation of the maxima and minima of the total temperature in the center of the wake vary 550 between a minimum of −15° to a maximum of −4° with respect to the inlet ambient temperature, 551 Figure 27. At the border of the wake the temperature raises considerably above the inlet temperature, Int. J. Turbomach. Propuls. Power 2020, 5, 10 20 of 55 552 while the time averaged temperature in the wake center is about −10°. Figure 27. Variation of total temperature maxima and minima through the wake; adapted from [51]. 553 Figure 27. Variation of total temperature maxima and minima through the wake; adapted from [51]. In 2004, Sieverding et al. [36] published very similar results for the turbine vane shown in Figure 17. 554 In 2004, Sieverding et al. [36] published very similar results for the turbine vane shown in Figure The wake traverse was performed at a trailing edge distance of only 2.5 d in direction of the tangent te 555 17. The wake traverse was performed at a trailing edge distance of only 2.5 in direction of the to the blade camber line, which forms an angle of 66 with the axial direction. The traverse is made 556 tangent to the blade camber line, which forms an angle of 66° with the axial direction. The traverse is normal to this tangent. 557 made normal to this tangent. The steady state total pressure and total temperature measurements are presented in Figure 28. Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 22 of 58 558 The steady state total pressure and total temperature measurements are presented in Figure 28. Similar to the results obtained at the NRC Canada, the wake center is characterized by a pronounced 559 Simila562 r to the resul The tss obta teady stiat ne t ed ota al pr t the NRC Ca essure and tota na l teda mp, eth rate w ure m ae ke asure center ments is char are present acter ed in ized by Figure 2 8a prono . unced total temperature drop of 3% of the inlet value of 290 K which corresponds to about9 , a variation 563 Similar to the results obtained at the NRC Canada, the wake center is characterized by a pronounced 560 total temperature drop of 3% of the inlet value of 290 K which corresponds to about −9°, a variation which is of the same order as that reported in Figure 27. On the borders of the wake, total temperature 564 total temperature drop of 3% of the inlet value of 290 K which corresponds to about −9°, a variation 561 which is of the same order as that reported in Figure 27. On the borders of the wake, total temperature 565 which is of the same order as that reported in Figure 27. On the borders of the wake, total temperature peaks in excess of the inlet temperature are also recorded. The mass integrated total temperature 566 peaks in excess of the inlet temperature are also recorded. The mass integrated total temperature 562 peaks in excess of the inlet temperature are also recorded. The mass integrated total temperature value across the wake (denoted with a <  >) should be such that <T > /< T > = 1, but lack of 02 01 567 value across the wake (denoted with a <∗ ) should be such that < /< = 1 , but lack of 563 value across the wake (denoted with a <∗ ) should be such that < /< = 1 , but lack of information on the local velocity did not allow to perform this integration. 568 information on the local velocity did not allow to perform this integration. 564 information on the local velocity did not allow to perform this integration. (a) Steady total pressure (b) Steady total temperature (a) Steady total pressure (b) Steady total temperature (c) Unsteady total pressure (d) Unsteady total temperature Figure 28. Steady and time varying phase lock averaged total pressure and total temperature through 570 Figure 28. Steady and time varying phase lock averaged total pressure and total temperature through Commented [M49]: Please add explanation for subgraph. turbine vane wake at M = 0.79; adapted from [36]. 2,is 571 turbine vane wake at =0.79; adapted from [36]. Commented [MM50R49]: Done. Please do not cut the For the measurement of the time varying temperature a fast 2 m cold wire probe, developed by 572 For the measurement of the time varying temperature a fast 2 μm cold wire probe, developed figure on two pages. 573 by Denos and Sieverding [54], was used. Numerical compensation allowed to extend the naturally Denos and Sieverding [54], was used. Numerical compensation allowed to extend the naturally low 574 low frequency response of the probe to much higher ranges for adequate restitution of the nearly frequency response of the probe to much higher ranges for adequate restitution of the nearly sinusoidal 575 sinusoidal temperature variation associated with the vortex shedding frequency of 7.6 kHz at a temperature variation associated with the vortex shedding frequency of 7.6 kHz at a downstream 576 downstream isentropic Mach number of =0.79 . As regards the total pressure variation 577 ⁄ , minimum values of 0.768 are reached in the wake center while at the wake border maximum isentropic Mach number of M = 0.79. As regards the total pressure variation p /p , minimum 2,is 02 01 578 values of 1.061 are recorded. As regards the total temperature the authors quote maximum and values of 0.768 are reached in the wake center while at the wake border maximum values of 1.061 579 minimum total temperature ratios of ⁄ = 1.046 and 0.96, respectively. With a =290 are recorded. As regards the total temperature the authors quote maximum and minimum total 580 the maximum total temperature variations are of the order of 24°, similar to those reported by 581 Carscallen et al. [51]. However, the flow conditions were different: =0.79 at VKI, versus 0.95 582 at NRC Canada, and a distance of the wake traverses with respect to the trailing edge of 2.5 diameters 583 at VKI, versus 5.76 at NRC. 584 5. Effect of Vortex Shedding on Blade Pressure Distribution 585 The previous section focused on the unsteady character of turbine blade wake flows, the 586 visualization of the von Kármán vortices through smoke visualizations, schlieren photographs and 587 interferometric techniques. The measurement of the instantaneous velocity fields using LDV and PIV 588 techniques allowed to determine the vorticity distribution and the measurement of the unsteady total Int. J. Turbomach. Propuls. Power 2020, 5, 10 21 of 55 temperature ratios of T /T = 1.046 and 0.96, respectively. With a T = 290 K the maximum 02 01 01 total temperature variations are of the order of 24 , similar to those reported by Carscallen et al. [51]. However, the flow conditions were di erent: M = 0.79 at VKI, versus 0.95 at NRC Canada, and a 2,is distance of the wake traverses with respect to the trailing edge of 2.5 diameters at VKI, versus 5.76 at NRC. 5. E ect of Vortex Shedding on Blade Pressure Distribution The previous section focused on the unsteady character of turbine blade wake flows, the visualization of the von Kármán vortices through smoke visualizations, schlieren photographs and interferometric techniques. The measurement of the instantaneous velocity fields using LDV and PIV techniques allowed to determine the vorticity distribution and the measurement of the unsteady total pressure and temperature distribution putting into evidence the energy separation e ect in the wakes due to the von Kármán vortices. Naturally the vortex shedding a ects also the trailing edge pressure distribution and, beyond that, the suction side pressure distribution. The following is entirely based on research work carried out at the VKI by the team of the lead author, who was the only one to measure with high spatial resolution the pressure distribution around the trailing edge of a turbine blade. 5.1. E ect on Trailing Edge Pressure Distribution The very large-scale turbine guide vane designed and tested at the von Kármán Institute with a trailing edge thickness of 15 mm did allow an innovative approach for obtaining a high spatial resolution for the pressure distribution around the trailing edge. Cicatelli and Sieverding [32], fitted the blade with a rotatable 20 mm long cylinder in the center of the blade (Figure 29). The cylinder was equipped with a single Kulite fast response pressure sensor side by side with an ordinary pneumatic pressure tapping. The pressure sensor was mounted underneath the trailing edge surface with a slot width of only 0.2 mm to the outside, the same width as the pressure tapping, reducing the angular sensing area to only 1.53 . To control any e ect of the rear facing step between the blade lip and the rotatable trailing edge, a second blade was equipped with additional pressure sensors placed at, and slightly up-stream of, the trailing edge. The time averaged base pressure distribution, non-dimensionalized by the inlet total pressure, is presented in Figure 30. The circles denote data obtained with the rotatable trailing edge cylinder on blade A, while the triangles are measured with pressure tappings on blade B (see Figure 29), except for the two points “a” and “e” which are taken from the pressure tappings positioned aside the rotating cylinder on blade A (see Figure 30, left panel). The flow approaching the trailing edge undergoes, both on the pressure and suction side, a strong acceleration before separating from the trailing edge circle. The authors attribute the asymmetry to di erences in the blade boundary layers and to the blade circulation, which, following Han and Cox [26], strengthens the pressure side vortex shedding. Compared to the downstream Mach number M = 0.4, the local peak numbers are as high 2,is as M = 0.49 and 0.47, respectively. These high over-expansions are incompatible with a steady state max boundary layer separation and are attributed to the e ect of the vortex shedding. Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 23 of 58 601 The time averaged base pressure distribution, non-dimensionalized by the inlet total pressure, 602 is presented in Figure 30. The circles denote data obtained with the rotatable trailing edge cylinder 603 on blade A, while the triangles are measured with pressure tappings on blade B (see Figure 29), except 604 for the two points “a” and “e” which are taken from the pressure tappings positioned aside the 605 rotating cylinder on blade A (see Figure 30, left panel). The flow approaching the trailing edge 606 undergoes, both on the pressure and suction side, a strong acceleration before separating from the 607 trailing edge circle. The authors attribute the asymmetry to differences in the blade boundary layers 608 and to the blade circulation, which, following Han and Cox [26], strengthens the pressure side vortex 609 shedding. Compared to the downstream Mach number =0.4, the local peak numbers are as 610 high as =0.49 and 0.47, respectively. These high over-expansions are incompatible with a Int. J. Turbomach. Propuls. Power 2020, 5, 10 22 of 55 611 steady state boundary layer separation and are attributed to the effect of the vortex shedding. (b) (a) Figure 29. Blade instrumented with rotatable trailing edge (a: blade A) and control blade (b: blade B); 612 Figure 29. Blade instrumented with rotatable trailing edge (a: blade A) and control blade (b: blade B); adapted from [32]. 613 adapted from [32]. Figure 30, right panel, presents the corresponding root mean square of the pressure signal. 614 Figure 30, right panel, presents the corresponding root mean square of the pressure signal. Maximum pressure fluctuations of the order of 8% occur near the locations of the pressure minima, i.e., 615 Maximum pressure fluctuations of the order of 8% occur near the locations of the pressure minima, close to the boundary layer separation points, while the RMS/Q drops to nearly half this value in the 616 i.e. close to the boundary layer separation points, while the RMS/Q drops to nearly half this value in central region Int.of J. Tu the rbomach. trailing Propuls. Pow edge er 2018 base. , 3, x FOIt R Pis EERalso REVIEworth W noting that the pressure fluctuations 24 of 58 a ect also 617 the central region of the trailing edge base. It is also worth noting that the pressure fluctuations affect the flow upstream of the trailing edge. In the center of the base region there is an extended constant 618 also the flow upstream of the trailing edge. In the center of the base region there is an extended 624 also the flow upstream of the trailing edge. In the center of the base region there is an extended pressure plateau (Figure 30). The base pressure coecient corresponding to this plateau agrees well 625 constant pressure plateau (Figure 30). The base pressure coefficient corresponding to this plateau 619 constant pressure plateau (Figure 30). The base pressure coefficient corresponding to this plateau 626 agrees well with the Sieverding’s base pressure correlation. with the Sieverding’s base pressure correlation. 620 agrees well with the Sieverding’s base pressure correlation. (a) (b) 627 Figure 30. Phase lock averaged trailing edge pressure distribution (a) and root mean square values of Figure 30. Phase lock averaged trailing edge pressure distribution (a) and root mean square values of 628 the pressure fluctuations (b) for the VKI turbine blade at =0.4; adapted from [32]. the pressure fluctuations (b) for the VKI turbine blade at M = 0.4; adapted from [32]. 2,is 629 The base pressure distribution changes dramatically at high subsonic downstream Mach The base pressure distribution (a) changes dramatically at high subsonic (bdownstr ) eam Mach numbers 630 numbers as illustrated by Sieverding et al. [21], Figure 31. The pressure distribution is characterized 631 by the presence of three minima: the two pressure minima associated with the over-expansion of the as illustrated by Sieverding et al. [21], Figure 31. The pressure distribution is characterized by the 632 suction and pressure side flows before separation from the trailing edge, and an additional minimum presence of three minima: the two pressure minima associated with the over-expansion of the suction 633 around the center of the trailing edge circle. The pressure minima related to the overexpansion from and pr634 essur esuction a side flows nd pressu befor re sides e separation are of the ord from er of the ⁄ trailing =0.52 fo edge, r both sides, and an i.e. additional the local peak Ma minimum ch around 635 numbers are close to 1. Contrary to the low Mach number flow condition of Figure 30, the the center of the trailing edge circle. The pressure minima related to the overexpansion from suction 636 recompression following the over-expansion does not lead to a pressure plateau but gives way to a and pressure sides are of the order of p/p = 0.52 for both sides, i.e., The local peak Mach numbers are 637 new strong pressure drop reaching a minimum of ⁄ =0.485 at +7°. This is the result of the 638 enrolment of the separating shear layers into a vortex right at the trailing edge; the vortex core close to 1. Contrary to the low Mach number flow condition of Figure 30, the recompression following 639 approaches the wake centerline and its distance to the trailing edge becomes less than half the trailing the over-expansion does not lead to a pressure plateau but gives way to a new strong pressure drop 640 edge diameter, see smoke visualization and interferogram in Figure 17 and Figure 19. reaching a minimum of p/p = 0.485 at +7 . This is the result of the enrolment of the separating shear layers into a vortex right at the trailing edge; the vortex core approaches the wake centerline and its distance to the trailing edge becomes less than half the trailing edge diameter, see smoke visualization and interferogram in Figures 17 and 19. (a) (b) 641 Figure 31. Steady state trailing edge pressure distribution (a) and phase locked average pressure fluctuation (b) 642 around trailing edge; adapted from [21]. 643 The phase lock averaged pressure fluctuations around the trailing edge in Figure 31 are 644 impressive. They are naturally highest near the separation points of the boundary layer from the Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 24 of 58 624 also the flow upstream of the trailing edge. In the center of the base region there is an extended 625 constant pressure plateau (Figure 30). The base pressure coefficient corresponding to this plateau 626 agrees well with the Sieverding’s base pressure correlation. (a) (b) 627 Figure 30. Phase lock averaged trailing edge pressure distribution (a) and root mean square values of 628 the pressure fluctuations (b) for the VKI turbine blade at =0.4; adapted from [32]. 629 The base pressure distribution changes dramatically at high subsonic downstream Mach 630 numbers as illustrated by Sieverding et al. [21], Figure 31. The pressure distribution is characterized 631 by the presence of three minima: the two pressure minima associated with the over-expansion of the 632 suction and pressure side flows before separation from the trailing edge, and an additional minimum 633 around the center of the trailing edge circle. The pressure minima related to the overexpansion from 634 suction and pressure sides are of the order of ⁄ =0.52 for both sides, i.e. the local peak Mach 635 numbers are close to 1. Contrary to the low Mach number flow condition of Figure 30, the 636 recompression following the over-expansion does not lead to a pressure plateau but gives way to a 637 new strong pressure drop reaching a minimum of ⁄ =0.485 at +7°. This is the result of the 638 enrolment of the separating shear layers into a vortex right at the trailing edge; the vortex core 639 approaches the wake centerline and its distance to the trailing edge becomes less than half the trailing Int. J. Turbomach. Propuls. Power 2020, 5, 10 23 of 55 640 edge diameter, see smoke visualization and interferogram in Figure 17 and Figure 19. (a) (b) 641 Figure 31. Steady state trailing edge pressure distribution (a) and phase locked average pressure fluctuation (b) Figure 31. Steady state trailing edge pressure distribution (a) and phase locked average pressure 642 around trailing edge; adapted from [21]. fluctuation (b) around trailing edge; adapted from [21]. 643 The phase lock averaged pressure fluctuations around the trailing edge in Figure 31 are 644 impressive. They are naturally highest near the separation points of the boundary layer from the The phase lock averaged pressure fluctuations around the trailing edge in Figure 31 are impressive. They are naturally highest near the separation points of the boundary layer from the trailing edge where maximum values of around 100% of the dynamic pressure (p p ) are recorded. The minimum 01 2 pressure in a given position corresponds to the maximum inward motion of the separating shear layer, the maximum pressure to the maximum outward motion of the separating shear layer. The maximum local instantaneous Mach number at the point of the most inward position can be as high as M = 1.25. max The authors assumed that the curvature driven supersonic trailing edge expansion is the real reason for the formation of the vortex so close to the trailing edge, with the entrainment of high-speed free stream fluid into the trailing edge base region. In the center of the trailing edge the fluctuations drop to 20% of the dynamic head. The authors provide also some interesting information on the evolution of the pressure signal on the trailing edge circle over one complete vortex shedding cycle. This is demonstrated in Figure 32 showing the evolution for the phase locked average pressure at the angular position of 60 on the pressure side of the trailing edge circle. A decrease of the pressure indicates an acceleration of the flow around the trailing edge i.e., The separating shear layer moves inwards, the vortex is in its formation phase. An increase of the pressure indicates on the contrary an outwards motion of the shear layer, the vortex is in its shedding phase. Surprisingly, the pressure rise time is much shorter than the pressure fall time, i.e., The time for the vortex formation is longer than that for the vortex shedding. The same was observed for the pressure evolution on the opposite side of the trailing edge, but of Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 25 of 58 course with 180 out of phase. Figure 32. Phase locked average pressure variation at trailing edge at an angular position of60 [21]. 657 Figure 32. Phase locked average pressure variation at trailing edge at an angular position of −60° [21]. The change of an isobaric pressure zone over an extended region at the base of the trailing edge at 658 The change of an isobaric pressure zone over an extended region at the base of the trailing edge an exit Mach number M = 0.4 to a highly non-uniform pressure distribution with a strong pressure 2,is 659 at an exit Mach number =0.4 to a highly non-uniform pressure distribution with a strong 660 pressure minimum at the center of the trailing edge circle at =0.79, did of course raise the 661 question about the evolution of the trailing edge pressure distribution over the entire Mach number 662 range, from low subsonic to transonic Mach numbers. To respond to this lack of information a 663 research program was carried out by Mateos Prieto [55] at VKI as part of his diploma thesis in 2003. 664 For various reasons, the data were not published at that time but only in 2015, as part of the paper of 665 Vagnoli et al. [56] on the prediction of unsteady turbine blade wake flow characteristics and 666 comparison with experimental data, see Figure 33. 667 Figure 33. Effect of downstream Mach number on trailing edge Mach number distribution [56]. 668 The figure puts clearly into evidence the effect of the vortex shedding on the trailing edge 669 pressure distribution. Up to about ≈0.65 the trailing edge base region is characterized by an 670 extended, nearly isobaric, pressure plateau which implies that the vortex formation occurs 671 sufficiently far downstream not to affect the trailing edge base region. With increasing Mach number, 672 the enrolment of the shear layers into vortices occurs gradually closer to the trailing edge causing an 673 increasingly non-uniform pressure distribution with a marked pressure minimum at the center of the 674 trailing edge. To characterize the degree of non-uniformity the authors define a factor : Int. J. Turbomach. Propuls. Power 2020, 5, 10 24 of 55 minimum Int. J. Turbomat achthe . Prop center uls. Power of the 2018trailing , 3, x FOR PE edge ER RE circle VIEW at M = 0.79, did of course raise the question 25 of about 58 2,is the evolution of the trailing edge pressure distribution over the entire Mach number range, from low subsonic to transonic Mach numbers. To respond to this lack of information a research program was carried out by Mateos Prieto [55] at VKI as part of his diploma thesis in 2003. For various reasons, the data were not published at that time but only in 2015, as part of the paper of Vagnoli et al. [56] on the prediction of unsteady turbine blade wake flow characteristics and comparison with experimental data, see Figure 33. The figure puts clearly into evidence the e ect of the vortex shedding on the trailing edge pressure distribution. Up to about M  0.65 the trailing edge base region is characterized by an extended, 2,is nearly isobaric, pressure plateau which implies that the vortex formation occurs suciently far downstream not to a ect the trailing edge base region. With increasing Mach number, the enrolment of the shear layers into vortices occurs gradually closer to the trailing edge causing an increasingly non-uniform pressure distribution with a marked pressure minimum at the center of the trailing edge. To characterize the degree of non-uniformity the authors define a factor Z: Z = p p /(p p ) (5) b,max b,min 01 2 657 Figure 32. Phase locked average pressure variation at trailing edge at an angular position of −60° [21]. where p is the maximum pressure following the recompression after the separation of the shear b,max 658 The change of an isobaric pressure zone over an extended region at the base of the trailing edge layer from the trailing edge and p the minimum pressure near the center of the trailing edge. b,min 659 at an exit Mach number =0.4 to a highly non-uniform pressure distribution with a strong The maximum degree of non-uniformity is reached at M = 0.93 with a Z value of 21%. At this 2,is 660 pressure minimum at the center of the trailing edge circle at =0.79, did of course raise the Mach number the minimum pressure reaches a value of p /p = 0.325 for a downstream pressure b,min 01 661 question about the evolution of the trailing edge pressure distribution over the entire Mach number p /p = 0.572. With further increase of the Mach number, Z starts to decrease rapidly. It decreases to 2 01 662 range, from low subsonic to transonic Mach numbers. To respond to this lack of information a Z = 12% at M = 0.99 and drops to zero at M = 1.01. For this Mach number the local trailing edge 2,is 2,is 663 research program was carried out by Mateos Prieto [55] at VKI as part of his diploma thesis in 2003. conditions are such that oblique shocks emerge from the region of the confluence of the suction and 664 For various reasons, the data were not published at that time but only in 2015, as part of the paper of pressure side shear layers and the vortex formation is delayed to after this region as shown e.g. In the 665 Vagnoli et al. [56] on the prediction of unsteady turbine blade wake flow characteristics and schlieren picture by Carscallen and Gostelow [39] at the NRC Canada in Figure 34, left, and another 666 comparison with experimental data, see Figure 33. schlieren picture taken at VKI in Figure 34, right (unpublished). Figure 33. E ect of downstream Mach number on trailing edge Mach number distribution [56]. 667 Figure 33. Effect of downstream Mach number on trailing edge Mach number distribution [56]. 668 The figure puts clearly into evidence the effect of the vortex shedding on the trailing edge 669 pressure distribution. Up to about ≈0.65 the trailing edge base region is characterized by an 670 extended, nearly isobaric, pressure plateau which implies that the vortex formation occurs 671 sufficiently far downstream not to affect the trailing edge base region. With increasing Mach number, 672 the enrolment of the shear layers into vortices occurs gradually closer to the trailing edge causing an 673 increasingly non-uniform pressure distribution with a marked pressure minimum at the center of the 674 trailing edge. To characterize the degree of non-uniformity the authors define a factor : Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 26 of 58 = ( − )/( − ) (5) , , 675 where is the maximum pressure following the recompression after the separation of the shear 676 layer from the trailing edge and the minimum pressure near the center of the trailing edge. 677 The maximum degree of non-uniformity is reached at =0.93 with a value of 21%. At this 678 Mach number the minimum pressure reaches a value of ⁄ = 0.325 for a downstream 679 pressure ⁄ = 0.572. With further increase of the Mach number, starts to decrease rapidly. It 680 decreases to = 12% at =0.99 and drops to zero at =1.01. For this Mach number the , , 681 local trailing edge conditions are such that oblique shocks emerge from the region of the confluence 682 of the suction and pressure side shear layers and the vortex formation is delayed to after this region 683 as shown e.g. in the schlieren picture by Carscallen and Gostelow [39] at the NRC Canada in Figure Int. J. Turbomach. Propuls. Power 2020, 5, 10 25 of 55 684 34, left, and another schlieren picture taken at VKI in Figure 34, right (unpublished). (a) (b) Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 27 of 58 Figure 34. At fully established oblique trailing edge shock system the vortex formation is delayed to 685 Figure 34. At fully established oblique trailing edge shock system the vortex formation is delayed to 693 As already pointed out at the end of Section 2 the departure from the generally assumed isobaric after the point of confluence of the blade shear layers. NRC blade (a) [39], VKI blade (b). 694 trailing edge base region may explain the differences of base pressure data published by different 686 after the point of confluence of the blade shear layers. NRC blade (a) [39], VKI blade (b). 695 authors at high subsonic/transonic downstream Mach numbers. The scatter between experimental 696 data from different research organizations may be partially due to the use of very different ratios of As already pointed out at the end of Section 2 the departure from the generally assumed isobaric 687 As already pointed out at the end of Section 2 the departure from the generally assumed isobaric 697 the diameter of the trailing edge pressure hole to the trailing edge diameter, / . Small / ratios trailing edge base region may explain the di erences of base pressure data published by di erent 688 trailing 698 edge m base region may ay lead to an overestimat exp ionl of ain thethe differenc base pressure effe ect s of base pr . Hence, base pessu ressure re d meaa sta published by urements should different authors at high subsonic/transonic downstream Mach numbers. The scatter between experimental 699 be taken with a / ratio as large as possible. 689 authors at high subsonic/transonic downstream Mach numbers. The scatter between experimental 700 The existence of an isobaric base pressure region for supersonic trailing edge flows, i.e. for blades data from di erent research organizations may be partially due to the use of very di erent ratios of the 690 data from different research organizations may be partially due to the use of very different ratios of 701 with a well-established oblique trailing edge shock system as those in Figure 34, was already known diameter of the trailing edge pressure hole to the trailing edge diameter, d/D. Small d/D ratios may 702 from flat plate model tests simulating the overhang section of convergent turbine blades with straight 691 the diameter of the trailing edge pressure hole to the trailing edge diameter, / . Small / ratios lead to703 an over reaestimation r suction side sin ofcthe e 1976 base [18], see pressur Figure e 3e 5 . Th ect. e teHence, sts were per base formed f pressur or a ga eumeasur ging anglements e = should be 692 may lead to an overestimation of the base pressure effect. Hence, base pressure measurements should 704 30°. The inclination of the tail board attached to the lower nozzle block allows to increase the taken with a d/D ratio as large as possible. 693 be taken with a / ratio as large as possible. 705 downstream Mach number which entails of course the displacement of the suction side shock The existence of an isobaric base pressure region for supersonic trailing edge flows, i.e., for blades 694 The existence 706 boundaof an isobar ry interaction alo ic nbase pressur g the blade suctie on region side towa fo rds th r sue personic t trailing edgr ea . i ling edge flows, i.e. for blades with a well-established oblique trailing edge shock system as those in Figure 34, was already known 707 The schlieren photograph on the right in Figure 34 shows the occurrence of so-called lip shocks 695 with a well-established oblique trailing edge shock system as those in Figure 34, was already known 708 at the separation of the shear layers from the trailing edge due to a slight overturning and a non- from flat plate model tests simulating the overhang section of convergent turbine blades with straight 696 from flat plate model tests simulating the overhang section of convergent turbine blades with straight 709 tangential separation of the flow from the trailing edge surface. In a later test series with a denser rear suction side since 1976 [18], see Figure 35. The tests were performed for a gauging angle = 30. 710 instrumentation of the trailing edge, Sieverding et al. [4] showed that the trailing edge shock strength 2 697 rear suction side since 1976 [18], see Figure 35. The tests were performed for a gauging angle = The inclination 711 was h ofothe wevtail er weak. boar In d Fattached igure 36 the pressure i to the lower ncrease nozzle ⁄ block across th allows e lip shto ock incr is presented ease the in downstream 698 30°. The inclination of the tail board attached to the lower nozzle block allows to increase the 712 function of the expansion ratio around the trailing edge ⁄ , where is the pressure before the Mach number which entails of course the displacement of the suction side shock boundary interaction 699 downstream Mach number which entails of course the displacement of the suction side shock 713 start of the expansion around the trailing edge and the pressure before the lip shock. All data are along the blade suction side towards the trailing edge. 714 within a bandwidth of ⁄ = 1.1 − 1.2. 700 boundary interaction along the blade suction side towards the trailing edge. 701 The schlieren photograph on the right in Figure 34 shows the occurrence of so-called lip shocks 702 at the separation of the shear layers from the trailing edge due to a slight overturning and a non- 703 tangential separation of the flow from the trailing edge surface. In a later test series with a denser 704 instrumentation of the trailing edge, Sieverding et al. [4] showed that the trailing edge shock strength 705 was however weak. In Figure 36 the pressure increase across the lip shock is presented in 706 function of the expansion ratio around the trailing edge ⁄ , where is the pressure before the 707 start of the expansion around the trailing edge and the pressure before the lip shock. All data are 708 within a bandwidth of ⁄ = 1.1 − 1.2. (a) (b) 715 Figure 35. Surface (a) and trailing edge (b) pressure distribution for flat plate model tests; adapted Figure 35. Surface (a) and trailing edge (b) pressure distribution for flat plate model tests; adapted Commented [M57]: Please add the add explanation for 716 from [18]. from [18]. subgraph(a, left) Commented [MM58R57]: Done. The schlieren photograph on the right in Figure 34 shows the occurrence of so-called lip shocks at the separation of the shear layers from the trailing edge due to a slight overturning and a non-tangential separation of the flow from the trailing edge surface. In a later test series with a denser instrumentation of the trailing edge, Sieverding et al. [4] showed that the trailing edge shock strength was however weak. In Figure 36 the pressure increase p /p across the lip shock is presented in function of the 3 2 expansion ratio around the trailing edge p /p , where p is the pressure before the start of the expansion 2 1 1 717 Figure 36. Strength of the trailing edge lip shocks; adapted from [4]. around the trailing edge and p the pressure before the lip shock. All data are within a bandwidth of 2 Commented [M59]: Please add explanation for subgraph. p /p = 1.1–1.2. 3 2 718 Raffel and Kost [57] performed similar large-scale tests on a flat plate with a slotted trailing edge Commented [MM60R59]: The graph and the sketch have to 719 to investigate the effect of trailing edge blowing on the formation of the trailing edge vortex street. be considered as a single entity. 720 Their trailing edge pressure distributions taken at zero coolant flow ejection for downstream Mach Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 27 of 58 693 As already pointed out at the end of Section 2 the departure from the generally assumed isobaric 694 trailing edge base region may explain the differences of base pressure data published by different 695 authors at high subsonic/transonic downstream Mach numbers. The scatter between experimental 696 data from different research organizations may be partially due to the use of very different ratios of 697 the diameter of the trailing edge pressure hole to the trailing edge diameter, / . Small / ratios 698 may lead to an overestimation of the base pressure effect. Hence, base pressure measurements should 699 be taken with a / ratio as large as possible. 700 The existence of an isobaric base pressure region for supersonic trailing edge flows, i.e. for blades 701 with a well-established oblique trailing edge shock system as those in Figure 34, was already known 702 from flat plate model tests simulating the overhang section of convergent turbine blades with straight 703 rear suction side since 1976 [18], see Figure 35. The tests were performed for a gauging angle = 704 30°. The inclination of the tail board attached to the lower nozzle block allows to increase the 705 downstream Mach number which entails of course the displacement of the suction side shock 706 boundary interaction along the blade suction side towards the trailing edge. 707 The schlieren photograph on the right in Figure 34 shows the occurrence of so-called lip shocks 708 at the separation of the shear layers from the trailing edge due to a slight overturning and a non- 709 tangential separation of the flow from the trailing edge surface. In a later test series with a denser 710 instrumentation of the trailing edge, Sieverding et al. [4] showed that the trailing edge shock strength 711 was however weak. In Figure 36 the pressure increase ⁄ across the lip shock is presented in 712 function of the expansion ratio around the trailing edge ⁄ , where is the pressure before the 713 start of the expansion around the trailing edge and the pressure before the lip shock. All data are 714 within a bandwidth of ⁄ = 1.1 − 1.2. (a) (b) 715 Figure 35. Surface (a) and trailing edge (b) pressure distribution for flat plate model tests; adapted Int. J. Turbomach. Propuls. Power 2020, 5, 10 26 of 55 Commented [M57]: Please add the add explanation for 716 from [18]. subgraph(a, left) Commented [MM58R57]: Done. 717 FiguFigure re 36. Stre 36. ngthStr of th ength e trailin of g ed the ge trailing lip shocks; adapt edge elip d fro shocks; m [4]. adapted from [4]. Commented [M59]: Please add explanation for subgraph. 718 Raffel and Kost [57] performed similar large-scale tests on a flat plate with a slotted trailing edge Commented [MM60R59]: The graph and the sketch have to Ra el and Kost [57] performed similar large-scale tests on a flat plate with a slotted trailing edge to 719 to investigate the effect of trailing edge blowing on the formation of the trailing edge vortex street. be considered as a single entity. investigate the e ect of trailing edge blowing on the formation of the trailing edge vortex street. Their 720 Their trailing edge pressure distributions taken at zero coolant flow ejection for downstream Mach trailing edge pressure distributions taken at zero coolant flow ejection for downstream Mach numbers of M = 1.01 1.45 confirm the existence of an isobaric pressure trailing edge base region, but the 2,is measurements are unfortunately not dense enough to extract consistent data about the strength of the lip shock. A few data allow to conclude that in their experiments the maximum lip shock strength is of the order of p /p = 1.08. 3 2 Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 28 of 58 721 numbers of = 1.01 − 1.45 confirm the existence of an isobaric pressure trailing edge base 5.2. E ect on Blade Suction Side Pressure Distribution 722 region, but the measurements are unfortunately not dense enough to extract consistent data about In 723 the discussion the strength of ofthe the lip schlier shock. en A fephotographs w data allow to cin oncl Figur ude tha e 18 t in their it was experi shown mentsthat the m the axim outwar um ds motion 724 lip shock strength is of the order of / =1.08. of the oscillating shear layers at the blade trailing edge does not only lead to large pressure fluctuations in the zone of separations, but it does also induce strong acoustic pressure waves travelling upstream 725 5.2. Effect on Blade Suction Side Pressure Distribution on both the suction and pressure side of the blade. To facilitate the understanding of the suction side 726 In the discussion of the schlieren photographs in Figure 18 it was shown that the outwards 727 motion of the oscillating shear layers at the blade trailing edge does not only lead to large pressure pressure fluctuations in Figure 37, the left photo of the schlieren pictures in Figure 18 is reproduced 728 fluctuations in the zone of separations, but it does also induce strong acoustic pressure waves at the right of the pressure curves. The wave Pi generated at the pressure side will interact with the 729 travelling upstream on both the suction and pressure side of the blade. To facilitate the understanding suction side of the neighboring blade causing significant unsteady pressure variations as measured by 730 of the suction side pressure fluctuations in Figure 37, the left photo of the schlieren pictures in Figure fast response 731 18 i pr s repro essur duced at e sensors the righ implemented t of the pressure curves between . The w the ave thr Pi ge oat nerated andat t the he p trailing ressure side w edge ill of this blade, 732 interact with the suction side of the neighboring blade causing significant unsteady pressure see Figure 37. The pressure wave P induced by the outwards motion of the pressure side shear 733 variations as measured by fast response pressure sensors implemented between the throat and the layer of the neighboring blade intersects the suction side between the sensors 3 and 4. It moves then 734 trailing edge of this blade, see Figure 37. The pressure wave induced by the outwards motion of 735 the pressure side shear layer of the neighboring blade intersects the suction side between the sensors successively upstream across the sensors 3 and 2. The signals are asymmetric, characterized by a sharp 736 3 and 4. It moves then successively upstream across the sensors 3 and 2. The signals are asymmetric, pressure rise followed by a slow decay. The amplitude of the pressure fluctuations is important with 737 characterized by a sharp pressure rise followed by a slow decay. The amplitude of the pressure Dp = 12% up to 15% of (p p ) at sensor 3, and10% at sensor 2, while the pressure signal is flat at 738 fluctuations is impo01 rtant with Δ = 12 % up to 15% of ( − ) at sensor 3, and 10% at sensor 739 2, while the pressure signal is flat at sensor 1 situated slightly up-stream of the geometric throat where sensor 1 situated slightly up-stream of the geometric throat where the blade Mach number reaches 740 the blade Mach number reaches =0.95. The pressure waves observed at sensor 4 and further M = 0.95. The pressure waves observed at sensor 4 and further downstream at sensors 5 and 6 are 2,is 741 downstream at sensors 5 and 6 are more sinusoidal in nature and of smaller amplitude. The authors more sinusoidal in nature and of smaller amplitude. The authors suggested that these fluctuations are 742 suggested that these fluctuations are likely to be caused by the downstream travelling vortices of the 743 neighboring blade. likely to be caused by the downstream travelling vortices of the neighboring blade. (b) (a) (c) 745 Figure 37. Unsteady pressure variations along rear suction side (a); schlieren photograph (b); kulite Commented [M61]: Please add explanation for subgraph. Figure 37. Unsteady pressure variations along rear suction side (a); schlieren photograph (b); kulite 746 sensor positioning (c). Adapted from [21]. sensor positioning (c). Adapted from [21]. Commented [MM62R61]: Done. 747 The periodicity of the pressure signal at position 7, slightly upstream of the trailing edge, is 748 rather poor and only phase lock averaging provides useful information on its periodic character. The 749 reason is most likely the result of a superposition of waves induced by the von Kármán vortices in 750 the wake of the neighboring blade and upstream travelling waves induced by the oscillation of the 751 suction side shear layer designated by “S” in the schlieren photographs. Right at the trailing edge, 752 position 11, we have, as expected, strong periodic signals associated with the oscillating shear layers. Int. J. Turbomach. Propuls. Power 2020, 5, 10 27 of 55 The periodicity of the pressure signal at position 7, slightly upstream of the trailing edge, is rather poor and only phase lock averaging provides useful information on its periodic character. The reason is most likely the result of a superposition of waves induced by the von Kármán vortices in the wake of the neighboring blade and upstream travelling waves induced by the oscillation of the suction side shear layer designated by “S” in the schlieren photographs. Right at the trailing edge, position 11, we have, as expected, strong periodic signals associated with the oscillating shear layers. 6. Turbine Trailing Edge Vortex Frequency Shedding Besides the importance of trailing edge vortex shedding for the wake mixing process and the trailing edge pressure distribution discussed before, vortex shedding deserves also special attention due to its importance as excitation for acoustic resonances and structural vibrations. Heinemann & Bütefisch [25], investigated 10 subsonic and transonic turbine cascades: two flat plate turbine tip sections, three mid-sections with nearly axial inlet (one blade tested with three di erent trailing edge thicknesses) and 3 high turning hub sections. The trailing edge thickness varied from 0.8% to 5%. The vortex shedding frequency was determined with an electronic-optical method developed at the DFVLR-AVA by Heinemann et al. [58]. The corresponding Strouhal numbers defined in (3) covered a wide range: 0.2  St  0.4 for a Reynolds number range based on trailing edge thickness and downstream isentropic velocity of 4 5 0.3 10  Re  1.6 10 . The Strouhal numbers for flows from cylinders over the same Reynolds Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 29 of 58 number range are of the order of St = 0.19 and 0.21 as shown in Figure 38. Figure 38. Strouhal numbers in sub-critical Reynolds number range for flow over cylinders; adapted 761 Figure 38. Strouhal numbers in sub-critical Reynolds number range for flow over cylinders; adapted from [59]. 762 from [59]. Figure 39 presents the Strouhal numbers for 3 of the 10 turbine blade sections investigated by 763 Figure 39 presents the Strouhal numbers for 3 of the 10 turbine blade sections investigated by Heinemann and Bütefisch [25]. The comparison with the flow across cylinders is limited to the subsonic 764 Heinemann and Bütefisch [25]. The comparison with the flow across cylinders is limited to the range. The Strouhal numbers for the flat plate tip section T2 are of the order of St = 0.2 in the Mach 765 subsonic range. The Strouhal numbers for the flat plate tip section T2 are of the order of = 0.2 in range M = 0.2 to 0.8 and therewith very close to the those of the flow over cylinders. On the other 2,is 766 the Mach range = 0.2 to 0.8 and therewith very close to the those of the flow over cylinders. On side, the hub section H2 with a high rear suction side curvature distinguishes itself by Strouhal numbers 767 the other side, the hub section H2 with a high rear suction side curvature distinguishes itself by as high as 0.38 0.3, with a decreasing tendency from M = 0.2 to 0.9. For the mid-section M2 with 2,is 768 Strouhal numbers as high as 0.38 – 0.3, with a decreasing tendency from = 0.2 to 0.9. For the low rear suction side turning, the authors report Strouhal numbers increasing from St = 0.22 to 0.29 for 769 mid-section M2 with low rear suction side turning, the authors report Strouhal numbers increasing a Mach range 0.2 to 0.8. 770 from = 0.22 to 0.29 for a Mach range 0.2 to 0.8. (a) steam turbine tip section T2 (b) steam turbine mid section M2 (c) gas turbine hub section H2 771 Figure 39. Strouhal number for 3 blade sections. : and based on isentropic downstream 772 velocity. : and based on homogeneous downstream velocity. Adapted from [25]. 773 Additional information on turbine blade trailing edge frequency measurements were published 774 by Sieverding [60] who used fast response pressure sensors implemented in the blade trailing edge 775 and in a total pressure probe positioned at short distance from the trailing edge, while Bryanston- 776 Cross and Camus [61] made use of a 20 MHz bandwidth digital correlator combined with 777 conventional schlieren optics. The Strouhal numbers of Sieverding’s rotor blade with a straight rear 778 suction side were in the lower part of the band width of the DFVLR-AVA data, while those of 779 Bryanston-Cross and Camus rotor blades with higher suction side curvature resided in the upper 780 part. 781 The large range of Strouhal numbers were possibly due to differences in the state of the 782 boundary layers at the point of separation. Besides that, the vortex shedding frequency does not 783 simply depend on the trailing edge thickness augmented by the boundary layer displacement Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 10 of 58 271 suction side for both blades are exactly the same, but contrary to the nearly constant Mach number 272 for the VKI blade downstream of the throat, the blade of Xu and Denton is characterized by a very 273 strong adverse pressure gradient in this region. As pointed out by the authors, this causes the suction 274 side boundary layer to be either separated or close to separation up-stream of the trailing edge. 275 Clearly, Sieverding’s correlation cannot deal with blade designs characterized by very strong adverse 276 pressure gradients on the rear suction side causing boundary layer separation before the trailing 277 edge. Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 29 of 58 Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 29 of 58 753 6. Turbine Trailing Edge Vortex Frequency Shedding 754 Besides the importance of trailing edge vortex shedding for the wake mixing process and the 755 trailing edge pressure distribution discussed before, vortex shedding deserves also special attention 756 due to its importance as excitation for acoustic resonances and structural vibrations. Heinemann & 757 Bütefisch [25], investigated 10 subsonic and transonic turbine cascades: two flat plate turbine tip 758 sections, three mid-sections with nearly axial inlet (one blade tested with three different trailing edge 759 thicknesses) and 3 high turning hub sections. The trailing edge thickness varied from 0.8% to 5%. The 760 vortex shedding frequency was determined with an electronic-optical method developed at the 761 DFVLR-AVA by Heinemann et al. [58]. 278 Figure 9. Comparison of blade Mach number distribution for blade RD of Xu and Denton [15], (solid 279 curve, =0.8, =8 10 ) with VKI blade (dashed curve, =0.8, =10 ). , , 280 The possible effect of boundary layer separation resulting from high rear suction side diffusion 281 resulting in high base pressures was also mentioned by Corriveau and Sjolander in 2004 [20], 282 comparing their nominal mid-loaded rotor blade HS1A, mentioned already before, with an aft- 761 Figure 38. Strouhal numbers in sub-critical Reynolds number range for flow over cylinders; adapted 283 loaded blade HS1C with an increase of the suction side unguided turning angle from 11.5° to 14.5°. 762 from [59]. 284 It appears that the increased turning angle could cause, in the transonic range, shock induced 285 boundary layer transition near the trailing edge with, as consequence, a sharp increase of the base 763 Figure 39 presents the Strouhal numbers for 3 of the 10 turbine blade sections investigated by 286 pressure, i.e. a sudden drop in the base pressure coefficient as seen in Figure 10. Note that the 764 Heinem 762 ann and B Figure 38. ütefi Str sc oh [2 uhal number 5]. The c s in sub-cr omparison w itical Reynolds num ith b the f er range for low flow a ov cross cyli er cylindersnders is li ; adapted mited to the 287 reported in the figure has been converted to − of the original data. 763 from [59]. 765 subsonic range. The Strouhal numbers for the flat plate tip section T2 are of the order of = 0.2 in 288 As regards the base pressure data by Deckers and Denton [13] for a low turning blade model 766 the Mach range = 0.2 to 0.8 and therewith very close to the those of the flow over cylinders. On 764 The co , rresponding Strouhal numbers defined in (3) covered a wide range: 0.2 0.4 for a 289 and Gostelow et al. [14] for a high turning nozzle guide vane, who report base pressure data far below 765 Reynolds number range based on trailing edge thickness and downstream isentropic velocity of 767 the other side, the hub section H2 with a high rear suction side curvature distinguishes itself by 766 0.3 10 1.6 10 . The Strouhal numbers for flows from cylinders over the same Reynolds 290 those of Sieverding’s BPC, their blade pressure distribution resembles that of the convergent/ 768 Strouhal numbers as high as 0.38 – 0.3, with a decreasing tendency from = 0.2 to 0.9. For the 767 number range are of the order of = 0.19 and 0.21 as shown in Figure 38. 291 divergent blade C in Figure 3 with a negative blade loading near the trailing edge which would 769 mid-section M2 with low rear suction side turning, the authors report Strouhal numbers increasing 768 Figure 39 presents the Strouhal numbers for 3 of the 10 turbine blade sections investigated by 292 explain the very low base pressures. In addition, the blade of Deckers and Denton has a blunt trailing 769 Heinemann and Bütefisch [25]. The comparison with the flow across cylinders is limited to the 770 from = 0.22 to 0.29 for a Mach range 0.2 to 0.8. 770 subsonic range. The Strouhal numbers for the flat plate tip section T2 are of the order of = 0.2 in 293 edge, and there is experimental evidence that, compared to a circular trailing edge, the base pressure 771 the Mach range = 0.2 to 0.8 and therewith very close to the those of the flow over cylinders. On 294 for blades with blunt trailing edge might be considerably lower. Sieverding and Heinemann [16] 772 the other side, the hub section H2 with a high rear suction side curvature distinguishes itself by 295 report for flat plate tests at moderate subsonic Mach numbers a drop of the base pressure coefficient 773 Strouhal numbers as high as 0.38 – 0.3, with a decreasing tendency from = 0.2 to 0.9. For the Int. J. Turbomach. Propuls. Power 2020, 5, 10 28 of 55 774 mid-section M2 with low rear suction side turning, the authors report Strouhal numbers increasing 296 by 11% for a plate with squared trailing edge compared to that with a circular trailing edge. 775 from = 0.22 to 0.29 for a Mach range 0.2 to 0.8. (a) steam turbine tip section T2 (b) steam turbine mid section M2 (c) gas turbine hub section H2 (a) steam turbine tip section T2 (b) steam turbine mid section M2 (c) gas turbine hub section H2 776 Figure 39. Strouhal number for 3 blade sections. : and based on isentropic downstream 771 Figure Figure 39. 39. Str Stro ouhal uhal number for 3 blade sect number for 3 blade sections. ions. : : St and and M based based on i on isentr sentr opic opic downstr downstream eam 297 Figure 10. Base pressure coefficient for mid-loaded (solid line) and aft2 -loaded (dashed line) rotor 777 velocity. : and based on homogeneous downstream velocity. Adapted from [25]. 772 velocity velocity .. : : St and and M based based on on homogeneous homogeneou downstr s downstream eam velocity velocity . Adapted . Adaptefr d om from [25 [25]. ]. 298 blade. Symbols: HS1A geom etry, HS1C geometry. Adapted from [20]. Additional information on turbine blade trailing edge frequency measurements were published 773 Additional information on turbine blade trailing edge frequency measurements were published by Sieverding [60] who used fast response pressure sensors implemented in the blade trailing edge and 774 by Sieverding [60] who used fast response pressure sensors implemented in the blade trailing edge in a total pressure probe positioned at short distance from the trailing edge, while Bryanston-Cross and 775 and in a total pressure probe positioned at short distance from the trailing edge, while Bryanston- Camus [61] made use of a 20 MHz bandwidth digital correlator combined with conventional schlieren 776 Cross and Camus [61] made use of a 20 MHz bandwidth digital correlator combined with optics. The Strouhal numbers of Sieverding’s rotor blade with a straight rear suction side were in the 777 conventional schlieren optics. The Strouhal numbers of Sieverding’s rotor blade with a straight rear lower part of the band width of the DFVLR-AVA data, while those of Bryanston-Cross and Camus 778 suction side were in the lower part of the band width of the DFVLR-AVA data, while those of rotor blades with higher suction side curvature resided in the upper part. 779 Bryanston-Cross and Camus rotor blades with higher suction side curvature resided in the upper The large range of Strouhal numbers were possibly due to di erences in the state of the boundary 780 part. layers at the point of separation. Besides that, the vortex shedding frequency does not simply depend 781 The large range of Strouhal numbers were possibly due to differences in the state of the on the trailing edge thickness augmented by the boundary layer displacement thickness, which, 782 boundary layers at the point of separation. Besides that, the vortex shedding frequency does not however, is in general not known, but rather on the e ective distance between the separating shear 783 simply depend on the trailing edge thickness augmented by the boundary layer displacement layers which could be significantly smaller than the trailing edge thickness. Patterson & Weingold [62], simulating a compressor airfoil trailing edge flow field on a flat plate, concluded that, compared to the e ective distance between the separating upper and lower shear layers, the state of the boundary layer before separation played a much more important role. The influence of the boundary layer state and of the e ective distance of the separating shear layers was specifically addressed in a series of cascade and flat plate tests investigated by Sieverding & Heinemann [16], at VKI and DLR. Figure 40 shows the blade surface isentropic Mach number distributions of a front loaded blade, with the particularity of a straight rear suction side (blade A), and a rear loaded blade (blade C), characterized by a high rear suction side turning angle, at a downstream Mach number of M  0.8. 2is Figure 40. Blade Mach number distributions for front and rear-loaded blades; adapted from [16]. Figure 40. Blade Mach number distributions for front and rear-loaded blades; adapted from [16]. The early suction side velocity peak on blade A will cause early boundary layer transition. On the contrary, considering the weak velocity peak on the rear suction side followed by a very moderate recompression, the suction side boundary layer of blade C is likely to be laminar at the trailing edge over a large range of Reynolds numbers. As regards the pressure sides of both blades, the strong Blade 𝜶 𝜶 g/c Ref 𝟏 𝟐 A 30° 22° 0.75 [5] B 60° 25° 0.75 VKI C 66° 18° 0.70 VKI D 156° 19.5° 0.85 [6] Figure 1. Blade profile losses versus isentropic outlet Mach number for four transonic turbines. (c) topology of wake vortex structure (a) turbine blade flat plate model. (b) schlieren photograph. behind a cylinder [29]. Figure 13. Vortex shedding at transonic exit flow conditions [30]. Int. J. Turbomach. Propuls. Power 2020, 5, 10 29 of 55 acceleration over most part of the surface is likely to guarantee laminar conditions at the trailing edge on both blades and trip wires had to be used to enforce transition and turbulent boundary layers at the trailing edge, if desired so. The blades were tested from low subsonic to high subsonic outlet Mach numbers. Due to the use of blow down and suction tunnels at VKI and DLR, respectively, the Reynolds number increases with Mach number as shown in Figure 41. Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 31 of 58 Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 31 of 58 810 Figure 41. Variation of Reynolds number in function of downstream Mach number; adapted from Figure 41. Variation of Reynolds number in function of downstream Mach number; adapted from [16]. Commented [MM65]: Please do not cut figure and caption 811 [16]. over two pages. The tests for the front-loaded blade A are presented in Figure 42. In case of forced transition on 812 The tests for the front-loaded blade A are presented in Figure 42. In case of forced transition on the pressure side through a trip wire at 24% of the chord length, the Strouhal number is nearly constant 813 the pressure side through a trip wire at 24% of the chord length, the Strouhal number is nearly 814 constant and roughly equal to ~0.195 over the entire Mach range. In absence of a trip wire, the 810 Figure 41. Variation of Reynolds number in function of downstream Mach number; adapted from and roughly equal to St  0.195 over the entire Mach range. In absence of a trip wire, the evolution Commented [MM65]: Please do not cut figure and caption 815 evolution of = ( ) is quite different. Starting from the low Mach number and Reynolds 811 [16]. over two pages. of St = St(M ) is quite di erent. Starting from the low Mach number and Reynolds number end, 2is 816 number end, the Strouhal number decreases from ~0.34 at =0.2 to ~0.26 at =0.53. the Str812 ouhal number The tesdecr ts for eases the front fr -loa om dedSt blad e 0.34 A areat prM esented = in0.2 Figu to re 42 St . In c 0.26 ase of forc at M ed transition = 0.53 on . At this point 817 At this point the drops suddenly to the level of all turbulent cases. This sudden change obviously 2is 2is 813 the pressure side through a trip wire at 24% of the chord length, the Strouhal number is nearly 818 indicates that boundary layer transition has taken place on the pressure side. The slow decrease the St drops suddenly to the level of all turbulent cases. This sudden change obviously indicates that 814 constant and roughly equal to ~0.195 over the entire Mach range. In absence of a trip wire, the 819 before the sudden jump points to a progressive change from a laminar to a transitional boundary boundary layer transition has taken place on the pressure side. The slow decrease before the sudden 815 820 evolutio layer whn of ich is obv = ( iously rela ) is ted qu to ite di the in ffecr reeasi nt. n St gar Rti eng yn f olds n rom t uh m e l ber. ow Mach number and Reynolds jump points 816 821 to number e a pr Caogr scn ad, t dessive e C he S wa trou s t change e hst al num ed wit b fr h er a om de circula cr a ea laminar se r s f tra rom ilin g~ ed to 0.3 g a 4 etransitional at D at L R an=0 d a squa .2boundary to red ~0.2 t6 raili at n layer g edge which =0 at VK .53. I is obviously 817 822 A over a t this poin range t th e = drops 0.2 to su 0.9 dden . The tw ly to the l o serie esv of el of test a diffe ll turbu red n lent cases. ot only by Th is thsudden eir trailin ch ga e n d g g ee ob ge v oim ou et sl ry y related to the increasing Reynolds number. 818 indicates that boundary layer transition has taken place on the pressure side. The slow decrease 823 but also, at the same Mach number, by a higher Reynolds number in the VKI tests, see Figure 41. Cascade C was tested with a circular trailing edge at DLR and a squared trailing edge at VKI over 819 before the sudden jump points to a progressive change from a laminar to a transitional boundary 824 Note, that in the case of the squared trailing edge the distance between the separating shear layers is a range 820 M la = yer 0.2 whto ich0.9. is obv The iously two related series to the in of cr test easin di g Re er yn ed olds n not um only ber. by their trailing edge geometry but 825 well defined. However, this is not the case for the rounded trailing edge in which case the distance 2,is 821 Cascade C was tested with a circular trailing edge at DLR and a squared trailing edge at VKI 826 should be in any way smaller. But one single test, at = 0.59, was run at VKI also with a rounded also, at the same Mach number, by a higher Reynolds number in the VKI tests, see Figure 41. Note, 822 over a range = 0.2 to 0.9. The two series of test differed not only by their trailing edge geometry 827 trailing edge to eliminate any bias between the tests at DLR and VKI. that in the case of the squared trailing edge the distance between the separating shear layers is well 823 but also, at the same Mach number, by a higher Reynolds number in the VKI tests, see Figure 41. 824 Note, that in the case of the squared trailing edge the distance between the separating shear layers is defined. However, this is not the case for the rounded trailing edge in which case the distance should 825 well defined. However, this is not the case for the rounded trailing edge in which case the distance be in any way smaller. But one single test, at M = 0.59, was run at VKI also with a rounded trailing 2,is 826 should be in any way smaller. But one single test, at = 0.59, was run at VKI also with a rounded edge to eliminate any bias between the tests at DLR and VKI. 827 trailing edge to eliminate any bias between the tests at DLR and VKI. 829 Figure 42. Strouhal number variation with downstream Mach number for cascade A; adapted from 830 [16]. 831 Figure 43 presents the Strouhal number for blade C both in function of the downstream Mach 832 number and the Reynolds number. Both data sets show a plateau of = 0.36 at low Mach number 833 and Reynolds number which is characteristic for a fully laminar trailing edge boundary layer 829 Figure 42. Strouhal number variation with downstream Mach number for cascade A; adapted from Figure 42. Strouhal number variation with downstream Mach number for cascade A; adapted from [16]. 834 separation. The Strouhal number starts to decrease with increasing Reynolds number, the drop of 830 [16]. 835 occurring earlier at = 0.35 × 10 for the squared trailing edge, instead of 0.6 10 for the circular 831 836 trailiFi ng e gud re g 4 e.3 At pre s = en 1ts .1 t × he 1 S 0 trou h th al e s nu quare mber d f tr oa r b ilin lad g edg e C b e dat otha i rea n fu cnc h a p tion lat oeau wit f the do hw n =st 0r.24 eam M . No acth e Figure 43 presents the Strouhal number for blade C both in function of the downstream Mach 832 837 n th uat mber the sing and the le r Re oun yded nolds trailin number. g edg Bo e test at th data s VK eI t sin sdi how cated a pb lat y a star in th eau of = 0e .36 grap at h is r low Mac ight in h li nu nem wi ber th number and the Reynolds number. Both data sets show a plateau of St = 0.36 at low Mach number and 833 and Reynolds number which is characteristic for a fully laminar trailing edge boundary layer Reynolds number which is characteristic for a fully laminar trailing edge boundary layer separation. 834 separation. The Strouhal number starts to decrease with increasing Reynolds number, the drop of 835 occurring earlier at = 0.35 × 10 for the squared trailing edge, instead of 0.6 10 for the circular The Strouhal number starts to decrease with increasing Reynolds number, the drop of St occurring 836 trailing edge. At = 1.1 × 10 the squared trailing edge data reach a plateau with = 0.24 . Note 6 6 earlier at Re = 0.35 10 for the squared trailing edge, instead of 0.6 10 for the circular trailing edge. 837 that the single rounded trailing edge test at VKI indicated by a star in the graph is right in line with At Re = 1.1 10 the squared trailing edge data reach a plateau with St = 0.24. Note that the single rounded trailing edge test at VKI indicated by a star in the graph is right in line with the squared Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 32 of 58 838 the squared trailing edge data. Extrapolating the DLR data to higher Reynolds number one may 839 expect that they will reach the plateau of = 0.24 at ≈ 1.1 → 1.2 10 . 840 Comparing the two curves in Figure 43 raises of course the question as to the reasons for the 841 differences between them. The possible influence of the different distance between the separating 842 shear layers was already mentioned before, but, if this would be the case, then the Strouhal number Int. J. Turbomach. Propuls. Power 2020, 5, 10 30 of 55 843 for the VKI tests with squared trailing edge should be higher than those of the DLR tests with 844 rounded trailing edge. There must be therefore a different reason. The key for the understanding 845 comes from flat plate tests presented in [16], see Figure 44, which showed that the difference of the trailing edge data. Extrapolating the DLR data to higher Reynolds number one may expect that they 846 Strouhal number between a full laminar and full turbulent flow conditions was much bigger for tests will reach the plateau of St = 0.24 at Re  1.1 ! 1.2 10 . 847 with rounded trailing edges than squared trailing edges, 30% instead of 13%. Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 32 of 58 (a) (b) 838 the squared trailing edge data. Extrapolating the DLR data to higher Reynolds number one may 849 Figure 43. Strouhal number variation with Mach number (a) and Reynolds number (b) for cascade C; 839 expect that they will reach the plateau of = 0.24 at ≈ 1.1 → 1.2 10 . Figure 43. Strouhal number variation with Mach number (a) and Reynolds number (b) for cascade Commente C; d [M66]: Please add explanation for subgraph. 850 adapted from [16]. 840 Comparing the two curves in Figure 43 raises of course the question as to the reasons for the adapted from [16]. Commented [MM67R66]: Done. 841 differences between them. The possible influence of the different distance between the separating 842 shear layers was already mentioned before, but, if this would be the case, then the Strouhal number Comparing the two curves in Figure 43 raises of course the question as to the reasons for the 843 for the VKI tests with squared trailing edge should be higher than those of the DLR tests with 844 rounded trailing edge. There must be therefore a different reason. The key for the understanding di erences between them. The possible influence of the di erent distance between the separating shear 845 comes from flat plate tests presented in [16], see Figure 44, which showed that the difference of the layers was already mentioned before, but, if this would be the case, then the Strouhal number for the 846 Strouhal number between a full laminar and full turbulent flow conditions was much bigger for tests VKI tests with squared trailing edge should be higher than those of the DLR tests with rounded trailing 847 with rounded trailing edges than squared trailing edges, 30% instead of 13%. edge. There must be therefore a di erent reason. The key for the understanding comes from flat plate tests presented in [16], see Figure 44, which showed that the di erence of the Strouhal number between a full laminar and full turbulent flow conditions was much bigger for tests with rounded trailing edges than squared trailing edges, 30% instead of 13%. (a) (b) This di erent behavior can be explained if one assumes that the shape of the trailing edge may 851 Figure 44. Strouhal numbers for vortex shedding from flat plates with rounded (a) and squared (b) strongly 852 a ect the traili evolution ng edges; adapt of ed from the shear [16]. layer, and that it is the state of the shear layer rather than that of the boundary layer which plays the most important role in the generation of the vortex street. 853 This different behavior can be explained if one assumes that the shape of the trailing edge may Of course, a sharp corner will not necessarily induce immediately full transition, but transition will 854 strongly affect the evolution of the shear layer, and that it is the state of the shear layer rather than 855 that of the boundary layer which plays the most important role in the generation of the vortex street. occur over a certain length, and(a) (b) this length a ects the length of the enrolment of the vortex and 856 Of course, a sharp corner will not necessarily induce immediately full transition, but transition will therewith its frequency. The transition length of the shear layer will be a ected by both the Reynolds 849 Figure 43. Strouhal number variation with Mach number (a) and Reynolds number (b) for cascade C; Commented [M66]: Please add explanation for subgraph. 857 occur over a certain length, and this length affects the length of the enrolment of the vortex and 850 adapted from [16]. number and the Mach number. 858 therewith its frequency. The transition length of the shear layer will be affected by both the Reynolds Commented [MM67R66]: Done. 859 number and the Mach number. 860 Contrary to the vortex shedding for subsonic flow conditions discussed above, where the 861 vortices are generated by the enrolment of the separating shear layers close to the blade trailing edge, 862 the situation changes with the emergence of oblique shocks from the region of the confluence of the 863 pressure and suction side shear layers for transonic outlet Mach numbers. In this case the vortex 864 formation is delayed to after this region as shown already in the schlieren pictures in Figure 34. (a) (b) 851 Figure 44. Strouhal numbers for vortex shedding from flat plates with rounded (a) and squared (b) Figure 44. Strouhal numbers for vortex shedding from flat plates with rounded (a) and squared 852 trailing edges; adapted from [16]. (b) trailing edges; adapted from [16]. 853 This different behavior can be explained if one assumes that the shape of the trailing edge may 854 strongly affect the evolution of the shear layer, and that it is the state of the shear layer rather than Contrary to the vortex shedding for subsonic flow conditions discussed above, where the 855 that of the boundary layer which plays the most important role in the generation of the vortex street. vortices are generated by the enrolment of the separating shear layers close to the blade trailing edge, 856 Of course, a sharp corner will not necessarily induce immediately full transition, but transition will 857 occur over a certain length, and this length affects the length of the enrolment of the vortex and the situation changes with the emergence of oblique shocks from the region of the confluence of the 858 therewith its frequency. The transition length of the shear layer will be affected by both the Reynolds pressure and suction side shear layers for transonic outlet Mach numbers. In this case the vortex 859 number and the Mach number. formation is delayed to after this region as shown already in the schlieren pictures in Figure 34. 860 Contrary to the vortex shedding for subsonic flow conditions discussed above, where the 861 vortices are generated by the enrolment of the separating shear layers close to the blade trailing edge, This is even more clearly demonstrated in Figure 45 presenting the evolution of the wake density 862 the situation changes with the emergence of oblique shocks from the region of the confluence of the gradients predicted with a LES by Vagnoli et al. [56], for the turbine blade shown in Figure 17, from 863 pressure and suction side shear layers for transonic outlet Mach numbers. In this case the vortex high subsonic to low supersonic outlet Mach numbers. For M = 1.05 the vortex shedding frequency 864 formation is delayed to after this region as shown already in the schlieren pictures in Figure 34. 2is Int. J. Turbomach. Propuls. Power 2020, 5, 10 31 of 55 Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 33 of 58 is not any more conditioned by the trailing edge thickness but by the distance between the feet of the Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 33 of 58 trailing edge shocks emanating from the region of the confluence of the two shear layers. 865 This is even more clearly demonstrated in Figure 45 presenting the evolution of the wake density 866 gradients predicted with a LES by Vagnoli et al. [56], for the turbine blade shown in Figure 17, from 867 high subsonic to low supersonic outlet Mach numbers. For =1.05 the vortex shedding 868 frequency is not any more conditioned by the trailing edge thickness but by the distance between the 869 feet of the trailing edge shocks emanating from the region of the confluence of the two shear layers. (a) =0.79 (b) =0.97 (c) =1.05 , , , (a) =0.79 (b) =0.97 (c) =1.05 , , , Figure 45. Change of wake density gradients from subsonic to supersonic exit Mach numbers as 865 Figure 45. Change of wake density gradients from subsonic to supersonic exit Mach numbers as 871 Figure 45. Change of wake density gradients from subsonic to supersonic exit Mach numbers as predicted by LES [56]. 866 predicted by LES [56]. Commented [M68]: Please add explanation for subgraph. 872 predicted by LES [56]. Commented [MM69R68]: Done. Consequently, one observes a sudden increase of the vortex shedding frequency as for example 867 Consequently, one observes a sudden increase of the vortex shedding frequency as for example 873 Consequently, one observes a sudden increase of the vortex shedding frequency as for example recorded 874by Carscallen recorded by Ca et rscal. allen [43 et a ],l. on [43]their , on their nozzle nozzle guide guide vane vane, , see Fi see gure 46. Figure 46. 868 recorded by Carscallen et al. [43], on their nozzle guide vane, see Figure 46. Figure 46. Strouhal number versus downstream Mach number; adapted from [43]. 875 Figure 46. Strouhal number versus downstream Mach number; adapted from [43]. 876 7. Advances in the Numerical Simulation of Unsteady Turbine Wake Characteristics 7. Advances in the Numerical Simulation of Unsteady Turbine Wake Characteristics 877 The numerical simulation of unsteady turbine wake flow is relatively young, and the first 869 Figure 46. Strouhal number versus downstream Mach number; adapted from [43]. The numerical simulation of unsteady turbine wake flow is relatively young, and the first 878 contributions appeared in the mid-80s. The decade 1980–1990 has in fact seen the final move from the contributions appeared in the mid-80s. The decade 1980–1990 has in fact seen the final move from 879 potential flow models to the Euler and Navier-Stokes equations whose numerical solutions were 870 7. Advances in the Numerical Simulation of Unsteady Turbine Wake Characteristics 880 tackled with new, revolutionary for the time, techniques. Those were also the years of the first vector the potential flow models to the Euler and Navier-Stokes equations whose numerical solutions were 881 and parallel super-computers capable of a few sustained gigaflops (CRAY YMP, IBM SP2, NEC SX- 871 tackled The numer with new ica , l sim revolutionary ulation of forunste the time, ady turbine w techniques. ake fl Those ow is rela were also tively you the yearsng, of the and the first vector first 882 3, to quote a few examples), and of the beginning of the massive availability of computing resources 872 contri and parallel buti 883 ons super o ab ppea eying -computers rM ed i oore n’s th lae mi w (tcapable ran d-80 sisto s. The decade 1980 r co of unt do a few ublin sustained g every t –19 wo y gigaflops 90 e ha ars).s Sin inc f (CRA ea tct seen the fi hen th Ye YMP progres , IBM nal ses h move f a SP2, ve NEC rom the SX-3, 884 been huge both on the numerical techniques and on the turbulence modelling side. Indeed, the most 873 potential to quote flo a few w models to examples), the Eule and of r an the d Navier beginning -Sto ofkes e the massive quations whose n availability um ofecomputing rical solutiorn esour s were ces 885 advanced option, that is the Direct Numerical Simulation (DNS) approach, where all turbulent scales 874 ta obeying ckled wiMoor th new, revolutiona e’s law (transistor ry focount r the tidoubling me, techniqu every es. Those w two years). ere also the Since then years of the the progr fir esses st vector have 886 are properly space-time resolved down to the dissipative one, has also recently entered the 875 and p beenahuge rallel both super-comput on the numerical ers capable of a techniques few su andst on ain the ed gigaflop turbulence s (CRAY YM modelling P, side. IBM SP2, NE Indeed, the C SX- most 887 turbomachinery community starting from the pioneering work of Jan Wissink in 2002 [63]. 888 Unfortunately, because of the very severe resolution requirements, there is still no DNS study of 876 3, advanced to quote a option, few ex that amp isles the ), Dir and ect ofNumerical the beginnin Simulation g of the ma (DNS) ssive appr avai oach, labilit wher y ofe comput all turbulent ing rescales source ar s e 889 turbine wake flow (TWF) at realistic Reynolds and Mach numbers, that is Re ~ 10 and high subsonic 877 obeying properly Moo space-time re’s law r (t esolved ransist down or count to the doubl dissipative ing every two yea one, has also rs). recently Since then the progresses ha entered the turbomachinery ve 890 and transonic outlet Mach numbers with shocked flow conditions, although improvements have 878 been huge bo community 891 starting th on the numerical techn been recenfr tlyom attained [ the pioneering 64]. With the iques de work velop and on m of ent o Jan fthe turbulenc high Wissink ly paralleliza in 2002 e b modelling le co[ des 63]. and th Unfortunately side. Inde e help of veryed, t , because he most of 892 large-scale computing hardware such a simulation is likely to appear soon, as the result of some 879 athe dvavery nced opti sever on, tha e resolution t is the Di requir rect ements, Numerica ther l Si emu is still lation (DN no DNS S) ap study proach, of turbine where al wake l turbu flow len (TWF) t scales at 880 are prope realistic Reynolds rly space and -time Mach resolved numbers, down to the that is Re ~d10 issipative and high onsubsonic e, has also r and transonic ecently enter outlet ed the Mach 881 turboma numbers chi with nery communi shocked flow ty st conditions, arting from the pi although o impr neering ovements work of have Ja be n en Wi rssin ecently k inattaine 2002 d[6[3] 64. ]. 882 Unfortun With theately, bec development ause of the ve of highlyryparallelizable severe resoluti codes on reand quiremen the help ts, there of very is still no large-scale DNS computing study of 883 tu har rbine w dwara eksuch e flow a (TW simulation F) at real is ist likely ic Reyn to olds appear and Mac soon, h num as the bers result , thatof is Re some ~ 10 cutting-edge and high sub scientific sonic 884 and tr researansonic outlet M ch. In the meantime, ach numbers w and within ithe th shocked foreseeable flow futur cond e, the itions, alt industrial hough improvement world and the designers s have 885 been recently interested in attained tangled [64]. aspects With of TWF the dev forelopment of h stage performance ighly par enhancement allelizable co will des certainly and the help of very run unsteady 886 lar flow ge-sca simulations le comput wher ing h e turbulence ardware su is ch handled a simuthr latiough on is advanced likely to ap modeling. pear soon, Many as t of he those result simulations of some 887 cut will ting-ed rely on ge scient in-house ific rese developed arch. In resear the meant ch codes ime, and anturbomachinery d within the forese oriented eable future, the commercialin packages, dustrial 888 world which, and indeed, the designe have impr rs interested in oved significantly tangled since aspect thes of TWF very first fo unsteady r stage perfor TWF mance enh simulation.an Yc et, ement there 889 wi arll e cert twoaar inl eas y ru wher n uensteady fl importantow challenges simulations still whe needre turbulenc to be satisfactorily e is han faced dled through before thead prvanced esently 890 modeling. Many of those simulations will rely on in-house developed research codes and 891 turbomachinery oriented commercial packages, which, indeed, have improved significantly since the 892 very first unsteady TWF simulation. Yet, there are two areas where important challenges still need to 893 be satisfactorily faced before the presently available (lower fidelity) computations could be Int. J. Turbomach. Propuls. Power 2020, 5, 10 32 of 55 available (lower fidelity) computations could be considered reliable and successful. They can be, loosely speaking, termed of numerical and modeling nature. We shall try to review both, in the context of the presently discussed unsteady turbine wake flow subject category, presenting a short overview of the available technologies. A more specialized review study on high-fidelity simulations as applied to turbomachinery components has recently been published by Sandberg et al. [65]. 7.1. Numerical Aspects Most of the available turbine wake flow computations have been obtained with eddy viscosity closures and structured grid technologies, although a few examples documenting the use of fully unstructured locally adaptive solvers are available [66,67]. In the structured context turbomachinery blades gridding is considered a relatively simple problem, and automated mesh generators of commercial nature producing appreciable quality multi-block grids, are available [68]. The geometrical factors most a ecting the grid smoothness are the cooling holes, the trailing edge shape, the sealing devices and the fillets. Of those the trailing edge thickness and its shape are the most important in TWF computations. Low and intermediate pressure turbines (LPT and IPT, respectively) have relatively sharp trailing edges, while the first and second stages of the high-pressure turbines (HPT), often because of cooling needs, have thicker trailing edges. Typically, the trailing edge thickness to chord ratio D/C, is a few percent in LPTs and IPTs, and may reach values of 10% or higher in some HPTs. Thus, the ratio of the trailing edge wet area to the total one may easily range from 1/200 to 1/20, having roughly estimated the blade wet area as twice the chord. Therefore, resolving the local curvature of the trailing edge area is extremely demanding in terms of blade surface grid, that is, in number of points on the blade wall. Curvature based node clustering may only partially alleviate this problem. In addition, preserving grid smoothness and orthogonality in the trailing edge area is dicult, if not impossible with H or C-type grids, even with elliptic grid generators relying on forcing functions [69]. Wrapping an O-type mesh around the blade is somewhat unavoidable, and in any event the use of a multi-block or multi-zone meshing is highly desirable. Unstructured hybrid meshes would also typically adopt a thin O mesh in the inner wall layer. Non-conformal interfaces of the patched or overlapped type would certainly enhance the grid quality, at the price of additional computational complexities and some local loss of accuracy occurring on the fine-to-coarse boundaries [70]. Local grid skewness accompanied by a potential lack of smoothness will pollute the numerical solution obtained with low-order methods, introducing spurious entropy generation largely a ecting the features of the vortex shedding flow. In those conditions, the base pressure is typically under-predicted as a consequence of the local flow turning and separation mismatch, with a higher momentum loss and an overall larger unphysical loss generation in the far wake. The impact of those grid distorted induced local errors on the quality of the solution is hard to quantitatively ascertain both a-priori and a-posteriori, and often grid refinement will not suce, as they frequently turn out to be order 1, rather than order h with h the mesh size and p the order of accuracy. Nominally second order schemes have in practice 1 < p < 2. In this context, higher order finite di erence and finite volume methods, together with the increasingly popular spectral-element methods, o er a valid alternative to standard low order methods [71–75]. This is especially true for those techniques capable of preserving the uniform accuracy over arbitrarily distorted meshes, a remarkable feature that may significantly relieve the grid generation constraints, besides o ering the opportunity to resolve a wider range of spatial and temporal scales with a smaller number of parameters compared to the so called second order methods (rarely returning p = 2 on curvilinear grids). The span of scales that needs to be resolved and the features of the coherent structures associated to the vortex shedding depend upon the blade Reynolds number, the Mach number (usually built with the isentropic downstream flow conditions) and the D/C ratio. This is equivalent to state that the Reynolds number formed with the momentum thickness of the turbulent boundary layer at the trailing edge (Re ) and the Reynolds number defined using the trailing edge thickness (Re ), are independent parameters. For thick trailing edge blades the vortex shedding is vigorous and the near wake development is governed by the suction and pressure Int. J. Turbomach. Propuls. Power 2020, 5, 10 33 of 55 side boundary layers which di er. Thus, the early stages of the asymmetric wake formation chiefly depend upon the local grid richness, the resolution of the turbulent boundary layers at the TE and the capabilities of the numerical method to properly describe their mixing process. Well-designed turbine blades operate with an equivalent di usion factor smaller than 0.5 yielding a /C ratio less than 1% according to Stewart correlation [76]. This e ectively means that the resolution to be adopted for the blade base area will have to scale like the product /C C/D which may be considerably less than one; in order words the base area region needs more points that those required to resolve the boundary layers at the trailing edge. Very few simulations have complied with this simple criterion as today. Compressibility e ects present additional numerical diculties, especially in scale resolving simulations. It is a known fact that transonic turbulent TWF calculations require the adoption of special numerical technologies capable to handle time varying discontinuous flow features like shock waves and slip lines without a ecting their physical evolution. Unfortunately, most of the numerical techniques with successful shock-capturing capabilities rely on a local reduction of the formal accuracy of the convection scheme whether or not based on a Riemann solver. Since at grid scale it is hard to distinguish discontinuities from turbulent eddies, and even more their mutual interaction, Total Variation Diminishing (TVD) and Total Variation Bounded (TVB) schemes [77–79] are considered too dissipative for turbulence resolving simulations, and they are generally disregarded. At present, in the framework of finite di erence and finite volume methods, there is scarce alternative to the adoption of the class of ENO (Essentially Non Oscillatory) [80–82] and WENO (Weighted Essentially Non Oscillatory) [83–87] schemes developed in the 90s. A possibility is o ered by the Discontinuous Galerkin (DG) methods [88]. The DG is a relatively new finite element technique relying on discontinuous basis functions, and typically on piecewise polynomials. The possibility of using discontinuous basis functions makes the method extremely flexible compared to standard finite element techniques, in as much arbitrary triangulations with multiple hanging nodes, free independent choice of the polynomial degree in each element and an extremely local data structure o ering terrific parallel eciencies are possible. In their native unstructured framework, opening the way to the simulation of complex geometries, h and p-adaptivity are readily obtained. The DG method has several interesting properties, and, because of the many degrees of freedom per element, it has been shown to require much coarser meshes to achieve the same error magnitudes when compared to Finite Volume Methods (FVM) and Finite Di erence Methods (FDM) of equal order of accuracy [89]. Yet, there seem to persist problems in the presence of strong shocks requiring the use of advanced non-linear limiters [90] that need to be solved. This is an area of intensive research that will soon change the scenario of the available computational methods for high fidelity compressible turbulence simulations. 7.2. Modeling Aspects The lowest fidelity level acceptable for TWF calculations is given by the Unsteady Reynolds Averaged Navier-Stokes Equations (URANS) or, better, Unsteady Favre Averaged Navier-Stokes Equations (UFANS) in the compressible domain. URANS have been extensively used in the turbomachinery field to solve blade-row interaction problems, with remarkable success [91,92]. The pre-requisite for a valid URANS (here used also in lieu of UFANS) is that the time scale of the resolved turbulence has to be much larger than that of the modeled one, that is to say the characteristic time used to form the base state should be suciently small compared to the time scale of the unsteady phenomena under investigation. This is often referred to as the spectral gap requirement of URANS [93]. Therefore, we should first ascertain if TWF calculations can be dealt with this technology, or else if a spectral gap exists. The analysis amounts at estimating the characteristic time  , or frequency f , vs vs of the wake vortex shedding, and compare it with that of the turbulent boundary layer at the trailing edge,  , or f . The wake vortex shedding frequency is readily estimated from: bl l bl 2,is f = St = f(geometry, Reynolds, Mach) vs te Int. J. Turbomach. Propuls. Power 2020, 5, 10 34 of 55 which has been shown to depend upon the turbine blade geometry and the flow regimes (see Figures 39, 42–44 and 46). For the turbulent boundary layers the characteristic frequency can be estimated, using inner scaling variables, as: bl with u = the friction velocity, and  the kinematic viscosity. Assuming the boundary layer to be fully turbulent from the leading edge, and using the zero pressure gradient incompressible flat plate correlation of Schlichting [59]: 2  0.059 C = = f ,x 1/5 Re one gets: 2 2 2 u u u / / 1/5 = C = 0.0295 Re . f ,x x At the turbine trailing edge x = C, and u = V so that: 2,is 2,is 4/5 f = 0.0295 Re bl 2,is Therefore, the ratio of the turbulent boundary layer characteristic frequency to the wake vortex shedding one is, roughly: 4/5 Re bl vs te 2,is = = 0.0295 (6) f  C St vs bl The explicit dependence of the Strouhal number upon the geometry term d /C is unknown, te although clear trends have been highlighted in the previous section. However, taking d /C  0.05 and te St  0.3 as reasonable values, Equation (6) returns: bl 4/5 0.005 Re (7) 2,is vs The estimates obtained from the above Equation are reported in Table 3, for a few Reynolds numbers. Table 3. Turbulent boundary layer to vortex shedding frequency ratio; Equation (7). 5 6 6 6 Re 5 10 10 2 10 3 10 2,is bl 180 310 550 760 vs From the above table it is readily inferred that, for the problem under investigation, a neat spectral gap exists, and, thus, URANS calculations can be carried out with some confidence. The results reported in the foregoing confirm that this is indeed the case. Formally, RANS are obtained from URANS dropping the linear unsteady terms, and, therefore, the closures developed for the steady form of the equations apply to the unsteady ones as well. Whether the abilities of the steady models broaden to the unsteady world is controversial, even though the limited available literature seem to indicate that this is rarely the case. A review of the existing RANS closures is out of the scope of the present work, and the relevant literature is too large to be cited here, even partially. In the turbomachinery field, turbulence and transition modelling problems have been extensively addressed over the past decades, and significant advances have been achieved [94–96]. Here, we will mainly stick to those models which have been applied in the TWF simulations presently reviewed. In the RANS context Eddy Viscosity Models (EVM) are by far more popular than Reynolds Stress Models (RSM), whether di erential (DRSM) or algebraic (ARSM). Part of the reasons are to be Int. J. Turbomach. Propuls. Power 2020, 5, 10 35 of 55 found with the relatively poor performance of DRS and ARS when compared to the computational e ort required to implement these models, especially for unsteady three-dimensional problems. Also, the prediction of pressure induced separation and, more in general, of separated shear layers is, admittedly, disappointing, so that the expectations of advancing the fidelity level attainable with EVM has been disattended. This explains why most of the engineering applications of RANS, and thus of URANS, are routinely based on EVM, and typically on algebraic [97], one equation [98] and two equations (k- of Jones and Launder [99], k-! of Wilcox [100], Shear Stress Transport (SST) of Menter [101]) formulations. In the foregoing we shall see that the TWF URANS computations reviewed herein all adopted the above closures. A few of those were based on the k-! model of Wilcox. This closure, and its SST variant, has gained considerable attention in the past two decades and it is widely used and frequently preferred to the k- models, as it is reported to perform better in transitional flows and in flows with adverse pressure gradients. Further, the model is numerically very stable, especially its low-Reynolds number version, and considered more “friendly” in coding and in the numerical integration process, than the k- competitors [100]. On the scale resolved simulations the scenario is rather di erent. Wall resolved Large Eddy Simulations (LES) are now recognized as una ordable for engineering applications because of the very stringent near wall resolution requirements and of the inability of all SGS models to account for the e ects of the near wall turbulence activity on the resolved large scales [102,103]. On the wall modeled side, the most successful approaches rely on hybrid URANS-LES blends, and in this framework the pioneering work of Philip Spalart and co-workers should be acknowledged [104,105]. Already 20 years ago this research group introduced the Detached Eddy Simulation (DES), a technique designed to describe the boundary layers with a URANS models and the rest of the flow, particularly the separated (detached) regions, with an LES. The switching or, better, the bridging between the two methods takes place in the so called “grey area” whose definition turned out to be critical, because of conceptual and/or inappropriate, though very frequent, user decisions. The latter are particularly related to the erroneous mesh sizes selected for the model to follow the URANS and the LES branches. Nevertheless, the original DES formulation su ered from intrinsic to the model deficiencies leading to the appearance of unphysical phenomena in thick boundary layers and thin separation regions. Those shortcomings appear when the mesh size in the tangent to the wall direction, i.e., parallel to it, D , becomes smaller than the boundary layer thickness , either as a consequence of a jj local grid refinement, or because of an adverse pressure gradient leading to a sudden rise of . In those instances, the local grid size, i.e., The filter width in most of the LES, is small enough for the DES length scale to fall in the LES mode, with an immediate local reduction of the eddy viscosity level far below the URANS one. The switching to the LES mode, however, is inappropriate because the super-grid Reynolds stresses do not have enough energy content to properly replace the modeled one, a consequence of the mesh coarseness. The decrease in the eddy viscosity, or else the stress depletion, reduces the wall friction and promotes an unphysical premature flow separation. This is the so-called Model Stress Depletion (MSD) phenomenon, leading to a kind of grid induced separation, which is not easy to tackle in engineering applications, because it entails the unknown relation between the flow to be simulated and the mesh spacing to be used. In recent years, however, two new models o ering remedies to the MSD phenomenon have been proposed, one by Philip Spalart and co-workers [106], the other by Florian Menter and co-workers [107]. Before proceeding any further, let us briefly mention the physical idea underlying the DES approach. In its original version based on the Spalart and Allmaras turbulence model [98] the length scale d used in the eddy viscosity is modified to be: d  min(d, C D) (8) DES where d is the distance from the wall, D a measure of the grid spacing (typically D  max(Dx,Dy,Dz) in a Cartesian mesh), and C a suitable constant of order 1. The URANS and the wall modeled DES ˜ ˜ LES modes are obtained when d  d and d  C D, respectively. The DES formulation based on the DES two equations Shear Stress Transport turbulence model of Menter [101] is similar. It is based on the Int. J. Turbomach. Propuls. Power 2020, 5, 10 36 of 55 introduction of a multiplier (the function F ) in the dissipation term of the k-equation of the k-! DES model which becomes: k!F DES with: F = max , 1 (9) DES C D DES In the above equations L is the turbulent length scale as predicted by the k-! model, = 0.09 the model equilibrium constant and C a calibration constant for the DES formulation: DES L = Both the DES-SA (DES based on the Spalart and Allmaras model) and the DES-SST (DES based on Menter ’s SST model) models su er from the premature grid induced separation occurrence previously discussed. To overcome the MSD phenomenon Menter and Kuntz [107] introduced the F blending SST functions that were designed to reduce the grid influence of the DES limiter (9) on the URANS part of the boundary layer that was “protected” from the limiter, that is, protected from an uncontrolled and undesired switch to the LES branch. This amounts to modify Equation (9) as follows: " # F = max (1 F ), 1 DES-SST-zonal SST C D DES with F selected from the blending functions of the SST model, whose argument is k/(!d), that SST is the ratio of the k-! turbulent length scale k/! and the distance from the wall d. The blending functions are 1 in the boundary layer and go to zero towards the edge. The proposal of Spalart et al. [106] termed DDES is similar to the DES-SST-zonal proposal of Menter et al. [107], and, while presented for the Spalart and Allmaras turbulence model it can be readily extended to any EVM. In the Spalart and Allmaras model a turbulence length scale is not solved for through a transport equation. It is instead built from the mean shear and the turbulent viscosity: t t r = = d p 2 2 (d) 2S S (d) ij ij with S = @U /@x + @U /@x /2 the rate of strain tensor,  the eddy viscosity and  the von Kàrmàn ij i j j i t constant. This quantity, actually a length scale squared, is 1 in the outer portion of the boundary layer and goes to zero towards its edge. The term  is often augmented of the molecular viscosity  to ensure that r remains positive in the inner layer. This dimensionless length scale squared is used in the following function: f = 1 tanh [8r ] d d reaching 1 in the LES region where L < d and 0 in the wall layer. It plays the role of 1 F in the t SST DES-SST-zonal model. Additional details on the design and calibration of the model constants can be found in [106]. The Delayed DES (DDES), a surrogate of the DES, is obtained replacing d in Equation (8) with the following expression: ( ) d  d f max 0, d C D (10) DES The URANS and the original DES model are retrieved when f = 0 and f = 1, respectively, d d ˜ ˜ corresponding to d  d and d  C D. This new formulation makes the length scale (10) depending DES on the resolved unsteady velocity field rather than on the grid solely. As the authors stated the model prevents the migration on the LES branch if the function f is close to zero, that is the current point is in the boundary layer as judged from the value of r . If the flow separates f increases and the LES mode d d is activated more rapidly than with the classical DES approach. As for DES this strategy, designed to Int. J. Turbomach. Propuls. Power 2020, 5, 10 37 of 55 tackle the MSD phenomenon, does not relieve the complexity of generating adequate grids, that is grids capable of properly resolving the energy containing scales of the LES area. Thus, unlike a proper grid assessment study is conducted, it will be dicult to judge the quality of those scale resolving models especially in the present context of TWF. 7.3. Achievements Unsteady turbine wake flow simulation is a relatively new subject and the very first pioneering works appeared in the mid-90s [66,108–110]. The reason is twofold; on one side the numerical and modelling capabilities were not yet ready to tackle the complexities of the physical problem, and on the other side, the lack of detailed experimental measurements discouraged any attempt to simulate the wake flow. This until the workshop held at the von Kàrmàn Institute in 1994 during a Lecture Series [37], where the first detailed time resolved experimental data of a thick trailing edge turbine blade where presented and proposed for experiment-to-code validation in an open fashion. The turbine geometry was also disclosed. As mentioned in Section 3 those tests referred to a low Mach, high Reynolds number case (M = 0.4, Re = 2 10 ). The numerical e orts of [108,110–113] addressing this test case and 2,is listed in Table 4, were devoted at ascertaining the capabilities of the state-of-the-art technologies to predict the main unsteady features of the flow, namely the wake vortex shedding frequency and the time averaged blade surface pressure distribution, particularly in the base region. Table 4. Available computations of the M = 0.4, Re = 2 10 VKI LS-94 turbine blade. 2,is 2 Numerical Space/Time Grid Authors Eqs. Grid Closure Method Discretization Density Manna Structured EVM (Baldwin & URANS CC-FVM 2nd/2nd 44k et al. [110] Multi-Block (H-O) Lomax [97]) Arnone EVM (Baldwin & URANS CC-FVM Structured C-grid 2nd/2nd 36k et al. [111] Lomax [97]) Sondak EVM (Deiwert URANS FDM Overset grids (H-O) 3nd/2nd 60k et al. [112] et al. [114]) Structured Ning et al. EVM (Roberts URANS CV-FVM Multi-Block 2nd/2nd 42k [113] [115]) (H-O-H-H) All of the above contributors solved the URANS with a Finite Volume (FVM) or Finite Di erence Method (FDM) and adopted simple algebraic closures. Both Cell Vertex (CV) and Cell Centered (CC) approaches where used. The more recent computations of Magagnato et al. [116] referred to a similar test case, though with rather di erent flow conditions, and will not be reviewed. Appropriate resolution of the trailing edge region and the adoption of O grids turned out to be essential to reproduce the basic features of the unsteady flow in a time averaged sense. The use of C grids with their severe skewing and distortion of the base region a ected the resolved flow physics and required computational and modelling tuning to fit the experiments. The time mean blade loading could be fairly accurately predicted (see Figure 47) by nearly all authors listed in Table 4, although discrepancies with the experiments and among the computations exist. They have been attributed to stream-tube contraction e ects and to the tripping wire installed on the pressure side at x/C = 0.61 in the experiments [112]. ax Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 10 of 58 271 suction side for both blades are exactly the same, but contrary to the nearly constant Mach number 272 for the VKI blade downstream of the throat, the blade of Xu and Denton is characterized by a very 273 strong adverse pressure gradient in this region. As pointed out by the authors, this causes the suction 274 side boundary layer to be either separated or close to separation up-stream of the trailing edge. 275 Clearly, Sieverding’s correlation cannot deal with blade designs characterized by very strong adverse 276 pressure gradients on the rear suction side causing boundary layer separation before the trailing 277 edge. 278 Figure 9. Comparison of blade Mach number distribution for blade RD of Xu and Denton [15], (solid 279 curve, =0.8, =8 10 ) with VKI blade (dashed curve, =0.8, =10 ). , , 280 The possible effect of boundary layer separation resulting from high rear suction side diffusion 281 resulting in high base pressures was also mentioned by Corriveau and Sjolander in 2004 [20], 282 comparing their nominal mid-loaded rotor blade HS1A, mentioned already before, with an aft- 283 loaded blade HS1C with an increase of the suction side unguided turning angle from 11.5° to 14.5°. 284 It appears that the increased turning angle could cause, in the transonic range, shock induced 285 boundary layer transition near the trailing edge with, as consequence, a sharp increase of the base 286 pressure, i.e. a sudden drop in the base pressure coefficient as seen in Figure 10. Note that the 287 reported in the figure has been converted to − of the original data. 288 As regards the base pressure data by Deckers and Denton [13] for a low turning blade model 289 and Gostelow et al. [14] for a high turning nozzle guide vane, who report base pressure data far below 290 those of Sieverding’s BPC, their blade pressure distribution resembles that of the convergent/ 291 divergent blade C in Figure 3 with a negative blade loading near the trailing edge which would 292 explain the very low base pressures. In addition, the blade of Deckers and Denton has a blunt trailing 293 edge, and there is experimental evidence that, compared to a circular trailing edge, the base pressure Int. J. Turbo Int. J. Turbo mm ach. ach. Propuls. Power Propuls. Power 2018 2018 , 3 , ,3 x FOR PE , x FOR PE ER RE ER RE VI VI EE W W 40 of 40 of 58 58 Int. J. Turbo Int. J. Turbo mm ach. ach. Propuls. Power Propuls. Power 2018 2018 , 3 , ,3 x FOR PE , x FOR PE ER RE ER RE VI VI EW EW 40 of 40 of 58 58 294 for blades with blunt trailing edge might be considerably lower. Sieverding and Heinemann [16] Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 40 of 58 295 report for flat plate tests at moderate subsonic Mach numbers a drop of the base pressure coefficient Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 40 of 58 296 by 11% for a plate with squared trailing edge compared to that with a circular trailing edge. 1139 stream-tube contraction effects and to the tripping wire installed on the pressure side at / =0.61 Int. J. Turbomach. Propuls. Power 2020, 5, 10 38 of 55 1140 in the experiments [112]. 1139 stream-tube contraction effects and to the tripping wire installed on the pressure side at / =0.61 1140 in the experiments [112]. 1135 1135 Figure 47. Figure 47. 1141 VKI LS94 turbine VKI LS94 turbine Figure 47. VKI blade, LS94 turbine blade, blade, =0 =0 .4.4 , , =0 .=2 4,=2 1=2 1 00 1case 0 case case . Ti .. Ti Ti m me m e m e mean m ean blad e blade an blad surface e s e s u isen u rfrf ac a tre is c opic e is e n en trto ro pp ici c 1135 1135 Figure 47. Figure 47. Figure VKI LS94 turbine VKI LS94 turbine 47. VKI LS94 turbine blade, blade, blade, M =0 =0 .4.= ,4 , 0.4,=2 Re =2 1 = 010 2case  case 10 . Ti . Ti case. m m e m eT m e ime an blad ean blad mean e s e s blade urf urf ac asurface e is ce is en et n rto isentr rp oi p ci c opic , , , , , 297 Figure 10. Base pressure coe 2,is fficient fo 2r mid-loaded (solid line) and aft-loaded (dashed line) rotor 1142 Mach number distribution. experiments [32]; [110]; [111]; [112]; 1136 1136 Mach number distribution. Mach number distribution. 1141 Figure 47. VKI LS94 turbine experime experime blade, nts nts [32]; =0[32]; .4, =2 10[110]; [110]; case. Time mean blade[111]; [111]; surface isentr[112]; opic[112]; 1136 1136 Mach number distribution. Mach number distribution. Mach number distribution. experime experime experimentsnts nts ,[32]; [32]; [32]; [110]; [110]; [110]; [111]; [[111]; 111]; [112]; [112[112]; ]; [113 ]. 298 blade. Symbols: HS1A geometry, HS1C geometry. Adapted from [20]. 1143 [113]. 1142 Mach number distribution. experiments [32]; [110]; [111]; [112]; 1137 1137 [113]. [113]. 1137 1137 [113]. [113]. 1143 [113]. The time averaged base pressure region was also fairly well reproduced by the available numerical 1144 The time averaged base pressure region was also fairly well reproduced by the available 1145 numerical data, although the differences among the computations and the experiments are generally 1138 1138 data, The time The time 1144 although av average erage The the time dd di b b averag aer asese ences pressu pressu ed ba among sere re pressure reg reg the ion was ion was region computations was also also also fairly w ffairly w airland y welle the ell r reproduced b ll r experiments eeproduced produced y the ava by by ar the ila e the b generally le available available larger 1138 1138 The time The time av av erage erage dd b b ase ase pressu pressu re re reg reg ion was ion was also also fairly w fairly w ell r ell r eproduced eproduced by by the the available available 1146 larger than those reported in Figure 48. Indeed, the underlying physics is more complex, as the 1145 numerical data, although the differences among the computations and the experiments are generally 1139 numerical data, although the differences among the computations and the experiments are generally 1139 1139 1139 numeri numeri numeri than ca ca ca l da those l da l da ta ta ta , al , r, eported al al though the dif though the dif though the dif in Figur fe fe f rences e erences 48 rences . Indeed, am a am m ong th ong th ong th the e co underlying e co e co mputations mputations mputations physics and the experiments are generally and the experiments are generally and the experiments are generally is more complex, as the presence 1147 presence of the two pressure and suction side sharp over-expansions at the locations of the boundary 1146 larger than those reported in Figure 48. Indeed, the underlying physics is more complex, as the 1140 1140 lar lar gg er th of er th the an t an t two hh ose reporte ose reporte pressure and dd in in suction Figur Figur ee side 48. In 48. In sharp deed, deed, over th th e unde -expansions e unde rly rly ing ing physic at physic the locations s s isis m m oo re complex re complex of the boundary , as , as t t hh e layer e 1140 1140 lar lar ger th ger th an t an t 1148 hh ose reporte ose reporte layer separation dd in in Figur su Figur ggests.e e 48. In 48. In deed, deed, thth e unde e unde rly rly ing ing physic physic s s isis m m ore complex ore complex , as , as t t hh e e 1147 presence of the two pressure and suction side sharp over-expansions at the locations of the boundary 1141 presence o 1148 f the two pressur layer separatione su and ggests. suction side sharp over-expansions at the locations of the boundary 1141 1141 1141 presence o presence o presence o separation f the two pressur ff the two pressur the two pressur suggests. e and e e and and suction suction suction side side side shar shar shar p ov p ov p ov er-expan er-expan er-expan sion sion sion s at the s at the s at the locations of the bo locations of the bo locations of the bo undary undary undary 1142 1142 lay lay ee r separ r separ aa tion tion su su ggests ggests . . 1142 1142 lay lay er separ er separ ation ation su su ggests ggests . . 1149 Figure 48. VKI LS94 turbine blade, =0.4, =2 10 case. Time mean base pressure Commented [M73]: This figure has not been referred to 1150 distribution. For symbols see Figure 47. 1Figure 149 48. VKI Figure 48. LS94 turbine VKI LS94 turbi blade,nM e blade,= 0.4, Re =0.4= , 2 =2  10 10 case. case. T Time ime mean mean bas base e pre pressur ssure e distribution. 2,is , 2 Commente within the d t [ ex M t 73]: of the This manuscr figure h ipta . P s n lease ot be add exp en referred lanation for to 1150 distribution. For symbols see Figure 47. For symbols see Figure 47. withi subgra n thp e h t.ex t of the manuscript. Please add explanation for 1151 The location and the magnitude of these two accelerations seem within the reach of the adopted 1152 closure, as well as the pressure plateau of the base region. The predicted base pressure coefficients subgraph. 1151 The location and the magnitude of these two accelerations seem within the reach of the adopted Commented [MM74R73]: You are right, the text has been The 1153location defined b and y Equa the tion magnitude (2) agree fairl of y wel these l with two the ex accelerations perimental value, as w seem ell as w within ith th the e one ob reach tained of the adopted 1152 closure, as well as the pressure plateau of the base region. The predicted base pressure coefficients corrected. The subgraph explanation is really not needed Commented [MM74R73]: You are right, the text has been 1154 from the VKI correlation [110]. The success of these simple models is attributed to the proper space- 1153 defined by Equation (2) agree fairly well with the experimental value, as well as with the one obtained closure, as well as the pressure plateau of the base region. The predicted base pressure coecients corr becau ected. The s se the grubgr aph and the a aph explan ssoci ation is ated ske realtch have ly not needed to be 1155 time resolution of the boundary layers at separation points in the trailing edge region. Again, this has 1154 from the VKI correlation [110]. The success of these simple m o dels is attributed to the proper space- 1143 Figure 48. VKI LS94 turbine blade, =0.4, =2 10 case. Time mean base pressure 1143 1143 1143 Figure 48. Figure 48. Figure 48. VKI LS94 turbine blade VKI LS94 turbine blade VKI LS94 turbine blade , ,, =0 =0 =0 .4.,4 .4 ,, =2 =2 =2 1 01 100 case. case. case. Tim Tim Tim e m ee m m ean base pre eean base pre an base pre sssusssre uure re defined by Equation (2) agree fairly well , , with , the experimental value, as well as with the one obtained 1156 been documented by Manna et al. [110] and by Sondak et al. [112] (see Figure 49) who could show a becau consi se t dered he gr aa s a ph and the a single entity ssociated sketch have to be 1155 time resolution of the boundary layers at separation points in the trailing edge region. Again, this has 1144 distribution. For symbols see Figure 47. 1144 1144 1144 distribu distribu distribu tion tion tion . F . F . F or sy o or sy r sy m m m bols b bols ols see see see F i F g Fiu ig g re 47. u ure 47. re 47. from the 1157 VKI more t correlation han satisf[act 110 or]. y agr The eement success of the com of these puted t simple ime averag models ed veloc is ity attribu profiles wit tedhto the measured the proper space-time 1156 been documented by Manna et al. [110] and by Sondak et al. [112] (see Figure 49) who could show a considered as a single entity 1158 one, both on the pressure and suction sides at 1.75 diameters upstream of the trailing edge (/ = 1157 more than satisfactory agreement of the computed time averaged velocity profiles with the measured resolution of the boundary layers at separation points in the trailing edge region. Again, this has been 1159 1.75 with = 0 at the trailing edge, and = ). The thinner pressure side boundary layer and the 1145 1145 1145 1145 The location The location The location The location 1158 on and the magn and the magn e, both and the magn and the magn on the pressure an itude o itude o itude o itude o d su ff these f these f these c these tion two acceler sides at two acceler two acceler two acceler 1.75 diaa m a t a eters upstr a t it ons se iti ons se ions se ons se em wit em wit eem wit a em wit m of th h h e h in the reach h in the reach trailin in the reach in the reach g edge (/of the adopte = of the adopte of the adopte of the adopte d d d d documented by Manna et al. [110] and by Sondak et al. [112] (see Figure 49) who could show a more 1160 blade circulation strengthening the pressure side vortex shedding were estimated to be the cause of 1159 1.75 with = 0 at the trailing edge, and = ). The thinner pressure side boundary layer and the 1146 1146 closu closu re, re, as w as w ee ll ll as t as t hh e p e p re re ssur ssur ee p p lat lat ee aa uu of t of t hh e b e b aa se se r r ee gion. The pr gion. The pr edicted base edicted base pressure coeffic pressure coeffic ients ients 1146 1146 closu closu re, re, as w as w ell ell as t as t hh e p e p re rssur essur e p e p lat lat ea eu au of t of t hh e b e b ase ase r r egion. The pr egion. The pr edicted base edicted base pressure coeffic pressure coeffic ients ients than satisfactory 1161 the hig agr her loca eement l over ex ofpan the siocomputed n at the trailing time edge averaged [32]. The very velocity consistentpr grid ref ofiles inem with ent stthe udy o measur f ed one, 1160 blade circulation strengthening the pressure side vortex shedding were estimated to be the cause of 1147 1147 1147 1147 defin defin defin defin ee d by Eq e e d by Eq d by Eq d by Eq uat uat uat uat ion ii on i on on (2 (2 (2 (2 ) a ) a ) a ) a g g r g g r ee fa r ee fa r ee fa ee fa ir ir ir ly wel ir ly wel ly wel ly wel l lwit l l wit wit wit h h h t h t h t t h h e ex h e ex e ex e ex perimental value, as we perimental value, as we perimental value, as we perimental value, as we ll ll ll as w ll as w as w as w ith the one obtained ii th the one obtained i th the one obtained th the one obtained 1162 [112] brought some improvements in the thinner and fuller pressure side boundary layers 1161 the higher local over expansion at the trailing edge [32]. The very consistent grid refinement study of both on the pressure and suction sides at 1.75 diameters upstream of the trailing edge (s/D = 1.75 1148 1148 from the VKI from the VKI 1162 correlation correlation [112] brough [110]. The t some [110]. The improv succ succ emess o en ess o ts ifn these th f these e thisi nsi n mple model emple model r and fuller sp sris essure si is at at trtibut ribut de ed t bed t oun odary la o t t hh e proper space e proper space yers - - 1148 1148 from the VKI from the VKI correlation correlation [110]. The [110]. The succ succ ess o ess o f these f these sisi mple model mple model s s is is at at trtibut ribut ed t ed t o t o t hh e proper space e proper space - - with s = 0 at the trailing edge, and D = d ). The thinner pressure side boundary layer and the blade te 1149 1149 1149 1149 ti ti ti me resoluti ti me resoluti me resoluti me resoluti on of on of on of on of the bounda the bounda the bounda the bounda ry ry ry ry la ll a l yers a a a yers a yers a yers a tt sepa t sepa t sepa sepa ra ra ra ra ti ti ti on ti on on on p p p p o o int o o int int int ss in t s in t s in t in t h h h e t h e t e t e t rr ai r ai rai ai ling ling ling ling edge r edge r edge r edge r ee gion. e e gion. gion. gion. Aga Aga Aga Aga in, t ii n, t in, t n, t h h h is h h is h is h is h aa s a a s s s circulation strengthening the pressure side vortex shedding were estimated to be the cause of the 1150 1150 been doc been doc uu me me nted by nted by M M aa nn nn a a et et al. al. [110] [110] and and by by Sond Sond ak et ak et al al . [11 . [11 22 ] (see ] (see Figu Figu re re 49 49 ) who coul ) who coul d show a d show a 1150 1150 been doc been doc uu me me nted by nted by M M an an nn a a et et al. al. [110] [110] and and by by Sond Sond ak et ak et al al . [11 . [11 2] (see 2] (see Figu Figu re re 49 49 ) who coul ) who coul d show a d show a higher local over expansion at the trailing edge [32]. The very consistent grid refinement study of [112] 1151 1151 1151 1151 more tha more tha more tha more tha n n n sa n sa sa sa tt it sfa it i sfa isfa sfa cc tory c tory ctory tory aa g a a g reement of g g reement of reement of reement of the computed ti the computed ti the computed ti the computed ti m m m m ee aver e e aver aver aver age age age age d d d ve d ve ve ve locity pro locity pro locity pro locity pro ff iles with the me f iles with the me files with the me iles with the me asured asured asured asured brought some improvements in the thinner and fuller pressure side boundary layers predictions. 1152 1152 one, bot one, bot hh on t on t hh e press e press uu re re aa nn d suct d suct ion ion sisi des at des at 1. 1. 75 75 di di amet amet ers ers upst upst re re am of t am of t hh e t e t ra ra ilin ilin g ed g ed ge ( ge ( // = =± ± 1152 1152 one, bot one, bot hh on t on t hh e press e press uu re re an an d suct d suct ion ion sisi des at des at 1. 1. 75 75 di di amet amet ers ers upst upst re ram of t eam of t hh e t e t ra rilin ailin g ed g ed ge ( ge ( // = = ±± It is no surprise that with a proper characterization of the boundary layers and of the base region, 1153 1153 1153 1153 1. 1. 1. 75 1. 75 75 75 wit wit wit wit h h h h = = = =0000 at at at at t t t h t h h e h e e t e t rt r ai tr ai rai ai lin lin lin lin g e g e g e g e d d d ge, d ge, ge, ge, an an an an d d d d = = = = ). T ). T ). T ). T h h h e thinner h e thinner e thinner e thinner pr pr pr pr essure essure essure essure sid sid sid sid ee b e e b b b o o undary o o undary undary undary laye laye laye laye r an r an r an r an d the d the d the d the the computed and measured losses agreed well. 1154 1154 blade blade circ circ ulat ulat ion stren ion stren gg thening the pr thening the pr essure essure side vo side vo rtrt ex shedd ex shedd ing ing were est were est imat imat ee d to be the c d to be the c aa use o use o f f 1154 1154 blade blade circ circ ulat ulat ion stren ion stren gthening the pr gthening the pr essure essure side vo side vo rtrt ex shedd ex shedd ing ing were est were est imat imat ed to be the c ed to be the c ause o ause o f f 1155 1155 1155 1155 tt h t h th e higher h e higher e higher e higher loc loc loc loc aa l a a over exp ll over exp l over exp over exp aa n a a n n sn s ion s ion sion ion at at at at t t h t t h h e t h e t e t e t rr a r a r ilin a a ilin ilin ilin g e g e g e g e d d d ge d ge ge ge [[ 3 [ 3 [ 2 3 3 2 ] 2 2 ] . The ] . The ]. The . The very c very c very c very c o o nsist o o nsist nsist nsist ee nt e e nt nt nt g g g g rr id r id rid id refin refin refin refin ee ment e e ment ment ment st st st st udy o udy o udy o udy o ff f f 1156 1156 [112 [112] brought some i ] brought some immprovements i provements inn the thi the thinne nner and r and fuller fuller pressu pressure re sid sidee bound boundaaryry lay layeersr s 1156 1156 [112 [112] brought some i ] brought some immprovements i provements inn the thi the thinne nner and r and fuller fuller pressu pressure re sid side bound e boundary ary lay layeresr s 1157 predictions. It is no surprise that with a proper characterization of the boundary layers and of the 1157 1157 1157 predi predi predi cti ccti ons. It i tions. It i ons. It i s no ss no no surprise tha surprise tha surprise tha t wi tt wi wi th th th a proper aa proper proper cha cha cha ra rrc aateriza ccteriza teriza titi on of tion of on of the boundary la the boundary la the boundary la yers yers yers an aan d of nd of d of the the the 1158 1158 base base region, t region, t hh e computed e computed and me and me asure asure dd losses agree losses agree dd well. well. 1158 1158 base base region, t region, t hh e computed e computed and me and me asure asure dd losses agree losses agree dd well. well. Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 41 of 58 Int. J. Turbomach. Propuls. Power 2020, 5, 10 39 of 55 1163 predictions. It is no surprise that with a proper characterization of the boundary layers and of the 1164 base region, the computed and measured losses agreed well. (a) (b) 1165 Figure 49. VKI LS94 turbine blade, =0.4, =2 10 case. Time mean velocity profiles on Figure 49. VKI LS94 turbine blade, M = 0.4, Re = 2 10 case. Time mean velocity profiles on 2,is 2 1166 pressure side (a) and suction side (b) at / = 1.75 . For symbols see Figure 47. pressure side (a) and suction side (b) at s/D = 1.75. For symbols see Figure 47. 1167 The correct prediction of the vortex shedding frequency within experimental uncertainty proved The 1168 corr ect to be pr m edi ore di ction fficult, of si the nce,vortex to this aim, th shedding e near wfr ake equency physics ha within s to be caexperimental ptured in terms ofuncertainty the large- proved 1169 scale coherent structures formation, development and propagation. This is probably outside the reach to be more dicult, since, to this aim, the near wake physics has to be captured in terms of the 1170 of any eddy viscosity closure, and most likely of the URANS approach. Also, it has been shown large-scale coherent structures formation, development and propagation. This is probably outside the 1171 experimentally that the dominant frequency does not appear as a single sharp amplitude peak in the reach of 1172 any eddy Fourie viscosity r transform, closur but rath e, erand as a sm most all size f likely reque of ncy the banURANS d-width [32]. approach. Also, it has been shown 1173 This is best seen with the help of Table 5, comparing the predicted Strouhal number with the experimentally that the dominant frequency does not appear as a single sharp amplitude peak in the 1174 experimental datum. Computations are assumed to report the dimensionless frequency in terms of Fourier transform, but rather as a small size frequency band-width [32]. 1175 isentropic exit velocity . The experimental Strouhal value of 0.27, has been rescaled using the 1176 nominal shedding frequency of 2.65 kHz and the isentropic velocity corresponding to the = This is best seen with the help of Table 5, comparing the predicted Strouhal number with the 1177 0.409 value (Cicatelli et al. [32]). Despite the use of the same simplistic closure the scatter is rather experimental datum. Computations are assumed to report the dimensionless frequency in terms of 1178 large both among the computations and with the experiments. The predicted Strouhal number of isentropic exit velocity V . The experimental Strouhal value of 0.27, has been rescaled using the 1179 Sondak et al. [12, 12is ] agrees perfectly with the experimental value. nominal shedding frequency of 2.65 kHz and the isentropic velocity corresponding to the M = 0.409 2,is 1180 Table 5. VKI LS94 turbine blade, =0.4, =2 10 case. Strouhal numbers. value (Cicatelli et al. [32]). Despite the use of the same simplistic closure the scatter is rather large Authors Method Closure Strouhal both among the computations and with the experiments. The predicted Strouhal number of Sondak Cicatelli et al. [32] Experiments / 0.285 et al. [112] agrees perfectly with the experimental value. Manna et al. [110] URANS EVM: Baldwin & Lomax [97] 0.253 Arnone et al. [111] URANS EVM: Baldwin & Lomax [97] 0.210 Sondak et al. [112] URANS EVM: Deiwert et al. [114] 6 0.285 Table 5. VKI LS94 turbine blade, M = 0.4, Re = 2 10 case. Strouhal numbers. 2,is 2 Ning et al. [113] URANS EVM: Roberts [115] 0.245 Authors Method Closure Strouhal 1181 Those results obtained at a relatively low Mach number pushed the VKI group to extend the 1182 experimental investigation into the high subsonic/transonic range in 2003 [21] and 2004 [36] as Cicatelli et al. [32] Experiments / 0.285 1183 already discussed in Section 3. This was a new breakthrough, as it offered once more, and again for Manna et al. [110] URANS EVM: Baldwin & Lomax [97] 0.253 1184 the first time, a set of highly resolved experimental data documenting the effects of compressibility Arnone et al. [111] URANS EVM: Baldwin & Lomax [97] 0.210 1185 on the unsteady wake formation and development process, throwing some considerable light on the Sondak et al. [112] URANS EVM: Deiwert et al. [114] 0.285 1186 relation between the base pressure distribution and the vortex shedding phenomenon. In the next Ning et al. [113] URANS EVM: Roberts [115] 0.245 1187 ten, fifteen years a number of research groups attempted to simulate this flow setup, mostly with 1188 higher fidelity approaches and the results were again rather satisfactory. The nominal Mach and 1189 Reynolds numbers were increased considerably ( =0.79, =2.8 10 ), and a variety of Those results obtained at a relatively low Mach number pushed the VKI group to extend the 1190 additional flow conditions including supersonic outlet regimes were tested, as discussed in § 5. Table 1191 6 summarizes the relevant contributions. experimental investigation into the high subsonic/transonic range in 2003 [21] and 2004 [36] as already Commented [MM75]: OK for displacing this text ahead of discussed in Section 3. This was a new breakthrough, as it o ered once more, and again for the first Table 6. time, a set of highly resolved experimental data documenting the e ects of compressibility on the unsteady wake formation and development process, throwing some considerable light on the relation between the base pressure distribution and the vortex shedding phenomenon. In the next ten, fifteen years a number of research groups attempted to simulate this flow setup, mostly with higher fidelity approaches and the results were again rather satisfactory. The nominal Mach and Reynolds numbers were increased considerably (M = 0.79, Re = 2.8 10 ), and a variety of additional flow conditions 2,is 2 including supersonic outlet regimes were tested, as discussed in Section 5. Table 6 summarizes the relevant contributions. Int. J. Turbomach. Propuls. Power 2020, 5, 10 40 of 55 Table 6. Available computations of the M = 0.79, Re = 2.8 10 VKI LS-94 turbine blade. 2,is 2 Numerical Space/Time Grid Authors Eqs. Grid Closure y Method Discretization Density EVM (Baldwin Structured Mokulys Lomax [97], Spalart URANS CC-FVM Multi-Block 2nd/2nd NA 5–10 et al. [117] and Allmaras [98], (O-H) Wilcox [100]) Structured Leonard URANS CC-FVM Multi-Block EVM (Wilcox [100]) 2nd/2nd 0.63 M 5 et al. [118] (H-O-H-H-H) Structured Leonard SRS (Smagorinsky LES CC-FVM Multi-Block 2nd/2nd 0.63 M 5 et al. [118] [119]) (H-O-H-H-H) Leonard SRS (Smagorinsky LES CV-FVM Unstructured 3nd/3nd 0.4 M 40 et al. [118] [119]) El-Gendi Structured SRS (Spalart et al. et al. DDES CV-FVM 2nd/2nd 4 M 1 (O) [104]) [120,121] Kopriva URANS CV-FEM Unstructured EVM (Wilcox [100]) 1st–2nd/2nd 1.3 M 1 et al. [67] (CFX) Vagnoli LES SRS (Wall dumped CC-FVM Unstructured 1st/2nd 2.53 M 0.4 et al. [56] (OpenFOAM) Smagorinsky [122]) Structured Wang et al. DDES SRS (Spalart et al. CC-FVM Multi-Block 2nd/2nd 4.3 M 0.4–1 [123] (Fluent) [104]) (H-O-H-H) For the structured meshes the number of nodes and the number of cells is similar. In the unstructured cases the di erence is rather large, and typically there is a factor 5 more cells than nodes. The URANS simulations should have been carried on a two-dimensional mesh, since there is no reason for transversal modes to develop with 2D inflow conditions in a perfectly cylindrical geometry extruded by some percentage of the chord in the spanwise direction. The URANS computations of Leonard et al. [118] and those of Kopriva et al. [67] were carried out on a 3D mesh obtained expanding the 2D domain in the third direction by a fraction of the chord length (5.7% in [118] and 8% in [67]). None of the authors discussed the appearance of spanwise modes in the URANS data. Conversely, the scale resolving simulations (LES and DDES) need to be carried out on a 3D domain, with a homogeneous spanwise direction, to allow for the appropriate description of the most relevant energy carrying turbulent eddies, which are inherently three dimensional in nature. Occasionally, some authors reported two dimensional pseudo-DDES and pseudo-LES, that is, unsteady computations obtained on a purely two dimensional mesh, none of which has been included in Table 6. On the resolution side, the URANS simulations of Kopriva et al. [67] seem to have gone through some grid refinement study, while those of Leonard et al. [118] did not. On the LES and DDES side the situation is far more involved. At a Reynolds number of about three million the grid point requirements for a wall resolved LES is about 5 10 [124] which is a couple of orders of magnitude higher than the most refined LES of Table 6. Thus, the very neat inertial subrange of this HPT flow is likely not to be resolved at all by any of the available simulations, and consequently the cut-o is poorly placed. These deficiencies will seriously impact on the quality of the simulations as they undermine the essential prerequisites upon which LES relies. For the DDES simulation this inconsistency is only partially relieved. Detached Eddy Simulation and similar hybrid URANS-LES approaches have somewhat met certain expectations, even though they are routinely overlooked as a means of achieving a LES-like quality at the cost of a URANS setup. Instead, DES and its evolved version DDES, should be categorized as Wall Modeled LES, and thus they can by no means be considered as a coarser grid version of LES [106]. In the present context modelling the boundary layer via URANS all the way down to the point of incipient separation will not return any of the key features the true turbulent boundary layer should possess to properly form the wake and determine its correct space time development. And relieving the Modeled Stress Depletion of DES by better addressing the URANS-LES migration in the grey area, will only Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 10 of 58 271 suction side for both blades are exactly the same, but contrary to the nearly constant Mach number 272 for the VKI blade downstream of the throat, the blade of Xu and Denton is characterized by a very 273 strong adverse pressure gradient in this region. As pointed out by the authors, this causes the suction 274 side boundary layer to be either separated or close to separation up-stream of the trailing edge. 275 Clearly, Sieverding’s correlation cannot deal with blade designs characterized by very strong adverse 276 pressure gradients on the rear suction side causing boundary layer separation before the trailing 277 edge. 278 Figure 9. Comparison of blade Mach number distribution for blade RD of Xu and Denton [15], (solid 279 curve, =0.8, =8 10 ) with VKI blade (dashed curve, =0.8, =10 ). , , 280 The possible effect of boundary layer separation resulting from high rear suction side diffusion 281 resulting in high base pressures was also mentioned by Corriveau and Sjolander in 2004 [20], 282 comparing their nominal mid-loaded rotor blade HS1A, mentioned already before, with an aft- 283 loaded blade HS1C with an increase of the suction side unguided turning angle from 11.5° to 14.5°. 284 It appears that the increased turning angle could cause, in the transonic range, shock induced 285 boundary layer transition near the trailing edge with, as consequence, a sharp increase of the base Int. J. Turbomach. Propuls. Power 2020, 5, 10 41 of 55 286 pressure, i.e. a sudden drop in the base pressure coefficient as seen in Figure 10. Note that the Int. J. Turbo Int. J. Turbo mm ach. ach. Propuls. Power Propuls. Power 2018 2018 , 3 , ,3 x FOR PE , x FOR PE ER RE ER RE VI VI EE W W 43 of 43 of 58 58 287 reported in the figure has been converted to − of the original data. partially alleviate the grid induced288 separation issue As reg of a these rds th hybrid e base pr methods. essure d All ata by in all, De the ckers an two DDES d Denton [13] for a low turning blade model 1216 addressing the URANS-LES migration in the grey area, will only partially alleviate the grid induced 1216 addressing the URANS-LES migration in the grey area, will only partially alleviate the grid induced 289 and Gostelow et al. [14] for a high turning nozzle guide vane, who report base pressure data far below simulations of El-Gendi et al. [120,121] and Wang et al. [123] are also to be considered as unresolved, 1217 1217 separ separ aa titon is ion is sue of sue of th th ee se hybrid se hybrid met met hh ods. Al ods. Al l in l in aa lll,l the two DDES , the two DDES sisi mula mula tions of El tions of El -G -G endi endi et et alal . . Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 43 of 58 because of the previously mentioned 290 cut-othose of misplacement. Sieverding’s BPC We shall return , their blad to thise point pressure later on. distribution resembles that of the convergent/ 1218 [120,121] and Wang et al. [123] are also to be considered as unresolved, because of the previously 1218 [120,121] and Wang et al. [123] are also to be considered as unresolved, because of the previously 1220 all the way down to the point of incipient separation will not return any of the key features the true 291 divergent blade C in Figure 3 with a negative blade loading near the trailing edge which would At this high subsonic regime, the experimental time mean blade pressure distribution already 1219 1219 mentioned cut-off m mentioned cut-off m isp isp lacement. We lacement. We sha sha lll return to thi l return to thi s poi s poi nn t la t la ter ter on. on. 1221 turbulent boundary layer should possess to properly form the wake and determine its correct space presented in Figure 17 in terms of 292 local isentr explain the ve opic Mach ry lo number w base p , reveals ressures. that the In flow additi is on, the b subsonic lad all e of Deckers and Denton has a blunt trailing 1220 At this high subsonic regime, the experimental time mean blade pressure distribution already 1220 At this high subsonic regime, the experimental time mean blade pressure distribution already 1222 time development. And relieving the Modeled Stress Depletion of DES by better addressing the 293 edge, and there is experimental evidence that, compared to a circular trailing edge, the base pressure around the blade. 1221 1221 present present 1223 ee d in Fig d in Fig URAN uu re 1 S-re 1 LES mi 77 in in grta e t te rms iorms n in th oo fe floc g loc rey a ar a l isent l isent ea, wilr l on opic ropic ly p M a M rtia aa ch num llch num y alleviatb e th b er, er, e r g r e rid i e vv en e al duced separation al s t s t hh at at t t hh e f e f low l issue ow is is sub sub sonic sonic al al l l Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 40 of 58 1224 of these hybrid methods. All in all, the two DDES simulations of El-Gendi et al. [120,121] and Wang 294 for blades with blunt trailing edge might be considerably lower. Sieverding and Heinemann [16] 1222 aroun The d the blade. computations compared in Figure 50 predict fairly well the continuous accelerationCommente of the d [MM80]: The reference order is correct see 1222 around the blade. Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 40 of 58 Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 40 of 58 1225 et al. [123] are also to be considered as unresolved, because of the previously mentioned cut-off 295 report for flat plate tests at moderate subsonic Mach Ta n ble u 6mbers a dro . p of the base pressure coefficient flow both on the suction (till the throat location at x/C =0.61) and on the pressure side (till the ax 1226 misplacement. We shall return to this point later on. 296 by 11% for a plate with squared trailing edge compared to that with a circular trailing edge. trailing edge). Also, the sudden deceleration from the throat to the trailing edge is well predicted by 1227 At this high subsonic regime, the experimental time mean blade pressure distribution already 1228 presented in Figure 17 in terms of local isentropic Mach number, reveals that the flow is subsonic all all simulations. 1229 around the blade. 1224 1224 Figure 50. Figure 50. VKI LS94 turbin VKI LS94 turbine blade, e blade, = =00 .79, .79, =2 =2.8.8 1 100 case. T case. Time mean blade ime mean blade surface surface 1231 Figure 50. 1135 VKI LS94 turbin Figure 47. e blade, , VKI LS94 turbine = 0.79, =2 .8 10 blade, case. Time me=0 an blade surface .4, =2 10 case. Time mean blade surface isentropic Figure 50. VKI LS94 turbine blade, 297 M , = , 0.79, Figure 10. Re = Ba 2.8 se pressu 10 case. re coe , Time fficient f mean or m blade id-loade surface d (solid line) and aft-loaded (dashed line) rotor 2,is 1232 isentropic Mach number distribution; eddy viscosity models. experiments [21]; Baldwin & 1135 Figure 47. VKI LS94 turbine 1225 1225 blade, isentropic Ma isentropic Ma =0 ch c.4 h nu , nu m m b=2 b er distribu er distribu 10 case tion; ed tion; ed . Time dy m dy visco e visco an blad sity sity m e s m o uo del rf del ac se is . s. e n ex ex tr perim o perim pic ents [21]; ents [21]; Baldwin Baldwin & & 1135 Figure 47. VKI LS94 turbine blade, =0.4, =2 10 case. Ti1136 me mean blade s Mach number distribution. urface isentropic experiments [32]; [110]; [111]; [112]; isentropic , Mach number distribution; 298 eddy viscosity blade. Symbol models. s: experiments HS1A geometry, [21]; HS1C ge Baldwin ometry. Adapted from [20]. 1233 Lomax [117]; Spalart & Allmaras [117]; Wilcox - [117]; Wilcox - [118]; 1136 Mach number distribution. 1226 1226 Lomax [117]; Lomax [117]; experiments [32]; Spalart & Spalart & [110]; Allmaras [117]; Allmaras [117]; [111]; Wilcox Wilcox [112]; - - [117]; [117]; Wilcox Wilcox - - [118]; [118]; 1136 Mach number distribution. experiments [32]; & Lomax[110]; [117]; 1137 Spalart[111]; &[113]. Allmaras[112]; [117]; Wilcox k-! [117]; Wilcox k-! [118]; 1234 Wilcox - [67]. 1137 [113]. 1227 1227 Wilcox Wilcox - - [67]. [67]. 1137 [113]. Wilcox k-! [67]. 1235 The computations compared in Figure 50 predict fairly well the continuous acceleration of the 1138 The time averaged base pressure region was also fairly well reproduced by the available 1236 flow both on the suction (till the throat location at / = 0.61) and on the pressure side (till the 1138 The time av 1228 erage 1228 d base Leonar The comput pressu The comput re d etreg al. aia ton was ions comp t [ions comp 118] and also ar later ar ed in fairly w ed in Kopriva F F ig ig ueu re ll r re et 50 e 50 al. produced pred pred [67ict ]ict have fa by fa irily rclearly ly the wel wel lavailable t l t demonstrated hh ee c c oo nt nt inuou inuou s s that ac ac celer celer a steady aa titon o ion o state f t f t hh e e 1138 The time averaged base pressure region was also fairly w 1139 ell reproduced numerica by l da the ta, alavailable though the differences among the computations and the experiments are generally 1237 trailing edge). Also, the sudden deceleration from the throat to the trailing edge is well predicted by 1229 solution flow both on the sucti will propose supersonic on (till the throa flow conditions t location and at / a normal = 0.61 shock ) anon d on t theh suction e pressure si side at de x/(tC ill the 1139 numerical data, al 1229 though the dif flow both on the sucti ferences among th on e co (timputations ll the throatand the experiments are generally location at / = 0.61) and on the pressure side (till the 1139 numerical data, although the differences among th 1e co 238 mputations all simulations. 1140 and the experiments are generally larger than those reported in Figure 48. Indeed, the underlying physic ax s is more complex, as the 1239 Leonard et al. [118] and later Kopriva et al. [67] have clearly demonstrated that a steady state 1140 larger than those reporte 1230 1230 d 0.61, tra in tra ili i Figur li ng edge). Al an ng edge). Al artefact e 48. In of so, the deed, so, the the wr sudden decel tsudden decel ong he unde modelling rlying ee ra ra physic which titi on on fr fom the rs disappears om the is mo throa re complex throa in t to the trai t to the trai the unsteady , as t lih li ng e ng edge is wel appr edge is wel oach (see l predi l predi Figur cted b cted b e 51y ).y 1140 larger than those reported in Figure 48. Indeed, the underlying 1141 physics presence o is more complex f the two pressur , as the e and suction side sharp over-expansions at the locations of the boundary 1240 solution will propose supersonic flow conditions and a normal shock on the suction side at / ≈ 1231 The all s trailing imulatiedge ons. induced unsteadiness, whose upstream propagation is significant (see Figure 37), 1141 presence of the two pressur 1231 al el s and imu suction lations. side sharp over-expansions at the locations of the boundary 1141 presence of the two pressure and suction side sharp over-expansion 1142 s at the lay locations of the bo er separation suundary ggests. 1241 0.61, an artefact of the wrong modelling which disappears in the unsteady approach (see Figure 51). 1142 layer separation su 1232 1232 ggests. causes Leonard Leonard the shock et et al to al . [ . [ flap 11 18 18 ] ] up and and and l l aa te tdown e r Kop r Kop r on iv riv a e the a e t al tstraight al . . [6 [6 77 ] h ] h ra ear a vv ee cle part cle aa rlr of y ly dem the dem suction oo nst nst rat rat ed t side, ed t hh at a at a phenomenon a stst ead ead yy st st at at e e 1142 layer separation suggests. 1242 The trailing edge induced unsteadiness, whose upstream propagation is significant (see Figure 37), 1243 causes the shock to flap up and down on the straight rear part of the suction side, a phenomenon that 1233 that solut causes ion wi all spatial propose supe smoothing rsonic offlow c the pr oessur nditions e discontinuity and a normal sh at the ock on t wall and he su the ction side disappearance at / of≈ 1233 solution will propose supersonic flow conditions and a normal shock on the suction side at / ≈ 1244 causes a spatial smoothing of the pressure discontinuity at the wall and the disappearance of the 1234 1234 the 0. 0. 61 61 supersonic , , an an art art ee fa fa ct pocket ct of t of t hh e wron in e wron a time g m g m averaged oo del del lin lin g g which d sense. which d In is ia s fact, a pp pp ee it ars ars is in in likely t t hh e un e un that st st eady the eady lack ap ap pp of ro ro a sharpness a ch ch (see (see Fi Fi gur of gur many ee 5 5 11 ).) . 1245 supersonic pocket in a time averaged sense. In fact, it is likely that the lack of sharpness of many 1235 transonic The trailing e experimentally dge inducmeasur ed unsteadine ed surface ss, whose pressur upstr e distributions eam propobtained agation is with sign slow ificant response (see Fig sensors, ure 37), 1235 The trailing edge induced unsteadiness, whose upstream propagation is significant (see Figure 37), 1246 transonic experimentally measured surface pressure distributions obtained with slow response 1247 sensors, is to be attributed to the implicit temporal averaging resulting from the unresolved 1236 1236 is ca cto a uu sbe e se s t sattributed t hh ee s h sh oo ck ck tto t oo fthe f la la pp u implicit u pp a a nn dd d temporal d oo w w nn o o nn t ave t hh ee s raging s tr ta ra ig ig hh r tesulting r t r ee aa r p r p aa rfr t r o tom o f t f t h the h ee s s u unr u ct cito esolved io nn s i sd id ee , unsteadiness. , a a p p hh ee nn oo m m ee nn oo n Eddy n t t hh aa t t 1248 unsteadiness. Eddy viscosity and scale resolving models (Figure 52) seem to yield comparable results 1237 viscosity causes a and spat scale ial sm resolving oothingmodels of the p (Fi re gur ssure e 52 di ) seem scontto inuit yield y at comparable the wall and results the in dis aatime ppear mean ance of sense the 1237 causes a spatial smoothing of the pressure discontinuity at the wall and the disappearance of the 1249 in a time mean sense all along the blade, while the proper prediction of the base flow appears more 1238 1238 all su su p along p ee rs ro so nthe n ici p c p blade, oo ck ck ee t i twhile i nn a a t t ithe m im ee a pr a v oper v ee ra ra gg e pr e dd ediction s e se nn se se . I . I n of n f f the aa ctc,tbase i , i t i t i s l s flow l ik ik ee ly appears ly t t hh aa t t t t hmor h ee l l aa c ek ccumbersome. k o o f s f s hh aa rp rp nn ee ss sY so et, o f m f m ther aa nn e yy 1250 cumbersome. Yet, there are appreciable differences among the computations, as well as with the 1251 experiments, in the leading edge area for 0< / <0.2, whose origin is unclear. Potential sources 1239 artransonic e appreciable experimentally di erences measur among the ed computations, surface pressuas re well distributions obtained as with the experiments, with in slow the r leading esponse 1239 transonic experimentally measured surface pressure distributions obtained with slow response 1252 of discrepancies are the inflow angle setting (purely axial) yielding some leading edge de-loading in 1240 1240 edge sensors sensors area , , is t is t foroo be at 0 be at < x/tCrtibut ribut <ed t ed t 0.2,owhose o t thhe i e im origin mplicit plicit is temp temp unclear oral aver oral aver . Potential ag aging resultin ing resultin sources of gg from the from the discrepanciunresolved unresolved es are the ax 1253 the experiments, the low Mach number effects on the accuracy of compressible flow solver not relying 1241 1241 inflow unsteadine unsteadine angle ss. ss. setting Edd Edd y v y v (pur iscos iscos ely itit y axial) y and and sc yielding sc ale ale re re solvin solvin some g mod g mod leading ee ls l (F s edge (F ig ig ure ure de-loading 5 5 22 ) ) seem t seem t in oo yi the yi eld eld experiments, comparab comparab le le the re re su low su ltlt s s 1254 on pre-conditioning techniques, larger relative errors of the pressure sensors in this incompressible 1255 flow region, some geometry effects. In the remaining part of the blade, trailing edge area excluded, 1242 1242 Mach in in a a ti ti me mean sense al number me mean sense al e ects on l along the bla l along the bla the accuracydof d e, whil e, whil compr e the e the essible proper predict proper predict flow solver ion of the b inot on of the b relying aa se flow on se flow pre-conditioning appears more appears more 1256 i.e. 0.2 < / <0.9 , the agreement among all computations and experiments is very good. 1243 1243 techniques, cumbersome. Yet, there ar cumbersome. Yet, there ar larger relative err e apprec e apprec ors ofiable the iable pr difference difference essure sensors s among the computations, s among the computations, in this incompressibleas we flow as we ll rll egion, as with the as with the some 1257 Surprisingly, the difficult region of the unguided turning in the rear part of the suction side (0.6 < 1244 1244 geometry experiments, experiments, 1258 e ects. / in the le in the le In <0the .8)adin w adin r hemaining erg e g edge area th edge area e shock w part avfor eof tu for rbul the 0< 0< en blade, t b // o unda trailing <0 ry layer <0.2.2 , whose ori , whose ori in edge teractar ion ea ocg excluded, cu g in rs, i in is unc is unc s well p li.e., e rle e a d a r. ic r. 0.2 ted Pot Pot in< e e nt x nt/ ial ial C sou sou <r0.9 ces rces , ax 1259 a time averaged sense by all closures. 1143 Figure 48. VKI LS94 turbine blade, =0.4, =2 10 case. Time mean base pressure 1245 1245 the of d of d agr iscr iscr eement ee panc panc ies among ies ar ar e t e t h all h ee in com in flfow low putations ang ang le le set set and ting ting experiments (pure (pure ly ly a a xx ia is ia l)very l) y y ield ield good. ing ing some some Surprisingly lea lea dd ing ing , edge edge the di de de -lo cult -lo aa d rd iegion ng ing in in 1143 Figure 48. 1143 VKI LS94 turbine blade Figure 48. VKI LS94 turbine blade , =0.4, =2, 1 0 case. =0 1144 .4 Tim , e=2 m 1e0an base pre distribu case.tion Tim . Fsse ou m r sy re e man base pre bols see Figuss re 47. ure 1246 1246 , of the experime the experime the unguided nts, the low Mach numbe nts, the low Mach numbe turning in the rear part r effects on th r effects on th of the suction e accur e accur sidea( a cy o 0.6 cy o< f f compressib x compressib /C < 0.8 le ) le wher flow so flow so e the lver n lver n shock oo t relyin t relyin wave g g ax 1144 distribu 1144 tion. For sym distribu bols see tion Fig . F ure 47. or symbols see Figure 47. 1247 1247 turbulent on pre-cond on pre-cond boundary itit ioning t ioning t layer ee chn chn interaction ique ique s, s, lar lar gg occurs, er r er r ee lat lat iis ve ive well error error prsedicted of t s of t hh e press e press in a time uu re senso re senso averaged rs in rs in sense t h th is incompre is incompre by all closur ssib ssib es. le le 1145 The location and the magnitude of these two accelerations seem within the reach of the adopted 1248 1248 flow flow region, some geometry effects. In region, some geometry effects. In t t hh e rem e rem aa in in in in gg p p aa rtrt of of t t hh ee b b lade, lade, t t rai rai ling ling e e dd ge ge are are aa excl excl uded uded , , 1145 The location and the magnitude of these two accelerations seem within the reach of the adopted 1145 The location and the magnitude of these two accelerations se 1146 em within the reach closure, as w of the adopte ell as the prd e ssure plateau of the base region. The predicted base pressure coefficients 1249 1249 i.e. i.e. 0.2 0.2< <// <0 <0.9.9, the ag , the agreement amo reement amonng all comp g all computations utations and and experiments is very experiments is very goo goodd. . 1146 closure, as well as the pressure plateau of the base region. The predicted base pressure coefficients 1146 closure, as well as the pressure plateau of the base region. The pr 1147 edicted base define pressure coeffic d by Equation (2 ients ) agree fairly well with the experimental value, as well as with the one obtained 1250 1250 SS uu rp rp riri sisi ng ng ly ly , ,the the d d iff iff iciu cu lt re lt re gg ion of ion of the the uu nn gg uu id id ee dd turni turni nn g g in in the the rea rea r p r p aa rt of the sucti rt of the sucti oo n si n si de ( de ( 0.6 0.6< < 1147 defined by Equation (2) agree fairly well with the experimental value, as well as with the one obtained 1147 defined by Equation (2) agree fairly well with the experimental value, as we 1148 from the VKI ll as with the one obtained correlation [110]. The success of these simple models is attributed to the proper space- 1251 1251 // <0 <0.8.8 ) whe ) whe re re t t hh e e sh sh ock w ock w aa ve ve t t uu rbulent rbulent boun boun dary dary la la yer yer int int ee ract ract ion ion occ occ uu rs, rs, i i s w s w ee lll pr l pr ee dd ict ict ee dd in in 1148 from the VKI correlation [110]. The success of these simple models is attributed to the proper space- 1148 from the VKI correlation [110]. The success of these simple model 1149 s is attribut time resoluti ed to the proper space on of the bounda - ry layers at separation points in the trailing edge region. Again, this has 1252 1252 a time a time aver aver ag ag ed sen ed sen se by se by all clo all clo sur sur es. es. 1149 time resolution of the boundary layers at separation points in the trailing edge region. Again, this has 1149 time resolution of the boundary layers at separation points in the t 1150 railing edge r been doc egion. ume Aga nted by in, this h Ma an s na et al. [110] and by Sondak et al. [112] (see Figure 49) who could show a 1150 been documented by Manna et al. [110] and by Sondak et al. [112] (see Figure 49) who could show a 1150 been documented by Manna et al. [110] and by Sondak et al. [112 1151 ] (see Figu more tha re 49) who coul n satisfad show a ctory agreement of the computed time averaged velocity profiles with the measured 1151 more than satisfactory agreement of the computed time averaged velocity profiles with the measured 1151 more than satisfactory agreement of the computed time averaged1152 velocity pro one, bot files with the me h on the press asured ure and suction sides at 1.75 diameters upstream of the trailing edge (/ = ± 1152 one, both on the pressure and suction sides at 1.75 diameters upstream of the trailing edge (/ = ± 1152 one, both on the pressure and suction sides at 1.75 diameters upst 1153 ream of t 1. h75 e t wit railin h g ed = 0ge ( at / t = he ± trailin g edge, and = ). The thinner pressure side boundary layer and the 1153 1.75 with = 0 at the trailing edge, and = ). The thinner pressure side boundary layer and the 1153 1.75 with = 0 at the trailing edge, and = ). The thinner pr1154 essure sid blade e boundary circulat laye ion stren r and the gthening the pressure side vortex shedding were estimated to be the cause of 1154 blade circulation strengthening the pressure side vortex shedding were estimated to be the cause of 1154 blade circulation strengthening the pressure side vortex shedding 1155 were est th imat e higher ed to be the c local over exp ause o afn sion at the trailing edge [32]. The very consistent grid refinement study of 1155 the higher local over expansion at the trailing edge [32]. The very consistent grid refinement study of 1155 the higher local over expansion at the trailing edge [32]. The very c 1156 onsistent[112 grid ] brought some i refinement study o mprovements i f n the thinner and fuller pressure side boundary layers 1156 [112] brought some improvements in the thinner and fuller pressure side boundary layers 1156 [112] brought some improvements in the thinner and fuller 1157 pressure predi sid ce ti bound ons. It ia s no ry lay surprise tha ers t with a proper characterization of the boundary layers and of the 1157 predictions. It i 1157 s no predi surprise tha ctions. It i t wi s no th surprise tha a proper cha t wi racth teriza a proper tion of cha the boundary la racterization of yers the boundary la and of the yers and of the 1158 base region, the computed and measured losses agreed well. 1158 base region, the computed and measured losses agreed well. 1158 base region, the computed and measured losses agreed well. Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 10 of 58 271 suction side for both blades are exactly the same, but contrary to the nearly constant Mach number 272 for the VKI blade downstream of the throat, the blade of Xu and Denton is characterized by a very 273 strong adverse pressure gradient in this region. As pointed out by the authors, this causes the suction 274 side boundary layer to be either separated or close to separation up-stream of the trailing edge. 275 Clearly, Sieverding’s correlation cannot deal with blade designs characterized by very strong adverse 276 pressure gradients on the rear suction side causing boundary layer separation before the trailing 277 edge. Int.Int. Int. J. Turb J. J. Turb Turb omach o om mach .ach Prop .. Prop Prop uls. P uls. P uls. P ower ower ower 2018 2018 2018 , 3, x FOR PE , , 3 3,, x FOR PE x FOR PE ER RE ER RE ER RE VIE VI VI W E E W W 44 of 44 of 44 of 58 58 58 Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 44 of 58 278 Figure 9. Comparison of blade Mach number distribution for blade RD of Xu and Denton [15], (solid Int. J. Turbomach. Propuls. Power 2020, 5, 10 42 of 55 279 curve, =0.8, =8 10 ) with VKI blade (dashed curve, =0.8, =10 ). , , Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 44 of 58 280 The possible effect of boundary layer separation resulting from high rear suction side diffusion 281 resulting in high base pressures was also mentioned by Corriveau and Sjolander in 2004 [20], 282 comparing their nominal mid-loaded rotor blade HS1A, mentioned already before, with an aft- Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 44 of 58 283 loaded blade HS1C with an increase of the suction side unguided turning angle from 11.5° to 14.5°. 284 It appears that the increased turning angle could cause, in the transonic range, shock induced 285 boundary layer transition near the trailing edge with, as consequence, a sharp increase of the base 286 pressure, i.e. a sudden drop in the base pressure coefficient as seen in Figure 10. Note that the 287 reported in the figure has been converted to − of the original data. 288 As regards the base pressure data by Deckers and Denton [13] for a low turning blade model 289 and Gostelow et al. [14] for a high turning nozzle guide vane, who report base pressure data far below 1253 1253 1253 1253 290 those of Sieverding’s BPC, their blad e pressure distribution resembles that of the convergent/ 291 divergent blade C in Figure 3 with a negative blade loading near the trailing edge which would 1254 1254 Figure 51. Figure 51. VK VKII L LSS--994 t 4 tuurrbbiinnee b bllaadde,e, = = 0 0.79, .79, = = 2 2.8 10 .8 10 cas casee.. Density Density grad gradient bas ient baseedd 1254 1254 Figure 51. Figure 51. VK VKI L I LSS-9-4 t 94 tuurbrbinine b e blaldade,e, = = 00 .79, .79, = = 22 .8 10 .8 10 cas case.e Density . Density grad gradient bas ient baseded ,, , , 292 explain the very low base pressures. In addition, the blade of Deckers and Denton has a blunt trailing 1255 1255 contou contours; ( rs; (a a) RA ) RAN NS S, , -- , ( , (b b) ) UR URANS, ANS, -- , ( , (c c) L ) LE ES, S, ( (d d) ) Ex Experim perime ents, nts, schli schlieren eren photograph. Adapte photograph. Adapted d 1255 1255 contou contou rs; ( rs; ( a) RA a) RA N N SS , , - - , ( , ( bb ) UR ) UR ANS, ANS, - - , ( , ( c) L c) L EE S, S, ( ( dd ) Ex ) Ex perim perim ents, ents, schli schli eren eren photograph. Adapte photograph. Adapte dd 293 edge, and there is experimental evidence that, compared to a circular trailing edge, the base pressure 1256 1256 from Leonard from Leonard et al. [118] et al. [118]. . 1256 1256 from Leonard from Leonard et al. [118] et al. [118] . . Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 40 of 58 294 for blades with blunt trailing edge might be considerably lower. Sieverding and Heinemann [16] 295 report for flat plate tests at moderate subsonic Mach numbers a drop of the base pressure coefficient Figure 51. VKI LS-94 turbine blade, M = 0.79, Re = 2.8 10 case. Density gradient based contours; 1254 Figure 51. VKI LS-94 turbine blad 2,e, is = 02 .79, = 2.8 10 case. Density gradient based 1261 Figure 51. VKI LS-94 turbine blade, = 0.79, =2.8 10 case. Density gradient based , , 296 by 11% for a plate with squared trailing edge compared to that with a circular trailing edge. (a 1)262 RANS, k-! contours , (b) URANS, ; (a) RANS, k-! - , , ( (cb )) URA LES,N (S, d) Experiments, - , (c) LES, (d) Expe schlie riment ren s, schli photograph. eren photograph. Ad Adapted apted from Leonard 1255 contours; (a) RANS, - , (b) URANS, - , (c) LES, (d) Experiments, schlieren photograph. Adapted 1263 from Leonard et al. [118]. et al. [118]. 1256 from Leonard et al. [118]. 1257 1257 Figure 52. Figure 52. VKI LS94 turbin VKI LS94 turbine blade, e blade, =0 =0..7799,, = = 2 2.8 10 .8 10 case. T case. Tiime mean blade me mean blade surface surface 1135 1257 1257 Figure 47. Figure 52. Figure 52. VKI LS94 turbine VKI LS94 turbin VKI LS94 turbin 1264 blade, Figure 52. e blade, e blade, VKI LS94 turbin =0 .4, =0 =0 =2 e blade, .7.7 99 , 1 ,0 = case = 2 = 2 .8 10 ..8 10 0 Ti .79, me m case. T =2 case. T ean blad .8 10ime mean blade icase. T me mean blade e suirf me a cm e is ean blade surface ent surface ro surface pic Figure 52. VKI LS94 turbine blade, , ,, M = 0.79, , Re = 2.8  10 case. Time mean blade surface isentropic , , 297 2,is Figure 10. 2 Base pressure coefficient for mid-loaded (solid line) and aft-loaded (dashed line) rotor 1265 isentropic Mach number distribution; scale resolving simulations. experiments [21]; DDES 1258 1258 isentropic Ma isentropic Mac ch h nu number distribu mber distribution; sca tion; scale re le resolv solving ing sim simu ulations. lations. ex experim perime ents [21]; nts [21]; DDES DDES 1136 1258 1258 Mach number distribution. isentropic Ma isentropic Ma ch ch nu nu mber distribu mber distribu experime tion; sca tion; scants le re le re[32]; solv solv ing ing sim sim[110]; uu lations. lations. ex [111]; ex perim perim ents [21]; ents [21]; [112]; DDES DDES Mach number distribution; scale resolving simulations. experiments [21]; DDES [120,121]; 298 blade. Symbols: HS1A geometry, HS1C geometry. Adapted from [20]. 1266 [120,121]; DDES [123]; structured LES [118]; unstructured LES [118]; 1259 1259 [120,121]; [120,121]; DD DDES [123] ES [123]; ; struc structtured LES [11 ured LES [1188]; ]; unstructure unstructured d LES [118] LES [118]; ; 1137 1259 1259 [113]. [120,121]; [120,121]; DD DD ES [123] ES [123] ; ; struc struc tured LES [11 tured LES [11 8]; 8]; unstructure unstructure dd LES [118] LES [118] ; ; DDES [123]; structured LES [118]; unstructured LES [118]; unstructured 1267 unstructured LES [56]. 1260 1260 u un nstru structu ctur red LES [56] ed LES [56].. 1260 1260 uu nn stru stru ctu ctu red LES [56] red LES [56] . . LES [56]. 1268 In the base flow region the scatter is instead remarkable, as shown in Figure 53 and Figure 54. 1138 The time averaged base pressure region was also fairly well reproduced by the available 1257 Figure 52. VKI LS94 turbine blade, =0.79, = 2.8 10 case. Time mean blade surface 1269 The physics of the time averaged base pressure, consisting of three pressure minima and two maxima, In the base flow region the scatter is instead remarkable, as shown in Figures 53 and 54. The physics 1261 1261 In the b In the ba ase se flow reg flow regiion on the sc the scatter atter is is in instead stead rem rema ark rkable, able, as show as shown in n in F Fiig gu ure re 53 53 and and F Fiig gu ure re 54. 54. 1261 1261 In the b In the b aa se se flow reg flow reg ion ion the sc the sc atter atter is is in in stead stead rem rem aa rk rk able, able, as show as show n in n in F F ig ig uu re re 53 53 and and F F ig ig uu re re 54. 54. 1139 numerical data, although the differences among the computations and the experiments are generally 1258 isentropic Ma 1270 has alre ch a nu dy b mber distribu een explained b tion; sca efore, and le re willsolv not be ing r sim epeat u ed lations. here. Wha t i ex s wor perim th me ents [21]; ntioning is that DDES of the time averaged base pressure, consisting of three pressure minima and two maxima, has already 1262 1262 1262 The physics o The physics o The physics o f the t f f the t the t ime i ime me ave ave ave rage r rage age d b d d a b b se a a pr se se pr pr essure, co essure, co essure, co nsisting o nsisting o nsisting o f three p f f three p three p ressure r ressure essure m m i m nima i inima nima an a d two an nd two d two ma ma ma xim xim xim a, a a,, 1262 The physics of the t 1271 ime ave the p rage hysica dl b ex aplana se pr tio essure, co n offered for nsisting o the disappe f three p arance of th ressure e pressure minima plateau a a n t d two the trailin ma g xim edge a, 1140 larger than those reported in Figure 48. Indeed, the underlying physics is more complex, as the 1259 [120,121]; DDES [123]; structured LES [118]; unstructured LES [118]; 1272 center at higher Mach number is thoroughly supported by the numerical results of Leonard et al. been explained before, and will not be repeated here. What is worth mentioning is that the physical 1263 1263 has has already b already been explaine een explained before d before, an , and d will not be will not be repeated here. What repeated here. What is wort is worth mentioning is th h mentioning is that at 1263 1263 has has already b already b een explaine een explaine d before d before , an , an dd will not be will not be repeated here. What repeated here. What is wort is wort h mentioning is th h mentioning is th at at 1141 presence of the two pressure and suction side sharp over-expansions at the locations of the boundary 1260 unstructured LES [56]. 1273 [118] and Kopriva et al. [67] (results not shown herein). In fact, when the simulations are performed explanation o ered for the disappearance of the pressure plateau at the trailing edge center at higher 1264 1264 1264 the p tth he p e p hysic h hysic ysic al al exp al exp exp lanllan at anion o at ation o ion o ffer ffer ffer ed ed ed for t for t for t he di h he e di di sap sap sap pep ar pe ean ar aran ce of the pressure ance of the pressure ce of the pressure plat plat plat eau eau eau at the tr at the tr at the tr ailing ailing ailing edg edg edg e e e 1264 the physical explanation offered for the disappearance of the pressure plateau at the trailing edge 1142 layer separation suggests. 1274 with a steady-state approach there is no sudden pressure drop originated by the enrolment of the Mach number is thoroughly supported by the numerical results of Leonard et al. [118] and Kopriva 1265 1265 c ce en ntte err a att h hiig gh he err M Ma ac ch h n nu um mb be err i is s t th ho orro ou ug gh hlly y s su uppor pportted by the numerical re ed by the numerical results of Leon sults of Leona ard rd et al. et al. 1265 1265 ce ce nn tete r a r a t h t h ig ig hh ee r M r M aa ch ch n n uu m m bb ee r i r i s t s t hh oo ro ro uu gg hh ly ly s s uu ppor ppor ted by the numerical re ted by the numerical re sults of Leon sults of Leon aa rd rd et al. et al. 1261 In the b 1275 a un se steady flow reg separating ion the sc shear lay atter ers inis to a in vstead ortex right a remta th rk e trailin able, g as show edge, and th n in e o F vier-ex gure pan 53 sioand ns Figure 54. et al. [1 67 276 ] (r esults occurrin not g atshown the separa her tion ein). pointIn s arfact, e followed when by the a masimulations rked and unphyar sice al r performed ecompressionwith leadina g steady-state 1266 1266 1266 [118 [[118 118 ] a] a n ] a d Kopri n nd Kopri d Kopri va et v va et a et al. [67 al al. [67 . [67 ] (resul ]] (resul (resul ts not ts not ts not shown he shown he shown he rein) r re ein) in) . In .. In In facffta a , whe c ctt, whe , whe n the s n n the s the s imu iimu mu lati la la ons ti tions ons are pe a ar re pe e pe rfor rf rfme or orme me d d d 1266 [118] and Kopriva et al. [67] (results not shown herein). In fact, when the simulations are performed 1262 The physics of the time averaged base pressure, consisting of three pressure minima and two maxima, 1277 to a nearly constant pressure zone. Conversely, all unsteady simulations reproduce, at least approach there is no sudden pressure drop originated by the enrolment of the unsteady separating 1267 1267 with a steady with a steady-state appro -state approa ach there is no ch there is no sudden sudden pre pres ssure drop or sure drop originated by th iginated by the enrolment of the e enrolment of the 1267 1267 with a steady with a steady -state appro -state appro aa ch there is no ch there is no sudden sudden pre pre ssure drop or ssure drop or iginated by th iginated by th e enrolment of the e enrolment of the 1263 has already been explained before, and will not be repeated here. What is worth mentioning is that 1278 qualitatively, the correct base pressure footprint. There is some scatter in the position of the shear layers into a vortex right at the trailing edge, and the over-expansions occurring at the separation 1268 1268 1268 unsteady unsteady unsteady sep sep sep araa ating rra ating ting she she she ar layer a arr layer layer s into ss into into a vortex r a vortex r a vortex r ight iight ght at the tr at the tr at the tr ailin ailin ailin g ed g ed g ed ge, an ge, an ge, an d td t d t he over-expansions h he over-expansions e over-expansions 1268 unsteady separa1 ting 279 she separ ar layer ating sh s ea into r layea vortex r rs as predicted ight by the at the tr eddy viscosity ailing ed closures, a ph ge, and t enomenon he over-expansions that is related to 1264 the physical explanation offered for the disappearance of the pressure plateau at the trailing edge 1280 the correct characterization of the turbulent boundary layers at the point of incipient separation. Both points are followed by a marked and unphysical recompression leading to a nearly constant pressure 1269 1269 occurrin occurring g at at t th he sep e sepa arat ratiion on point points s are are fo followed llowed by by a a marked marked an and d unphysi unphysic ca all re recompression compression lea lead ding ing 1269 1269 occurrin occurrin gg at at t t hh e sep e sep aa rat rat ion ion point point s sare are fo fo llowed llowed by by a a marked marked an an d d unphysi unphysi ca ca l re l re compression compression lea lea dd ing ing 1265 center at higher Mach number is thoroughly supported by the numerical results of Leonard et al. 1281 experiments and computations have shown in fact that there is little or no motion of the separation zone. Conversely, all unsteady simulations reproduce, at least qualitatively, the correct base pressure 1270 1270 1270 to a near ttoo a near a near ly ly ly const const const antaa p nt nt p p ressur rreessur ssur e zone ee zone zone . C. . oC C nversely oonversely nversely , al, , l un al all un l un stead st stead ead y sim yy sim sim ulat uulat ilat ons iions ons rep rep rep roduce rroduce oduce , at ,, le at at le le ast ast ast 1270 to a nearly constant pressure zone. Conversely, all unsteady simulations reproduce, at least 1266 [118] and Kopriva et al. [67] (results not shown herein). In fact, when the simulations are performed 1282 point along the blade surface so that the position of the over-expansion is neat both in a time averaged footprint. There is some scatter in the position of the separating shear layers as predicted by the 1271 1271 qu qual alit itat ative ivelly, y, t thhe correct e correct base pres base pressur suree foot footprint print.. There i There iss some sca some scatter in the posi tter in the positi tion of on of the the 1271 1271 qu qualalititatative ively, ly, t thhe correct e correct 1283 an base pres d i base pres nstantaneousur s sur see nse e foot foot . Conprint vprint ersel. y ,. There i thThere i e intensitsy of some sca s some sca the over-exp tter in the posi tter in the posi ansion strongly depti ends up tion of on of on the the the 1267 with a steady-state approach there is no sudden pressure drop originated by the enrolment of the eddy viscosity closures, a phenomenon that is related to the correct characterization of the turbulent 1272 1272 1272 separ separ separ ating sh a ating sh ting sh ear lay ear lay ear lay ers as predicted e ers as predicted rs as predicted by the by the by the eddy eddy eddy visc visc visc osity clo osity clo osity clo sur s ses, a phenomen ur ures, a phenomen es, a phenomen on that is re on that is re on that is re lated to lated to lated to 1272 separating shear layers as predicted by the eddy viscosity closures, a phenomenon that is related to 1268 unsteady separating shear layers into a vortex right at the trailing edge, and the over-expansions boundary layers at the point of incipient separation. Both experiments and computations have shown 1273 1273 the correct cha the correct char ra ac cteriz teriza at tion of ion of the turb the turbulent bounda ulent boundary l ry la ayers a yers at t the the poi poin nt of t of i in nci cip pi ient sep ent sepa ara rat ti ion. Both on. Both 1273 1273 the correct cha the correct cha ra ra cteriz cteriz aa tion of tion of the turb the turb ulent bounda ulent bounda ry l ry l aa yers a yers a t the t the poi poi nn t of t of i n in ci ci pp ient sep ient sep aa ra ra titon. Both ion. Both 1269 occurring at the separation points are followed by a marked and unphysical recompression leading in fact that there is little or no motion of the separation point along the blade surface so that the 1274 1274 1274 1274 experi experi experi experi ments ments ments ments aa nn a d comp ad comp n nd comp d comp utati utati utati utati oo ns ha ns ha o ons ha ns ha ve ve ve ve shown i shown i shown i shown i nn fa n n fa ct tha fa fa ct tha ct tha ct tha t there t there tt there there isi littl sii littl s s littl littl e or no moti e or no moti e or no moti e or no moti on of on of on of on of the the the the sep sep sep sep aa ra ra a ati ra ra ti on ti on tion on 1270 to a nearly constant pressure zone. Conversely, all unsteady simulations reproduce, at least position of the over-expansion is neat both in a time averaged and instantaneous sense. Conversely, 1275 1275 poi poin nt a t al long the bl ong the bla ad de surfa e surfac ce so tha e so that t the p the po osi siti tion of on of the the over- over-e expa xpan nsion i sion is s nea neat t both i both in n a ti a time me a av vera erag ged ed 1275 1275 poi poi nn t a t a long the bl long the bl aa dd e surfa e surfa ce so tha ce so tha t the p t the p oo si si titi on of on of the the over- over- ee xpa xpa nn sion i sion i s nea s nea t both i t both i nn a ti a ti me me aa vv era era gg ed ed 1271 qualitatively, the correct base pressure footprint. There is some scatter in the position of the the intensity of the over-expansion strongly depends upon the pitchwise flapping motion of the shear 1276 1276 1276 1276 and and and and inst inst inst inst antan antan antan antan ee ous sense. C ous sense. C e eous sense. C ous sense. C oo nversely, th nversely, th o onversely, th nversely, th e intensity e intensity e intensity e intensity oo f the f o o the f f the the over-exp over-exp over-exp over-exp ansion strongly ansion strongly ansion strongly ansion strongly depend depend depend depend s up s up s s up up on on on on the the the the 1272 separating shear layers as predicted by the eddy viscosity closures, a phenomenon that is related to layers, which, as shown by the experiments, is vigorous. This is necessarily smeared by the Reynolds 1273 the correct characterization of the turbulent boundary layers at the point of incipient separation. Both 1143 Figure 48. averaging VKI LS94 turbine blade and by the time, averaging. =0.4, The =2 small 10 range case.of Tim scales e m resolved ean base pre by thesseddy ure viscosity closures 1274 experiments and computations have shown in fact that there is little or no motion of the separation 1144 distribution is . F causing or symbols thesee lar F ge igu discr re 47. epancies between the computations and the experiments. 1275 point along the blade surface so that the position of the over-expansion is neat both in a time averaged 1276 and instantaneous sense. Conversely, the intensity of the over-expansion strongly depends upon the 1145 The location and the magnitude of these two accelerations seem within the reach of the adopted 1146 closure, as well as the pressure plateau of the base region. The predicted base pressure coefficients 1147 defined by Equation (2) agree fairly well with the experimental value, as well as with the one obtained 1148 from the VKI correlation [110]. The success of these simple models is attributed to the proper space- 1149 time resolution of the boundary layers at separation points in the trailing edge region. Again, this has 1150 been documented by Manna et al. [110] and by Sondak et al. [112] (see Figure 49) who could show a 1151 more than satisfactory agreement of the computed time averaged velocity profiles with the measured 1152 one, both on the pressure and suction sides at 1.75 diameters upstream of the trailing edge (/ = ± 1153 1.75 with = 0 at the trailing edge, and = ). The thinner pressure side boundary layer and the 1154 blade circulation strengthening the pressure side vortex shedding were estimated to be the cause of 1155 the higher local over expansion at the trailing edge [32]. The very consistent grid refinement study of 1156 [112] brought some improvements in the thinner and fuller pressure side boundary layers 1157 predictions. It is no surprise that with a proper characterization of the boundary layers and of the 1158 base region, the computed and measured losses agreed well. Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 45 of 58 1284 pitchwise flapping motion of the shear layers, which, as shown by the experiments, is vigorous. This 1285 is necessarily smeared by the Reynolds averaging and by the time averaging. The small range of 1286 scales resolved by the eddy viscosity closures is causing the large discrepancies between the 1287 computations and the experiments. Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 45 of 58 1284 pitchwise flapping motion of the shear layers, which, as shown by the experiments, is vigorous. This 1288 Figure 53. VKI LS94 turbine blade, = 0.79, =2.8 ∙ 10 case. Time mean base pressure 1285 is necessarily smeared by the Reynolds averaging and by the time averaging. The small range of 1289 distribution; eddy viscosity models. For symbols see Figure 50. Int. J. Turbomach. 1286 scales resol Propuls. Power ved b 2020 y th,e e 5, 10 ddy viscosity closures is causing the large discrepancies between the 43 of 55 1 1287 290 comp Rema utatio rns ka a bly nd th the e sa em xpe e c rliosu men re, ts. im plemented in a similar numerical technology returns very large 1291 scatters in the time averaged base pressure region (Leonard et al. [118] and Kopriva et al. [67]), a 1292 phenomenon that should be traced back to the inadequate grid resolution, both in the normal to the 1293 wall and in the streamwise direction of nearly all computations. None of the presented simulations 1294 did undergo a consistent grid refinement study in an unsteady sense, and the effects of the lack of 1295 resolution are evident from the improper prediction of the near trailing edge pressure data, that is 1296 the region at / 2 . As a matter of fact, only one out the three - contributions has an adequate 1297 first cell y value [67], and has attempted to investigate the effects of the grid size in an unstructured 1298 approach. The authors claimed that the coarsest grid achieved grid convergence, but, on account of 1299 the adopted technology, this conclusion is uncertain. Scale resolving simulations presented in Figure 1300 54, produce significant improvements in the base pressure distribution predictions, and the quality 1301 of the LES and DDES data should be considered comparable, despite the differences in modelling 1302 and grid densities, the latter playing a key role. The general trend is to under-predict the pressure 1303 level, while the shape of the wall signal, with its characteristic peak-valley structure, is well 1304 represented by all simulations. Inspection of the boundary layer profiles extracted one diameter Figure 1288 53. VKI Figure 53. LS94 VK turbine I LS94 turbine blade, blM ade, = = 0.79, 0.79,Re =2 =.82.8 ∙ 10 c10 ase. T case. ime meTan base ime mean pressubase re pressure 2,is 2 1305 upstream of the trailing edge circle on both sides of the blades is helpful to understand the scatter in 1289 distribution; eddy viscosity models. For symbols see Figure 50. distribution; eddy viscosity models. For symbols see Figure 50. 1306 the base pressure data. Those data are presented later on. Commented [MM81]: The reference to the two figures 1290 Remarkably the same closure, implemented in a similar numerical technology returns very large causing the problem has been eliminated. 1291 scatters in the time averaged base pressure region (Leonard et al. [118] and Kopriva et al. [67]), a 1292 phenomenon that should be traced back to the inadequate grid resolution, both in the normal to the 1293 wall and in the streamwise direction of nearly all computations. None of the presented simulations 1294 did undergo a consistent grid refinement study in an unsteady sense, and the effects of the lack of 1295 resolution are evident from the improper prediction of the near trailing edge pressure data, that is 1296 the region at / 2 . As a matter of fact, only one out the three - contributions has an adequate 1297 first cell y value [67], and has attempted to investigate the effects of the grid size in an unstructured 1298 approach. The authors claimed that the coarsest grid achieved grid convergence, but, on account of 1299 the adopted technology, this conclusion is uncertain. Scale resolving simulations presented in Figure 1300 54, produce significant improvements in the base pressure distribution predictions, and the quality 1301 of the LES and DDES data should be considered comparable, despite the differences in modelling 1302 and grid densities, the latter playing a key role. The general tr end is to under-predict the pressure Figure 54. VKI LS94 turbine blade, M = 0.79, Re = 2.8 10 case. Time mean base pressure 1307 Figure 54. VKI LS94 turbine blade, = 0.79, 2 =2.8 10 case. Time mean base pressure 2,is 1303 level, while the shape of the wall signal, with its characteristic peak-valley structure, is well 1308 distribution; scale resolving simulations. For symbols see Figure 52. distribution; scale resolving simulations. For symbols see Figure 52. 1304 represented by all simulations. Inspection of the boundary layer profiles extracted one diameter 1305 upstream of the trailing edge circle on both sides of the blades is helpful to understand the scatter in Remarkably the same closure, implemented in a similar numerical technology returns very 1306 the base pressure data. Those data are presented later on. Commented [MM81]: The reference to the two figures large scatters in the time averaged base pressure region (Leonard et al. [118] and Kopriva et al. [67]), causing the problem has been eliminated. a phenomenon that should be traced back to the inadequate grid resolution, both in the normal to the wall and in the streamwise direction of nearly all computations. None of the presented simulations did undergo a consistent grid refinement study in an unsteady sense, and the e ects of the lack of resolution are evident from the improper prediction of the near trailing edge pressure data, that is the region at S/D 2. As a matter of fact, only one out the three k-! contributions has an adequate first cell y value [67], and has attempted to investigate the e ects of the grid size in an unstructured approach. The authors claimed that the coarsest grid achieved grid convergence, but, on account of the adopted technology, this conclusion is uncertain. Scale resolving simulations presented in Figure 54, produce significant improvements in the base 1307 Figure 54. VKI LS94 turbine blade, = 0.79, =2.8 10 case. Time mean base pressure pressure distribution predictions, and the quality of the LES and DDES data should be considered 1308 distribution; scale resolving simulations. For symbols see Figure 52. comparable, despite the di erences in modelling and grid densities, the latter playing a key role. The general trend is to under-predict the pressure level, while the shape of the wall signal, with its characteristic peak-valley structure, is well represented by all simulations. Inspection of the boundary layer profiles extracted one diameter upstream of the trailing edge circle on both sides of the blades is helpful to understand the scatter in the base pressure data. Those data are presented later on. Before proceeding with the analysis of the boundary layers, let us briefly discuss the numerical results of Vagnoli et al. [56], whose simulations are the only one documenting the capabilities of scale resolving simulations to cope with the diculties associated to the base flow prediction in the transonic regime, all the way up to mildly supersonic exit Mach numbers. Those data are reported in Figure 55, where some of the experimental data already presented in Figure 33 (see Section 5), are compared with the LES results obtained with the numerical setup and technology previously described. The agreement is, generally speaking, good at all Mach numbers. The shape of the static pressure traces and level of the base pressure is fairly well captured, although discrepancies exist. At M = 0.79 and M = 0.97 the peak-valley structure of the pressure signal with the neat 2,is 2,is pressure minimum at the center of the trailing edge is essentially reproduced, and the position of the separating shear layers is reasonable. The maximum di erences appear to be in order of 10%. Int. J. Turbo Int. J. Turbo Int. J. Turbo m m m ach. ach. ach. Propuls. Power Propuls. Power Propuls. Power 2018 2018 2018 , , , 3 33 ,, x FOR PE x FOR PE , x FOR PE ER RE ER RE ER RE VI VI VI E E E W W W 46 of 46 of 46 of 58 58 58 1301 Figure 54. VKI LS94 turbine blade, = 0.79, =2.8 10 case. Time mean base pressure 1301 1301 Figure 54. Figure 54. VKI LS94 turbine VKI LS94 turbine blade blade, , = =0 0 .79, .79, =2 =2.8.8 1 100 case. case. Tim Timee m meean an base pres base pressu sure re ,, , 1302 distribution; scale resolving simulations. For symbols see Figure 52. 1302 1302 distribu distribu tion tion ; ; sc sc ale reso ale reso lving lving si si m m u u lations. F lations. F o o r r sy sy m m b b ols s ols s ee F ee F ii g g u u re 52. re 52. 1303 1303 1303 Bef Bef Befo o ore proceedi re proceedi re proceeding wi ng wi ng with the a th the a th the an n na a al llysis of the boun ysis of the boun ysis of the bound d da a ary ry ry lay lay laye eers rs rs, let , let , let us us us brief brief briefl lly d y d y di iisc sc scus us uss t s t s th h he numer e numer e numeri iic cca a al ll 1304 1304 1304 resul resul resul t tt s of Va s of Va s of Va gnol gnol gnol i et i et i et a a a l ll . [5 . [5 . [5 6] 6] 6] , whose si , whose si , whose si mula mula mula ti ti ti ons ons ons a a a r rr e e e t t t h h h e e e only only only one one one doc doc doc u u u ment ment ment ing ing ing t t t h h h e e e capabi capabi capabi lit lit lit ies ies ies of of of sc sc sc al al al e e e 1305 1305 1305 resol resol resolv vviiin nng g g simula simula simulati titions to cope wi ons to cope wi ons to cope with the dif th the dif th the diffffiiicccul ul ulti tities es es associ associ associa aattted ed ed to the ba to the ba to the base fl se fl se flow predi ow predi ow predicccti tition i on i on in nn the the the 1306 1306 1306 tra tra tra n n n soni soni soni c c c regime, al regime, al regime, al l ll the way up to mi the way up to mi the way up to mi ldly supersoni ldly supersoni ldly supersoni c cc exit Mach n exit Mach n exit Mach n u u u mbers. Those mbers. Those mbers. Those data data data are r are r are r e e e po po po rted in rted in rted in 1307 1307 1307 Fig Fig Figu u ure re re 55, w 55, w 55, wh h here some ere some ere some of of of the experime the experime the experimental d ntal d ntal da aata ta ta al al alre re ready ady ady presente presente presented in d in d in Figur Figur Figure ee 33 33 33 (see (see (see Section Section Section 5), 5), 5), are are are 1308 1308 1308 compa compa comparrred with the LES resul ed with the LES resul ed with the LES resulttts obtai s obtai s obtain nned wi ed wi ed with th th the the the numeri numeri numerica ca cal setup a l setup a l setup an nnd d d technol technol technolo oogy previou gy previou gy previousssly ly ly 1309 described. The agreement is, generally speaking, good at all Mach numbers. The shape of the static 1309 1309 describe described. The agreement d. The agreement is, is, gener generaally lly speakin speaking g, goo , good d at all M at all Maach ch number numbers. Th s. The sh e shape ape of the stat of the static ic Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 46 of 58 1310 1310 1310 pressu pressu pressu re t re t re t r rr ac ac ac es and es and es and leve leve leve l o l o l o f ff t t t h h h e base pr e base pr e base pr e e e ssur ssur ssur e e e is is is f f f a a a ir ir ir ly ly ly well c well c well c a a a pt pt pt ured ured ured , a , a , a l ll t tt h h h ough d ough d ough d i ii s ss c cc repanci repanci repanci e e e s ex s ex s ex ist ist ist ... At At At 1309 Before proceeding with the analysis of the boundary layers, let us briefly discuss the numerical 1311 =0.79 and =0.97 the peak-valley structure of the pressure signal with the neat pressure 1311 1311 =0 =0.7 .799 an and d =0 =0.9 .977 tth hee p peeak-v ak-val alley ley st stru ruct cture o ure off t th hee p prres essure signal wi sure signal with the nea th the neatt pressure pressure , , , , , , 1310 results of Vagnoli et al. [56], whose simulations are the only one documenting the capabilities of scale 1312 1312 1312 minimum minimum minimum at at at the center the center the center of the trailin of the trailin of the trailing gg edge edge edge is is is essentially repr essentially repr essentially reproduced, an oduced, an oduced, and dd the position the position the position of the of the of the 1311 resolving simulations to cope with the difficulties associated to the base flow prediction in the 1313 separating shear layers is reasonable. The maximum differences appear to be in order of 10%. When 1313 1313 separ separa 1 at312 tiing ng sh sh tear rear anso la la niyers cyers regi is me, a is r re ll th eason ason e wable aable y up. Th to . Th mildl e maxim e maxim y supersu o u n m m ic difference exit Mac difference h numb s s appear to be in appear to be in ers. Those data are r order o e order o ported inf f 10% 10%.. When When 1313 Figure 55, where some of the experimental data already presented in Figure 33 (see Section 5), are Commented [M82]: Figures should be cited in numerical 1314 1314 1314 the Mach number is inc the Mach number is inc the Mach number is incrrrease ease eased to d to d to = = =1 1 1 .047 .047 .047 th th the degree o e degree o e degree offf non-un non-un non-uniform iform iformiiity of the pressure ty of the pressure ty of the pressure ,, , 1314 compared with the LES results obtained with the numerical setup and technology previously order. 1315 distribution, quantified through the parameter (see Equation (5)) reduces drastically, ending in a 1315 1315 di distri strib bu uti tion, on, q qu uaan nti tifi fied through the p ed through the paara rameter meter ((ssee Eq ee Equa uati tion (5 on (5)) )) redu reduce ces dr s drast astiica call lly, end y, endiing ng in a in a Int. J. Turbomach. Propuls. Power 2020, 5, 10 44 of 55 1315 described. The agreement is, generally speaking, good at all Mach numbers. The shape of the static 1316 1316 1316 pressure plateau. The disappearanc pressure plateau. The disappearanc pressure plateau. The disappearanc 1316 pressure traces and level of the b e ee of of a of se pthe enrollme the enrollme rthe enrollme essure is fairly well c n n nt of the she t of the she t of the she aptured, altha aa ou r lay r lay r lay gh discr e eers rs rs ep into vortices in t into vortices in t into vortices in t ancies exist. At h h he base e base e base Commented [MM83R82]: The text has been corrected. 1317 =0.79 and =0.97 the peak-valley structure of the pressure signal with the neat pressure , , 1317 region characterizing the lower Mach numbers cases, and the effects of the shock patterns delaying 1317 1317 region ch region char arac acterizing the lower Mach terizing the lower Mach n nu umbers c mbers caase ses, a s, an nd the ef d the effects of fects of the shock pa the shock patterns del tterns delaayi ying ng 1318 minimum at the center of the trailing edge is essentially reproduced, and the position of the When the Mach number is increased to M = 1.047 the degree of non-uniformity of the pressure 2,is 1318 1318 1318 ttth h he v e v e vo oort rt rtex fo ex fo ex form rm rmat at ation dow ion dow ion down n nst st stream ream ream t t th h he t e t e tr rrai ai ailin lin ling edg g edg g edge ee ap ap app p pe eear ar ars v s v s ve eery ry ry well well well p p pr rredict edict edicte eed, d, d, at at at le le least ast ast in in in a t a t a tiiim m me ee 1319 separating shear layers is reasonable. The maximum differences appear to be in order of 10%. When distribution, quantified through the parameter Z (see Equation (5)) reduces drastically, ending in a 1319 averaged 1320 sen the se. Those Mach numb are er i indeed rem s increased ato rkable = results, 1.047 thstill repre e degree of sn ent on-iunif ng the ormitstate- y of the pr of-the-art essure in the field. 1319 1319 aver aver aged aged sen sen ss e. Those e. Those are are indeed rem indeed rem a a rr k k able able results, results, still repre still repre ss ent ent ii ng the ng the state- state- of-the-art of-the-art in t in t h h e field e field .. pressure plateau. The disappearance of the enrollment of the shear layers into vortices in the base 1321 distribution, quantified through the parameter (see Equation (5)) reduces drastically, ending in a 1322 pressure plateau. The disappearance of the enrollment of the shear layers into vortices in the base region characterizing the lower Mach numbers cases, and the e ects of the shock patterns delaying the 1323 region characterizing the lower Mach numbers cases, and the effects of the shock patterns delaying vortex formation downstream the trailing edge appears very well predicted, at least in a time averaged 1324 the vortex formation downstream the trailing edge appears very well predicted, at least in a time sense. Those are indeed remarkable results, still representing the state-of-the-art in the field. 1325 averaged sense. Those are indeed remarkable results, still representing the state-of-the-art in the field. 1320 1320 1320 Figure 55. Figure 55. Figure 55. VK VK VKI I I LS94 LS94 LS94 turbine blade. Comparison of turbine blade. Comparison of turbine blade. Comparison of nume nume numerical and expe rical and expe rical and experimental time rimental time rimental time mean base mean base mean base Figure 1326 55. VKI Figure 55. LS94VK turbine I LS94 turbine blade. blade Comparison . Comparison of ofnume numerical rical and eand xperimen experimental tal time mean time base mean base 1327 pressure distribution at transonic exit flow conditions. Symbols—experiments, lines—computations 1321 1321 1321 pressu pressu pressure di re di re distri stri stribu bu bution at trans tion at trans tion at transo o onic ex nic ex nic exit it it flow flow flow c cco o onditions nditions nditions. Sy . Sy . Sym m mb b bols—ex ols—ex ols—experim perim perime eents, nts, nts, lines— lines— lines—c cco o om m mpu pu putations tations tations pressure distribution at transonic exit flow conditions. Symbols—experiments, lines—computations 1328 (LES). =0.79, = 0.97, =1.05. Adapted from [56]. , , , 1322 1322 1322 (LES) (LES) (LES).. . =0 =0 =0..7 7 .79 99 ,, , = = =0 0 0 .97, .97, .97, =1 =1 =1..0 0 .05 55 . Adapted . Adapted . Adapted from [56]. from [56]. from [56]. (LES). M , = 0.79 M, = 0.97, M , = 1.05. Adapted from [56]. , ,2,is ,2, , is , 2, , is 1329 Returning now to the numerical prediction of the boundary layer profiles at the trailing edge, 1330 Figure 56 shows that the eddy viscosity simulations differ considerably, both on the pressure and Returning now to the numerical prediction of the boundary layer profiles at the trailing edge, 1323 1323 1323 R R Re eeturni turni turnin n ng now to the num g now to the num g now to the nume eeri ri rica ca calll predi predi predic cctttiiion of th on of th on of the bo e bo e boundary undary undary layer layer layer profiles profiles profiles at th at th at the trailin e trailin e trailing ed g ed g edg g ge ee,,, 1331 suction sides. Again, the two - of models of Mokulys et al. [117] and Kopriva et al. [67], disagree Figure 56 shows that the eddy viscosity simulations di er considerably, both on the pressure and 1324 Figure 56 shows that the eddy viscosity simulations differ considerably, both on the pressure and 1324 1324 Fig Figu ure re 56 56 sho show ws that the s that the e eddy v ddy viiscos scosity ity sim simu ulation lations s differ differ consid considerably, both erably, both on the pressure on the pressure and and 1332 to some considerable extent. The results of Kopriva et al. [67] are closer to the measurements, and suction sides. Again, the two k-! of models of Mokulys et al. [117] and Kopriva et al. [67], disagree to 1325 1325 1325 suction suction suction sides. sides. sides. Ag Ag Again ain ain,,, the t the t the tw w wo o o - -- o o of ff m m mo o od d de eels o ls o ls of ff Mok Mok Moku u uly ly lys et s et s et al al al. [ . [ . [1 1 117 17 17] ] ] a a an n nd Kop d Kop d Kopr rriv iv iva e a e a et tt al al al. [ . [ . [6 6 67] 7] 7], d , d , di iis ssa a agre gre gree ee 1333 similar to the Baldwin and Lomax values of [117]. This last agreement seems fortuitous, and probably some 1 considerable 334 related t extent. o the insu The fficient results grid resolu of Kopriva tion of Mo et al. kuly [s et a 67] ar l. [117 e closer ]. The already to the measur mention ements, ed grid and similar 1326 to some considerable extent. The results of Kopriva et al. [67] are closer to the measurements, and 1326 1326 ttoo some con some conssider iderable able ext exteent nt. The r . The reesu sult lts o s off Kopr Kopriv iva a et al et al. [6 . [67] a 7] arre cl e closer to the measurements, oser to the measurements, aan nd d 1335 refinement study of Kopriva et al. [67] is based on three unstructured grids characterized by element to the Baldwin and Lomax values of [117]. This last agreement seems fortuitous, and probably related 1327 1327 1327 si si si mil mil mil a a a r to the r to the r to the Ba Ba Ba ldwi ldwi ldwi n a n a n a n n n d d d Loma Loma Loma x v x v x v a a a lues of lues of lues of [1 [1 [1 17 17 17 ]. ]. ]. Th Th Th is last is last is last agreemen agreemen agreemen t t t seems fortuitous, and seems fortuitous, and seems fortuitous, and pro pro pro b b b ably ably ably 1336 edge length change in the wake region of approximately 15–20%. Results presented in their study 1328 rel to a the ted t insu o t  he ins cientu grid fficient resolution grid reso of lut Mokulys ion of Mok et al.u[lys 117].et a The l. [alr 117 eady ]. The mentioned already ment grid ione refinement d grid 1328 1328 rel relaatteed t d too t thhe ins e insuuffffic icient ient grid grid reso resolut lutiion of on of Mok Mokuulys lys et et a all. . [[111177]].. The The alr alreeaaddy ment y mentione ioned gri d gridd 1337 refer to near wake time averaged pressure data collected through a traverse across the wake in the 1338 direction normal to the tangent to the camber line at the trailing edge. The traverse is 2.5 trailing edge study of Kopriva et al. [67] is based on three unstructured grids characterized by element edge length 1329 1329 1329 r rr e e e f ffi iine ne ne m m m e e e n n n t tt s s st tt u u ud d d y y y o o o f ff K K K o o o p p p r rr i ii v v v a a a e e e t tt a a a l ll... [ [ [ 6 6 6 7 7 7] ]] i i i s ss b b b a a a s sse e e d o d o d o n n n thr thr thr ee un ee un ee un structur structur structur ed gr ed gr ed gr id id id s ch s ch s ch ar ar ar acterized acterized acterized by by by e e e l llement ement ement 1339 diameters downstream the trailing edge itself. Since velocity and rate of strains in the boundary layers 1330 edge change len in gth c thehwake ange in region the wake of appr reoximately gion of appr 15–20%. oximatel Results y 15–20 pr %. esented Resultin s presented i their studynr thei eferr to study near 1330 1330 edge edge len lenggth c th ch hange ange in in the the wake wake re region gion of of appr approxi oxim matel ately y 15 15–20 –20%. %. R Reesul sultts presented i s presented in n thei their r study study 1340 are known to be more sensitive quantities than pressure, and on account of the convection scheme wake time averaged pressure data collected through a traverse across the wake in the direction normal 1331 1331 1331 refer refer refer t t to o o ne ne near ar ar wake wake wake t t tiiim m me ee av av average erage eraged d d p p pr rress ess essu u ure re re dat dat data aa col col collllected through ected through ected through a aa tra tra trav v verse erse erse a aac ccross the wa ross the wa ross the wake ke ke iiin n n the the the 1341 adopted in the solver, which is based on a blend of second order central differencing and first order 1342 upwinding, the achievement of grid independence with the coarsest mesh is uncertain. Yet, their - 1332 di torecti the o tangent n norma to l to the ta the camber ngent to th line at e c the amber line trailing at edge. the trailing The trav ederse ge. The tr is 2.5aver trailing se is 2.5 tr edgeailing ed diameters ge 1332 1332 di di recti recti o o n norma n norma ll to the ta to the ta ngent to th ngent to th e e c c a a mber line mber line at at the trailing the trailing ed ed ge. The tr ge. The tr aver aver se se is is 2.5 tr 2.5 tr ailing ed ailing ed g g e e 1343 simulation is by far the best eddy viscosity result available as today. It is not a coincidence that the downstream the trailing edge itself. Since velocity and rate of strains in the boundary layers are known 1333 1333 1333 diameter diameter diameter s d s d s d o o o wnstream the wnstream the wnstream the trailing trailing trailing edge edge edge itself. Since ve itself. Since ve itself. Since ve locit locit locit y y y and and and r r r a a a t tt e e e of of of st st st r rr a a a in in in s s s in in in t tt h h h e bound e bound e bound a a a ry ry ry lay lay lay e e e rs rs rs 1344 appropriate resolution of the boundary layers at the point of incipient separation warrants a more 1334 a to re known to be more sensi be more sensitive quantities tive qua thann pr titi essur es tha e, n and pressure, an on account d on of account of the the convection scheme convection sc adopted hem in e 1334 1334 aarre known to be more sensi e known to be more sensi 1345 than satisfactory base pre ttiiv ve qua e qua ssure re n nt gio tiiti ti n es tha es tha predicti n n on. pr pr essure, an essure, and on d on account of the account of the convection sc convection schem hemee the solver, which is based on a blend of second order central di erencing and first order upwinding, 1335 1335 1335 adopt adopt adopte eed d d in in in t t th h he so e so e solver, lver, lver, wh wh whic ic ich i h i h is ss bas bas base eed d d on on on a b a b a blllend end end of of of sec sec seco o ond order central differen nd order central differen nd order central differencing an cing an cing and d d first first first order order order 1336 upwind the achievement ing, the achieveme of grid independence nt of grid ind with ependence wi the coarsest th the coa mesh is rsest mesh i uncertain. s Y uncerta et, theirin. Yet, thei k-! simulation r - 1336 1336 upwind upwinding, t ing, t h he achieveme e achievemen nt of grid ind t of grid indeependence wi pendence with the coa th the coarrsest mesh i sest mesh iss uncerta uncertaiin. Yet, thei n. Yet, their r -- is by far the best eddy viscosity result available as today. It is not a coincidence that the appropriate 1337 1337 1337 s ss i ii mu mu mu lat lat lat i ii on on on is is is by fa by fa by fa r t r t r t h h h e bes e bes e bes t tt eddy visco eddy visco eddy visco s ss i ii t tt y resu y resu y resu lt lt lt av av av ai ai ai l ll a a a b b b l ll e e e as t as t as t o o o d d d a a a y. y. y. It It It is not is not is not a co a co a co i ii n n n cidence t cidence t cidence t h h h at at at t tt h h h e e e resolution of the boundary layers at the point of incipient separation warrants a more than satisfactory base pressure region prediction. Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 47 of 58 (a) (b) 1346 Figure 56. VKI LS94 turbine blade, = 0.79, =2.8 10 case 6. Time mean velocity profiles on Figure 56. VKI LS94 turbine blade, M = 0.79, Re = 2.8 10 case. Time mean velocity profiles on 2,is 2 1347 pressure (a) and suction (b) side of the blade at ⁄ = 1.75 . Eddy viscosity models. For symbols see pressure (a) and suction (b) side of the blade at S/D = 1.75. Eddy viscosity models. For symbols see 1348 Figure 50. Figure 50. 1349 The scale resolving simulations here exhibit the largest differences, Figure 57. The two DDESs Commented [MM84]: Text has been amended. 1350 of El-Gendi et al. [120,121] and Wang et al. [123] predict remarkably well the suction and pressure 1351 side velocity profiles. Conversely the two LESs of Leonard et al. [118] and Vagnoli et al. [56], 1352 completely miss both profiles. There is a factor 10 in the number of grid nodes between the two 1353 DDESs and the LES of Leonard et al. [118], and a factor of 2 for that of Vagnoli et al. [56]. Furthermore, 1354 the inner layer of the LESs is either bypassed (first y at 5 or 40) or fully unresolved (spacing of 48 1355 wall units along the blade height, in [56]). The major shortcoming of wall resolving LESs is precisely 1356 the inability of all subgrid scale models to reproduce the effects of the dynamics of the low speed 1357 streaks, their growth, breakdown and the wall turbulence generation process [95,102,124]. The 1358 consequence of this shortcoming is that the only successful wall resolving LESs are those whose inner 1359 layer resolution is sufficient to describe to some extent the streaks dynamics. The requirements are 1360 rather severe, since these near wall coherent structures have a typical length of 1000 wall units, a 1361 width of 30, while their average lateral spacing is in the order of 100 wall units [103,124]. They are 1362 responsible for the sweep and ejection phenomena, the inward/outward motion (with respect to the 1363 wall) of high energy fluid lumps, and therefore they are energy carrying structures. Their appropriate 1364 numerical resolution usually qualified in terms of mesh spacing in inner coordinates, is rather 1365 demanding, and also heavily depends upon the accuracy of the numerical procedure used to solve 1366 the governing equations. (a) (b) 1367 Figure 57. VKI LS94 turbine blade, = 0.79, =2.8 10 case. Time mean velocity profiles on 1368 pressure (a) and suction (b) side of the blade at ⁄ = 1.75 . Scale resolving models. For symbols 1369 see Figure 52. Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 47 of 58 Int. J. Turbomach. Propuls. Power 2020, 5, 10 45 of 55 (a) (b) The scale resolving simulations here exhibit the largest di erences, Figure 57. The two DDESs of 1346 Figure 56. VKI LS94 turbine blade, = 0.79, =2.8 10 case. Time mean velocity profiles on El-Gendi et al. [120,121] and Wang et al. [123] predict remarkably well the suction and pressure side 1347 pressure (a) and suction (b) side of the blade at ⁄ = 1.75 . Eddy viscosity models. For symbols see velocity profiles. Conversely the two LESs of Leonard et al. [118] and Vagnoli et al. [56], completely 1348 Figure 50. miss both profiles. There is a factor 10 in the number of grid nodes between the two DDESs and the 1349 The scale resolving simulations here exhibit the largest differences, Figure 57. The two DDESs Commented [MM84]: Text has been amended. LES of Leonard et al. [118], and a factor of 2 for that of Vagnoli et al. [56]. Furthermore, the inner 1350 of El-Gendi et al. [120,121] and Wang et al. [123] predict remarkably well the suction and pressure layer of 1351 the LESs side ve islocity profile either bypassed s. Conversel (first y the ytwo LESs at 5 or of 40) Leo or narfully d et alunr . [118 esolved ] and Vagno (spacing li et al. [of 56], 48 wall units 1352 completely miss both profiles. There is a factor 10 in the number of grid nodes between the two along the blade height, in [56]). The major shortcoming of wall resolving LESs is precisely the inability 1353 DDESs and the LES of Leonard et al. [118], and a factor of 2 for that of Vagnoli et al. [56]. Furthermore, of all subgrid scale models to reproduce the e ects of the dynamics of the low speed streaks, their 1354 the inner layer of the LESs is either bypassed (first y at 5 or 40) or fully unresolved (spacing of 48 1355 wall units along the blade height, in [56]). The major shortcoming of wall resolving LESs is precisely growth, breakdown and the wall turbulence generation process [95,102,124]. The consequence of this 1356 the inability of all subgrid scale models to reproduce the effects of the dynamics of the low speed shortcoming is that the only successful wall resolving LESs are those whose inner layer resolution is 1357 streaks, their growth, breakdown and the wall turbulence generation process [95,102,124]. The sucient to describe to some extent the streaks dynamics. The requirements are rather severe, since 1358 consequence of this shortcoming is that the only successful wall resolving LESs are those whose inner 1359 layer resolution is sufficient to describe to some extent the streaks dynamics. The requirements are these near wall coherent structures have a typical length of 1000 wall units, a width of 30, while their 1360 rather severe, since these near wall coherent structures have a typical length of 1000 wall units, a average lateral spacing is in the order of 100 wall units [103,124]. They are responsible for the sweep 1361 width of 30, while their average lateral spacing is in the order of 100 wall units [103,124]. They are and ejection phenomena, the inward/outward motion (with respect to the wall) of high energy fluid 1362 responsible for the sweep and ejection phenomena, the inward/outward motion (with respect to the 1363 wall) of high energy fluid lumps, and therefore they are energy carrying structures. Their appropriate lumps, and therefore they are energy carrying structures. Their appropriate numerical resolution 1364 numerical resolution usually qualified in terms of mesh spacing in inner coordinates, is rather usually qualified in terms of mesh spacing in inner coordinates, is rather demanding, and also heavily 1365 demanding, and also heavily depends upon the accuracy of the numerical procedure used to solve depends upon the accuracy of the numerical procedure used to solve the governing equations. 1366 the governing equations. (a) (b) 1367 Figure 57. VKI LS94 turbine blade, = 0.79, =2.8 10 cas6e. Time mean velocity profiles on Figure 57. VKI LS94 turbine blade, M = 0.79, Re = 2.8 10 case. Time mean velocity profiles on 2,is 2 1368 pressure (a) and suction (b) side of the blade at ⁄ = 1.75 . Scale resolving models. For symbols pressure (a) and suction (b) side of the blade at S/D = 1.75. Scale resolving models. For symbols see 1369 see Figure 52. Figure 52. For higher order methods, viz those with spectral error decay, they can be estimated to be + + Dx  50–100, Dy  10–20 in the streamwise and spanwise directions, respectively, while in the normal to the wall direction there should be some 10–20 points in the first 30 wall units. These requirements are not respected by any of the two LESs clearly highlighting the inability of the SGS model to provide the correct energy contribution of the sub-grid scales to the super-grid one; it is also no surprise that the two DDESs perform better than the two LESs, thanks to the properly modelled (via k-!) inner wall layer. Indeed, their suction and pressure side boundary layer predictions are by far the most accurate among the available data. This is clearly shown in Figure 57. Also, the first order time integration scheme of Vagnoli et al. [56] is inadequate for a scale resolving simulation requiring a minimum time accuracy of order two. The benefit of the considerably more refined DDES meshes, allowing for the resolution of larger number of turbulent scales, should become evident elsewhere. We next compare in Figure 58 the wake shape as predicted by the available closures. The comparison is based on a wake traverse located at 2.5 trailing edge diameters downstream the trailing edge itself, as already previously described. The prediction of a turbulent wake behind a turbine blade is a rather challenging task which is complicated by the trailing edge bluntness promoting the shedding of large-scale vortex structures. Essential for the correct prediction of the wake formation and development is the proper description of the boundary layers at the point of incipient separation. At the current Reynolds number, the scale separation is huge, and the boundary between modelled and resolved scales is uncertain, so that the extent of the grey area and the filter width may become a Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 48 of 58 1370 For higher order methods, viz those with spectral error decay, they can be estimated to be Δ ≈ 1371 50-100, Δ ≈10-20 in the streamwise and spanwise directions, respectively, while in the normal 1372 to the wall direction there should be some 10–20 points in the first 30 wall units. 1373 These requirements are not respected by any of the two LESs clearly highlighting the inability 1374 of the SGS model to provide the correct energy contribution of the sub-grid scales to the super-grid 1375 one; it is also no surprise that the two DDESs perform better than the two LESs, thanks to the properly 1376 modelled (via - ) inner wall layer. Indeed, their suction and pressure side boundary layer 1377 predictions are by far the most accurate among the available data. This is clearly shown in Figure 57. 1378 Also, the first order time integration scheme of Vagnoli et al. [56] is inadequate for a scale resolving 1379 simulation requiring a minimum time accuracy of order two. The benefit of the considerably more 1380 refined DDES meshes, allowing for the resolution of larger number of turbulent scales, should 1381 become evident elsewhere. 1382 We next compare in Figure 58 the wake shape as predicted by the available closures. The 1383 comparison is based on a wake traverse located at 2.5 trailing edge diameters downstream the trailing 1384 edge itself, as already previously described. The prediction of a turbulent wake behind a turbine 1385 blade is a rather challenging task which is complicated by the trailing edge bluntness promoting the 1386 shedding of large-scale vortex structures. Essential for the correct prediction of the wake formation Int. J. Turbomach. Propuls. Power 2020, 5, 10 46 of 55 1387 and development is the proper description of the boundary layers at the point of incipient separation. 1388 At the current Reynolds number, the scale separation is huge, and the boundary between modelled 1389 and resolved scales is uncertain, so that the extent of the grey area and the filter width may become 1390 a concern. Yet all closures seem capable to reproduce the essential features of the large-scale concern. Yet all closures seem capable to reproduce the essential features of the large-scale unsteadiness 1391 unsteadiness associated with the vortex shedding process; the agreement is a little more than associated with the vortex shedding process; the agreement is a little more than qualitative. This is 1392 qualitative. This is best seen in Figure 58 comparing the numerical total pressure profiles with the best seen 1393in Figur experim e e 58 ntal comparing data. the numerical total pressure profiles with the experimental data. (a) (b) 1394 Figure 58. VKI LS94 turbine blade, = 0.79, =2.8 10 case 6 . Time mean total pressure wake Figure 58. VKI LS94 turbine blade, M = 0.79, Re = 2.8 10 case. Time mean total pressure wake 2,is 2 1395 traverses at 2.5 diameters downstream the trailing edge. (a): eddy viscosity models, (b): scale resolving traverses at 2.5 diameters downstream the trailing edge. (a): eddy viscosity models, (b): scale resolving 1396 simulations. For symbols see Figure 50 and Figure 52. simulations. For symbols see Figures 50 and 52. 1397 While the wake width seems fairly well predicted by all closures, the wake velocity deficit is not, 1398 by some appreciable quantities. Surprisingly, the - results of Mokulys et al. [117] look better than While the wake width seems fairly well predicted by all closures, the wake velocity deficit is 1399 those of Kopriva et al. [67] despite the grid refinement study of the latter and the superior agreement not, by some appreciable quantities. Surprisingly, the k-! results of Mokulys et al. [117] look better 1400 in terms of boundary layer features on both sides of the blade. The DDES of El-Gendi et al. [120,121] 1401 is by far the worst of all simulations in terms of closeness to the experiments. This is surprising given than those of Kopriva et al. [67] despite the grid refinement study of the latter and the superior 1402 the good quality of the other results extracted from the same simulation. The authors discuss in some agreement in terms of boundary layer features on both sides of the blade. The DDES of El-Gendi 1403 details the potential reasons for those discrepancies, addressing numerical issues, turbulence et al. [120,121] is by far the worst of all simulations in terms of closeness to the experiments. This is 1404 modelling issues and grid size effects. Unfortunately, the analysis was inconclusive, and a more in- 1405 depth inspection of the data would have been necessary to identify the root reasons for the deviations surprising given the good quality of the other results extracted from the same simulation. The authors 1406 documented in Figure 58. As previously detailed in this section, a DDES is characterized by three discuss in some details the potential reasons for those discrepancies, addressing numerical issues, turbulence modelling issues and grid size e ects. Unfortunately, the analysis was inconclusive, and a more in-depth inspection of the data would have been necessary to identify the root reasons for the deviations documented in Figure 58. As previously detailed in this section, a DDES is characterized by three zones, namely a URANS, a LES and a hybrid one, and the extent of the latter dominates to some remarkable extent the quality of the whole simulation. The in-depth analysis of the spatial distribution of the f function (see Equation (10)) (or equivalently of the F in the DES-SST-zonal model) would d SST have been of great help to identify the responsibilities of the turbulence modelling and of the filter width. What can be conjectured here, is that at the location of the wake traverses the DDES simulation is in the grey area, or, worse, in the LES one with a too large filter width. Conversely, in the base region and all around the blade in the boundary layers, the URANS mode is properly working. This can be inferred from the nearly identical boundary layer profiles as predicted by the k-! results of Kopriva et al. [67], and the DDES data of El-Gendi et al. [120,121], both of which qualified through an identical eddy viscosity model in the wall region (see Figures 56 and 57). Thus, while the very near wake and the base region features heavily depend upon the characteristics of the boundary layers at the point of incipient separation, already a few diameters downstream the trailing edge the dynamics of the vortex shedding formation scheme is too complicate for an eddy viscosity closure as well as for an unresolved LES. The total temperature results reported in Figure 59 are similar to the total pressure ones. All models reproduce approximately well the occurrence of the Eckert-Weiss e ect, with its characteristic flow heating (respectively cooling) at the wake edges (respectively center). The magnitude of the positive and negative (compared to the inlet value) total temperature peaks, as well as their locations is only marginally well predicted by the eddy viscosity closures, while some improvements can be appreciated in the DDES of El-Gendi et al. [120,121]. Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 49 of 58 1407 zones, namely a URANS, a LES and a hybrid one, and the extent of the latter dominates to some 1408 remarkable extent the quality of the whole simulation. The in-depth analysis of the spatial 1409 distribution of the function (see Equation (10)) (or equivalently of the in the DES-SST-zonal 1410 model) would have been of great help to identify the responsibilities of the turbulence modelling and 1411 of the filter width. What can be conjectured here, is that at the location of the wake traverses the DDES 1412 simulation is in the grey area, or, worse, in the LES one with a too large filter width. Conversely, in 1413 the base region and all around the blade in the boundary layers, the URANS mode is properly 1414 working. This can be inferred from the nearly identical boundary layer profiles as predicted by the 1415 - results of Kopriva et al. [67], and the DDES data of El-Gendi et al. [120,121], both of which 1416 qualified through an identical eddy viscosity model in the wall region (see Figure 56 and Figure 57). 1417 Thus, while the very near wake and the base region features heavily depend upon the characteristics 1418 of the boundary layers at the point of incipient separation, already a few diameters downstream the 1419 trailing edge the dynamics of the vortex shedding formation scheme is too complicate for an eddy 1420 viscosity closure as well as for an unresolved LES. 1421 The total temperature results reported in Figure 59 are similar to the total pressure ones. All 1422 models reproduce approximately well the occurrence of the Eckert-Weiss effect, with its characteristic 1423 flow heating (respectively cooling) at the wake edges (respectively center). The magnitude of the 1424 positive and negative (compared to the inlet value) total temperature peaks, as well as their locations 1425 is only marginally well predicted by the eddy viscosity closures, while some improvements can be Int. J. Turbomach. Propuls. Power 2020, 5, 10 47 of 55 1426 appreciated in the DDES of El-Gendi et al. [120,121]. (a) (b) 1427 Figure 59. VKI LS94 turbine blade, = 0.79, =2.8 10 cas6 e. Time mean total temperature Figure 59. VKI LS94 turbine blade, M = 0.79, Re = 2.8 10 case. Time mean total temperature 2,is 2 1428 traverse at 2.5 diameters downstream the trailing edge. (a): eddy viscosity models, (b): scale resolving traverse at 2.5 diameters downstream the trailing edge. (a): eddy viscosity models, (b): scale resolving 1429 simulations. For symbols see Figure 50 and Figure 52. simulations. For symbols see Figures 50 and 52. 1430 Finally, the Strouhal numbers as predicted by all numerical models are presented in Table 7. Finally, the Strouhal numbers as predicted by all numerical models are presented in Table 7. 1431 Table 7. VKI LS94 turbine blade, = 0.79, =2.8 10 case. Strouhal numbers. Authors Method Closure Strouhal Table 7. VKI LS94 turbine blade, M = 0.79, Re = 2.8 10 case. Strouhal numbers. 2,is 2 Sieverding et al. [21] Experiments / 0.219 Mokulys et al. [117] URANS Baldwin and Lomax [97] 0.206 Authors Method Closure Strouhal Mokulys et al. [117] URANS Spalart and Allmaras [98] 0.177 Mokulys et al. [117] URANS Wilcox - [100] 0.199 Sieverding et al. [21] Experiments / 0.219 Leonard et al. [118] URANS Wilcox - [100] 0.276 Mokulys et al. [117] URANS Baldwin and Lomax [97] 0.206 Kopriva et al. [67] URANS Wilcox - [100] 0.212 Mokulys et al. [117] URANS Spalart and Allmaras [98] 0.177 El Gendi et al. [120,121] DDES Spalart et al. [104] 0.215 Mokulys et al. [117] URANS Wilcox k-! [100] 0.199 Wang et al. [123] DDES Spalart et al. [104] 0.216 Leonard et al. [118] URANS Wilcox k-! [100] 0.276 Leonard et al. [118] LES Smagorinsky [119] 0.228 Kopriva et al. [67] URANS Wilcox k-! [100] 0.212 El Gendi et al. [120,121] DDES Spalart et al. [104] 0.215 Wang et al. [123] DDES Spalart et al. [104] 0.216 Leonard et al. [118] LES Smagorinsky [119] 0.228 Vagnoli et al. [56] LES Wall damped Smagorinsky [122] 0.220 Recall that the proper evaluation of the vortex shedding frequency requires a correct modelling of the near wake mixing process, that is of the interaction between the unsteady separating shear layers [38]. The di erences between the experiments and the EVM solutions are definitely larger than those pertaining to the SRS, all of which predict rather well the dominant shedding frequency. However, on account of the complexity and cost of the SRS the results obtained with the simple EVM closures are to be considered appealing. Inspection of the higher pressure modes in the near wake, both in terms of amplitude and phase, would probably underline larger di erences and discrepancies. 8. Conclusions This review manuscript has addressed in full details the flow peculiarities occurring at the trailing edge of steam and gas turbine blades, with the help of experimental and numerical data. The study is started presenting the achievements of the 40 years old VKI base pressure correlation as applied to old and new turbine blades. While the simple architecture of the formula returns satisfactorily base pressure estimates and thus loss predictions for conventional turbine blade designs, the correlation appears to fail in cases of blade designs characterized by very strong adverse pressure gradients on the rear suction side causing possibly boundary layer separation before the trailing edge. An additional weakness of the correlation resides in the fact that all experimental base pressure data are recorded by a single pressure tap in the blade trailing edges which implies the assumption of an isobaric trailing edge base pressure. This assumption is unfortunately only valid for low subsonic and supersonic Mach numbers as demonstrated recently by large scale cascade experiments. Indeed, about twenty years after the publication of the base pressure correlation, experiments carried out at the von Kàrmàn Institute on large scale turbine blades both at subsonic and transonic Int. J. Turbomach. Propuls. Power 2020, 5, 10 48 of 55 outlet Mach numbers allowed major advances in the understanding of the mechanism of vortex formation and shedding in the near trailing edge wake region. Thanks to the large size of the test article, specifically designed for providing time resolved data at high spatial resolution, it has been shown that the flow approaching the trailing edge undergoes a strong acceleration both on the pressure and suction side, before leaving the blade. Two over-expansions of di erent strength because of the di erences in the boundary layers state and of the blade circulation, have been documented and attributed to the e ects of the vortex shedding. At those locations remarkable pressure fluctuations occur, reaching 80% of the outlet dynamic pressure. While at subsonic flow conditions the central trailing edge base region exhibits a rather constant pressure area, at higher Mach numbers the base pressure is characterized by the appearance of a steadily growing pressure minimum which, at the transition from a normal to an oblique trailing edge shock system, does give suddenly again way to an isobaric region. A physically consistent explanation of the departure from the assumed isobaric trailing edge base region has been proposed, and the implications with the VKI correlation outlined. The dynamic of the shear layers has also been identified as the root cause of the formation of the acoustic wave systems occurring in the trailing edge region and their impact on the rear suction side pressure distribution. The energy separation phenomenon, since long known to occur in cylinder flow, has been documented to also exist at the exit of transonic uncooled stator blades, causing major concerns for the mechanical integrity of the following blade row when invested by uneven total temperature distributions. Important achievements were obtained by the Canadian research group of the NRC who first measured time resolved pressure and total temperature distribution in the wake of transonic turbine blades. The data, corroborated by successive experiments, highlighted the relation between the vortex street formation and propagation with the energy separation phenomenon. High resolution experimental data were released for code-to-experiment validation and the outcome of the available simulations, presented in a dedicated section, has been discussed at length. The turbine trailing edge frequency features, as measured on a number of blades, have been analyzed and their relations with the geometry, the boundary layer state at the point of incipient separation and the governing dimensionless parameters, clarified. In spite of the considerable progress made so far for a better understanding of unsteady trailing edge flows and their e ects on the blade performance, there is clearly room for further experimental research on unsteady trailing edge flows. The main objective should be the conception and preparation of additional large-scale cascade tests allowing high resolution spatial and temporal measurements. New benchmark test cases would then be available for experiment-to-experiment and code-to-experiment validation. The benchmark test cases presented in this paper were characterized by turbulent boundary layers on both suction and pressure sides at the point of separation from the trailing edge. It would be certainly interesting to dispose of a large-scale test case with mixed turbulent/laminar (suction side/pressure side) trailing edge flow conditions. It would also be desirable to apply high resolution fast optical measurement techniques to determine the time varying wake velocity field for the evaluation of the rate of strain and the vorticity tensors. Long time and phase averaged turbulence data will naturally come out, thus enhancing the actual knowledge of the wake mixing process. This may ultimately require the measurement of the three-dimensional time-varying velocity field. It would also be highly desirable to have more test data for the downstream evolution of the wake total temperature profile, the knowledge of which is of prime importance for the evaluation of the mechanical integrity of the downstream blade row. The reduction of the trailing edge vortex intensity and therewith the profile losses by appropriate trailing edge shaping, as e.g. elliptic trailing edges, deserves certainly further attention. Again, large scale test set up will be needed to highlight the di erences in the wake mixing process. On the numerical side the progresses achieved over 40 years of unsteady turbine wake flow computations have been impressive. This is equally due to the advances in numerical methods and modelling concepts. The authors have put together all available computations of the VKI LS94 turbine Int. J. Turbomach. Propuls. Power 2020, 5, 10 49 of 55 blade, whose geometry and experimental data have been previously presented. Within the bounds of the limited published material, a few concluding remarks on the ability of the adopted turbulence closures can be put forward. While the freely available turbine geometry is relatively simple, the flow conditions are not, mainly because of the large Reynolds number. Very few of the URANS contributions did achieve grid convergence in the sense of the local truncation error, in order to confine those errors at values smaller than the modelling ones. The problem of the inadequate number of numerical parameters becomes particularly o ending for the scale resolving simulations (DES, DDES and LES) for which the interaction between the space-time numerical integration procedure and the turbulence closure is known to be warring, especially when implicit filtering and low order methods are used. In addition, and unlike URANS, the resolution requirements are far more stringent, and hard to satisfy. It has been shown that the URANS calculations presently reviewed comply with the spectral gap requirement, and, therefore, the expectations of predictivity are legitimate. In fact, although a systematic grid convergence study was rarely achieved, the general quality of the numerical solutions obtained with eddy viscosity models can be rated satisfactory. Algebraic, one equation and two equations models proved capable to predict reasonably well the time averaged blade pressure distribution, even in the dicult base region, both in the subsonic and transonic regimes. Time averaged boundary layer profiles in the near trailing edge region, and even more, wake features are more problematic, especially in the high Mach number cases. Particularly, the total pressure and total temperature profiles did not go further than a qualitative agreement with the experiments, although the energy separation phenomenon was correctly represented. Scale resolving simulations improved the predictivity level of the URANS, but not as expected. Most of the deficiencies have been traced back to an inadequate sub-grid filter positioning often causing severe deviations from the experiments. The hybrid simulations were more performant than the pure LESs, mainly because of the larger number of parameters of the former. Boundary layers and even more the near wake region were poorly predicted by the scale resolving simulations, partly because of the already mentioned inertial subrange reproduction failure, a consequence of the insucient spatial resolution, and partly because of known SGS limitations, that is their inability to provide the appropriate energy contribution of the unresolved scales to the resolved ones in regions of strong shears. The unsteady features of the flow have not been fully exploited, and thus the judgment on their quality is uncertain. The Strouhal number was reasonably well predicted by all closure. What appears to be needed to improve the quality of the available high-fidelity TWF simulations is a more detailed and conscious selection of the spatial resolution. At high Reynolds number and even more in transonic flow conditions, this turns out to be the most dicult objective to comply with, especially because in the DES/LES world there is no equivalent of the grid convergence concept routinely applied in the URANS world to isolate the modelling errors. DES/LES have an indissoluble cut-o placement - modelling error relationship which is dicult to identify, especially when the cut-o , that is the filter width, is implicitly defined by the mesh size. In those instances, the mixture of numerical and modelling errors cannot be unraveled. Also, the presence of an inertial subrange, a pre-requisite for the correct application of the LES concept, is dicult to ascertain a-priori. Nevertheless, to acquire more credibility, the future class of numerical computations will have to provide more and more details of the resolved turbulence, presenting spectra, spatial correlations and stress tensor components of the computed fields at key locations. Those data will hopefully convince the reader of the quality of the simulations and give more confidence in the collected data. Hybrid methods will have to systematically o er quantitative details of the boundaries of the so-called grey area, to give a precise idea of what and where was modeled and resolved by the simulations. Databases of scale resolving simulations respecting certain properly defined quality criteria should be made openly available to the whole turbomachinery community for code-to-code and code-to-experiment validation. Author Contributions: Conceptualization, methodology, formal analysis, investigation, data curation, writing—original draft preparation, writing—review and editing: C.S. and M.M. All authors have read and agreed to the published version of the manuscript. Int. J. Turbomach. Propuls. Power 2020, 5, 10 50 of 55 Funding: This research received no external funding. The APC were a orded by the Dipartimento di Ingegneria Industriale of the Università degli Studi di Napoli Federico II. Conflicts of Interest: The authors declare no conflict of interest. List of Symbols Roman Greek letters c chord , flow angles from tangential direction C skin friction coecient , gauging angles from tangential direction c specific heat at constant pressure  trailing edge wedge angle c specific heat at constant volume D di erence operator, grid spacing cp base pressure coecient  rear suction side turning angle d distance from the wall " turbulent dissipation d , D trailing edge thickness  momentum thickness te f frequency  kinematic viscosity g pitch  turbulent viscosity von Kàrmàn constant ! vorticity, specific dissipation k thermal conductivity, c /c  density p v M Mach number  stress tensor ij o throat, gauging angle  characteristic time p pressure  wall shear stress PR pressure ratio  loss coecient Q dynamic pressure R perfect gas constant Subscripts Re Reynolds number 0 stagnation value r recovery coecient 1 at blade inlet s specific entropy 2 at blade outlet S rate of strain tensor ax in the axial direction ij St, S Strouhal number bl boundary layer t time b at the base of the profile T temperature 1 at infinity u, v velocity components is isentropic value u friction velocity n, t in the normal, tangential direction V velocity magnitude References 1. 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Wang, S.; Wen, F.; Zhang, S.; Zhang, S.; Zhou, X. Influence of Trailing Boundary Layer Velocity Profiles on Wake Vortex Formation in a High Subsonic Turbine Cascade. Proc. IMechE Part A J. Power Energy 2018, 233, 186–198. [CrossRef] 124. Choi, H.; Moin, P. Grid-Point Requirements for Large Eddy Simulation: Chapman’s Estimates Revisited. Phys. Fluids 2012, 24. [CrossRef] © 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY-NC-ND) license (http://creativecommons.org/licenses/by-nc-nd/4.0/). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png International Journal of Turbomachinery, Propulsion and Power Multidisciplinary Digital Publishing Institute

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International Journal of Turbomachinery Propulsion and Power Review 1 2 , Claus Sieverding and Marcello Manna * Turbomachinery and Propulsion Department, von Kàrmàn Institute for Fluid Dynamics, Chaussée de Waterloo 72, 1640 Rhode-St-Genèse, Belgium; sieverding@vki.ac.be Dipartimento di Ingegneria Industriale, Università degli Studi di Napoli Federico II, via Claudio 21, 80125 Napoli, Italy * Correspondence: marcello.manna@unina.it; Tel.: +39-081-768-3287 Received: 15 February 2020; Accepted: 11 May 2020; Published: 20 May 2020 Abstract: The paper presents a state-of-the-art review of turbine trailing edge flows, both from an experimental and numerical point of view. With the help of old and recent high-resolution time resolved data, the main advances in the understanding of the essential features of the unsteady wake flow are collected and homogenized. Attention is paid to the energy separation phenomenon occurring in turbine wakes, as well as to the e ects of the aerodynamic parameters chiefly influencing the features of the vortex shedding. Achievements in terms of unsteady numerical simulations of turbine wake flow characterized by vigorous vortex shedding are also reviewed. Whenever possible the outcome of a detailed code-to-code and code-to-experiments validation process is presented and discussed, on account of the adopted numerical method and turbulence closure. Keywords: turbine wake flow; vortex shedding; base pressure correlation; energy separation; numerical simulation 1. Introduction The first time the lead author came in touch with the problematic of turbine trailing edge flows was in 1965 when, as part of his diploma thesis, which consisted mainly in the measurement of the boundary layer development around a very large scale HP steam turbine nozzle blade, he measured with a very thin pitot probe a static pressure at the trailing edge significantly below the downstream static pressure. This negative pressure di erence explained the discrepancy between the losses obtained from downstream wake traverses and the sum of the losses based on the momentum thickness of the blade boundary layers and the losses induced by the sudden expansion at the trailing edge. Pursuing his curriculum at the von Kármán Institute the author was soon in charge of building a small transonic turbine cascade tunnel with a test section of 150 50 mm, the C2 facility, which was intensively used for cascade testing for industry and in-house designed transonic bladings for gas and steam turbine application. These tests allowed systematic measurements of the base pressure as part of the blade pressure distribution for a large number of cascades which were first presented at the occasion of a Lecture Series held at the von Kàrmàn Institute (VKI) in 1976 and led to the publication of the well-known VKI base pressure correlation published in 1980. This correlation has served ever since for comparison with new base pressure data obtained in other research labs. Among these let us already mention in particular the investigations carried out on several turbine blades at the University of Cambridge, published in 1988, at the University of Carlton, published between 2001 and 2004, and at the Moscow Power Institute, published between 2014 and 2018. In parallel to these steady state measurements, the arrival of short duration flow visualizations and the development of fast measurement techniques in the 1970’s allowed to put into evidence the existence of the von Kármán vortex streets in the wakes of turbine blades. Pioneering work was performed at the DLR Göttingen in the mid-1970’s, with systematic flow visualizations revealing the Int. J. Turbomach. Propuls. Power 2020, 5, 10; doi:10.3390/ijtpp5020010 www.mdpi.com/journal/ijtpp Int. J. Turbomach. Propuls. Power 2020, 5, 10 2 of 55 existence of von Kármán vortices on a large number of turbine cascades in the mid-seventies. This was the beginning of an intense research on the e ect of vortex shedding on the trailing edge base pressure. A major breakthrough was achieved in the frame of two European research projects. The first one, initiated in 1992, Experimental and Numerical Investigation of Time Varying Wakes Behind Turbine Blades (BRITE/EURAM CT92-0048, 1992–1996) included very large-scale cascade tests in a new VKI cascade facility with a much larger test section allowing the testing of a 280 mm chord blade in a three bladed cascade at a moderate subsonic Mach number, M = 0.4, with emphasis on flow visualizations and 2,is detailed unsteady trailing edge pressure measurements. The VKI tests were completed by low speed tests at the University of Genoa on the same large-scale profile for unsteady wake measurements using LDV. In the follow-up project Turbulence Modelling of Unsteady Flows on Flat Plate and Turbine Cascades in 1996 (BRITE/EURAM CT96-0143, 1996-1999) VKI extended the blade pressure measurements on a 50% reduced four bladed cascade model to a high subsonic Mach number, M = 0.79. Both programs not 2,is only contributed to an improved understanding of unsteady trailing edge wake flow characteristics, of their e ect on the rear blade surface and on the trailing edge pressure distribution, but also o ered unique test cases for the validation of unsteady Navier-Stokes flow solvers. A special and unexpected result of the research on unsteady turbine blade wakes was the discovery of energy separation in the wake leading to non-negligible total temperature variations within the wake. This e ect was known from steady state tests on cylindrical bodies since the early 1940’s, but its first discovery in a turbine cascade was made at the NRAC, National Research Aeronautical Laboratory of Canada, in the mid-1990s within the framework of tests on the performance of a nozzle vane cascade at transonic outlet Mach numbers. The experimental results of the total temperature distribution in the wake of cascade at supersonic outlet Mach number served many researchers, in particular from the University of Leicester, for elaborating on the e ect of energy separation. The paper starts with the evaluation of the VKI base pressure correlation (Section 2) in view of new experiments. This is followed with a review of the advances in the understanding of unsteady trailing edge wake flows (Section 3), the observation and explanation of energy separation in turbine blade wakes (Section 4), the e ect of vortex shedding on the blade pressure distribution (Section 5) and the e ect of Mach number and boundary layer state on the vortex shedding frequency (Section 6). This experimental part is complemented with a review of the numerical methods and modelling concepts as applied to the simulation of unsteady turbine wake characteristics using advanced Navier-Stokes solvers. Available numerical data documenting significant vortex shedding a ecting the turbine performance even in a time averaged sense, are collected and compared on a code-to-code and code-to-experiments basis in Section 7. 2. Turbine Trailing Edge Base Pressure Traupel [1], was probably the first to present in his book Thermische Turbomaschinen, a detailed analysis of the profile loss mechanism for turbine blades at subsonic flows conditions. The total losses comprised three terms: the boundary losses including the downstream mixing losses for infinitely thin trailing edges, the loss due to the sudden expansion at the trailing edge (Carnot shock) for a blade with finite trailing edge thickness d taking into account the trailing edge blockage e ect and a third term te which did take into account that the static pressure at the trailing edge di ered from the average static pressure between the pressure side (PS) and the suction side (SS) trailing edges across one pitch. Thus, the profile loss coecient  reads: 0 1 B C B te C B C = 2 Q + B C sin ( ) + k d (1) p 2 te @ A 1 d te where: Q + Q ss ps Q = ( ) g sin 2 Int. J. Turbomach. Propuls. Power 2020, 5, 10 3 of 55 is the dimensionless average momentum thickness, and: te d = te g sin( ) the dimensionless thickness of the trailing edge. The constant k appearing at the right-hand-side of Equation (1) depends on the ratio: te d = te Q + Q ss ps that is, k = 0.1 for d = 2.5 and k = 0.2 for d = 7, while a linear variation of k is used for 2.5 < d < 7. te te te Terms containing squares and products of Q + Q /d were considered to be negligible. ss ps te Most researchers are, however, more familiar with a similar analysis of the loss mechanism by Denton [2], who introduced in the loss coecient expression  , the term cp d quantifying the trailing p b te edge base pressure contribution, with: p p 2 b cp = (2) 1/2V re f For commodity may be taken as the isentropic downstream velocity . However, there For commodity V may be taken as the isentropic downstream velocity V , . However, there re f 2,is was a big uncertainty as regards the magnitude of this term, although it appeared that it could was a big uncertainty as regards the magnitude of this term, although it appeared that it could become become very important in the transonic range and explain the presence of a strong local loss very important in the transonic range and explain the presence of a strong local loss maximum as maximum as demonstrated in Figure 1, which presents a few examples of early transonic cascades demonstrated in Figure 1, which presents a few examples of early transonic cascades measurements measurements performed at VKI and the DLR. (performance of VKI blades B and C are unpublished). performed at VKI and the DLR. Pioneering experimental research concerning the evolution of the turbine trailing edge base Pioneering experimental research concerning the evolution of the turbine trailing edge base pressure from subsonic to supersonic outlet flow conditions was carried out at the von Kármán pressure from subsonic to supersonic outlet flow conditions was carried out at the von Kármán Institute. Institute. In 1976, at the occasion of the VKI Lecture Series Transonic Flows in Axial Turbines, In 1976, at the occasion of the VKI Lecture Series Transonic Flows in Axial Turbines, Sieverding presented Sieverding presented base pressure data for eight different cascades for gas and steam turbine blade base pressure data for eight di erent cascades for gas and steam turbine blade profiles over a wide profiles over a wide range of Mach numbers [3] and in 1980 Sieverding et al. [4] published a base range of Mach numbers [3] and in 1980 Sieverding et al. [4] published a base pressure correlation (also pressure correlation (also referred to as BPC) based on a total of 16 blade profiles. referred to as BPC) based on a total of 16 blade profiles. Blade g/c Ref A 30° 22° 0.75 [5] B 60° 25° 0.75 VKI C 66° 18° 0.70 VKI D 156° 19.5° 0.85 [6] Figure 1. Blade profile losses versus isentropic outlet Mach number for four transonic turbines. Blade A Figure 1. Blade profile losses versus isentropic outlet Mach number for four transonic turbines. Blade data from [5], blade B and C unpublished data from VKI, blade D data from [6]. A data from [5], blade B and C unpublished data from VKI, blade D data from [6]. Int. J. Turbomach. Propuls. Power 2020, 5, 10 4 of 55 All tests were performed with cascades containing typically 8 blades and care was taken to ensure in all cases, and over the whole Mach range, a good periodicity. The latter was quantified to be 3%, in the supersonic range, in terms of the maximum di erence between the pitch-wise averaged Mach number (based on 10 wall pressure tappings per pitch) of each of the three central passages and the mean value computed over the same three passages. The correlation covered blades with a wide range of cascade parameters, as outlined in Table 1: Table 1. Parameters range for Sieverding’s correlation. Parameter Symbol Value Pitch to Chord Ratio g/c 0.32–0.84 Trailing edge thickness to throat ratio d /o 0.04–0.16 te Inlet flow angle 45 –156 Outlet flow angle 18 –34 Trailing edge wedge angle  2 –16 te Rear suction side turning angle " 0 –18 Of all cascade parameters only the rear suction side turning angle " and the trailing edge wedge angle  appeared to correlate convincingly the available data, although the latter were insucient to te di erentiate their respective influence. In fact, in many blade designs both parameters are closely linked to each other and, for two thirds of all convergent blades with convex rear suction side, both " and te were of the same order of magnitude. For this reason, it was decided to use the mean value (" +  )/2 te as parameter. The relation p /p = f(p /p ), is graphically presented in Figure 2. The curves cover b 01 s2 01 a range from M  0.6 to M  1.5, but flow conditions characterized by a suction side shock 2,is 2,is interference with the trailing edge wake region are not considered. Comparing the experiments with the correlation (results not shown herein), it turned out that 80% of all data fall within a bandwidth Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 5 of 65 5% and 96% within10%. Figure 2. Sieverding’s base pressure correlation; solid lines (resp. dashed lines) denote convergent 129 Figure 2. Sieverding’s base pressure correlation; solid lines (resp. dashed lines) denote convergent blades (resp. convergent-divergent blades) [4]. 130 blades (resp. convergent-divergent blades) [4]. An explanation for the significance of " for the trailing edge base pressure is seen in Figure 3, 131 An explanation for the significance of for the trailing edge base pressure is seen in Figure 3, presenting the blade velocity distribution for two convergent blades with di erent rear suction side 132 presenting the blade velocity distribution for two convergent blades with different rear suction side turning angles of " = 20 and 4.5 , blade A and B, together with a convergent/divergent blade with an 133 turning angles of = 20° and 4.5°, blade A and B, together with a convergent/divergent blade with internal passage area increase of A/A = 1.05, blade C. The curves end at x/c = 0.95 because beyond, 134 an internal passage area increase of / =1.05, blade C. The curves end at / = 0.95 because the pressure distribution is influenced by the acceleration around the trailing edge. 135 beyond, the pressure distribution is influenced by the acceleration around the trailing edge. 136 The rear suction side turning angle ε has a remarkable effect on the pressure difference across 137 the blade near the trailing edge. For blade A one observes a strong difference between the SS and PS 138 isentropic Mach numbers, respectively pressures, while the difference is very small for blade B. On 139 the contrary, for blade C the pressure side curve crosses the SS curve well ahead of the trailing edge 140 and the PS isentropic Mach number near the trailing edge exceeds considerably that of the SS. The 141 base pressure is function of the blade pressure difference upstream of the trailing edge. Blade A Convergent blade ε = 20° Blade B Convergent blade ε = 4.5° Blade C Convergent divergent blade (A/A*) = 1.05 142 Figure 3. Surface isentropic Mach number distribution for two convergent and one 143 convergent/divergent blades at =0.9, based on data from [3]. 144 It is also worthwhile mentioning that also plays an important role for the optimum blade 145 design in function of the outlet Mach number. Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 5 of 65 129 Figure 2. Sieverding’s base pressure correlation; solid lines (resp. dashed lines) denote convergent 130 blades (resp. convergent-divergent blades) [4]. 131 An explanation for the significance of for the trailing edge base pressure is seen in Figure 3, 132 presenting the blade velocity distribution for two convergent blades with different rear suction side 133 turning angles of = 20° and 4.5°, blade A and B, together with a convergent/divergent blade with Int. J. Turbomach. Propuls. Power 2020, 5, 10 5 of 55 134 an internal passage area increase of / =1.05, blade C. The curves end at / = 0.95 because 135 beyond, the pressure distribution is influenced by the acceleration around the trailing edge. 136 The rear suction side turning angle ε has a remarkable effect on the pressure difference across The rear suction side turning angle " has a remarkable e ect on the pressure di erence across 137 the blade near the trailing edge. For blade A one observes a strong difference between the SS and PS the blade near the trailing edge. For blade A one observes a strong di erence between the SS and PS 138 isentropic Mach numbers, respectively pressures, while the difference is very small for blade B. On isentropic Mach numbers, respectively pressures, while the di erence is very small for blade B. On the 139 the contrary, for blade C the pressure side curve crosses the SS curve well ahead of the trailing edge contrary, for blade C the pressure side curve crosses the SS curve well ahead of the trailing edge and 140 and the PS isentropic Mach number near the trailing edge exceeds considerably that of the SS. The the PS isentropic Mach number near the trailing edge exceeds considerably that of the SS. The base 141 base pressure is function of the blade pressure difference upstream of the trailing edge. pressure is function of the blade pressure di erence upstream of the trailing edge. Blade A Convergent blade ε = 20° Blade B Convergent blade ε = 4.5° Blade C Convergent divergent blade (A/A*) = 1.05 Figure 3. Surface isentropic Mach number distribution for two convergent and one convergent/divergent 142 Figure 3. Surface isentropic Mach number distribution for two convergent and one blades at M = 0.9, based on data from [3]. 143 convergent/divergent blades at =0.9, based on data from [3]. 2,is It is also worthwhile mentioning that " also plays an important role for the optimum blade design 144 It is also worthwhile mentioning that also plays an important role for the optimum blade in function of the outlet Mach number. Figure 4 presents design recommendations for the rear suction 145 design in function of the outlet Mach number. side curvature with increasing Mach number from subsonic to low supersonic Mach numbers as successfully used at VKI. (a) (b) (c) (d) Figure 4. Recommended values of " (a) and l/L (b) for the design of the blade rear suction side for increasing outlet Mach numbers. Full (c) and close-up (d) view of the parameters characterizing the turbine geometry. Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 6 of 58 (a) (b) Int. J. Turbomach. Propuls. Power 2020, 5, 10 6 of 55 The rear suction turning angle " for convergent blades should decrease with increasing Mach number reaching a minimum of 4 at M  1.3 (maximum Mach number for convergent blades). 2,is Note that similar trends can be derived from the loss correlation by Craig and Cox [7]. They showed that in order to minimize the blade profile losses the rear suction side curvature, expressed by the ratio g/e, where g represents the pitch(c and) (d) e the radius of a circular arc appr oximating the rear suction side curvature, should decrease with increasing Mach number. 154 Figure 4. Recommended values of (a) and / (b) for the design of the blade rear suction side for Commented [M17]: Please add explanation for subgraph and 155 increasing outlet Mach numbers. Full (c) and close-up (d) view of the parameters characterizing the For a given rear suction side angle " the designer is free as regards the evolution of the surface add “a,b,c,d” in the figure. Please unify the all figures which 156 turbine geometry. angle from the throat to the trailing edge. It appears to be a good design practice to subdivide the rear have subgraph like figure 51. 157 For a given rear suction side angle ε the designer is free as regards the evolution of the surface suction side length L into two parts, a first part along which the blade angle asymptotically decreases to Commented [MM18R17]: Done. 158 angle from the throat to the trailing edge. It appears to be a good design practice to subdivide the the value of the trailing edge angle, followed by a second entirely straight part of length l, see Figure 4. 159 rear suction side length into two parts, a first part along which the blade angle asymptotically With increasing outlet Mach number, the length of the straight part, that is the ratio l/L increases, but it 160 decreases to the value of the trailing edge angle, followed by a second entirely straight part of length 161 , see Figure 4. With increasing outlet Mach number, the length of the straight part, that is the ratio does never extend up to the throat. 162 / increases, but it does never extend up to the throat. For calculating the trailing edge losses induced by the di erence between the base pressure and 163 For calculating the trailing edge losses induced by the difference between the base pressure and the downstream pressure, Fabry & Sieverding [8], presented the data for the convergent blades in 164 the downstream pressure, Fabry & Sieverding [8], presented the data for the convergent blades in 165 Figure 2 in terms of the base pressure coefficient , defined by Equation (2), see Figure 5. Figure 2 in terms of the base pressure coecient cp , defined by Equation (2), see Figure 5. 167 Figure 5. Base Figupr re 5. essur Base e prcoe essu re c cients oefficiecorr nts coesponding rresponding toto the the base base pressu pr re c essur urves of Fi e curves gure 2 of [8]. Figure 2 [8]. Since the base pressure losses are proportional to the base pressure coecient cp , the curves give immediately an idea of the strong variation of the profile losses in the transonic range. As regards the low Mach number range, the contribution of the base pressure loss is implicitly taken into account by all loss correlations. Therefore the base pressure loss is not to be added straight away to the profile losses as predicted for example with the methods by Traupel [1] and Craig and Cox [7] but rather as a di erence with respect to the profile losses at M = 0.7: 2,is te = cp cp bp b b,M =0.7 2,is g sin( ) Martelli and Boretti [9], used the VKI base pressure correlation for verifying a simple procedure to compute losses in transonic turbine cascades. The surface static pressure distribution for a given downstream Mach number is obtained from an inviscid time marching flow calculation. An integral boundary layer calculation is used to calculate the momentum thickness at the trailing edge before separation. The trailing edge shocks are calculated using the base pressure correlation. Two examples are shown in Figure 6. Calculation of eight blades showed that 80% of the predicted losses were within the range of the experimental uncertainty. Besides the data reported by Sieverding et al. in [4,6], the only authors who published recently a systematic investigation of the e ect of the rear suction side curvature on the base pressure were Granovskij et al. Of the Moscow Power Institute [10]. The authors investigated 4 moderately loaded rotor blades (g/c = 0.73, d /o = 0.12,  85,  22) with di erent unguided turning angles (" = 2 te 1 2 to 16 ) in the frame of the optimization of cooled gas turbine blades. A direct comparison with the VKI base pressure correlation is dicult because the authors omitted to indicate the trailing edge wedge Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 7 of 58 168 Since the base pressure losses are proportional to the base pressure coefficient , the curves Int. J. Turbomach. Propuls. Power 2020, 5, 10 7 of 55 169 give immediately an idea of the strong variation of the profile losses in the transonic range. As regards 170 the low Mach number range, the contribution of the base pressure loss is implicitly taken into account 171 by all loss correlations. Therefore the base pressure loss is not to be added straight away to the profile 172 losses as predicted for example with the methods by Traupel [1] and Craig and Cox [7] but rather as angle  . Nevertheless, a comparison appeared to be useful. Figure 7 presents the comparison, after te 173 a difference with respect to the profile losses at =0.7: conversion, of the base pressure coecient: Commented [M19]: Please confirm and relayout the = − , . ( ) sin p p b 2 number of equation in order.. 174 Martelli and Boretti [9], used the VKcp I base pres = sure correlation for verifying a simple procedure p p 02 2 175 to compute losses in transonic turbine cascades. The surface static pressure distribution for a given Commented [MM20R19]: No need to add an eq. 176 downstream Mach number is obtained from an inviscid time marching flow calculation. number here, and wherever it was not inserted in the used by Granovskij et al. [10], to the base pressure coecient (2) based on V , used by Fabry and 177 An integral boundary layer calculation is used to calculate the momentum thi2, cknes is s at the original manuscript Sieverding at VKI [8]. The data of Granovskij et al. [10] (dashed lines) confirm globally the overall 178 trailing edge before separation. The trailing edge shocks are calculated using the base pressure 179 correlation. Two examples are shown in Figure 6. Calculation of eight blades showed that 80% of the trends of the VKI base pressure correlation (solid lines). However, the peaks in the transonic range are 180 predicted losses were within the range of the experimental uncertainty. much more pronounced. (a) (b) 182 Figure 6. Example of profile loss prediction for transonic turbine cascade, adapted from [9]; (a) low Commented [M21]: Please add explanation for subgraph Figure 6. Example of profile loss prediction for transonic turbine cascade, adapted from [9]; (a) low 183 pressure steam turbine tip section, (b) high pressure gas turbine guide vane. Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 8 of 58 pressure steam turbine tip section, (b) high pressure gas turbine guide vane. Commented [MM22R21]: Done 184 Besides the data reported by Sieverding et al. in [6] and [4], the only authors who published 185 recently a systematic investigation of the effect of the rear suction side curvature on the base pressure 186 were Granovskij et al. of the Moscow Power Institute [10]. The authors investigated 4 moderately 187 loaded rotor blades ( =0.73, / = 0.12 , ≈85°, ≈22°) with different unguided turning 188 angles (= 2° to 16°) in the frame of the optimization of cooled gas turbine blades. A direct 189 comparison with the VKI base pressure correlation is difficult because the authors omitted to indicate 190 the trailing edge wedge angle . Nevertheless, a comparison appeared to be useful. Figure 7 191 presents the comparison, after conversion, of the base pressure coefficient: 192 used by Granovskij et al. [10], to the base pressure coefficient (2) based on , used by Fabry and 193 Sieverding at VKI [8]. The data of Granovskij et al. [10] (dashed lines) confirm globally the overall 194 trends of the VKI base pressure correlation (solid lines). However, the peaks in the transonic range 195 are much more pronounced. 196 Figure 7. Comparison of Granovskij’s base pressure data (dashed lines) with the VKI base pressure Figure 7. Comparison of Granovskij’s base pressure data (dashed lines) with the VKI base pressure 197 correlation (solid lines). correlation (solid lines). 198 Also, cascade data reported by Dvorak et al. in 1978 [11] on a low pressure steam turbine rotor Also, cascade data reported by Dvorak et al. In 1978 [11] on a low pressure steam turbine rotor 199 tip section, and by Jouini et al. in 2001 [12] for a relatively high turning rotor blade (∆ = 110° , and a 200 smaller pitch to chord ratio / = 0.73 ), are in fair agreement with the VKI base pressure correlation, tip section, and by Jouini et al. In 2001 [12] for a relatively high turning rotor blade (D = 110, and a 201 although the latter authors state that below / =0.45, their data drop below those of the BPC. smaller pitch to chord ratio g/c = 0.73), are in fair agreement with the VKI base pressure correlation, 202 However, some other cascade measurements deviate very significantly from the VKI curves. although the latter authors state that below p /p = 0.45, their data drop below those of the BPC. 203 Deckers and Denton [13], for a low turning blade model and Gostelow et al. [14] for a high turning 2 01 204 nozzle guide vane, report base pressure data far below those of Sieverding’s BPC, while Xu and However, some other cascade measurements deviate very significantly from the VKI curves. 205 Denton [15], for a very highly loaded HP gas turbine rotor blade (∆ = 124° and / = 0.84 ) report Deckers and Denton [13], for a low turning blade model and Gostelow et al. [14] for a high turning 206 base pressure data far above those of the BPC. nozzle guide vane, report base pressure data far below those of Sieverding’s BPC, while Xu and 207 The simplicity of Sieverding’s base pressure correlation was often criticized because it was felt 208 that aspects as important as the state of the boundary layer, the ratio of boundary layer momentum Denton [15], for a very highly loaded HP gas turbine rotor blade (D = 124 and g/c = 0.84) report 209 to trailing edge thickness and the trailing edge blockage effects (trailing edge thickness to throat base pressure data far above those of the BPC. 210 opening) should play an important role. The simplicity of Sieverding’s base pressure correlation was often criticized because it was felt 211 As regards the state of the boundary layer and its thickness, tests on a flat plate model at 212 moderate subsonic Mach numbers in a strongly convergent channel by Sieverding and Heinemann that aspects as important as the state of the boundary layer, the ratio of boundary layer momentum to 213 [16], showed that the difference of the base pressure for laminar and turbulent flow conditions was trailing edge thickness and the trailing edge blockage e ects (trailing edge thickness to throat opening) 214 only of the order of 1.5–2% of the dynamic head of the flow before separation from the trailing edge. should215 play an For t important he case of sup role. ersonic trailing edge flows, Carriere [17], demonstrated, that for turbulent 216 boundary layers the base pressure would increase with increasing momentum thickness. On the 217 contrary, supersonic flat plate model tests simulating the overhang section of convergent turbine 218 cascades with straight rear suction sides showed that for fully expanded flow along the suction side 219 (limit loading condition) an increase of the ratio of the boundary layer momentum to the trailing edge 220 thickness by a factor of two, obtained roughening the blade surface, did not affect the base pressure, 221 Sieverding et al. [18]. Note, that for both the smooth and rough surface the boundary layer was 222 turbulent. Similarly, roughening the blade surface in case of shock boundary layer interactions on the 223 blade suction side did not affect the base pressure as compared to the smooth blade, Sieverding and 224 Heinemann [16]. However, a comparison of the base pressure for the same Mach numbers before 225 separation at the trailing edge for a fully expanding flow and a flow with shock boundary layer 226 interaction on the suction side before the TE showed an increase of the base pressure by 10–25 % in 227 case of shock interaction before the TE. Since it was shown before that an increase of the momentum 228 thickness did not affect the base pressure, the difference may be attributed to (a) different total 229 pressures due to shock losses for the shock interference curve, (b) differences in the boundary layer 230 shape factor and (c) differences in pressure gradients in stream-wise direction in the near wake 231 region. 232 A systematic investigation of possible effects of changes in shape factor and boundary layer 233 momentum thickness on the base pressure in cascades is difficult. Hence, the investigations are Int. J. Turbomach. Propuls. Power 2020, 5, 10 8 of 55 As regards the state of the boundary layer and its thickness, tests on a flat plate model at moderate subsonic Mach numbers in a strongly convergent channel by Sieverding and Heinemann [16], showed that the di erence of the base pressure for laminar and turbulent flow conditions was only of the order of 1.5–2% of the dynamic head of the flow before separation from the trailing edge. For the case of supersonic trailing edge flows, Carriere [17], demonstrated, that for turbulent boundary layers the base pressure would increase with increasing momentum thickness. On the contrary, supersonic flat plate model tests simulating the overhang section of convergent turbine cascades with straight rear suction sides showed that for fully expanded flow along the suction side (limit loading condition) an increase of the ratio of the boundary layer momentum to the trailing edge thickness by a factor of two, obtained roughening the blade surface, did not a ect the base pressure, Sieverding et al. [18]. Note, that for both the smooth and rough surface the boundary layer was turbulent. Similarly, roughening the blade surface in case of shock boundary layer interactions on the blade suction side did not a ect the base pressure as compared to the smooth blade, Sieverding and Heinemann [16]. However, a comparison of the base pressure for the same Mach numbers before separation at the trailing edge for a fully expanding flow and a flow with shock boundary layer interaction on the suction side before the TE showed an increase of the base pressure by 10–25 % in case of shock interaction before the TE. Since it was shown before that an increase of the momentum thickness did not a ect the base pressure, the di erence may be attributed to (a) di erent total pressures due to shock losses for the shock interference curve, (b) di erences in the boundary layer shape factor and (c) di erences in pressure gradients in stream-wise direction in the near wake region. A systematic investigation of possible e ects of changes in shape factor and boundary layer momentum thickness on the base pressure in cascades is dicult. Hence, the investigations are mostly confined to variations of the incidence angle which, via a modification of the blade velocity distribution, should have an impact on both the shape factor and the boundary layer momentum thickness. Based on linear transonic cascade tests on two high turning rotor blades Jouini et al. [19] at Carlton University, (blade HS1A: g/c = 0.73, d /o = 0.082, = 39.5, = 31,  = 6, " = 11.5; blade HS1B is similar to te 1 2 te HS1A, but with less loading on the front side and = 29) concluded that discrepancies in the base region did not appear to be strongly related to changes of the inlet angle by14.5, however in broad terms the weakest base pressure drop in the transonic range were obtained for high positive incidence. Similarly, experiments at VKI on a high turning rotor blade (g/c = 0.49, d /o = 0.082, = 45, = 28, te 1 2 = 10, " = 10) did not show any e ect on the base pressure for incidence angle changes of10 [3]. te In conclusion it appears that for conventional blade designs, changes in the boundary layer thickness alone, as induced by incidence variations, do not a ect significantly the base pressure. Therefore, we need to look for possible other influence factors. Figure 3 showed that the e ect of the blade rear suction side blade turning angle " on the base pressure was in fact function of the pressure di erence across the blade near the trailing edge. Inversely, one should be able to deduct from the rear blade loading the tendency of the base pressure. The higher the blade loading at the trailing edge, the higher the base pressure. Corollary, a low or even negative blade loading near the trailing edge causes increasingly lower base pressures. This might help in explaining the large di erences with respect to the BPC as found by Xu and Denton [15] on one side and Deckers et al. [13] and Gostelow et al. [14], mentioned before, on the other side. To illustrate this, Figure 8 presents the base pressure data of Xu and Denton [15] for three of a family of four very highly loaded gas turbine rotor blades with a blade turning angle of D = 124 and a pitch-to-chord g/c = 0.84, tested with three di erent trailing edge thicknesses. The blades are referred to as blade RD, for the datum blade, and blades DN and DK for changes of 0.5 and 1.5 times the trailing edge thickness with respect to the datum case. The base pressures are overall much higher than those of the BPC which are indicated in the figure by the dashed line for a mean value of (" +  )/2 = 9. te A possible explanation for the large di erences is given by comparing the blade Mach number distribution of the datum blade with that of a VKI blade with a (" +  )/2 = 16 taken from [6], te Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 9 of 58 234 mostly confined to variations of the incidence angle which, via a modification of the blade velocity 235 distribution, should have an impact on both the shape factor and the boundary layer momentum 236 thickness. Based on linear transonic cascade tests on two high turning rotor blades Jouini et al. [19] 237 at Carlton University, (blade HS1A: / =0.73, / = 0.082, = 39.5° , = 31° , =6°, = 238 11.5°; blade HS1B is similar to HS1A, but with less loading on the front side and = 29°) concluded 239 that discrepancies in the base region did not appear to be strongly related to changes of the inlet angle 240 by 14.5° , however in broad terms the weakest base pressure drop in the transonic range were 241 obtained for high positive incidence. Similarly, experiments at VKI on a high turning rotor blade 242 (/ =0.49, / = 0.082, =45°, = 28°, = 10°, = 10°) did not show any effect on the base 243 pressure for incidence angle changes of 10° [3]. 244 In conclusion it appears that for conventional blade designs, changes in the boundary layer 245 thickness alone, as induced by incidence variations, do not affect significantly the base pressure. 246 Therefore, we need to look for possible other influence factors. 247 Figure 3 showed that the effect of the blade rear suction side blade turning angle ε on the base 248 pressure was in fact function of the pressure difference across the blade near the trailing edge. 249 Inversely, one should be able to deduct from the rear blade loading the tendency of the base pressure. 250 The higher the blade loading at the trailing edge, the higher the base pressure. Corollary, a low or 251 even negative blade loading near the trailing edge causes increasingly lower base pressures. This 252 might help in explaining the large differences with respect to the BPC as found by Xu and Denton 253 [15] on one side and Deckers et al. [13] and Gostelow et al. [14], mentioned before, on the other side. 254 To illustrate this, Figure 8 presents the base pressure data of Xu and Denton [15] for three of a 255 family of four very highly loaded gas turbine rotor blades with a blade turning angle of ∆ = 124° 256 and a pitch-to-chord / = 0.84 , tested with three different trailing edge thicknesses. The blades are 257 referred to as blade RD, for the datum blade, and blades DN and DK for changes of 0.5 and 1.5 times 258 the trailing edge thickness with respect to the datum case. Int. J. Turbomach. Propuls. Power 2020, 5, 10 9 of 55 259 The base pressures are overall much higher than those of the BPC which are indicated in the 260 figure by the dashed line for a mean value of ( + )/2 = 9°. 261 A possible explanation for the large differences is given by comparing the blade Mach number 262 distribution of the datum blade with that of a VKI blade with a ( + )/2 = 16° taken from [6], see see Figure 9. To enable the comparison, the blade Mach number distribution of Xu & Denton (solid line) 263 Figure 9. To enable the comparison, the blade Mach number distribution of Xu & Denton (solid line) presented originally in function of the axial chord x/c , had to be replotted in function of x/c. ax 264 presented originally in function of the axial chord / , had to be replotted in function of / . The The comparison is done for an isentropic outlet Mach number M = 0.8. 265 comparison is done for an isentropic outlet Mach number =0.8. , 2,is Commented [M23]: Please add explanation for subgraph Figure 267 8. Base Figurpr e 8. essur Base pre e vssu ariation re variatifor on for bl blades ades of of Xu & Xu & Den Denton; ton; blade R blade D datum RD cas datum e, blade D case, K thicblade k DK thick 268 trailing edge, blade DN thin trailing edge. Adapted from [15]. trailing edge, blade DN thin trailing edge. Adapted from [15]. Commented [MM24R23]: This is not a subgraph, and the geometries are well explained by the caption. Letter 269 Note that the geometric throat for the Xu & Denton blade is situated at / ≈ 0.34 , while for the Note that the geometric throat for the Xu & Denton blade is situated at x/c  0.34, while for the 270 VKI blade at / = 0.5 . At the trailing edge, the Mach number difference between pressure and referencing is inappropriate. VKI blade at x/c = 0.5. At the trailing edge, the Mach number di erence between pressure and suction Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 10 of 58 side for both blades are exactly the same, but contrary to the nearly constant Mach number for the VKI blade downstream of the throat, the blade of Xu and Denton is characterized by a very strong adverse 271 suction side for both blades are exactly the same, but contrary to the nearly constant Mach number 272 for the VKI blade downstream of the throat, the blade of Xu and Denton is characterized by a very pressure gradient in this region. As pointed out by the authors, this causes the suction side boundary 273 strong adverse pressure gradient in this region. As pointed out by the authors, this causes the suction layer to be either separated or close to separation up-stream of the trailing edge. Clearly, Sieverding’s 274 side boundary layer to be either separated or close to separation up-stream of the trailing edge. correlation cannot deal with blade designs characterized by very strong adverse pressure gradients on 275 Clearly, Sieverding’s correlation cannot deal with blade designs characterized by very strong adverse 276 pressure gradients on the rear suction side causing boundary layer separation before the trailing the rear suction side causing boundary layer separation before the trailing edge. 277 edge. Figure 9. Comparison of blade Mach number distribution for blade RD of Xu and Denton [15], 278 Figure 9. Comparison of blade Mach number distribution for blade RD of Xu and Denton [15], (solid 5 6 279 curve, =0.8, =8 10 ) with VKI blade (dashed curve, =0.8, =10 ). (solid curve, M = , 0.8, Re = 8 10 ) with VKI blade (dashed , curve, M = 0.8, Re = 10 ). Commented [M25]: Please add explanation for subgraph. 2,is 2 2,is 2 280 The possible effect of boundary layer separation resulting from high rear suction side diffusion Commented [MM26R25]: This is not a subgraph, and the The possible e ect of boundary layer separation resulting from high rear suction side di usion 281 resulting in high base pressures was also mentioned by Corriveau and Sjolander in 2004 [20], geometries are well explained by the caption. Letter resulting in high base pressures was also mentioned by Corriveau and Sjolander in 2004 [20], comparing 282 comparing their nominal mid-loaded rotor blade HS1A, mentioned already before, with an aft- referencing is inappropriate. 283 loaded blade HS1C with an increase of the suction side unguided turning angle from 11.5° to 14.5°. their nominal mid-loaded rotor blade HS1A, mentioned already before, with an aft- loaded blade HS1C 284 It appears that the increased turning angle could cause, in the transonic range, shock induced with an increase of the suction side unguided turning angle from 11.5 to 14.5 . It appears that the 285 boundary layer transition near the trailing edge with, as consequence, a sharp increase of the base increased 286 turning pressure, i.e angle . a sudde could n cause, drop in tin he b the ase p transonic ressure coeff range, icient as seen shock in induced Figure 10. N boundary ote that the layer transition 287 reported in the figure has been converted to − of the original data. near the trailing edge with, as consequence, a sharp increase of the base pressure, i.e., a sudden drop in 288 As regards the base pressure data by Deckers and Denton [13] for a low turning blade model the base pressure coecient as seen in Figure 10. Note that the cp reported in the figure has been 289 and Gostelow et al. [14] for a high turning nozzle guide vane, who report base pressure data far below converted 290 to those o cp of f Si the everdi original ng’s BPC data. , their blade pressure distribution resembles that of the convergent/ 291 divergent blade C in Figure 3 with a negative blade loading near the trailing edge which would As regards the base pressure data by Deckers and Denton [13] for a low turning blade model and 292 explain the very low base pressures. In addition, the blade of Deckers and Denton has a blunt trailing Gostelow et al. [14] for a high turning nozzle guide vane, who report base pressure data far below 293 edge, and there is experimental evidence that, compared to a circular trailing edge, the base pressure those of 294 Siever for ding’s blades with BPC, blunt their trablade iling edpr ge mi essur ght be co e distribution nsiderably lo rwer. esembles Sieverding an that of d Hei thenem conver ann [16] gent /divergent 295 report for flat plate tests at moderate subsonic Mach numbers a drop of the base pressure coefficient blade C in Figure 3 with a negative blade loading near the trailing edge which would explain the very 296 by 11% for a plate with squared trailing edge compared to that with a circular trailing edge. low base pressures. In addition, the blade of Deckers and Denton has a blunt trailing edge, and there is This image cannot currently be displayed. Commented [M27]: Please add explanation for subgraph. Commented [MM28R27]: This is not a subgraph, and the 297 Figure 10. Base pressure coefficient for mid-loaded (solid line) and aft-loaded (dashed line) rotor geometries are well explained by the caption. Letter 298 blade. Symbols: HS1A geometry, HS1C geometry. Adapted from [20]. referencing is inappropriate Int. J. Turbo Int. J. Turbo m m ach. ach. Propuls. Power Propuls. Power 2018 2018 , , 33 , x FOR PE , x FOR PE ER RE ER RE VI VI EE W W 10 of 10 of 58 58 271 271 suction suction side side for both blade for both blade ss are ex are ex actly actly t t hh e same, e same, bb uu t t contrary to th contrary to th e nearly constant Mach e nearly constant Mach number number 272 for the VKI blade downstream of the throat, the blade of Xu and Denton is characterized by a very 272 for the VKI blade downstream of the throat, the blade of Xu and Denton is characterized by a very 273 273 strong adver strong adver ss e pressure gr e pressure gr adient in this adient in this region. region. As As poi poi nn ted out by the a ted out by the a u u thors, thi thors, thi ss causes the sucti causes the sucti oo n n 274 side boundary layer to be either separated or close to separation up-stream of the trailing edge. 274 side boundary layer to be either separated or close to separation up-stream of the trailing edge. 275 275 C C lear lear ly, ly, Si Si ev ev ee rr ding’ ding’ ss cor cor rr el el at at ion cann ion cann ot ot dea dea l wit l wit hh b b la la d d e des e des igns ch igns ch ar ar act act ee ri ri zed b zed b yy v v ee ry st ry st rong rong adv adv ee rs rs e e 276 pressure gradients on the rear suction side causing boundary layer separation before the trailing 276 pressure gradients on the rear suction side causing boundary layer separation before the trailing 277 277 edge. edge. 278 278 Figure 9. Figure 9. Comparison of bla Comparison of bla dd e Mach number distri e Mach number distri bution f bution f oo r blade RD of r blade RD of Xu and Denton Xu and Denton [ [ 15], ( 15], ( ss olid olid 279 279 cu cu rve, rve, =0 =0.8.8, , =8 =8 1 100) with V ) with V K K I bla I bla dd e e (dash (dash ee d curve d curve , , =0 =0.8.8, , =1 =100). ). , , , , 280 280 The possib The possib le le effect o effect o f boun f boun dary dary la la yer yer se se paration paration res res uu lting lting from h from h igh re igh re ar s ar s uu ctio ctio n sid n sid ee d d iffus iffus ion ion 281 281 resul resultitn ing i g inn hi high b gh baase pressures was se pressures was aalso ment lso mentioioned by Corri ned by Corrivveeaauu aannd d Sjola Sjolannder i der inn 2004 2004 [20] [20], , 282 282 com compparing aring t thheir nom eir nomin inaal m l mid- id-lo loaded aded rot rotoor b r blad lade e HS HS1A, m 1A, meent ntio ioned a ned alre lready ady b beefore, fore, wit withh an an aft aft- - 283 283 load load ed ed bl bl ade ade HS HS 1C w 1C w itih th a a nn inc inc rr ea ea se se of of t h th e s e s uu ct ct ion ion si si de de ung ung uu ide ide dd t u tu rning rning angl angl ee fr fr om om 11 11 .5 .5 °° t t oo 1 1 44 .5 .5 °° . . 284 284 ItIt ap apppeear ars t s thhatat t thhe e incre increaased t sed tuurning rning angl angle co e coul uldd cause, in cause, in the transon the transonic ic range, range, shock shock induced induced 285 285 bounda bounda ry ry la la yer tra yer tra nn si si titon ion nea nea rr the tra the tra iliilng edge wi ing edge wi th th , , as conse as conse qq ue ue nce, nce, a sh a sh arp arp increase o increase o f the base f the base 286 286 pressu pressu re, re, i.e i.e . .a s a s uu dden dden dro dro pp in the b in the b aa se se pressu pressu re coe re coe ffic ffic ient as s ient as s ee ee nn in in Fig Fig uu re re 10. 10. Note that t Note that t hh e e 287 287 reported in reported in the figur the figur ee h h aa s b s b een converte een converte d to d to − − of of t t hh e o e o rr igin igin aa l d l d aa ta ta . . 288 288 As As reg reg aa rds th rds th e base pr e base pr essure d essure d aa ta by ta by De De ckers an ckers an d De d De nton [13] nton [13] for for a a low low turnin turnin g blade g blade mod mod ee l l 289 289 and G and G oo st st ee low low et et al al . [ . [ 11 4] 4] for for a hi a hi gh t gh t u u rn rn in in g noz g noz zz le le guide va guide va ne, who report ba ne, who report ba se pressure da se pressure da ta ta far below far below 290 290 those of those of Siev Sieverding’s BPC erding’s BPC, their blad , their bladee pressure pressure ddistribution r istribution reesembles th sembles that of the conv at of the convergent/ ergent/ 291 divergent blade C in Figure 3 with a negative blade loading near the trailing edge which would 291 divergent blade C in Figure 3 with a negative blade loading near the trailing edge which would 292 292 explain the ve explain the ve ry lo ry lo w base p w base p rr essures. essures. In In addi addi ti ti on, the b on, the b la la d d e of e of Deckers Deckers aa nn d Denton has d Denton has aa blunt blunt tra tra ili ili ng ng 293 edge, and there is experimental evidence that, compared to a circular trailing edge, the base pressure 293 edge, and there is experimental evidence that, compared to a circular trailing edge, the base pressure Int. J. Turbomach. Propuls. Power 2020, 5, 10 10 of 55 294 294 fo fo r b r b lades with bl lades with blunt tra unt trailiilng edge m ing edge m ight be co ight be co ns ns ide ide rr ably ably lower. lower. Siever Siever ding ding a a nn d d He He inem inem aa nn nn [1 [1 6] 6] 295 report for flat plate tests at moderate subsonic Mach numbers a drop of the base pressure coefficient 295 report for flat plate tests at moderate subsonic Mach numbers a drop of the base pressure coefficient experimental evidence that, compared to a circular trailing edge, the base pressure for blades with 296 296 bb yy 1 1 11 % % for for a a pp lat lat e w e w itih th s s qq uar uar ee d t d t rr ai ai ling ling edge edge com com pp ar ar ed t ed t oo t t hh at at wit wit hh a c a c irc irc ul ul ar t ar t rr ai ai ling ling edg edg ee . . blunt trailing edge might be considerably lower. Sieverding and Heinemann [16] report for flat plate tests at moderate subsonic Mach numbers a drop of the base pressure coecient by 11% for a plate with squared trailing edge compared to that with a circular trailing edge. Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 13 of 65 298 Figure 10. Base pressure coefficient for mid-loaded (solid line) and aft-loaded (dashed line) rotor 297 297 Figure 10. Figure 10. Ba Ba se se pressu pressu re coe re coe fficient f fficient f oo r m r m id-loade id-loade dd ( ( ss oli oli dd line) an line) an d aft d aft -loaded -loaded (da (da ss hed line) rot hed line) rot oo r r Figure 10. Base pressure coecient for mid-loaded (solid line) and aft-loaded (dashed line) rotor blade. 299 blade. Symbols: HS1A geometry, HS1C geometry. Adapted from [20]. 298 298 bl bl ad ad e. Symbol e. Symbol Symbols: s:s: HS1A geom HS1A HS1A geom geometry etry, etry, , HS1C HS1C ge HS1C ge geometry ometry. Adapt ometry. Adapt . Adapted ee d from [20] d from [20] from [20]. . . 300 It is important to remember that the measurement of the base pressure carried out with a single It is important to remember that the measurement of the base pressure carried out with a single 301 pressure tapping in the trailing edge base region implies the assumption of an isobaric trailing edge 302 pressure distribution. However, in 2003 Sieverding et al. [21] demonstrated that at high subsonic pressure tapping in the trailing edge base region implies the assumption of an isobaric trailing edge 303 Mach numbers the pressure distribution could be highly non-uniform with a marked pressure pressure distribution. However, in 2003 Sieverding et al. [21] demonstrated that at high subsonic Mach 304 minimum at the center of the trailing edge base, as will be shown later in Section 5. Under these numbers the pressure distribution could be highly non-uniform with a marked pressure minimum at 305 conditions it is likely that the base pressure measured with a single pressure hole does not reflect the 306 true mean pressure. In addition, the measured pressure would depend on the ratio of the pressure the center of the trailing edge base, as will be shown later in Section 5. Under these conditions it is likely 307 hole to trailing edge diameter / , which is typically in the range / = 0.15 − 0.50 . This fact was that the base pressure measured with a single pressure hole does not reflect the true mean pressure. 308 also recognized by Jouini et al. [12], who mentioned the difficulties for obtaining representative In addition, the measured pressure would depend on the ratio of the pressure hole to trailing edge 309 trailing edge base pressures measurements: “It should also be noted that at high Mach numbers the base 310 pressure varies considerably with location on the trailing edge and the single tap gives a somewhat limited diameter d/D, which is typically in the range d/D = 0.15–0.50. This fact was also recognized by Jouini 311 picture of the base pressure behavior”. It is probably correct to say that differences between experimental et al. [12], who mentioned the diculties for obtaining representative trailing edge base pressures 312 base pressure data and the base pressure correlation may at least partially be attributed to the use of measurements: “It should also be noted that at high Mach numbers the base pressure varies considerably with 313 different pressure hole to trailing edge diameters / by the various researchers. 314 Finally, it is important to mention that the trailing edge pressure is sensitive to the trailing edge location on the trailing edge and the single tap gives a somewhat limited picture of the base pressure behavior”. 315 shape as demonstrated by El Gendi et al. [22] who showed with the help of high fidelity simulation It is probably correct to say that di erences between experimental base pressure data and the base 316 that the base pressure for blades with elliptic trailing edges was higher than for blades with circular pressure correlation may at least partially be attributed to the use of di erent pressure hole to trailing 317 trailing edges. Melzer and Pullan [23] proved experimentally that designing blades with elliptical 318 trailing edges improved the blade performance. The reason is that an elliptic trailing edge reduces edge diameters d/D by the various researchers. 319 not only the wake width but causes also an increase of the base pressure compared to that of blades Finally, it is important to mention that the trailing edge pressure is sensitive to the trailing edge 320 with a circular trailing edge. This suggests that inaccuracies in the machining of blades with thin shape as demonstrated by El Gendi et al. [22] who showed with the help of high fidelity simulation 321 trailing edges could easily lead to deviations from the designed circular trailing edge shape and thus 322 contribute to the differences in the base pressure. that the base pressure for blades with elliptic trailing edges was higher than for blades with circular trailing edges. Melzer and Pullan [23] proved experimentally that designing blades with elliptical 323 3. Unsteady Trailing Edge Wake Flow trailing edges improved the blade performance. The reason is that an elliptic trailing edge reduces 324 The mixing process of the wake behind turbine blades has been viewed for a long time as a not only the wake width but causes also an increase of the base pressure compared to that of blades 325 steady state process although it was well known that the separation of the boundary layers at the 326 trailing edge is a highly unsteady phenomenon which leads to the formation of large coherent with a circular trailing edge. This suggests that inaccuracies in the machining of blades with thin 327 structures, known as the von Kármán vortex street. The unsteady character of turbine blade wakes is trailing edges could easily lead to deviations from the designed circular trailing edge shape and thus 328 best illustrated by flow visualizations. contribute to the di erences in the base pressure. 329 Lawaczeck and Heinemann [24], and Heinemann and Bütefisch [25], were probably the first to 330 perform some systematic schlieren visualizations on transonic flat plate and cascades with different 331 trailing edge thicknesses using a flash light of 20 nano-seconds only, and deriving from the photos 3. Unsteady Trailing Edge Wake Flow 332 the vortex shedding frequencies and Strouhal numbers. The schlieren picture in Figure 11 shows 333 impressively that the shedding of each vortex from the trailing edge generates a pressure wave which The mixing process of the wake behind turbine blades has been viewed for a long time as a steady 334 travels upstream. state process although it was well known that the separation of the boundary layers at the trailing edge is a highly unsteady phenomenon which leads to the formation of large coherent structures, known as the von Kármán vortex street. The unsteady character of turbine blade wakes is best illustrated by flow visualizations. Lawaczeck and Heinemann [24], and Heinemann and Bütefisch [25], were probably the first to perform some systematic schlieren visualizations on transonic flat plate and cascades with di erent trailing edge thicknesses using a flash light of 20 nano-seconds only, and deriving from the photos the vortex shedding frequencies and Strouhal numbers. The schlieren picture in Figure 11 shows impressively that the shedding of each vortex from the trailing edge generates a pressure wave which travels upstream. Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 11 of 58 299 It is important to remember that the measurement of the base pressure carried out with a single 300 pressure tapping in the trailing edge base region implies the assumption of an isobaric trailing edge 301 pressure distribution. However, in 2003 Sieverding et al. [21] demonstrated that at high subsonic 302 Mach numbers the pressure distribution could be highly non-uniform with a marked pressure 303 minimum at the center of the trailing edge base, as will be shown later in Section 5. Under these 304 conditions it is likely that the base pressure measured with a single pressure hole does not reflect the 305 true mean pressure. In addition, the measured pressure would depend on the ratio of the pressure 306 hole to trailing edge diameter / , which is typically in the range / = 0.15 − 0.50 . This fact was 307 also recognized by Jouini et al. [12], who mentioned the difficulties for obtaining representative 308 trailing edge base pressures measurements: “It should also be noted that at high Mach numbers the base 309 pressure varies considerably with location on the trailing edge and the single tap gives a somewhat limited 310 picture of the base pressure behavior”. It is probably correct to say that differences between experimental 311 base pressure data and the base pressure correlation may at least partially be attributed to the use of 312 different pressure hole to trailing edge diameters / by the various researchers. 313 Finally, it is important to mention that the trailing edge pressure is sensitive to the trailing edge 314 shape as demonstrated by El Gendi et al. [22] who showed with the help of high fidelity simulation 315 that the base pressure for blades with elliptic trailing edges was higher than for blades with circular 316 trailing edges. Melzer and Pullan [23] proved experimentally that designing blades with elliptical 317 trailing edges improved the blade performance. The reason is that an elliptic trailing edge reduces 318 not only the wake width but causes also an increase of the base pressure compared to that of blades 319 with a circular trailing edge. This suggests that inaccuracies in the machining of blades with thin 320 trailing edges could easily lead to deviations from the designed circular trailing edge shape and thus 321 contribute to the differences in the base pressure. 322 3. Unsteady Trailing Edge Wake Flow 323 The mixing process of the wake behind turbine blades has been viewed for a long time as a 324 steady state process although it was well known that the separation of the boundary layers at the 325 trailing edge is a highly unsteady phenomenon which leads to the formation of large coherent 326 structures, known as the von Kármán vortex street. The unsteady character of turbine blade wakes is 327 best illustrated by flow visualizations. 328 Lawaczeck and Heinemann [24], and Heinemann and Bütefisch [25], were probably the first to 329 perform some systematic schlieren visualizations on transonic flat plate and cascades with different 330 trailing edge thicknesses using a flash light of 20 nano-seconds only, and deriving from the photos 331 the vortex shedding frequencies and Strouhal numbers. The schlieren picture in Figure 11 shows 332 impressively that the shedding of each vortex from the trailing edge generates a pressure wave which Int. J. Turbomach. Propuls. Power 2020, 5, 10 11 of 55 333 travels upstream. Figure 11. Schlieren picture of turbine rotor blade wake at M = 0.8 [24]. 334 Figure 11. Schlieren picture of turbine rotor blade wake at 2,is =0.8 [24]. Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 12 of 58 In 1982 Han and Cox [26], performed smoke visualizations on a very large-scale nozzle blade at 335 In 1982 Han and Cox [26], performed smoke visualizations on a very large-scale noz zle blade at low speed (Figure 12). The authors found much sharper and well-defined contours of the vortices 336 low speed (Figure 12). The authors found much sharper and well-defined contours of the vortices 337 from the pressure side and concluded that this implied stronger vortex shedding from this side and Figure 40. Blade Mach number distributions for front and rear-loaded blades; adapted from [16]. from the pressure side and concluded that this implied stronger vortex shedding from this side and 338 attributed this to the circulation around the blade. attributed this to the circulation around the blade. Figure 12. Smoke visualization of the vortex shedding from a low speed nozzle blade [26]. 339 Figure 12. Smoke visualization of the vortex shedding from a low speed nozzle blade [26]. Beretta-Piccoli [27] (reported by Bölcs and Sari [28]) was possibly the first to use interferometry to 340 Beretta-Piccoli [27] (reported by Bölcs and Sari [28]) was possibly the first to use interferometry visualize the vortex formation at the blunt trailing edge of a blade at transonic flow conditions. 341 to visualize the vortex formation at the blunt trailing edge of a blade at transonic flow conditions. Besides the problem of time resolution for measuring high frequency phenomena, there was also 342 Besides the problem of time resolution for measuring high frequency phenomena, there was also the problem of spatial resolution for resolving the vortex structures behind the usually rather thin Blade 𝜶 𝜶 Ref g/c 343 the problem of spatial resolution for resolving the vortex structures behind the usually rather thin turbine blade trailing edges. First tests on a large scale flat late model simulating the overhang section A 30° 22° 0.75 [5] 344 turbine blade trailing edges. First tests on a large scale flat late model simulating the overhang section of a cascade allowed to visualize impressively details of the vortex shedding at transonic outlet Mach B 60° 25° 0.75 VKI 345 of a cascade allowed to visualize impressively details of the vortex shedding at transonic outlet Mach number (Figure 13a,b). C 66° 18° 0.70 VKI 346 number (Figure 13a,b). Following Hussain and Hayakawa [29], the wake vortex structures can be described by a set of D 156° 19.5° 0.85 [6] centers which characterize the location of a peak of coherent span-wise vortices and saddles located between the coherent vorticity structures and defined by a minimum of coherent span-wise vorticity. Figure 1. Blade profile losses versus isentropic outlet Mach number for four transonic turbines. The successive span-wise vortices are connected by ribs, that are longitudinal smaller scale vortices of alternating signs. (c) topology of wake vortex structure (a) turbine blade flat plate model. (b) schlieren photograph. behind a cylinder [29]. 348 Figure 13. Vortex shedding at transonic exit flow conditions [30]. 349 Following Hussain and Hayakawa [29], the wake vortex structur (c) topo es can logy be of w descr ake vor ibed by tex struct a set ure of (a) turbine blade flat plate model. (b) schlieren photograph. 350 centers which characterize the location of a peak of coherent span-wise vort behind ice a s cyli and nde sr a [dd 29]les . located 351 between the coherent vorticity structures and defined by a minimum of coherent span-wise vorticity. Figure 13. Vortex shedding at transonic exit flow conditions [30]. Figure 13. Vortex shedding at transonic exit flow conditions [30]. 352 The successive span-wise vortices are connected by ribs, that are longitudinal smaller scale vortices 353 of alternating signs. 354 A significant progress was made in the 1990’s in the frame of two European Research Projects, 355 i.e. Experimental and Numerical Investigations of Time Varying Wakes behind Turbine Blades 356 (BRITE/EURAM CT-92-0048) and Turbulence Modeling for Unsteady Flows in Axial Turbines 357 (BRITE/EURAM CT-96-0143) in which the von Kármán Institute, the University of Genoa and the 358 ONERA of Lille used short duration flow visualizations and fast response instrumentation in 359 combination with large scale blade models to improve the understanding of the formation of the 360 vortical structures at the turbine blade trailing edges and their impact on the unsteady wake flow 361 characteristics. Results of these research projects are reported by Ubaldi et al. [31], Cicatelli and 362 Sieverding [32], Desse [33], Sieverding et al. [34], Ubaldi and Zunino [35] and Sieverding et al. [21,36]. 363 The large-scale turbine guide vane used in these experiments was designed at VKI (Table 2) and 364 released in 1994 [37]. The blade design features a front-loaded blade with an overall low suction side 365 turning in the overhang section and, in particular, a straight rear suction side from halfway 366 downstream of the throat, Figure 14. Due to mass flow restrictions in the VKI blow down facility, the 367 three-bladed cascade with a chord length = 280 mm was limited to investigations at a relatively Int. J. Turbomach. Propuls. Power 2020, 5, 10 12 of 55 A significant progress was made in the 1990’s in the frame of two European Research Projects, i.e., Experimental and Numerical Investigations of Time Varying Wakes behind Turbine Blades (BRITE/EURAM CT-92-0048) and Turbulence Modeling for Unsteady Flows in Axial Turbines (BRITE/EURAM CT-96-0143) in which the von Kármán Institute, the University of Genoa and the ONERA of Lille used short duration flow visualizations and fast response instrumentation in combination with large scale blade models to improve the understanding of the formation of the vortical structures at the turbine blade trailing edges and their impact on the unsteady wake flow characteristics. Results of these research projects are reported by Ubaldi et al. [31], Cicatelli and Sieverding [32], Desse [33], Sieverding et al. [34], Ubaldi and Zunino [35] and Sieverding et al. [21,36]. The large-scale turbine guide vane used in these experiments was designed at VKI (Table 2) and released in 1994 [37]. The blade design features a front-loaded blade with an overall low suction side turning in the overhang section and, in particular, a straight rear suction side from halfway downstream of the throat, Figure 14. Due to mass flow restrictions in the VKI blow down facility, the three-bladed cascade with a chord length c = 280 mm was limited to investigations at a relatively low subsonic outlet Mach number of M = 0.4. The suction side boundary layer undergoes natural transition at 2,is Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 13 of 58 x/c  0.6. On the pressure side the boundary layer was tripped at x/c  0.61. The boundary ax ax 368 three-bladed cascade with a chord length = 280 mm was limited to investigations at a relatively layers at the trailing edge with shape factors H of 1.64 and 1.41 for the pressure and suction sides 369 low subsonic outlet Mach number of =0.4. The suction side boundary layer undergoes natural respectively, were clearly turbulent. The schlieren photographs in Figure 14 were taken with a Nanolite 370 transition at / ~0.6. On the pressure side the boundary layer was tripped at / ~0.61. The spark source, with Dt = 20 10 s. The dominant vortex shedding frequency was 2.65 kHz and the 371 boundary layers at the trailing edge with shape factors of 1.64 and 1.41 for the pressure and 372 suction sides respectively, were clearly turbulent. The schlieren photographs in Figure 14 were taken corresponding Strouhal number, defined as: 373 with a Nanolite spark source, with ∆ = 20 10 . The dominant vortex shedding frequency was 374 2.65 kHz and the corresponding Strouhal number, defined as: f d vs te St = (3) = (3) 2,is 375 was = 0.27 . was St = 0.27. (a) (b) (c) 377 Figure 14. Very large-scale turbine nozzle guide vane (VKI LS-94); =0.4, =2 x 10 case. (a) 6 Figure 14. Very large-scale turbine nozzle guide vane (VKI LS-94); M = 0.4, Re = 2 10 case. 2,is 2 378 test section, (b) surface isentropic Mach number distribution, (c) schlieren photographs. Adapted (a) test section, (b) surface isentropic Mach number distribution, (c) schlieren photographs. Adapted 379 from [32]. from [32]. 380 Figure 14c presents two instances in time of the vortex shedding process. The left flow 381 visualization shows the enrolment of the pressure side shear layer into a vortex, the right one the Figure 14c presents two instances in time of the vortex shedding process. The left flow visualization 382 formation of the suction side vortex. Note that the pressure side vortex appears to be much stronger shows the enrolment of the pressure side shear layer into a vortex, the right one the formation of the 383 than the suction side one, which confirms the observations made by Han and Cox [26]. suction side vortex. Note that the pressure side vortex appears to be much stronger than the suction 384 Table 2. VKI LS-94 large scale nozzle blade geometric characteristics [32]. side one, which confirms the observations made by Han and Cox [26]. Parameter Symbol Value Gerrard [38], describes the vortex formation for the flow behind a cylinder as follows, Figure 15. Chord c 280 mm The growing vortex (A) is fed by the circulation existing in the upstream shear layer until the vortex is Pitch to chord ratio / 0.73 Blade aspect ratio ℎ/ 0.7 strong enough to entrain fluid from the opposite shear layer bearing vorticity of the opposite circulation. Stagger angle −49.83° When the quantity of entrained fluid is sucient to cut o the supply of circulation to the growing Trailing edge thickness to throat ratio 0.053 Trailing edge wedge angle 7.5° Gauging angle (arcsin( /) ) 19.1° 385 Gerrard [38], describes the vortex formation for the flow behind a cylinder as follows, Figure 15. 386 The growing vortex (A) is fed by the circulation existing in the upstream shear layer until the vortex 387 is strong enough to entrain fluid from the opposite shear layer bearing vorticity of the opposite Int. J. Turbomach. Propuls. Power 2020, 5, 10 13 of 55 vortex—the opposite vorticity of the fluid in both shear layers cancel each other—then the vortex is shed o . Table 2. VKI LS-94 large scale nozzle blade geometric characteristics [32]. Parameter Symbol Value Chord c 280 mm Pitch to chord ratio g/c 0.73 Blade aspect ratio h/c 0.7 Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 14 of 58 Stagger angle 49.83 Trailing edge thickness to throat ratio d /o 0.053 te Trailing edge wedge angle  7.5 388 the growing vortex—the opposite vorticity of the fluid in both shear layer te s cancel each other—then Gauging angle (arcsin(o/g)) 19.1 389 the vortex is shed off. 390 Figure 15. Vortex formation mechanism; adapted from [38]. Figure 15. Vortex formation mechanism; adapted from [38]. 391 Contrary Contrary to t to the he bl blow ow down down tunnel a tunnel at t VKI, the Is VKI, the Istituto tituto di M di Macchine acchine e e Sistemi Sistemi Ener Energetici (I getici (ISME) SME) at the University of Genoa used a continuous running low speed wind tunnel. Miniature cross-wire 392 at the University of Genoa used a continuous running low speed wind tunnel. Miniature cross-wire 393 hot-wir hot-wire pro e probe be and a and a four four-beam -beam laser laser Doppler Doppler velo velocimeter cimeter are are used used for the for the measur measurement ements s of the of the unsteady wake. An example of the instantaneous patterns of the ensemble averaged periodic wake 394 unsteady wake. An example of the instantaneous patterns of the ensemble averaged periodic wake 395 characteristics characteristics is is present presented ed in F in Figur igure e 16 16.. A det A detailed ailed d description escription is is given by given by Uba Ubaldi ldi and Z and Zunino unino [3 [35 5]]. . The streamwise periodic component of the velocity, U U in Figure 16 (upper left), shows asymmetric 396 The streamwise periodic component of the velocity, − in Figure 16 (upper left), shows s s periodic patterns of alternating positive and negative velocity components issued from the pressure to 397 asymmetric periodic patterns of alternating positive and negative velocity components issued from the suction side. As already shown schematically in Figure 13, saddle points separating groups of 398 the pressure to the suction side. As already shown schematically in Figure 13, saddle points four cores, are located along the wake center line. On the contrary, the periodic parts of the transverse 399 separating groups of four cores, are located along the wake center line. On the contrary, the periodic component U U (upper right) appear as cores of positive and negative values, approximately 400 parts of the trnansver n se component − (upper right) appear as cores of positive and negative centered in the wake which alternate, enlarging in streamwise direction. The combination of the two 401 values, approximately centered in the wake which alternate, enlarging in streamwise direction. The velocity components give rise to the rolling up of the periodic flow into a row of vortices rotating in 402 combination of the two velocity components give rise to the rolling up of the periodic flow into a row opposite direction as shown by the velocity vector plots (lower left). 403 of vortices rotating in opposite direction as shown by the velocity vector plots (lower left). As illustrated by Gerrard [38] (see Figure 15), the vortex formation is driven by the vorticity 404 As illustrated by Gerrard [38] (see Figure 15), the vortex formation is driven by the vorticity in in the suction and pressure side boundary layers. The vorticity terms ! e and ! in the wake have 405 the suction and pressure side boundary layers. The vorticity terms and in the wake have been been determined taking respectively the curl of the phase averaged and time averaged velocity field: 406 determined taking respectively the curl of the phase averaged and time averaged velocity field: = e e @U @U @U @U s n s n ! e = and ! = , Figure 16 (lower right). The local maxima and minima and 407 − and = − , Figure 16 (lower right). The local maxima and minima and saddle @n @s @n @s saddle regions (the points where the vorticity changes its sign) define the location, extension, rotation 408 regions (the points where the vorticity changes its sign) define the location, extension, rotation and and intensity of the vortices. 409 intensity of the vortices. With increasing downstream Mach number, the vortices become much more intense as demonstrated in Figure 17 on a half scale model of the blade already presented in Figure 14 and Table 2, at an outlet Mach number M = 0.79 in a four bladed cascade, Sieverding et al. [36]. Contrary to 2,is schlieren photographs which visualize density changes, the smoke visualizations in Figure 17 show the instantaneous flow patterns and are therefore particular well suited to visualize the enrolment of the vortices. A close look at the vortex structures reveals that the distances between successive vortices change. In fact, the distance between a pressure side vortex and a suction side vortex is always smaller than the distance between two successive pressure side vortices. A possible reason is that the pressure side vortex plays a dominant role and exerts an attraction on the suction side vortex as already found by Han and Cox [26]. Int. J. Turbomach. Propuls. Power 2020, 5, 10 14 of 55 Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 15 of 58 (a) (b) (c) (d) 411 Figure Figure 16. 16. Inst Instantaneous antaneous realization of the realization of the ensemble ensemble aver averag aged streamwise velocity ( ed streamwise velocity (a a), ), transversal transversal velocity (b), velocity vector (c) and vorticity patterns (d) of the periodic flow [35]. 412 velocity (b), velocity vector (c) and vorticity patterns (d) of the periodic flow [35]. 413 With increasing downstream Mach number, the vortices become much more intense as 414 demonstrated in Figure 17 on a half scale model of the blade already presented in Figure 14 and Table 415 2, at an outlet Mach number =0.79 in a four bladed cascade, Sieverding et al. [36]. Contrary to 416 schlieren photographs which visualize density changes, the smoke visualizations in Figure 17 show 417 the instantaneous flow patterns and are therefore particular well suited to visualize the enrolment of 418 the vortices. A close look at the vortex structures reveals that the distances between successive 419 vortices change. In fact, the distance between a pressure side vortex and a suction side vortex is 420 always smaller than the distance between two successive pressure side vortices. A possible reason is 421 that the pressure side vortex plays a dominant role and exerts an attraction on the suction side vortex 422 as already found by Han and Cox [26]. (a) (b) (c) Figure 17. VKI LS94 turbine blade, M = 0.79, Re = 2.8 10 case. (a) four blades cascade, (b) surface 2,is isentropic Mach number distribution and (c) smoke visualizations. Adapted from [36]. The vortex formation and subsequent shedding is accompanied by large angle fluctuations of the separating shear layers which does not only lead to large pressure fluctuations in the zone of separations but also induces strong acoustic waves. The latter travel upstream on both the pressure and suction side as shown in the corresponding schlieren photographs obtained this time with a continuous light source, a high speed rotating drum and rotating prism camera from ONERA with a maximum frame rate of 35,000 frames per second (see Figure 18), as reported by Sieverding et al. [21]. (a) (b) (c) Int. J. Turbomach. Propuls. Power 2020, 5, 10 15 of 55 In image 1 of Figure 18 the suction side shear layer has reached its farthest inward position and Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 16 of 58 the local pressure just upstream of the separation point has reached its minimum value. Conversely, on the pressure side the separating shear layer has reached its most outward position. A pressure 423 Figure 17. VKI LS94 turbine blade, = 0.79, = 2.8 x 10 case. (a) four blades cascade, (b) wave denoted Pi originates from the point where the boundary layer separates from the trailing edge. 424 surface isentropic Mach number distribution and (c) smoke visualizations. Adapted from [36]. Upstream of Pi is the pressure wave from the previous cycle. It interferes with the suction side of 425 the neighboring The vortex fo blade rmation fromand wher subse e it is quent shedding reflected. In image is ac4companied b of Figure 18,yt large he suction angleside fluctuations shear layer of 426 is that e separ its most ating outwar shear dlaposition. yers which A pr does essur not e only wavelea originates d to large pr at the essu point re flof uct separation, uations in t denoted he zone o Si.f 427 The separ pr aessur tions but also in e wave further duce upstr s strong eamaco is due ustic w to the aves pr . eviou The la stter travel cycle. On upstrea the pressur m on both the pressure e side the pressure 428 wave and suction Pi extends side now as shown to the in the suction corre sidesponding of the neighboring schlieren photogra blade. The phs wave obtaiinterfer ned this ence time point with ofa 429 the continuous light source, a previous cycle has moved high sp up-str eed ro eam. tating dr It can ther um and rotating efore be expected prism ca thatmera the suction from ONERA side pressur with a e 430 distribution maximum frnear ame ra the te of 35 throat,0 r00 f egion ram ises per second ( highly unsteady see . Figure 18), as reported by Sieverding et al. [21]. Figure 18. Schlieren photographs of vortex shedding at two instances in time; M = 0.79, Re = 431 Figure 18. Schlieren photographs of vortex shedding at two instances in time; 2,is = 0.79, 2 = 2.8 10 [21]. 432 2.8 x 10 [21]. Holographic interferometric density measurements, performed at VKI at M = 0.79 by Sieverding 2,is 433 In image 1 of Figure 18 the suction side shear layer has reached its farthest inward position and Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 17 of 58 et al. [36], give further information about the formation and the shedding process of the von Kármán 434 the local pressure just upstream of the separation point has reached its minimum value. Conversely, vortices. The reference density is evaluated from pressure measurements with a fast response needle 451 The interferogram in Figure 19 shows the suction side vortex (upper blade surface) in its out 435 on the pressure side the separating shear layer has reached its most outward position. A pressure 452 most outward position i.e. at the start of the shedding phase. On the pressure side the density patterns static pressure probe positioned just outside of the wake assuming the total temperature to be constant 436 wave denoted Pi originates from the point where the boundary layer separates from the trailing edge. 453 point to the start of the formation of a new pressure side vortex. The pressure side vortex of the outside the wake. 437 Upstream of Pi is the pressure wave from the previous cycle. It interferes with the suction side of the 454 previous cycle is situated at a trailing edge distance of ≈2. This vortex is defined by ten The interferogram in Figure 19 shows the suction side vortex (upper blade surface) in its out most 455 fringes. With a relative density change between two successive fringes of ∆( ) = 0.0184 the total 438 neighboring blade from where it is reflected. In image 4 of Figure 18, the suction side shear layer is 456 relative density change from the outside to the vortex center is ∆( ) = 0.184. The minimum in outward position i.e., at the start of the shedding phase. On the pressur e side the density patterns point 439 at its most outward position. A pressure wave originates at the point of separation, denoted Si. The 457 the vortex center is = 0.552 compared to an isentropic downstream static to total density ratio to the start of the formation of a new pressure side vortex. The pressure side vortex of the previous 440 pressure wave further upstream is due to the previous cycle. On the pressure side the pressure wave 458 of ⁄ =0.745. cycle is situated at a trailing edge distance of x/d  2. This vortex is defined by ten fringes. With a te 459 Based on a large number of tests with holographic interferometry and white light interferometry, 441 Pi extends now to the suction side of the neighboring blade. The wave interference point of the 460 see Desse [33], Figure 19 shows the variation of the vortex density minima non-dimensionalized by relative density change between two successive fringes of D(/ ) = 0.0184 the total relative density 442 previous cycle has moved up-stream. It can therefore be expected that the suction side pressure 461 the upstream total density / , in function of the trailing edge distance / . There are two distinct change from the outside to the vortex center is D(/ ) = 0.184. The minimum in the vortex center is 443 distribution near the throat region is highly unsteady. 462 regions for the evolution of the vortex minima: a rapid linear density rise-up to distance / = 1.7 / = 0.552 compared to an isentropic downstream static to total density ratio of  / = 0.745. 0 463 followed by a much slower rise further downstream. 2 01 444 Holographic interferometric density measurements, performed at VKI at =0.79 by 445 Sieverding et al. [36], give further information about the formation and the shedding process of the 446 von Kármán vortices. The reference density is evaluated from pressure measurements with a fast 447 response needle static pressure probe positioned just outside of the wake assuming the total 448 temperature to be constant outside the wake. 449 The interferogram in Figure 19 shows the suction side vortex (upper blade surface) in its out 450 most outward position i.e. at the start of the shedding phase. On the pressure side the density patterns 451 point to the start of the formation of a new pressure side vortex. The pressure side vortex of the 452 previous cycle is situated at a trailing edge distance of ⁄ ≈2. This vortex is defined by ten 453 fringes. With a relative density change between two successive fringes of ∆( ⁄ ) = 0.0184 the total (a) (b) 454 relative density change from the outside to the vortex center is ∆( ⁄ ) = 0.184. The minimum in 465 Figure 19. Instantaneous density distribution (a) and variation of density minima with trailing edge 455 the vortex ce Figure 19. nter is Instantaneous ⁄ = density 0.552 comp distribu ared to a tion (a)n and isentropi variation c downstrea of density minim m stata ic with to tota trailing l densi edge ty Commente ratio d [M39]: Can it be in terms of (a)? 466 distance (b) at = 0.79, =2.8 x 10 ; adapted from [36]. Please check all figure captions format. distance (b) at M = 0.79, Re = 2.8 10 ; adapted from [36]. 456 of ⁄ = 0.745. 2 2,is 467 Comparing the vortex formation at =0.4 and 0.79 shows that with increasing Mach 457 Based on a large number of tests with holographi , c interferometry and white light interferometry, Commented [MM40R39]: Done 468 number the vortices form much closer to the trailing edge. This tendency goes crescendo with further 458 see Desse [33], Figure 19 shows the variation of the vortex density minima non-dimensionalized by 469 increase of the downstream Mach number as already shown in Figure 13 where normal shocks 459 the upstream total density / , in function of the trailing edge distance / . There are two distinct 470 oscillate close to the trailing edge forward and backward with the alternating shedding of the 471 vortices. A further increase of the outlet flow leads gradually to the formation of an oblique shock 472 system at the convergence of the separating shear layers at short distance behind the trailing edge, 473 causing a delay of the vortex formation to this region as demonstrated by Carscallen and Gostelow 474 [39], in the high speed cascade facility of the NRC Canada. The high speed schlieren pictures revealed 475 some very unusual types of wake vortex patterns as shown in Figure 20. Besides the regular von 476 Kármán vortex street (left), the authors visualized other vortex patterns, such as e.g. couples or 477 doublets, on the right. In other moments in time they observed what they called hybrid or random 478 or no patterns. The schlieren photos in Figure 20 show the existence of an unexpected shock 479 emanating from the trailing edge pressure side at the beginning of the trailing edge circle. 480 Questioning Bill Carscallen [40] recently about the origin of this shock it appeared that the shock was 481 simply due to an inaccuracy in the blade manufacturing of the trailing edge circle. Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 17 of 58 460 regions for the evolution of the vortex minima: a rapid linear density rise-up to distance / = 1.7 461 followed by a much slower rise further downstream. Int. J. Turbomach. Propuls. Power 2020, 5, 10 16 of 55 Based on a large number of tests with holographic interferometry and white light interferometry, (a) (b) see Desse [33], Figure 19 shows the variation of the vortex density minima non-dimensionalized by the upstream total density / , in function of the trailing edge distance x/D. There are two distinct 462 Figure 19. Instantaneous density distribution (a) and variation of density minima with trailing edge regions for the evolution of the vortex minima: a rapid linear density rise-up to distance x/D = 1.7 463 distance (b) at =0.79, = 2.8 x 10 ; adapted from [36]. followed by a much slower rise further downstream. Comparing the vortex formation at M = 0.4 and 0.79 shows that with increasing Mach number 464 Comparing the vortex formation at =0.4 and 0.79 shows that with increasing Mach 2,is the vortices form much closer to the trailing edge. This tendency goes crescendo with further increase of 465 number the vortices form much closer to the trailing edge. This tendency goes crescendo with further the downstream Mach number as already shown in Figure 13 where normal shocks oscillate close to the 466 increase of the downstream Mach number as already shown in Figure 13 where normal shocks trailing edge forward and backward with the alternating shedding of the vortices. A further increase 467 oscillate close to the trailing edge forward and backward with the alternating shedding of the of the outlet flow leads gradually to the formation of an oblique shock system at the convergence of 468 vortices. A further increase of the outlet flow leads gradually to the formation of an oblique shock the separating shear layers at short distance behind the trailing edge, causing a delay of the vortex 469 system at the convergence of the separating shear layers at short distance behind the trailing edge, formation to this region as demonstrated by Carscallen and Gostelow [39], in the high speed cascade 470 causing a delay of the vortex formation to this region as demonstrated by Carscallen and Gostelow facility of the NRC Canada. The high speed schlieren pictures revealed some very unusual types 471 [39], in the high speed cascade facility of the NRC Canada. The high speed schlieren pictures revealed of wake vortex patterns as shown in Figure 20. Besides the regular von Kármán vortex street (left), 472 some very unusual types of wake vortex patterns as shown in Figure 20. Besides the regular von the authors visualized other vortex patterns, such as e.g. couples or doublets, on the right. In other 473 Kármán vortex street (left), the authors visualized other vortex patterns, such as e.g. couples or moments in time they observed what they called hybrid or random or no patterns. The schlieren 474 doublets, on the right. In other moments in time they observed what they called hybrid or random 475 photos or no p inatterns. The sch Figure 20 showlieren photos the existencein Figure of an unexpected 20 show the exi shock emanating stence of fran u om the nexpe trailing cted sedge hock pressure side at the beginning of the trailing edge circle. Questioning Bill Carscallen [40] recently 476 emanating from the trailing edge pressure side at the beginning of the trailing edge circle. 477 about Questi the oning origin Billof Ca this rsca shock llen [4it 0]appear recented ly about that the the shock origin was of tsimply his shock it due appe to an ar inaccuracy ed that the in sh the ock w blade as manufacturing of the trailing edge circle. 478 simply due to an inaccuracy in the blade manufacturing of the trailing edge circle. (a) (b) (c) 479 Figure Figure 20. 20. Occurrence of different vortex patter Occurrence of di erent vortex patters s in in wak wake e of of t transonic ransonic bla blade de at at M = =1 1.07 .07.. ( (a a) ) re regular gular 2,is 480 vortex vortex street street,, ( (b b)) cou couples, ples, ( (cc)) doublets doublets [[39]. 39]. The question whether in distinction of the conventional von Kármán vortex street, a double 481 The question whether in distinction of the conventional von Kármán vortex street, a double row row vortex street of unequal vortex strength may exist, was treated by Sun in 1983 [41]. Figure 21 482 vortex street of unequal vortex strength may exist, was treated by Sun in 1983 [41]. Figure 21 presents presents an example of a double row vortex street with unequal vortex strength and vortex distances. 483 an example of a double row vortex street with unequal vortex strength and vortex distances. The The Int. J.author Turbomach demonstrated . Propuls. Power 2018 that, such 3, x FOR PE configurations ER REVIEW ar e basically unstable. 18 of 58 484 author demonstrated that such configurations are basically unstable. Figure 21. An example of vortex street with unequal vortex strengths and unequal vortex spacings; 485 Figure 21. An example of vortex street with unequal vortex strengths and unequal vortex spacings; adapted from [41]. 486 adapted from [41]. 487 4. Energy Separation in the Turbine Blade Wakes 488 In the course of a joint research program between the National Research Council of Canada and 489 Pratt & Whitney Canada of the flow through an annular transonic nozzle guide vane in the 1980s 490 certain experiments revealed a non-uniform total temperature distribution downstream of the 491 uncooled blades, Carscallen and Oosthuizen [42]. Considering the importance of the existence of a 492 non-uniform total temperature distribution at the exit of uncooled stator blade row for the 493 aerothermal aspects of the downstream rotor, the Gasdynamics Laboratory of the National Research 494 Council, Canada decided to build a continuously running suction type large scale planar cascade 495 tunnel (chord length 175.3 mm, turning angle 76°, trailing edge diameter 6.35 mm) and launched an 496 extensive research program aiming at the understanding of the mechanism causing the occurrence 497 of total temperature variations downstream of a fixed blade row, determine their magnitude and 498 evaluate their significance for the design of the downstream rotor. Downstream traverses with copper 499 constantan thermocouples reported by Carscallen et al. [43] in 1996 showed that the total temperature 500 contours correlated perfectly with the total pressure wake profiles, Figure 22. (b) Total pressure coefficient (c) Total temperature difference (a) Test section 502 Figure 22. Total pressure coefficient and temperature contours downstream of a nozzle guide vane at 503 a pressure ratio PR = 1.9; adapted from [42]. 504 In the wake center the total temperature dropped significantly below the inlet total temperature 505 while higher values were recorded near the border of the wake. The differences increased with Mach 506 number and reached a maximum at sonic outlet conditions. The question was then to elucidate the 507 reasons for these temperature variations. The research on flows across cylinders was already more Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 18 of 58 485 Figure 21. An example of vortex street with unequal vortex strengths and unequal vortex spacings; 486 adapted from [41]. Int. J. Turbomach. Propuls. Power 2020, 5, 10 17 of 55 487 4. Energy Separation in the Turbine Blade Wakes 4. Energy Separation in the Turbine Blade Wakes 488 In the course of a joint research program between the National Research Council of Canada and In the course of a joint research program between the National Research Council of Canada and 489 Pratt & Whitney Canada of the flow through an annular transonic nozzle guide vane in the 1980s Pratt & Whitney Canada of the flow through an annular transonic nozzle guide vane in the 1980s 490 certain experiments revealed a non-uniform total temperature distribution downstream of the certain experiments revealed a non-uniform total temperature distribution downstream of the uncooled 491 uncooled blades, Carscallen and Oosthuizen [42]. Considering the importance of the existence of a blades, Carscallen and Oosthuizen [42]. Considering the importance of the existence of a non-uniform 492 non-uniform total temperature distribution at the exit of uncooled stator blade row for the total temperature distribution at the exit of uncooled stator blade row for the aerothermal aspects 493 aerothermal aspects of the downstream rotor, the Gasdynamics Laboratory of the National Research of the downstream rotor, the Gasdynamics Laboratory of the National Research Council, Canada 494 Council, Canada decided to build a continuously running suction type large scale planar cascade decided to build a continuously running suction type large scale planar cascade tunnel (chord length 495 tunnel (chord length 175.3 mm, turning angle 76°, trailing edge diameter 6.35 mm) and launched an 175.3 mm, turning angle 76 , trailing edge diameter 6.35 mm) and launched an extensive research 496 extensive research program aiming at the understanding of the mechanism causing the occurrence program aiming at the understanding of the mechanism causing the occurrence of total temperature 497 of total temperature variations downstream of a fixed blade row, determine their magnitude and variations downstream of a fixed blade row, determine their magnitude and evaluate their significance 498 evaluate their significance for the design of the downstream rotor. Downstream traverses with copper for the design of the downstream rotor. Downstream traverses with copper constantan thermocouples 499 constantan thermocouples reported by Carscallen et al. [43] in 1996 showed that the total temperature reported by Carscallen et al. [43] in 1996 showed that the total temperature contours correlated perfectly 500 contours correlated perfectly with the total pressure wake profiles, Figure 22. with the total pressure wake profiles, Figure 22. (b) Total pressure coefficient (c) Total temperature difference (a) Test section Figure 22. Total pressure coecient and temperature contours downstream of a nozzle guide vane at a 502 Figure 22. Total pressure coefficient and temperature contours downstream of a nozzle guide vane at pressure ratio PR = 1.9; adapted from [42]. 503 a pressure ratio PR = 1.9; adapted from [42]. In the wake center the total temperature dropped significantly below the inlet total temperature 504 In the wake center the total temperature dropped significantly below the inlet total temperature while higher values were recorded near the border of the wake. The di erences increased with Mach 505 while higher values were recorded near the border of the wake. The differences increased with Mach number and reached a maximum at sonic outlet conditions. The question was then to elucidate the 506 number and reached a maximum at sonic outlet conditions. The question was then to elucidate the reasons for these temperature variations. The research on flows across cylinders was already more 507 reasons for these temperature variations. The research on flows across cylinders was already more advanced in this respect. Measurements of the temperature distribution around a cylinder for a flow normal to the axis of the cylinder, performed at the Aeronautical Institute of Braunschweig in the late 1930’s and reported by Eckert and Weise [44] in 1943, showed that the recovery temperature at the base of the cylinder reduced below the true (static) temperature of the incoming flow, so that the recovery factor: T T r = T T attained negative values in the base region (see Figure 23). The authors suspected that the low values were possibly due to the intermittent separation of vortices from the cylinder. These results were confirmed by Ryan [45] in 1951 at Ackeret’s Institute in Zürich who clearly related this low temperature to the periodic vortex shedding behind the cylinder as Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 19 of 58 Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 19 of 58 (b) Total pressure coefficient 508 advanced in this respect. Measurements of the temperature distribution around a cylinder for a flow 509 normal to the axis of the cylinder, performed at the Aeronautical Institute of Braunschweig in the late 510 1930’s and reported by Eckert and Weise [44] in 1943, showed that the recovery temperature at the 511 base of the cylinder reduced below the true (static) temperature of the incoming flow, so that the 512 recovery factor: (c) Total temperature difference 513 attained negative values in the base region (see Figure 23). (a) Test section 506 Figure 22. Total pressure coefficient and temperature contours downstream of a nozzle guide vane at Commented [M43]: Please add explanation for subgraph. 507 a pressure ratio PR = 1.9; adapted from [42]. Commented [MM44R43]: Done. Int. J. Turbomach. 508 Propuls. In th Power e wak 2020 e ce,nt5e , r 10 the total temperature dropped significantly below the inlet total temperature 18 of 55 509 while higher values were recorded near the border of the wake. The differences increased with Mach 510 number and reached a maximum at sonic outlet conditions. The question was then to elucidate the 511 reasons for these temperature variations. The research on flows across cylinders was already more cause for the energy separation in the fluctuating wake. He also noticed that the energy separation 512 advanced in this respect. Measurements of the temperature distribution around a cylinder for a flow was particularly large when a strong sound was generated by the flow. 513 normal to the axis of the cylinder, performed at the Aeronautical Institute of Braunschweig in the late 514 1930’s and reported by Eckert and Weise [44] in 1943, showed that the recovery temperature at the The existence of a low temperature field at the base of a cylinder was also observed by Sieverding 515 base of the cylinder reduced below the true (static) temperature of the incoming flow, so that the in 1985, who used an infrared camera to visualize through a germanium window in the side wall of a 516 recovery factor: blow down wind tunnel the wall temperature field around a 15 mm diameter cylinder at M = 0.4, 514 Figure 23. Evolution of the recovery factor in the azimuthal direction from the stagnation point see Figure 24. Unfortunately, due to a lack of time it was not possible to determine the absolute 515 (= 0 °) to the rear side of the cylinder ( = 180° ); adapted from [44]. temperature values. 517 attained negative values in the base region (see Figure 23). 516 The authors suspected that the low values were possibly due to the intermittent separation of 517 vortices from the cylinder. These results were confirmed by Ryan [45] in 1951 at Ackeret’s Institute 518 in Zürich who clearly related this low temperature to the periodic vortex shedding behind the 519 cylinder as cause for the energy separation in the fluctuating wake. He also noticed that the energy 520 separation was particularly large when a strong sound was generated by the flow. 521 The existence of a low temperature field at the base of a cylinder was also observed by Sieverding in 522 1985, who used an infrared camera to visualize through a germanium window in the side wall of a 523 blow down wind tunnel the wall temperature field around a 15 mm diameter cylinder at = 0.4 , 524 see Figure 24. Unfortunately, due to a lack of time it was not possible to determine the absolute 518 Figure 23. Evolution of the recovery factor in the azimuthal direction from the stagnation point Figure 23. Evolution of the recovery factor r in the azimuthal direction from the stagnation point 525 temperature values. 519 ( =0°) to the rear side of the cylinder ( = 180° ); adapted from [44]. Commented [M45]: Please add explanation for subgraph. ( = 0 ) to the rear side of the cylinder ( = 180); adapted from [44]. 520 The authors suspected that the low values were possibly due to the intermittent separation of Commented [MM46R45]: These two pictures have to be 521 vortices from the cylinder. These results were confirmed by Ryan [45] in 1951 at Ackeret’s Institute considered as a single entity. 522 in Zürich who clearly related this low temperature to the periodic vortex shedding behind the Figure 24. Side wall temperature field around a cylinder recorded by an infrared camera. 526 Figure 24. Side wall temperature field around a cylinder recorded by an infrared camera. Eckert [46] explained the mechanism of energy separation along a flow path with the help of the 527 Eckert [46] explained the mechanism of energy separation along a flow path with the help of the unsteady energy equation: 528 unsteady energy equation: @p DT @ @T @ c = + k +   (4) p i ij Dt @t @x @x @x i i j |{z} | {z } | {z } (a) (b) (c) The change of the total temperature with time depends on: (a) the partial derivative of the pressure with time, (b) on the energy transport due to heat conduction between regions of di erent temperatures and (c) on the work due to viscous stresses between regions of di erent velocities. As regards the flow behind blu bodies the two latter terms are considered small compared the pressure gradient term and Equation (4) then reduces to: DT @p c = Dt @t The occurrence of total temperature variations in the vortex streets behind cylinders was e.g. extensively described by Kurosaka et al. [47], Ng et al. [48] and Sunduram et al. [49]. The progress in the understanding of the mechanism was boosted with the arrival of fast temperature probes as for example the dual sensor thin film platinum resistance thermometer probe developed by Buttsworth and Jones [50] in 1996 at Oxford. Using their technique Carscallen et al. [51,52] were the first to measure the time varying total pressure and temperature in the wake of their turbine Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 20 of 58 = + + (4) () () () 529 The change of the total temperature with time depends on: (a) the partial derivative of the pressure 530 with time, (b) on the energy transport due to heat conduction between regions of different 531 temperatures and (c) on the work due to viscous stresses between regions of different velocities. As 532 regards the flow behind bluff bodies the two latter terms are considered small compared the pressure 533 gradient term and Equation (4) then reduces to: 534 The occurrence of total temperature variations in the vortex streets behind cylinders was e.g. 535 extensively described by Kurosaka et al. [47], Ng et al. [48] and Sunduram et al. [49]. 536 The progress in the understanding of the mechanism was boosted with the arrival of fast 537 temperature probes as for example the dual sensor thin film platinum resistance thermometer probe 538 developed by Buttsworth and Jones [50] in 1996 at Oxford. Using their technique Carscallen et al. Int. J. Turbomach. Propuls. Power 2020, 5, 10 19 of 55 539 [51,52] were the first to measure the time varying total pressure and temperature in the wake of their 540 turbine vane. Figure 25 presents the results for an isentropic outlet Mach number =0.95 and a 541 vortex shedding frequency of the order of 10 kHz. The probe traverse plane was normal to the wake vane. Figure 25 presents the results for an isentropic outlet Mach number M = 0.95 and a vortex 2,is 542 at a distance of 5.76 trailing edge diameters from the vane trailing edge. In a later paper concerning shedding frequency of the order of 10 kHz. The probe traverse plane was normal to the wake at a 543 the same cascade, Gostelow and Rona [53], published also the corresponding entropy variations from distance of 5.76 trailing edge diameters from the vane trailing edge. In a later paper concerning the 544 the Gibb’s relation: same cascade, Gostelow and Rona [53], published also the corresponding entropy variations from the Gibb’s relation: T 02 − = 02− s s = c ln R ln 2 1 p T p 01 01 545 The results are presented in Figure 26. The results are presented in Figure 26. (a) (b) Figure 25. Contour plots of phase averaged total pressure (a) and total temperature (b) behind turbine 546 Figure 25. Contour plots of phase averaged total pressure (a) and total temperature (b) behind turbine Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 21 of 58 vane; adapted from [51]. 547 vane; adapted from [51]. Figure 26. Time resolved measurements of entropy increase [53]. 548 Figure 26. Time resolved measurements of entropy increase [53]. The variation of the maxima and minima of the total temperature in the center of the wake vary 549 The variation of the maxima and minima of the total temperature in the center of the wake vary between a minimum of 15 to a maximum of 4 with respect to the inlet ambient temperature, 550 between a minimum of −15° to a maximum of −4° with respect to the inlet ambient temperature, Figure 27. At the border of the wake the temperature raises considerably above the inlet temperature, 551 Figure 27. At the border of the wake the temperature raises considerably above the inlet temperature, while the time averaged temperature in the wake center is about10 . 552 while the time averaged temperature in the wake center is about −10°. 553 Figure 27. Variation of total temperature maxima and minima through the wake; adapted from [51]. 554 In 2004, Sieverding et al. [36] published very similar results for the turbine vane shown in Figure 555 17. The wake traverse was performed at a trailing edge distance of only 2.5 in direction of the 556 tangent to the blade camber line, which forms an angle of 66° with the axial direction. The traverse is 557 made normal to this tangent. 558 The steady state total pressure and total temperature measurements are presented in Figure 28. 559 Similar to the results obtained at the NRC Canada, the wake center is characterized by a pronounced 560 total temperature drop of 3% of the inlet value of 290 K which corresponds to about −9°, a variation 561 which is of the same order as that reported in Figure 27. On the borders of the wake, total temperature 562 peaks in excess of the inlet temperature are also recorded. The mass integrated total temperature 563 value across the wake (denoted with a <∗ ) should be such that < /< = 1 , but lack of 564 information on the local velocity did not allow to perform this integration. (a) Steady total pressure (b) Steady total temperature Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 21 of 58 548 Figure 26. Time resolved measurements of entropy increase [53]. 549 The variation of the maxima and minima of the total temperature in the center of the wake vary 550 between a minimum of −15° to a maximum of −4° with respect to the inlet ambient temperature, 551 Figure 27. At the border of the wake the temperature raises considerably above the inlet temperature, Int. J. Turbomach. Propuls. Power 2020, 5, 10 20 of 55 552 while the time averaged temperature in the wake center is about −10°. Figure 27. Variation of total temperature maxima and minima through the wake; adapted from [51]. 553 Figure 27. Variation of total temperature maxima and minima through the wake; adapted from [51]. In 2004, Sieverding et al. [36] published very similar results for the turbine vane shown in Figure 17. 554 In 2004, Sieverding et al. [36] published very similar results for the turbine vane shown in Figure The wake traverse was performed at a trailing edge distance of only 2.5 d in direction of the tangent te 555 17. The wake traverse was performed at a trailing edge distance of only 2.5 in direction of the to the blade camber line, which forms an angle of 66 with the axial direction. The traverse is made 556 tangent to the blade camber line, which forms an angle of 66° with the axial direction. The traverse is normal to this tangent. 557 made normal to this tangent. The steady state total pressure and total temperature measurements are presented in Figure 28. Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 22 of 58 558 The steady state total pressure and total temperature measurements are presented in Figure 28. Similar to the results obtained at the NRC Canada, the wake center is characterized by a pronounced 559 Simila562 r to the resul The tss obta teady stiat ne t ed ota al pr t the NRC Ca essure and tota na l teda mp, eth rate w ure m ae ke asure center ments is char are present acter ed in ized by Figure 2 8a prono . unced total temperature drop of 3% of the inlet value of 290 K which corresponds to about9 , a variation 563 Similar to the results obtained at the NRC Canada, the wake center is characterized by a pronounced 560 total temperature drop of 3% of the inlet value of 290 K which corresponds to about −9°, a variation which is of the same order as that reported in Figure 27. On the borders of the wake, total temperature 564 total temperature drop of 3% of the inlet value of 290 K which corresponds to about −9°, a variation 561 which is of the same order as that reported in Figure 27. On the borders of the wake, total temperature 565 which is of the same order as that reported in Figure 27. On the borders of the wake, total temperature peaks in excess of the inlet temperature are also recorded. The mass integrated total temperature 566 peaks in excess of the inlet temperature are also recorded. The mass integrated total temperature 562 peaks in excess of the inlet temperature are also recorded. The mass integrated total temperature value across the wake (denoted with a <  >) should be such that <T > /< T > = 1, but lack of 02 01 567 value across the wake (denoted with a <∗ ) should be such that < /< = 1 , but lack of 563 value across the wake (denoted with a <∗ ) should be such that < /< = 1 , but lack of information on the local velocity did not allow to perform this integration. 568 information on the local velocity did not allow to perform this integration. 564 information on the local velocity did not allow to perform this integration. (a) Steady total pressure (b) Steady total temperature (a) Steady total pressure (b) Steady total temperature (c) Unsteady total pressure (d) Unsteady total temperature Figure 28. Steady and time varying phase lock averaged total pressure and total temperature through 570 Figure 28. Steady and time varying phase lock averaged total pressure and total temperature through Commented [M49]: Please add explanation for subgraph. turbine vane wake at M = 0.79; adapted from [36]. 2,is 571 turbine vane wake at =0.79; adapted from [36]. Commented [MM50R49]: Done. Please do not cut the For the measurement of the time varying temperature a fast 2 m cold wire probe, developed by 572 For the measurement of the time varying temperature a fast 2 μm cold wire probe, developed figure on two pages. 573 by Denos and Sieverding [54], was used. Numerical compensation allowed to extend the naturally Denos and Sieverding [54], was used. Numerical compensation allowed to extend the naturally low 574 low frequency response of the probe to much higher ranges for adequate restitution of the nearly frequency response of the probe to much higher ranges for adequate restitution of the nearly sinusoidal 575 sinusoidal temperature variation associated with the vortex shedding frequency of 7.6 kHz at a temperature variation associated with the vortex shedding frequency of 7.6 kHz at a downstream 576 downstream isentropic Mach number of =0.79 . As regards the total pressure variation 577 ⁄ , minimum values of 0.768 are reached in the wake center while at the wake border maximum isentropic Mach number of M = 0.79. As regards the total pressure variation p /p , minimum 2,is 02 01 578 values of 1.061 are recorded. As regards the total temperature the authors quote maximum and values of 0.768 are reached in the wake center while at the wake border maximum values of 1.061 579 minimum total temperature ratios of ⁄ = 1.046 and 0.96, respectively. With a =290 are recorded. As regards the total temperature the authors quote maximum and minimum total 580 the maximum total temperature variations are of the order of 24°, similar to those reported by 581 Carscallen et al. [51]. However, the flow conditions were different: =0.79 at VKI, versus 0.95 582 at NRC Canada, and a distance of the wake traverses with respect to the trailing edge of 2.5 diameters 583 at VKI, versus 5.76 at NRC. 584 5. Effect of Vortex Shedding on Blade Pressure Distribution 585 The previous section focused on the unsteady character of turbine blade wake flows, the 586 visualization of the von Kármán vortices through smoke visualizations, schlieren photographs and 587 interferometric techniques. The measurement of the instantaneous velocity fields using LDV and PIV 588 techniques allowed to determine the vorticity distribution and the measurement of the unsteady total Int. J. Turbomach. Propuls. Power 2020, 5, 10 21 of 55 temperature ratios of T /T = 1.046 and 0.96, respectively. With a T = 290 K the maximum 02 01 01 total temperature variations are of the order of 24 , similar to those reported by Carscallen et al. [51]. However, the flow conditions were di erent: M = 0.79 at VKI, versus 0.95 at NRC Canada, and a 2,is distance of the wake traverses with respect to the trailing edge of 2.5 diameters at VKI, versus 5.76 at NRC. 5. E ect of Vortex Shedding on Blade Pressure Distribution The previous section focused on the unsteady character of turbine blade wake flows, the visualization of the von Kármán vortices through smoke visualizations, schlieren photographs and interferometric techniques. The measurement of the instantaneous velocity fields using LDV and PIV techniques allowed to determine the vorticity distribution and the measurement of the unsteady total pressure and temperature distribution putting into evidence the energy separation e ect in the wakes due to the von Kármán vortices. Naturally the vortex shedding a ects also the trailing edge pressure distribution and, beyond that, the suction side pressure distribution. The following is entirely based on research work carried out at the VKI by the team of the lead author, who was the only one to measure with high spatial resolution the pressure distribution around the trailing edge of a turbine blade. 5.1. E ect on Trailing Edge Pressure Distribution The very large-scale turbine guide vane designed and tested at the von Kármán Institute with a trailing edge thickness of 15 mm did allow an innovative approach for obtaining a high spatial resolution for the pressure distribution around the trailing edge. Cicatelli and Sieverding [32], fitted the blade with a rotatable 20 mm long cylinder in the center of the blade (Figure 29). The cylinder was equipped with a single Kulite fast response pressure sensor side by side with an ordinary pneumatic pressure tapping. The pressure sensor was mounted underneath the trailing edge surface with a slot width of only 0.2 mm to the outside, the same width as the pressure tapping, reducing the angular sensing area to only 1.53 . To control any e ect of the rear facing step between the blade lip and the rotatable trailing edge, a second blade was equipped with additional pressure sensors placed at, and slightly up-stream of, the trailing edge. The time averaged base pressure distribution, non-dimensionalized by the inlet total pressure, is presented in Figure 30. The circles denote data obtained with the rotatable trailing edge cylinder on blade A, while the triangles are measured with pressure tappings on blade B (see Figure 29), except for the two points “a” and “e” which are taken from the pressure tappings positioned aside the rotating cylinder on blade A (see Figure 30, left panel). The flow approaching the trailing edge undergoes, both on the pressure and suction side, a strong acceleration before separating from the trailing edge circle. The authors attribute the asymmetry to di erences in the blade boundary layers and to the blade circulation, which, following Han and Cox [26], strengthens the pressure side vortex shedding. Compared to the downstream Mach number M = 0.4, the local peak numbers are as high 2,is as M = 0.49 and 0.47, respectively. These high over-expansions are incompatible with a steady state max boundary layer separation and are attributed to the e ect of the vortex shedding. Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 23 of 58 601 The time averaged base pressure distribution, non-dimensionalized by the inlet total pressure, 602 is presented in Figure 30. The circles denote data obtained with the rotatable trailing edge cylinder 603 on blade A, while the triangles are measured with pressure tappings on blade B (see Figure 29), except 604 for the two points “a” and “e” which are taken from the pressure tappings positioned aside the 605 rotating cylinder on blade A (see Figure 30, left panel). The flow approaching the trailing edge 606 undergoes, both on the pressure and suction side, a strong acceleration before separating from the 607 trailing edge circle. The authors attribute the asymmetry to differences in the blade boundary layers 608 and to the blade circulation, which, following Han and Cox [26], strengthens the pressure side vortex 609 shedding. Compared to the downstream Mach number =0.4, the local peak numbers are as 610 high as =0.49 and 0.47, respectively. These high over-expansions are incompatible with a Int. J. Turbomach. Propuls. Power 2020, 5, 10 22 of 55 611 steady state boundary layer separation and are attributed to the effect of the vortex shedding. (b) (a) Figure 29. Blade instrumented with rotatable trailing edge (a: blade A) and control blade (b: blade B); 612 Figure 29. Blade instrumented with rotatable trailing edge (a: blade A) and control blade (b: blade B); adapted from [32]. 613 adapted from [32]. Figure 30, right panel, presents the corresponding root mean square of the pressure signal. 614 Figure 30, right panel, presents the corresponding root mean square of the pressure signal. Maximum pressure fluctuations of the order of 8% occur near the locations of the pressure minima, i.e., 615 Maximum pressure fluctuations of the order of 8% occur near the locations of the pressure minima, close to the boundary layer separation points, while the RMS/Q drops to nearly half this value in the 616 i.e. close to the boundary layer separation points, while the RMS/Q drops to nearly half this value in central region Int.of J. Tu the rbomach. trailing Propuls. Pow edge er 2018 base. , 3, x FOIt R Pis EERalso REVIEworth W noting that the pressure fluctuations 24 of 58 a ect also 617 the central region of the trailing edge base. It is also worth noting that the pressure fluctuations affect the flow upstream of the trailing edge. In the center of the base region there is an extended constant 618 also the flow upstream of the trailing edge. In the center of the base region there is an extended 624 also the flow upstream of the trailing edge. In the center of the base region there is an extended pressure plateau (Figure 30). The base pressure coecient corresponding to this plateau agrees well 625 constant pressure plateau (Figure 30). The base pressure coefficient corresponding to this plateau 619 constant pressure plateau (Figure 30). The base pressure coefficient corresponding to this plateau 626 agrees well with the Sieverding’s base pressure correlation. with the Sieverding’s base pressure correlation. 620 agrees well with the Sieverding’s base pressure correlation. (a) (b) 627 Figure 30. Phase lock averaged trailing edge pressure distribution (a) and root mean square values of Figure 30. Phase lock averaged trailing edge pressure distribution (a) and root mean square values of 628 the pressure fluctuations (b) for the VKI turbine blade at =0.4; adapted from [32]. the pressure fluctuations (b) for the VKI turbine blade at M = 0.4; adapted from [32]. 2,is 629 The base pressure distribution changes dramatically at high subsonic downstream Mach The base pressure distribution (a) changes dramatically at high subsonic (bdownstr ) eam Mach numbers 630 numbers as illustrated by Sieverding et al. [21], Figure 31. The pressure distribution is characterized 631 by the presence of three minima: the two pressure minima associated with the over-expansion of the as illustrated by Sieverding et al. [21], Figure 31. The pressure distribution is characterized by the 632 suction and pressure side flows before separation from the trailing edge, and an additional minimum presence of three minima: the two pressure minima associated with the over-expansion of the suction 633 around the center of the trailing edge circle. The pressure minima related to the overexpansion from and pr634 essur esuction a side flows nd pressu befor re sides e separation are of the ord from er of the ⁄ trailing =0.52 fo edge, r both sides, and an i.e. additional the local peak Ma minimum ch around 635 numbers are close to 1. Contrary to the low Mach number flow condition of Figure 30, the the center of the trailing edge circle. The pressure minima related to the overexpansion from suction 636 recompression following the over-expansion does not lead to a pressure plateau but gives way to a and pressure sides are of the order of p/p = 0.52 for both sides, i.e., The local peak Mach numbers are 637 new strong pressure drop reaching a minimum of ⁄ =0.485 at +7°. This is the result of the 638 enrolment of the separating shear layers into a vortex right at the trailing edge; the vortex core close to 1. Contrary to the low Mach number flow condition of Figure 30, the recompression following 639 approaches the wake centerline and its distance to the trailing edge becomes less than half the trailing the over-expansion does not lead to a pressure plateau but gives way to a new strong pressure drop 640 edge diameter, see smoke visualization and interferogram in Figure 17 and Figure 19. reaching a minimum of p/p = 0.485 at +7 . This is the result of the enrolment of the separating shear layers into a vortex right at the trailing edge; the vortex core approaches the wake centerline and its distance to the trailing edge becomes less than half the trailing edge diameter, see smoke visualization and interferogram in Figures 17 and 19. (a) (b) 641 Figure 31. Steady state trailing edge pressure distribution (a) and phase locked average pressure fluctuation (b) 642 around trailing edge; adapted from [21]. 643 The phase lock averaged pressure fluctuations around the trailing edge in Figure 31 are 644 impressive. They are naturally highest near the separation points of the boundary layer from the Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 24 of 58 624 also the flow upstream of the trailing edge. In the center of the base region there is an extended 625 constant pressure plateau (Figure 30). The base pressure coefficient corresponding to this plateau 626 agrees well with the Sieverding’s base pressure correlation. (a) (b) 627 Figure 30. Phase lock averaged trailing edge pressure distribution (a) and root mean square values of 628 the pressure fluctuations (b) for the VKI turbine blade at =0.4; adapted from [32]. 629 The base pressure distribution changes dramatically at high subsonic downstream Mach 630 numbers as illustrated by Sieverding et al. [21], Figure 31. The pressure distribution is characterized 631 by the presence of three minima: the two pressure minima associated with the over-expansion of the 632 suction and pressure side flows before separation from the trailing edge, and an additional minimum 633 around the center of the trailing edge circle. The pressure minima related to the overexpansion from 634 suction and pressure sides are of the order of ⁄ =0.52 for both sides, i.e. the local peak Mach 635 numbers are close to 1. Contrary to the low Mach number flow condition of Figure 30, the 636 recompression following the over-expansion does not lead to a pressure plateau but gives way to a 637 new strong pressure drop reaching a minimum of ⁄ =0.485 at +7°. This is the result of the 638 enrolment of the separating shear layers into a vortex right at the trailing edge; the vortex core 639 approaches the wake centerline and its distance to the trailing edge becomes less than half the trailing Int. J. Turbomach. Propuls. Power 2020, 5, 10 23 of 55 640 edge diameter, see smoke visualization and interferogram in Figure 17 and Figure 19. (a) (b) 641 Figure 31. Steady state trailing edge pressure distribution (a) and phase locked average pressure fluctuation (b) Figure 31. Steady state trailing edge pressure distribution (a) and phase locked average pressure 642 around trailing edge; adapted from [21]. fluctuation (b) around trailing edge; adapted from [21]. 643 The phase lock averaged pressure fluctuations around the trailing edge in Figure 31 are 644 impressive. They are naturally highest near the separation points of the boundary layer from the The phase lock averaged pressure fluctuations around the trailing edge in Figure 31 are impressive. They are naturally highest near the separation points of the boundary layer from the trailing edge where maximum values of around 100% of the dynamic pressure (p p ) are recorded. The minimum 01 2 pressure in a given position corresponds to the maximum inward motion of the separating shear layer, the maximum pressure to the maximum outward motion of the separating shear layer. The maximum local instantaneous Mach number at the point of the most inward position can be as high as M = 1.25. max The authors assumed that the curvature driven supersonic trailing edge expansion is the real reason for the formation of the vortex so close to the trailing edge, with the entrainment of high-speed free stream fluid into the trailing edge base region. In the center of the trailing edge the fluctuations drop to 20% of the dynamic head. The authors provide also some interesting information on the evolution of the pressure signal on the trailing edge circle over one complete vortex shedding cycle. This is demonstrated in Figure 32 showing the evolution for the phase locked average pressure at the angular position of 60 on the pressure side of the trailing edge circle. A decrease of the pressure indicates an acceleration of the flow around the trailing edge i.e., The separating shear layer moves inwards, the vortex is in its formation phase. An increase of the pressure indicates on the contrary an outwards motion of the shear layer, the vortex is in its shedding phase. Surprisingly, the pressure rise time is much shorter than the pressure fall time, i.e., The time for the vortex formation is longer than that for the vortex shedding. The same was observed for the pressure evolution on the opposite side of the trailing edge, but of Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 25 of 58 course with 180 out of phase. Figure 32. Phase locked average pressure variation at trailing edge at an angular position of60 [21]. 657 Figure 32. Phase locked average pressure variation at trailing edge at an angular position of −60° [21]. The change of an isobaric pressure zone over an extended region at the base of the trailing edge at 658 The change of an isobaric pressure zone over an extended region at the base of the trailing edge an exit Mach number M = 0.4 to a highly non-uniform pressure distribution with a strong pressure 2,is 659 at an exit Mach number =0.4 to a highly non-uniform pressure distribution with a strong 660 pressure minimum at the center of the trailing edge circle at =0.79, did of course raise the 661 question about the evolution of the trailing edge pressure distribution over the entire Mach number 662 range, from low subsonic to transonic Mach numbers. To respond to this lack of information a 663 research program was carried out by Mateos Prieto [55] at VKI as part of his diploma thesis in 2003. 664 For various reasons, the data were not published at that time but only in 2015, as part of the paper of 665 Vagnoli et al. [56] on the prediction of unsteady turbine blade wake flow characteristics and 666 comparison with experimental data, see Figure 33. 667 Figure 33. Effect of downstream Mach number on trailing edge Mach number distribution [56]. 668 The figure puts clearly into evidence the effect of the vortex shedding on the trailing edge 669 pressure distribution. Up to about ≈0.65 the trailing edge base region is characterized by an 670 extended, nearly isobaric, pressure plateau which implies that the vortex formation occurs 671 sufficiently far downstream not to affect the trailing edge base region. With increasing Mach number, 672 the enrolment of the shear layers into vortices occurs gradually closer to the trailing edge causing an 673 increasingly non-uniform pressure distribution with a marked pressure minimum at the center of the 674 trailing edge. To characterize the degree of non-uniformity the authors define a factor : Int. J. Turbomach. Propuls. Power 2020, 5, 10 24 of 55 minimum Int. J. Turbomat achthe . Prop center uls. Power of the 2018trailing , 3, x FOR PE edge ER RE circle VIEW at M = 0.79, did of course raise the question 25 of about 58 2,is the evolution of the trailing edge pressure distribution over the entire Mach number range, from low subsonic to transonic Mach numbers. To respond to this lack of information a research program was carried out by Mateos Prieto [55] at VKI as part of his diploma thesis in 2003. For various reasons, the data were not published at that time but only in 2015, as part of the paper of Vagnoli et al. [56] on the prediction of unsteady turbine blade wake flow characteristics and comparison with experimental data, see Figure 33. The figure puts clearly into evidence the e ect of the vortex shedding on the trailing edge pressure distribution. Up to about M  0.65 the trailing edge base region is characterized by an extended, 2,is nearly isobaric, pressure plateau which implies that the vortex formation occurs suciently far downstream not to a ect the trailing edge base region. With increasing Mach number, the enrolment of the shear layers into vortices occurs gradually closer to the trailing edge causing an increasingly non-uniform pressure distribution with a marked pressure minimum at the center of the trailing edge. To characterize the degree of non-uniformity the authors define a factor Z: Z = p p /(p p ) (5) b,max b,min 01 2 657 Figure 32. Phase locked average pressure variation at trailing edge at an angular position of −60° [21]. where p is the maximum pressure following the recompression after the separation of the shear b,max 658 The change of an isobaric pressure zone over an extended region at the base of the trailing edge layer from the trailing edge and p the minimum pressure near the center of the trailing edge. b,min 659 at an exit Mach number =0.4 to a highly non-uniform pressure distribution with a strong The maximum degree of non-uniformity is reached at M = 0.93 with a Z value of 21%. At this 2,is 660 pressure minimum at the center of the trailing edge circle at =0.79, did of course raise the Mach number the minimum pressure reaches a value of p /p = 0.325 for a downstream pressure b,min 01 661 question about the evolution of the trailing edge pressure distribution over the entire Mach number p /p = 0.572. With further increase of the Mach number, Z starts to decrease rapidly. It decreases to 2 01 662 range, from low subsonic to transonic Mach numbers. To respond to this lack of information a Z = 12% at M = 0.99 and drops to zero at M = 1.01. For this Mach number the local trailing edge 2,is 2,is 663 research program was carried out by Mateos Prieto [55] at VKI as part of his diploma thesis in 2003. conditions are such that oblique shocks emerge from the region of the confluence of the suction and 664 For various reasons, the data were not published at that time but only in 2015, as part of the paper of pressure side shear layers and the vortex formation is delayed to after this region as shown e.g. In the 665 Vagnoli et al. [56] on the prediction of unsteady turbine blade wake flow characteristics and schlieren picture by Carscallen and Gostelow [39] at the NRC Canada in Figure 34, left, and another 666 comparison with experimental data, see Figure 33. schlieren picture taken at VKI in Figure 34, right (unpublished). Figure 33. E ect of downstream Mach number on trailing edge Mach number distribution [56]. 667 Figure 33. Effect of downstream Mach number on trailing edge Mach number distribution [56]. 668 The figure puts clearly into evidence the effect of the vortex shedding on the trailing edge 669 pressure distribution. Up to about ≈0.65 the trailing edge base region is characterized by an 670 extended, nearly isobaric, pressure plateau which implies that the vortex formation occurs 671 sufficiently far downstream not to affect the trailing edge base region. With increasing Mach number, 672 the enrolment of the shear layers into vortices occurs gradually closer to the trailing edge causing an 673 increasingly non-uniform pressure distribution with a marked pressure minimum at the center of the 674 trailing edge. To characterize the degree of non-uniformity the authors define a factor : Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 26 of 58 = ( − )/( − ) (5) , , 675 where is the maximum pressure following the recompression after the separation of the shear 676 layer from the trailing edge and the minimum pressure near the center of the trailing edge. 677 The maximum degree of non-uniformity is reached at =0.93 with a value of 21%. At this 678 Mach number the minimum pressure reaches a value of ⁄ = 0.325 for a downstream 679 pressure ⁄ = 0.572. With further increase of the Mach number, starts to decrease rapidly. It 680 decreases to = 12% at =0.99 and drops to zero at =1.01. For this Mach number the , , 681 local trailing edge conditions are such that oblique shocks emerge from the region of the confluence 682 of the suction and pressure side shear layers and the vortex formation is delayed to after this region 683 as shown e.g. in the schlieren picture by Carscallen and Gostelow [39] at the NRC Canada in Figure Int. J. Turbomach. Propuls. Power 2020, 5, 10 25 of 55 684 34, left, and another schlieren picture taken at VKI in Figure 34, right (unpublished). (a) (b) Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 27 of 58 Figure 34. At fully established oblique trailing edge shock system the vortex formation is delayed to 685 Figure 34. At fully established oblique trailing edge shock system the vortex formation is delayed to 693 As already pointed out at the end of Section 2 the departure from the generally assumed isobaric after the point of confluence of the blade shear layers. NRC blade (a) [39], VKI blade (b). 694 trailing edge base region may explain the differences of base pressure data published by different 686 after the point of confluence of the blade shear layers. NRC blade (a) [39], VKI blade (b). 695 authors at high subsonic/transonic downstream Mach numbers. The scatter between experimental 696 data from different research organizations may be partially due to the use of very different ratios of As already pointed out at the end of Section 2 the departure from the generally assumed isobaric 687 As already pointed out at the end of Section 2 the departure from the generally assumed isobaric 697 the diameter of the trailing edge pressure hole to the trailing edge diameter, / . Small / ratios trailing edge base region may explain the di erences of base pressure data published by di erent 688 trailing 698 edge m base region may ay lead to an overestimat exp ionl of ain thethe differenc base pressure effe ect s of base pr . Hence, base pessu ressure re d meaa sta published by urements should different authors at high subsonic/transonic downstream Mach numbers. The scatter between experimental 699 be taken with a / ratio as large as possible. 689 authors at high subsonic/transonic downstream Mach numbers. The scatter between experimental 700 The existence of an isobaric base pressure region for supersonic trailing edge flows, i.e. for blades data from di erent research organizations may be partially due to the use of very di erent ratios of the 690 data from different research organizations may be partially due to the use of very different ratios of 701 with a well-established oblique trailing edge shock system as those in Figure 34, was already known diameter of the trailing edge pressure hole to the trailing edge diameter, d/D. Small d/D ratios may 702 from flat plate model tests simulating the overhang section of convergent turbine blades with straight 691 the diameter of the trailing edge pressure hole to the trailing edge diameter, / . Small / ratios lead to703 an over reaestimation r suction side sin ofcthe e 1976 base [18], see pressur Figure e 3e 5 . Th ect. e teHence, sts were per base formed f pressur or a ga eumeasur ging anglements e = should be 692 may lead to an overestimation of the base pressure effect. Hence, base pressure measurements should 704 30°. The inclination of the tail board attached to the lower nozzle block allows to increase the taken with a d/D ratio as large as possible. 693 be taken with a / ratio as large as possible. 705 downstream Mach number which entails of course the displacement of the suction side shock The existence of an isobaric base pressure region for supersonic trailing edge flows, i.e., for blades 694 The existence 706 boundaof an isobar ry interaction alo ic nbase pressur g the blade suctie on region side towa fo rds th r sue personic t trailing edgr ea . i ling edge flows, i.e. for blades with a well-established oblique trailing edge shock system as those in Figure 34, was already known 707 The schlieren photograph on the right in Figure 34 shows the occurrence of so-called lip shocks 695 with a well-established oblique trailing edge shock system as those in Figure 34, was already known 708 at the separation of the shear layers from the trailing edge due to a slight overturning and a non- from flat plate model tests simulating the overhang section of convergent turbine blades with straight 696 from flat plate model tests simulating the overhang section of convergent turbine blades with straight 709 tangential separation of the flow from the trailing edge surface. In a later test series with a denser rear suction side since 1976 [18], see Figure 35. The tests were performed for a gauging angle = 30. 710 instrumentation of the trailing edge, Sieverding et al. [4] showed that the trailing edge shock strength 2 697 rear suction side since 1976 [18], see Figure 35. The tests were performed for a gauging angle = The inclination 711 was h ofothe wevtail er weak. boar In d Fattached igure 36 the pressure i to the lower ncrease nozzle ⁄ block across th allows e lip shto ock incr is presented ease the in downstream 698 30°. The inclination of the tail board attached to the lower nozzle block allows to increase the 712 function of the expansion ratio around the trailing edge ⁄ , where is the pressure before the Mach number which entails of course the displacement of the suction side shock boundary interaction 699 downstream Mach number which entails of course the displacement of the suction side shock 713 start of the expansion around the trailing edge and the pressure before the lip shock. All data are along the blade suction side towards the trailing edge. 714 within a bandwidth of ⁄ = 1.1 − 1.2. 700 boundary interaction along the blade suction side towards the trailing edge. 701 The schlieren photograph on the right in Figure 34 shows the occurrence of so-called lip shocks 702 at the separation of the shear layers from the trailing edge due to a slight overturning and a non- 703 tangential separation of the flow from the trailing edge surface. In a later test series with a denser 704 instrumentation of the trailing edge, Sieverding et al. [4] showed that the trailing edge shock strength 705 was however weak. In Figure 36 the pressure increase across the lip shock is presented in 706 function of the expansion ratio around the trailing edge ⁄ , where is the pressure before the 707 start of the expansion around the trailing edge and the pressure before the lip shock. All data are 708 within a bandwidth of ⁄ = 1.1 − 1.2. (a) (b) 715 Figure 35. Surface (a) and trailing edge (b) pressure distribution for flat plate model tests; adapted Figure 35. Surface (a) and trailing edge (b) pressure distribution for flat plate model tests; adapted Commented [M57]: Please add the add explanation for 716 from [18]. from [18]. subgraph(a, left) Commented [MM58R57]: Done. The schlieren photograph on the right in Figure 34 shows the occurrence of so-called lip shocks at the separation of the shear layers from the trailing edge due to a slight overturning and a non-tangential separation of the flow from the trailing edge surface. In a later test series with a denser instrumentation of the trailing edge, Sieverding et al. [4] showed that the trailing edge shock strength was however weak. In Figure 36 the pressure increase p /p across the lip shock is presented in function of the 3 2 expansion ratio around the trailing edge p /p , where p is the pressure before the start of the expansion 2 1 1 717 Figure 36. Strength of the trailing edge lip shocks; adapted from [4]. around the trailing edge and p the pressure before the lip shock. All data are within a bandwidth of 2 Commented [M59]: Please add explanation for subgraph. p /p = 1.1–1.2. 3 2 718 Raffel and Kost [57] performed similar large-scale tests on a flat plate with a slotted trailing edge Commented [MM60R59]: The graph and the sketch have to 719 to investigate the effect of trailing edge blowing on the formation of the trailing edge vortex street. be considered as a single entity. 720 Their trailing edge pressure distributions taken at zero coolant flow ejection for downstream Mach Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 27 of 58 693 As already pointed out at the end of Section 2 the departure from the generally assumed isobaric 694 trailing edge base region may explain the differences of base pressure data published by different 695 authors at high subsonic/transonic downstream Mach numbers. The scatter between experimental 696 data from different research organizations may be partially due to the use of very different ratios of 697 the diameter of the trailing edge pressure hole to the trailing edge diameter, / . Small / ratios 698 may lead to an overestimation of the base pressure effect. Hence, base pressure measurements should 699 be taken with a / ratio as large as possible. 700 The existence of an isobaric base pressure region for supersonic trailing edge flows, i.e. for blades 701 with a well-established oblique trailing edge shock system as those in Figure 34, was already known 702 from flat plate model tests simulating the overhang section of convergent turbine blades with straight 703 rear suction side since 1976 [18], see Figure 35. The tests were performed for a gauging angle = 704 30°. The inclination of the tail board attached to the lower nozzle block allows to increase the 705 downstream Mach number which entails of course the displacement of the suction side shock 706 boundary interaction along the blade suction side towards the trailing edge. 707 The schlieren photograph on the right in Figure 34 shows the occurrence of so-called lip shocks 708 at the separation of the shear layers from the trailing edge due to a slight overturning and a non- 709 tangential separation of the flow from the trailing edge surface. In a later test series with a denser 710 instrumentation of the trailing edge, Sieverding et al. [4] showed that the trailing edge shock strength 711 was however weak. In Figure 36 the pressure increase ⁄ across the lip shock is presented in 712 function of the expansion ratio around the trailing edge ⁄ , where is the pressure before the 713 start of the expansion around the trailing edge and the pressure before the lip shock. All data are 714 within a bandwidth of ⁄ = 1.1 − 1.2. (a) (b) 715 Figure 35. Surface (a) and trailing edge (b) pressure distribution for flat plate model tests; adapted Int. J. Turbomach. Propuls. Power 2020, 5, 10 26 of 55 Commented [M57]: Please add the add explanation for 716 from [18]. subgraph(a, left) Commented [MM58R57]: Done. 717 FiguFigure re 36. Stre 36. ngthStr of th ength e trailin of g ed the ge trailing lip shocks; adapt edge elip d fro shocks; m [4]. adapted from [4]. Commented [M59]: Please add explanation for subgraph. 718 Raffel and Kost [57] performed similar large-scale tests on a flat plate with a slotted trailing edge Commented [MM60R59]: The graph and the sketch have to Ra el and Kost [57] performed similar large-scale tests on a flat plate with a slotted trailing edge to 719 to investigate the effect of trailing edge blowing on the formation of the trailing edge vortex street. be considered as a single entity. investigate the e ect of trailing edge blowing on the formation of the trailing edge vortex street. Their 720 Their trailing edge pressure distributions taken at zero coolant flow ejection for downstream Mach trailing edge pressure distributions taken at zero coolant flow ejection for downstream Mach numbers of M = 1.01 1.45 confirm the existence of an isobaric pressure trailing edge base region, but the 2,is measurements are unfortunately not dense enough to extract consistent data about the strength of the lip shock. A few data allow to conclude that in their experiments the maximum lip shock strength is of the order of p /p = 1.08. 3 2 Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 28 of 58 721 numbers of = 1.01 − 1.45 confirm the existence of an isobaric pressure trailing edge base 5.2. E ect on Blade Suction Side Pressure Distribution 722 region, but the measurements are unfortunately not dense enough to extract consistent data about In 723 the discussion the strength of ofthe the lip schlier shock. en A fephotographs w data allow to cin oncl Figur ude tha e 18 t in their it was experi shown mentsthat the m the axim outwar um ds motion 724 lip shock strength is of the order of / =1.08. of the oscillating shear layers at the blade trailing edge does not only lead to large pressure fluctuations in the zone of separations, but it does also induce strong acoustic pressure waves travelling upstream 725 5.2. Effect on Blade Suction Side Pressure Distribution on both the suction and pressure side of the blade. To facilitate the understanding of the suction side 726 In the discussion of the schlieren photographs in Figure 18 it was shown that the outwards 727 motion of the oscillating shear layers at the blade trailing edge does not only lead to large pressure pressure fluctuations in Figure 37, the left photo of the schlieren pictures in Figure 18 is reproduced 728 fluctuations in the zone of separations, but it does also induce strong acoustic pressure waves at the right of the pressure curves. The wave Pi generated at the pressure side will interact with the 729 travelling upstream on both the suction and pressure side of the blade. To facilitate the understanding suction side of the neighboring blade causing significant unsteady pressure variations as measured by 730 of the suction side pressure fluctuations in Figure 37, the left photo of the schlieren pictures in Figure fast response 731 18 i pr s repro essur duced at e sensors the righ implemented t of the pressure curves between . The w the ave thr Pi ge oat nerated andat t the he p trailing ressure side w edge ill of this blade, 732 interact with the suction side of the neighboring blade causing significant unsteady pressure see Figure 37. The pressure wave P induced by the outwards motion of the pressure side shear 733 variations as measured by fast response pressure sensors implemented between the throat and the layer of the neighboring blade intersects the suction side between the sensors 3 and 4. It moves then 734 trailing edge of this blade, see Figure 37. The pressure wave induced by the outwards motion of 735 the pressure side shear layer of the neighboring blade intersects the suction side between the sensors successively upstream across the sensors 3 and 2. The signals are asymmetric, characterized by a sharp 736 3 and 4. It moves then successively upstream across the sensors 3 and 2. The signals are asymmetric, pressure rise followed by a slow decay. The amplitude of the pressure fluctuations is important with 737 characterized by a sharp pressure rise followed by a slow decay. The amplitude of the pressure Dp = 12% up to 15% of (p p ) at sensor 3, and10% at sensor 2, while the pressure signal is flat at 738 fluctuations is impo01 rtant with Δ = 12 % up to 15% of ( − ) at sensor 3, and 10% at sensor 739 2, while the pressure signal is flat at sensor 1 situated slightly up-stream of the geometric throat where sensor 1 situated slightly up-stream of the geometric throat where the blade Mach number reaches 740 the blade Mach number reaches =0.95. The pressure waves observed at sensor 4 and further M = 0.95. The pressure waves observed at sensor 4 and further downstream at sensors 5 and 6 are 2,is 741 downstream at sensors 5 and 6 are more sinusoidal in nature and of smaller amplitude. The authors more sinusoidal in nature and of smaller amplitude. The authors suggested that these fluctuations are 742 suggested that these fluctuations are likely to be caused by the downstream travelling vortices of the 743 neighboring blade. likely to be caused by the downstream travelling vortices of the neighboring blade. (b) (a) (c) 745 Figure 37. Unsteady pressure variations along rear suction side (a); schlieren photograph (b); kulite Commented [M61]: Please add explanation for subgraph. Figure 37. Unsteady pressure variations along rear suction side (a); schlieren photograph (b); kulite 746 sensor positioning (c). Adapted from [21]. sensor positioning (c). Adapted from [21]. Commented [MM62R61]: Done. 747 The periodicity of the pressure signal at position 7, slightly upstream of the trailing edge, is 748 rather poor and only phase lock averaging provides useful information on its periodic character. The 749 reason is most likely the result of a superposition of waves induced by the von Kármán vortices in 750 the wake of the neighboring blade and upstream travelling waves induced by the oscillation of the 751 suction side shear layer designated by “S” in the schlieren photographs. Right at the trailing edge, 752 position 11, we have, as expected, strong periodic signals associated with the oscillating shear layers. Int. J. Turbomach. Propuls. Power 2020, 5, 10 27 of 55 The periodicity of the pressure signal at position 7, slightly upstream of the trailing edge, is rather poor and only phase lock averaging provides useful information on its periodic character. The reason is most likely the result of a superposition of waves induced by the von Kármán vortices in the wake of the neighboring blade and upstream travelling waves induced by the oscillation of the suction side shear layer designated by “S” in the schlieren photographs. Right at the trailing edge, position 11, we have, as expected, strong periodic signals associated with the oscillating shear layers. 6. Turbine Trailing Edge Vortex Frequency Shedding Besides the importance of trailing edge vortex shedding for the wake mixing process and the trailing edge pressure distribution discussed before, vortex shedding deserves also special attention due to its importance as excitation for acoustic resonances and structural vibrations. Heinemann & Bütefisch [25], investigated 10 subsonic and transonic turbine cascades: two flat plate turbine tip sections, three mid-sections with nearly axial inlet (one blade tested with three di erent trailing edge thicknesses) and 3 high turning hub sections. The trailing edge thickness varied from 0.8% to 5%. The vortex shedding frequency was determined with an electronic-optical method developed at the DFVLR-AVA by Heinemann et al. [58]. The corresponding Strouhal numbers defined in (3) covered a wide range: 0.2  St  0.4 for a Reynolds number range based on trailing edge thickness and downstream isentropic velocity of 4 5 0.3 10  Re  1.6 10 . The Strouhal numbers for flows from cylinders over the same Reynolds Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 29 of 58 number range are of the order of St = 0.19 and 0.21 as shown in Figure 38. Figure 38. Strouhal numbers in sub-critical Reynolds number range for flow over cylinders; adapted 761 Figure 38. Strouhal numbers in sub-critical Reynolds number range for flow over cylinders; adapted from [59]. 762 from [59]. Figure 39 presents the Strouhal numbers for 3 of the 10 turbine blade sections investigated by 763 Figure 39 presents the Strouhal numbers for 3 of the 10 turbine blade sections investigated by Heinemann and Bütefisch [25]. The comparison with the flow across cylinders is limited to the subsonic 764 Heinemann and Bütefisch [25]. The comparison with the flow across cylinders is limited to the range. The Strouhal numbers for the flat plate tip section T2 are of the order of St = 0.2 in the Mach 765 subsonic range. The Strouhal numbers for the flat plate tip section T2 are of the order of = 0.2 in range M = 0.2 to 0.8 and therewith very close to the those of the flow over cylinders. On the other 2,is 766 the Mach range = 0.2 to 0.8 and therewith very close to the those of the flow over cylinders. On side, the hub section H2 with a high rear suction side curvature distinguishes itself by Strouhal numbers 767 the other side, the hub section H2 with a high rear suction side curvature distinguishes itself by as high as 0.38 0.3, with a decreasing tendency from M = 0.2 to 0.9. For the mid-section M2 with 2,is 768 Strouhal numbers as high as 0.38 – 0.3, with a decreasing tendency from = 0.2 to 0.9. For the low rear suction side turning, the authors report Strouhal numbers increasing from St = 0.22 to 0.29 for 769 mid-section M2 with low rear suction side turning, the authors report Strouhal numbers increasing a Mach range 0.2 to 0.8. 770 from = 0.22 to 0.29 for a Mach range 0.2 to 0.8. (a) steam turbine tip section T2 (b) steam turbine mid section M2 (c) gas turbine hub section H2 771 Figure 39. Strouhal number for 3 blade sections. : and based on isentropic downstream 772 velocity. : and based on homogeneous downstream velocity. Adapted from [25]. 773 Additional information on turbine blade trailing edge frequency measurements were published 774 by Sieverding [60] who used fast response pressure sensors implemented in the blade trailing edge 775 and in a total pressure probe positioned at short distance from the trailing edge, while Bryanston- 776 Cross and Camus [61] made use of a 20 MHz bandwidth digital correlator combined with 777 conventional schlieren optics. The Strouhal numbers of Sieverding’s rotor blade with a straight rear 778 suction side were in the lower part of the band width of the DFVLR-AVA data, while those of 779 Bryanston-Cross and Camus rotor blades with higher suction side curvature resided in the upper 780 part. 781 The large range of Strouhal numbers were possibly due to differences in the state of the 782 boundary layers at the point of separation. Besides that, the vortex shedding frequency does not 783 simply depend on the trailing edge thickness augmented by the boundary layer displacement Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 10 of 58 271 suction side for both blades are exactly the same, but contrary to the nearly constant Mach number 272 for the VKI blade downstream of the throat, the blade of Xu and Denton is characterized by a very 273 strong adverse pressure gradient in this region. As pointed out by the authors, this causes the suction 274 side boundary layer to be either separated or close to separation up-stream of the trailing edge. 275 Clearly, Sieverding’s correlation cannot deal with blade designs characterized by very strong adverse 276 pressure gradients on the rear suction side causing boundary layer separation before the trailing 277 edge. Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 29 of 58 Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 29 of 58 753 6. Turbine Trailing Edge Vortex Frequency Shedding 754 Besides the importance of trailing edge vortex shedding for the wake mixing process and the 755 trailing edge pressure distribution discussed before, vortex shedding deserves also special attention 756 due to its importance as excitation for acoustic resonances and structural vibrations. Heinemann & 757 Bütefisch [25], investigated 10 subsonic and transonic turbine cascades: two flat plate turbine tip 758 sections, three mid-sections with nearly axial inlet (one blade tested with three different trailing edge 759 thicknesses) and 3 high turning hub sections. The trailing edge thickness varied from 0.8% to 5%. The 760 vortex shedding frequency was determined with an electronic-optical method developed at the 761 DFVLR-AVA by Heinemann et al. [58]. 278 Figure 9. Comparison of blade Mach number distribution for blade RD of Xu and Denton [15], (solid 279 curve, =0.8, =8 10 ) with VKI blade (dashed curve, =0.8, =10 ). , , 280 The possible effect of boundary layer separation resulting from high rear suction side diffusion 281 resulting in high base pressures was also mentioned by Corriveau and Sjolander in 2004 [20], 282 comparing their nominal mid-loaded rotor blade HS1A, mentioned already before, with an aft- 761 Figure 38. Strouhal numbers in sub-critical Reynolds number range for flow over cylinders; adapted 283 loaded blade HS1C with an increase of the suction side unguided turning angle from 11.5° to 14.5°. 762 from [59]. 284 It appears that the increased turning angle could cause, in the transonic range, shock induced 285 boundary layer transition near the trailing edge with, as consequence, a sharp increase of the base 763 Figure 39 presents the Strouhal numbers for 3 of the 10 turbine blade sections investigated by 286 pressure, i.e. a sudden drop in the base pressure coefficient as seen in Figure 10. Note that the 764 Heinem 762 ann and B Figure 38. ütefi Str sc oh [2 uhal number 5]. The c s in sub-cr omparison w itical Reynolds num ith b the f er range for low flow a ov cross cyli er cylindersnders is li ; adapted mited to the 287 reported in the figure has been converted to − of the original data. 763 from [59]. 765 subsonic range. The Strouhal numbers for the flat plate tip section T2 are of the order of = 0.2 in 288 As regards the base pressure data by Deckers and Denton [13] for a low turning blade model 766 the Mach range = 0.2 to 0.8 and therewith very close to the those of the flow over cylinders. On 764 The co , rresponding Strouhal numbers defined in (3) covered a wide range: 0.2 0.4 for a 289 and Gostelow et al. [14] for a high turning nozzle guide vane, who report base pressure data far below 765 Reynolds number range based on trailing edge thickness and downstream isentropic velocity of 767 the other side, the hub section H2 with a high rear suction side curvature distinguishes itself by 766 0.3 10 1.6 10 . The Strouhal numbers for flows from cylinders over the same Reynolds 290 those of Sieverding’s BPC, their blade pressure distribution resembles that of the convergent/ 768 Strouhal numbers as high as 0.38 – 0.3, with a decreasing tendency from = 0.2 to 0.9. For the 767 number range are of the order of = 0.19 and 0.21 as shown in Figure 38. 291 divergent blade C in Figure 3 with a negative blade loading near the trailing edge which would 769 mid-section M2 with low rear suction side turning, the authors report Strouhal numbers increasing 768 Figure 39 presents the Strouhal numbers for 3 of the 10 turbine blade sections investigated by 292 explain the very low base pressures. In addition, the blade of Deckers and Denton has a blunt trailing 769 Heinemann and Bütefisch [25]. The comparison with the flow across cylinders is limited to the 770 from = 0.22 to 0.29 for a Mach range 0.2 to 0.8. 770 subsonic range. The Strouhal numbers for the flat plate tip section T2 are of the order of = 0.2 in 293 edge, and there is experimental evidence that, compared to a circular trailing edge, the base pressure 771 the Mach range = 0.2 to 0.8 and therewith very close to the those of the flow over cylinders. On 294 for blades with blunt trailing edge might be considerably lower. Sieverding and Heinemann [16] 772 the other side, the hub section H2 with a high rear suction side curvature distinguishes itself by 295 report for flat plate tests at moderate subsonic Mach numbers a drop of the base pressure coefficient 773 Strouhal numbers as high as 0.38 – 0.3, with a decreasing tendency from = 0.2 to 0.9. For the Int. J. Turbomach. Propuls. Power 2020, 5, 10 28 of 55 774 mid-section M2 with low rear suction side turning, the authors report Strouhal numbers increasing 296 by 11% for a plate with squared trailing edge compared to that with a circular trailing edge. 775 from = 0.22 to 0.29 for a Mach range 0.2 to 0.8. (a) steam turbine tip section T2 (b) steam turbine mid section M2 (c) gas turbine hub section H2 (a) steam turbine tip section T2 (b) steam turbine mid section M2 (c) gas turbine hub section H2 776 Figure 39. Strouhal number for 3 blade sections. : and based on isentropic downstream 771 Figure Figure 39. 39. Str Stro ouhal uhal number for 3 blade sect number for 3 blade sections. ions. : : St and and M based based on i on isentr sentr opic opic downstr downstream eam 297 Figure 10. Base pressure coefficient for mid-loaded (solid line) and aft2 -loaded (dashed line) rotor 777 velocity. : and based on homogeneous downstream velocity. Adapted from [25]. 772 velocity velocity .. : : St and and M based based on on homogeneous homogeneou downstr s downstream eam velocity velocity . Adapted . Adaptefr d om from [25 [25]. ]. 298 blade. Symbols: HS1A geom etry, HS1C geometry. Adapted from [20]. Additional information on turbine blade trailing edge frequency measurements were published 773 Additional information on turbine blade trailing edge frequency measurements were published by Sieverding [60] who used fast response pressure sensors implemented in the blade trailing edge and 774 by Sieverding [60] who used fast response pressure sensors implemented in the blade trailing edge in a total pressure probe positioned at short distance from the trailing edge, while Bryanston-Cross and 775 and in a total pressure probe positioned at short distance from the trailing edge, while Bryanston- Camus [61] made use of a 20 MHz bandwidth digital correlator combined with conventional schlieren 776 Cross and Camus [61] made use of a 20 MHz bandwidth digital correlator combined with optics. The Strouhal numbers of Sieverding’s rotor blade with a straight rear suction side were in the 777 conventional schlieren optics. The Strouhal numbers of Sieverding’s rotor blade with a straight rear lower part of the band width of the DFVLR-AVA data, while those of Bryanston-Cross and Camus 778 suction side were in the lower part of the band width of the DFVLR-AVA data, while those of rotor blades with higher suction side curvature resided in the upper part. 779 Bryanston-Cross and Camus rotor blades with higher suction side curvature resided in the upper The large range of Strouhal numbers were possibly due to di erences in the state of the boundary 780 part. layers at the point of separation. Besides that, the vortex shedding frequency does not simply depend 781 The large range of Strouhal numbers were possibly due to differences in the state of the on the trailing edge thickness augmented by the boundary layer displacement thickness, which, 782 boundary layers at the point of separation. Besides that, the vortex shedding frequency does not however, is in general not known, but rather on the e ective distance between the separating shear 783 simply depend on the trailing edge thickness augmented by the boundary layer displacement layers which could be significantly smaller than the trailing edge thickness. Patterson & Weingold [62], simulating a compressor airfoil trailing edge flow field on a flat plate, concluded that, compared to the e ective distance between the separating upper and lower shear layers, the state of the boundary layer before separation played a much more important role. The influence of the boundary layer state and of the e ective distance of the separating shear layers was specifically addressed in a series of cascade and flat plate tests investigated by Sieverding & Heinemann [16], at VKI and DLR. Figure 40 shows the blade surface isentropic Mach number distributions of a front loaded blade, with the particularity of a straight rear suction side (blade A), and a rear loaded blade (blade C), characterized by a high rear suction side turning angle, at a downstream Mach number of M  0.8. 2is Figure 40. Blade Mach number distributions for front and rear-loaded blades; adapted from [16]. Figure 40. Blade Mach number distributions for front and rear-loaded blades; adapted from [16]. The early suction side velocity peak on blade A will cause early boundary layer transition. On the contrary, considering the weak velocity peak on the rear suction side followed by a very moderate recompression, the suction side boundary layer of blade C is likely to be laminar at the trailing edge over a large range of Reynolds numbers. As regards the pressure sides of both blades, the strong Blade 𝜶 𝜶 g/c Ref 𝟏 𝟐 A 30° 22° 0.75 [5] B 60° 25° 0.75 VKI C 66° 18° 0.70 VKI D 156° 19.5° 0.85 [6] Figure 1. Blade profile losses versus isentropic outlet Mach number for four transonic turbines. (c) topology of wake vortex structure (a) turbine blade flat plate model. (b) schlieren photograph. behind a cylinder [29]. Figure 13. Vortex shedding at transonic exit flow conditions [30]. Int. J. Turbomach. Propuls. Power 2020, 5, 10 29 of 55 acceleration over most part of the surface is likely to guarantee laminar conditions at the trailing edge on both blades and trip wires had to be used to enforce transition and turbulent boundary layers at the trailing edge, if desired so. The blades were tested from low subsonic to high subsonic outlet Mach numbers. Due to the use of blow down and suction tunnels at VKI and DLR, respectively, the Reynolds number increases with Mach number as shown in Figure 41. Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 31 of 58 Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 31 of 58 810 Figure 41. Variation of Reynolds number in function of downstream Mach number; adapted from Figure 41. Variation of Reynolds number in function of downstream Mach number; adapted from [16]. Commented [MM65]: Please do not cut figure and caption 811 [16]. over two pages. The tests for the front-loaded blade A are presented in Figure 42. In case of forced transition on 812 The tests for the front-loaded blade A are presented in Figure 42. In case of forced transition on the pressure side through a trip wire at 24% of the chord length, the Strouhal number is nearly constant 813 the pressure side through a trip wire at 24% of the chord length, the Strouhal number is nearly 814 constant and roughly equal to ~0.195 over the entire Mach range. In absence of a trip wire, the 810 Figure 41. Variation of Reynolds number in function of downstream Mach number; adapted from and roughly equal to St  0.195 over the entire Mach range. In absence of a trip wire, the evolution Commented [MM65]: Please do not cut figure and caption 815 evolution of = ( ) is quite different. Starting from the low Mach number and Reynolds 811 [16]. over two pages. of St = St(M ) is quite di erent. Starting from the low Mach number and Reynolds number end, 2is 816 number end, the Strouhal number decreases from ~0.34 at =0.2 to ~0.26 at =0.53. the Str812 ouhal number The tesdecr ts for eases the front fr -loa om dedSt blad e 0.34 A areat prM esented = in0.2 Figu to re 42 St . In c 0.26 ase of forc at M ed transition = 0.53 on . At this point 817 At this point the drops suddenly to the level of all turbulent cases. This sudden change obviously 2is 2is 813 the pressure side through a trip wire at 24% of the chord length, the Strouhal number is nearly 818 indicates that boundary layer transition has taken place on the pressure side. The slow decrease the St drops suddenly to the level of all turbulent cases. This sudden change obviously indicates that 814 constant and roughly equal to ~0.195 over the entire Mach range. In absence of a trip wire, the 819 before the sudden jump points to a progressive change from a laminar to a transitional boundary boundary layer transition has taken place on the pressure side. The slow decrease before the sudden 815 820 evolutio layer whn of ich is obv = ( iously rela ) is ted qu to ite di the in ffecr reeasi nt. n St gar Rti eng yn f olds n rom t uh m e l ber. ow Mach number and Reynolds jump points 816 821 to number e a pr Caogr scn ad, t dessive e C he S wa trou s t change e hst al num ed wit b fr h er a om de circula cr a ea laminar se r s f tra rom ilin g~ ed to 0.3 g a 4 etransitional at D at L R an=0 d a squa .2boundary to red ~0.2 t6 raili at n layer g edge which =0 at VK .53. I is obviously 817 822 A over a t this poin range t th e = drops 0.2 to su 0.9 dden . The tw ly to the l o serie esv of el of test a diffe ll turbu red n lent cases. ot only by Th is thsudden eir trailin ch ga e n d g g ee ob ge v oim ou et sl ry y related to the increasing Reynolds number. 818 indicates that boundary layer transition has taken place on the pressure side. The slow decrease 823 but also, at the same Mach number, by a higher Reynolds number in the VKI tests, see Figure 41. Cascade C was tested with a circular trailing edge at DLR and a squared trailing edge at VKI over 819 before the sudden jump points to a progressive change from a laminar to a transitional boundary 824 Note, that in the case of the squared trailing edge the distance between the separating shear layers is a range 820 M la = yer 0.2 whto ich0.9. is obv The iously two related series to the in of cr test easin di g Re er yn ed olds n not um only ber. by their trailing edge geometry but 825 well defined. However, this is not the case for the rounded trailing edge in which case the distance 2,is 821 Cascade C was tested with a circular trailing edge at DLR and a squared trailing edge at VKI 826 should be in any way smaller. But one single test, at = 0.59, was run at VKI also with a rounded also, at the same Mach number, by a higher Reynolds number in the VKI tests, see Figure 41. Note, 822 over a range = 0.2 to 0.9. The two series of test differed not only by their trailing edge geometry 827 trailing edge to eliminate any bias between the tests at DLR and VKI. that in the case of the squared trailing edge the distance between the separating shear layers is well 823 but also, at the same Mach number, by a higher Reynolds number in the VKI tests, see Figure 41. 824 Note, that in the case of the squared trailing edge the distance between the separating shear layers is defined. However, this is not the case for the rounded trailing edge in which case the distance should 825 well defined. However, this is not the case for the rounded trailing edge in which case the distance be in any way smaller. But one single test, at M = 0.59, was run at VKI also with a rounded trailing 2,is 826 should be in any way smaller. But one single test, at = 0.59, was run at VKI also with a rounded edge to eliminate any bias between the tests at DLR and VKI. 827 trailing edge to eliminate any bias between the tests at DLR and VKI. 829 Figure 42. Strouhal number variation with downstream Mach number for cascade A; adapted from 830 [16]. 831 Figure 43 presents the Strouhal number for blade C both in function of the downstream Mach 832 number and the Reynolds number. Both data sets show a plateau of = 0.36 at low Mach number 833 and Reynolds number which is characteristic for a fully laminar trailing edge boundary layer 829 Figure 42. Strouhal number variation with downstream Mach number for cascade A; adapted from Figure 42. Strouhal number variation with downstream Mach number for cascade A; adapted from [16]. 834 separation. The Strouhal number starts to decrease with increasing Reynolds number, the drop of 830 [16]. 835 occurring earlier at = 0.35 × 10 for the squared trailing edge, instead of 0.6 10 for the circular 831 836 trailiFi ng e gud re g 4 e.3 At pre s = en 1ts .1 t × he 1 S 0 trou h th al e s nu quare mber d f tr oa r b ilin lad g edg e C b e dat otha i rea n fu cnc h a p tion lat oeau wit f the do hw n =st 0r.24 eam M . No acth e Figure 43 presents the Strouhal number for blade C both in function of the downstream Mach 832 837 n th uat mber the sing and the le r Re oun yded nolds trailin number. g edg Bo e test at th data s VK eI t sin sdi how cated a pb lat y a star in th eau of = 0e .36 grap at h is r low Mac ight in h li nu nem wi ber th number and the Reynolds number. Both data sets show a plateau of St = 0.36 at low Mach number and 833 and Reynolds number which is characteristic for a fully laminar trailing edge boundary layer Reynolds number which is characteristic for a fully laminar trailing edge boundary layer separation. 834 separation. The Strouhal number starts to decrease with increasing Reynolds number, the drop of 835 occurring earlier at = 0.35 × 10 for the squared trailing edge, instead of 0.6 10 for the circular The Strouhal number starts to decrease with increasing Reynolds number, the drop of St occurring 836 trailing edge. At = 1.1 × 10 the squared trailing edge data reach a plateau with = 0.24 . Note 6 6 earlier at Re = 0.35 10 for the squared trailing edge, instead of 0.6 10 for the circular trailing edge. 837 that the single rounded trailing edge test at VKI indicated by a star in the graph is right in line with At Re = 1.1 10 the squared trailing edge data reach a plateau with St = 0.24. Note that the single rounded trailing edge test at VKI indicated by a star in the graph is right in line with the squared Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 32 of 58 838 the squared trailing edge data. Extrapolating the DLR data to higher Reynolds number one may 839 expect that they will reach the plateau of = 0.24 at ≈ 1.1 → 1.2 10 . 840 Comparing the two curves in Figure 43 raises of course the question as to the reasons for the 841 differences between them. The possible influence of the different distance between the separating 842 shear layers was already mentioned before, but, if this would be the case, then the Strouhal number Int. J. Turbomach. Propuls. Power 2020, 5, 10 30 of 55 843 for the VKI tests with squared trailing edge should be higher than those of the DLR tests with 844 rounded trailing edge. There must be therefore a different reason. The key for the understanding 845 comes from flat plate tests presented in [16], see Figure 44, which showed that the difference of the trailing edge data. Extrapolating the DLR data to higher Reynolds number one may expect that they 846 Strouhal number between a full laminar and full turbulent flow conditions was much bigger for tests will reach the plateau of St = 0.24 at Re  1.1 ! 1.2 10 . 847 with rounded trailing edges than squared trailing edges, 30% instead of 13%. Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 32 of 58 (a) (b) 838 the squared trailing edge data. Extrapolating the DLR data to higher Reynolds number one may 849 Figure 43. Strouhal number variation with Mach number (a) and Reynolds number (b) for cascade C; 839 expect that they will reach the plateau of = 0.24 at ≈ 1.1 → 1.2 10 . Figure 43. Strouhal number variation with Mach number (a) and Reynolds number (b) for cascade Commente C; d [M66]: Please add explanation for subgraph. 850 adapted from [16]. 840 Comparing the two curves in Figure 43 raises of course the question as to the reasons for the adapted from [16]. Commented [MM67R66]: Done. 841 differences between them. The possible influence of the different distance between the separating 842 shear layers was already mentioned before, but, if this would be the case, then the Strouhal number Comparing the two curves in Figure 43 raises of course the question as to the reasons for the 843 for the VKI tests with squared trailing edge should be higher than those of the DLR tests with 844 rounded trailing edge. There must be therefore a different reason. The key for the understanding di erences between them. The possible influence of the di erent distance between the separating shear 845 comes from flat plate tests presented in [16], see Figure 44, which showed that the difference of the layers was already mentioned before, but, if this would be the case, then the Strouhal number for the 846 Strouhal number between a full laminar and full turbulent flow conditions was much bigger for tests VKI tests with squared trailing edge should be higher than those of the DLR tests with rounded trailing 847 with rounded trailing edges than squared trailing edges, 30% instead of 13%. edge. There must be therefore a di erent reason. The key for the understanding comes from flat plate tests presented in [16], see Figure 44, which showed that the di erence of the Strouhal number between a full laminar and full turbulent flow conditions was much bigger for tests with rounded trailing edges than squared trailing edges, 30% instead of 13%. (a) (b) This di erent behavior can be explained if one assumes that the shape of the trailing edge may 851 Figure 44. Strouhal numbers for vortex shedding from flat plates with rounded (a) and squared (b) strongly 852 a ect the traili evolution ng edges; adapt of ed from the shear [16]. layer, and that it is the state of the shear layer rather than that of the boundary layer which plays the most important role in the generation of the vortex street. 853 This different behavior can be explained if one assumes that the shape of the trailing edge may Of course, a sharp corner will not necessarily induce immediately full transition, but transition will 854 strongly affect the evolution of the shear layer, and that it is the state of the shear layer rather than 855 that of the boundary layer which plays the most important role in the generation of the vortex street. occur over a certain length, and(a) (b) this length a ects the length of the enrolment of the vortex and 856 Of course, a sharp corner will not necessarily induce immediately full transition, but transition will therewith its frequency. The transition length of the shear layer will be a ected by both the Reynolds 849 Figure 43. Strouhal number variation with Mach number (a) and Reynolds number (b) for cascade C; Commented [M66]: Please add explanation for subgraph. 857 occur over a certain length, and this length affects the length of the enrolment of the vortex and 850 adapted from [16]. number and the Mach number. 858 therewith its frequency. The transition length of the shear layer will be affected by both the Reynolds Commented [MM67R66]: Done. 859 number and the Mach number. 860 Contrary to the vortex shedding for subsonic flow conditions discussed above, where the 861 vortices are generated by the enrolment of the separating shear layers close to the blade trailing edge, 862 the situation changes with the emergence of oblique shocks from the region of the confluence of the 863 pressure and suction side shear layers for transonic outlet Mach numbers. In this case the vortex 864 formation is delayed to after this region as shown already in the schlieren pictures in Figure 34. (a) (b) 851 Figure 44. Strouhal numbers for vortex shedding from flat plates with rounded (a) and squared (b) Figure 44. Strouhal numbers for vortex shedding from flat plates with rounded (a) and squared 852 trailing edges; adapted from [16]. (b) trailing edges; adapted from [16]. 853 This different behavior can be explained if one assumes that the shape of the trailing edge may 854 strongly affect the evolution of the shear layer, and that it is the state of the shear layer rather than Contrary to the vortex shedding for subsonic flow conditions discussed above, where the 855 that of the boundary layer which plays the most important role in the generation of the vortex street. vortices are generated by the enrolment of the separating shear layers close to the blade trailing edge, 856 Of course, a sharp corner will not necessarily induce immediately full transition, but transition will 857 occur over a certain length, and this length affects the length of the enrolment of the vortex and the situation changes with the emergence of oblique shocks from the region of the confluence of the 858 therewith its frequency. The transition length of the shear layer will be affected by both the Reynolds pressure and suction side shear layers for transonic outlet Mach numbers. In this case the vortex 859 number and the Mach number. formation is delayed to after this region as shown already in the schlieren pictures in Figure 34. 860 Contrary to the vortex shedding for subsonic flow conditions discussed above, where the 861 vortices are generated by the enrolment of the separating shear layers close to the blade trailing edge, This is even more clearly demonstrated in Figure 45 presenting the evolution of the wake density 862 the situation changes with the emergence of oblique shocks from the region of the confluence of the gradients predicted with a LES by Vagnoli et al. [56], for the turbine blade shown in Figure 17, from 863 pressure and suction side shear layers for transonic outlet Mach numbers. In this case the vortex high subsonic to low supersonic outlet Mach numbers. For M = 1.05 the vortex shedding frequency 864 formation is delayed to after this region as shown already in the schlieren pictures in Figure 34. 2is Int. J. Turbomach. Propuls. Power 2020, 5, 10 31 of 55 Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 33 of 58 is not any more conditioned by the trailing edge thickness but by the distance between the feet of the Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 33 of 58 trailing edge shocks emanating from the region of the confluence of the two shear layers. 865 This is even more clearly demonstrated in Figure 45 presenting the evolution of the wake density 866 gradients predicted with a LES by Vagnoli et al. [56], for the turbine blade shown in Figure 17, from 867 high subsonic to low supersonic outlet Mach numbers. For =1.05 the vortex shedding 868 frequency is not any more conditioned by the trailing edge thickness but by the distance between the 869 feet of the trailing edge shocks emanating from the region of the confluence of the two shear layers. (a) =0.79 (b) =0.97 (c) =1.05 , , , (a) =0.79 (b) =0.97 (c) =1.05 , , , Figure 45. Change of wake density gradients from subsonic to supersonic exit Mach numbers as 865 Figure 45. Change of wake density gradients from subsonic to supersonic exit Mach numbers as 871 Figure 45. Change of wake density gradients from subsonic to supersonic exit Mach numbers as predicted by LES [56]. 866 predicted by LES [56]. Commented [M68]: Please add explanation for subgraph. 872 predicted by LES [56]. Commented [MM69R68]: Done. Consequently, one observes a sudden increase of the vortex shedding frequency as for example 867 Consequently, one observes a sudden increase of the vortex shedding frequency as for example 873 Consequently, one observes a sudden increase of the vortex shedding frequency as for example recorded 874by Carscallen recorded by Ca et rscal. allen [43 et a ],l. on [43]their , on their nozzle nozzle guide guide vane vane, , see Fi see gure 46. Figure 46. 868 recorded by Carscallen et al. [43], on their nozzle guide vane, see Figure 46. Figure 46. Strouhal number versus downstream Mach number; adapted from [43]. 875 Figure 46. Strouhal number versus downstream Mach number; adapted from [43]. 876 7. Advances in the Numerical Simulation of Unsteady Turbine Wake Characteristics 7. Advances in the Numerical Simulation of Unsteady Turbine Wake Characteristics 877 The numerical simulation of unsteady turbine wake flow is relatively young, and the first 869 Figure 46. Strouhal number versus downstream Mach number; adapted from [43]. The numerical simulation of unsteady turbine wake flow is relatively young, and the first 878 contributions appeared in the mid-80s. The decade 1980–1990 has in fact seen the final move from the contributions appeared in the mid-80s. The decade 1980–1990 has in fact seen the final move from 879 potential flow models to the Euler and Navier-Stokes equations whose numerical solutions were 870 7. Advances in the Numerical Simulation of Unsteady Turbine Wake Characteristics 880 tackled with new, revolutionary for the time, techniques. Those were also the years of the first vector the potential flow models to the Euler and Navier-Stokes equations whose numerical solutions were 881 and parallel super-computers capable of a few sustained gigaflops (CRAY YMP, IBM SP2, NEC SX- 871 tackled The numer with new ica , l sim revolutionary ulation of forunste the time, ady turbine w techniques. ake fl Those ow is rela were also tively you the yearsng, of the and the first vector first 882 3, to quote a few examples), and of the beginning of the massive availability of computing resources 872 contri and parallel buti 883 ons super o ab ppea eying -computers rM ed i oore n’s th lae mi w (tcapable ran d-80 sisto s. The decade 1980 r co of unt do a few ublin sustained g every t –19 wo y gigaflops 90 e ha ars).s Sin inc f (CRA ea tct seen the fi hen th Ye YMP progres , IBM nal ses h move f a SP2, ve NEC rom the SX-3, 884 been huge both on the numerical techniques and on the turbulence modelling side. Indeed, the most 873 potential to quote flo a few w models to examples), the Eule and of r an the d Navier beginning -Sto ofkes e the massive quations whose n availability um ofecomputing rical solutiorn esour s were ces 885 advanced option, that is the Direct Numerical Simulation (DNS) approach, where all turbulent scales 874 ta obeying ckled wiMoor th new, revolutiona e’s law (transistor ry focount r the tidoubling me, techniqu every es. Those w two years). ere also the Since then years of the the progr fir esses st vector have 886 are properly space-time resolved down to the dissipative one, has also recently entered the 875 and p beenahuge rallel both super-comput on the numerical ers capable of a techniques few su andst on ain the ed gigaflop turbulence s (CRAY YM modelling P, side. IBM SP2, NE Indeed, the C SX- most 887 turbomachinery community starting from the pioneering work of Jan Wissink in 2002 [63]. 888 Unfortunately, because of the very severe resolution requirements, there is still no DNS study of 876 3, advanced to quote a option, few ex that amp isles the ), Dir and ect ofNumerical the beginnin Simulation g of the ma (DNS) ssive appr avai oach, labilit wher y ofe comput all turbulent ing rescales source ar s e 889 turbine wake flow (TWF) at realistic Reynolds and Mach numbers, that is Re ~ 10 and high subsonic 877 obeying properly Moo space-time re’s law r (t esolved ransist down or count to the doubl dissipative ing every two yea one, has also rs). recently Since then the progresses ha entered the turbomachinery ve 890 and transonic outlet Mach numbers with shocked flow conditions, although improvements have 878 been huge bo community 891 starting th on the numerical techn been recenfr tlyom attained [ the pioneering 64]. 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Inde e help of veryed, t , because he most of 892 large-scale computing hardware such a simulation is likely to appear soon, as the result of some 879 athe dvavery nced opti sever on, tha e resolution t is the Di requir rect ements, Numerica ther l Si emu is still lation (DN no DNS S) ap study proach, of turbine where al wake l turbu flow len (TWF) t scales at 880 are prope realistic Reynolds rly space and -time Mach resolved numbers, down to the that is Re ~d10 issipative and high onsubsonic e, has also r and transonic ecently enter outlet ed the Mach 881 turboma numbers chi with nery communi shocked flow ty st conditions, arting from the pi although o impr neering ovements work of have Ja be n en Wi rssin ecently k inattaine 2002 d[6[3] 64. ]. 882 Unfortun With theately, bec development ause of the ve of highlyryparallelizable severe resoluti codes on reand quiremen the help ts, there of very is still no large-scale DNS computing study of 883 tu har rbine w dwara eksuch e flow a (TW simulation F) at real is ist likely ic Reyn to olds appear and Mac soon, h num as the bers result , thatof is Re some ~ 10 cutting-edge and high sub scientific sonic 884 and tr researansonic outlet M ch. In the meantime, ach numbers w and within ithe th shocked foreseeable flow futur cond e, the itions, alt industrial hough improvement world and the designers s have 885 been recently interested in attained tangled [64]. aspects With of TWF the dev forelopment of h stage performance ighly par enhancement allelizable co will des certainly and the help of very run unsteady 886 lar flow ge-sca simulations le comput wher ing h e turbulence ardware su is ch handled a simuthr latiough on is advanced likely to ap modeling. pear soon, Many as t of he those result simulations of some 887 cut will ting-ed rely on ge scient in-house ific rese developed arch. In resear the meant ch codes ime, and anturbomachinery d within the forese oriented eable future, the commercialin packages, dustrial 888 world which, and indeed, the designe have impr rs interested in oved significantly tangled since aspect thes of TWF very first fo unsteady r stage perfor TWF mance enh simulation.an Yc et, ement there 889 wi arll e cert twoaar inl eas y ru wher n uensteady fl importantow challenges simulations still whe needre turbulenc to be satisfactorily e is han faced dled through before thead prvanced esently 890 modeling. Many of those simulations will rely on in-house developed research codes and 891 turbomachinery oriented commercial packages, which, indeed, have improved significantly since the 892 very first unsteady TWF simulation. Yet, there are two areas where important challenges still need to 893 be satisfactorily faced before the presently available (lower fidelity) computations could be Int. J. Turbomach. Propuls. Power 2020, 5, 10 32 of 55 available (lower fidelity) computations could be considered reliable and successful. They can be, loosely speaking, termed of numerical and modeling nature. We shall try to review both, in the context of the presently discussed unsteady turbine wake flow subject category, presenting a short overview of the available technologies. A more specialized review study on high-fidelity simulations as applied to turbomachinery components has recently been published by Sandberg et al. [65]. 7.1. Numerical Aspects Most of the available turbine wake flow computations have been obtained with eddy viscosity closures and structured grid technologies, although a few examples documenting the use of fully unstructured locally adaptive solvers are available [66,67]. In the structured context turbomachinery blades gridding is considered a relatively simple problem, and automated mesh generators of commercial nature producing appreciable quality multi-block grids, are available [68]. The geometrical factors most a ecting the grid smoothness are the cooling holes, the trailing edge shape, the sealing devices and the fillets. Of those the trailing edge thickness and its shape are the most important in TWF computations. Low and intermediate pressure turbines (LPT and IPT, respectively) have relatively sharp trailing edges, while the first and second stages of the high-pressure turbines (HPT), often because of cooling needs, have thicker trailing edges. Typically, the trailing edge thickness to chord ratio D/C, is a few percent in LPTs and IPTs, and may reach values of 10% or higher in some HPTs. Thus, the ratio of the trailing edge wet area to the total one may easily range from 1/200 to 1/20, having roughly estimated the blade wet area as twice the chord. Therefore, resolving the local curvature of the trailing edge area is extremely demanding in terms of blade surface grid, that is, in number of points on the blade wall. Curvature based node clustering may only partially alleviate this problem. In addition, preserving grid smoothness and orthogonality in the trailing edge area is dicult, if not impossible with H or C-type grids, even with elliptic grid generators relying on forcing functions [69]. Wrapping an O-type mesh around the blade is somewhat unavoidable, and in any event the use of a multi-block or multi-zone meshing is highly desirable. Unstructured hybrid meshes would also typically adopt a thin O mesh in the inner wall layer. Non-conformal interfaces of the patched or overlapped type would certainly enhance the grid quality, at the price of additional computational complexities and some local loss of accuracy occurring on the fine-to-coarse boundaries [70]. Local grid skewness accompanied by a potential lack of smoothness will pollute the numerical solution obtained with low-order methods, introducing spurious entropy generation largely a ecting the features of the vortex shedding flow. In those conditions, the base pressure is typically under-predicted as a consequence of the local flow turning and separation mismatch, with a higher momentum loss and an overall larger unphysical loss generation in the far wake. The impact of those grid distorted induced local errors on the quality of the solution is hard to quantitatively ascertain both a-priori and a-posteriori, and often grid refinement will not suce, as they frequently turn out to be order 1, rather than order h with h the mesh size and p the order of accuracy. Nominally second order schemes have in practice 1 < p < 2. In this context, higher order finite di erence and finite volume methods, together with the increasingly popular spectral-element methods, o er a valid alternative to standard low order methods [71–75]. This is especially true for those techniques capable of preserving the uniform accuracy over arbitrarily distorted meshes, a remarkable feature that may significantly relieve the grid generation constraints, besides o ering the opportunity to resolve a wider range of spatial and temporal scales with a smaller number of parameters compared to the so called second order methods (rarely returning p = 2 on curvilinear grids). The span of scales that needs to be resolved and the features of the coherent structures associated to the vortex shedding depend upon the blade Reynolds number, the Mach number (usually built with the isentropic downstream flow conditions) and the D/C ratio. This is equivalent to state that the Reynolds number formed with the momentum thickness of the turbulent boundary layer at the trailing edge (Re ) and the Reynolds number defined using the trailing edge thickness (Re ), are independent parameters. For thick trailing edge blades the vortex shedding is vigorous and the near wake development is governed by the suction and pressure Int. J. Turbomach. Propuls. Power 2020, 5, 10 33 of 55 side boundary layers which di er. Thus, the early stages of the asymmetric wake formation chiefly depend upon the local grid richness, the resolution of the turbulent boundary layers at the TE and the capabilities of the numerical method to properly describe their mixing process. Well-designed turbine blades operate with an equivalent di usion factor smaller than 0.5 yielding a /C ratio less than 1% according to Stewart correlation [76]. This e ectively means that the resolution to be adopted for the blade base area will have to scale like the product /C C/D which may be considerably less than one; in order words the base area region needs more points that those required to resolve the boundary layers at the trailing edge. Very few simulations have complied with this simple criterion as today. Compressibility e ects present additional numerical diculties, especially in scale resolving simulations. It is a known fact that transonic turbulent TWF calculations require the adoption of special numerical technologies capable to handle time varying discontinuous flow features like shock waves and slip lines without a ecting their physical evolution. Unfortunately, most of the numerical techniques with successful shock-capturing capabilities rely on a local reduction of the formal accuracy of the convection scheme whether or not based on a Riemann solver. Since at grid scale it is hard to distinguish discontinuities from turbulent eddies, and even more their mutual interaction, Total Variation Diminishing (TVD) and Total Variation Bounded (TVB) schemes [77–79] are considered too dissipative for turbulence resolving simulations, and they are generally disregarded. At present, in the framework of finite di erence and finite volume methods, there is scarce alternative to the adoption of the class of ENO (Essentially Non Oscillatory) [80–82] and WENO (Weighted Essentially Non Oscillatory) [83–87] schemes developed in the 90s. A possibility is o ered by the Discontinuous Galerkin (DG) methods [88]. The DG is a relatively new finite element technique relying on discontinuous basis functions, and typically on piecewise polynomials. The possibility of using discontinuous basis functions makes the method extremely flexible compared to standard finite element techniques, in as much arbitrary triangulations with multiple hanging nodes, free independent choice of the polynomial degree in each element and an extremely local data structure o ering terrific parallel eciencies are possible. In their native unstructured framework, opening the way to the simulation of complex geometries, h and p-adaptivity are readily obtained. The DG method has several interesting properties, and, because of the many degrees of freedom per element, it has been shown to require much coarser meshes to achieve the same error magnitudes when compared to Finite Volume Methods (FVM) and Finite Di erence Methods (FDM) of equal order of accuracy [89]. Yet, there seem to persist problems in the presence of strong shocks requiring the use of advanced non-linear limiters [90] that need to be solved. This is an area of intensive research that will soon change the scenario of the available computational methods for high fidelity compressible turbulence simulations. 7.2. Modeling Aspects The lowest fidelity level acceptable for TWF calculations is given by the Unsteady Reynolds Averaged Navier-Stokes Equations (URANS) or, better, Unsteady Favre Averaged Navier-Stokes Equations (UFANS) in the compressible domain. URANS have been extensively used in the turbomachinery field to solve blade-row interaction problems, with remarkable success [91,92]. The pre-requisite for a valid URANS (here used also in lieu of UFANS) is that the time scale of the resolved turbulence has to be much larger than that of the modeled one, that is to say the characteristic time used to form the base state should be suciently small compared to the time scale of the unsteady phenomena under investigation. This is often referred to as the spectral gap requirement of URANS [93]. Therefore, we should first ascertain if TWF calculations can be dealt with this technology, or else if a spectral gap exists. The analysis amounts at estimating the characteristic time  , or frequency f , vs vs of the wake vortex shedding, and compare it with that of the turbulent boundary layer at the trailing edge,  , or f . The wake vortex shedding frequency is readily estimated from: bl l bl 2,is f = St = f(geometry, Reynolds, Mach) vs te Int. J. Turbomach. Propuls. Power 2020, 5, 10 34 of 55 which has been shown to depend upon the turbine blade geometry and the flow regimes (see Figures 39, 42–44 and 46). For the turbulent boundary layers the characteristic frequency can be estimated, using inner scaling variables, as: bl with u = the friction velocity, and  the kinematic viscosity. Assuming the boundary layer to be fully turbulent from the leading edge, and using the zero pressure gradient incompressible flat plate correlation of Schlichting [59]: 2  0.059 C = = f ,x 1/5 Re one gets: 2 2 2 u u u / / 1/5 = C = 0.0295 Re . f ,x x At the turbine trailing edge x = C, and u = V so that: 2,is 2,is 4/5 f = 0.0295 Re bl 2,is Therefore, the ratio of the turbulent boundary layer characteristic frequency to the wake vortex shedding one is, roughly: 4/5 Re bl vs te 2,is = = 0.0295 (6) f  C St vs bl The explicit dependence of the Strouhal number upon the geometry term d /C is unknown, te although clear trends have been highlighted in the previous section. However, taking d /C  0.05 and te St  0.3 as reasonable values, Equation (6) returns: bl 4/5 0.005 Re (7) 2,is vs The estimates obtained from the above Equation are reported in Table 3, for a few Reynolds numbers. Table 3. Turbulent boundary layer to vortex shedding frequency ratio; Equation (7). 5 6 6 6 Re 5 10 10 2 10 3 10 2,is bl 180 310 550 760 vs From the above table it is readily inferred that, for the problem under investigation, a neat spectral gap exists, and, thus, URANS calculations can be carried out with some confidence. The results reported in the foregoing confirm that this is indeed the case. Formally, RANS are obtained from URANS dropping the linear unsteady terms, and, therefore, the closures developed for the steady form of the equations apply to the unsteady ones as well. Whether the abilities of the steady models broaden to the unsteady world is controversial, even though the limited available literature seem to indicate that this is rarely the case. A review of the existing RANS closures is out of the scope of the present work, and the relevant literature is too large to be cited here, even partially. In the turbomachinery field, turbulence and transition modelling problems have been extensively addressed over the past decades, and significant advances have been achieved [94–96]. Here, we will mainly stick to those models which have been applied in the TWF simulations presently reviewed. In the RANS context Eddy Viscosity Models (EVM) are by far more popular than Reynolds Stress Models (RSM), whether di erential (DRSM) or algebraic (ARSM). Part of the reasons are to be Int. J. Turbomach. Propuls. Power 2020, 5, 10 35 of 55 found with the relatively poor performance of DRS and ARS when compared to the computational e ort required to implement these models, especially for unsteady three-dimensional problems. Also, the prediction of pressure induced separation and, more in general, of separated shear layers is, admittedly, disappointing, so that the expectations of advancing the fidelity level attainable with EVM has been disattended. This explains why most of the engineering applications of RANS, and thus of URANS, are routinely based on EVM, and typically on algebraic [97], one equation [98] and two equations (k- of Jones and Launder [99], k-! of Wilcox [100], Shear Stress Transport (SST) of Menter [101]) formulations. In the foregoing we shall see that the TWF URANS computations reviewed herein all adopted the above closures. A few of those were based on the k-! model of Wilcox. This closure, and its SST variant, has gained considerable attention in the past two decades and it is widely used and frequently preferred to the k- models, as it is reported to perform better in transitional flows and in flows with adverse pressure gradients. Further, the model is numerically very stable, especially its low-Reynolds number version, and considered more “friendly” in coding and in the numerical integration process, than the k- competitors [100]. On the scale resolved simulations the scenario is rather di erent. Wall resolved Large Eddy Simulations (LES) are now recognized as una ordable for engineering applications because of the very stringent near wall resolution requirements and of the inability of all SGS models to account for the e ects of the near wall turbulence activity on the resolved large scales [102,103]. On the wall modeled side, the most successful approaches rely on hybrid URANS-LES blends, and in this framework the pioneering work of Philip Spalart and co-workers should be acknowledged [104,105]. Already 20 years ago this research group introduced the Detached Eddy Simulation (DES), a technique designed to describe the boundary layers with a URANS models and the rest of the flow, particularly the separated (detached) regions, with an LES. The switching or, better, the bridging between the two methods takes place in the so called “grey area” whose definition turned out to be critical, because of conceptual and/or inappropriate, though very frequent, user decisions. The latter are particularly related to the erroneous mesh sizes selected for the model to follow the URANS and the LES branches. Nevertheless, the original DES formulation su ered from intrinsic to the model deficiencies leading to the appearance of unphysical phenomena in thick boundary layers and thin separation regions. Those shortcomings appear when the mesh size in the tangent to the wall direction, i.e., parallel to it, D , becomes smaller than the boundary layer thickness , either as a consequence of a jj local grid refinement, or because of an adverse pressure gradient leading to a sudden rise of . In those instances, the local grid size, i.e., The filter width in most of the LES, is small enough for the DES length scale to fall in the LES mode, with an immediate local reduction of the eddy viscosity level far below the URANS one. The switching to the LES mode, however, is inappropriate because the super-grid Reynolds stresses do not have enough energy content to properly replace the modeled one, a consequence of the mesh coarseness. The decrease in the eddy viscosity, or else the stress depletion, reduces the wall friction and promotes an unphysical premature flow separation. This is the so-called Model Stress Depletion (MSD) phenomenon, leading to a kind of grid induced separation, which is not easy to tackle in engineering applications, because it entails the unknown relation between the flow to be simulated and the mesh spacing to be used. In recent years, however, two new models o ering remedies to the MSD phenomenon have been proposed, one by Philip Spalart and co-workers [106], the other by Florian Menter and co-workers [107]. Before proceeding any further, let us briefly mention the physical idea underlying the DES approach. In its original version based on the Spalart and Allmaras turbulence model [98] the length scale d used in the eddy viscosity is modified to be: d  min(d, C D) (8) DES where d is the distance from the wall, D a measure of the grid spacing (typically D  max(Dx,Dy,Dz) in a Cartesian mesh), and C a suitable constant of order 1. The URANS and the wall modeled DES ˜ ˜ LES modes are obtained when d  d and d  C D, respectively. The DES formulation based on the DES two equations Shear Stress Transport turbulence model of Menter [101] is similar. It is based on the Int. J. Turbomach. Propuls. Power 2020, 5, 10 36 of 55 introduction of a multiplier (the function F ) in the dissipation term of the k-equation of the k-! DES model which becomes: k!F DES with: F = max , 1 (9) DES C D DES In the above equations L is the turbulent length scale as predicted by the k-! model, = 0.09 the model equilibrium constant and C a calibration constant for the DES formulation: DES L = Both the DES-SA (DES based on the Spalart and Allmaras model) and the DES-SST (DES based on Menter ’s SST model) models su er from the premature grid induced separation occurrence previously discussed. To overcome the MSD phenomenon Menter and Kuntz [107] introduced the F blending SST functions that were designed to reduce the grid influence of the DES limiter (9) on the URANS part of the boundary layer that was “protected” from the limiter, that is, protected from an uncontrolled and undesired switch to the LES branch. This amounts to modify Equation (9) as follows: " # F = max (1 F ), 1 DES-SST-zonal SST C D DES with F selected from the blending functions of the SST model, whose argument is k/(!d), that SST is the ratio of the k-! turbulent length scale k/! and the distance from the wall d. The blending functions are 1 in the boundary layer and go to zero towards the edge. The proposal of Spalart et al. [106] termed DDES is similar to the DES-SST-zonal proposal of Menter et al. [107], and, while presented for the Spalart and Allmaras turbulence model it can be readily extended to any EVM. In the Spalart and Allmaras model a turbulence length scale is not solved for through a transport equation. It is instead built from the mean shear and the turbulent viscosity: t t r = = d p 2 2 (d) 2S S (d) ij ij with S = @U /@x + @U /@x /2 the rate of strain tensor,  the eddy viscosity and  the von Kàrmàn ij i j j i t constant. This quantity, actually a length scale squared, is 1 in the outer portion of the boundary layer and goes to zero towards its edge. The term  is often augmented of the molecular viscosity  to ensure that r remains positive in the inner layer. This dimensionless length scale squared is used in the following function: f = 1 tanh [8r ] d d reaching 1 in the LES region where L < d and 0 in the wall layer. It plays the role of 1 F in the t SST DES-SST-zonal model. Additional details on the design and calibration of the model constants can be found in [106]. The Delayed DES (DDES), a surrogate of the DES, is obtained replacing d in Equation (8) with the following expression: ( ) d  d f max 0, d C D (10) DES The URANS and the original DES model are retrieved when f = 0 and f = 1, respectively, d d ˜ ˜ corresponding to d  d and d  C D. This new formulation makes the length scale (10) depending DES on the resolved unsteady velocity field rather than on the grid solely. As the authors stated the model prevents the migration on the LES branch if the function f is close to zero, that is the current point is in the boundary layer as judged from the value of r . If the flow separates f increases and the LES mode d d is activated more rapidly than with the classical DES approach. As for DES this strategy, designed to Int. J. Turbomach. Propuls. Power 2020, 5, 10 37 of 55 tackle the MSD phenomenon, does not relieve the complexity of generating adequate grids, that is grids capable of properly resolving the energy containing scales of the LES area. Thus, unlike a proper grid assessment study is conducted, it will be dicult to judge the quality of those scale resolving models especially in the present context of TWF. 7.3. Achievements Unsteady turbine wake flow simulation is a relatively new subject and the very first pioneering works appeared in the mid-90s [66,108–110]. The reason is twofold; on one side the numerical and modelling capabilities were not yet ready to tackle the complexities of the physical problem, and on the other side, the lack of detailed experimental measurements discouraged any attempt to simulate the wake flow. This until the workshop held at the von Kàrmàn Institute in 1994 during a Lecture Series [37], where the first detailed time resolved experimental data of a thick trailing edge turbine blade where presented and proposed for experiment-to-code validation in an open fashion. The turbine geometry was also disclosed. As mentioned in Section 3 those tests referred to a low Mach, high Reynolds number case (M = 0.4, Re = 2 10 ). The numerical e orts of [108,110–113] addressing this test case and 2,is listed in Table 4, were devoted at ascertaining the capabilities of the state-of-the-art technologies to predict the main unsteady features of the flow, namely the wake vortex shedding frequency and the time averaged blade surface pressure distribution, particularly in the base region. Table 4. Available computations of the M = 0.4, Re = 2 10 VKI LS-94 turbine blade. 2,is 2 Numerical Space/Time Grid Authors Eqs. Grid Closure Method Discretization Density Manna Structured EVM (Baldwin & URANS CC-FVM 2nd/2nd 44k et al. [110] Multi-Block (H-O) Lomax [97]) Arnone EVM (Baldwin & URANS CC-FVM Structured C-grid 2nd/2nd 36k et al. [111] Lomax [97]) Sondak EVM (Deiwert URANS FDM Overset grids (H-O) 3nd/2nd 60k et al. [112] et al. [114]) Structured Ning et al. EVM (Roberts URANS CV-FVM Multi-Block 2nd/2nd 42k [113] [115]) (H-O-H-H) All of the above contributors solved the URANS with a Finite Volume (FVM) or Finite Di erence Method (FDM) and adopted simple algebraic closures. Both Cell Vertex (CV) and Cell Centered (CC) approaches where used. The more recent computations of Magagnato et al. [116] referred to a similar test case, though with rather di erent flow conditions, and will not be reviewed. Appropriate resolution of the trailing edge region and the adoption of O grids turned out to be essential to reproduce the basic features of the unsteady flow in a time averaged sense. The use of C grids with their severe skewing and distortion of the base region a ected the resolved flow physics and required computational and modelling tuning to fit the experiments. The time mean blade loading could be fairly accurately predicted (see Figure 47) by nearly all authors listed in Table 4, although discrepancies with the experiments and among the computations exist. They have been attributed to stream-tube contraction e ects and to the tripping wire installed on the pressure side at x/C = 0.61 in the experiments [112]. ax Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 10 of 58 271 suction side for both blades are exactly the same, but contrary to the nearly constant Mach number 272 for the VKI blade downstream of the throat, the blade of Xu and Denton is characterized by a very 273 strong adverse pressure gradient in this region. As pointed out by the authors, this causes the suction 274 side boundary layer to be either separated or close to separation up-stream of the trailing edge. 275 Clearly, Sieverding’s correlation cannot deal with blade designs characterized by very strong adverse 276 pressure gradients on the rear suction side causing boundary layer separation before the trailing 277 edge. 278 Figure 9. Comparison of blade Mach number distribution for blade RD of Xu and Denton [15], (solid 279 curve, =0.8, =8 10 ) with VKI blade (dashed curve, =0.8, =10 ). , , 280 The possible effect of boundary layer separation resulting from high rear suction side diffusion 281 resulting in high base pressures was also mentioned by Corriveau and Sjolander in 2004 [20], 282 comparing their nominal mid-loaded rotor blade HS1A, mentioned already before, with an aft- 283 loaded blade HS1C with an increase of the suction side unguided turning angle from 11.5° to 14.5°. 284 It appears that the increased turning angle could cause, in the transonic range, shock induced 285 boundary layer transition near the trailing edge with, as consequence, a sharp increase of the base 286 pressure, i.e. a sudden drop in the base pressure coefficient as seen in Figure 10. Note that the 287 reported in the figure has been converted to − of the original data. 288 As regards the base pressure data by Deckers and Denton [13] for a low turning blade model 289 and Gostelow et al. [14] for a high turning nozzle guide vane, who report base pressure data far below 290 those of Sieverding’s BPC, their blade pressure distribution resembles that of the convergent/ 291 divergent blade C in Figure 3 with a negative blade loading near the trailing edge which would 292 explain the very low base pressures. In addition, the blade of Deckers and Denton has a blunt trailing 293 edge, and there is experimental evidence that, compared to a circular trailing edge, the base pressure Int. J. Turbo Int. J. Turbo mm ach. ach. Propuls. Power Propuls. Power 2018 2018 , 3 , ,3 x FOR PE , x FOR PE ER RE ER RE VI VI EE W W 40 of 40 of 58 58 Int. J. Turbo Int. J. Turbo mm ach. ach. Propuls. Power Propuls. Power 2018 2018 , 3 , ,3 x FOR PE , x FOR PE ER RE ER RE VI VI EW EW 40 of 40 of 58 58 294 for blades with blunt trailing edge might be considerably lower. Sieverding and Heinemann [16] Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 40 of 58 295 report for flat plate tests at moderate subsonic Mach numbers a drop of the base pressure coefficient Int. J. Turbomach. Propuls. Power 2018, 3, x FOR PEER REVIEW 40 of 58 296 by 11% for a plate with squared trailing edge compared to that with a circular trailing edge. 1139 stream-tube contraction effects and to the tripping wire installed on the pressure side at / =0.61 Int. J. Turbomach. Propuls. Power 2020, 5, 10 38 of 55 1140 in the experiments [112]. 1139 stream-tube contraction effects and to the tripping wire installed on the pressure side at / =0.61 1140 in the experiments [112]. 1135 1135 Figure 47. Figure 47. 1141 VKI LS94 turbine VKI LS94 turbine Figure 47. VKI blade, LS94 turbine blade, blade, =0 =0 .4.4 , , =0 .=2 4,=2 1=2 1 00 1case 0 case case . Ti .. Ti Ti m me m e m e mean m ean blad e blade an blad surface e s e s u isen u rfrf ac a tre is c opic e is e n en trto ro pp ici c 1135 1135 Figure 47. Figure 47. Figure VKI LS94 turbine VKI LS94 turbine 47. VKI LS94 turbine blade, blade, blade, M =0 =0 .4.= ,4 , 0.4,=2 Re =2 1 = 010 2case  case 10 . Ti . Ti case. m m e m eT m e ime an blad ean blad mean e s e s blade urf urf ac asurface e is ce is en et n rto isentr rp oi p ci c opic , , , , , 297 Figure 10. Base pressure coe 2,is fficient fo 2r mid-loaded (solid line) and aft-loaded (dashed line) rotor 1142 Mach number distribution. experiments [32]; [110]; [111]; [112]; 1136 1136 Mach number distribution. Mach number distribution. 1141 Figure 47. VKI LS94 turbine experime experime blade, nts nts [32]; =0[32]; .4, =2 10[110]; [110]; case. Time mean blade[111]; [111]; surface isentr[112]; opic[112]; 1136 1136 Mach number distribution. Mach number distribution. Mach number distribution. experime experime experimentsnts nts ,[32]; [32]; [32]; [110]; [110]; [110]; [111]; [[111]; 111]; [112]; [112[112]; ]; [113 ]. 298 blade. Symbols: HS1A geometry, HS1C geometry. Adapted from [20]. 1143 [113]. 1142 Mach number distribution. experiments [32]; [110]; [111]; [112]; 1137 1137 [113]. [113]. 1137 1137 [113]. [113]. 1143 [113]. The time averaged base pressure region was also fairly well reproduced by the available numerical 1144 The time averaged base pressure region was also fairly well reproduced by the available 1145 numerical data, although the differences among the computations and the experiments are generally 1138 1138 data, The time The time 1144 although av average erage The the time dd di b b averag aer asese ences pressu pressu ed ba among sere re pressure reg reg the ion was ion was region computations was also also also fairly w ffairly w airland y welle the ell r reproduced b ll r experiments eeproduced produced y the ava by by ar the ila e the b generally le available available larger 1138 1138 The time The time av av erage erage dd b b ase ase pressu pressu re re reg reg ion was ion was also also fairly w fairly w ell r ell r eproduced eproduced by by the the available available 1146 larger than those reported in Figure 48. Indeed, the underlying physics is more complex, as the 1145 numerical data, although the differences among the computations and the experiments are generally 1139 numerical data, although the differences among the computations and the experiments are generally 1139 1139 1139 numeri numeri numeri than ca ca ca l da those l da l da ta ta ta , al , r, eported al al though the dif though the dif though the dif in Figur fe fe f rences e erences 48 rences . Indeed, am a am m ong th ong th ong th the e co underlying e co e co mputations mputations mputations physics and the experiments are generally and the experiments are generally and the experiments are generally is more complex, as the presence 1147 presence of the two pressure and suction side sharp over-expansions at the locations of the boundary 1146 larger than those reported in Figure 48. Indeed, the underlying physics is more complex, as the 1140 1140 lar lar gg er th of er th the an t an t two hh ose reporte ose