可压缩壁湍流的时间转捩预测方法和粗糙度对流动的影响

本文主要采用数值模拟方法，研究了可压缩流动转捩预测和粗糙壁面湍流的 基本统计特性以及能量传输粗糙效应和马赫数效应。

首先，本文在约束大涡模拟方法（Constrained Large-Eddy Simulation，CLES）中引入基于形状因数的间歇因子，改进后的约束大涡模拟方法可以准确预测可压缩槽道流动的转捩起始点和转捩过程，且预测效果显著好于基于 Dynamic Smagorinsky Model （DSM）的大涡模拟方法（Large-Eddy Simulation, LES）。同时，改进后的 CLES 方法保持了 CLES 在湍流预测中的优势，可以准确预测平均速度型、平均温度型、总雷诺应力和总热流。另外，改进后的 CLES 在湍流结构捕捉中也有良好性能，壁面附近高低速条带的分布、条带长度和宽度等特征与直接数值模拟（Direct Numerical Simulation，DNS）结果一致，好于基于 DSM 的 LES 结果。

其次，本文采用对动量方程和能量方程连续两次积分的方法，对可压缩时间模式转捩过程中的壁面摩阻系数 𝐶_𝑓 和热流系数 𝐵_𝑞 进行了分解。结果表明，转捩过程中 𝐶_𝑓 的主导机制为雷诺切应力、平均速度对时间的导数和平均速度对流；𝐵_𝑞 的主导机制为湍流传热、总能量在时间尺度的变化和粘性应力作用。当马赫数增大到 3.0 时，分子传热项和平均对流项也变得重要。进一步地，在 Ma = 1.5 时，𝐶_𝑓 的过冲是由平均速度时间导数在转捩后期的快速增长引起，而在 Ma = 3.0 时，𝐶_𝑓 的过冲则是由平均速度时间导数和空间导数在转捩后期的快速增长引起，这与前人推测的过冲由雷诺应力主导的结论显著不同。𝐵_𝑞 的过冲在低马赫数时由总能量在时间尺度的快速变化引起，而马赫数为 3.0 时，𝐵_𝑞 的过冲是由总能量在时间尺度的变化和平均对流项两项共同引起的，这与之前认为高超声速边界层中 𝐵_𝑞 的过 冲由粘性耗散引起的结论不同。另外，本文使用同样方法对 CLES 流场进行壁面摩阻和热流系数分解，并与 DNS 结果进行了比较。结果表明，CLES 中 𝐶_𝑓^{𝐶𝐿𝐸𝑆} 和 𝐵_𝑞^{𝐶𝐿𝐸𝑆} 的过冲预测不足主要是由于 CLES 与 DNS 在平均速度梯度和总能量梯度 上有显著偏差造成的，而湍流切应力和湍流传热对于转捩过程中过冲预测不足的影响较小。 最后，本文利用直接数值模拟方法，研究了粗糙高度和马赫数对可压缩粗糙壁湍流的统计特性和能量传输机制的影响。结果显示，所有算例的粗糙函数和粗糙雷诺数均满足对数律。而雷诺相似系数和粗糙雷诺数之间的关系则与马赫数有关， 当流动为亚音速时，二者呈线性关系，当流动为超音速时，二者呈对数关系。粗糙壁面的存在会显著改变流场中的压缩区和膨胀区分布，当流动为亚音速时，在粗 糙背风坡上方形成压缩区，在粗糙迎风坡上方形成膨胀区；当流动为超音速时，在 流场中出现明显的激波，在激波作用下，下半槽道呈现出压缩区和膨胀区交替分布的特征，而上半槽道除激波外则主要为膨胀区。本文还系统地研究了粗糙高度和马赫数如何影响可压缩波浪粗糙壁湍流中动能和内能之间的能量传输。全场平均结果表明，粘性作用是主要的能量传输机制。粘性主导的能量传输项受粗糙高度影响较大，而受马赫数影响较小。然而，由于粗糙壁面的存在，流场中压力呈现出正负交替变化的特征。这使得压力相关项在局部区域产生了非常强烈的能量传输。但在面平均后，由于正负相互抵消，它们在平均剖面上的作用就变得次要。当流动为亚音速时，压力相关项的极值集中在粗糙壁面附近；而当流动为超音速时， 压力相关项的极值也会出现在流场中心区域或上光滑壁面附近，这主要是流场中强激波带来的影响。 总之，本文对粗糙高度和马赫数在亚音速和超音速粗糙比湍流中的影响进行了全面的研究。更重要的是，本文揭示了时间转捩中壁面摩擦和传热的机理，并 提出了一种改进的 CLES 方法，可以准确预测可压缩流动的转捩。

[1] REYNOLDS O. An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels [J]. Philosophical Transactions of the Royal society of London, 1883(174): 935-982.
[2] ORR W M. The stability or instability of the steady motions of a perfect liquid and of a viscous liquid. Part II: A viscous liquid[C]//Proceedings of the Royal Irish Academy. Section A: Mathematical and Physical Sciences: volume 27. 1907: 69-138.
[3] SOMMERFELD A. Ein Beitrag zur hydrodynamischen Erkldrung der turbulent Flussigkeitsbewegung[C]//Atti del IV Congresso Internazionale dei Matematici: volume 3. Roma, 1908: 116-124.
[4] SCHLICHTING H, GERSTEN K. Boundary-layer theory, Ninth[M]. Verlag Berlin Heidelberg:Springer, 2016.
[5] GRANVILLE P S. The frictional resistance and turbulent boundary layer of rough surfaces[J].Journal of Ship Research, 1958, 2(04): 52-74.
[6] BUSHNELL D M. Hypersonic flight experimentation-status and shortfalls[J].Future Aerospace Technology in the Service of the Alliance, 1997, 3.
[7] BERTIN J J, CUMMINGS R M. Critical hypersonic aerothermodynamic phenomena[J]. Annual Review of Fluid Mechanics, 2006, 38(1): 129-157.
[8] CHEN X, FU S. Progress in the research of hypersonic and highenthalpy boundary layer instabilities and transition[J]. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(11): 2937-2957.
[9] SLOTNICK J P, KHODADOUST A, ALONSO J, et al. CFD vision 2030 study: a path to revolutionary computational aerosciences[R]. The National Aeronautics and Space Administration, 2014.
[10] CHEN J, TU G, ZHANG Y, et al. Hypersnonic boundary layer transition: what we know, where shall we go[J]. Acta Aerodynamica Sinica, 2017, 35(3): 311-337.
[11] HAGEN G H L. Über den einfluss der temperatur auf die bewegung des wassers in röhren[M]. Berlin: Druckerei der Königl. akademie der wissenschaften, 1854: 17-98.
[12] DARCY H. Recherches expérimentales relatives au mouvement de l’eau dans les tuyaux: volume 1[M]. Paris: Mallet-Bachelier, 1857: 268.
[13] KADIVAR M, TORMEY D, MCGRANAGHAN G. A review on turbulent flow over rough surfaces: Fundamentals and theories[J]. International Journal of Thermofluids, 2021, 10: 100077.
[14] DANBERG J E, SIGAL A. Analysis of turbulent boundary-layer over rough surfaces with application to projectile aerodynamics[R]. Army Ballistic Research Lab Aberdeen Proving Ground MD, 1988.
[15] DOMEL A G, SAADAT M, WEAVER J C, et al. Shark skin-inspired designs that improve aerodynamic performance[J]. Journal of the Royal Society Interface, 2018, 15(139): 20170828.
[16] LI P, GUO D, HUANG X. Heat transfer enhancement, entropy generation and temperature uniformity analyses of shark-skin bionic modified microchannel heat sink[J]. International Journal of Heat and Mass Transfer, 2020, 146: 118846.
[17] SOLEIMANI S, ECKELS S. A review of drag reduction and heat transfer enhancement by riblet surfaces in closed and open channel flow[J]. International Journal of Thermofluids, 2021, 9: 100053.
[18] SEEHAUSEN H, GILGE P, KELLERSMANN A, et al. Numerical study of stage roughness variations in a high pressure compressor[J]. International Journal of Gas Turbine, Propulsion and Power Systems, 2020, 11(3): 16-25.
[19] RAUPACH M, THOM A S. Turbulence in and above plant canopies[J]. Annual Review of Fluid Mechanics, 1981, 13(1): 97-129.
[20] FINNIGAN J. Turbulence in plant canopies[J]. Annual Review of Fluid Mechanics, 2000, 32(1): 519-571.
[21] COCEAL O, BELCHER S E. A canopy model of mean winds through urban areas[J]. Quarterly Journal of the Royal Meteorological Society, 2004, 130(599): 1349-1372.
[22] CHENG H, CASTRO I P. Near wall flow over urban-like roughness[J]. Boundary-Layer Meteorology, 2002, 104: 229-259.
[23] ZHANG W, ZHU X, YANG X I, et al. Evidence for Raupach et al.’s mixing-layer analogy in deep homogeneous urban-canopy flows[J]. Journal of Fluid Mechanics, 2022, 944: A46.
[24] STRIPF M, SCHULZ A, BAUER H J. Modeling of rough-wall boundary layer transition and heat transfer on turbine airfoils[J]. Journal of Turbomachinery, 2008, 130(2).
[25] BONS J P. A review of surface roughness effects in gas turbines[J]. Journal of Turbomachinery, 2010, 132(2).
[26] FOROOGHI P, WEIDENLENER A, MAGAGNATO F, et al. DNS of momentum and heat transfer over rough surfaces based on realistic combustion chamber deposit geometries[J]. International Journal of Heat and Fluid Flow, 2018, 69: 83-94.
[27] BALAN C, FRANCO J M. Influence of the geometry on the transient and steady flow of lubricating greases[J]. Tribology Transactions, 2001, 44(1): 53-58.
[28] MCHALE J P, GARIMELLA S V. Nucleate boiling from smooth and rough surfaces–Part 1:Fabrication and characterization of an optically transparent heater–sensor substrate with controlled surface roughness[J]. Experimental Thermal and Fluid Science, 2013, 44: 456-467.
[29] CHAMORRO L P, PORTÉ-AGEL F. A wind-tunnel investigation of wind-turbine wakes:boundary-layer turbulence effects[J]. Boundary-Layer Meteorology, 2009, 132: 129-149.
[30] LIU C, HU C. CFD simulation of a floating wind turbine platform in rough sea conditions[C]// The Twenty-fourth International Ocean and Polar Engineering Conference. OnePetro, 2014.
[31] CHUNG D, HUTCHINS N, SCHULTZ M P, et al. Predicting the drag of rough surfaces[J]. Annual Review of Fluid Mechanics, 2021, 53(1): 439-471.
[32] ZHANG W, WAN M, XIA Z, et al. Constrained large-eddy simulation of turbulent flow over rough walls[J]. Physical Review Fluids, 2021, 6(4): 044602.
[33] PRANDTL L. Über Flüssigkeitsbewegung beisehr kleiner Reibung[C]//Verhandlungen des III. Internationalen Mathematiker Kongresses: volume 2. Heidelberg, 1904: 484-491.
[34] ZHONG X, WANG X. Direct numerical simulation on the receptivity, instability, and transition of hypersonic boundary layers[J]. Annual Review of Fluid Mechanics, 2012, 44(1): 527-561.
[35] FEDOROV A V, RYZHOV A A, SOUDAKOV V G, et al. Receptivity of a high-speed boundary layer to temperature spottiness[J]. Journal of Fluid Mechanics, 2013, 722: 533-553.
[36] REMPFER D. Low-dimensional modeling and numerical simulation of transition in simple shear flows[J]. Annual Review of Fluid Mechanics, 2003, 35(1): 229-265.
[37] BORODULIN V, IVANOV A, KACHANOV Y, et al. Receptivity coefficients at excitation of cross-flow waves by free-stream vortices in the presence of surface roughness[J]. Journal of Fluid Mechanics, 2013, 716: 487-527.
[38] SCHMID P J, HENNINGSON D S, JANKOWSKI D. Stability and transition in shear flows [J]. Applied Mechanics Reviews, 2002, 55(3): B57-B59.
[39] 周恒, 赵耕夫. 流动稳定性[M]. 北京: 国防工业出版社, 2004.
[40] HERBERT T. Secondary instability of boundary layers[J]. Annual Review of Fluid Mechanics, 1988, 20(1): 487-526.
[41] LIN L, WU Y. Theoretical analysis of vorticity in a hairpin vortex in the viscous sublayer of a laminar boundary layer[J]. European Journal of Mechanics - B/Fluids, 2022, 94: 106-120.
[42] LANDAU L D. On the problem of turbulence[C]//Doklady Akademii Nauk USSR: volume 44.1944: 311.
[43] STUART J T. On the non-linear mechanics of wave disturbances in stable and unstable parallel flows Part 1. The basic behaviour in plane Poiseuille flow[J]. Journal of Fluid Mechanics, 1960, 9(3): 353-370.
[44] 周恒, 李骊. 流体层流运动稳定性中的 Ляпунов 方法[J]. 应用数学和力学, 1983, 4.
[45] 周恒, 尤学一. 流动稳定性弱非线性理论中的问题及其改进[J]. 力学学报, 1993, 25(5): 515-528.
[46] 周恒, 藤村薰. 流动稳定性弱非线性理论的进一步改进[J]. 中国科学: A 辑, 1997, 27(12):1111-1118.
[47] GASTER M. On the effects of boundary-layer growth on flow stability[J]. Journal of Fluid Mechanics, 1974, 66(3): 465-480.
[48] BERTOLOTTI F, HERBERT T. Analysis of the linear stability of compressible boundary layers using the PSE[J]. Theoretical and Computational Fluid Dynamics, 1991, 3(2): 117-124.
[49] HERBERT T, STUCKERT G, LIN N. Method for transition prediction in high-speed boundary layers: volume 93[M]. Flight Dynamics Directorate, Wright Laboratory, Air Force Materiel Command, 1993.
[50] HERBERT T. Parabolized stability equations[J]. Annual Review of Fluid Mechanics, 1997, 29 (1): 245-283.51] LEES L, LIN C C. Investigation of the stability of the laminar boundary layer in a compressible fluid: number 1115[M]. National Advisory Committee for Aeronautics, 1946.
[52] MACK L M. Boundary-layer linear stability theory[R]. California Inst of Tech Pasadena Jet Propulsion Lab, 1984.
[53] MALIK M R. Prediction and control of transition in supersonic and hypersonic boundary layers [J]. AIAA Journal, 1989, 27(11): 1487-1493.
[54] MALIK M R. Boundary-layer transition prediction toolkit[M]//Computational Fluid Dynamics Review 1998: (In 2 Volumes). World Scientific, 1998: 869-890.
[55] REED H L, SARIC W S, ARNAL D. Linear stability theory applied to boundary layers[J].Annual Review of Fluid Mechanics, 1996, 28(1): 389-428.
[56] GAPONOV S A, MASLOV A A. Development of disturbances in compressible flows[M].Nauka, Novosibirsk, 1980.
[57] MACK L M. Boundary-layer stability theory[M]. Jet Propulsion Laboratory, 1969.
[58] FEDOROV A. Transition and stability of high-speed boundary layers[J]. Annual Review of Fluid Mechanics, 2011, 43: 79-95.
[59] ARNAL D. Boundary layer transition: predictions based on linear theory[R]. In Special Course on Progress in Transition Modelling, AGARD, 1994.
[60] MORKOVIN M V. Transition at hypersonic speeds[R]. The National Aeronautics and Space Administration, 1987.
[61] RESHOTKO E. Transition issues for atmospheric entry[J]. Journal of Spacecraft and Rockets, 2008, 45(2): 161-164.
[62] FEDOROV A, TUMIN A. Branching of discrete modes in high-speed boundary layers and terminology issues[C]//40th Fluid Dynamics Conference and Exhibit. 2010: 5003.
[63] KLEISER L, ZANG T A. Numerical simulation of transition in wall-bounded shear flows[J].Annual Review of Fluid Mechanics, 1991, 23(1): 495-537.
[64] FASEL H F. Instability and transition in boundary layers: direct numerical simulations[C]// IUTAM Symposium on One Hundred Years of Boundary Layer Research: Proceedings of the IUTAM Symposium held at DLR-Göttingen, Germany, August 12-14, 2004. Springer, 2006: 257-267.
[65] PIROZZOLI S. Numerical methods for high-speed flows[J]. Annual Review of Fluid Mechanics, 2011, 43(1): 163-194.
[66] BALAKUMAR P. Receptivity of a supersonic boundary layer to acoustic disturbances[J].AIAA Journal, 2009, 47(5): 1069-1078.
[67] BALAKUMAR P, KEGERISE M A. Receptivity of hypersonic boundary layers over straight and flared cones[J]. AIAA Journal, 2015, 53(8): 2097-2109.
[68] JOHNSEN E, LARSSON J, BHAGATWALA A V, et al. Assessment of high-resolution methods for numerical simulations of compressible turbulence with shock waves[J]. Journal of Computational Physics, 2010, 229(4): 1213-1237.
[69] RAWAT P S, ZHONG X. On high-order shock-fitting and front-tracking schemes for numerical simulation of shock–disturbance interactions[J]. Journal of Computational Physics, 2010, 229 (19): 6744-6780.
[70] MORETTI G. Thirty-six years of shock fitting[J]. Computers & Fluids, 2002, 31(4-7): 719-723.
[71] ZHONG X. High-order finite-difference schemes for numerical simulation of hypersonic boundary-layer transition[J]. Journal of Computational Physics, 1998, 144(2): 662-709.
[72] WANG X, ZHONG X. Numerical simulations on mode S growth over feltmetal and regular porous coatings of a Mach 5.92 flow[C]//49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition. 2011: 375.
[73] IYER P S, MAHESH K. High-speed boundary-layer transition induced by a discrete roughness element[J]. Journal of Fluid Mechanics, 2013, 729: 524-562.
[74] WANG X, ZHONG X. Receptivity of a hypersonic flat-plate boundary layer to three-dimensional surface roughness[J]. Journal of Spacecraft and Rockets, 2008, 45(6): 1165-1175.
[75] DUAN L, WANG X, ZHONG X. A high-order cut-cell method for numerical simulation of hypersonic boundary-layer instability with surface roughness[J]. Journal of Computational Physics, 2010, 229(19): 7207-7237.
[76] WANG Y, YANG Y, YANG G, et al. DNS study on vortex and vorticity in late boundary layer transition[J]. Communications in Computational Physics, 2017, 22(2): 441-459.
[77] KOSINOV A, MASLOV A, SHEVELKOV S. Experiments on the stability of supersonic laminar boundary layers[J]. Journal of Fluid Mechanics, 1990, 219: 621-633.
[78] THUMM A, WOLZ W, FASEL H. Numerical simulation of spatially growing three-dimensional disturbance waves in compressible boundary layers[C]//Laminar-Turbulent Transition: IUTAM Symposium Toulouse/France September 11–15, 1989. Springer, 1990: 303-308.
[79] BESTEK H, THUMM A, FASEL H. Direct numerical simulation of the three-dimensional breakdown to turbulence in compressible boundary layers[C]//Thirteenth International Conference on Numerical Methods in Fluid Dynamics: Proceedings of the Conference Held at the Consiglio Nazionale delle Ricerche Rome, Italy, 6–10 July 1992. Springer, 2005: 145-149.
[80] JIANG L, CHOUDHARI M, CHANG C L, et al. Numerical simulations of laminar-turbulent transition in supersonic boundary layer[C]//36th AIAA Fluid Dynamics Conference and Exhibit. 2006: 3224.
[81] MUPPIDI S, MAHESH K. DNS of transition in supersonic boundary layers[C]//40th Fluid Dynamics Conference and Exhibit. 2010: 4440.
[82] MAYER C S, WERNZ S, FASEL H F. Numerical investigation of the nonlinear transition regime in a Mach 2 boundary layer[J]. Journal of Fluid Mechanics, 2011, 668: 113-149.
[83] KRISHNAN L, SANDHAM N. Effect of Mach number on the structure of turbulent spots[J].Journal of Fluid Mechanics, 2006, 566: 225-234.
[84] JOCKSCH A, KLEISER L. Growth of turbulent spots in high-speed boundary layers on a flat plate[J]. International Journal of Heat and Fluid Flow, 2008, 29(6): 1543-1557.
[85] MAYER C S J, LAIBLE A C, FASEL H F. Numerical investigation of wave packets in a mach3.5 cone boundary layer[J]. AIAA Journal, 2011, 49(1): 67-86.
[86] SIVASUBRAMANIAN J, FASEL H. Transition initiated by a localized disturbance in a hypersonic flat-plate boundary layer[C]//49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition. 2011: 374.
[87] SMITH A M O. Transition, pressure gradient and stability theory[J]. Douglas Aircraft Co., Report ES26388, 1956.
[88] 周恒, 苏彩虹, 张永明. 超声速/高超声速边界层的转捩机理及预测[M]. 北京: 科学出版社, 2015.
[89] ARNAL D, CASALIS G. Laminar-turbulent transition prediction in three-dimensional flows [J]. Progress in Aerospace Sciences, 2000, 36(2): 173-191.
[90] SU C, ZHOU H. Transition prediction of a hypersonic boundary layer over a cone at small angle of attack—with the improvement of 𝑒 𝑁 method[J]. Science in China Series G: Physics, Mechanics and Astronomy, 2009, 52(1): 115-123.
[91] SU C, ZHOU H. Transition prediction for supersonic and hypersonic boundary layers on a cone with angle of attack[J]. Science in China Series G: Physics, Mechanics and Astronomy, 2009, 52(8): 1223-1232.
[92] SU C, ZHOU H. Transition prediction of the supersonic boundary layer on a cone under the consideration of receptivity to slow acoustic waves[J]. Science China Physics, Mechanics and Astronomy, 2011, 54: 1875-1882.
[93] KING R A. Three-dimensional boundary-layer transition on a cone at Mach 3.5[J]. Experiments in Fluids, 1992, 13(5): 305-314.
[94] LI X, FU D, MA Y. Direct numerical simulation of hypersonic boundary layer transition over a blunt cone with a small angle of attack[J]. Physics of Fluids, 2010, 22(2): 025105.
[95] LAU K Y. Hypersonic boundary-layer transition: application to high-speed vehicle design[J]. Journal of Spacecraft and Rockets, 2008, 45(2): 176-183.
[96] PARK C. Injection-induced turbulence in stagnation-point boundary layers[J]. AIAA Journal, 1984, 22(2): 219-225.
[97] BERRY S, KING R, KEGERISE M, et al. Orbiter boundary layer transition prediction tool enhancements[C]//48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition. 2010: 246.
[98] MAYLE R, SCHULZ A. The path to predicting bypass transition[C]//Turbo Expo: Power for Land, Sea, and Air: volume 78729. American Society of Mechanical Engineers, 1996: V001T01A065.
[99] DHAWAN S, NARASIMHA R. Some properties of boundary layer flow during the transition from laminar to turbulent motion[J]. Journal of Fluid Mechanics, 1958, 3(4): 418-436.
[100] LIBBY P A. On the prediction of intermittent turbulent flows[J]. Journal of Fluid Mechanics, 1975, 68(2): 273-295.
[101] LIBBY P A. Prediction of the intermittent turbulent wake of a heated cylinder[J]. Physics of Fluids, 1976, 19(4): 494-501.
[102] CHO J R, CHUNG M K. A K—𝜀—𝛾 equation turbulence model[J]. Journal of Fluid Mechanics, 1992, 237: 301-322.
[103] LAUNDER B E, SHARMA B I. Application of the energy-dissipation model of turbulence to the calculation of flow near a spinning disc[J]. Letters in Heat and Mass Transfer, 1974, 1(2): 131-137.
[104] STEELANT J, DICK E. Modelling of bypass transition with conditioned Navier–Stokes equations coupled to an intermittency transport equation[J]. International Journal for Numerical Methods in Fluids, 1996, 23(3): 193-220.
[105] SUZEN Y, HUANG P. An intermittency transport equation for modeling flow transition[C]// 38th Aerospace Sciences Meeting and Exhibit. 2000: 287.
[106] SUZEN Y, HUANG P. Modeling of flow transition using an intermittency transport equation [J]. Journal of Fluids Engineering, 2000, 122(2): 273-284.
[107] SUZEN Y B, HUANG P, HULTGREN L S, et al. Predictions of separated and transitional boundary layers under low-pressure turbine airfoil conditions using an intermittency transport equation[J]. Journal of Turbomachinery, 2003, 125(3): 455-464.
[108] DICK E, KUBACKI S. Transition models for turbomachinery boundary layer flows: A review [J]. International Journal of Turbomachinery, Propulsion and Power, 2017, 2(2).
[109] MENTER F R, LANGTRY R B, LIKKI S R, et al. A correlation-based transition model using local variables—part I: Model formulation[J]. Journal of Turbomachinery, 2004: 413-422.
[110] LANGTRY R B, MENTER F R, LIKKI S R, et al. A correlation-based transition Model using local variables: Part II —test cases and industrial applications[C]//Proceedings of the ASME Turbo Expo 2004: Power for Land, Sea, and Air: volume 4. 2004: 69-79.
[111] LANGTRY R B, MENTER F R. Correlation-based transition modeling for unstructured parallelized computational fluid dynamics codes[J]. AIAA Journal, 2009, 47(12): 2894-2906.
[112] KRAUSE M, BEHR M, BALLMANN J. Modeling of transition effects in hypersonic intake flows using a correlation-based intermittency model[C]//15th AIAA International Space Planes and Hypersonic Systems and Technologies Conference. 2008: 2598.
[113] CHENG G, NICHOLS R, NEROORKAR K, et al. Validation and assessment of turbulence transition models[C]//47th AIAA Aerospace Sciences Meeting including The New Horizons Forum and Aerospace Exposition. 2009: 1141.
[114] ZHANG X D, GAO Z H. Numerical discussions on complete empirical correlation in Langtry’ s transition model[J]. Applied Mathematics and Mechanics, 2010, 31: 575-584.
[115] BENSASSI K, LANI A, RAMBAUD P. Numerical investigations of local correlation-based transition model in hypersonic flows[C]//42nd AIAA Fluid Dynamics Conference and Exhibit. 2012: 3151.
[116] MAO R, YAN P, QIANG X. Application of PSE analysis method with transitional turbulence model on hypersonic flows[C]//20th AIAA Computational Fluid Dynamics Conference. 2011: 3981.
[117] WALTERS D K, LEYLEK J H. A new model for boundary layer transition using a single-point RANS approach[J]. Journal of Turbomachinery, 2004, 126(1): 193-202.
[118] WALTERS D K, COKLJAT D. A three-equation eddy-viscosity model for Reynolds-averaged Navier–Stokes simulations of transitional flow[J]. Journal of Fluids Engineering, 2008, 130 (12).
[119] LOPEZ M, WALTERS D K. Prediction of transitional and fully turbulent flow using an alternative to the laminar kinetic energy approach[J]. Journal of Turbulence, 2016, 17(3): 253-273.
[120] CUTRONE L, De Palma P, PASCAZIO G, et al. Predicting transition in two- and three-dimensional separated flows[J]. International Journal of Heat and Fluid Flow, 2008, 29(2): 504-526.
[121] CHOUDHRY A, ARJOMANDI M, KELSO R. A study of long separation bubble on thick airfoils and its consequent effects[J]. International Journal of Heat and Fluid Flow, 2015, 52: 84-96.
[122] SANDERS D D, O’BRIEN W F, SONDERGAARD R, et al. Predicting separation and transitional flow in turbine blades at low reynolds numbers—part I: Development of prediction methodology[J]. Journal of Turbomachinery, 2010, 133(3).
[123] SANDERS D D, O’BRIEN W F, SONDERGAARD R, et al. Predicting separation and transitional flow in turbine blades at low reynolds numbers—Part II: The application to a highly separated turbine blade cascade geometry[J]. Journal of Turbomachinery, 2010, 133(3).
[124] 赵瑞, 周玲, 阎超, 等. 湍流与转捩数值模拟方法[M]. 北京: 北京理工大学出版社, 2019.
[125] QIN Y, YAN C, HAO Z, et al. A laminar kinetic energy transition model appropriate for hypersonic flow heat transfer[J]. International Journal of Heat and Mass Transfer, 2017, 107: 1054-1064.
[126] LIU Z, LU Y, WANG S, et al. Physics-based model for boundary layer transition prediction in a wide speed range[J]. Chinese Journal of Aeronautics, 2022, 35(9): 143-159.
[127] WANG L, FU S. Development of an intermittency equation for the modeling of the supersonic/hypersonic boundary layer flow transition[J]. Flow, Turbulence and Combustion, 2011, 87: 165-187.
[128] MENTER F. Zonal two equation 𝑘−𝜔 turbulence models for aerodynamic flows[C]//23rd Fluid Dynamics, Plasma dynamics, and Lasers Conference. 1993.
[129] WANG L, FU S. Modelling flow transition in a hypersonic boundary layer with Reynolds averaged Navier-Stokes approach[J]. Science in China Series G: Physics, Mechanics and Astronomy, 2009, 52(5): 768-774.
[130] WANG L, FU S, CARNARIUS A, et al. A modular RANS approach for modelling laminar turbulent transition in turbomachinery flows[J]. International Journal of Heat and Fluid Flow, 2012, 34: 62-69.
[131] FU S, WANG L. RANS modeling of high-speed aerodynamic flow transition with consideration of stability theory[J]. Progress in Aerospace Sciences, 2013, 58: 36-59.
[132] PIOMELLI U, ZANG T A, SPEZIALE C G, et al. On the large-eddy simulation of transitional wall-bounded flows[J]. Physics of Fluids, 1990, 2(2): 257-265.
[133] HORIUTI K. On the use of SGS modeling in the simulation of transition in plane channel flow [J]. Journal of the Physical Society of Japan, 1986, 55(5): 1528-1541.
[134] DUCROS F, COMTE P, LESIEUR M. Large-eddy simulation of transition to turbulence in a boundary layer developing spatially over a flat plate[J]. Journal of Fluid Mechanics, 1996, 326: 1-36.
[135] HUAI X, JOSLIN R D, PIOMELLI U. Large-eddy simulation of transition to turbulence in boundary layers[J]. Theoretical and Computational Fluid Dynamics, 1997, 9: 149-163.
[136] SAYADI T, MOIN P. A comparative study of subgrid scale models for the prediction of transition in turbulent boundary layers[J]. Center for Turbulence Research Annual Research Briefs, 2010: 237-247.
[137] SAYADI T, MOIN P. Large eddy simulation of controlled transition to turbulence[J]. Physics of Fluids, 2012, 24(11): 114103.
[138] YU C, HONG R, XIAO Z, et al. Subgrid-scale eddy viscosity model for helical turbulence[J].Physics of Fluids, 2013, 25(9): 095101.
[139] ZHOU H, LI X, QI H, et al. Subgrid-scale model for large-eddy simulation of transition and turbulence in compressible flows[J]. Physics of Fluids, 2019, 31(12): 125118.
[140] CHEN S, XIA Z, PEI S, et al. Reynolds-stress-constrained large-eddy simulation of wallbounded turbulent flows[J]. Journal of Fluid Mechanics, 2012, 703: 1-28.
[141] TESSICINI F, LI N, LESCHZINER M. Simulation of separation from curved surfaces with combined LES and RANS schemes[C]//Complex Effects in Large Eddy Simulations. Springer Berlin Heidelberg, 2007: 305-324.
[142] XIA Z, SHI Y, HONG R, et al. Constrained large-eddy simulation of separated flow in a channel with streamwise-periodic constrictions[J]. Journal of Turbulence, 2013, 14(1): 1-21.
[143] ZHAO Y, XIA Z, SHI Y, et al. Constrained large-eddy simulation of laminar-turbulent transition in channel flow[J]. Physics of Fluids, 2014, 26: 095103.
[144] JIANG Z, XIAO Z, SHI Y, et al. Constrained large-eddy simulation of wall-bounded compressible turbulent flows[J]. Physics of Fluids, 2013, 25: 106102.
[145] WANG X, XIAO Z. Transition-based constrained large-eddy simulation method with application to an ultrahigh-lift low-pressure turbine cascade flow[J]. Journal of Fluid Mechanics, 2022, 941: A22.
[146] KAYS W M, CRAWFORD M E, WEIGAND B. Convective heat and mass transfer: volume 4 [M]. McGraw-Hill New York, 1980.
[147] SCHULTZ M, FLACK K. Outer layer similarity in fully rough turbulent boundary layers[J].Experiments in Fluids, 2005, 38: 328-340.
[148] NIKURADSE J. Stromungsgesetze in rauhen Rohren[J]. Vdi-Forschungsheft, 1933, 361: 1.
[149] IOSELEVICH V, PILIPENKO V. Logarithmic velocity profile for flow of a weak polymer solution near a rough surface[C]//Soviet Physics Doklady: volume 18. 1974: 790.
[150] LIGRANI P M, MOFFAT R J. Structure of transitionally rough and fully rough turbulent boundary layers[J]. Journal of Fluid Mechanics, 1986, 162: 69-98.
[151] CLAUSER F H. The turbulent boundary layer[J]. Advances in Applied Mechanics, 1956, 4:1-51.
[152] SCHULTZ M, FLACK K. The rough-wall turbulent boundary layer from the hydraulically smooth to the fully rough regime[J]. Journal of Fluid Mechanics, 2007, 580: 381-405.
[153] Cebeci T, Bradshaw P. Momentum transfer in boundary layers[M]. Washington, 1977.
[154] SCHLICHTING H. Experimentelle untersuchungen zum rauhigkeitsproblem[J]. IngenieurArchiv, 1936, 7(1): 1-34.
[155] COLEBROOK C F, WHITE C M. Experiments with fluid friction in roughened pipes[J]. Proceedings of the Royal Society of London. Series A-Mathematical and Physical Sciences, 1937, 161(906): 367-381.
[156] BOTROS K. Experimental investigation into the relationship between the roughness height in use with Nikuradse or Colebrook roughness functions and the internal wall roughness profile for commercial steel pipes[J]. Journal of Fluids Engineering, 2016, 138(8).
[157] YUAN J, PIOMELLI U. Estimation and prediction of the roughness function on realistic surfaces[J]. Journal of Turbulence, 2014, 15(6): 350-365.
[158] FLACK K A, SCHULTZ M P. Review of hydraulic roughness scales in the fully rough regime [J]. Journal of Fluids Engineering, 2010, 132(4).
[159] THAKKAR M, BUSSE A, SANDHAM N. Surface correlations of hydrodynamic drag for transitionally rough engineering surfaces[J]. Journal of Turbulence, 2017, 18(2): 138-169.
[160] FOROOGHI P, STROH A, MAGAGNATO F, et al. Toward a universal roughness correlation [J]. Journal of Fluids Engineering, 2017, 139(12).
[161] NAPOLI E, ARMENIO V, DE MARCHIS M. The effect of the slope of irregularly distributed roughness elements on turbulent wall-bounded flows[J]. Journal of Fluid Mechanics, 2008, 613: 385-394.
[162] AGHAEI JOUYBARI M, YUAN J, BRERETON G J, et al. Data-driven prediction of the equivalent sand-grain height in rough-wall turbulent flows[J]. Journal of Fluid Mechanics, 2021, 912: A8.
[163] PERRY A E, SCHOFIELD W H, JOUBERT P N. Rough wall turbulent boundary layers[J].Journal of Fluid Mechanics, 1969, 37(2): 383-413.
[164] COLEMAN S, NIKORA V I, MCLEAN S, et al. Spatially averaged turbulent flow over square ribs[J]. Journal of Engineering Mechanics, 2007, 133(2): 194-204.
[165] DJENIDI L, ANTONIA R, ANSELMET F. LDA measurements in a turbulent boundary layer over a d-type rough wall[J]. Experiments in Fluids, 1994, 16(5): 323-329.
[166] JIMÉNEZ J. Turbulent flows over rough walls[J]. Annual Review of Fluid Mechanics, 2004, 36: 173-196.
[167] CHOI K S, FUJISAWA N. Possibility of drag reduction using d-type roughness[J]. Applied Scientific Research, 1993, 50: 315-324.
[168] GHADDAR N, MAGEN M, MIKIC B, et al. Numerical investigation of incompressible flow in grooved channels. Part 2. Resonance and oscillatory heat-transfer enhancement[J]. Journal of Fluid Mechanics, 1986, 168: 541-567.
[169] MACDONALD M, CHAN L, CHUNG D, et al. Turbulent flow over transitionally rough surfaces with varying roughness densities[J]. Journal of Fluid Mechanics, 2016, 804: 130-161.
[170] LEONARDI S, ORLANDI P, ANTONIA R A. Properties of d-and k-type roughness in a turbulent channel flow[J]. Physics of Fluids, 2007, 19(12): 125101.
[171] HAMA F R. Boundary Layer characteristics for smooth and rough surfaces[J]. Transactions, 1954, 62: 333.
[172] SCHULTZ M P, FLACK K A. Turbulent boundary layers on a systematically varied rough wall [J]. Physics of Fluids, 2009, 21(1): 015104.
[173] FLACK K A, SCHULTZ M P. Roughness effects on wall-bounded turbulent flows[J]. Physics of Fluids, 2014, 26(10): 101305.
[174] OKE T R. Street design and urban canopy layer climate[J]. Energy and Buildings, 1988, 11 (1-3): 103-113.
[175] MACDONALD R, GRIFFITHS R, HALL D. An improved method for the estimation of surface roughness of obstacle arrays[J]. Atmospheric Environment, 1998, 32(11): 1857-1864.
[176] LEONARDI S, CASTRO I P. Channel flow over large cube roughness: a direct numerical simulation study[J]. Journal of Fluid Mechanics, 2010, 651: 519-539.
[177] DIRLING R, JR. A method for computing roughwall heat transfer rates on reentry nosetips [C]//8th Thermophysics Conference. 1973: 763.
[178] WAIGH D, KIND R. Improved aerodynamic characterization of regular three-dimensional roughness[J]. AIAA Journal, 1998, 36(6): 1117-1119.
[179] CHAN L, MACDONALD M, CHUNG D, et al. A systematic investigation of roughness height and wavelength in turbulent pipe flow in the transitionally rough regime[J]. Journal of Fluid Mechanics, 2015, 771: 743-777.
[180] MA G Z, XU C X, SUNG H J, et al. Scaling of rough-wall turbulence by the roughness height and steepness[J]. Journal of Fluid Mechanics, 2020, 900: R7.
[181] CLAUSER F H. The Structure of Turbulent Shear Flow[J]. Nature, 1957, 179: 60-60.
[182] PERRY A E, CHONG M S. On the mechanism of wall turbulence[J]. Journal of Fluid Mechanics, 1982, 119: 173-217.
[183] RAUPACH M R, ANTONIA R A, RAJAGOPALAN S. Rough-wall turbulent boundary layers [J]. Applied Mechanics Reviews, 1991, 44(1): 1-25.
[184] SHOCKLING M, ALLEN J, SMITS A. Roughness effects in turbulent pipe flow[J]. Journal of Fluid Mechanics, 2006, 564: 267-285.
[185] KUNKEL G J, MARUSIC I. Study of the near-wall-turbulent region of the high-Reynoldsnumber boundary layer using an atmospheric flow[J]. Journal of Fluid Mechanics, 2006, 548: 375-402.
[186] FLACK K, SCHULTZ M, CONNELLY J. Examination of a critical roughness height for outer layer similarity[J]. Physics of Fluids, 2007, 19(9): 095104.
[187] KROGSTADT P Å, ANTONIA R. Surface roughness effects in turbulent boundary layers[J].Experiments in Fluids, 1999, 27(5): 450-460.
[188] TACHIE M, BERGSTROM D, BALACHANDAR R. Rough wall turbulent boundary layers in shallow open channel flow[J]. Journal of Fluids Engineering, 2000, 122(3): 533-541.
[189] LEONARDI S, ORLANDI P, SMALLEY R, et al. Direct numerical simulations of turbulent channel flow with transverse square bars on one wall[J]. Journal of Fluid Mechanics, 2003, 491: 229-238.
[190] EKOTO I W, BOWERSOX R D, BEUTNER T, et al. Supersonic boundary layers with periodic surface roughness[J]. AIAA Journal, 2008, 46(2): 486-497.
[191] MODESTI D, SATHYANARAYANA S, SALVADORE F, et al. Direct numerical simulation of supersonic turbulent flows over rough surfaces[J]. Journal of Fluid Mechanics, 2022, 942.
[192] SHENG ZHANG Y, BI W T, HUSSAIN F, et al. A generalized Reynolds analogy for compressible wall-bounded turbulent flows[J]. Journal of Fluid Mechanics, 2013, 739: 392 - 420.
[193] AGHAEI-JOUYBARI M, YUAN J, LI Z, et al. Supersonic turbulent flows over sinusoidal rough walls[J]. Journal of Fluid Mechanics, 2023, 956: A3.
[194] SPALART P R. Direct simulation of a turbulent boundary layer up to 𝑅 𝜃 = 1410[J]. Journal of Fluid Mechanics, 1988, 187: 61-98.
[195] MANSOUR N N, KIM J, MOIN P. Reynolds-stress and dissipation-rate budgets in a turbulent channel flow[J]. Journal of Fluid Mechanics, 1988, 194: 15-44.
[196] MIURA H, KIDA S. Acoustic energy exchange in compressible turbulence[J]. Physics of Fluids, 1995, 7(7): 1732-1742.
[197] WANG J, YANG Y, SHI Y, et al. Cascade of kinetic energy in three-dimensional compressible turbulence[J]. Physical Review Letters, 2013, 110(21): 214505.
[198] XU D, WANG J, WAN M, et al. Compressibility effect in hypersonic boundary layer with isothermal wall condition[J]. Physical Review Fluids, 2021, 6: 054609.
[199] ORLANDI P. Turbulent kinetic energy production and flow structures in flows past smooth and rough walls[J]. Journal of Fluid Mechanics, 2019, 866: 897-928.
[200] TONICELLO N, LODATO G, VERVISCH L. Turbulence kinetic energy transfers in direct numerical simulation of shock-wave–turbulence interaction in a compression/expansion ramp [J]. Journal of Fluid Mechanics, 2022, 935: A31.
[201] VYAS M A, YODER D A, GAITONDE D V. Reynolds-stress budgets in an impinging shockwave/boundary-layer interaction[J]. AIAA Journal, 2019, 57(11): 4698-4714.
[202] SUN Z, ZHU Y, HU Y, et al. Direct numerical simulation of a fully developed compressible wall turbulence over a wavy wall[J]. Journal of Turbulence, 2018, 19: 105 - 72.
[203] BATCHELOR G K. Cambridge Mathematical Library: An introduction to fluid dynamics[M]. Cambridge University Press, 2000.
[204] SUTHERLAND W. LII. The viscosity of gases and molecular force[J]. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 1893, 36(223): 507-531.
[205] BOSE S, PARK G I. Wall-modeled large-Eddy simulation for complex turbulent flows.[J].Annual Review of Fluid Mechanics, 2018, 50: 535-561.
[206] MENEVEAU C, KATZ J. Scale-invariance and turbulence models for large-eddy simulation [J]. Annual Review of Fluid Mechanics, 2000, 32: 1-32.
[207] SPALART P R. Detached-eddy simulation[J]. Annual Review of Fluid Mechanics, 2009, 41: 181-202.
[208] YOSHIZAWA A. Statistical theory for compressible turbulent shear flows, with the application to subgrid modeling[J]. Physics of Fluids, 1986, 29(7): 2152-2164.
[209] MOIN P, SQUIRES K, CABOT W, et al. A dynamic subgrid-scale model for compressible turbulence and scalar transport[J]. Physics of Fluids, 1991, 3(11): 2746-2757.
[210] WILCOX D C, et al. Turbulence modeling for CFD: volume 2[M]. Canada: DCW industries La Canada, CA, 1998.
[211] BALDWIN B, LOMAX H. Thin-layer approximation and algebraic model for separated turbulent flows[C]//16th Aerospace Sciences Meeting. 1978: 257.
[212] MENTER F, RUMSEY C. Assessment of two-equation turbulence models for transonic flows [C]//Fluid Dynamics Conference. 1994: 2343.
[213] HONG R, XIA Z, SHI Y, et al. Constrained Large-Eddy Simulation of compressible flow past a circular cylinder[J]. Communications in Computational Physics, 2014, 15(2): 388-421.
[214] SIMON F, DECK S, GUILLEN P, et al. Numerical simulation of the compressible mixing layer past an axisymmetric trailing edge[J]. Journal of Fluid Mechanics, 2007, 591: 215-253.
[215] SPALART P, ALLMARAS S. A one-equation turbulence model for aerodynamic flows[C]// 30th Aerospace Sciences Meeting and Exhibit. 1992: 439.
[216] LIANG X, LI X. DNS of a spatially evolving hypersonic turbulent boundary layer at Mach 8 [J]. Science China Physics, Mechanics and Astronomy, 2013, 56: 1408-1418.
[217] LIANG X, LI X. Direct numerical simulation on mach number and wall temperature effects in the turbulent flows of flat-plate boundary layer[J]. Communications in Computational Physics, 2015, 17: 189-212.
[218] SHE Z, ZOU H Y, JUAN XIAO M, et al. Prediction of compressible turbulent boundary layer via a symmetry-based length model[J]. Journal of Fluid Mechanics, 2018, 857: 449 - 468.
[219] XU D, WANG J, WAN M, et al. Effect of wall temperature on the kinetic energy transfer in a hypersonic turbulent boundary layer[J]. Journal of Fluid Mechanics, 2021, 929.
[220] XU D, WANG J, CHEN S. Skin-friction and heat-transfer decompositions in hypersonic transitional and turbulent boundary layers[J]. Journal of Fluid Mechanics, 2022, 941.
[221] JIANG G S, SHU C W. Efficient Implementation of Weighted ENO Schemes[J]. Journal of Computational Physics, 1996, 126(1): 202-228.
[222] 李新亮, 马延文, 傅德薰. 迎风紧致格式的混淆误差分析及其同谱方法的比较[J]. 计算物 理, 2002, 19(4): 7.
[223] SHU C W, OSHER S. Efficient implementation of essentially non-oscillatory shock-capturing schemes,II[J]. Journal of Computational Physics, 1989, 83: 32-78.
[224] LENORMAND E, SAGAUT P, PHUOC L T. Large eddy simulation of subsonic and supersonic channel flow at moderate Reynolds number[J]. International Journal for Numerical Methods in Fluids, 2000, 32: 369-406.
[225] COLEMAN G N, KIM J, MOSER R D. A numerical study of turbulent supersonic isothermal wall channel flow[J]. Journal of Fluid Mechanics, 1995, 305: 159–183.
[226] MORINISHI Y, TAMANO S, NAKABAYASHI K. Direct numerical simulation of compressible turbulent channel flow between adiabatic and isothermal walls[J]. Journal of Fluid Mechanics, 2004, 502: 273 - 308.
[227] VAN DRIEST E R. Turbulent boundary layer in compressible fluids[J]. Journal of Spacecraft and Rockets, 2003, 40: 1012-1028.
[228] BRUN C, BOIARCIUC M P, HABERKORN M, et al. Large eddy simulation of compressible channel flow[J]. Theoretical and Computational Fluid Dynamics, 2008, 22: 189-212.
[229] FUKAGATA K, IWAMOTO K, KASAGI N. Contribution of Reynolds stress distribution to the skin friction in wall-bounded flows[J]. Physics of Fluids, 2002, 14(11): L73-L76.
[230] GOMEZ T, FLUTET V, SAGAUT P. Contribution of Reynolds stress distribution to the skin friction in compressible turbulent channel flows[J]. Physical Review E, 2009, 79: 035301.
[231] MEHDI F, WHITE C M. Integral form of the skin friction coefficient suitable for experimental data[J]. Experiments in Fluids, 2011, 50(1): 43-51.
[232] MEHDI F, JOHANSSON T G, WHITE C M, et al. On determining wall shear stress in spatially developing two-dimensional wall-bounded flows[J]. Experiments in Fluids, 2014, 55(1): 1656.
[233] RENARD N, DECK S. A theoretical decomposition of mean skin friction generation into physical phenomena across the boundary layer[J]. Journal of Fluid Mechanics, 2016, 790: 339-367.
[234] LI W, FAN Y, MODESTI D, et al. Decomposition of the mean skin-friction drag in compressible turbulent channel flows[J]. Journal of Fluid Mechanics, 2019, 875: 101–123.
[235] FAN Y, LI W, PIROZZOLI S. Decomposition of the mean friction drag in zero-pressuregradient turbulent boundary layers[J]. Physics of Fluids, 2019, 31(8): 086105.
[236] ZHANG P, XIA Z. Contribution of viscous stress work to wall heat flux in compressible turbulent channel flows[J]. Physical Review E, 2020, 102(4): 043107.
[237] WENZEL C, GIBIS T, KLOKER M. About the influences of compressibility, heat transfer and pressure gradients in compressible turbulent boundary layers[J]. Journal of Fluid Mechanics, 2022, 930.
[238] BRANDT L, HENNINGSON D S. Transition of streamwise streaks in zero-pressure-gradient boundary layers[J]. Journal of Fluid Mechanics, 2002, 472: 229-261.
[239] PIROZZOLI S, GRASSO F, GATSKI T. Direct numerical simulation and analysis of a spatially evolving supersonic turbulent boundary layer at M= 2.25[J]. Physics of Fluids, 2004, 16(3): 530-545.
[240] FRANKO K J, LELE S K. Breakdown mechanisms and heat transfer overshoot in hypersonic zero pressure gradient boundary layers[J]. Journal of Fluid Mechanics, 2013, 730: 491–532.
[241] HORVATH T, BERRY S, HOLLIS B, et al. Boundary layer transition on slender cones in conventional and low disturbance Mach 6 wind tunnels[C]//32nd AIAA Fluid Dynamics Conference and Exhibit. 2002: 2743.
[242] MAYER C S, VON TERZI D A, FASEL H F. Direct numerical simulation of complete transition to turbulence via oblique breakdown at Mach 3[J]. Journal of Fluid Mechanics, 2011, 674: 5-42.
[243] 傅德薰, 马延文, 李新亮, 等. 可压缩湍流直接数值模拟[M]. 科学出版社, 2010.
[244] OBAYASHI S, KUWAHARA K, FUJII K, et al. Improvements in efficiency and reliability for Navier-Stokes computations using the LU-ADI factorization algorithm[C]//24th Aerospace Sciences Meeting. Reno, Nevada: American Institute of Aeronautics and Astronautics, 1986:338.
[245] FOROOGHI P, STRIPF M, FROHNAPFEL B. A systematic study of turbulent heat transfer over rough walls[J]. International Journal of Heat and Mass Transfer, 2018.
[246] GANJU S, BAILEY S C C, BREHM C. Amplitude and wavelength scaling of sinusoidal roughness effects in turbulent channel flow at fixed 𝑹𝒆 𝝉 = 𝟕𝟐𝟎[J]. Journal of Fluid Mechanics, 2022,937.
[247] NIKORA V I, STOESSER T, CAMERON S M, et al. Friction factor decomposition for rough wall flows: theoretical background and application to open-channel flows[J]. Journal of Fluid Mechanics, 2019, 872: 626-664.
[248] ANDERSON J D. Modern compressible flow: with historical perspective: volume 12[M]. New York, USA: McGraw-Hill, 1990.
[249] TOWNSEND A A. The structure of turbulent shear flow. 2nd edition[M]. Cambridge: Cambridge University Press, 1976.
[250] TRETTEL A J, LARSSON J. Mean velocity scaling for compressible wall turbulence with heat transfer[J]. Physics of Fluids, 2016, 28: 026102.
[251] VOLPIANI P S, IYER P S, PIROZZOLI S, et al. Data-driven compressibility transformation for turbulent wall layers[J]. Physical Review Fluids, 2020, 5: 052602.
[252] JACKSON P. On the displacement height in the logarithmic velocity profile[J]. Journal of Fluid Mechanics, 1981, 111: 15-25.
[253] ABDERRAHAMAN-ELENA N, FAIRHALL C T, GARCIA-MAYORAL R. Modulation of near-wall turbulence in the transitionally rough regime[J]. Journal of Fluid Mechanics, 2019, 865: 1042-1071.
[254] POPE S B. Turbulent flows[M]. Cambridge University Press, 2000.
[255] TOUBER E, LESCHZINER M A. Near-wall streak modification by spanwise oscillatory wall motion and drag-reduction mechanisms[J]. Journal of Fluid Mechanics, 2012, 693: 150-200.
[256] JIMéNEZ J. Coherent structures in wall-bounded turbulence[J]. Journal of Fluid Mechanics, 2018, 842: P1.
[257] JIMéNEZ J. The streaks of wall-bounded turbulence need not be long[J]. Journal of Fluid Mechanics, 2022, 945: R3.
[258] HUANG J, DUAN L, CHOUDHARI M M. Direct numerical simulation of hypersonic turbulent boundary layers: effect of spatial evolution and Reynolds number[J]. Journal of Fluid Mechanics, 2022, 937: A3.
[259] MITTAL A, GIRIMAJI S S. Mathematical framework for analysis of internal energy dynamics and spectral distribution in compressible turbulent flows[J]. Physical Review Fluids, 2019, 4 (4): 042601.
[260] HUANG P, COLEMAN G N, BRADSHAW P. Compressible turbulent channel flows: DNS results and modelling[J]. Journal of Fluid Mechanics, 1995, 305: 185-218.
[261] ALUIE H, LI S, LI H. Conservative cascade of kinetic energy in compressible turbulence[J].The Astrophysical Journal Letters, 2012, 751(2): L29.