PSE在可压缩边界层中扰动演化和超音速边界层二次失稳中的应用
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摘要
本文用抛物化稳定性方程研究可压缩平板边界层中扰动演化。首先推导出线性和非线性抛物化稳定性方程。在此基础之上,用线性PSE研究了可压缩边界层中小扰动的演化,及基本流的非平行性对可压缩边界层中性曲线的影响;然后用非线性PSE研究了可压缩边界层中有限幅值扰动的演化,并与DNS结果做了对比,证实了结果的可靠性;最后以非线性PSE为工具研究了超音速边界层二次失稳问题。共得到以下结果:
(1)对三种典型小扰动,即亚音速边界层中T-S波、超音速边界层中第一模态和第二模态T-S波,在大雷诺数情况下,用线性抛物化稳定性方程在平行流和非平行流中得到的扰动线性演化均与线性稳定性理论所得很接近。
(2)使用线性抛物化稳定性方程寻找可压缩边界层中二维小扰动的中性曲线,并与空间模式线性稳定性理论所得进行比较,发现非平行性对中性曲线的影响。无论是在亚音速还是超音速边界层中,非平行性的影响在临界雷诺数处比较明显,具体是一使得临界雷诺数变小,二使得临界雷诺数处增长扰动的频率范围变得更大。在大雷诺数处,由于流动近似平行,非平行性对边界层稳定性的影响很弱。
(3)对三种典型扰动,即亚音速边界层中的有限幅值T-S波、超音速边界层中有限幅值的第一模态和第二模态T-S波,用非线性抛物化稳定性方程得到的扰动演化均与空间模式直接数值模拟所得基本一致,包括平均流修正、基本扰动和高次谐波的幅值大小和剖面形状。因而非线性PSE可用以研究可压缩边界层内的有限幅值扰动演化。
(4)用非线性PSE方法研究马赫数4.5的超音速边界层中的二次失稳机制。结果显示,不论基本波是第一模态还是第二模态的T-S波,二次失稳机制都起作用。三维亚谐波的放大率随其展向波数和二维基本波幅值的变化关系与不可压缩边界层中所得类似。但是,即使二维基本波的幅值大到2%的量级,三维亚谐波的最大放大率仍远小于最不稳定的第二模态二维T-S波的放大率。因此,二次失稳应该不是导致超音速边界层转捩的主要因素。
In this paper, parabolized stability equations (PSE) approach is used to investigate the evolution of disturbances in compressible boundary layers on flat plates. Firstly, the linear and nonlinear parabolized stability equations are derived. On this basis, the linear PSE is used to study the evolution of small amplitude disturbances in compressible boundary layers and the nonparallel effects of the basic flow to the neutral curves of compressible boundary layers. Then, the nonlinear PSE is employed to study the evolution of finite amplitude disturbances in compressible boundary layers, and the results are compared with those obtained by DNS so that its reliability is confirmed. Finally, the nonlinear PSE is applied as a tool to investigate the problem of secondary instability in supersonic boundary layers. The following conclusions are drawn:
(1) For three typical small amplitude disturbances, i.e. the T-S wave in subsonic boundary layers and the first mode and second mode T-S waves in supersonic boundary layers, their linear evolutions computed by linear PSE agree almost perfectly with those obtained by linear stability theory for both parallel and nonparallel basic flows, provided the Reynolds number is large.
(2) Linear PSE is applied to search the neutral curves of 2-D small amplitude disturbances in compressible boundary layers, and the results are compared with those obtained by LST so that the nonparallel effects to neutral curves are found. No matter the boundary layer is subsonic or supersonic, nonparallel effects are obvious at the critical Reynolds number, such that the critical Reynolds number is reduced and the frequency range of unstable disturbances is broadened at the critical Reynolds number. At large Reynolds number, as the basic flow is closer to parallel flow, the nonparallel effect to boundary layer stability becomes insignificant.
(3) For three typical disturbances, i.e. the finite amplitude T-S wave in subsonic boundary layers and the finite amplitude first mode and second mode T-S waves in supersonic boundary layers, their evolutions obtained by using nonlinear PSE agree reasonably well with those obtained by spatial mode direct numerical simulations, including the amplitudes and shapes of the mean flow distortion, the fundamental disturbance and higher harmonics. Consequently, nonlinear PSE can be applied to investigate the evolution of finite amplitude disturbances in compressible boundary layers.
(4) The nonlinear PSE method is applied to study the secondary instability mechanism in a supersonic boundary layer with Mach number 4.5. The result shows that the mechanism of secondary instability does work, no matter the fundamental wave is first mode or second mode T-S wave. The variation of the growth rates of the 3-D sub-harmonic wave against its span-wise wave number and the amplitude of the 2-D fundamental wave is found to be similar to those found in incompressible boundary layers. But even as the amplitude of the 2-D fundamental wave is as large as at the order of 2%, the maximum growth rate of the 3-D sub-harmonic is still much smaller than the growth rate of the most unstable second mode 2-D T-S wave. Consequently, secondary instability is unlikely the main cause leading to transition in supersonic boundary layers.
(1)对三种典型小扰动,即亚音速边界层中T-S波、超音速边界层中第一模态和第二模态T-S波,在大雷诺数情况下,用线性抛物化稳定性方程在平行流和非平行流中得到的扰动线性演化均与线性稳定性理论所得很接近。
(2)使用线性抛物化稳定性方程寻找可压缩边界层中二维小扰动的中性曲线,并与空间模式线性稳定性理论所得进行比较,发现非平行性对中性曲线的影响。无论是在亚音速还是超音速边界层中,非平行性的影响在临界雷诺数处比较明显,具体是一使得临界雷诺数变小,二使得临界雷诺数处增长扰动的频率范围变得更大。在大雷诺数处,由于流动近似平行,非平行性对边界层稳定性的影响很弱。
(3)对三种典型扰动,即亚音速边界层中的有限幅值T-S波、超音速边界层中有限幅值的第一模态和第二模态T-S波,用非线性抛物化稳定性方程得到的扰动演化均与空间模式直接数值模拟所得基本一致,包括平均流修正、基本扰动和高次谐波的幅值大小和剖面形状。因而非线性PSE可用以研究可压缩边界层内的有限幅值扰动演化。
(4)用非线性PSE方法研究马赫数4.5的超音速边界层中的二次失稳机制。结果显示,不论基本波是第一模态还是第二模态的T-S波,二次失稳机制都起作用。三维亚谐波的放大率随其展向波数和二维基本波幅值的变化关系与不可压缩边界层中所得类似。但是,即使二维基本波的幅值大到2%的量级,三维亚谐波的最大放大率仍远小于最不稳定的第二模态二维T-S波的放大率。因此,二次失稳应该不是导致超音速边界层转捩的主要因素。
In this paper, parabolized stability equations (PSE) approach is used to investigate the evolution of disturbances in compressible boundary layers on flat plates. Firstly, the linear and nonlinear parabolized stability equations are derived. On this basis, the linear PSE is used to study the evolution of small amplitude disturbances in compressible boundary layers and the nonparallel effects of the basic flow to the neutral curves of compressible boundary layers. Then, the nonlinear PSE is employed to study the evolution of finite amplitude disturbances in compressible boundary layers, and the results are compared with those obtained by DNS so that its reliability is confirmed. Finally, the nonlinear PSE is applied as a tool to investigate the problem of secondary instability in supersonic boundary layers. The following conclusions are drawn:
(1) For three typical small amplitude disturbances, i.e. the T-S wave in subsonic boundary layers and the first mode and second mode T-S waves in supersonic boundary layers, their linear evolutions computed by linear PSE agree almost perfectly with those obtained by linear stability theory for both parallel and nonparallel basic flows, provided the Reynolds number is large.
(2) Linear PSE is applied to search the neutral curves of 2-D small amplitude disturbances in compressible boundary layers, and the results are compared with those obtained by LST so that the nonparallel effects to neutral curves are found. No matter the boundary layer is subsonic or supersonic, nonparallel effects are obvious at the critical Reynolds number, such that the critical Reynolds number is reduced and the frequency range of unstable disturbances is broadened at the critical Reynolds number. At large Reynolds number, as the basic flow is closer to parallel flow, the nonparallel effect to boundary layer stability becomes insignificant.
(3) For three typical disturbances, i.e. the finite amplitude T-S wave in subsonic boundary layers and the finite amplitude first mode and second mode T-S waves in supersonic boundary layers, their evolutions obtained by using nonlinear PSE agree reasonably well with those obtained by spatial mode direct numerical simulations, including the amplitudes and shapes of the mean flow distortion, the fundamental disturbance and higher harmonics. Consequently, nonlinear PSE can be applied to investigate the evolution of finite amplitude disturbances in compressible boundary layers.
(4) The nonlinear PSE method is applied to study the secondary instability mechanism in a supersonic boundary layer with Mach number 4.5. The result shows that the mechanism of secondary instability does work, no matter the fundamental wave is first mode or second mode T-S wave. The variation of the growth rates of the 3-D sub-harmonic wave against its span-wise wave number and the amplitude of the 2-D fundamental wave is found to be similar to those found in incompressible boundary layers. But even as the amplitude of the 2-D fundamental wave is as large as at the order of 2%, the maximum growth rate of the 3-D sub-harmonic is still much smaller than the growth rate of the most unstable second mode 2-D T-S wave. Consequently, secondary instability is unlikely the main cause leading to transition in supersonic boundary layers.
引文
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[6]黄章峰,曹伟,周恒.超音速平板边界层转捩中层流突变为湍流的机理——时间模式.中国科学, G辑, 2005, 35(5):537-547
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[28] Ditez A J. Local boundary-layer receptivity to a convected free-stream disturbance. Journal of Fluid Mechanics,1999,378:291-317
[29] Wu X. On local boundary-layer receptivity to vortical disturbances in the free stream. Journal of Fluid Mechanics,2001,449:373-393
[30]张永明,周恒.自由流中涡扰动的边界层感受性的数值研究.应用数学和力学,2005,26(5):505-511
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[2] Herbert Th. Subharmonic three-dimensional disturbances in unstable plane shear flow. AIAA Paper,1983,No.83-1759
[3] Herbert Th. Analysis of the subharmonic route to transition in boundary layers. AIAA Paper,1984,No.84-0009
[4] Herbert Th. Secondary instability of boundary layers. Ann. Rev. Fluid Mech.,1988,20:487-526
[5]王新军,罗纪生,周恒.平面槽道流中层流-湍流转捩的"breakdown"过程的内在机理.中国科学, G辑, 2005, 35(1):71-78
[6]黄章峰,曹伟,周恒.超音速平板边界层转捩中层流突变为湍流的机理——时间模式.中国科学, G辑, 2005, 35(5):537-547
[7]曹伟,黄章峰,周恒.超音速平板边界层转捩中层流突变为湍流的机理研究.应用数学和力学,2006,27(4):379-386
[8] Kleiser L, Zang T A. Numerical simulation of transition in wall-bounded shear flows. Annu. Rev. Fluid Mech., 1991, 23:495-537
[9] Rist U, Fasel H. Direct numerical simulation of controlled transition in a flat-plate boundary layer. J. Fluid Mech., 1995, 298:211-248
[10] Williams D R, Fasel H, Hama F R. Experimental determination of the three-dimensional vorticity field in the boundary layer transition process. J. Fluid Mech., 1984, 149:179-203
[11] Borodulin V L, Gaponenko V R, Kachanov Yu S, et al. Late-stage transitional boundary layer structures direct numerical simulation and experiment. Theoret. Comput. Fluid Dynamics, 2002, 15:317-337
[12]是勋刚,陈启生.槽道转捩的直接数值模拟.水动力学研究与进展, 1994, 9(6):666-675
[13] Dinavahi S P G, Pruett C D, Zang T A. Direct numerical simulation and data analysis of a Mach 4.5 transitional boundary-layer flow. Physics of Fluids, 1994, 6(3):1323-1330
[14] Johnson H B, Seipp T G, Candler G V. Numerical study of hypersonic reacting boundary layer transition on cones. Physics of Fluids, 1998, 10(10):2676-2685
[15]傅德熏,马延文,张林波.可压缩混合层流动转捩到湍流的直接数值模拟.中国科学, A辑, 2000, 30(2):161-168
[16]李新亮,傅德熏,马延文.可压缩钝楔边界层转捩到湍流的直接数值模拟.中国科学, G辑, 2004, 34(4):466-480
[17] Schneider S P. Flight data for boundary-layer transition at hypersonic and supersonic speeds. J. Spacecraft. Rockets, 1999, 36(1):8-20
[18] Bountin D A, Sidorenko A A, and Shiplyuk A N. Development of natural disturbances in a hypersonic boundary layer on a sharp cone. Journal of Applied Mechanics and Technical Physics, 2001, 42(1):57-62
[19] Schneider S P. Effects of high-speed tunnel noise on laminar-turbulent transition. J. Spacecraft. Rockets, 2001, 38(3):323-334
[20] Schneider S P. Hypersonic laminar-turbulent transition on circular cones and scramjet forebodies. Progress in Aerospace Sciences, 2004, 40(1):1-50
[21] Cebeci T, Shao J P, Chen H H, et al. The preferred approach for calculating transition by stability theory. An International Conference on Boundary and Interior Layers-Computational and Asymptotic Methods,2004,July 5-7
[22] Morkovin M V. On the many faces of transition. Wells C S. Ed. Vicious Drag Reduction,New York:Plenum,1969,1-31
[23] Reshotko E. Boundary-layer stability and transition. Annual Review of Fluid Mechanics,1976,8:311-349
[24] Kerschen E. Boundary layer receptivity. AIAA Paper,1989,No.89-1109
[25] Zavol’skii N A, Reutov V P, Rybushkina G V. Generation of Tollmien-Schlichting waves via scattering of acoustic and votex perturbations in boundary layer on a wavy surface. Journal of Applied Mechanics and Technical Physics,1983,24:355-361
[26] Luo J S, Zhou H. On the generation of Tollmien-Schlichting waves in the boundary layer of a flat plate by the disturbances in the free stream.Proceedings of the Royal Society of London, Series A,1987,413:351-367
[27] Wu X. Generation of Tollmien-Schlichting waves by convecting gusts interacting with sound. Journal of Fluid Mechanics,1999,397:285-316
[28] Ditez A J. Local boundary-layer receptivity to a convected free-stream disturbance. Journal of Fluid Mechanics,1999,378:291-317
[29] Wu X. On local boundary-layer receptivity to vortical disturbances in the free stream. Journal of Fluid Mechanics,2001,449:373-393
[30]张永明,周恒.自由流中涡扰动的边界层感受性的数值研究.应用数学和力学,2005,26(5):505-511
[31] Tollmien W.über die Entstehung der Turbulenz. Nachr. Ges. G?ttingen Math.-Phys. Kl.,1929,21-44
[32] Schlichting H. Zur Entstehung der Turbulenz bei der Plattenstr?mung. Nachr. Ges. G?ttingen Math.-Phys. Kl.,1933,181-208
[33] Liepmann H W. Investigations on laminar boundary-layer stability and transition on curved boundaries. NACA Adv. Conf. Rep.,1943,3H30
[34] Schubauer G B, Skramstad H K. Laminar boundary-layer oscillations and transition on a flat plate. J. Res. Natl. Bur. Stand.,1947,38:251-292, Reprint of NACA Adv. Conf. Rep., 1943
[35] Lin C C. The theory of hydrodynamic stability. Cambridge:Cambridge University Press,1955
[36] Lees L, Lin C C. Investigation of the stability of the laminar boundary layer in a compressible fluid. NACA,1964,TN-1115
[37] Mack L M. Boundary layer linear stability theory. AGARD Rep.,1984,709
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