基于空间模式的平板边界层层流到湍流转捩的研究
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摘要
本文基于空间模式的角度,采用直接数值模拟(DNS)的方法,应用MPI并行技术,模拟了不可压缩平板边界层层流到湍流转捩的过程,继续研究了转捩过程中“breakdown”过程的内在机理,重点考察了平均流剖面修正在层流转捩的“breakdown”过程中所起的作用。
流动的雷诺数选定为732,基本流为层流Blasius相似性解,计算域入口扰动形式为一个二维T-S波和一对对称的三维T-S波。控制方程为直角坐标系下无量纲化的扰动形式Navier-Stokes方程,时间离散采用三阶精度混合显隐相结合的分裂格式,空间离散则采用高精度紧致有限差分逼近与Fourier伪谱方法相结合的方法。法向采用非等距网格坐标变换,出口边界条件采用嵌边函数法。应用具体算例,验证了本文并行程序不仅符合线性理论,而且与串行计算结果一致。按照扰动幅值的大小,本文计算了亚谐转捩的三种情况,随后从壁面摩擦系数、平均脉动动能、谱空间扰动演化、y-z平面速度矢量分布、平均速度剖面的演化、流动稳定性分析、边界层的各种厚度及形状因子的演化等方面对计算结果进行了分析,得出以下结论:
1.不同的初始扰动幅值,转捩发生在下游不同的位置。如果三维扰动的幅值较小,只有在“breakdown”前的大约一、两个二维流向波长的范围内有明显三维扰动,且展向尺度基本上与入口处的三维波的展向尺度一致。转捩后更高的高次谐波被快速激发,扰动的展向尺度变得更小,同时平均流修正也增大的很快。
2.在“breakdown”的前期,用当地层流边界层排移厚度无量纲化的壁面摩擦系数是相同的。在“breakdown”的后期,壁面摩擦系数基本上增大到同样的大小。“breakdown”的过程为120个计算域入口处层流边界层排移厚度。
3.平均脉动动能随空间演化的规律与壁面摩擦系数类似,但先于壁面摩擦系数开始快速增长。总的脉动能量可以达到0.05左右,其中流向脉动能量占五分之三,法向和展向脉动能量各占五分之一。另外,初始扰动越小,转捩越靠下游,各个方向的平均脉动能量都越大,但平均脉动能量快速增长区间的大小是基本一致的。
4.不同初始扰动引起转捩,直至“breakdown”前,二维波的幅值都是不一样的,但平均流修正开始快速增长的位置是相同的,平均流修正快速增长的过程也是大致相同的。平均流修正开始快速增长的位置与二维波开始不规则的位置是一致的。在这个位置有规则的三维波达到最大,约为0.05左右,然后三维波开始变得无规则。这一期间三维波增长很快,也许是流动稳定性的二次稳定性在起作用。在转捩前,二维扰动幅值的演化与流动稳定性线性理论的预测基本符合,波长也与入口处二维波的波长近似相等,从某种意义上讲,这也许为转捩预测中的EN方法提供了理论依据。
5.对沿流向不同位置的平均流速度进行的稳定性分析表明,沿着流动方向,不稳定区域先逐渐减小,缩小到很小以后,再逐渐的增大到最大范围,然后再一次逐渐减小,趋于很小。不稳定区域开始逐渐增大的位置对应于壁面摩擦系数开始快速增长的位置,不稳定区域达到最大的位置大致位于壁面摩擦系数曲线快速增长段的1/3处,不稳定区域再一次逐渐减小,趋于零的位置对应于壁面摩擦系数达到最大的位置。当层流中的扰动幅值增大后,由于非线性作用将修正平均流剖面。当平均流剖面被修正到一定程度,不稳定性区域会扩得很大,这将导致更多的谐波被快速激发并增长,开始了“breakdown”过程。在“breakdown”过程中,由于大量谐波快速增长,导致扰动能量快速增长。扰动能量的快速增长又进一步加速了平均流剖面的快速修正。这样相互的快速增长使得流动快速地转变为湍流。这里的结论与其他流动中的研究结果是一致的。
6.在转捩过程中,排移厚度的增长比相应的层流排移厚度的增长要慢。
In this paper, direct numerical simulation (DNS) was adopted for the laminar-turbulent transition of an incompressible boundary layer on a flat plate based on the spatial model with MPI technology. Studies on breakdown mechanism of laminar-turbulent transition was carried on sequentially. The priority was given to the effects of the modification of the mean flow profile on the process of breakdown in laminar-turbulent transition.
The Reynolds number was taken as 732. The laminar Blasius similarity solution was taken as the basic flow. One two-dimensional T-S wave and two symmetrical three-dimensional T-S waves were taken as the disturbers at the entrance. Under Cartesian coordinate, the non-dimensional perturbation equations were taken as the governing equations. A third-order precision mixed explicit-implicit scheme was employed for time discretization and a combination method of high precision compact finite difference with the Fourier pseudo-spectral expansion was implemented for spatial discretization. The method of non-uniform meshes was applied in normal direction and the fringe method was applied in dealing with outlet boundary condition. By a simple case, the parallel programs were not only validated to be consistent with the linear theory, but also the results got by one computer.
According to the magnitude of amplification, we simulated three cases of subharmonic transition. Then we anaylized the results from the following aspects: skin friction coefficient、mean fluctuation kinetic energy、disturbance velocity evolution in Fourier space、velocity vector at y-z plane、evolution of mean velocity profile、stability analysis、thicknesses of boundary and shape factor. From the analysis, we drew the following conclusions:
1. Disturbance with different initial amplitude led to different transition positions at backward position. If the amplitudes of three-dimensional disturbers were smaller, obvious three-dimensional disturbances only began to exist at a length of one or two two-dimensional wavelengths in front of the breakdown, and the magnitudes of waves in z-direction approximate to the length of three-dimensional waves which were added at the entrance. Some higher frequency harmonic waves came into exist rapidly after the process of transition, the magnitudes of waves in z-direction became smaller, meanwhile, the modification of mean flow increased rapidly.
2. In the prophase of breakdown, The skin friction coefficient non-dimensionized by local laminar boundary layer displacement thickness was equal at every point along x-direction. In the anaphase of breakdown, skin friction coefficient increased to the same value for the cases considered, the length of breakdown process was about 120 laminar boundary layer displacement thickness of the entrance.
3. The evolution of mean fluctuation kinetic energy along x-direction was analogous with the one of skin friction coefficient, but it started to increase before the skin friction coefficient did. The total fluctuation kinetic energy could reach about 0.05, of which 0.03 was contributed by u, and the other 0.02 was contributed by v and w equally. In addition, the smaller the initial disturbers were, the longer distance from the entrance transition started and the bigger the mean fluctuation kinetic energy was along every direction, but the lengths of the regions that the mean fluctuation kinetic energy increased rapidly were almost the same.
4. Different initial disturbers all caused transition, the amplitudes of two-dimensional waves were not the same until the breakdown, but the positions where mean flow was modified rapidly were the same, the processes that the modification of mean flow increased rapidly were almost the same. The position that modification of mean flow increased rapidly agreed with the one that two-dimensional wave began to be ruleless. At that position, the regular three-dimensional waves reached their maximum values of about 0.05, then the three-dimensional waves began to be ruleless. During this process, the three-dimensional waves increased rapidly, it may because the second stability of stability began to work. Before the transition, the amplitude evolution of two-dimensional disturber agreed with the linear stability theory, and the wavelength approximated to the one of the two-dimensional wave added at the entrance, to some extent, this might offer theory basis for EN method in transition forecasting.
5. We analysised mean flow velocity stability for different positions along x direction,it indicated that the instability zone became smaller and smaller along x direction at first, when being very small, it began to increase to the maximum range, then smaller it became again, and to be very small. The position where instability zone began to increase corresponded to the one that skin friction coefficient began to increase rapidly, the position where instability zone reached it’s maximum located at the 1/3 station of skin friction coefficient increased rapidly. The instability zone decreased again, the position that it was to be zero corresponded to the one where skin friction coefficient reached the maximum. When the disturber’s amplitude increased in laminar flow, the nonlinear effect would modify the mean flow profile. When the mean flow profile was modified to certain extent, the instability zone would enlarge to be of great magnitude, more harmonic waves came into exist and they increased rapidly, then breakdown started. In the process of breakdown, because of the rapid increase of many harmonic waves, disturber’s energy increased rapidly. In return, the rapidly increased energy accelerated the mean flow profile modification rapidly. All of these mutually rapid increases made the flow turn into turbulent rapidly. The results of this thesis were consistent with other investigations.
6. In the process of transiton, the increase of displace thickness was slower than the one of the corresponding laminar displace thickness.
流动的雷诺数选定为732,基本流为层流Blasius相似性解,计算域入口扰动形式为一个二维T-S波和一对对称的三维T-S波。控制方程为直角坐标系下无量纲化的扰动形式Navier-Stokes方程,时间离散采用三阶精度混合显隐相结合的分裂格式,空间离散则采用高精度紧致有限差分逼近与Fourier伪谱方法相结合的方法。法向采用非等距网格坐标变换,出口边界条件采用嵌边函数法。应用具体算例,验证了本文并行程序不仅符合线性理论,而且与串行计算结果一致。按照扰动幅值的大小,本文计算了亚谐转捩的三种情况,随后从壁面摩擦系数、平均脉动动能、谱空间扰动演化、y-z平面速度矢量分布、平均速度剖面的演化、流动稳定性分析、边界层的各种厚度及形状因子的演化等方面对计算结果进行了分析,得出以下结论:
1.不同的初始扰动幅值,转捩发生在下游不同的位置。如果三维扰动的幅值较小,只有在“breakdown”前的大约一、两个二维流向波长的范围内有明显三维扰动,且展向尺度基本上与入口处的三维波的展向尺度一致。转捩后更高的高次谐波被快速激发,扰动的展向尺度变得更小,同时平均流修正也增大的很快。
2.在“breakdown”的前期,用当地层流边界层排移厚度无量纲化的壁面摩擦系数是相同的。在“breakdown”的后期,壁面摩擦系数基本上增大到同样的大小。“breakdown”的过程为120个计算域入口处层流边界层排移厚度。
3.平均脉动动能随空间演化的规律与壁面摩擦系数类似,但先于壁面摩擦系数开始快速增长。总的脉动能量可以达到0.05左右,其中流向脉动能量占五分之三,法向和展向脉动能量各占五分之一。另外,初始扰动越小,转捩越靠下游,各个方向的平均脉动能量都越大,但平均脉动能量快速增长区间的大小是基本一致的。
4.不同初始扰动引起转捩,直至“breakdown”前,二维波的幅值都是不一样的,但平均流修正开始快速增长的位置是相同的,平均流修正快速增长的过程也是大致相同的。平均流修正开始快速增长的位置与二维波开始不规则的位置是一致的。在这个位置有规则的三维波达到最大,约为0.05左右,然后三维波开始变得无规则。这一期间三维波增长很快,也许是流动稳定性的二次稳定性在起作用。在转捩前,二维扰动幅值的演化与流动稳定性线性理论的预测基本符合,波长也与入口处二维波的波长近似相等,从某种意义上讲,这也许为转捩预测中的EN方法提供了理论依据。
5.对沿流向不同位置的平均流速度进行的稳定性分析表明,沿着流动方向,不稳定区域先逐渐减小,缩小到很小以后,再逐渐的增大到最大范围,然后再一次逐渐减小,趋于很小。不稳定区域开始逐渐增大的位置对应于壁面摩擦系数开始快速增长的位置,不稳定区域达到最大的位置大致位于壁面摩擦系数曲线快速增长段的1/3处,不稳定区域再一次逐渐减小,趋于零的位置对应于壁面摩擦系数达到最大的位置。当层流中的扰动幅值增大后,由于非线性作用将修正平均流剖面。当平均流剖面被修正到一定程度,不稳定性区域会扩得很大,这将导致更多的谐波被快速激发并增长,开始了“breakdown”过程。在“breakdown”过程中,由于大量谐波快速增长,导致扰动能量快速增长。扰动能量的快速增长又进一步加速了平均流剖面的快速修正。这样相互的快速增长使得流动快速地转变为湍流。这里的结论与其他流动中的研究结果是一致的。
6.在转捩过程中,排移厚度的增长比相应的层流排移厚度的增长要慢。
In this paper, direct numerical simulation (DNS) was adopted for the laminar-turbulent transition of an incompressible boundary layer on a flat plate based on the spatial model with MPI technology. Studies on breakdown mechanism of laminar-turbulent transition was carried on sequentially. The priority was given to the effects of the modification of the mean flow profile on the process of breakdown in laminar-turbulent transition.
The Reynolds number was taken as 732. The laminar Blasius similarity solution was taken as the basic flow. One two-dimensional T-S wave and two symmetrical three-dimensional T-S waves were taken as the disturbers at the entrance. Under Cartesian coordinate, the non-dimensional perturbation equations were taken as the governing equations. A third-order precision mixed explicit-implicit scheme was employed for time discretization and a combination method of high precision compact finite difference with the Fourier pseudo-spectral expansion was implemented for spatial discretization. The method of non-uniform meshes was applied in normal direction and the fringe method was applied in dealing with outlet boundary condition. By a simple case, the parallel programs were not only validated to be consistent with the linear theory, but also the results got by one computer.
According to the magnitude of amplification, we simulated three cases of subharmonic transition. Then we anaylized the results from the following aspects: skin friction coefficient、mean fluctuation kinetic energy、disturbance velocity evolution in Fourier space、velocity vector at y-z plane、evolution of mean velocity profile、stability analysis、thicknesses of boundary and shape factor. From the analysis, we drew the following conclusions:
1. Disturbance with different initial amplitude led to different transition positions at backward position. If the amplitudes of three-dimensional disturbers were smaller, obvious three-dimensional disturbances only began to exist at a length of one or two two-dimensional wavelengths in front of the breakdown, and the magnitudes of waves in z-direction approximate to the length of three-dimensional waves which were added at the entrance. Some higher frequency harmonic waves came into exist rapidly after the process of transition, the magnitudes of waves in z-direction became smaller, meanwhile, the modification of mean flow increased rapidly.
2. In the prophase of breakdown, The skin friction coefficient non-dimensionized by local laminar boundary layer displacement thickness was equal at every point along x-direction. In the anaphase of breakdown, skin friction coefficient increased to the same value for the cases considered, the length of breakdown process was about 120 laminar boundary layer displacement thickness of the entrance.
3. The evolution of mean fluctuation kinetic energy along x-direction was analogous with the one of skin friction coefficient, but it started to increase before the skin friction coefficient did. The total fluctuation kinetic energy could reach about 0.05, of which 0.03 was contributed by u, and the other 0.02 was contributed by v and w equally. In addition, the smaller the initial disturbers were, the longer distance from the entrance transition started and the bigger the mean fluctuation kinetic energy was along every direction, but the lengths of the regions that the mean fluctuation kinetic energy increased rapidly were almost the same.
4. Different initial disturbers all caused transition, the amplitudes of two-dimensional waves were not the same until the breakdown, but the positions where mean flow was modified rapidly were the same, the processes that the modification of mean flow increased rapidly were almost the same. The position that modification of mean flow increased rapidly agreed with the one that two-dimensional wave began to be ruleless. At that position, the regular three-dimensional waves reached their maximum values of about 0.05, then the three-dimensional waves began to be ruleless. During this process, the three-dimensional waves increased rapidly, it may because the second stability of stability began to work. Before the transition, the amplitude evolution of two-dimensional disturber agreed with the linear stability theory, and the wavelength approximated to the one of the two-dimensional wave added at the entrance, to some extent, this might offer theory basis for EN method in transition forecasting.
5. We analysised mean flow velocity stability for different positions along x direction,it indicated that the instability zone became smaller and smaller along x direction at first, when being very small, it began to increase to the maximum range, then smaller it became again, and to be very small. The position where instability zone began to increase corresponded to the one that skin friction coefficient began to increase rapidly, the position where instability zone reached it’s maximum located at the 1/3 station of skin friction coefficient increased rapidly. The instability zone decreased again, the position that it was to be zero corresponded to the one where skin friction coefficient reached the maximum. When the disturber’s amplitude increased in laminar flow, the nonlinear effect would modify the mean flow profile. When the mean flow profile was modified to certain extent, the instability zone would enlarge to be of great magnitude, more harmonic waves came into exist and they increased rapidly, then breakdown started. In the process of breakdown, because of the rapid increase of many harmonic waves, disturber’s energy increased rapidly. In return, the rapidly increased energy accelerated the mean flow profile modification rapidly. All of these mutually rapid increases made the flow turn into turbulent rapidly. The results of this thesis were consistent with other investigations.
6. In the process of transiton, the increase of displace thickness was slower than the one of the corresponding laminar displace thickness.
引文
[1] 张涵信, 周恒. 流体力学的基础研究. 世界科技研究与发展, 2001, 23(1):15~18
[2] 周恒. 新世纪对流体力学提出的要求. 自然科学进展, 2000, 10(6):491~494
[3] 郑哲敏, 周恒, 张涵信, 等. 21 世纪初的力学发展趋势. 力学进展, 1995, 25(4):433~441
[4] Gad-el-Hak. Flow Control. Cambridge: The University of Cambridge, 2000, 104～105
[5] Thibert J. J., Reneaux J., Schmitt V., Onera activities in drag reduction, Congr. Int. Counc. Aeron. Sci. (ICAS), 1990, 17th, 1053～1064
[6] 周恒, 赵耕夫. 流动稳定性. 北京: 国防工业出版社, 2004, 1~255
[7] Blackwelder, R. F. Analogies between transition and turbulent boundary layer. Phys. Fluid, 1983, 26(1): 280
[8] 是勋刚(主编). 湍流. 天津: 天津大学出版社, 1994, 213~230
[9] 周恒, 赵耕夫. KB 方法在流动稳定性问题中的应用. 力学学报, 1982, 14:217~225
[10] Herbert, T., Nonlinear stability of parallel flows by high-order amplitude expansions, 1980, AIAA, JL18: 243～248
[11] Zhou H. On the nonlinear theory of stability of plane Poseuille fow in the subcritical range. Proceeding of the Royal Society of London, Series A, 1982, 381:407～418
[12] 周恒, 李骊. Orr-Sommerfeld 方程的特征值问题及展开定理. 应用数学和力学. 1981, 2(3):295~305
[13] 周恒, 李骊. 流体层流运动稳定性中的 Liapounoff 方法. 应用数学和力学. 1983, 4(3):453~462
[14] 周恒. 流动稳定性弱非线性理论中幅值方程的一个必要的修正. 科学通报, 1990, 35(16):1228~1230
[15] 周恒. 流动稳定发生弱非线性理论的重新考虑. 应用数学和力学. 1991, 12(3):203~208
[16] 周恒. 流动稳定性理论中空间模式不稳定波能量方程中的一个问题. 力学学报, 1991, 23(1):116~118
[17] 周恒, 尤学一. 论流动稳定性中的弱非线性理论. 中国科学 A 辑, 1992, A(6):615~622
[18] Zhou H, You X Y. On the problems in the weakly nonlinear theory of hydrodynamic stability and its improvements. Acta. Mechanica Sinica (English Edition), 1993, 9(1):1～12
[19] 周恒, 藤村薰. 流动稳定性弱非线性理论的进一步改进. 中国科学 A 辑, 1997, 27(12):1111~1118
[20] Luo J S, Zhou H. A theoretical investigation of the development of stationary crossflow vortices in the boundary layer on a swept wing. Acta. Mechanica Sinica (English Edition), 1998, 14(2):97～103
[21] Saric W S, Reed H L. Kerschen E J. Boundary layer receptivity to freestream disturbances. Annu. Rev. Fluid Mech., 2002, 34:291～319
[22] Alzin L B, Polyakov N F. Acoustic generation of Tollmien-Schlichting waves over local unevenness of surface immersed in streams. Preprint 17, Akad. Nauk USSR, Siberian Div, Inst. Theor. Appl. Mech., Novosibirsk, 1979
[23] Goldstein M E, Scattering of acoustic waves into Tollmien-Schlichting waves by small streamwise variations in surface geometry. J.Fluid Mech. 1985, 154: 509～529
[24] Wiegel M, Wlezien R. Acoustic receptivity of laminar boundary layers over wavy walls. AIAA Paper, No. 93～3280
[25] Rogler H L, Reshotko E. Disturbances in a boundary layer introduced by a low intensity array of vortices. SIAM J. Appl. Maths, 1975, 28: 431～462
[26] Zavol’skii, Reutov N A, Rybushkina G V. Generation of Tollmien-Schlichting waves via scattering of acoustic and vortex perturbations in boundary layer on a wavy surface. J. Appl. Mech. Tech. Phys. 1983, 24: 355～361
[27] Kerschen E J. Boundary layer receptivity theory. Appl. Mech. Rev. 1990, 43: 152～157
[28] 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. Proc. R. Soc. Lond, 1987, A413: 351～367
[29] Wu X. Generation of Tollmien-Schlichting waves by convecting gusts interacting with sound. J. Fluid Meth, 1999, 397: 285～316
[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] Ditez A J. Local boundary-layer receptivity to a convected free-stream disturbance. J. Fluid Mech, 1999, 378: 291～317
[32] Wu X. On local boundary-layer receptivity to vertical disturbances in the free stream. J. Fluid Mech, 2001, 449:373～393
[33] Saric W S. Physical description of boundary layer transition, experimental evidence. In AGARD-CP-793, 1994
[34] Mack L M., Transition prediction and linear stability theory, AGARD Conf. No.224, Paris, 1977
[35] Bippes H., Experiment on transition in three dimensional accelerated boundary layer flows, Proc. R. A. S. Boundary Layer Transition and control, Camridge, 1991
[36] Radeztsky RH Jr, Reibert MS, Saric WS., Development of stability crossflow vortices on a swept wing, 1994, AIAA Pap. No94-2373
[37] 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
[38] 李存标. 转捩和湍流研究的最新进展. 流体力学实验与测量. 1998, 12(1):8~27
[39] Schneider S P. Flight data for boundary-layer transition at hypersonic and supersonic speeds. J. Spacecraft. Rockets, 1999, 36(1):8～20
[40] Klebanoff P. S., Tidstrom K. D., Sargent L. M., The three-dimensional nature of boundary-layer instability. J.Fluid Mech. 1962, 12: 1～34
[41] Kachanov Y. S., Levchenko V.Y., The resonant interaction of disturbances at laminar-turbulent transition in a boundary layer. J.Fluid Mech. 1984, 138: 209～247
[42] Kachanov Y. S., On the resonant nature of the breakdown of a laminar boundary layer. J. Fluid Mech. 1987, 184: 43～74
[43] Dryganets S V, Kachanov Y S, Levchenko V Y, et all. Resonant flow randomization in K-regime of boundary layer transition. J. Appl. Mech. Tech. Phys., 1990, 31(2):239～249
[44] Kachanov Y S. Physical mechanisms of laminar-boundary-layer transition. Annu. Rev. Fluid Mech., 1994, 26:411～482
[45] Bake S, Fernholz H, Kachanov Y S. Resemblance of K- and N-regimes of boundary-layer transition at late stages. Eur. J. Mech. B/Fluids, 2000, 19(1):1～22
[46] Herbert, Th., Secondary instability of plane channel flow to subharmonic three-dimensional disturbances. Phys. Fluids, 1983, 26: 871～874
[47] Herbert, Th., Subharmonic three-dimensional disturbances in unstable plane shear flow. AIAA Paper, 1983, No.83-1759
[48] Herbert, Th., Analysis of the subharmonic route to transition in boundary layers, AIAA Paper, 1984, No.84-0009
[49] 赵耕夫, 周恒. 平面 Poiseuille 流中弱非线性理论和二次失稳理论的关系. 应用数学和力学. 1988, 9(8): 571~576
[50] Kleiser L, Zang T A. Numerical simulation of transition in wall-bounded shear flows. Annu. Rev. Fluid Mech., 1991, 23:495～537
[51] Rist U, Fasel H. Direct numerical simulation of controlled transition in a flat-plate boundary layer. J. Fluid Mech., 1995, 298:211～248
[52] 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
[53] Borodulin V L, Gaponenko V R, Kachanov Y S, et al. Late-stage transitional boundary layer structures direct numerical simulation and experiment. Theoret. Comput. Fluid Dynamics, 2002, 15:317～337
[54] Chaoqun Liu., Zhining Liu., Multigrid mapping and box relaxation for simulation of the whole process of flow transition in 3d boundary layers. J. Comput. Phy. 1995, 119: 325～341
[55] Saric W.S., Kzlov V.V., Levchenko C.Y., AIAA Paper, 1984, No.84-0007(unpublished)
[56] Wang Z, Yeo K.S, Khoo B.C. Spatial direct numerical simulation of transitional boundary layer over compliant surfaces. Computers & Fluids, 2005, 34:1062～1095
[57] Bhaganagar K., Rempfer D., Lumley J. Direct numerical simulation of spatial transition to turbulence using fourth-order vertical velocity second-order vertical vorticity formulation. J. Comput. Phy., 2002,180: 200～228
[58] Berlin S., Lundbladh A., Henningson D., Spatial simulations of oblique transition in a boundary layer. Phys. Fluids, 1994, 6(6): 1949～1951
[59] Rai M. M., Moin P., Direct numerical simulation of transition and turbulence in a spatially evolving boundary layer. J. Comput. Phy. 1993, 109: 169～192
[60] Markus J. K., A robust high-resolution split-type compact fd scheme for spatial direct numerical simulation of boundary-layer transition. Applied Scientific Research, 1998, 59: 353～377
[61] Fasel H. F., Rist U. Konzelmann U., Numerical investigation of the three-dimensional development in boundary-layer transition. AIAA, 1990, (1):29～37
[62] Fasel H., Konzelmann U., Non-parallel stability of a flat-plate boundary layer using the complete navier-stokes equations. J. Fluid Mech. 1990, 221: 311～347
[63] Rist U., Kachanov Y. S., Numerical and experimental investigation of the K-regime of boundary-layer transition. Proc. IUTAM-Symp. Laminar-Turbulent Transition, Sendai, Japan, Springer-Verlag, 1995, 405～412
[64] Rist U., DNS of boundary-layer instability and transition using the spatial approach. Transitional Boundary Layers in Aeronautics, Royal Netherlands Academy of Arts and Sciences. Amsterdam, Elsevier, 1996, 99～111
[65] Meyer D., Rist U., Wagner S., DNS of the generation of secondary λ-vortices in a transitional boundary layer. Advances in Turbulence VII, Kluwer, 1998: 97～100
[66] Maucher U., Rist U., Wagner S., A refined method for DNS of transition in interacting boundary layers. AIAA, 98-2435:1～11
[67] Meyer D. G. W., Rist U., Borodulin V. I., Gaponenko V. R., Late-stage transitional boundary-layer structures. Direct numerical simulation and experiment. Laminar-Turbulent Transition, Proc. IUTAM-Symp Sedona, Sept. 13-17, 1999, Springer-Verlag, 2000,167～172
[68] 陈健, 崔桂香, 许春晓等. 空间发展圆管转捩的直接数值模拟. 力学学报, 2003, 35(1): 6～13
[69] 陈健, 崔桂香, 许春晓等. 圆管流动的二次转捩. 工程力学, 2003, 20(5): 37～41
[70] 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
[71] 王新军. 转捩中“breakdown”的内在机理及周期湍流场的涡粘系数[天津大学博士论文]. 2005
[72] 唐洪涛 不可压缩平板边界层从层流突变为湍流的机理及湍流特性[天津大学博士论文]. 2007
[73] 黄章峰. 超音速平板边界层从层流到湍流的转捩机理及湍流特性[天津大学博士论文]. 2006
[74] 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. 应用数学与力学, 2006, 27(4): 379～386
[75] 南京工学院, 华中工学院, 重庆大学, 天津大学合编. 粘性流体力学. 高等教育出版社, 1987: 130, 328
[76] 章梓雄, 董曾南. 粘性流体力学. 北京: 清华大学出版社, 1998, 125～249
[77] 陈懋章. 粘性流体动力学基础. 北京: 高等教育出版社, 2004, 192～372
[78] H. 史里希廷. 边界层理论(上册).北京: 科学出版社. 1988, 141～172
[79] H. 史里希廷. 边界层理论(下册).北京: 科学出版社. 1988, 519～669
[80] Karniadakis G. E., Israeli M., Orszag S. A., High-order splitting methods for the incompressible navier-stokes equations. J. Comput. Phy. 1991, 97:414～443
[81] 傅德薰, 马延文. 平面混合流拟序结构的直接数值模拟. 中国科学 A 辑, 1996, 26(7):659~664
[82] Lele S K. Compact finite difference schemes with spectral-like resolution. J Comp Phys, 1992, 103: 16～43
[83] 傅德薰、马延文. 计算流体力学. 北京: 高等教育出版社. 2002,118
[84] Hegeman L A, Young D M. Applied iterative methods. New York: Wiley, 1981
[85] 苏彩虹. 嵌边法出流条件在可压缩流直接数值模拟中的应用[天津大学硕士论文]. 2005
[86] Lundbladh, A., Schmid, P.J. Berlin, S. & Henningson, D. S. Simulation of bypass transition for spatially evolving disturbances. In AGARDCP-551, 1994, 18: 1～13
[87] 都志辉. 高性能计算并行编程技术—MPI 并行程序设计. 清华大学出版社. 2001.
[88] 白中治. 并行矩阵多分裂迭代算法的收敛速度与发散速度比较. 工程数学学报, 1994, 11(1): 99～102
[89] 白中治. 并行矩阵多分裂块松弛迭代算法. 计算数学, 1995, 3: 238～252
[90] 白中治. 异步并行非线性对称 Gauss-Seidel 迭代算法. 计算数学, 1998, 20(2): 187～200
[91] 毛承雄, 陆继明, 樊俊等. 并行仿真计算中的新交替迭代算法. 电力系统及其自动化学报. 1999, 11(3): 18～22
[92] 金承日, 丁效华, 张少太. 双曲型方程的有限差分并行迭代算法. 哈尔滨工业大学学报, 2002, 34(3): 340～343
[93] 陈凌, 陈云霞, 殷新春等. MPP 上的并行松弛迭代算法. 中国科学技术大学学报. 2002, 32(6): 732～737
[94] 向淑晃, 张生雷. 并行多分裂块松弛 TOR 迭代算法的收敛性. 云南大学学报(自然科学版). 2004, 26(1): 30～34
[95] 李春光, 徐成贤. 确定 SOR 最佳松弛因子的一个实用算法. 计算力学学报, 2002, 19(3): 199～302
[96] 蒋家羚, 王勇. 最优超松弛因子的一种确定方法及其在裂纹计算中的应用. 机械强度, 2002, 24(1): 133～135
[97] Frederic Ducros, Pierre Comte, Marcel Lesieur. Large-eddy simulation of transition to turbulence in a boundary layer developing spatially over a flat plate. J. Fluid Mech. 1996, 326:1～36
[98] 郭辉, 连淇祥, Yury S. Kachanov 等 边界层转捩过程三维扰动的发展和演化. 北京航空航天大学学报, 2003, 29(6): 501～504
[99] 黄章峰. 超音速边界层扰动演化的数值模拟研究[天津大学硕士论文]. 2003
[100] 吕祥翠. T-S 波与边界层尾缘相互作用的数值研究[天津大学硕士论文]. 2004
[101] 郭大为. 带壁面抽吸的槽道湍流大涡模拟[天津大学硕士论文]. 2005
[102] 刘惠枝, 舒宏纪. 边界层理论. 北京: 人民交通出版社, 1991
[103] 张东明. 湍流近壁区相干结构生成机理的研究[天津大学博士论文]. 2001
[104] 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
[105] 吕祥翠, 罗纪生, 宋志军. T-S 波与边界层尾缘相互作用的数值研究. 中国民航学院学报, 2006, 24(1): 1～6
[106] 李存标. 关于转捩边界层中流向涡的产生. 物理学报, 2001, 50(1):182～184
[107] 陆金甫, 关治. 偏微分方程数值解法. 北京: 清华大学出版社, 1987, 217～238
[108] E.O.布赖姆. 快速富里叶变换. 上海: 上海科学技术出版社, 1979, 1～33, 100～121, 165～183
[109] 冷建华. 傅立叶变换. 北京: 清华大学出版社, 2004, 1～23
[110] Douglas J, Gunn J E. A general formation of alternative direction methods: Part I, Parabolic and hyperbolic problems. Numeriche Methematik, 1964(6): 428～453
[111] Street, C. L. & Macaraeg, M. G. Spectral multi-domain for large-scale fluid dynamic simulations. Appl.Numer.Math. 1990(6):123～139
[112] Kloker, M., Konzelmann, U. & Fasel, H. Outflow boundary conditions for spatial Navier-Stokes simulations of transition boundary layers. AIAA J. 1993, 31(4): 620～628
[113] Spalart, P. R. & Watmuff, J. H. Experimental and numerical study of a turbulent boundary layer with pressure gradients. J. Fluid Mech. 1993, 249: 337～371
[2] 周恒. 新世纪对流体力学提出的要求. 自然科学进展, 2000, 10(6):491~494
[3] 郑哲敏, 周恒, 张涵信, 等. 21 世纪初的力学发展趋势. 力学进展, 1995, 25(4):433~441
[4] Gad-el-Hak. Flow Control. Cambridge: The University of Cambridge, 2000, 104～105
[5] Thibert J. J., Reneaux J., Schmitt V., Onera activities in drag reduction, Congr. Int. Counc. Aeron. Sci. (ICAS), 1990, 17th, 1053～1064
[6] 周恒, 赵耕夫. 流动稳定性. 北京: 国防工业出版社, 2004, 1~255
[7] Blackwelder, R. F. Analogies between transition and turbulent boundary layer. Phys. Fluid, 1983, 26(1): 280
[8] 是勋刚(主编). 湍流. 天津: 天津大学出版社, 1994, 213~230
[9] 周恒, 赵耕夫. KB 方法在流动稳定性问题中的应用. 力学学报, 1982, 14:217~225
[10] Herbert, T., Nonlinear stability of parallel flows by high-order amplitude expansions, 1980, AIAA, JL18: 243～248
[11] Zhou H. On the nonlinear theory of stability of plane Poseuille fow in the subcritical range. Proceeding of the Royal Society of London, Series A, 1982, 381:407～418
[12] 周恒, 李骊. Orr-Sommerfeld 方程的特征值问题及展开定理. 应用数学和力学. 1981, 2(3):295~305
[13] 周恒, 李骊. 流体层流运动稳定性中的 Liapounoff 方法. 应用数学和力学. 1983, 4(3):453~462
[14] 周恒. 流动稳定性弱非线性理论中幅值方程的一个必要的修正. 科学通报, 1990, 35(16):1228~1230
[15] 周恒. 流动稳定发生弱非线性理论的重新考虑. 应用数学和力学. 1991, 12(3):203~208
[16] 周恒. 流动稳定性理论中空间模式不稳定波能量方程中的一个问题. 力学学报, 1991, 23(1):116~118
[17] 周恒, 尤学一. 论流动稳定性中的弱非线性理论. 中国科学 A 辑, 1992, A(6):615~622
[18] Zhou H, You X Y. On the problems in the weakly nonlinear theory of hydrodynamic stability and its improvements. Acta. Mechanica Sinica (English Edition), 1993, 9(1):1～12
[19] 周恒, 藤村薰. 流动稳定性弱非线性理论的进一步改进. 中国科学 A 辑, 1997, 27(12):1111~1118
[20] Luo J S, Zhou H. A theoretical investigation of the development of stationary crossflow vortices in the boundary layer on a swept wing. Acta. Mechanica Sinica (English Edition), 1998, 14(2):97～103
[21] Saric W S, Reed H L. Kerschen E J. Boundary layer receptivity to freestream disturbances. Annu. Rev. Fluid Mech., 2002, 34:291～319
[22] Alzin L B, Polyakov N F. Acoustic generation of Tollmien-Schlichting waves over local unevenness of surface immersed in streams. Preprint 17, Akad. Nauk USSR, Siberian Div, Inst. Theor. Appl. Mech., Novosibirsk, 1979
[23] Goldstein M E, Scattering of acoustic waves into Tollmien-Schlichting waves by small streamwise variations in surface geometry. J.Fluid Mech. 1985, 154: 509～529
[24] Wiegel M, Wlezien R. Acoustic receptivity of laminar boundary layers over wavy walls. AIAA Paper, No. 93～3280
[25] Rogler H L, Reshotko E. Disturbances in a boundary layer introduced by a low intensity array of vortices. SIAM J. Appl. Maths, 1975, 28: 431～462
[26] Zavol’skii, Reutov N A, Rybushkina G V. Generation of Tollmien-Schlichting waves via scattering of acoustic and vortex perturbations in boundary layer on a wavy surface. J. Appl. Mech. Tech. Phys. 1983, 24: 355～361
[27] Kerschen E J. Boundary layer receptivity theory. Appl. Mech. Rev. 1990, 43: 152～157
[28] 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. Proc. R. Soc. Lond, 1987, A413: 351～367
[29] Wu X. Generation of Tollmien-Schlichting waves by convecting gusts interacting with sound. J. Fluid Meth, 1999, 397: 285～316
[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] Ditez A J. Local boundary-layer receptivity to a convected free-stream disturbance. J. Fluid Mech, 1999, 378: 291～317
[32] Wu X. On local boundary-layer receptivity to vertical disturbances in the free stream. J. Fluid Mech, 2001, 449:373～393
[33] Saric W S. Physical description of boundary layer transition, experimental evidence. In AGARD-CP-793, 1994
[34] Mack L M., Transition prediction and linear stability theory, AGARD Conf. No.224, Paris, 1977
[35] Bippes H., Experiment on transition in three dimensional accelerated boundary layer flows, Proc. R. A. S. Boundary Layer Transition and control, Camridge, 1991
[36] Radeztsky RH Jr, Reibert MS, Saric WS., Development of stability crossflow vortices on a swept wing, 1994, AIAA Pap. No94-2373
[37] 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
[38] 李存标. 转捩和湍流研究的最新进展. 流体力学实验与测量. 1998, 12(1):8~27
[39] Schneider S P. Flight data for boundary-layer transition at hypersonic and supersonic speeds. J. Spacecraft. Rockets, 1999, 36(1):8～20
[40] Klebanoff P. S., Tidstrom K. D., Sargent L. M., The three-dimensional nature of boundary-layer instability. J.Fluid Mech. 1962, 12: 1～34
[41] Kachanov Y. S., Levchenko V.Y., The resonant interaction of disturbances at laminar-turbulent transition in a boundary layer. J.Fluid Mech. 1984, 138: 209～247
[42] Kachanov Y. S., On the resonant nature of the breakdown of a laminar boundary layer. J. Fluid Mech. 1987, 184: 43～74
[43] Dryganets S V, Kachanov Y S, Levchenko V Y, et all. Resonant flow randomization in K-regime of boundary layer transition. J. Appl. Mech. Tech. Phys., 1990, 31(2):239～249
[44] Kachanov Y S. Physical mechanisms of laminar-boundary-layer transition. Annu. Rev. Fluid Mech., 1994, 26:411～482
[45] Bake S, Fernholz H, Kachanov Y S. Resemblance of K- and N-regimes of boundary-layer transition at late stages. Eur. J. Mech. B/Fluids, 2000, 19(1):1～22
[46] Herbert, Th., Secondary instability of plane channel flow to subharmonic three-dimensional disturbances. Phys. Fluids, 1983, 26: 871～874
[47] Herbert, Th., Subharmonic three-dimensional disturbances in unstable plane shear flow. AIAA Paper, 1983, No.83-1759
[48] Herbert, Th., Analysis of the subharmonic route to transition in boundary layers, AIAA Paper, 1984, No.84-0009
[49] 赵耕夫, 周恒. 平面 Poiseuille 流中弱非线性理论和二次失稳理论的关系. 应用数学和力学. 1988, 9(8): 571~576
[50] Kleiser L, Zang T A. Numerical simulation of transition in wall-bounded shear flows. Annu. Rev. Fluid Mech., 1991, 23:495～537
[51] Rist U, Fasel H. Direct numerical simulation of controlled transition in a flat-plate boundary layer. J. Fluid Mech., 1995, 298:211～248
[52] 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
[53] Borodulin V L, Gaponenko V R, Kachanov Y S, et al. Late-stage transitional boundary layer structures direct numerical simulation and experiment. Theoret. Comput. Fluid Dynamics, 2002, 15:317～337
[54] Chaoqun Liu., Zhining Liu., Multigrid mapping and box relaxation for simulation of the whole process of flow transition in 3d boundary layers. J. Comput. Phy. 1995, 119: 325～341
[55] Saric W.S., Kzlov V.V., Levchenko C.Y., AIAA Paper, 1984, No.84-0007(unpublished)
[56] Wang Z, Yeo K.S, Khoo B.C. Spatial direct numerical simulation of transitional boundary layer over compliant surfaces. Computers & Fluids, 2005, 34:1062～1095
[57] Bhaganagar K., Rempfer D., Lumley J. Direct numerical simulation of spatial transition to turbulence using fourth-order vertical velocity second-order vertical vorticity formulation. J. Comput. Phy., 2002,180: 200～228
[58] Berlin S., Lundbladh A., Henningson D., Spatial simulations of oblique transition in a boundary layer. Phys. Fluids, 1994, 6(6): 1949～1951
[59] Rai M. M., Moin P., Direct numerical simulation of transition and turbulence in a spatially evolving boundary layer. J. Comput. Phy. 1993, 109: 169～192
[60] Markus J. K., A robust high-resolution split-type compact fd scheme for spatial direct numerical simulation of boundary-layer transition. Applied Scientific Research, 1998, 59: 353～377
[61] Fasel H. F., Rist U. Konzelmann U., Numerical investigation of the three-dimensional development in boundary-layer transition. AIAA, 1990, (1):29～37
[62] Fasel H., Konzelmann U., Non-parallel stability of a flat-plate boundary layer using the complete navier-stokes equations. J. Fluid Mech. 1990, 221: 311～347
[63] Rist U., Kachanov Y. S., Numerical and experimental investigation of the K-regime of boundary-layer transition. Proc. IUTAM-Symp. Laminar-Turbulent Transition, Sendai, Japan, Springer-Verlag, 1995, 405～412
[64] Rist U., DNS of boundary-layer instability and transition using the spatial approach. Transitional Boundary Layers in Aeronautics, Royal Netherlands Academy of Arts and Sciences. Amsterdam, Elsevier, 1996, 99～111
[65] Meyer D., Rist U., Wagner S., DNS of the generation of secondary λ-vortices in a transitional boundary layer. Advances in Turbulence VII, Kluwer, 1998: 97～100
[66] Maucher U., Rist U., Wagner S., A refined method for DNS of transition in interacting boundary layers. AIAA, 98-2435:1～11
[67] Meyer D. G. W., Rist U., Borodulin V. I., Gaponenko V. R., Late-stage transitional boundary-layer structures. Direct numerical simulation and experiment. Laminar-Turbulent Transition, Proc. IUTAM-Symp Sedona, Sept. 13-17, 1999, Springer-Verlag, 2000,167～172
[68] 陈健, 崔桂香, 许春晓等. 空间发展圆管转捩的直接数值模拟. 力学学报, 2003, 35(1): 6～13
[69] 陈健, 崔桂香, 许春晓等. 圆管流动的二次转捩. 工程力学, 2003, 20(5): 37～41
[70] 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
[71] 王新军. 转捩中“breakdown”的内在机理及周期湍流场的涡粘系数[天津大学博士论文]. 2005
[72] 唐洪涛 不可压缩平板边界层从层流突变为湍流的机理及湍流特性[天津大学博士论文]. 2007
[73] 黄章峰. 超音速平板边界层从层流到湍流的转捩机理及湍流特性[天津大学博士论文]. 2006
[74] 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. 应用数学与力学, 2006, 27(4): 379～386
[75] 南京工学院, 华中工学院, 重庆大学, 天津大学合编. 粘性流体力学. 高等教育出版社, 1987: 130, 328
[76] 章梓雄, 董曾南. 粘性流体力学. 北京: 清华大学出版社, 1998, 125～249
[77] 陈懋章. 粘性流体动力学基础. 北京: 高等教育出版社, 2004, 192～372
[78] H. 史里希廷. 边界层理论(上册).北京: 科学出版社. 1988, 141～172
[79] H. 史里希廷. 边界层理论(下册).北京: 科学出版社. 1988, 519～669
[80] Karniadakis G. E., Israeli M., Orszag S. A., High-order splitting methods for the incompressible navier-stokes equations. J. Comput. Phy. 1991, 97:414～443
[81] 傅德薰, 马延文. 平面混合流拟序结构的直接数值模拟. 中国科学 A 辑, 1996, 26(7):659~664
[82] Lele S K. Compact finite difference schemes with spectral-like resolution. J Comp Phys, 1992, 103: 16～43
[83] 傅德薰、马延文. 计算流体力学. 北京: 高等教育出版社. 2002,118
[84] Hegeman L A, Young D M. Applied iterative methods. New York: Wiley, 1981
[85] 苏彩虹. 嵌边法出流条件在可压缩流直接数值模拟中的应用[天津大学硕士论文]. 2005
[86] Lundbladh, A., Schmid, P.J. Berlin, S. & Henningson, D. S. Simulation of bypass transition for spatially evolving disturbances. In AGARDCP-551, 1994, 18: 1～13
[87] 都志辉. 高性能计算并行编程技术—MPI 并行程序设计. 清华大学出版社. 2001.
[88] 白中治. 并行矩阵多分裂迭代算法的收敛速度与发散速度比较. 工程数学学报, 1994, 11(1): 99～102
[89] 白中治. 并行矩阵多分裂块松弛迭代算法. 计算数学, 1995, 3: 238～252
[90] 白中治. 异步并行非线性对称 Gauss-Seidel 迭代算法. 计算数学, 1998, 20(2): 187～200
[91] 毛承雄, 陆继明, 樊俊等. 并行仿真计算中的新交替迭代算法. 电力系统及其自动化学报. 1999, 11(3): 18～22
[92] 金承日, 丁效华, 张少太. 双曲型方程的有限差分并行迭代算法. 哈尔滨工业大学学报, 2002, 34(3): 340～343
[93] 陈凌, 陈云霞, 殷新春等. MPP 上的并行松弛迭代算法. 中国科学技术大学学报. 2002, 32(6): 732～737
[94] 向淑晃, 张生雷. 并行多分裂块松弛 TOR 迭代算法的收敛性. 云南大学学报(自然科学版). 2004, 26(1): 30～34
[95] 李春光, 徐成贤. 确定 SOR 最佳松弛因子的一个实用算法. 计算力学学报, 2002, 19(3): 199～302
[96] 蒋家羚, 王勇. 最优超松弛因子的一种确定方法及其在裂纹计算中的应用. 机械强度, 2002, 24(1): 133～135
[97] Frederic Ducros, Pierre Comte, Marcel Lesieur. Large-eddy simulation of transition to turbulence in a boundary layer developing spatially over a flat plate. J. Fluid Mech. 1996, 326:1～36
[98] 郭辉, 连淇祥, Yury S. Kachanov 等 边界层转捩过程三维扰动的发展和演化. 北京航空航天大学学报, 2003, 29(6): 501～504
[99] 黄章峰. 超音速边界层扰动演化的数值模拟研究[天津大学硕士论文]. 2003
[100] 吕祥翠. T-S 波与边界层尾缘相互作用的数值研究[天津大学硕士论文]. 2004
[101] 郭大为. 带壁面抽吸的槽道湍流大涡模拟[天津大学硕士论文]. 2005
[102] 刘惠枝, 舒宏纪. 边界层理论. 北京: 人民交通出版社, 1991
[103] 张东明. 湍流近壁区相干结构生成机理的研究[天津大学博士论文]. 2001
[104] 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
[105] 吕祥翠, 罗纪生, 宋志军. T-S 波与边界层尾缘相互作用的数值研究. 中国民航学院学报, 2006, 24(1): 1～6
[106] 李存标. 关于转捩边界层中流向涡的产生. 物理学报, 2001, 50(1):182～184
[107] 陆金甫, 关治. 偏微分方程数值解法. 北京: 清华大学出版社, 1987, 217～238
[108] E.O.布赖姆. 快速富里叶变换. 上海: 上海科学技术出版社, 1979, 1～33, 100～121, 165～183
[109] 冷建华. 傅立叶变换. 北京: 清华大学出版社, 2004, 1～23
[110] Douglas J, Gunn J E. A general formation of alternative direction methods: Part I, Parabolic and hyperbolic problems. Numeriche Methematik, 1964(6): 428～453
[111] Street, C. L. & Macaraeg, M. G. Spectral multi-domain for large-scale fluid dynamic simulations. Appl.Numer.Math. 1990(6):123～139
[112] Kloker, M., Konzelmann, U. & Fasel, H. Outflow boundary conditions for spatial Navier-Stokes simulations of transition boundary layers. AIAA J. 1993, 31(4): 620～628
[113] Spalart, P. R. & Watmuff, J. H. Experimental and numerical study of a turbulent boundary layer with pressure gradients. J. Fluid Mech. 1993, 249: 337～371
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