Study on Film Cooling Performance of Round Hole Embedded in Different Shaped Craters and Trenches

Xiaojun Wu; Xin Du; Chunhua Wang

doi:10.3390/aerospace8060147

Study on Film Cooling Performance of Round Hole Embedded in Different Shaped Craters and Trenches

Wu, Xiaojun;Du, Xin;Wang, Chunhua 2021-05-25 00:00:00 aerospace Article Study on Film Cooling Performance of Round Hole Embedded in Different Shaped Craters and Trenches 1 1 1 , 2 , Xiaojun Wu , Xin Du and Chunhua Wang * AECC Shenyang Engine Research Institute, No. 1 Wanlian Road, Shenyang 110015, China; 18002492948@163.com (X.W.); dxknight@163.com (X.D.) College of Energy and Power Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China * Correspondence: chunhuawang@nuaa.edu.cn Abstract: Film cooling effectiveness can be improved signiﬁcantly by embedding a round hole in trenches or craters. In this study, ﬁlm cooling performances of a transverse trench, W-shaped trench and elliptic trench were compared and analyzed in detail. The CFD models for trench ﬁlm cooling were established and validated via the experimental results. Inside the transverse trench, a pair of recirculating vortices is formed, which promotes the coolant spreading in a lateral direction. The decrease of trench width and increase of trench depth both improve the ﬁlm cooling effectiveness of the transverse trench. For the W-shaped trench, the guide effect of the corner angle further improves the lateral spreading capability of coolant and generates higher cooling effectiveness than a transverse trench with the same depth and width. The ﬂow characteristics of the elliptic trench are similar to that of the round hole, and the kidney vortex pair takes a dominant role in the ﬂow ﬁelds downstream of the coolant exit. Accordingly, the elliptic trench generates the worst cooling performance in these shaped trenches. The increase of trench depth and decrease of trench width both result in an increase Citation: Wu, X.; Du, X.; Wang, C. of the discharge coefﬁcient for trench ﬁlm cooling. For the W-shaped trench, the increase of the corner Study on Film Cooling Performance angle causes a decrease of the discharge coefﬁcient. For the elliptic trench, the discharge coefﬁcient of Round Hole Embedded in increases with the decrease of the elliptic aspect ratio (major axis/minor axis). Different Shaped Craters and Trenches. Aerospace 2021, 8, 147. Keywords: film cooling; shaped trench; CFD; adiabatic film cooling effectiveness; discharge coefficient https://doi.org/10.3390/ aerospace8060147 Academic Editor: Qiang Zhang 1. Introduction To increase output efﬁciency, gas turbines are usually operated at the temperature Received: 16 April 2021 higher than the maximum allowable value for materials. To avoid the thermal damage of Accepted: 17 May 2021 the airfoil, various cooling strategies such as ﬁlm cooling, impingement cooling and pin ﬁns Published: 25 May 2021 are applied. In the external ﬁlm cooling process, coolant air extracted from the compressor is ejected through inclined holes and forms a coolant ﬁlm on the airfoil surface [1]. The Publisher’s Note: MDPI stays neutral coolant ﬁlm not only cools the airfoil surface, but also reduces the heat ﬂow from the with regard to jurisdictional claims in mainstream to hot-section surface. However, for a traditional inclined cylindrical hole, published maps and institutional afﬁl- the coolant injection tends to separate from the downstream wall, especially at high jet iations. momentum, and the spanwise coverage of the coolant is also weak. This causes low area-averaged ﬁlm cooling effectiveness [2,3]. To improve the spanwise coverage of coolant, one approach is to change the hole geometry from round to shaped outlet [4,5]. The transition from cylindrical inlet to shaped Copyright: © 2021 by the authors. outlet results in a decrease of outlet coolant momentum. Flow deceleration in the hole Licensee MDPI, Basel, Switzerland. diffuser section also promotes the spanwise spreading of coolant, which improves cooling This article is an open access article performance. Accordingly, to get the same cooling effectiveness, the coolant consumption distributed under the terms and conditions of the Creative Commons decreases. Over the past 40+ years, many different outlet shapes for ﬁlm cooling holes Attribution (CC BY) license (https:// have been proposed [2,3]. However, because of the limitation of manufacturing conditions, creativecommons.org/licenses/by/ shaped hole technology has not been widely used in practice. 4.0/). Aerospace 2021, 8, 147. https://doi.org/10.3390/aerospace8060147 https://www.mdpi.com/journal/aerospace Aerospace 2021, 8, 147 2 of 17 A certain conﬁguration by embedding a round hole in a transverse trench generates similar cooling performance to shaped holes. As trench conﬁguration can be fabricated via thermal barrier coating (TBC), trench ﬁlm cooling is more practical compared with other shaped holes [6]. Moreover, transverse trenches with the compound angle injection of a round hole or shaped hole can further improve ﬁlm cooling performance [7,8]. A trench study on a ﬂat plate was performed by Bunker [9]. He reported that, at high jet momentum, the coolant can still attach closely to the surface. This result is conﬁrmed by Harrison and Bogard [10] and Lu et al. [11–13]. The CFD and experimental results from Lu et al. [11–13] showed that, compared with a round hole, higher ﬁlm effectiveness and heat ﬂux reduction can be generated by embedding a round hole in a trench. Moreover, the heat transfer coefﬁcient does not change obviously after introducing a trench. Trench ﬁlm cooling on the vane surface was studied by Waye and Board [14]. As the trench width decreases, the cooling effectiveness increases. Even at high jet momentum, the mainstream can suppress the coolant jet ejected from the trench on the vane surface effectively. The inﬂuences of trench geometries on temperature distribution on a TBC coated vane were investigated by Davidson et al. [15]. Their results showed that, compared with round holes, trenches and craters can generate much better coolant coverage, however, temperature at the interface of the vane and TBC only shows a slight decrease. Lee and Kim [16] also conducted parametric studies on trench ﬁlm cooling. A trench height of 1 diameter and trench width of 2 diameter of the round hole generates the highest cooling effectiveness both at low and high jet momentums. Moreover, reverse injection of coolant can improve the ﬁlm cooling performance. Besides transverse trenches, some other shaped trenches are also proposed and have been proved to generate higher ﬁlm cooling effectiveness than traditional round holes [17]. Lu et al. [18] tested ﬁlm the cooling effectiveness of a round hole embedded in the elliptic crater. Their experimental results show that, compared with round holes, crater ﬁlm cooling generates a higher heat transfer coefﬁcient and cooling effectiveness. Dorrington et al. [19] also concluded the crater hole generates lower cooling effectiveness than the trench conﬁgu- ration, but higher effectiveness than cylindrical holes. Kross and Pﬁtzner [20,21] found that placing a tetrahedral element upstream of the trench can improve the cooling performance by reducing the coolant-mainstream mixture within the trench and improving lateral coolant spreading. Wei et al. [22] and Zhang et al. [23] developed double- and sine-wave trenches, and the inﬂuences of wave geometries on cooling effectiveness were studied. In the present study, a systematic parametric study is performed for trench or crater ﬁlm cooling. CFD models for a transverse trench, W-shaped trench and elliptic trench were established and validated using the experimental results. The ﬂow mechanisms and cooling performances with different trench shapes were analyzed in detail. 2. Computational Model 2.1. Computational Domain As illustrated in Figure 1, the computational domain consists of a mainstream channel, coolant channel, cylindrical hole and shaped trench. The cylindrical hole has an inclination angle of 30 and diameter of 5.0 mm. The total height including the cylindrical section and shaped trench, h , is 3d. The width of the mainstream and coolant channel is 3.0d. The coordinate origin locates at the trench exit center, and the x, y and z axes correspond to streamwise, spanwise and vertical direction, respectively. Three kinds of shaped trenches, a transverse trench, W-shaped trench and elliptic trench, were investigated. The geometries of the trench such as depth (h), width (w), corner angle (a) and axis length (D , D ) are x y deﬁned in Figure 1. The changing interval of these parameters are listed in Table 1. Aerospace 2021, 8, 147 3 of 17 Aerospace 2021, 8, x FOR PEER REVIEW 3 of 18 (a) 3D-view (b) Front view (c) Transverse trench (d) W-shaped trench (e) Elliptic trench Figure 1. Computational domain and geometric variables for trench holes in the present study. Figure 1. Computational domain and geometric variables for trench holes in the present study. Table 1. Table 1.Changing interval of Changing interval of ttr rench geometries. ench geometries. Trench Type Symbol Changing Interval Trench Type Symbol Changing Interval W 2.5~3.5d W 2.5~3.5d Transverse trench Transverse trench h h 0. 0.25~1.8 25~1.8dd W 1.2~2.4d W 1.2~2.4d W-sh W-shaped aped tre trench nch h h 0. 0.25~1.8 25~1.8dd a 40~80 α 40~80° D 1.2~3.2d Dxx 1.2~3.2d Elliptic trench D 1.2~3.2d Elliptic trench Dy 1.2~3.2d h 0.25~1.8d h 0.25~1.8d 2.2. Performance Evaluation Parameters for Film Cooling 2.2. Performance Evaluation Parameters for Film Cooling Adiabatic cooling effectiveness is an important index to evaluate ﬁlm cooling perfor- Adiabatic cooling effectiveness is an important index to evaluate film cooling perfor- mance, and can be calculated by: mance, and can be calculated by: T T ¥ T - T ad,w ∞ ad,w h (x, y) = (1) η (x, y) = (1) ad,loc ad,loc T T ¥ T - T c ∞ c Z △y/2 1 Dy/2 η (x) = η (x, y)dy h (x) = h (x, y)dy (2 (2) ) ad,lat ad, loc ad,lat ad, loc △y Dy D -△ y/ y/2 2 x 2 1 2 h = h (x)dx (3) ad,av ad, lat η = η (x) dx (3) ad,av Dx ad, lat ∆x where T is the adiabatic wall temperature, and T and T are the temperature of c ¥ ad,w where Tad,w is the adiabatic wall temperature, and Tc and T∞ are the temperature of the the coolant and mainstream. The subscripts ‘loc’, ‘lat’ and ‘av’ denote the local, laterally coolant and mainstream. The subscripts ‘loc’, ‘lat’ and ‘av’ denote the local, laterally aver- aged and area-averaged value, respectively. In the present study, △y = 3d, x1 = 2d and x2 = 20d. The discharge coefficient, Cd, is defined as Aerospace 2021, 8, 147 4 of 17 averaged and area-averaged value, respectively. In the present study, 4y = 3d, x = 2d and x = 20d. The discharge coefﬁcient, C , is deﬁned as C = q (4) Aerospace 2021, 8, x FOR PEER REVIEW 4 of 18 A 2r (P P c c,in c,out where m is the coolant mass, A is the cross section area of the round hole, P* and c c C,in P are the inlet total pressure of secondary ﬂow and the static pressure downstream c,out d A 2ρ P* - P ) c c,in c,out of the trench. (4) where mc is the coolant mass, Ac is the cross section area of the round hole, P*C,in and 2.3. Boundary Condition and Solution Method Pc,outare the inlet total pressure of secondary flow and the static pressure downstream of At the mainstream inlet, the velocity proﬁle with a 1/7 power law and the boundary the trench. thickness (d ) of 0.125d were speciﬁed. This is the same as that in the experimental 2.3. Boundary Condition and Solution Method conditions. The mainstream mean velocity is 20 m/s, and the temperature is 353 K. The At the mainstream inlet, the velocity profile with a 1/7 power law and the boundary turbulent intensity and length scale is 4% and 0.4d respectively. The coolant temperature thickness (δ99) of 0.125d were specified. This is the same as that in the experimental condi- is 300 K. The turbulent parameters at the coolant inlet are the same as the mainstream tions. The mainstream mean velocity is 20m/s, and the temperature is 353K. The turbulent inlet. Because of a low Mach number (<0.3), incompressible ideal gas is used. The top intensity and length scale is 4% and 0.4δ99 respectively. The coolant temperature is 300K. surface of the mainstream channel was set as a free boundary. The spanwise surfaces of the The turbulent parameters at the coolant inlet are the same as the mainstream inlet. Because mainstream and coolant channel were set as periodic boundaries. Other surfaces were set of a low Mach number (<0.3), incompressible ideal gas is used. The top surface of the as a non-slip adiabatic wall. In the present study, the density ratio is 1.176, the blowing mainstream channel was set as a free boundary. The spanwise surfaces of the mainstream 2 2 ratio (M = r u /r u ) is 0.5~3.0 and the momentum ratio (I = r u /r u ) is 0.21~7.65. and coolant channel were set as periodic boundaries. Other surfaces were set as a non-slip c c ¥ ¥ c c ¥ ¥ adiabatic wall. In the present study, the density ratio is 1.176, the blowing ratio (M = ANSYS Fluent is applied for solutions of governing equations. According to Ref. [10], 2 2 ρcuc/ρ∞u∞) is 0.5~3.0 and the momentum ratio (I = ρcuc /ρ∞u∞ ) is 0.21~7.65. realizable k-" equations with enhanced wall treatment are suitable for trench ﬁlm cooling. ANSYS Fluent is applied for solutions of governing equations. According to Ref. [10], The momentum, energy and turbulent equations were solved using a second-order upwind realizable k-ε equations with enhanced wall treatment are suitable for trench film cooling. scheme. The gradient and pressure interpolation were performed using a least squares The momentum, energy and turbulent equations were solved using a second-order up- cell-based scheme and second-order scheme, respectively. The convergence criteria include: wind scheme. The gradient and pressure interpolation were performed using a least (1) the mass balance error is smaller than 10 , (2) the normalized residuals are smaller squares cell-based scheme and second-order scheme, respectively. The convergence crite- 6 2 −6 ria inc than 10 lude: (1) the m , and (3) ass b theavariation lance error is smaller th of local adiabatic an 10 , (2 ef ) the norma fectiveness lized residu is smaller als a than re 10 . −6 −2 smaller than 10 , and (3) the variation of local adiabatic effectiveness is smaller than 10 . Structural meshes are created with ICEM software. As shown in Figure 2, near the Structural meshes are created with ICEM software. As shown in Figure 2, near the walls, the grid points are clustered. On the ﬂat plate, the wall-normal size of the ﬁrst- walls, the grid points are clustered. On the flat plate, the wall-normal size of the first-layer layer grid is 4z = 0.003d, which corresponds to z 1. In the wall-normal direction, grid is △z = 0.003d, which corresponds to z ≈ 1. In the wall-normal direction, the stretching the stretching factor is smaller than 1.2 in the near-wall region. Grid independence tests factor is smaller than 1.2 in the near-wall region. Grid independence tests were carried out were carried out to determine the optimal grid number. Taking the transverse trench with to determine the optimal grid number. Taking the transverse trench with w = 2.2d and h = 0. w5= d as 2.2 an example, the grid test result is show d and h = 0.5d as an example, the grid n in Figure 3. The calculation results do not test result is shown in Figure 3. The calculation change obviously as the grid number exceeds 1,896,323. results do not change obviously as the grid number exceeds 1,896,323. (a) Front view (b) Transverse trench (c) W-shaped trench (d) Elliptic trench Figure 2. Grids used in the present study. Figure 2. Grids used in the present study. Aerospace 2021, 8, x FOR PEER REVIEW 5 of 18 Aerospace 2021, 8, 147 5 of 17 Aerospace 2021, 8, x FOR PEER REVIEW 5 of 18 0.50 713,956 0.50 0.45 1,245,014 713,956 1,896,323 0.45 1,245,014 0.40 2,533,437 1,896,323 0.40 2,533,437 0.35 0.35 0.30 0.30 0.25 0.25 0.20 0.20 0.15 0.15 0 2 4 6 8 10 121416182022 0 2 4 6 8 10 121416182022 x/d x/d Figure 3. Gird independent test results. Figure 3. Gird independent test results. Figure 3. Gird independent test results. 3. Experimental Validation 3. Experimental Validation 3. Experimental Validation Figure 4 shows the experimental system. After being heated to 80 °C, the mainstream Figure 4 shows the experimental system. After being heated to 80 C, the mainstream from an Fig air ure co 4 sho mpressor ws the experimental s passes through a rectifie ystem. Af r section ter being hea and then enters the test ted to 80 °C, the mai secti nstrea ons. m from an air compressor passes through a rectiﬁer section and then enters the test sections. The cros from an air s sec co tion s mpressor ize o f passes the mthrough ainstrea a rectifie m channe r section l is 174m an m d then enters the test ×80mm, and the size o secti f t oh ns. e The cross section size of the mainstream channel is 174 mm 80 mm, and the size of the The cross section size of the mainstream channel is 174mm×80mm, and the size of the coolant channel is 64mm × 40mm. The inlet velocity and temperature are the same as the coolant channel is 64mm 40mm. The inlet velocity and temperature are the same as the coolant channel is 64mm × 40mm. The inlet velocity and temperature are the same as the CFD model. The boundary layer thickness and turbulent parameters for the mainstream CFD model. The boundary layer thickness and turbulent parameters for the mainstream CFD model. The boundary layer thickness and turbulent parameters for the mainstream inlet were measured at x/d = −15 using a hot wire anemometer (StreamLine Pro). The inlet were measured at x/d = 15 using a hot wire anemometer (StreamLine Pro). The inlet were measured at x/d = −15 using a hot wire anemometer (StreamLine Pro). The boundary layer thickness (δ99) is 0.125d. The turbulent intensity and length scale is 4% and boundary layer thickness (d ) is 0.125d. The turbulent intensity and length scale is 4% 0. boundar 4δ99. The test pl y layer thickn ate, wiess ( th a therma δ99) is 0.125 l conducti d. The t vi uty rbu ol fent 0.3 W/ inte(m·K nsity and ), is made o lengthf r sc u abber wood. le is 4% and and 0.4d . The test plate, with a thermal conductivity of 0.3 W/(mK), is made of rubber The size 0.4δ99. The test pl s of the hole ate, wi and th ta therma he trench l conducti are the same vity o as f 0.the computat 3 W/(m·K), is ional mo made of r del. ubber wood. The hole wood. The sizes of the hole and the trench are the same as the computational model. The sizes of the hole and the trench are the same as the computational model. The hole pitch is 3.0d. The transverse trench (w = 3.2d, h = 1d), the W-shaped trench (w = 1.7d, h = 1d The hole pitch is 3.0d. The transverse trench (w = 3.2d, h = 1d), the W-shaped trench pitch is 3.0d. The transverse trench (w = 3.2d, h = 1d), the W-shaped trench (w = 1.7d, h = 1d and α = 60°) and the elliptic trench (Dx = 2.2d, Dy = 2.8d and h = 1d) are tested in the present (w = 1.7d, h = 1d and a = 60 ) and the elliptic trench (D = 2.2d, D = 2.8d and h = 1d) are x y and α = 60°) and the elliptic trench (Dx = 2.2d, Dy = 2.8d and h = 1d) are tested in the present study. Viewing though CaF2-infrared glass, an infrared thermography system is applied tested in the present study. Viewing though CaF -infrared glass, an infrared thermography for temperature measurem study. Viewing though Caent of the flat F2-infrared glass plate , an wi infr th black pa ared therm int coa ogra ting. The emissi phy system is ap vity of plied system is applied for temperature measurement of the ﬂat plate with black paint coating. tfor temperature measurem he black paint is about 0.97ent of the flat . The infrared plate thermography (M with black pa ag in32HF model) pr t coating. The emissi oduced b vity of y The emissivity of the black paint is about 0.97. The infrared thermography (Mag32HF the black paint is about 0.97. The infrared thermography (Mag32HF model) produced by Magnity Electronics Co. Ltd. (Shanghai, China) has a test range of −20~300 °C and an ac- model) produced by Magnity Electronics Co. Ltd. (Shanghai, China) has a test range of Magnity Electronics Co. Ltd. (Shanghai, China) has a test range of −20~300 °C and an ac- curacy of ±1 °C. The calibration of the infrared measurement was performed according to 20~300 C and an accuracy of 1 C. The calibration of the infrared measurement was curacy of ±1 °C. The calibration of the infrared measurement was performed according to the temperature measured via thermocouples within the plate. Detail calibration pro- performed according to the temperature measured via thermocouples within the plate. the temperature measured via thermocouples within the plate. Detail calibration pro- cesses are introduced in Ref. [24]. Detail calibration processes are introduced in Ref. [24]. cesses are introduced in Ref. [24]. Infrared camera Temperature probe Infrared camera Temperature probe Valve Valve Baffles Test section section Baffles Primary flow Flow meter Test section Heater section Primary flow Flow meter Heater Flow meter Valve Coolant flow Flow meter Valve Coolant flow (a) Experimental system (b) Test plates (a) Experimental system (b) Test plates Figure 4. Experimental system in the present study. Figure 4. Experimental system in the present study. Figure 4. Experimental system in the present study. To better compare the CFD and experimental results, the heat conduction effect inside To better compare the CFD and experimental results, the heat conduction effect in- the ﬁlm-cooling To better com plate pare t is h taken e CFD an into account, d experim and entthe al result thermal s, the heat co conductivity nduction is 0.3W/(m effect in- K). side the film-cooling plate is taken into account, and the thermal conductivity is 0.side The 3W/ t top (m he ·K and fi ). lm The t lower -coolin op surfaces and g plat lower s e ofis the tua solid rken face int plate s of t o h wer account, a e sol e coupled id plat nd the therma e we with re c theoup ﬂuid led w l conducti phase. ith tThe he fl vity i span- uids wise surfaces of the plate were set as periodic boundaries. Figure 5 shows the distributions p0. h3W/ ase. The (m·Ksp ). anwi The tse s op and urfaclower s es of the p urflace ates were of th set e sol as p ide p riod late we ic bore c undar oup ies. led w Figu itre h 5 thshows e fluid p of hase. the The overall spanwi cooling se sef ur fectiveness faces of the p [h late were = (T set as p T e )/( riod T ic b oTun )]dar onithe es. F cooling igure 5 surface. shows the distributions of the overall cooling effectiveness ¥ w [ηoverall¥ = (T∞c − Tw)/(T∞ − Tc)] on the overall Inside the trench, the difference between the CFD and the experimental results is somewhat the distributions of the overall cooling effectiveness [ηoverall = (T∞ − Tw)/(T∞ − Tc)] on the cooling surface. Inside the trench, the difference between the CFD and the experimental cooling surface. Inside the trench, the difference between the CFD and the experimental η ad,lat ad,lat Aerospace 2021, 8, 147 6 of 17 Aerospace 2021, 8, x FOR PEER REVIEW 6 of 18 results is somewhat obvious, especially for the transverse trench at a high blowing ratio. obvious, especially for the transverse trench at a high blowing ratio. It illustrates that the It illustrates that the present CFD model overestimated the spreading capability of coolant present CFD model overestimated the spreading capability of coolant inside the trench, inside the trench, which results in better cooling performance. Figure 6 shows a quantita- which results in better cooling performance. Figure 6 shows a quantitative comparison tive comparison between the experimental and CFD results. The experimental data for the between the experimental and CFD results. The experimental data for the round hole is round hole is from Ref. [25]. At M = 0.5, the mean relative error for the round hole, trans- from Ref. [25]. At M = 0.5, the mean relative error for the round hole, transverse trench, verse trench, W-shaped trench and elliptic trench is about 9.4%, 7.7%, 8.5% and 10.6%. At W-shaped trench and elliptic trench is about 9.4%, 7.7%, 8.5% and 10.6%. At M = 1.5, the M = 1.5, the mean relative error for the round hole, transverse trench, W-shaped trench mean relative error for the round hole, transverse trench, W-shaped trench and elliptic and elliptic trench is about 16.4%, 10.2%, 12.2% and 13.9%. Overall, the CFD results agree trench is about 16.4%, 10.2%, 12.2% and 13.9%. Overall, the CFD results agree well with well with the experimental results. the experimental results. Exp CFD (a) Transverse trench (w = 3.0d, h = 1.0d) Exp CFD (b) W−shaped trench (w = 1.7d, h = 1d, α = 60°) Exp CFD (c) Elliptic trench (Dx = 2.2d, Dy = 2.8d, h = 1d) Figure 5. Distributions of overall cooling effectiveness on the cooling surface. Figure 5. Distributions of overall cooling effectiveness on the cooling surface. Aerospace 2021, 8, 147 7 of 17 Aerospace 2021, 8, x FOR PEER REVIEW 7 of 18 0.9 0.9 0.8 0.8 M=0.5-EXP[25] M=0.5-EXP M=0.5-CFD 0.7 0.7 M=0.5-CFD M=1.5-EXP[25] M=1.5-EXP 0.6 M=1.5-CFD 0.6 M=1.5-CFD 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0.0 0.0 0 2 4 6 8 101214 161820 0 2 4 6 8 101214 16 1820 x/d x/d (a) Round hole (b) Transverse trench (w = 3.0d, h = 1d) 0.9 0.7 0.8 M=0.5-EXP 0.6 0.7 M=0.5-CFD M=0.5-EXP M=1.5-EXP 0.5 0.6 M=0.5-CFD M=1.5-CFD M=1.5-EXP 0.4 0.5 M=1.5-CFD 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0.0 0.0 0 2 4 6 8 101214 161820 0 2 4 6 8 101214 16 1820 x/d x/d (c) W-shaped trench (w = 1.7d, h = 1d, α = 60°) (d) Elliptic trench (Dx = 2.2d, Dy = 2.8d, h = 1d) Figure 6. Quantitative comparison between CFD results and experimental results. Figure 6. Quantitative comparison between CFD results and experimental results. 4. CFD Results Analysis 4. CFD Results Analysis 4.1. Transverse Trench 4.1. Transverse Trench Figure 7 shows the variation of laterally averaged adiabatic cooling effectiveness Figure 7 shows the variation of laterally averaged adiabatic cooling effectiveness (h ) with the streamwise distance (x/d). For the round hole, h shows a continued ad,lat ad,lat (ηad,lat) with the streamwise distance (x/d). For the round hole, ηad,lat shows a continued decline as x/d increases at a low blowing ratio, but has a slight increase in the far ﬁeld region decline as x/d increases at a low blowing ratio, but has a slight increase in the far field due to the reattachment of a separated coolant jet at a high blowing ratio. For the transverse region due to the reattachment of a separated coolant jet at a high blowing ratio. For the trench, h decreases constantly with the increase of x/d even at a high blowing ratio. ad,lat transverse trench, ηad,lat decreases constantly with the increase of x/d even at a high blow- The optimal blowing ratio for the round hole is about 0.5. However, for trench-ﬁlm ing ratio. The optimal blowing ratio for the round hole is about 0.5. However, for trench- cooling in the present case, the optimal blowing ratio is between 1.0~1.5. Figure 8a,b film cooling in the present case, the optimal blowing ratio is between 1.0~1.5. Figure 8a,b show the streamline distributions for ﬁlm cooling of the transverse trench and round hole show the streamline distributions for film cooling of the transverse trench and round hole at M = 1.5, respectively, and the background color represents gas temperature. The most at M = 1.5, respectively, and the background color represents gas temperature. The most typical feature for trench ﬁlm cooling is that a pair of recirculating vortices is formed inside typical feature for trench film cooling is that a pair of recirculating vortices is formed in- the trench. The entrainment of recirculating vortices promotes the spreading of coolant side the trench. The entrainment of recirculating vortices promotes the spreading of cool- in the lateral direction [16,26]. The existence of the trench also increases the actual jet exit ant in the lateral direction [16,26]. The existence of the trench also increases the actual jet area, reduces the actual jet velocity and mitigates the jet detachment downstream of the exit area, reduces the actual jet velocity and mitigates the jet detachment downstream of hole. In the ﬂow ﬁelds downstream of the round hole, a pair of kidney vortices (also called the hole. In the flow fields downstream of the round hole, a pair of kidney vortices (also countered rotating vortices) dominate the ﬂow ﬁeld and promote the mixture between called countered rotating vortices) dominate the flow field and promote the mixture be- mainstream and coolant [27]. For trench ﬁlm cooling, beside kidney vortices, a pair of tween mainstream and coolant [27]. For trench film cooling, beside kidney vortices, a pair anti-kidney vortices can be observed. The anti-kidney vortices with the opposite rotating of anti-kidney vortices can be observed. The anti-kidney vortices with the opposite rotat- direction of kidney vortices mitigate the detachment of coolant jet and improve the ﬁlm ing direction of kidney vortices mitigate the detachment of coolant jet and improve the cooling performance [28]. Overall, compared with a round hole, a trench hole generates film cooling performance [28]. Overall, compared with a round hole, a trench hole gener- better cooling performance, especially at a high blowing ratio. Moreover, the results from ates better cooling performance, especially at a high blowing ratio. Moreover, the results Lu et al. [11–13] show that the heat transfer coefﬁcient does not change obviously after from Lu et al. [11–13] show that the heat transfer coefficient does not change obviously introducing the trench. after introducing the trench. overall, lat overall, lat overall, lat overall, lat Aerospace Aerospace 2021 2021 , 8 , , x FO 8, 147R PEER REVIEW 8 of 8 of 18 17 Aerospace 2021, 8, x FOR PEER REVIEW 8 of 18 0.8 0.4 0.8 0.4 M=0.5 M=0.5 M=0.5 M=0.5 0.7 M=1 0.7 M=1 M=1 M=1 M=1.5 M=1.5 M=1.5 M=1.5 0.6 0.3 0.6 0.3 M=2 M=2 M=2 M=2 M=3 0.5 M=3 0.5 η η ad,lat ad,lat ad,lat ad,lat 0.4 0.2 0.2 0.4 0.3 0.3 0.1 0.2 0.1 0.2 0.1 0.1 0.0 0.0 0.0 0.0 02468 10 12 14 16 18 20 22 02468 10 12 14 16 18 20 22 0 2 4 6 8 10 12 14 16 18 20 22 0 2 4 6 8 10 12 14 16 18 20 22 x/d x/d x/d x/d (a) Round hole (b) Transverse trench (w = 3d, h = 0.75d) (a) Round hole (b) Transverse trench (w = 3d, h = 0.75d) Figure 7. Variation of laterally averaged adiabatic cooling effectiveness with streamwise distance. Figure 7. Variation of laterally averaged adiabatic cooling effectiveness with streamwise distance. Figure 7. Variation of laterally averaged adiabatic cooling effectiveness with streamwise distance. (a) Transverse trench (w = 3d, h = 0.75d) (a) Transverse trench (w = 3d, h = 0.75d) (b) Round hole (b) Round hole Figure 8. Streamline distributions for film cooling of round hole and transverse trench at M = 1.5. Figure 8. Figure 8.Strea Streamline mline di distributions stributions for for film ﬁlm co cooling oling of round of round hole and transverse trench hole and transverse trench at M at = 1 M = .5. 1.5. (KVP: kidney vortex pair). (KVP: kidney vortex pair). (KVP: kidney vortex pair). Fig Figur ure e9 sho 9 shows ws th the e vari variation ation of loc of local al ad adiabatic iabatic cool cooling ing eff effectiveness ectiveness ((ηhad,loc) wi ) with th the the Figure 9 shows the variation of local adiabatic cooling effectiveness (η ad,loc ad,loc) with the spanwise distance ( y/d ). For the round hole, ηad,loc has the maximum value at y/d = 0, and spanwise distance (jy/dj). For the round hole, h has the maximum value at y/d = 0, spanwise distance ( y/d ). For the round hole, ηad,loc ad,loc has the maximum value at y/d = 0, and then decre and thenadecr ses sh eases arply wit sharply h the incre with the ase incr of ease y/d . ofηad,loc jy/ d in tj. hhe centerline in theregion centerline ( y/dr < egion 1) ad,loc then decreases sharply with the increase of y/d . ηad,loc in the centerline region ( y/d < 1) (jy/dj < 1) decreases by increasing the blowing ratio from 0.5 to 2.0. However, the effects decreases by increasing the blowing ratio from 0.5 to 2.0. However, the effects of the blow- decreases by increasing the blowing ratio from 0.5 to 2.0. However, the effects of the blow- of the blowing ratio on h at jy/dj > 1 is unobvious. For the transverse trench, at a low ing ratio on ηad,loc at y/d > 1 is unobvious. For the transverse trench, at a low blowing ad,loc ing ratio on ηad,loc at y/d > 1 is unobvious. For the transverse trench, at a low blowing blowing ratio (M = 0.5), the distribution of h in the lateral direction is similar to that for ratio (M = 0.5), the distribution of ηad,loc in the lateral direction is similar to that for the ad,loc ratio (M = 0.5), the distribution of ηad,loc in the lateral direction is similar to that for the the round hole. However, at a high blowing ratio, as the lateral distance increases, h round hole. However, at a high blowing ratio, as the lateral distance increases, ηad,locad,loc in- round hole. However, at a high blowing ratio, as the lateral distance increases, ηad,loc in- increases ﬁrstly ((jy/dj < 0.5), then decreases (0.5 <jy/dj < 1.2), and shows a slight increase creases firstly (( y/d < 0.5), then decreases (0.5 < y/d < 1.2), and shows a slight increase creases firstly (( y/d < 0.5), then decreases (0.5 < y/d < 1.2), and shows a slight increase ﬁnally ( y/d > 1.2). Figure 10 shows the distributions of ﬂow ﬁelds on the exit plane of j j finally ( y/d > 1.2). Figure 10 shows the distributions of flow fields on the exit plane of finally ( y/d > 1.2). Figure 10 shows the distributions of flow fields on the exit plane of holes; the background color represents the vertical velocity, and the arrow represents the holes; the background color represents the vertical velocity, and the arrow represents the holes; the background color represents the vertical velocity, and the arrow represents the clockwise or anti-clockwise rotation direction of the vortex. Compared with the transverse clockwise or anti-clockwise rotation direction of the vortex. Compared with the transverse clockwise or anti-clockwise rotation direction of the vortex. Compared with the transverse trench, the vertical velocity on the exit surface of the round hole distributes more uniformly trench, the vertical velocity on the exit surface of the round hole distributes more uni- trench, the vertical velocity on the exit surface of the round hole distributes more uni- and is of higher value. It results in the detachment of coolant jet immediately downstream formly and is of higher value. It results in the detachment of coolant jet immediately formly and is of higher value. It results in the detachment of coolant jet immediately of the round hole at a high blowing ratio. For the transverse trench, a pair of vortices is Aerospace 2021, 8, x FOR PEER REVIEW 9 of 18 Aerospace 2021, 8, x FOR PEER REVIEW 9 of 18 Aerospace 2021, 8, 147 9 of 17 downstream of the round hole at a high blowing ratio. For the transverse trench, a pair of downstream of the round hole at a high blowing ratio. For the transverse trench, a pair of vortices is formed, and the entrainment of the vortex pair promotes the lateral spreading vortices is formed, and form the ed, an entrainment d the entrainment of the vortex of th pair e vortex promotes pair promote the lateral s the lateral spreadingspread of coolant ing of coolant inside the trench. It results in the wavy distribution of vertical velocity in the of inside coolathe nt in trsi ench. de the trench. It resul It results in thetwavy s in the diwa stribution vy distribut of vertical ion of vertical velocity in the velocity in the lateral lateral direction. The wave crest locates at y/d = 0.0 where ηad,loc has a local minimum value lat dir eection. ral direct The ion. The wave wave crest locates crest loc atay te /s da= t y 0.0 /d = 0. wher 0 wh e h ere ηad,loc has ha a local s a local minimum va minimum value lue at a ad,loc at a high blowing ratio. at a high high blowing blowing r ratio. atio. 0.9 0.9 0.8 M=0.5 0.8 0.8 M=0.5 0.8 M=1 0.7 M=1 0.7 M=1.5 0.7 0.7 M=1.5 M=2 0.6 0.6 0.6 M=2 ad,loc η 0.6 η 0.5 ad,loc ad,loc 0.5 η ad,loc 0.5 M=0.5 0.4 0.5 M=0.5 0.4 M=1 0.3 M=1 0.4 M=1.5 0.3 0.4 M=1.5 0.2 M=2 0.2 0.3 M=2 M=3 0.1 0.3 M=3 0.1 0.0 0.2 0.0 0.2 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 y/d y/d y/d y/d (a) Round hole (b) Transverse trench (w = 3d, h = 0.75d) (a) Round hole (b) Transverse trench (w = 3d, h = 0.75d) Figure 9. Variation of local adiabatic film cooling effectiveness with spanwise distance on the cross Figure 9. Variation of local adiabatic ﬁlm cooling effectiveness with spanwise distance on the cross Figure 9. Variation of local adiabatic film cooling effectiveness with spanwise distance on the cross section of x/d = 2.2. section of x/d = 2.2. section of x/d = 2.2. (a) Round hole (b) Transverse trench (w = 3d, h = 0.75d) (a) Round hole (b) Transverse trench (w = 3d, h = 0.75d) Figure 10. Distribution of flow fields on the exit plane of the film cooling hole at M = 1.5 (unit: Figure 10. Distribution of flow fields on the exit plane of the film cooling hole at M = 1.5 (unit: Figure 10. Distribution of ﬂow ﬁelds on the exit plane of the ﬁlm cooling hole at M = 1.5 (unit: m/s). m/s). m/s). Figure 11a shows the effects of trench width (W/d) and depth (h/d) on the area-averaged Figure 11a shows the effects of trench width (W/d) and depth (h/d) on the area-aver- adiabatic Figure ﬁlm 11a sh cooling ows the e effectiveness ffects of trench w (h , 2 < idth ( x/d < W20). /d) an The d depth ( deep trh ench /d) on gives the a the rea highest -aver- ad,av aged adiabatic film cooling effectiveness (ηad,av, 2 < x/d < 20). The deep trench gives the aged adiabatic film cooling effectiveness (ηad,av, 2 < x/d < 20). The deep trench gives the h , and h decreases with the decrease of trench depth. For a shallow trench, the ad,av ad,av highest ηad,av, and ηad,av decreases with the decrease of trench depth. For a shallow trench, highest coolantη trajectory ad,av, and η is ad,a har v decr dly af ease fected s with th by the e decre trench, ase o and f trench depth. For a the pronounced ﬂow shallo separation w trench, still the coolant trajectory is hardly affected by the trench, and the pronounced flow separation the cool takes place ant tra downstr jectory eam is ha of rdl the y ahole. ffected b Inside y the trench, the deep tr and the pronounced ench, the recirculating flow ﬂow sepa reduces ration still takes place downstream of the hole. Inside the deep trench, the recirculating flow still take the coolant s place downstre penetration into am of the hole. Insi the mainstream and de the imprdeep trench, oves the coolant the recirc uniformity ulating flow . Accord- reduces the coolant penetration into the mainstream and improves the coolant uniformity. ingly, the coolant ejected from the deep trench exhibits better covering performance on the reduces the coolant penetration into the mainstream and improves the coolant uniformity. Accordingly, the coolant ejected from the deep trench exhibits better covering perfor- cooling surface downstream of the trench. For most cases, the ﬁlm cooling performance Accordingly, the coolant ejected from the deep trench exhibits better covering perfor- mance on the cooling surface downstream of the trench. For most cases, the film cooling can be improved effectively by reducing the trench width. For a narrow trench, because of mance on the cooling surface downstream of the trench. For most cases, the film cooling performance can be improved effectively by reducing the trench width. For a narrow the effect of recirculating ﬂow, the mainstream cannot enter the trench, which enhances performance can be improved effectively by reducing the trench width. For a narrow trench, because of the effect of recirculating flow, the mainstream cannot enter the trench, the ﬁlm cooling effectiveness. For a wide trench, due to the mainstream intursion, the trench, because of the effect of recirculating flow, the mainstream cannot enter the trench, which enhances the film cooling effectiveness. For a wide trench, due to the mainstream recirculation of the coolant inside the trench is mitigated, which results in the decrease which enhances the film cooling effectiveness. For a wide trench, due to the mainstream intursion, the recirculation of the coolant inside the trench is mitigated, which results in of cooling effectiveness [12,16]. Compared with trench width, the effect of trench depth intursion, the recirculation of the coolant inside the trench is mitigated, which results in the decrease of cooling effectiveness [12,16]. Compared with trench width, the effect of on cooling effectiveness is more pronounced. Figure 11b shows the effects of geometric the decrease of cooling effectiveness [12,16]. Compared with trench width, the effect of trench depth on cooling effectiveness is more pronounced. Figure 11b shows the effects of parameters on the discharge coefﬁcient (C ). The increase of h/d reduces the mixing loss trench depth on cooling effectiveness is more d pronounced. Figure 11b shows the effects of geometric parameters on the discharge coefficient (Cd). The increase of h/d reduces the between the mainstream and coolant jet, and the decrease of w/d results in the increase of geometric parameters on the discharge coefficient (Cd). The increase of h/d reduces the mixing loss between the mainstream and coolant jet, and the decrease of w/d results in the the actual blowing ratio. Thus, C increases with the increase of h/d but the decrease of mixing loss between the mainstream and coolant jet, and the decrease of w/d results in the w/d. Overall, a narrower and deeper trench generates better ﬁlm cooling performance. Aerospace 2021, 8, x FOR PEER REVIEW 10 of 18 Aerospace 2021, 8, x FOR PEER REVIEW 10 of 18 Aerospace 2021, 8, 147 10 of 17 increase of the actual blowing ratio. Thus, Cd increases with the increase of h/d but the increase of the actual blowing ratio. Thus, Cd increases with the increase of h/d but the decrease of w/d. Overall, a narrower and deeper trench generates better film cooling per- decrease of w/d. Overall, a narrower and deeper trench generates better film cooling per- formance. formance. W=2.5d, M=0.5 0.68 W=2.5d, M=0.5 0.68 0.5 0.5 W=2.5d, M=1.5 W=2.5d, M=1.5 W=3.0d, M=0.5 W=3.0d, M=0.5 0.66 0.66 W=3.0d, M=1.5 W=3.0d, M=1.5 0.4 0.4 W=3.5d, M=0.5 W=3.5d, M=0.5 0.64 0.64 W=3.5d, M=1.5 W=3.5d, M=1.5 0.3 0.3 0.62 0.62 w=2.5d, M=0.5 w=2.5d, M=0.5 0.60 0.60 w=2.5d, M=1.5 w=2.5d, M=1.5 0.2 0.2 w=3.0d, M=0.5 w=3.0d, M=0.5 0.58 0.58 w=3.0d, M=1.5 w=3.0d, M=1.5 0.1 0.1 w=3.5d, M=0.5 w=3.5d, M=0.5 0.56 0.56 w=3.5d, M=1.5 w=3.5d, M=1.5 0.54 0.0 0.54 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.2 0.4 0.6 0.8 1.0 1.2 1. 04 .2 0.4 0.6 0.8 1.0 1.2 1.4 0.2 0.4 0.6 0.8 1.0 1.2 1.4 h/d h/d h/d h/d (a) Area-averaged adiabatic cooling effectiveness (b) Discharge coefficient (a) Area-averaged adiabatic cooling effectiveness (b) Discharge coefficient Figure 11. Effects of trench depth and width on film cooling performance for transverse trench. Figure 11. Effects Figure of trench 11. Ef de fects pth and width of trench depth on film and cool width ing perform on ﬁlm ance f cooling or transverse performance trench. for transverse trench. 4.2. W-Shaped Trench 4.2. W-Shaped Trench 4.2. W-Shaped Trench Figure 12a shows the variation of laterally averaged adiabatic film cooling effective- Figure 12a shows the variation of laterally averaged adiabatic ﬁlm cooling effectiveness Figure 12a shows the variation of laterally averaged adiabatic film cooling effective- ness with the streamwise distance. As the blowing ratio increases, ηad,lat increases firstly, with the streamwise distance. As the blowing ratio increases, h increases ﬁrstly, ness with the streamwise distance. As the blowing ratio increases, ηad,lat ad,lat increases firstly, and then decreases. The optimal blowing ratio for the W-shaped trench is about 1.5. Com- and then decreases. The optimal blowing ratio for the W-shaped trench is about 1.5. and then decreases. The optimal blowing ratio for the W-shaped trench is about 1.5. Com- pared with the transverse trench, the cooling performance for the W-shaped trench is bet- Compared with the transverse trench, the cooling performance for the W-shaped trench is pared with the transverse trench, the cooling performance for the W-shaped trench is bet- ter. Figure 12b shows the variation of local adiabatic film cooling effectiveness with the better. Figure 12b shows the variation of local adiabatic ﬁlm cooling effectiveness with the ter. Figure 12b shows the variation of local adiabatic film cooling effectiveness with the spanwise distance. At a low blowing ratio (M = 0.5), as y/d increases, ηad,loc decreases spanwise spanwise dis distance. tance. At At a lo a low w blow blowing ing ratio ( ratioM (M = = 0.5) 0.5 , a ),sas yj /y d/ incre dj incr ase eases, s, ηad,loc h decre decr ases eases ad,loc firstly, and then increases. ηad,loc at M = 0.5 has a maximum value at y/d = 0.0 and a mini- fir ﬁrstly stly, ,and and then inc then incr reases. eases.ηad,loc h at M at = 0.5 has a ma M = 0.5 has ximum val a maximum ue at value y/d = 0. at 0 an y/d a d = m 0.0 iniand - a ad,loc mum value at y/d = 0.75. At a moderate blowing ratio (M = 1.5), ηad,loc decreases slightly mum va minimum lue value at y/dat = jy 0. / 75 dj. At = 0.75. a moAt derat a moderate e blowing r blowing atio (M = ratio 1.5), (ηM ad,loc =d 1.5), ecreases sligh h decr tly eases ad,loc as y/d increases from 0.0 to 1.0, and then shows a sharp decrease as y/d exceeds 1.0. as y/d increases from 0.0 to 1.0, and then shows a sharp decrease as y/d exceeds 1.0. slightly asjy/dj increases from 0.0 to 1.0, and then shows a sharp decrease asjy/dj exceeds At a high blowing ratio (M = 3.0), the changing trend of ηad,loc is similar to that at a moder- At a h 1.0. At igh a blo high wing r blowing atio (M ratio = 3.0 (M ), th =e3.0), changin the g changing trend of η tr ad,l end oc isof sim hilar tois th similar at at a m to oder- that at a ad,loc ate blowing ratio. However, ηad,loc at a high blowing ratio has a local minimum value at y/d ate blowing ratio. However, ηad,loc at a high blowing ratio has a local minimum value at y/d moderate blowing ratio. However, h at a high blowing ratio has a local minimum value = 0.0. Figure 13 shows ad,loc the streamline distributions for film cooling of the W-shaped trench, = 0.0. Figure 13 shows the streamline distributions for film cooling of the W-shaped trench, at y/d = 0.0. Figure 13 shows the streamline distributions for ﬁlm cooling of the W-shaped and the background color represents the gas temperature. At a low blowing ratio, pro- and the background color represents the gas temperature. At a low blowing ratio, pro- trench, and the background color represents the gas temperature. At a low blowing ratio, nounced recirculating flow can be observed inside the trench and promotes the lateral nounced recirculating flow can be observed inside the trench and promotes the lateral pronounced recir sprea culating ding of ﬂow cool can ant. At be observed a high blowi inside ng ra the tio, the coola trench and nt jet exhi promote bitss sl the ight deta lateralchment in spreading of coolant. At a high blowing ratio, the coolant jet exhibits slight detachment in the region of x/d = 2.0, but the coverage performance of coolant is still much better than spreading of coolant. At a high blowing ratio, the coolant jet exhibits slight detachment the region of x/d = 2.0, but the coverage performance of coolant is still much better than the transverse trench and the round hole, especially in the far-field region (x/d > 5.0). Sim- in the region of x/d = 2.0, but the coverage performance of coolant is still much better the transverse trench and the round hole, especially in the far-field region (x/d > 5.0). Sim- ilar to the transverse trench, the kidney vortex pair and anti-kidney vortex pair exist sim- than the transverse trench and the round hole, especially in the far-ﬁeld region (x/d > 5.0). ilar to the transverse trench, the kidney vortex pair and anti-kidney vortex pair exist sim- ultaneously downstream of the W-shaped trench. The anti-kidney vortices improve cool- Similar to the transverse trench, the kidney vortex pair and anti-kidney vortex pair exist ultaneously downstream ing performance, while the of the W-shaped trench. The kidney vortic anti-kidne es degrad y v e co ortices oling perform improve cool- ance. However, com- simultaneously downstream of the W-shaped trench. The anti-kidney vortices improve ing performance, while the pared with the tra kidney vortic nsverse trench, the W- es degrade cooling sha perform ped trench exhi ance. Howev bits ea r, com- stronger anti-kidney cooling performance, while the kidney vortices degrade cooling performance. However, vortex pair and weaker kidney vortex pair. pared with the transverse trench, the W-shaped trench exhibits a stronger anti-kidney compared with the transverse trench, the W-shaped trench exhibits a stronger anti-kidney vortex pair and weaker kidney vortex pair. vortex pair and weaker kidney vortex pair. 0.9 0.9 M=0.5 0.8 M=1 0.9 0.9 0.8 M=0.5 M=1.5 0.7 0.8 M=1 M=2 0.6 0.8 0.7 M=1.5 M=3 0.7 ad,lat M=2 0.5 ad,loc 0.6 0.6 M=3 0.7 η 0.4 η M=0.5 ad,lat 0.5 ad,loc 0.5 M=1 0.6 0.3 0.4 M=1.5 M=0.5 0.2 M=2 0.4 0.5 M=1 0.3 M=3 0.1 M=1.5 0.2 0.3 0.4 M=2 0.0 0 2468 10 12 14 16 18 20 22 M=3 0.1 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 0.3 x/d 0.0 y/d 0 2468 10 12 14 16 18 20 22 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 x/d (a) Laterally averaged value (b) Local value on the cross section of x/d = 2.2 y/d (a) Laterally averaged value (b) Local value on the cross section of x/d = 2.2 Figure 12. Distributions of adiabatic ﬁlm cooling effectiveness in different directions for W-shaped trench (w = 1.2d, h = 0.75d, a = 60 ). ad, av ad, av d Aerospace 2021, 8, x FOR PEER REVIEW 11 of 18 Aerospace 2021, 8, 147 11 of 17 Figure 12. Distributions of adiabatic film cooling effectiveness in different directions for W-shaped trench (w = 1.2d, h = 0.75d, α = 60°). (a) M = 0.5 (b) M = 1.5 Figure 13. Streamline distributions for film cooling of W−shaped trench (w = 1.2d, h = 0.75d, α = Figure 13. Streamline distributions for ﬁlm cooling of Wshaped trench (w = 1.2d, h = 0.75d, a = 60 ). 60°). Figure 14a shows the effects of trench width and depth on the area-averaged adiabatic Figure 14a shows the effects of trench width and depth on the area-averaged adia- ﬁlm cooling effectiveness (h , 2 < x/d < 20). For the narrow trench, because of the ad,av batic film cooling effectiveness (ηad,av, 2 < x/d < 20). For the narrow trench, because of the effect of recirculating ﬂow, the mainstream cannot enter the trench, which enhances the effect of recirculating flow, the mainstream cannot enter the trench, which enhances the ﬁlm cooling effectiveness. For the wide trench, due to the mainstream intursion, the film cooling effectiveness. For the wide trench, due to the mainstream intursion, the recir- recirculation of the coolant inside the trench is mitigated, which results in the decrease of culation of the coolant inside the trench is mitigated, which results in the decrease of cool- cooling effectiveness. Compared with trench width, the inﬂuence of h/d on ﬁlm cooling ing effectiveness. Compared with trench width, the influence of h/d on film cooling per- performance is much more obvious, especially at a high blowing ratio. At a high blowing formance ratio (M = is 1.5), much more h incr o eases bvious, from esabout pecially 0.15 at to a hig 0.5 h with blow the ing incr raease tio. At of ha h /difr gh om blow 0.25ing to ad,av ratio ( 1.3. AtM a = 1. low5blowi ), ηad,a ng v inc ratio, reases f h rom also abincr outeases 0.15 twith o 0.5 the witincr h th ease e incr ofease o trench f h depth, /d from however 0.25 to, ad,av the variation amplitude is weaker compared to that at a high blowing ratio. Figure 14b 1.3. At a low blowing ratio, ηad,av also increases with the increase of trench depth, however, t shows he varthe iation effects amp of littr ude ench is width weaker and com depth pared on tthe o thdischar at at a high ge coef bﬁcient. lowing For ratia o.deep Figutr re ench, 14b the distribution of coolant velocity at the trench exit is uniform, and the area of the high shows the effects of trench width and depth on the discharge coefficient. For a deep trench, speed zone is smaller compared with the shallow trench (shown in Figure 15), which the distribution of coolant velocity at the trench exit is uniform, and the area of the high reduces mixing loss between the mainstream and coolant jet. For the narrow trench, the jet speed zone is smaller compared with the shallow trench (shown in Figure 15), which re- Aerospace 2021, 8, x FOR PEER REVIEW 12 of 18 velocity is higher, which results in the increase of ﬂow loss. Thus, the increase of trench duces mixing loss between the mainstream and coolant jet. For the narrow trench, the jet depth and width both cause the increase of the discharge coefﬁcient. velocity is higher, which results in the increase of flow loss. Thus, the increase of trench depth and width both cause the increase of the discharge coefficient. 0.6 0.70 w=1.2d, M=0.5 w=1.2d, M=1.5 0.5 w=1.9d, M=0.5 0.68 w=1.9d, M=1.5 0.4 0.66 0.3 0.64 w=1.2d, M=0.5 w=1.2d, M=1.5 w=1.8d, M=0.5 w=2.4d, M=0.5 0.2 0.62 w=1.8d, M=1.5 w=2.4d, M=1.5 w=2.4d, M=0.5 w=2.4d, M=1.5 0.1 0.60 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.2 0.4 0.6 0.8 1.0 1.2 1.4 h/d h/d (a) Area-averaged adiabatic cooling effectiveness (b) Discharge coefficient Figure 14. Effects of trench depth and width on film cooling performance for W-shaped trench (α = 60°). Figure 14. Effects of trench depth and width on film cooling performance for W-shaped trench (a = 60 ). (a) w = 1.2d, h = 1.2d, α = 60° (b) w = 1.2d, h = 1.8d, α = 60° Figure 15. Distribution of flow fields on the exit plane of W−shaped trench at M = 1.5 (Unit: m/s). Figure 16a shows the effect of corner angle on the area-averaged adiabatic film cool- ing effectiveness. For a high corner angle, the lateral spreading inside the trench cannot be affected effectively by the W-structure. In fact, as the corner angle approaches 180°, the W-shaped trench turns into a transverse trench, and the guide effect of the corner angle disappears. Conversely, a small corner angle promotes lateral spreading of coolant and improves distribution uniformity of the coolant velocity at the trench exit, which results in high cooling effectiveness. Figure 16b shows the effect of corner angle on the discharge coefficient. Cd decreases with the increase of the corner angle. It is because that high corner angle causes the decrease of the trench exit area, which results in high flow loss. However, because the changing interval of α is small (40~60 ), the effects of the corner angle on the cooling effectiveness and discharge coefficient are not very obvious in the present study. ad,av Aerospace 2021, 8, x FOR PEER REVIEW 12 of 18 0.6 0.70 w=1.2d, M=0.5 w=1.2d, M=1.5 w=1.9d, M=0.5 0.5 0.68 w=1.9d, M=1.5 0.4 0.66 0.3 0.64 w=1.2d, M=0.5 w=1.2d, M=1.5 w=1.8d, M=0.5 w=2.4d, M=0.5 0.2 0.62 w=1.8d, M=1.5 w=2.4d, M=1.5 w=2.4d, M=0.5 w=2.4d, M=1.5 0.1 0.60 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.2 0.4 0.6 0.8 1.0 1.2 1.4 h/d h/d Aerospace 2021, 8, 147 12 of 17 (a) Area-averaged adiabatic cooling effectiveness (b) Discharge coefficient Figure 14. Effects of trench depth and width on film cooling performance for W-shaped trench (α = 60°). (a) w = 1.2d, h = 1.2d, α = 60° (b) w = 1.2d, h = 1.8d, α = 60° Figure 15. Distribution of flow fields on the exit plane of W−shaped trench at M = 1.5 (Unit: m/s). Figure 15. Distribution of ﬂow ﬁelds on the exit plane of Wshaped trench at M = 1.5 (Unit: m/s). Figure 16a shows the effect of corner angle on the area-averaged adiabatic ﬁlm cooling Figure 16a shows the effect of corner angle on the area-averaged adiabatic film cool- effectiveness. For a high corner angle, the lateral spreading inside the trench cannot be ing effectiveness. For a high corner angle, the lateral spreading inside the trench cannot affected effectively by the W-structure. In fact, as the corner angle approaches 180 , the be affected effectively by the W-structure. In fact, as the corner angle approaches 180°, the W-shaped trench turns into a transverse trench, and the guide effect of the corner angle W-shaped trench turns into a transverse trench, and the guide effect of the corner angle disappears. Conversely, a small corner angle promotes lateral spreading of coolant and disappears. Conversely, a small corner angle promotes lateral spreading of coolant and improves distribution uniformity of the coolant velocity at the trench exit, which results in improves distribution uniformity of the coolant velocity at the trench exit, which results high cooling effectiveness. Figure 16b shows the effect of corner angle on the discharge in high cooling effectiveness. Figure 16b shows the effect of corner angle on the discharge coefﬁcient. C decreases with the increase of the corner angle. It is because that high corner coefficient. Cd decreases with the increase of the corner angle. It is because that high corner angle causes the decrease of the trench exit area, which results in high ﬂow loss. However, Aerospace 2021 angle c , 8, x FO aR P uses t EER RE he decre VIEW ase of the trench exit area, which results in high flow loss. However, 13 of 18 because the changing interval of a is small (40~60 ), the effects of the corner angle on the because the changing interval of α is small (40~60 ), the effects of the corner angle on the cooling effectiveness and discharge coefﬁcient are not very obvious in the present study. cooling effectiveness and discharge coefficient are not very obvious in the present study. 0.7 0.70 w=1.2d, h=0.3d, M=0.5 w=1.2d, h=0.3d, M=1.5 0.6 0.68 w=1.8d, h=0.9d, M=0.5 w=1.8d, h=0.9d, M=1.5 0.5 0.66 ad,av 0.4 w=1.2d, h=0.3d, M=0.5 w=1.2d, h=0.3d, M=1.5 0.64 w=1.8d, h=0.9d, M=0.5 0.3 w=1.8d, h=0.9d, M=1.5 0.62 0.2 0.60 0.1 40 50 60 70 80 40 50 60 70 80 α (°) α (°) (a) Area−averaged adiabatic cooling effectiveness (b) Discharge coefficient Figure 16. Effects of corner angle on film cooling performance for W−shaped trench. Figure 16. Effects of corner angle on ﬁlm cooling performance for Wshaped trench. 4.3. Elliptical Trench 4.3. Elliptical Trench Figure 17a shows the variation of laterally averaged adiabatic film cooling effective- Figure 17a shows the variation of laterally averaged adiabatic ﬁlm cooling effectiveness ness with the streamwise distance. At a low blowing ratio, ηad,lat shows continued decrease with the streamwise distance. At a low blowing ratio, h shows continued decrease ad,lat with the increase of x/d. At a high blowing ratio, as x/d increases, ηad,lat decreases firstly, with the increase of x/d. At a high blowing ratio, as x/d increases, h decreases ﬁrstly, ad,lat and then increases. The rebound of ηad,lat in the far-field region can be attributed to the and then increases. The rebound of h in the far-ﬁeld region can be attributed to the ad,lat reattachment of the coolant jet. Figure 17b shows the variation of local adiabatic film cool- reattachment of the coolant jet. Figure 17b shows the variation of local adiabatic ﬁlm ing effectiveness with the lateral distance. ηad,loc decreases with the increase of y/d and cooling effectiveness with the lateral distance. h decreases with the increase of y/d j j ad,loc has a maximum value at y/d = 0. The changing trend of ηad,loc for the elliptical trench is and has a maximum value at y/d = 0. The changing trend of h for the elliptical similar to that for the round hole, but different from that f ad,loc or the transverse and W-shaped trench is similar to trench. Figure 18 that for the r ound shows the strea hole, but mldif ine fer distri entbfr uti om ons that for fifor lm cooling f the transverse or the ellipti and c trench, W-shaped trench. and the back Figure 18 ground co shows the lorstr represents g eamline distributions as temperature. Compare for ﬁlm cooling d with the transver for the se trench and the W-shaped trench, the secondary flow inside the elliptic trench is unobvi- elliptic trench, and the background color represents gas temperature. Compared with the ous, and lateral spreading of coolant is also weak in the elliptic trench. The kidney vortex transverse trench and the W-shaped trench, the secondary ﬂow inside the elliptic trench is pair takes the dominant role on the cross section downstream of the elliptic trench, while the anti-kidney vortex pair cannot be formed. The entrainment effect of the kidney vortex pair results in the detachment of coolant downstream of the hole and promotes the mix- ture between mainstream and coolant. Thus, the elliptic trench generates lower cooling effectiveness than the transverse and W-shaped trench. However, the scale of the kidney vortex pair for the elliptic trench is smaller than that for the round hole, accordingly, the cooling effectiveness of the elliptic trench is higher than that with the round hole. In gen- eral, the flow characteristics for the elliptical trench are very similar to those for the round hole, but different from the transverse trench and W-shaped trench. 0.5 0.9 M=0.5 M=0.5 0.8 M=1 M=1 0.4 M=1.5 0.7 M=1.5 M=2 M=2 0.6 η M=2.5 0.3 ad,lat ad,loc 0.5 M=3 0.4 0.2 0.3 0.2 0.1 0.1 0.0 0.0 0 2 4 6 8 10 12 14 16 18 20 22 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 x/d y/d (a) Laterally averaged value (b) Local value on the cross section of x/d = 2.2 Figure 17. Distributions of adiabatic film cooling effectiveness for elliptic trench (Dx = 2.4d, Dy = 1.2d, h = 0.75d). ad,av Aerospace 2021, 8, x FOR PEER REVIEW 13 of 18 0.7 0.70 w=1.2d, h=0.3d, M=0.5 w=1.2d, h=0.3d, M=1.5 0.6 0.68 w=1.8d, h=0.9d, M=0.5 w=1.8d, h=0.9d, M=1.5 0.5 0.66 ad,av 0.4 w=1.2d, h=0.3d, M=0.5 w=1.2d, h=0.3d, M=1.5 0.64 w=1.8d, h=0.9d, M=0.5 0.3 w=1.8d, h=0.9d, M=1.5 0.62 0.2 0.60 0.1 40 50 60 70 80 40 50 60 70 80 α (°) α (°) (a) Area−averaged adiabatic cooling effectiveness (b) Discharge coefficient Figure 16. Effects of corner angle on film cooling performance for W−shaped trench. 4.3. Elliptical Trench Figure 17a shows the variation of laterally averaged adiabatic film cooling effective- ness with the streamwise distance. At a low blowing ratio, ηad,lat shows continued decrease with the increase of x/d. At a high blowing ratio, as x/d increases, ηad,lat decreases firstly, and then increases. The rebound of ηad,lat in the far-field region can be attributed to the reattachment of the coolant jet. Figure 17b shows the variation of local adiabatic film cool- ing effectiveness with the lateral distance. ηad,loc decreases with the increase of y/d and has a maximum value at y/d = 0. The changing trend of ηad,loc for the elliptical trench is similar to that for the round hole, but different from that for the transverse and W-shaped Aerospace 2021, 8, 147 13 of 17 trench. Figure 18 shows the streamline distributions for film cooling for the elliptic trench, and the background color represents gas temperature. Compared with the transverse trench and the W-shaped trench, the secondary flow inside the elliptic trench is unobvi- unobvious, and lateral spreading of coolant is also weak in the elliptic trench. The kidney ous, and lateral spreading of coolant is also weak in the elliptic trench. The kidney vortex vortex pair takes pair ta the kes the domi dominant rna ole nt role on the cross se on the cross section downstr ction dow eam nstream o of the elliptic f the elliptic trench, trench, while while the anti-kidney vortex pair cannot be formed. The entrainment effect of the kidney the anti-kidney vortex pair cannot be formed. The entrainment effect of the kidney vortex vortex pair results pair resul in the ts in the deta detachment chment of of cool coolant do ant downstr wnstream of the hole and promotes the mix- eam of the hole and promotes the mixture between mainstream and coolant. Thus, the elliptic trench generates lower ture between mainstream and coolant. Thus, the elliptic trench generates lower cooling cooling effectiveness effectiveness than than the transverse and the transverse and W-shaped W-shaped trenc trench. However h. However, t , the scale he scale of th of the e kidney kidney vortex pair for the elliptic trench is smaller than that for the round hole, accordingly, vortex pair for the elliptic trench is smaller than that for the round hole, accordingly, the the cooling ef cooling fectiveness effect of iveness the elliptic of thetr el ench liptic is trhigher ench isthan higher tha that with n tha the t wi round th the round hole. In gen- hole. In general, the ﬂow characteristics for the elliptical trench are very similar to those for the eral, the flow characteristics for the elliptical trench are very similar to those for the round round hole, but different from the transverse trench and W-shaped trench. hole, but different from the transverse trench and W-shaped trench. 0.5 0.9 M=0.5 M=0.5 0.8 M=1 M=1 0.4 M=1.5 0.7 M=1.5 M=2 M=2 0.6 η M=2.5 0.3 ad,lat η ad,loc 0.5 M=3 0.4 0.2 0.3 0.2 0.1 0.1 0.0 0.0 0 2 4 6 8 10 12 14 16 18 20 22 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 x/d y/d (a) Laterally averaged value (b) Local value on the cross section of x/d = 2.2 Aerospace 2021, 8, x FOR PEER REVIEW 14 of 18 Figure 17. Distributions of adiabatic film cooling effectiveness for elliptic trench (Dx = 2.4d, Dy = 1.2d, h = 0.75d). Figure 17. Distributions of adiabatic ﬁlm cooling effectiveness for elliptic trench (D = 2.4d, D = 1.2d, x y h = 0.75d). (a) M = 0.5 x/d=10 x/d=2 x/d=5 (b) M = 1.5 Figure 18. Streamline distributions for film cooling of elliptic trench (Dx = 2.4d, Dy = 1.2d, h = 0.75d). Figure 18. Streamline distributions for ﬁlm cooling of elliptic trench (D = 2.4d, D = 1.2d, h = 0.75d). x y Figure 19a shows the effects of axis length of the elliptical trench (Dx and Dy) on the Figure 19a shows the effects of axis length of the elliptical trench (D and D ) on the x y area-averaged adiabatic film cooling effectiveness (ηad,av, 2 < x/d < 20). For small Dy, ηad,av area-averaged adiabatic ﬁlm cooling effectiveness (h , 2 < x/d < 20). For small D , ad,av increases with the rise of Dx. However, for large Dy, this changing trend becomes wholly h increases with the rise of D . However, for large D , this changing trend becomes ad,av x y opposite, and ηad,av decreases with the increase of Dx. It illustrates that there is an optimal wholly opposite, and h decreases with the increase of D . It illustrates that there is an ad,av exit area of the elliptical trench, and a too high and low exit area both deteriorate the cool- ing performance. If the exit area is higher than the optimal value, the actual jet velocity is too low, and the mainstream can penetrate into the trench. If the exit area is small, the actual jet velocity has high momentum, and shows a detachment effect from the wall downstream of the hole. Figure 20 shows the streamline distributions on the exit planes of elliptic trenches. Compared with the W-shaped trench and transverse trench, the streamlines for the elliptic trench are smoother, and the recirculating vortex pair cannot be formed inside the trench. Figure 19b shows the effects of axis length on discharge co- efficient. The variation trend of Cd with Dx for large Dy is contrary to that for small Dy. For Dy = 2.5d, Cd increases with the increases of Dx. However, for Dy = 1.5d, Cd decreases as Dx increases. In general, as Dx is close to Dy, the flow loss is relatively low. (a) Area−averaged adiabatic cooling effectiveness (b) Discharge coefficient Aerospace 2021, 8, x FOR PEER REVIEW 14 of 18 (a) M = 0.5 x/d=10 x/d=2 x/d=5 (b) M = 1.5 Figure 18. Streamline distributions for film cooling of elliptic trench (Dx = 2.4d, Dy = 1.2d, h = 0.75d). Aerospace 2021, 8, 147 14 of 17 Figure 19a shows the effects of axis length of the elliptical trench (Dx and Dy) on the area-averaged adiabatic film cooling effectiveness (ηad,av, 2 < x/d < 20). For small Dy, ηad,av increases with the rise of Dx. However, for large Dy, this changing trend becomes wholly opposite, and ηad,av decreases with the increase of Dx. It illustrates that there is an optimal optimal exit area of the elliptical trench, and a too high and low exit area both deteriorate exit area of the elliptical trench, and a too high and low exit area both deteriorate the cool- the cooling performance. If the exit area is higher than the optimal value, the actual jet ing performance. If the exit area is higher than the optimal value, the actual jet velocity is velocity is too low, and the mainstream can penetrate into the trench. If the exit area is too low, and the mainstream can penetrate into the trench. If the exit area is small, the small, the actual jet velocity has high momentum, and shows a detachment effect from actual jet velocity has high momentum, and shows a detachment effect from the wall the wall downstream of the hole. Figure 20 shows the streamline distributions on the exit downstream of the hole. Figure 20 shows the streamline distributions on the exit planes planes of elliptic trenches. Compared with the W-shaped trench and transverse trench, the of elliptic trenches. Compared with the W-shaped trench and transverse trench, the streamlines for the elliptic trench are smoother, and the recirculating vortex pair cannot streamlines for the elliptic trench are smoother, and the recirculating vortex pair cannot be formed inside the trench. Figure 19b shows the effects of axis length on discharge be formed inside the trench. Figure 19b shows the effects of axis length on discharge co- coefﬁcient. The variation trend of C with D for large D is contrary to that for small D . efficient. The variation trend of Cd with Dx for large Dy is contrary to that for small Dy. For x y y For D = 2.5d, C Dy = incr 2.5eases d, Cd increases with with the increases the increases of of D . However Dx. Howe , forver, for D = 1.5 Dd y = , C 1.5decr d, Cdeases decreases as Dx y d x y d increases. In general, as Dx is close to Dy, the flow loss is relatively low. as D increases. In general, as D is close to D , the ﬂow loss is relatively low. x x y Aerospace 2021, 8, x FOR PEER REVIEW 15 of 18 (a) Area−averaged adiabatic cooling effectiveness (b) Discharge coefficient Figure 19. Effects of axis length on film cooling performance for elliptic trench. Figure 19. Effects of axis length on ﬁlm cooling performance for elliptic trench. (a) Dx = 2.2d, Dy = 2.8d, h = 0.9d (b) Dx = 3.2d, Dy = 2.2d, h = 0.9d Figure 20. Streamline distribution on the exit plane of elliptic trench at M = 0.5 (Unit: m/s). Figure 20. Streamline distribution on the exit plane of elliptic trench at M = 0.5 (Unit: m/s). Figure 21 compares the distribution of adiabatic film cooling effectiveness on the wall Figure 21 compares the distribution of adiabatic ﬁlm cooling effectiveness on the wall for differentfo tr r di ench fferent shapes. trench sha At a low pes. At blowing a low b ratio, lowi the ng ra ﬂowtio, the fl separation ow sep of the aracoolant tion of the coolant downstream downstream of these four of these kinds four of holes kinds of ho is unobvious, les is unobvio and the us, and the cooling perform cooling performance of ance of the the W-shaped trench is the best, while the cooling effectiveness of the round hole is the W-shaped trench is the best, while the cooling effectiveness of the round hole is the lowest. lowest. As the blowing ratio increases to 1.5, the coolant from the round hole and elliptic As the blowing ratio increases to 1.5, the coolant from the round hole and elliptic trench trench shows shows deta detachment chment f from the rom the wa wall, but ll, but the epi the epileptic letr pti ench c trench genera generates better tes better cool cooling ing perfor- performance than the round hole. At this blowing ratio, the coverage performance of the mance than the round hole. At this blowing ratio, the coverage performance of the W- W-shaped trench is the best. At a very high blowing ratio, the coolant from these four shaped trench is the best. At a very high blowing ratio, the coolant from these four kinds kinds of holes exhibits detachment from the wall, however, the transverse trench and of holes exhibits detachment from the wall, however, the transverse trench and W-shaped trench still generate high cooling effectiveness. In general, the W-shaped trench generates the highest cooling effectiveness, while the cooling performance of the elliptic trench is the worst. M = 0.5 M = 1.5 M = 2.5 (a) Round hole (b) Transverse trench (w = 3d, h = 0.75d) Aerospace 2021, 8, x FOR PEER REVIEW 15 of 18 Figure 19. Effects of axis length on film cooling performance for elliptic trench. (a) Dx = 2.2d, Dy = 2.8d, h = 0.9d (b) Dx = 3.2d, Dy = 2.2d, h = 0.9d Figure 20. Streamline distribution on the exit plane of elliptic trench at M = 0.5 (Unit: m/s). Figure 21 compares the distribution of adiabatic film cooling effectiveness on the wall for different trench shapes. At a low blowing ratio, the flow separation of the coolant downstream of these four kinds of holes is unobvious, and the cooling performance of the W-shaped trench is the best, while the cooling effectiveness of the round hole is the lowest. As the blowing ratio increases to 1.5, the coolant from the round hole and elliptic trench shows detachment from the wall, but the epileptic trench generates better cooling perfor- mance than the round hole. At this blowing ratio, the coverage performance of the W- Aerospace 2021, 8, 147 15 of 17 shaped trench is the best. At a very high blowing ratio, the coolant from these four kinds of holes exhibits detachment from the wall, however, the transverse trench and W-shaped trench still generate high cooling effectiveness. In general, the W-shaped trench generates the highest cooling effectiveness, while the cooling performance of the elliptic trench is W-shaped trench still generate high cooling effectiveness. In general, the W-shaped trench the worst. generates the highest cooling effectiveness, while the cooling performance of the elliptic trench is the worst. M = 0.5 M = 1.5 M = 2.5 Aerospace 2021, 8, x FOR PEER REVIEW 16 of 18 (a) Round hole (b) Transverse trench (w = 3d, h = 0.75d) M = 0.5 M = 1.5 M = 2.5 (c) W−shaped trench (w = 1.2d, h = 0.75d, α = 60°) (d) Elliptic trench (Dx = 2.4d, Dy = 2d, h = 0.75d) Figure 21. Distribution of adiabatic ﬁlm cooling effectiveness on the wall for different trench shapes. Figure 21. Distribution of adiabatic film cooling effectiveness on the wall for different trench shapes. 5. Conclusions 5. Conclusions The ﬁlm cooling performances of the transverse trench, W-shaped trench and elliptic The film cooling performances of the transverse trench, W-shaped trench and elliptic trench were investigated using the CFD method. The inﬂuences of blowing ratio and trench were investigated using the CFD method. The influences of blowing ratio and ge- geometric parameters on the discharge coefﬁcient and ﬁlm cooling effectiveness were ometric parameters on the discharge coefficient and film cooling effectiveness were dis- discussed. Some useful conclusions are listed below: cussed. Some useful conclusions are listed below: (1) Inside the transverse trench, a pair of recirculating vortices is formed, which promotes (1) Inside the transverse trench, a pair of recirculating vortices is formed, which pro- the lateral spreading of coolant. Downstream of the transverse trench, a kidney vortex motes the lateral spreading of coolant. Downstream of the transverse trench, a kidney pair and anti-vortex pair are formed simultaneously. The increase of trench depth and vortex pair and anti-vortex pair are formed simultaneously. The increase of trench the decrease of trench width can both cause increases of ﬁlm cooling effectiveness. depth and the decrease of trench width can both cause increases of film cooling ef- (2) Inside the W-shaped trench, the existence of a corner angle further promotes the fectiveness. coolant spreading in the lateral direction and generates higher ﬁlm cooling effec- (2) Inside the W-shaped trench, the existence of a corner angle further promotes the cool- tiveness than the transverse trench. Similar to the transverse trench, the increase ant spreading in the lateral direction and generates higher film cooling effectiveness of trench depth and the decrease of trench width both result in the increase of than the transverse trench. Similar to the transverse trench, the increase of trench cooling effectiveness. depth and the decrease of trench width both result in the increase of cooling effec- tiveness. (3) For the elliptic trench, the flow characteristics are very similar to the round hole, and the kidney vortex pair is the largest-scale vortex structure. There exists an optimal exit area ratio (the exit area of elliptic trench/the exit area of round hole) for the ellip- tic trench, and too large or small an exit area ratio can lead to the degradation of film cooling effectiveness. The elliptic trench generates higher film cooling effectiveness than the round hole, but lower effectiveness than the transverse trench and W-shaped trench. (4) As the blowing ratio increases, the discharge coefficient increases firstly and then keeps stable. The increase of trench depth and decrease of trench width results in the increase of the discharge coefficient for the transverse trench. For the W-shaped trench, the increase of the corner angle causes the decrease of the discharge coeffi- cient. For the elliptic trench, the discharge coefficient increases with the decrease of the elliptic aspect ratio (major axis/minor axis). Author Contributions: Conceptualization, C.W. and X.W.; methodology, X.W.; writing—original draft preparation, C.W; writing—review and editing, X.D and C.W.; funding acquisition, C.W. All authors have read and agreed to the published version of the manuscript. Funding: This research was supported by the Fundamental Research Funds for the Central Univer- sities (grant No: NS2020013) and National Science and Technology Major Project of China (grant No: 2017-III-0011-0037). Institutional Review Board Statement: Not applicable. Aerospace 2021, 8, 147 16 of 17 (3) For the elliptic trench, the ﬂow characteristics are very similar to the round hole, and the kidney vortex pair is the largest-scale vortex structure. There exists an optimal exit area ratio (the exit area of elliptic trench/the exit area of round hole) for the elliptic trench, and too large or small an exit area ratio can lead to the degradation of ﬁlm cooling effectiveness. The elliptic trench generates higher ﬁlm cooling effec- tiveness than the round hole, but lower effectiveness than the transverse trench and W-shaped trench. (4) As the blowing ratio increases, the discharge coefﬁcient increases ﬁrstly and then keeps stable. The increase of trench depth and decrease of trench width results in the increase of the discharge coefﬁcient for the transverse trench. For the W-shaped trench, the increase of the corner angle causes the decrease of the discharge coefﬁcient. For the elliptic trench, the discharge coefﬁcient increases with the decrease of the elliptic aspect ratio (major axis/minor axis). Author Contributions: Conceptualization, C.W. and X.W.; methodology, X.W.; writing—original draft preparation, C.W.; writing—review and editing, X.D. and C.W.; funding acquisition, C.W. All authors have read and agreed to the published version of the manuscript. Funding: This research was supported by the Fundamental Research Funds for the Central Universi- ties (grant No: NS2020013) and National Science and Technology Major Project of China (grant No: 2017-III-0011-0037). Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Data available on request due to restrictions e.g., privacy or ethical. Conﬂicts of Interest: The authors declare no conﬂict of interest. Nomenclature A Cross section area of round hole (m ) C Discharge coefﬁcient (-) d Diameter of round hole (mm) D Axis Length of elliptic crater (mm) h Trench depth (mm) M Blowing ratio (-) m Coolant mass ﬂow rate (kg/s) P* Inlet total pressure of secondary ﬂow (Pa) C,in P Static pressure downstream of trench (Pa) C,out T Temperature (K) w Trench width (mm) x, y and z Streamwise, spanwise and vertical direction Greek symbols a Corner angle of W-shaped trench ( ) h Cooling effectiveness (-) r Gas density (kg/m ) d Boundary layer thickness of mainstream inlet (mm) q Dimensionless temperature [=(T T )/(T T )] c ¥ c subscript w Wall ¥ Mainstream c Coolant ad At adiabatic condition av Area-averaged value loc Local value lat Laterally averaged value overall At the condition considering heat conduction x, y and z Streamwise, spanwise and vertical component Aerospace 2021, 8, 147 17 of 17 References 1. Bogard, D.G.; Thole, K.A. Gas turbine ﬁlm cooling. J. Propuls. Power 2006, 22, 249–270. 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Li, J.; Ren, J.; Jiang, H.D. Film cooling performance of the embedded holes in trenches with compound angles. In Proceedings of the ASME Turbo Expo 2010, Glasgow, Scotland, UK, 14–18 June 2010; pp. 1415–1424, ASME Paper No. GT2010-22337. 9. Bunker, R.S. Film cooling effectiveness due to discrete holes within a transverse surface slot. In Proceedings of the ASME Turbo Expo 2002, Amsterdam, The Netherlands, 3–6 June 2002; ASME Paper No. GT2002-30178. 10. Harrison, K.L.; Bogard, D.G. CFD predictions of ﬁlm cooling adiabatic effectiveness for cylindrical holes embed in nar- row and wide transverse trenches. In Proceedings of the ASME Turbo Expo 2007, Montreal, QC, Canada, 14–17 May 2007; ASME Paper No. GT2007-28005. 11. Lu, Y.; Nasir, H.; Ekkad, S.V. Film cooling from a row of holes embedded in transverse slots. In Proceedings of the ASME Turbo Expo 2005, Reno, NV, USA, 6–9 June 2005; ASME paper No. GT2005-68598. 12. Lu, Y.; Dhungel, A.; Ekkad, S.V.; Bunker, R.S. Effect of trench width and depth on ﬁlm cooling from cylindrical holes embedded in trenches. ASME J. Turbomach. 2009, 131, 011003. [CrossRef] 13. Lu, Y.; Ekkad, S.V.; Bunker, R.S. Trench ﬁlm cooling: Effect of trench downstream edge and hole spacing. In Proceedings of the ASME Turbo Expo 2008, Berlin, Germany, 9–13 June 2008; ASME Paper No. GT2008-50606. 14. Waye, S.K.; Bogard, D.G. High resolution ﬁlm cooling effectiveness measurements of axial holes embedded in a transverse trench with various trench conﬁgurations. In Proceedings of the ASME Turbo Expo 2006, Barcelona, Spain, 8–11 May 2006; ASME Paper No. GT2006-90226. 15. Davidson, F.T.; KistenMacher, D.A.; Bogard, D.G. Film Cooling With a Thermal Barrier Coating: Round Holes, Craters, and Trenches. ASME J. Turbomach. 2014, 136, 041007. [CrossRef] 16. Lee, K.D.; Kim, K.Y. Film cooling performance of cylindrical holes embedded in a tranverse trench. Numer. Heat Transf. Part A Appl. 2014, 65, 127–143. [CrossRef] 17. Oguntade, H.I.; Andrews, G.E.; Burns, A.D.; Ingham, D.B.; Pourkashanian, M. Improved trench ﬁlm cooling with shaped trench outlets. ASME J. Turbomach. 2013, 135, 021009. [CrossRef] 18. Lu, Y.; Dhungel, A.; Ekkad, S.V.; Bunker, R.S. Film cooling measurements for cratered cylindrical inclined holes. ASME J. Turbomach. 2009, 131, 011005. [CrossRef] 19. Dorrington, J.R.; Bogard, D.G.; Bunker, R.S. Film Effectiveness Performance for Coolant Holes Imbedded in Various Shallow Trench and Crater Depressions. In Proceedings of the ASME Turbo Expo 2007, Montreal, QC, Canada, 14–17 May 2007; ASME Paper No. GT2007-27992. 20. Kross, B.; Pﬁtzner, M. Numerical and Experimental Investigation of the Film Cooling Effectiveness in a Novel Trench Conﬁgura- tion. In Proceedings of the ASME Turbo Expo 2012, Copenhagen, Denmark, 11–15 June 2012; ASME Paper No. GT2012-68125. 21. Wang, C.H.; Sun, X.K.; Fan, F.S.; Zhang, J.Z. Study on trench ﬁlm cooling on turbine vane by large-eddy simulation. Numer. Heat Transf. Part A Appl. 2020, 78, 338–358. [CrossRef] 22. Wei, J.S.; Zhu, H.R.; Liu, C.L.; Song, H.; Liu, C.; Meng, T. Experimental study on the ﬁlm cooling characteristics of the cylindrical holes embedded in sine-wave shaped trench. In Proceedings of the ASME Turbo Expo 2016, Seoul, Korea, 13–17 June 2016; ASME Paper No. GT2016-56856. 23. Zhang, B.L.; Zhang, L.; Zhu, H.R.; Wei, J.S.; Fu, Z.Y. Numerical study on the inﬂuence of trench width on ﬁlm cooling characteristics of double-wave trench. In Proceedings of the ASME Turbo Expo 2017, Charlotte, VA, USA, 26–30 June 2017; ASME Paper No. GT2017-63552. 24. Fan, F.S. Large Eddy Simulation and Experimental Study on Film Cooling Flow and Heat Transfer Performance of Shallow Trenches. Master ’s Thesis, Nanjing Universtiy of Aeronautics and Astronautics, Nanjing, China, 2020. 25. Fan, F.S.; Wang, C.H.; Feng, H.K.; Zhang, J.Z. Large eddy simulation of ﬁlm cooling from a shallow trench hole. J. Propuls. Technol. 2020, 41, 830–839. 26. Renze, P.; Schroder, W.; Meinke, M. Large eddy simulation of ﬁlm cooling ﬂow ejected in a shallow cavity. In Proceedings of the ASME Turbo Expo 2008, Berlin, Germany, 9–13 June 2008; ASME paper No. GT2008-50120. 27. Mahesh, K. The interaction of jets with crossﬂow. Annu. Rev. Fluid Mech. 2013, 45, 379–407. [CrossRef] 28. Haven, B.A.; Yamagata, D.K.; Kurosaka, M.; Yamawaki, S.; Maya, T. Anti-kidney pair of vortices in shaped holes and their inﬂuence on ﬁlm cooling effectiveness. In Proceedings of the ASME Turbo Expo 2010, Glasgow, Scotland, 14–18 June 2010; ASME paper No. 97-GT-045. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Aerospace Multidisciplinary Digital Publishing Institute http://www.deepdyve.com/lp/multidisciplinary-digital-publishing-institute/study-on-film-cooling-performance-of-round-hole-embedded-in-different-QF0Lw15AvN

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ISSN: 2226-4310
DOI: 10.3390/aerospace8060147
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Abstract

aerospace Article Study on Film Cooling Performance of Round Hole Embedded in Different Shaped Craters and Trenches 1 1 1 , 2 , Xiaojun Wu , Xin Du and Chunhua Wang * AECC Shenyang Engine Research Institute, No. 1 Wanlian Road, Shenyang 110015, China; 18002492948@163.com (X.W.); dxknight@163.com (X.D.) College of Energy and Power Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China * Correspondence: chunhuawang@nuaa.edu.cn Abstract: Film cooling effectiveness can be improved signiﬁcantly by embedding a round hole in trenches or craters. In this study, ﬁlm cooling performances of a transverse trench, W-shaped trench and elliptic trench were compared and analyzed in detail. The CFD models for trench ﬁlm cooling were established and validated via the experimental results. Inside the transverse trench, a pair of recirculating vortices is formed, which promotes the coolant spreading in a lateral direction. The decrease of trench width and increase of trench depth both improve the ﬁlm cooling effectiveness of the transverse trench. For the W-shaped trench, the guide effect of the corner angle further improves the lateral spreading capability of coolant and generates higher cooling effectiveness than a transverse trench with the same depth and width. The ﬂow characteristics of the elliptic trench are similar to that of the round hole, and the kidney vortex pair takes a dominant role in the ﬂow ﬁelds downstream of the coolant exit. Accordingly, the elliptic trench generates the worst cooling performance in these shaped trenches. The increase of trench depth and decrease of trench width both result in an increase Citation: Wu, X.; Du, X.; Wang, C. of the discharge coefﬁcient for trench ﬁlm cooling. For the W-shaped trench, the increase of the corner Study on Film Cooling Performance angle causes a decrease of the discharge coefﬁcient. For the elliptic trench, the discharge coefﬁcient of Round Hole Embedded in increases with the decrease of the elliptic aspect ratio (major axis/minor axis). Different Shaped Craters and Trenches. Aerospace 2021, 8, 147. Keywords: film cooling; shaped trench; CFD; adiabatic film cooling effectiveness; discharge coefficient https://doi.org/10.3390/ aerospace8060147 Academic Editor: Qiang Zhang 1. Introduction To increase output efﬁciency, gas turbines are usually operated at the temperature Received: 16 April 2021 higher than the maximum allowable value for materials. To avoid the thermal damage of Accepted: 17 May 2021 the airfoil, various cooling strategies such as ﬁlm cooling, impingement cooling and pin ﬁns Published: 25 May 2021 are applied. In the external ﬁlm cooling process, coolant air extracted from the compressor is ejected through inclined holes and forms a coolant ﬁlm on the airfoil surface [1]. The Publisher’s Note: MDPI stays neutral coolant ﬁlm not only cools the airfoil surface, but also reduces the heat ﬂow from the with regard to jurisdictional claims in mainstream to hot-section surface. However, for a traditional inclined cylindrical hole, published maps and institutional afﬁl- the coolant injection tends to separate from the downstream wall, especially at high jet iations. momentum, and the spanwise coverage of the coolant is also weak. This causes low area-averaged ﬁlm cooling effectiveness [2,3]. To improve the spanwise coverage of coolant, one approach is to change the hole geometry from round to shaped outlet [4,5]. The transition from cylindrical inlet to shaped Copyright: © 2021 by the authors. outlet results in a decrease of outlet coolant momentum. Flow deceleration in the hole Licensee MDPI, Basel, Switzerland. diffuser section also promotes the spanwise spreading of coolant, which improves cooling This article is an open access article performance. Accordingly, to get the same cooling effectiveness, the coolant consumption distributed under the terms and conditions of the Creative Commons decreases. Over the past 40+ years, many different outlet shapes for ﬁlm cooling holes Attribution (CC BY) license (https:// have been proposed [2,3]. However, because of the limitation of manufacturing conditions, creativecommons.org/licenses/by/ shaped hole technology has not been widely used in practice. 4.0/). Aerospace 2021, 8, 147. https://doi.org/10.3390/aerospace8060147 https://www.mdpi.com/journal/aerospace Aerospace 2021, 8, 147 2 of 17 A certain conﬁguration by embedding a round hole in a transverse trench generates similar cooling performance to shaped holes. As trench conﬁguration can be fabricated via thermal barrier coating (TBC), trench ﬁlm cooling is more practical compared with other shaped holes [6]. Moreover, transverse trenches with the compound angle injection of a round hole or shaped hole can further improve ﬁlm cooling performance [7,8]. A trench study on a ﬂat plate was performed by Bunker [9]. He reported that, at high jet momentum, the coolant can still attach closely to the surface. This result is conﬁrmed by Harrison and Bogard [10] and Lu et al. [11–13]. The CFD and experimental results from Lu et al. [11–13] showed that, compared with a round hole, higher ﬁlm effectiveness and heat ﬂux reduction can be generated by embedding a round hole in a trench. Moreover, the heat transfer coefﬁcient does not change obviously after introducing a trench. Trench ﬁlm cooling on the vane surface was studied by Waye and Board [14]. As the trench width decreases, the cooling effectiveness increases. Even at high jet momentum, the mainstream can suppress the coolant jet ejected from the trench on the vane surface effectively. The inﬂuences of trench geometries on temperature distribution on a TBC coated vane were investigated by Davidson et al. [15]. Their results showed that, compared with round holes, trenches and craters can generate much better coolant coverage, however, temperature at the interface of the vane and TBC only shows a slight decrease. Lee and Kim [16] also conducted parametric studies on trench ﬁlm cooling. A trench height of 1 diameter and trench width of 2 diameter of the round hole generates the highest cooling effectiveness both at low and high jet momentums. Moreover, reverse injection of coolant can improve the ﬁlm cooling performance. Besides transverse trenches, some other shaped trenches are also proposed and have been proved to generate higher ﬁlm cooling effectiveness than traditional round holes [17]. Lu et al. [18] tested ﬁlm the cooling effectiveness of a round hole embedded in the elliptic crater. Their experimental results show that, compared with round holes, crater ﬁlm cooling generates a higher heat transfer coefﬁcient and cooling effectiveness. Dorrington et al. [19] also concluded the crater hole generates lower cooling effectiveness than the trench conﬁgu- ration, but higher effectiveness than cylindrical holes. Kross and Pﬁtzner [20,21] found that placing a tetrahedral element upstream of the trench can improve the cooling performance by reducing the coolant-mainstream mixture within the trench and improving lateral coolant spreading. Wei et al. [22] and Zhang et al. [23] developed double- and sine-wave trenches, and the inﬂuences of wave geometries on cooling effectiveness were studied. In the present study, a systematic parametric study is performed for trench or crater ﬁlm cooling. CFD models for a transverse trench, W-shaped trench and elliptic trench were established and validated using the experimental results. The ﬂow mechanisms and cooling performances with different trench shapes were analyzed in detail. 2. Computational Model 2.1. Computational Domain As illustrated in Figure 1, the computational domain consists of a mainstream channel, coolant channel, cylindrical hole and shaped trench. The cylindrical hole has an inclination angle of 30 and diameter of 5.0 mm. The total height including the cylindrical section and shaped trench, h , is 3d. The width of the mainstream and coolant channel is 3.0d. The coordinate origin locates at the trench exit center, and the x, y and z axes correspond to streamwise, spanwise and vertical direction, respectively. Three kinds of shaped trenches, a transverse trench, W-shaped trench and elliptic trench, were investigated. The geometries of the trench such as depth (h), width (w), corner angle (a) and axis length (D , D ) are x y deﬁned in Figure 1. The changing interval of these parameters are listed in Table 1. Aerospace 2021, 8, 147 3 of 17 Aerospace 2021, 8, x FOR PEER REVIEW 3 of 18 (a) 3D-view (b) Front view (c) Transverse trench (d) W-shaped trench (e) Elliptic trench Figure 1. Computational domain and geometric variables for trench holes in the present study. Figure 1. Computational domain and geometric variables for trench holes in the present study. Table 1. Table 1.Changing interval of Changing interval of ttr rench geometries. ench geometries. Trench Type Symbol Changing Interval Trench Type Symbol Changing Interval W 2.5~3.5d W 2.5~3.5d Transverse trench Transverse trench h h 0. 0.25~1.8 25~1.8dd W 1.2~2.4d W 1.2~2.4d W-sh W-shaped aped tre trench nch h h 0. 0.25~1.8 25~1.8dd a 40~80 α 40~80° D 1.2~3.2d Dxx 1.2~3.2d Elliptic trench D 1.2~3.2d Elliptic trench Dy 1.2~3.2d h 0.25~1.8d h 0.25~1.8d 2.2. Performance Evaluation Parameters for Film Cooling 2.2. Performance Evaluation Parameters for Film Cooling Adiabatic cooling effectiveness is an important index to evaluate ﬁlm cooling perfor- Adiabatic cooling effectiveness is an important index to evaluate film cooling perfor- mance, and can be calculated by: mance, and can be calculated by: T T ¥ T - T ad,w ∞ ad,w h (x, y) = (1) η (x, y) = (1) ad,loc ad,loc T T ¥ T - T c ∞ c Z △y/2 1 Dy/2 η (x) = η (x, y)dy h (x) = h (x, y)dy (2 (2) ) ad,lat ad, loc ad,lat ad, loc △y Dy D -△ y/ y/2 2 x 2 1 2 h = h (x)dx (3) ad,av ad, lat η = η (x) dx (3) ad,av Dx ad, lat ∆x where T is the adiabatic wall temperature, and T and T are the temperature of c ¥ ad,w where Tad,w is the adiabatic wall temperature, and Tc and T∞ are the temperature of the the coolant and mainstream. The subscripts ‘loc’, ‘lat’ and ‘av’ denote the local, laterally coolant and mainstream. The subscripts ‘loc’, ‘lat’ and ‘av’ denote the local, laterally aver- aged and area-averaged value, respectively. In the present study, △y = 3d, x1 = 2d and x2 = 20d. The discharge coefficient, Cd, is defined as Aerospace 2021, 8, 147 4 of 17 averaged and area-averaged value, respectively. In the present study, 4y = 3d, x = 2d and x = 20d. The discharge coefﬁcient, C , is deﬁned as C = q (4) Aerospace 2021, 8, x FOR PEER REVIEW 4 of 18 A 2r (P P c c,in c,out where m is the coolant mass, A is the cross section area of the round hole, P* and c c C,in P are the inlet total pressure of secondary ﬂow and the static pressure downstream c,out d A 2ρ P* - P ) c c,in c,out of the trench. (4) where mc is the coolant mass, Ac is the cross section area of the round hole, P*C,in and 2.3. Boundary Condition and Solution Method Pc,outare the inlet total pressure of secondary flow and the static pressure downstream of At the mainstream inlet, the velocity proﬁle with a 1/7 power law and the boundary the trench. thickness (d ) of 0.125d were speciﬁed. This is the same as that in the experimental 2.3. Boundary Condition and Solution Method conditions. The mainstream mean velocity is 20 m/s, and the temperature is 353 K. The At the mainstream inlet, the velocity profile with a 1/7 power law and the boundary turbulent intensity and length scale is 4% and 0.4d respectively. The coolant temperature thickness (δ99) of 0.125d were specified. This is the same as that in the experimental condi- is 300 K. The turbulent parameters at the coolant inlet are the same as the mainstream tions. The mainstream mean velocity is 20m/s, and the temperature is 353K. The turbulent inlet. Because of a low Mach number (<0.3), incompressible ideal gas is used. The top intensity and length scale is 4% and 0.4δ99 respectively. The coolant temperature is 300K. surface of the mainstream channel was set as a free boundary. The spanwise surfaces of the The turbulent parameters at the coolant inlet are the same as the mainstream inlet. Because mainstream and coolant channel were set as periodic boundaries. Other surfaces were set of a low Mach number (<0.3), incompressible ideal gas is used. The top surface of the as a non-slip adiabatic wall. In the present study, the density ratio is 1.176, the blowing mainstream channel was set as a free boundary. The spanwise surfaces of the mainstream 2 2 ratio (M = r u /r u ) is 0.5~3.0 and the momentum ratio (I = r u /r u ) is 0.21~7.65. and coolant channel were set as periodic boundaries. Other surfaces were set as a non-slip c c ¥ ¥ c c ¥ ¥ adiabatic wall. In the present study, the density ratio is 1.176, the blowing ratio (M = ANSYS Fluent is applied for solutions of governing equations. According to Ref. [10], 2 2 ρcuc/ρ∞u∞) is 0.5~3.0 and the momentum ratio (I = ρcuc /ρ∞u∞ ) is 0.21~7.65. realizable k-" equations with enhanced wall treatment are suitable for trench ﬁlm cooling. ANSYS Fluent is applied for solutions of governing equations. According to Ref. [10], The momentum, energy and turbulent equations were solved using a second-order upwind realizable k-ε equations with enhanced wall treatment are suitable for trench film cooling. scheme. The gradient and pressure interpolation were performed using a least squares The momentum, energy and turbulent equations were solved using a second-order up- cell-based scheme and second-order scheme, respectively. The convergence criteria include: wind scheme. The gradient and pressure interpolation were performed using a least (1) the mass balance error is smaller than 10 , (2) the normalized residuals are smaller squares cell-based scheme and second-order scheme, respectively. The convergence crite- 6 2 −6 ria inc than 10 lude: (1) the m , and (3) ass b theavariation lance error is smaller th of local adiabatic an 10 , (2 ef ) the norma fectiveness lized residu is smaller als a than re 10 . −6 −2 smaller than 10 , and (3) the variation of local adiabatic effectiveness is smaller than 10 . Structural meshes are created with ICEM software. As shown in Figure 2, near the Structural meshes are created with ICEM software. As shown in Figure 2, near the walls, the grid points are clustered. On the ﬂat plate, the wall-normal size of the ﬁrst- walls, the grid points are clustered. On the flat plate, the wall-normal size of the first-layer layer grid is 4z = 0.003d, which corresponds to z 1. In the wall-normal direction, grid is △z = 0.003d, which corresponds to z ≈ 1. In the wall-normal direction, the stretching the stretching factor is smaller than 1.2 in the near-wall region. Grid independence tests factor is smaller than 1.2 in the near-wall region. Grid independence tests were carried out were carried out to determine the optimal grid number. Taking the transverse trench with to determine the optimal grid number. Taking the transverse trench with w = 2.2d and h = 0. w5= d as 2.2 an example, the grid test result is show d and h = 0.5d as an example, the grid n in Figure 3. The calculation results do not test result is shown in Figure 3. The calculation change obviously as the grid number exceeds 1,896,323. results do not change obviously as the grid number exceeds 1,896,323. (a) Front view (b) Transverse trench (c) W-shaped trench (d) Elliptic trench Figure 2. Grids used in the present study. Figure 2. Grids used in the present study. Aerospace 2021, 8, x FOR PEER REVIEW 5 of 18 Aerospace 2021, 8, 147 5 of 17 Aerospace 2021, 8, x FOR PEER REVIEW 5 of 18 0.50 713,956 0.50 0.45 1,245,014 713,956 1,896,323 0.45 1,245,014 0.40 2,533,437 1,896,323 0.40 2,533,437 0.35 0.35 0.30 0.30 0.25 0.25 0.20 0.20 0.15 0.15 0 2 4 6 8 10 121416182022 0 2 4 6 8 10 121416182022 x/d x/d Figure 3. Gird independent test results. Figure 3. Gird independent test results. Figure 3. Gird independent test results. 3. Experimental Validation 3. Experimental Validation 3. Experimental Validation Figure 4 shows the experimental system. After being heated to 80 °C, the mainstream Figure 4 shows the experimental system. After being heated to 80 C, the mainstream from an Fig air ure co 4 sho mpressor ws the experimental s passes through a rectifie ystem. Af r section ter being hea and then enters the test ted to 80 °C, the mai secti nstrea ons. m from an air compressor passes through a rectiﬁer section and then enters the test sections. The cros from an air s sec co tion s mpressor ize o f passes the mthrough ainstrea a rectifie m channe r section l is 174m an m d then enters the test ×80mm, and the size o secti f t oh ns. e The cross section size of the mainstream channel is 174 mm 80 mm, and the size of the The cross section size of the mainstream channel is 174mm×80mm, and the size of the coolant channel is 64mm × 40mm. The inlet velocity and temperature are the same as the coolant channel is 64mm 40mm. The inlet velocity and temperature are the same as the coolant channel is 64mm × 40mm. The inlet velocity and temperature are the same as the CFD model. The boundary layer thickness and turbulent parameters for the mainstream CFD model. The boundary layer thickness and turbulent parameters for the mainstream CFD model. The boundary layer thickness and turbulent parameters for the mainstream inlet were measured at x/d = −15 using a hot wire anemometer (StreamLine Pro). The inlet were measured at x/d = 15 using a hot wire anemometer (StreamLine Pro). The inlet were measured at x/d = −15 using a hot wire anemometer (StreamLine Pro). The boundary layer thickness (δ99) is 0.125d. The turbulent intensity and length scale is 4% and boundary layer thickness (d ) is 0.125d. The turbulent intensity and length scale is 4% 0. boundar 4δ99. The test pl y layer thickn ate, wiess ( th a therma δ99) is 0.125 l conducti d. The t vi uty rbu ol fent 0.3 W/ inte(m·K nsity and ), is made o lengthf r sc u abber wood. le is 4% and and 0.4d . The test plate, with a thermal conductivity of 0.3 W/(mK), is made of rubber The size 0.4δ99. The test pl s of the hole ate, wi and th ta therma he trench l conducti are the same vity o as f 0.the computat 3 W/(m·K), is ional mo made of r del. ubber wood. The hole wood. The sizes of the hole and the trench are the same as the computational model. The sizes of the hole and the trench are the same as the computational model. The hole pitch is 3.0d. The transverse trench (w = 3.2d, h = 1d), the W-shaped trench (w = 1.7d, h = 1d The hole pitch is 3.0d. The transverse trench (w = 3.2d, h = 1d), the W-shaped trench pitch is 3.0d. The transverse trench (w = 3.2d, h = 1d), the W-shaped trench (w = 1.7d, h = 1d and α = 60°) and the elliptic trench (Dx = 2.2d, Dy = 2.8d and h = 1d) are tested in the present (w = 1.7d, h = 1d and a = 60 ) and the elliptic trench (D = 2.2d, D = 2.8d and h = 1d) are x y and α = 60°) and the elliptic trench (Dx = 2.2d, Dy = 2.8d and h = 1d) are tested in the present study. Viewing though CaF2-infrared glass, an infrared thermography system is applied tested in the present study. Viewing though CaF -infrared glass, an infrared thermography for temperature measurem study. Viewing though Caent of the flat F2-infrared glass plate , an wi infr th black pa ared therm int coa ogra ting. The emissi phy system is ap vity of plied system is applied for temperature measurement of the ﬂat plate with black paint coating. tfor temperature measurem he black paint is about 0.97ent of the flat . The infrared plate thermography (M with black pa ag in32HF model) pr t coating. The emissi oduced b vity of y The emissivity of the black paint is about 0.97. The infrared thermography (Mag32HF the black paint is about 0.97. The infrared thermography (Mag32HF model) produced by Magnity Electronics Co. Ltd. (Shanghai, China) has a test range of −20~300 °C and an ac- model) produced by Magnity Electronics Co. Ltd. (Shanghai, China) has a test range of Magnity Electronics Co. Ltd. (Shanghai, China) has a test range of −20~300 °C and an ac- curacy of ±1 °C. The calibration of the infrared measurement was performed according to 20~300 C and an accuracy of 1 C. The calibration of the infrared measurement was curacy of ±1 °C. The calibration of the infrared measurement was performed according to the temperature measured via thermocouples within the plate. Detail calibration pro- performed according to the temperature measured via thermocouples within the plate. the temperature measured via thermocouples within the plate. Detail calibration pro- cesses are introduced in Ref. [24]. Detail calibration processes are introduced in Ref. [24]. cesses are introduced in Ref. [24]. Infrared camera Temperature probe Infrared camera Temperature probe Valve Valve Baffles Test section section Baffles Primary flow Flow meter Test section Heater section Primary flow Flow meter Heater Flow meter Valve Coolant flow Flow meter Valve Coolant flow (a) Experimental system (b) Test plates (a) Experimental system (b) Test plates Figure 4. Experimental system in the present study. Figure 4. Experimental system in the present study. Figure 4. Experimental system in the present study. To better compare the CFD and experimental results, the heat conduction effect inside To better compare the CFD and experimental results, the heat conduction effect in- the ﬁlm-cooling To better com plate pare t is h taken e CFD an into account, d experim and entthe al result thermal s, the heat co conductivity nduction is 0.3W/(m effect in- K). side the film-cooling plate is taken into account, and the thermal conductivity is 0.side The 3W/ t top (m he ·K and fi ). lm The t lower -coolin op surfaces and g plat lower s e ofis the tua solid rken face int plate s of t o h wer account, a e sol e coupled id plat nd the therma e we with re c theoup ﬂuid led w l conducti phase. ith tThe he fl vity i span- uids wise surfaces of the plate were set as periodic boundaries. Figure 5 shows the distributions p0. h3W/ ase. The (m·Ksp ). anwi The tse s op and urfaclower s es of the p urflace ates were of th set e sol as p ide p riod late we ic bore c undar oup ies. led w Figu itre h 5 thshows e fluid p of hase. the The overall spanwi cooling se sef ur fectiveness faces of the p [h late were = (T set as p T e )/( riod T ic b oTun )]dar onithe es. F cooling igure 5 surface. shows the distributions of the overall cooling effectiveness ¥ w [ηoverall¥ = (T∞c − Tw)/(T∞ − Tc)] on the overall Inside the trench, the difference between the CFD and the experimental results is somewhat the distributions of the overall cooling effectiveness [ηoverall = (T∞ − Tw)/(T∞ − Tc)] on the cooling surface. Inside the trench, the difference between the CFD and the experimental cooling surface. Inside the trench, the difference between the CFD and the experimental η ad,lat ad,lat Aerospace 2021, 8, 147 6 of 17 Aerospace 2021, 8, x FOR PEER REVIEW 6 of 18 results is somewhat obvious, especially for the transverse trench at a high blowing ratio. obvious, especially for the transverse trench at a high blowing ratio. It illustrates that the It illustrates that the present CFD model overestimated the spreading capability of coolant present CFD model overestimated the spreading capability of coolant inside the trench, inside the trench, which results in better cooling performance. Figure 6 shows a quantita- which results in better cooling performance. Figure 6 shows a quantitative comparison tive comparison between the experimental and CFD results. The experimental data for the between the experimental and CFD results. The experimental data for the round hole is round hole is from Ref. [25]. At M = 0.5, the mean relative error for the round hole, trans- from Ref. [25]. At M = 0.5, the mean relative error for the round hole, transverse trench, verse trench, W-shaped trench and elliptic trench is about 9.4%, 7.7%, 8.5% and 10.6%. At W-shaped trench and elliptic trench is about 9.4%, 7.7%, 8.5% and 10.6%. At M = 1.5, the M = 1.5, the mean relative error for the round hole, transverse trench, W-shaped trench mean relative error for the round hole, transverse trench, W-shaped trench and elliptic and elliptic trench is about 16.4%, 10.2%, 12.2% and 13.9%. Overall, the CFD results agree trench is about 16.4%, 10.2%, 12.2% and 13.9%. Overall, the CFD results agree well with well with the experimental results. the experimental results. Exp CFD (a) Transverse trench (w = 3.0d, h = 1.0d) Exp CFD (b) W−shaped trench (w = 1.7d, h = 1d, α = 60°) Exp CFD (c) Elliptic trench (Dx = 2.2d, Dy = 2.8d, h = 1d) Figure 5. Distributions of overall cooling effectiveness on the cooling surface. Figure 5. Distributions of overall cooling effectiveness on the cooling surface. Aerospace 2021, 8, 147 7 of 17 Aerospace 2021, 8, x FOR PEER REVIEW 7 of 18 0.9 0.9 0.8 0.8 M=0.5-EXP[25] M=0.5-EXP M=0.5-CFD 0.7 0.7 M=0.5-CFD M=1.5-EXP[25] M=1.5-EXP 0.6 M=1.5-CFD 0.6 M=1.5-CFD 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0.0 0.0 0 2 4 6 8 101214 161820 0 2 4 6 8 101214 16 1820 x/d x/d (a) Round hole (b) Transverse trench (w = 3.0d, h = 1d) 0.9 0.7 0.8 M=0.5-EXP 0.6 0.7 M=0.5-CFD M=0.5-EXP M=1.5-EXP 0.5 0.6 M=0.5-CFD M=1.5-CFD M=1.5-EXP 0.4 0.5 M=1.5-CFD 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0.0 0.0 0 2 4 6 8 101214 161820 0 2 4 6 8 101214 16 1820 x/d x/d (c) W-shaped trench (w = 1.7d, h = 1d, α = 60°) (d) Elliptic trench (Dx = 2.2d, Dy = 2.8d, h = 1d) Figure 6. Quantitative comparison between CFD results and experimental results. Figure 6. Quantitative comparison between CFD results and experimental results. 4. CFD Results Analysis 4. CFD Results Analysis 4.1. Transverse Trench 4.1. Transverse Trench Figure 7 shows the variation of laterally averaged adiabatic cooling effectiveness Figure 7 shows the variation of laterally averaged adiabatic cooling effectiveness (h ) with the streamwise distance (x/d). For the round hole, h shows a continued ad,lat ad,lat (ηad,lat) with the streamwise distance (x/d). For the round hole, ηad,lat shows a continued decline as x/d increases at a low blowing ratio, but has a slight increase in the far ﬁeld region decline as x/d increases at a low blowing ratio, but has a slight increase in the far field due to the reattachment of a separated coolant jet at a high blowing ratio. For the transverse region due to the reattachment of a separated coolant jet at a high blowing ratio. For the trench, h decreases constantly with the increase of x/d even at a high blowing ratio. ad,lat transverse trench, ηad,lat decreases constantly with the increase of x/d even at a high blow- The optimal blowing ratio for the round hole is about 0.5. However, for trench-ﬁlm ing ratio. The optimal blowing ratio for the round hole is about 0.5. However, for trench- cooling in the present case, the optimal blowing ratio is between 1.0~1.5. Figure 8a,b film cooling in the present case, the optimal blowing ratio is between 1.0~1.5. Figure 8a,b show the streamline distributions for ﬁlm cooling of the transverse trench and round hole show the streamline distributions for film cooling of the transverse trench and round hole at M = 1.5, respectively, and the background color represents gas temperature. The most at M = 1.5, respectively, and the background color represents gas temperature. The most typical feature for trench ﬁlm cooling is that a pair of recirculating vortices is formed inside typical feature for trench film cooling is that a pair of recirculating vortices is formed in- the trench. The entrainment of recirculating vortices promotes the spreading of coolant side the trench. The entrainment of recirculating vortices promotes the spreading of cool- in the lateral direction [16,26]. The existence of the trench also increases the actual jet exit ant in the lateral direction [16,26]. The existence of the trench also increases the actual jet area, reduces the actual jet velocity and mitigates the jet detachment downstream of the exit area, reduces the actual jet velocity and mitigates the jet detachment downstream of hole. In the ﬂow ﬁelds downstream of the round hole, a pair of kidney vortices (also called the hole. In the flow fields downstream of the round hole, a pair of kidney vortices (also countered rotating vortices) dominate the ﬂow ﬁeld and promote the mixture between called countered rotating vortices) dominate the flow field and promote the mixture be- mainstream and coolant [27]. For trench ﬁlm cooling, beside kidney vortices, a pair of tween mainstream and coolant [27]. For trench film cooling, beside kidney vortices, a pair anti-kidney vortices can be observed. The anti-kidney vortices with the opposite rotating of anti-kidney vortices can be observed. The anti-kidney vortices with the opposite rotat- direction of kidney vortices mitigate the detachment of coolant jet and improve the ﬁlm ing direction of kidney vortices mitigate the detachment of coolant jet and improve the cooling performance [28]. Overall, compared with a round hole, a trench hole generates film cooling performance [28]. Overall, compared with a round hole, a trench hole gener- better cooling performance, especially at a high blowing ratio. Moreover, the results from ates better cooling performance, especially at a high blowing ratio. Moreover, the results Lu et al. [11–13] show that the heat transfer coefﬁcient does not change obviously after from Lu et al. [11–13] show that the heat transfer coefficient does not change obviously introducing the trench. after introducing the trench. overall, lat overall, lat overall, lat overall, lat Aerospace Aerospace 2021 2021 , 8 , , x FO 8, 147R PEER REVIEW 8 of 8 of 18 17 Aerospace 2021, 8, x FOR PEER REVIEW 8 of 18 0.8 0.4 0.8 0.4 M=0.5 M=0.5 M=0.5 M=0.5 0.7 M=1 0.7 M=1 M=1 M=1 M=1.5 M=1.5 M=1.5 M=1.5 0.6 0.3 0.6 0.3 M=2 M=2 M=2 M=2 M=3 0.5 M=3 0.5 η η ad,lat ad,lat ad,lat ad,lat 0.4 0.2 0.2 0.4 0.3 0.3 0.1 0.2 0.1 0.2 0.1 0.1 0.0 0.0 0.0 0.0 02468 10 12 14 16 18 20 22 02468 10 12 14 16 18 20 22 0 2 4 6 8 10 12 14 16 18 20 22 0 2 4 6 8 10 12 14 16 18 20 22 x/d x/d x/d x/d (a) Round hole (b) Transverse trench (w = 3d, h = 0.75d) (a) Round hole (b) Transverse trench (w = 3d, h = 0.75d) Figure 7. Variation of laterally averaged adiabatic cooling effectiveness with streamwise distance. Figure 7. Variation of laterally averaged adiabatic cooling effectiveness with streamwise distance. Figure 7. Variation of laterally averaged adiabatic cooling effectiveness with streamwise distance. (a) Transverse trench (w = 3d, h = 0.75d) (a) Transverse trench (w = 3d, h = 0.75d) (b) Round hole (b) Round hole Figure 8. Streamline distributions for film cooling of round hole and transverse trench at M = 1.5. Figure 8. Figure 8.Strea Streamline mline di distributions stributions for for film ﬁlm co cooling oling of round of round hole and transverse trench hole and transverse trench at M at = 1 M = .5. 1.5. (KVP: kidney vortex pair). (KVP: kidney vortex pair). (KVP: kidney vortex pair). Fig Figur ure e9 sho 9 shows ws th the e vari variation ation of loc of local al ad adiabatic iabatic cool cooling ing eff effectiveness ectiveness ((ηhad,loc) wi ) with th the the Figure 9 shows the variation of local adiabatic cooling effectiveness (η ad,loc ad,loc) with the spanwise distance ( y/d ). For the round hole, ηad,loc has the maximum value at y/d = 0, and spanwise distance (jy/dj). For the round hole, h has the maximum value at y/d = 0, spanwise distance ( y/d ). For the round hole, ηad,loc ad,loc has the maximum value at y/d = 0, and then decre and thenadecr ses sh eases arply wit sharply h the incre with the ase incr of ease y/d . ofηad,loc jy/ d in tj. hhe centerline in theregion centerline ( y/dr < egion 1) ad,loc then decreases sharply with the increase of y/d . ηad,loc in the centerline region ( y/d < 1) (jy/dj < 1) decreases by increasing the blowing ratio from 0.5 to 2.0. However, the effects decreases by increasing the blowing ratio from 0.5 to 2.0. However, the effects of the blow- decreases by increasing the blowing ratio from 0.5 to 2.0. However, the effects of the blow- of the blowing ratio on h at jy/dj > 1 is unobvious. For the transverse trench, at a low ing ratio on ηad,loc at y/d > 1 is unobvious. For the transverse trench, at a low blowing ad,loc ing ratio on ηad,loc at y/d > 1 is unobvious. For the transverse trench, at a low blowing blowing ratio (M = 0.5), the distribution of h in the lateral direction is similar to that for ratio (M = 0.5), the distribution of ηad,loc in the lateral direction is similar to that for the ad,loc ratio (M = 0.5), the distribution of ηad,loc in the lateral direction is similar to that for the the round hole. However, at a high blowing ratio, as the lateral distance increases, h round hole. However, at a high blowing ratio, as the lateral distance increases, ηad,locad,loc in- round hole. However, at a high blowing ratio, as the lateral distance increases, ηad,loc in- increases ﬁrstly ((jy/dj < 0.5), then decreases (0.5 <jy/dj < 1.2), and shows a slight increase creases firstly (( y/d < 0.5), then decreases (0.5 < y/d < 1.2), and shows a slight increase creases firstly (( y/d < 0.5), then decreases (0.5 < y/d < 1.2), and shows a slight increase ﬁnally ( y/d > 1.2). Figure 10 shows the distributions of ﬂow ﬁelds on the exit plane of j j finally ( y/d > 1.2). Figure 10 shows the distributions of flow fields on the exit plane of finally ( y/d > 1.2). Figure 10 shows the distributions of flow fields on the exit plane of holes; the background color represents the vertical velocity, and the arrow represents the holes; the background color represents the vertical velocity, and the arrow represents the holes; the background color represents the vertical velocity, and the arrow represents the clockwise or anti-clockwise rotation direction of the vortex. Compared with the transverse clockwise or anti-clockwise rotation direction of the vortex. Compared with the transverse clockwise or anti-clockwise rotation direction of the vortex. Compared with the transverse trench, the vertical velocity on the exit surface of the round hole distributes more uniformly trench, the vertical velocity on the exit surface of the round hole distributes more uni- trench, the vertical velocity on the exit surface of the round hole distributes more uni- and is of higher value. It results in the detachment of coolant jet immediately downstream formly and is of higher value. It results in the detachment of coolant jet immediately formly and is of higher value. It results in the detachment of coolant jet immediately of the round hole at a high blowing ratio. For the transverse trench, a pair of vortices is Aerospace 2021, 8, x FOR PEER REVIEW 9 of 18 Aerospace 2021, 8, x FOR PEER REVIEW 9 of 18 Aerospace 2021, 8, 147 9 of 17 downstream of the round hole at a high blowing ratio. For the transverse trench, a pair of downstream of the round hole at a high blowing ratio. For the transverse trench, a pair of vortices is formed, and the entrainment of the vortex pair promotes the lateral spreading vortices is formed, and form the ed, an entrainment d the entrainment of the vortex of th pair e vortex promotes pair promote the lateral s the lateral spreadingspread of coolant ing of coolant inside the trench. It results in the wavy distribution of vertical velocity in the of inside coolathe nt in trsi ench. de the trench. It resul It results in thetwavy s in the diwa stribution vy distribut of vertical ion of vertical velocity in the velocity in the lateral lateral direction. The wave crest locates at y/d = 0.0 where ηad,loc has a local minimum value lat dir eection. ral direct The ion. The wave wave crest locates crest loc atay te /s da= t y 0.0 /d = 0. wher 0 wh e h ere ηad,loc has ha a local s a local minimum va minimum value lue at a ad,loc at a high blowing ratio. at a high high blowing blowing r ratio. atio. 0.9 0.9 0.8 M=0.5 0.8 0.8 M=0.5 0.8 M=1 0.7 M=1 0.7 M=1.5 0.7 0.7 M=1.5 M=2 0.6 0.6 0.6 M=2 ad,loc η 0.6 η 0.5 ad,loc ad,loc 0.5 η ad,loc 0.5 M=0.5 0.4 0.5 M=0.5 0.4 M=1 0.3 M=1 0.4 M=1.5 0.3 0.4 M=1.5 0.2 M=2 0.2 0.3 M=2 M=3 0.1 0.3 M=3 0.1 0.0 0.2 0.0 0.2 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 y/d y/d y/d y/d (a) Round hole (b) Transverse trench (w = 3d, h = 0.75d) (a) Round hole (b) Transverse trench (w = 3d, h = 0.75d) Figure 9. Variation of local adiabatic film cooling effectiveness with spanwise distance on the cross Figure 9. Variation of local adiabatic ﬁlm cooling effectiveness with spanwise distance on the cross Figure 9. Variation of local adiabatic film cooling effectiveness with spanwise distance on the cross section of x/d = 2.2. section of x/d = 2.2. section of x/d = 2.2. (a) Round hole (b) Transverse trench (w = 3d, h = 0.75d) (a) Round hole (b) Transverse trench (w = 3d, h = 0.75d) Figure 10. Distribution of flow fields on the exit plane of the film cooling hole at M = 1.5 (unit: Figure 10. Distribution of flow fields on the exit plane of the film cooling hole at M = 1.5 (unit: Figure 10. Distribution of ﬂow ﬁelds on the exit plane of the ﬁlm cooling hole at M = 1.5 (unit: m/s). m/s). m/s). Figure 11a shows the effects of trench width (W/d) and depth (h/d) on the area-averaged Figure 11a shows the effects of trench width (W/d) and depth (h/d) on the area-aver- adiabatic Figure ﬁlm 11a sh cooling ows the e effectiveness ffects of trench w (h , 2 < idth ( x/d < W20). /d) an The d depth ( deep trh ench /d) on gives the a the rea highest -aver- ad,av aged adiabatic film cooling effectiveness (ηad,av, 2 < x/d < 20). The deep trench gives the aged adiabatic film cooling effectiveness (ηad,av, 2 < x/d < 20). The deep trench gives the h , and h decreases with the decrease of trench depth. For a shallow trench, the ad,av ad,av highest ηad,av, and ηad,av decreases with the decrease of trench depth. For a shallow trench, highest coolantη trajectory ad,av, and η is ad,a har v decr dly af ease fected s with th by the e decre trench, ase o and f trench depth. For a the pronounced ﬂow shallo separation w trench, still the coolant trajectory is hardly affected by the trench, and the pronounced flow separation the cool takes place ant tra downstr jectory eam is ha of rdl the y ahole. ffected b Inside y the trench, the deep tr and the pronounced ench, the recirculating flow ﬂow sepa reduces ration still takes place downstream of the hole. Inside the deep trench, the recirculating flow still take the coolant s place downstre penetration into am of the hole. Insi the mainstream and de the imprdeep trench, oves the coolant the recirc uniformity ulating flow . Accord- reduces the coolant penetration into the mainstream and improves the coolant uniformity. ingly, the coolant ejected from the deep trench exhibits better covering performance on the reduces the coolant penetration into the mainstream and improves the coolant uniformity. Accordingly, the coolant ejected from the deep trench exhibits better covering perfor- cooling surface downstream of the trench. For most cases, the ﬁlm cooling performance Accordingly, the coolant ejected from the deep trench exhibits better covering perfor- mance on the cooling surface downstream of the trench. For most cases, the film cooling can be improved effectively by reducing the trench width. For a narrow trench, because of mance on the cooling surface downstream of the trench. For most cases, the film cooling performance can be improved effectively by reducing the trench width. For a narrow the effect of recirculating ﬂow, the mainstream cannot enter the trench, which enhances performance can be improved effectively by reducing the trench width. For a narrow trench, because of the effect of recirculating flow, the mainstream cannot enter the trench, the ﬁlm cooling effectiveness. For a wide trench, due to the mainstream intursion, the trench, because of the effect of recirculating flow, the mainstream cannot enter the trench, which enhances the film cooling effectiveness. For a wide trench, due to the mainstream recirculation of the coolant inside the trench is mitigated, which results in the decrease which enhances the film cooling effectiveness. For a wide trench, due to the mainstream intursion, the recirculation of the coolant inside the trench is mitigated, which results in of cooling effectiveness [12,16]. Compared with trench width, the effect of trench depth intursion, the recirculation of the coolant inside the trench is mitigated, which results in the decrease of cooling effectiveness [12,16]. Compared with trench width, the effect of on cooling effectiveness is more pronounced. Figure 11b shows the effects of geometric the decrease of cooling effectiveness [12,16]. Compared with trench width, the effect of trench depth on cooling effectiveness is more pronounced. Figure 11b shows the effects of parameters on the discharge coefﬁcient (C ). The increase of h/d reduces the mixing loss trench depth on cooling effectiveness is more d pronounced. Figure 11b shows the effects of geometric parameters on the discharge coefficient (Cd). The increase of h/d reduces the between the mainstream and coolant jet, and the decrease of w/d results in the increase of geometric parameters on the discharge coefficient (Cd). The increase of h/d reduces the mixing loss between the mainstream and coolant jet, and the decrease of w/d results in the the actual blowing ratio. Thus, C increases with the increase of h/d but the decrease of mixing loss between the mainstream and coolant jet, and the decrease of w/d results in the w/d. Overall, a narrower and deeper trench generates better ﬁlm cooling performance. Aerospace 2021, 8, x FOR PEER REVIEW 10 of 18 Aerospace 2021, 8, x FOR PEER REVIEW 10 of 18 Aerospace 2021, 8, 147 10 of 17 increase of the actual blowing ratio. Thus, Cd increases with the increase of h/d but the increase of the actual blowing ratio. Thus, Cd increases with the increase of h/d but the decrease of w/d. Overall, a narrower and deeper trench generates better film cooling per- decrease of w/d. Overall, a narrower and deeper trench generates better film cooling per- formance. formance. W=2.5d, M=0.5 0.68 W=2.5d, M=0.5 0.68 0.5 0.5 W=2.5d, M=1.5 W=2.5d, M=1.5 W=3.0d, M=0.5 W=3.0d, M=0.5 0.66 0.66 W=3.0d, M=1.5 W=3.0d, M=1.5 0.4 0.4 W=3.5d, M=0.5 W=3.5d, M=0.5 0.64 0.64 W=3.5d, M=1.5 W=3.5d, M=1.5 0.3 0.3 0.62 0.62 w=2.5d, M=0.5 w=2.5d, M=0.5 0.60 0.60 w=2.5d, M=1.5 w=2.5d, M=1.5 0.2 0.2 w=3.0d, M=0.5 w=3.0d, M=0.5 0.58 0.58 w=3.0d, M=1.5 w=3.0d, M=1.5 0.1 0.1 w=3.5d, M=0.5 w=3.5d, M=0.5 0.56 0.56 w=3.5d, M=1.5 w=3.5d, M=1.5 0.54 0.0 0.54 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.2 0.4 0.6 0.8 1.0 1.2 1. 04 .2 0.4 0.6 0.8 1.0 1.2 1.4 0.2 0.4 0.6 0.8 1.0 1.2 1.4 h/d h/d h/d h/d (a) Area-averaged adiabatic cooling effectiveness (b) Discharge coefficient (a) Area-averaged adiabatic cooling effectiveness (b) Discharge coefficient Figure 11. Effects of trench depth and width on film cooling performance for transverse trench. Figure 11. Effects Figure of trench 11. Ef de fects pth and width of trench depth on film and cool width ing perform on ﬁlm ance f cooling or transverse performance trench. for transverse trench. 4.2. W-Shaped Trench 4.2. W-Shaped Trench 4.2. W-Shaped Trench Figure 12a shows the variation of laterally averaged adiabatic film cooling effective- Figure 12a shows the variation of laterally averaged adiabatic ﬁlm cooling effectiveness Figure 12a shows the variation of laterally averaged adiabatic film cooling effective- ness with the streamwise distance. As the blowing ratio increases, ηad,lat increases firstly, with the streamwise distance. As the blowing ratio increases, h increases ﬁrstly, ness with the streamwise distance. As the blowing ratio increases, ηad,lat ad,lat increases firstly, and then decreases. The optimal blowing ratio for the W-shaped trench is about 1.5. Com- and then decreases. The optimal blowing ratio for the W-shaped trench is about 1.5. and then decreases. The optimal blowing ratio for the W-shaped trench is about 1.5. Com- pared with the transverse trench, the cooling performance for the W-shaped trench is bet- Compared with the transverse trench, the cooling performance for the W-shaped trench is pared with the transverse trench, the cooling performance for the W-shaped trench is bet- ter. Figure 12b shows the variation of local adiabatic film cooling effectiveness with the better. Figure 12b shows the variation of local adiabatic ﬁlm cooling effectiveness with the ter. Figure 12b shows the variation of local adiabatic film cooling effectiveness with the spanwise distance. At a low blowing ratio (M = 0.5), as y/d increases, ηad,loc decreases spanwise spanwise dis distance. tance. At At a lo a low w blow blowing ing ratio ( ratioM (M = = 0.5) 0.5 , a ),sas yj /y d/ incre dj incr ase eases, s, ηad,loc h decre decr ases eases ad,loc firstly, and then increases. ηad,loc at M = 0.5 has a maximum value at y/d = 0.0 and a mini- fir ﬁrstly stly, ,and and then inc then incr reases. eases.ηad,loc h at M at = 0.5 has a ma M = 0.5 has ximum val a maximum ue at value y/d = 0. at 0 an y/d a d = m 0.0 iniand - a ad,loc mum value at y/d = 0.75. At a moderate blowing ratio (M = 1.5), ηad,loc decreases slightly mum va minimum lue value at y/dat = jy 0. / 75 dj. At = 0.75. a moAt derat a moderate e blowing r blowing atio (M = ratio 1.5), (ηM ad,loc =d 1.5), ecreases sligh h decr tly eases ad,loc as y/d increases from 0.0 to 1.0, and then shows a sharp decrease as y/d exceeds 1.0. as y/d increases from 0.0 to 1.0, and then shows a sharp decrease as y/d exceeds 1.0. slightly asjy/dj increases from 0.0 to 1.0, and then shows a sharp decrease asjy/dj exceeds At a high blowing ratio (M = 3.0), the changing trend of ηad,loc is similar to that at a moder- At a h 1.0. At igh a blo high wing r blowing atio (M ratio = 3.0 (M ), th =e3.0), changin the g changing trend of η tr ad,l end oc isof sim hilar tois th similar at at a m to oder- that at a ad,loc ate blowing ratio. However, ηad,loc at a high blowing ratio has a local minimum value at y/d ate blowing ratio. However, ηad,loc at a high blowing ratio has a local minimum value at y/d moderate blowing ratio. However, h at a high blowing ratio has a local minimum value = 0.0. Figure 13 shows ad,loc the streamline distributions for film cooling of the W-shaped trench, = 0.0. Figure 13 shows the streamline distributions for film cooling of the W-shaped trench, at y/d = 0.0. Figure 13 shows the streamline distributions for ﬁlm cooling of the W-shaped and the background color represents the gas temperature. At a low blowing ratio, pro- and the background color represents the gas temperature. At a low blowing ratio, pro- trench, and the background color represents the gas temperature. At a low blowing ratio, nounced recirculating flow can be observed inside the trench and promotes the lateral nounced recirculating flow can be observed inside the trench and promotes the lateral pronounced recir sprea culating ding of ﬂow cool can ant. At be observed a high blowi inside ng ra the tio, the coola trench and nt jet exhi promote bitss sl the ight deta lateralchment in spreading of coolant. At a high blowing ratio, the coolant jet exhibits slight detachment in the region of x/d = 2.0, but the coverage performance of coolant is still much better than spreading of coolant. At a high blowing ratio, the coolant jet exhibits slight detachment the region of x/d = 2.0, but the coverage performance of coolant is still much better than the transverse trench and the round hole, especially in the far-field region (x/d > 5.0). Sim- in the region of x/d = 2.0, but the coverage performance of coolant is still much better the transverse trench and the round hole, especially in the far-field region (x/d > 5.0). Sim- ilar to the transverse trench, the kidney vortex pair and anti-kidney vortex pair exist sim- than the transverse trench and the round hole, especially in the far-ﬁeld region (x/d > 5.0). ilar to the transverse trench, the kidney vortex pair and anti-kidney vortex pair exist sim- ultaneously downstream of the W-shaped trench. The anti-kidney vortices improve cool- Similar to the transverse trench, the kidney vortex pair and anti-kidney vortex pair exist ultaneously downstream ing performance, while the of the W-shaped trench. The kidney vortic anti-kidne es degrad y v e co ortices oling perform improve cool- ance. However, com- simultaneously downstream of the W-shaped trench. The anti-kidney vortices improve ing performance, while the pared with the tra kidney vortic nsverse trench, the W- es degrade cooling sha perform ped trench exhi ance. Howev bits ea r, com- stronger anti-kidney cooling performance, while the kidney vortices degrade cooling performance. However, vortex pair and weaker kidney vortex pair. pared with the transverse trench, the W-shaped trench exhibits a stronger anti-kidney compared with the transverse trench, the W-shaped trench exhibits a stronger anti-kidney vortex pair and weaker kidney vortex pair. vortex pair and weaker kidney vortex pair. 0.9 0.9 M=0.5 0.8 M=1 0.9 0.9 0.8 M=0.5 M=1.5 0.7 0.8 M=1 M=2 0.6 0.8 0.7 M=1.5 M=3 0.7 ad,lat M=2 0.5 ad,loc 0.6 0.6 M=3 0.7 η 0.4 η M=0.5 ad,lat 0.5 ad,loc 0.5 M=1 0.6 0.3 0.4 M=1.5 M=0.5 0.2 M=2 0.4 0.5 M=1 0.3 M=3 0.1 M=1.5 0.2 0.3 0.4 M=2 0.0 0 2468 10 12 14 16 18 20 22 M=3 0.1 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 0.3 x/d 0.0 y/d 0 2468 10 12 14 16 18 20 22 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 x/d (a) Laterally averaged value (b) Local value on the cross section of x/d = 2.2 y/d (a) Laterally averaged value (b) Local value on the cross section of x/d = 2.2 Figure 12. Distributions of adiabatic ﬁlm cooling effectiveness in different directions for W-shaped trench (w = 1.2d, h = 0.75d, a = 60 ). ad, av ad, av d Aerospace 2021, 8, x FOR PEER REVIEW 11 of 18 Aerospace 2021, 8, 147 11 of 17 Figure 12. Distributions of adiabatic film cooling effectiveness in different directions for W-shaped trench (w = 1.2d, h = 0.75d, α = 60°). (a) M = 0.5 (b) M = 1.5 Figure 13. Streamline distributions for film cooling of W−shaped trench (w = 1.2d, h = 0.75d, α = Figure 13. Streamline distributions for ﬁlm cooling of Wshaped trench (w = 1.2d, h = 0.75d, a = 60 ). 60°). Figure 14a shows the effects of trench width and depth on the area-averaged adiabatic Figure 14a shows the effects of trench width and depth on the area-averaged adia- ﬁlm cooling effectiveness (h , 2 < x/d < 20). For the narrow trench, because of the ad,av batic film cooling effectiveness (ηad,av, 2 < x/d < 20). For the narrow trench, because of the effect of recirculating ﬂow, the mainstream cannot enter the trench, which enhances the effect of recirculating flow, the mainstream cannot enter the trench, which enhances the ﬁlm cooling effectiveness. For the wide trench, due to the mainstream intursion, the film cooling effectiveness. For the wide trench, due to the mainstream intursion, the recir- recirculation of the coolant inside the trench is mitigated, which results in the decrease of culation of the coolant inside the trench is mitigated, which results in the decrease of cool- cooling effectiveness. Compared with trench width, the inﬂuence of h/d on ﬁlm cooling ing effectiveness. Compared with trench width, the influence of h/d on film cooling per- performance is much more obvious, especially at a high blowing ratio. At a high blowing formance ratio (M = is 1.5), much more h incr o eases bvious, from esabout pecially 0.15 at to a hig 0.5 h with blow the ing incr raease tio. At of ha h /difr gh om blow 0.25ing to ad,av ratio ( 1.3. AtM a = 1. low5blowi ), ηad,a ng v inc ratio, reases f h rom also abincr outeases 0.15 twith o 0.5 the witincr h th ease e incr ofease o trench f h depth, /d from however 0.25 to, ad,av the variation amplitude is weaker compared to that at a high blowing ratio. Figure 14b 1.3. At a low blowing ratio, ηad,av also increases with the increase of trench depth, however, t shows he varthe iation effects amp of littr ude ench is width weaker and com depth pared on tthe o thdischar at at a high ge coef bﬁcient. lowing For ratia o.deep Figutr re ench, 14b the distribution of coolant velocity at the trench exit is uniform, and the area of the high shows the effects of trench width and depth on the discharge coefficient. For a deep trench, speed zone is smaller compared with the shallow trench (shown in Figure 15), which the distribution of coolant velocity at the trench exit is uniform, and the area of the high reduces mixing loss between the mainstream and coolant jet. For the narrow trench, the jet speed zone is smaller compared with the shallow trench (shown in Figure 15), which re- Aerospace 2021, 8, x FOR PEER REVIEW 12 of 18 velocity is higher, which results in the increase of ﬂow loss. Thus, the increase of trench duces mixing loss between the mainstream and coolant jet. For the narrow trench, the jet depth and width both cause the increase of the discharge coefﬁcient. velocity is higher, which results in the increase of flow loss. Thus, the increase of trench depth and width both cause the increase of the discharge coefficient. 0.6 0.70 w=1.2d, M=0.5 w=1.2d, M=1.5 0.5 w=1.9d, M=0.5 0.68 w=1.9d, M=1.5 0.4 0.66 0.3 0.64 w=1.2d, M=0.5 w=1.2d, M=1.5 w=1.8d, M=0.5 w=2.4d, M=0.5 0.2 0.62 w=1.8d, M=1.5 w=2.4d, M=1.5 w=2.4d, M=0.5 w=2.4d, M=1.5 0.1 0.60 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.2 0.4 0.6 0.8 1.0 1.2 1.4 h/d h/d (a) Area-averaged adiabatic cooling effectiveness (b) Discharge coefficient Figure 14. Effects of trench depth and width on film cooling performance for W-shaped trench (α = 60°). Figure 14. Effects of trench depth and width on film cooling performance for W-shaped trench (a = 60 ). (a) w = 1.2d, h = 1.2d, α = 60° (b) w = 1.2d, h = 1.8d, α = 60° Figure 15. Distribution of flow fields on the exit plane of W−shaped trench at M = 1.5 (Unit: m/s). Figure 16a shows the effect of corner angle on the area-averaged adiabatic film cool- ing effectiveness. For a high corner angle, the lateral spreading inside the trench cannot be affected effectively by the W-structure. In fact, as the corner angle approaches 180°, the W-shaped trench turns into a transverse trench, and the guide effect of the corner angle disappears. Conversely, a small corner angle promotes lateral spreading of coolant and improves distribution uniformity of the coolant velocity at the trench exit, which results in high cooling effectiveness. Figure 16b shows the effect of corner angle on the discharge coefficient. Cd decreases with the increase of the corner angle. It is because that high corner angle causes the decrease of the trench exit area, which results in high flow loss. However, because the changing interval of α is small (40~60 ), the effects of the corner angle on the cooling effectiveness and discharge coefficient are not very obvious in the present study. ad,av Aerospace 2021, 8, x FOR PEER REVIEW 12 of 18 0.6 0.70 w=1.2d, M=0.5 w=1.2d, M=1.5 w=1.9d, M=0.5 0.5 0.68 w=1.9d, M=1.5 0.4 0.66 0.3 0.64 w=1.2d, M=0.5 w=1.2d, M=1.5 w=1.8d, M=0.5 w=2.4d, M=0.5 0.2 0.62 w=1.8d, M=1.5 w=2.4d, M=1.5 w=2.4d, M=0.5 w=2.4d, M=1.5 0.1 0.60 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.2 0.4 0.6 0.8 1.0 1.2 1.4 h/d h/d Aerospace 2021, 8, 147 12 of 17 (a) Area-averaged adiabatic cooling effectiveness (b) Discharge coefficient Figure 14. Effects of trench depth and width on film cooling performance for W-shaped trench (α = 60°). (a) w = 1.2d, h = 1.2d, α = 60° (b) w = 1.2d, h = 1.8d, α = 60° Figure 15. Distribution of flow fields on the exit plane of W−shaped trench at M = 1.5 (Unit: m/s). Figure 15. Distribution of ﬂow ﬁelds on the exit plane of Wshaped trench at M = 1.5 (Unit: m/s). Figure 16a shows the effect of corner angle on the area-averaged adiabatic ﬁlm cooling Figure 16a shows the effect of corner angle on the area-averaged adiabatic film cool- effectiveness. For a high corner angle, the lateral spreading inside the trench cannot be ing effectiveness. For a high corner angle, the lateral spreading inside the trench cannot affected effectively by the W-structure. In fact, as the corner angle approaches 180 , the be affected effectively by the W-structure. In fact, as the corner angle approaches 180°, the W-shaped trench turns into a transverse trench, and the guide effect of the corner angle W-shaped trench turns into a transverse trench, and the guide effect of the corner angle disappears. Conversely, a small corner angle promotes lateral spreading of coolant and disappears. Conversely, a small corner angle promotes lateral spreading of coolant and improves distribution uniformity of the coolant velocity at the trench exit, which results in improves distribution uniformity of the coolant velocity at the trench exit, which results high cooling effectiveness. Figure 16b shows the effect of corner angle on the discharge in high cooling effectiveness. Figure 16b shows the effect of corner angle on the discharge coefﬁcient. C decreases with the increase of the corner angle. It is because that high corner coefficient. Cd decreases with the increase of the corner angle. It is because that high corner angle causes the decrease of the trench exit area, which results in high ﬂow loss. However, Aerospace 2021 angle c , 8, x FO aR P uses t EER RE he decre VIEW ase of the trench exit area, which results in high flow loss. However, 13 of 18 because the changing interval of a is small (40~60 ), the effects of the corner angle on the because the changing interval of α is small (40~60 ), the effects of the corner angle on the cooling effectiveness and discharge coefﬁcient are not very obvious in the present study. cooling effectiveness and discharge coefficient are not very obvious in the present study. 0.7 0.70 w=1.2d, h=0.3d, M=0.5 w=1.2d, h=0.3d, M=1.5 0.6 0.68 w=1.8d, h=0.9d, M=0.5 w=1.8d, h=0.9d, M=1.5 0.5 0.66 ad,av 0.4 w=1.2d, h=0.3d, M=0.5 w=1.2d, h=0.3d, M=1.5 0.64 w=1.8d, h=0.9d, M=0.5 0.3 w=1.8d, h=0.9d, M=1.5 0.62 0.2 0.60 0.1 40 50 60 70 80 40 50 60 70 80 α (°) α (°) (a) Area−averaged adiabatic cooling effectiveness (b) Discharge coefficient Figure 16. Effects of corner angle on film cooling performance for W−shaped trench. Figure 16. Effects of corner angle on ﬁlm cooling performance for Wshaped trench. 4.3. Elliptical Trench 4.3. Elliptical Trench Figure 17a shows the variation of laterally averaged adiabatic film cooling effective- Figure 17a shows the variation of laterally averaged adiabatic ﬁlm cooling effectiveness ness with the streamwise distance. At a low blowing ratio, ηad,lat shows continued decrease with the streamwise distance. At a low blowing ratio, h shows continued decrease ad,lat with the increase of x/d. At a high blowing ratio, as x/d increases, ηad,lat decreases firstly, with the increase of x/d. At a high blowing ratio, as x/d increases, h decreases ﬁrstly, ad,lat and then increases. The rebound of ηad,lat in the far-field region can be attributed to the and then increases. The rebound of h in the far-ﬁeld region can be attributed to the ad,lat reattachment of the coolant jet. Figure 17b shows the variation of local adiabatic film cool- reattachment of the coolant jet. Figure 17b shows the variation of local adiabatic ﬁlm ing effectiveness with the lateral distance. ηad,loc decreases with the increase of y/d and cooling effectiveness with the lateral distance. h decreases with the increase of y/d j j ad,loc has a maximum value at y/d = 0. The changing trend of ηad,loc for the elliptical trench is and has a maximum value at y/d = 0. The changing trend of h for the elliptical similar to that for the round hole, but different from that f ad,loc or the transverse and W-shaped trench is similar to trench. Figure 18 that for the r ound shows the strea hole, but mldif ine fer distri entbfr uti om ons that for fifor lm cooling f the transverse or the ellipti and c trench, W-shaped trench. and the back Figure 18 ground co shows the lorstr represents g eamline distributions as temperature. Compare for ﬁlm cooling d with the transver for the se trench and the W-shaped trench, the secondary flow inside the elliptic trench is unobvi- elliptic trench, and the background color represents gas temperature. Compared with the ous, and lateral spreading of coolant is also weak in the elliptic trench. The kidney vortex transverse trench and the W-shaped trench, the secondary ﬂow inside the elliptic trench is pair takes the dominant role on the cross section downstream of the elliptic trench, while the anti-kidney vortex pair cannot be formed. The entrainment effect of the kidney vortex pair results in the detachment of coolant downstream of the hole and promotes the mix- ture between mainstream and coolant. Thus, the elliptic trench generates lower cooling effectiveness than the transverse and W-shaped trench. However, the scale of the kidney vortex pair for the elliptic trench is smaller than that for the round hole, accordingly, the cooling effectiveness of the elliptic trench is higher than that with the round hole. In gen- eral, the flow characteristics for the elliptical trench are very similar to those for the round hole, but different from the transverse trench and W-shaped trench. 0.5 0.9 M=0.5 M=0.5 0.8 M=1 M=1 0.4 M=1.5 0.7 M=1.5 M=2 M=2 0.6 η M=2.5 0.3 ad,lat ad,loc 0.5 M=3 0.4 0.2 0.3 0.2 0.1 0.1 0.0 0.0 0 2 4 6 8 10 12 14 16 18 20 22 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 x/d y/d (a) Laterally averaged value (b) Local value on the cross section of x/d = 2.2 Figure 17. Distributions of adiabatic film cooling effectiveness for elliptic trench (Dx = 2.4d, Dy = 1.2d, h = 0.75d). ad,av Aerospace 2021, 8, x FOR PEER REVIEW 13 of 18 0.7 0.70 w=1.2d, h=0.3d, M=0.5 w=1.2d, h=0.3d, M=1.5 0.6 0.68 w=1.8d, h=0.9d, M=0.5 w=1.8d, h=0.9d, M=1.5 0.5 0.66 ad,av 0.4 w=1.2d, h=0.3d, M=0.5 w=1.2d, h=0.3d, M=1.5 0.64 w=1.8d, h=0.9d, M=0.5 0.3 w=1.8d, h=0.9d, M=1.5 0.62 0.2 0.60 0.1 40 50 60 70 80 40 50 60 70 80 α (°) α (°) (a) Area−averaged adiabatic cooling effectiveness (b) Discharge coefficient Figure 16. Effects of corner angle on film cooling performance for W−shaped trench. 4.3. Elliptical Trench Figure 17a shows the variation of laterally averaged adiabatic film cooling effective- ness with the streamwise distance. At a low blowing ratio, ηad,lat shows continued decrease with the increase of x/d. At a high blowing ratio, as x/d increases, ηad,lat decreases firstly, and then increases. The rebound of ηad,lat in the far-field region can be attributed to the reattachment of the coolant jet. Figure 17b shows the variation of local adiabatic film cool- ing effectiveness with the lateral distance. ηad,loc decreases with the increase of y/d and has a maximum value at y/d = 0. The changing trend of ηad,loc for the elliptical trench is similar to that for the round hole, but different from that for the transverse and W-shaped Aerospace 2021, 8, 147 13 of 17 trench. Figure 18 shows the streamline distributions for film cooling for the elliptic trench, and the background color represents gas temperature. Compared with the transverse trench and the W-shaped trench, the secondary flow inside the elliptic trench is unobvi- unobvious, and lateral spreading of coolant is also weak in the elliptic trench. The kidney ous, and lateral spreading of coolant is also weak in the elliptic trench. The kidney vortex vortex pair takes pair ta the kes the domi dominant rna ole nt role on the cross se on the cross section downstr ction dow eam nstream o of the elliptic f the elliptic trench, trench, while while the anti-kidney vortex pair cannot be formed. The entrainment effect of the kidney the anti-kidney vortex pair cannot be formed. The entrainment effect of the kidney vortex vortex pair results pair resul in the ts in the deta detachment chment of of cool coolant do ant downstr wnstream of the hole and promotes the mix- eam of the hole and promotes the mixture between mainstream and coolant. Thus, the elliptic trench generates lower ture between mainstream and coolant. Thus, the elliptic trench generates lower cooling cooling effectiveness effectiveness than than the transverse and the transverse and W-shaped W-shaped trenc trench. However h. However, t , the scale he scale of th of the e kidney kidney vortex pair for the elliptic trench is smaller than that for the round hole, accordingly, vortex pair for the elliptic trench is smaller than that for the round hole, accordingly, the the cooling ef cooling fectiveness effect of iveness the elliptic of thetr el ench liptic is trhigher ench isthan higher tha that with n tha the t wi round th the round hole. In gen- hole. In general, the ﬂow characteristics for the elliptical trench are very similar to those for the eral, the flow characteristics for the elliptical trench are very similar to those for the round round hole, but different from the transverse trench and W-shaped trench. hole, but different from the transverse trench and W-shaped trench. 0.5 0.9 M=0.5 M=0.5 0.8 M=1 M=1 0.4 M=1.5 0.7 M=1.5 M=2 M=2 0.6 η M=2.5 0.3 ad,lat η ad,loc 0.5 M=3 0.4 0.2 0.3 0.2 0.1 0.1 0.0 0.0 0 2 4 6 8 10 12 14 16 18 20 22 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 x/d y/d (a) Laterally averaged value (b) Local value on the cross section of x/d = 2.2 Aerospace 2021, 8, x FOR PEER REVIEW 14 of 18 Figure 17. Distributions of adiabatic film cooling effectiveness for elliptic trench (Dx = 2.4d, Dy = 1.2d, h = 0.75d). Figure 17. Distributions of adiabatic ﬁlm cooling effectiveness for elliptic trench (D = 2.4d, D = 1.2d, x y h = 0.75d). (a) M = 0.5 x/d=10 x/d=2 x/d=5 (b) M = 1.5 Figure 18. Streamline distributions for film cooling of elliptic trench (Dx = 2.4d, Dy = 1.2d, h = 0.75d). Figure 18. Streamline distributions for ﬁlm cooling of elliptic trench (D = 2.4d, D = 1.2d, h = 0.75d). x y Figure 19a shows the effects of axis length of the elliptical trench (Dx and Dy) on the Figure 19a shows the effects of axis length of the elliptical trench (D and D ) on the x y area-averaged adiabatic film cooling effectiveness (ηad,av, 2 < x/d < 20). For small Dy, ηad,av area-averaged adiabatic ﬁlm cooling effectiveness (h , 2 < x/d < 20). For small D , ad,av increases with the rise of Dx. However, for large Dy, this changing trend becomes wholly h increases with the rise of D . However, for large D , this changing trend becomes ad,av x y opposite, and ηad,av decreases with the increase of Dx. It illustrates that there is an optimal wholly opposite, and h decreases with the increase of D . It illustrates that there is an ad,av exit area of the elliptical trench, and a too high and low exit area both deteriorate the cool- ing performance. If the exit area is higher than the optimal value, the actual jet velocity is too low, and the mainstream can penetrate into the trench. If the exit area is small, the actual jet velocity has high momentum, and shows a detachment effect from the wall downstream of the hole. Figure 20 shows the streamline distributions on the exit planes of elliptic trenches. Compared with the W-shaped trench and transverse trench, the streamlines for the elliptic trench are smoother, and the recirculating vortex pair cannot be formed inside the trench. Figure 19b shows the effects of axis length on discharge co- efficient. The variation trend of Cd with Dx for large Dy is contrary to that for small Dy. For Dy = 2.5d, Cd increases with the increases of Dx. However, for Dy = 1.5d, Cd decreases as Dx increases. In general, as Dx is close to Dy, the flow loss is relatively low. (a) Area−averaged adiabatic cooling effectiveness (b) Discharge coefficient Aerospace 2021, 8, x FOR PEER REVIEW 14 of 18 (a) M = 0.5 x/d=10 x/d=2 x/d=5 (b) M = 1.5 Figure 18. Streamline distributions for film cooling of elliptic trench (Dx = 2.4d, Dy = 1.2d, h = 0.75d). Aerospace 2021, 8, 147 14 of 17 Figure 19a shows the effects of axis length of the elliptical trench (Dx and Dy) on the area-averaged adiabatic film cooling effectiveness (ηad,av, 2 < x/d < 20). For small Dy, ηad,av increases with the rise of Dx. However, for large Dy, this changing trend becomes wholly opposite, and ηad,av decreases with the increase of Dx. It illustrates that there is an optimal optimal exit area of the elliptical trench, and a too high and low exit area both deteriorate exit area of the elliptical trench, and a too high and low exit area both deteriorate the cool- the cooling performance. If the exit area is higher than the optimal value, the actual jet ing performance. If the exit area is higher than the optimal value, the actual jet velocity is velocity is too low, and the mainstream can penetrate into the trench. If the exit area is too low, and the mainstream can penetrate into the trench. If the exit area is small, the small, the actual jet velocity has high momentum, and shows a detachment effect from actual jet velocity has high momentum, and shows a detachment effect from the wall the wall downstream of the hole. Figure 20 shows the streamline distributions on the exit downstream of the hole. Figure 20 shows the streamline distributions on the exit planes planes of elliptic trenches. Compared with the W-shaped trench and transverse trench, the of elliptic trenches. Compared with the W-shaped trench and transverse trench, the streamlines for the elliptic trench are smoother, and the recirculating vortex pair cannot streamlines for the elliptic trench are smoother, and the recirculating vortex pair cannot be formed inside the trench. Figure 19b shows the effects of axis length on discharge be formed inside the trench. Figure 19b shows the effects of axis length on discharge co- coefﬁcient. The variation trend of C with D for large D is contrary to that for small D . efficient. The variation trend of Cd with Dx for large Dy is contrary to that for small Dy. For x y y For D = 2.5d, C Dy = incr 2.5eases d, Cd increases with with the increases the increases of of D . However Dx. Howe , forver, for D = 1.5 Dd y = , C 1.5decr d, Cdeases decreases as Dx y d x y d increases. In general, as Dx is close to Dy, the flow loss is relatively low. as D increases. In general, as D is close to D , the ﬂow loss is relatively low. x x y Aerospace 2021, 8, x FOR PEER REVIEW 15 of 18 (a) Area−averaged adiabatic cooling effectiveness (b) Discharge coefficient Figure 19. Effects of axis length on film cooling performance for elliptic trench. Figure 19. Effects of axis length on ﬁlm cooling performance for elliptic trench. (a) Dx = 2.2d, Dy = 2.8d, h = 0.9d (b) Dx = 3.2d, Dy = 2.2d, h = 0.9d Figure 20. Streamline distribution on the exit plane of elliptic trench at M = 0.5 (Unit: m/s). Figure 20. Streamline distribution on the exit plane of elliptic trench at M = 0.5 (Unit: m/s). Figure 21 compares the distribution of adiabatic film cooling effectiveness on the wall Figure 21 compares the distribution of adiabatic ﬁlm cooling effectiveness on the wall for differentfo tr r di ench fferent shapes. trench sha At a low pes. At blowing a low b ratio, lowi the ng ra ﬂowtio, the fl separation ow sep of the aracoolant tion of the coolant downstream downstream of these four of these kinds four of holes kinds of ho is unobvious, les is unobvio and the us, and the cooling perform cooling performance of ance of the the W-shaped trench is the best, while the cooling effectiveness of the round hole is the W-shaped trench is the best, while the cooling effectiveness of the round hole is the lowest. lowest. As the blowing ratio increases to 1.5, the coolant from the round hole and elliptic As the blowing ratio increases to 1.5, the coolant from the round hole and elliptic trench trench shows shows deta detachment chment f from the rom the wa wall, but ll, but the epi the epileptic letr pti ench c trench genera generates better tes better cool cooling ing perfor- performance than the round hole. At this blowing ratio, the coverage performance of the mance than the round hole. At this blowing ratio, the coverage performance of the W- W-shaped trench is the best. At a very high blowing ratio, the coolant from these four shaped trench is the best. At a very high blowing ratio, the coolant from these four kinds kinds of holes exhibits detachment from the wall, however, the transverse trench and of holes exhibits detachment from the wall, however, the transverse trench and W-shaped trench still generate high cooling effectiveness. In general, the W-shaped trench generates the highest cooling effectiveness, while the cooling performance of the elliptic trench is the worst. M = 0.5 M = 1.5 M = 2.5 (a) Round hole (b) Transverse trench (w = 3d, h = 0.75d) Aerospace 2021, 8, x FOR PEER REVIEW 15 of 18 Figure 19. Effects of axis length on film cooling performance for elliptic trench. (a) Dx = 2.2d, Dy = 2.8d, h = 0.9d (b) Dx = 3.2d, Dy = 2.2d, h = 0.9d Figure 20. Streamline distribution on the exit plane of elliptic trench at M = 0.5 (Unit: m/s). Figure 21 compares the distribution of adiabatic film cooling effectiveness on the wall for different trench shapes. At a low blowing ratio, the flow separation of the coolant downstream of these four kinds of holes is unobvious, and the cooling performance of the W-shaped trench is the best, while the cooling effectiveness of the round hole is the lowest. As the blowing ratio increases to 1.5, the coolant from the round hole and elliptic trench shows detachment from the wall, but the epileptic trench generates better cooling perfor- mance than the round hole. At this blowing ratio, the coverage performance of the W- Aerospace 2021, 8, 147 15 of 17 shaped trench is the best. At a very high blowing ratio, the coolant from these four kinds of holes exhibits detachment from the wall, however, the transverse trench and W-shaped trench still generate high cooling effectiveness. In general, the W-shaped trench generates the highest cooling effectiveness, while the cooling performance of the elliptic trench is W-shaped trench still generate high cooling effectiveness. In general, the W-shaped trench the worst. generates the highest cooling effectiveness, while the cooling performance of the elliptic trench is the worst. M = 0.5 M = 1.5 M = 2.5 Aerospace 2021, 8, x FOR PEER REVIEW 16 of 18 (a) Round hole (b) Transverse trench (w = 3d, h = 0.75d) M = 0.5 M = 1.5 M = 2.5 (c) W−shaped trench (w = 1.2d, h = 0.75d, α = 60°) (d) Elliptic trench (Dx = 2.4d, Dy = 2d, h = 0.75d) Figure 21. Distribution of adiabatic ﬁlm cooling effectiveness on the wall for different trench shapes. Figure 21. Distribution of adiabatic film cooling effectiveness on the wall for different trench shapes. 5. Conclusions 5. Conclusions The ﬁlm cooling performances of the transverse trench, W-shaped trench and elliptic The film cooling performances of the transverse trench, W-shaped trench and elliptic trench were investigated using the CFD method. The inﬂuences of blowing ratio and trench were investigated using the CFD method. The influences of blowing ratio and ge- geometric parameters on the discharge coefﬁcient and ﬁlm cooling effectiveness were ometric parameters on the discharge coefficient and film cooling effectiveness were dis- discussed. Some useful conclusions are listed below: cussed. Some useful conclusions are listed below: (1) Inside the transverse trench, a pair of recirculating vortices is formed, which promotes (1) Inside the transverse trench, a pair of recirculating vortices is formed, which pro- the lateral spreading of coolant. Downstream of the transverse trench, a kidney vortex motes the lateral spreading of coolant. Downstream of the transverse trench, a kidney pair and anti-vortex pair are formed simultaneously. The increase of trench depth and vortex pair and anti-vortex pair are formed simultaneously. The increase of trench the decrease of trench width can both cause increases of ﬁlm cooling effectiveness. depth and the decrease of trench width can both cause increases of film cooling ef- (2) Inside the W-shaped trench, the existence of a corner angle further promotes the fectiveness. coolant spreading in the lateral direction and generates higher ﬁlm cooling effec- (2) Inside the W-shaped trench, the existence of a corner angle further promotes the cool- tiveness than the transverse trench. Similar to the transverse trench, the increase ant spreading in the lateral direction and generates higher film cooling effectiveness of trench depth and the decrease of trench width both result in the increase of than the transverse trench. Similar to the transverse trench, the increase of trench cooling effectiveness. depth and the decrease of trench width both result in the increase of cooling effec- tiveness. (3) For the elliptic trench, the flow characteristics are very similar to the round hole, and the kidney vortex pair is the largest-scale vortex structure. There exists an optimal exit area ratio (the exit area of elliptic trench/the exit area of round hole) for the ellip- tic trench, and too large or small an exit area ratio can lead to the degradation of film cooling effectiveness. The elliptic trench generates higher film cooling effectiveness than the round hole, but lower effectiveness than the transverse trench and W-shaped trench. (4) As the blowing ratio increases, the discharge coefficient increases firstly and then keeps stable. The increase of trench depth and decrease of trench width results in the increase of the discharge coefficient for the transverse trench. For the W-shaped trench, the increase of the corner angle causes the decrease of the discharge coeffi- cient. For the elliptic trench, the discharge coefficient increases with the decrease of the elliptic aspect ratio (major axis/minor axis). Author Contributions: Conceptualization, C.W. and X.W.; methodology, X.W.; writing—original draft preparation, C.W; writing—review and editing, X.D and C.W.; funding acquisition, C.W. All authors have read and agreed to the published version of the manuscript. Funding: This research was supported by the Fundamental Research Funds for the Central Univer- sities (grant No: NS2020013) and National Science and Technology Major Project of China (grant No: 2017-III-0011-0037). Institutional Review Board Statement: Not applicable. Aerospace 2021, 8, 147 16 of 17 (3) For the elliptic trench, the ﬂow characteristics are very similar to the round hole, and the kidney vortex pair is the largest-scale vortex structure. There exists an optimal exit area ratio (the exit area of elliptic trench/the exit area of round hole) for the elliptic trench, and too large or small an exit area ratio can lead to the degradation of ﬁlm cooling effectiveness. The elliptic trench generates higher ﬁlm cooling effec- tiveness than the round hole, but lower effectiveness than the transverse trench and W-shaped trench. (4) As the blowing ratio increases, the discharge coefﬁcient increases ﬁrstly and then keeps stable. The increase of trench depth and decrease of trench width results in the increase of the discharge coefﬁcient for the transverse trench. For the W-shaped trench, the increase of the corner angle causes the decrease of the discharge coefﬁcient. For the elliptic trench, the discharge coefﬁcient increases with the decrease of the elliptic aspect ratio (major axis/minor axis). Author Contributions: Conceptualization, C.W. and X.W.; methodology, X.W.; writing—original draft preparation, C.W.; writing—review and editing, X.D. and C.W.; funding acquisition, C.W. All authors have read and agreed to the published version of the manuscript. Funding: This research was supported by the Fundamental Research Funds for the Central Universi- ties (grant No: NS2020013) and National Science and Technology Major Project of China (grant No: 2017-III-0011-0037). Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Data available on request due to restrictions e.g., privacy or ethical. Conﬂicts of Interest: The authors declare no conﬂict of interest. Nomenclature A Cross section area of round hole (m ) C Discharge coefﬁcient (-) d Diameter of round hole (mm) D Axis Length of elliptic crater (mm) h Trench depth (mm) M Blowing ratio (-) m Coolant mass ﬂow rate (kg/s) P* Inlet total pressure of secondary ﬂow (Pa) C,in P Static pressure downstream of trench (Pa) C,out T Temperature (K) w Trench width (mm) x, y and z Streamwise, spanwise and vertical direction Greek symbols a Corner angle of W-shaped trench ( ) h Cooling effectiveness (-) r Gas density (kg/m ) d Boundary layer thickness of mainstream inlet (mm) q Dimensionless temperature [=(T T )/(T T )] c ¥ c subscript w Wall ¥ Mainstream c Coolant ad At adiabatic condition av Area-averaged value loc Local value lat Laterally averaged value overall At the condition considering heat conduction x, y and z Streamwise, spanwise and vertical component Aerospace 2021, 8, 147 17 of 17 References 1. Bogard, D.G.; Thole, K.A. Gas turbine ﬁlm cooling. J. Propuls. Power 2006, 22, 249–270. 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Aerospace – Multidisciplinary Digital Publishing Institute

Published: May 25, 2021

Keywords: film cooling; shaped trench; CFD; adiabatic film cooling effectiveness; discharge coefficient

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Study on Film Cooling Performance of Round Hole Embedded in Different Shaped Craters and Trenches

Study on Film Cooling Performance of Round Hole Embedded in Different Shaped Craters and Trenches

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Study on Film Cooling Performance of Round Hole Embedded in Different Shaped Craters and Trenches

Study on Film Cooling Performance of Round Hole Embedded in Different Shaped Craters and Trenches

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