International Journal of Turbomachinery, Propulsion and Power
, Volume 5 (2) – May 20, 2020

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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 eects 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 dierence 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 eect 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 eect on the rear blade surface and on the trailing edge pressure distribution, but also oered 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 eect 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 eect 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 eect of vortex shedding on the blade pressure distribution (Section 5) and the eect 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 aecting 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 eect and a third term te which did take into account that the static pressure at the trailing edge diered 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 dierent 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 dierence 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 dierentiate 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 dierent 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 eect on the pressure dierence 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 dierence 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 dierence 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 dierence 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 dierence 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 dierence 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 eect 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 dierent 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 eects (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 dierence 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 aect 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 aect 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 aect the base pressure, the dierence may be attributed to (a) dierent total pressures due to shock losses for the shock interference curve, (b) dierences in the boundary layer shape factor and (c) dierences in pressure gradients in stream-wise direction in the near wake region. A systematic investigation of possible eects 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 eect 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 aect signiﬁcantly the base pressure. Therefore, we need to look for possible other inﬂuence factors. Figure 3 showed that the eect of the blade rear suction side blade turning angle " on the base pressure was in fact function of the pressure dierence 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 dierences 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 dierent 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 dierences 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 dierence 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 eect of boundary layer separation resulting from high rear suction side diusion 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 dierences 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 dierent 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 dierences 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 dierent 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 dierent 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 dierences 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 dierent temperatures and (c) on the work due to viscous stresses between regions of dierent 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 dierent: 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. Eect 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 eect in the wakes due to the von Kármán vortices. Naturally the vortex shedding aects 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. Eect 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 eect 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 dierences 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 eect 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 aect 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 eect 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 aect 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. Eect 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 dierences of base pressure data published by dierent 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 dierent research organizations may be partially due to the use of very dierent 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 Rael 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 eect 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. Eect 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 dierent 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 dierences 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 eective 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 eective 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 eective 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 dierent. 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 dierences between them. The possible inﬂuence of the dierent 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 dierent reason. The key for the understanding comes from ﬂat plate tests presented in [16], see Figure 44, which showed that the dierence 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 dierent 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 aects 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 aected 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 aecting 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 aecting 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 dierence and ﬁnite volume methods, together with the increasingly popular spectral-element methods, oer 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 oering 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 dier. 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 diusion factor smaller than 0.5 yielding a /C ratio less than 1% according to Stewart correlation [76]. This eectively 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 eects 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 aecting 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 dierence 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 oered 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 oering 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 Dierence 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 dierential (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 eort 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 dierent. Wall resolved Large Eddy Simulations (LES) are now recognized as unaordable for engineering applications because of the very stringent near wall resolution requirements and of the inability of all SGS models to account for the eects 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 suered 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 oering 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 suer 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 eorts 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 Dierence 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 dierent 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 aected 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 eects 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