基于间断有限元的可压缩混合层数值模拟及其结构系综分析
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摘要
湍流因为具有多尺度、多自由度的内部复杂性和多样产生环境的外部复杂性两大复杂性而成为尚未解决的世纪难题,其中一个重要的流动类型是自由剪切湍流。可压缩混合层作为自由剪切湍流的一种,在科学研究上具有重要的理论价值而且在工程实际中具有广泛的应用。它是由两股具有不同速度的平行可压缩气流相互作用所形成,也称作可压缩剪切层、超声速混合层和超声速剪切层等。
本论文针对可压缩混合层流动,从流动显示和热线测量、数值模拟和结构系综理论三个角度对流场中的结构及其演化形态、不同物理过程以及压缩效应的定量刻画展开研究。
对于Mc=0.107和Mc=0.474的可压缩混合层平面激光米氏散射(PLMS)的实验结果,我们基于结构系综的思想,提出了等相位的结构提取方法。根据该方法我们提取了瞬时流场中的斜条纹结构;将斜条纹结构在样本空间进行系综平均得到了斜条纹的空间分布及其斜率沿流向的变化;在两个流动参数下,斜条纹结构的斜率随压缩性变化不明显,因此,我们认为斜条纹结构主要是高速气流在大尺度结构附近受到扰动而形成从混合层核心区域向外传播的马赫波和相干结构曲面效应造成的压缩\膨胀波系组成。基于流动显示的灰度图像,我们计算了混合层的厚度和增长率,得到了与文献相符合结果。流动显示图像的一维灰度信号能谱在对数坐标下存在斜率为- 5/3的惯性区,反映了流动中的被动标量具有多尺度行为。
在本文的研究中,我们发展了可压缩流动的间断有限元(DG)计算方法,并首次将DG方法用于可压缩湍流的研究,开展了可压缩混合层流动的二维和三维数值模拟。对于二维时间发展的混合层(Mc=0.2-0.8),我们统计了涡结构的几何特征及涡心轨迹在结构对并中的演化过程,分析了压缩性效应对对并过程的影响;在流场中具有相似的流动结构条件下,我们分析了可压缩耗散和压力胀量项的压缩性效应。对于二维空间发展的混合层(Mc=0.2,0.4.0.6,0.8),数值模拟得到的时序脉动信号具有与热线测量的复合信号具有相似的特征;在充分发展区域,混合层的核心的时序脉动信号的频谱在对数坐标下存在斜率为-4的惯性区。通过采用将涡量、速度散度和压力梯度相结合的流动显示技术,我们观察到了相干结构附近的压缩和膨胀区域,分析了压缩效应对大尺度结构的影响。数值模拟结果表明,在充分发展区域,混合层流动满足很好的自相似特征和线性的混合层厚度增长趋势,这也体现了当前计算的可靠性。利用Favre形式的平均动量方程和湍流雷诺应力方程,我们研究了流场中动量和能量的输运平衡情况,分析了其中不同物理过程的作用,并与文献结果进行了比较。在三维流动中,流动结构及其之间的相互作用更加丰富:在流动初始区域主要是展向涡卷结构,在下游由于展向不稳定性和拉伸机制流场中出现了肋状结构和发卡涡森林结构,这与零压力梯度的空间发展边界层流动相似。与二维结果相比较,三维流动的自相似性更为突出,一些高阶统计量也满足自相似性。动量方程的输运平衡表明,压力梯度项把流向的动量转移到了横向;而在雷诺应力方程中主要是产生项从平均流场中获取能量,与湍流扩散、压力扩散、粘性扩散和粘性耗散作用相平衡。
通过流场中的结构来预测平均流场的物理量一直都是流体力学研究的一个努力方向,在混合层流动中存在良好的拟序结构,这为我们建立基于结构的湍流封闭模型提供了可能。在本论文中,我们根据结构系综动力学的理论思想,对二维可压缩混合层的数值模拟结果进行了结构系综分析,通过将平均流场的控制方程分解成状态函数、微分算子和未封闭项三个部分,提出了基于连接状态函数和未封闭项的序函数来实现平均控制方程封闭的湍流模型框架。我们认为序函数的空间变化行为反映了流场中结构的统计信息,它源自传统的流动结构而又升华了传统的流动结构。通过学习数值模拟结果中的经验数据,我们对方程中出现的序函数的空间变化行为和随对流马赫数的变化特征进行了分析,完成了时间发展混合层中序函数的经验规律模拟,为完成可压缩流动在结构系综意义下的封闭,推动和建立新一代可压缩流动的湍流模型奠定了基础。
Turbulence is a century-old problem for its internal multi-scale and multi-freedom complexity and various burn environment complexities in daily life and engineering applications, in which one important kind of flow is free shear turbulence. Compressible mixing layers, as one typical kind of free shear turbulence, is resulted by interaction between two parallel compressible free streams with different flow velocities. It not only has theoretical meaning for it contains fundamental physical problems such as instability and vortex interaction, but also has large engineering applications in engine and supersonic combustion.
In this thesis, from the view of experiments, numerical simulation and structural ensemble dynamic theory, we try to give a comprehensive study of compressible mixing layers, and obtain the quantitative description of compressibility effect.
For the PLMS visualization results for planar compressible mixing layers with Mc=0.107 and Mc=0.474, an“equal phase method”is introduced to pick out the“oblique wave”structures in the instantaneous results of visualization. After an ensemble average for all the oblique wave structures in sample space, the variation of their slope along stream direction is obtained. The oblique wave structures are mainly composed of the Mach waves of high speed free stream that generated near the large scale structures, for their slope change little with compressibility. In addition, the visualization thickness and growth rate that normalized with the counterpart in incompressible cases of mixing layers are calculated, the results agree reference well. An interesting thing in the results of visualization is that a k=-5/3 slope is found in the energy spectrum of one dimensional gray scale signal along the stream direction, which means the multi-scale characteristics of the passive scalar field.
Discontinuous Galerkin method is the first time used to do compressible turbulence study for its convenience to handle strong discontinuous in flow field and high accuracy in smooth region, with which compressible mixing layers in two and three dimensions are simulated. With the statistical information of the geometry characteristics of vortexes and their vortex center trajectories in the case of Mc=0.2-0.8, the compressibility effect on vortex merging process in two dimensional temporal mixing layers is analyzed. It is found that the merging process in high Mc cases is different from that of low Mc cases with a vortex center“slip”phenomenon. While for the case of two dimensional spatially developing mixing layers, we obtain the time sequence of velocity fluctuations, which have similar characteristics as the signals obtained in hot-wire experiments. In fully developed region, the energy spectrum of velocity fluctuations in the center of mixing layers has an inertial region with the slope k=-4 in log-log frame. To visualize the compressibility effect on large scale structures in scompressible mixing layers, a hybrid visualization technique that is composed of vorticity, divergence of velocity and pressure gradient is utilized, with which expansion and compressed zones around large scale structures are shown clearly, as Mc increasing the compressed regions shrink till a shocklet is formed. In three dimensions case, Q2 is used instead of vorticity to identify structures. It is found that span-wise roller structures dominate the field in the initial region of mixing layer, rib structures, stream-wise vorticities and the forest of hairpin structures are resulted down-stream for vortex stretch mechanism and complex interaction among structures. The self-similarity in fully developed mixing layers is excellent both in our two and three dimensional numerical results, even some high order statistical variables such as skewness and flatness in three dimensional cases, which verify the credibility of current simulations. To analyze the function of different physical process in the flow, the control equations of momentum and turbulence Reynolds stress in Favre averaged form are used. It is found that, the function of pressure gradient is to transport momentum from stream direction to vertical direction; while for the case of turbulence Reynolds stress equation, the production gains energy from averaged field and feeds the turbulence diffusion, pressure diffusion, viscosity diffusion and viscosity dissipation.
There is a long time dream to predict mean flow field with structures in turbulence study, the well organized coherent structures in mixing layers make it is possible here. Recently, structural ensemble dynamic (SED) theory introduced by 2009 She et al. sheds light on the closure problem based on structures in turbulence study. In this theory, the averaged control equations are decomposed into state functions, differential calculator and unknown terms, after introduce order functions, which is the bridge that connect state functions and unknown terms, the control equations is closed. Here the meaning of order functions is to describe the statistical information of structures, which is from traditional flow structures but goes further. In this thesis, the frame of turbulence model based on SED is configured. With the study of numerical results, we analyze the spatial variation of order functions emerged in current control equations, obtain the empirical laws for some order functions in temporal mixing layers, and find the self-similarity of order functions in spatially developing mixing layers, which together makes it is possible to construct the next generate turbulence models based on SED for compressible mixing layers in the near future.
本论文针对可压缩混合层流动,从流动显示和热线测量、数值模拟和结构系综理论三个角度对流场中的结构及其演化形态、不同物理过程以及压缩效应的定量刻画展开研究。
对于Mc=0.107和Mc=0.474的可压缩混合层平面激光米氏散射(PLMS)的实验结果,我们基于结构系综的思想,提出了等相位的结构提取方法。根据该方法我们提取了瞬时流场中的斜条纹结构;将斜条纹结构在样本空间进行系综平均得到了斜条纹的空间分布及其斜率沿流向的变化;在两个流动参数下,斜条纹结构的斜率随压缩性变化不明显,因此,我们认为斜条纹结构主要是高速气流在大尺度结构附近受到扰动而形成从混合层核心区域向外传播的马赫波和相干结构曲面效应造成的压缩\膨胀波系组成。基于流动显示的灰度图像,我们计算了混合层的厚度和增长率,得到了与文献相符合结果。流动显示图像的一维灰度信号能谱在对数坐标下存在斜率为- 5/3的惯性区,反映了流动中的被动标量具有多尺度行为。
在本文的研究中,我们发展了可压缩流动的间断有限元(DG)计算方法,并首次将DG方法用于可压缩湍流的研究,开展了可压缩混合层流动的二维和三维数值模拟。对于二维时间发展的混合层(Mc=0.2-0.8),我们统计了涡结构的几何特征及涡心轨迹在结构对并中的演化过程,分析了压缩性效应对对并过程的影响;在流场中具有相似的流动结构条件下,我们分析了可压缩耗散和压力胀量项的压缩性效应。对于二维空间发展的混合层(Mc=0.2,0.4.0.6,0.8),数值模拟得到的时序脉动信号具有与热线测量的复合信号具有相似的特征;在充分发展区域,混合层的核心的时序脉动信号的频谱在对数坐标下存在斜率为-4的惯性区。通过采用将涡量、速度散度和压力梯度相结合的流动显示技术,我们观察到了相干结构附近的压缩和膨胀区域,分析了压缩效应对大尺度结构的影响。数值模拟结果表明,在充分发展区域,混合层流动满足很好的自相似特征和线性的混合层厚度增长趋势,这也体现了当前计算的可靠性。利用Favre形式的平均动量方程和湍流雷诺应力方程,我们研究了流场中动量和能量的输运平衡情况,分析了其中不同物理过程的作用,并与文献结果进行了比较。在三维流动中,流动结构及其之间的相互作用更加丰富:在流动初始区域主要是展向涡卷结构,在下游由于展向不稳定性和拉伸机制流场中出现了肋状结构和发卡涡森林结构,这与零压力梯度的空间发展边界层流动相似。与二维结果相比较,三维流动的自相似性更为突出,一些高阶统计量也满足自相似性。动量方程的输运平衡表明,压力梯度项把流向的动量转移到了横向;而在雷诺应力方程中主要是产生项从平均流场中获取能量,与湍流扩散、压力扩散、粘性扩散和粘性耗散作用相平衡。
通过流场中的结构来预测平均流场的物理量一直都是流体力学研究的一个努力方向,在混合层流动中存在良好的拟序结构,这为我们建立基于结构的湍流封闭模型提供了可能。在本论文中,我们根据结构系综动力学的理论思想,对二维可压缩混合层的数值模拟结果进行了结构系综分析,通过将平均流场的控制方程分解成状态函数、微分算子和未封闭项三个部分,提出了基于连接状态函数和未封闭项的序函数来实现平均控制方程封闭的湍流模型框架。我们认为序函数的空间变化行为反映了流场中结构的统计信息,它源自传统的流动结构而又升华了传统的流动结构。通过学习数值模拟结果中的经验数据,我们对方程中出现的序函数的空间变化行为和随对流马赫数的变化特征进行了分析,完成了时间发展混合层中序函数的经验规律模拟,为完成可压缩流动在结构系综意义下的封闭,推动和建立新一代可压缩流动的湍流模型奠定了基础。
Turbulence is a century-old problem for its internal multi-scale and multi-freedom complexity and various burn environment complexities in daily life and engineering applications, in which one important kind of flow is free shear turbulence. Compressible mixing layers, as one typical kind of free shear turbulence, is resulted by interaction between two parallel compressible free streams with different flow velocities. It not only has theoretical meaning for it contains fundamental physical problems such as instability and vortex interaction, but also has large engineering applications in engine and supersonic combustion.
In this thesis, from the view of experiments, numerical simulation and structural ensemble dynamic theory, we try to give a comprehensive study of compressible mixing layers, and obtain the quantitative description of compressibility effect.
For the PLMS visualization results for planar compressible mixing layers with Mc=0.107 and Mc=0.474, an“equal phase method”is introduced to pick out the“oblique wave”structures in the instantaneous results of visualization. After an ensemble average for all the oblique wave structures in sample space, the variation of their slope along stream direction is obtained. The oblique wave structures are mainly composed of the Mach waves of high speed free stream that generated near the large scale structures, for their slope change little with compressibility. In addition, the visualization thickness and growth rate that normalized with the counterpart in incompressible cases of mixing layers are calculated, the results agree reference well. An interesting thing in the results of visualization is that a k=-5/3 slope is found in the energy spectrum of one dimensional gray scale signal along the stream direction, which means the multi-scale characteristics of the passive scalar field.
Discontinuous Galerkin method is the first time used to do compressible turbulence study for its convenience to handle strong discontinuous in flow field and high accuracy in smooth region, with which compressible mixing layers in two and three dimensions are simulated. With the statistical information of the geometry characteristics of vortexes and their vortex center trajectories in the case of Mc=0.2-0.8, the compressibility effect on vortex merging process in two dimensional temporal mixing layers is analyzed. It is found that the merging process in high Mc cases is different from that of low Mc cases with a vortex center“slip”phenomenon. While for the case of two dimensional spatially developing mixing layers, we obtain the time sequence of velocity fluctuations, which have similar characteristics as the signals obtained in hot-wire experiments. In fully developed region, the energy spectrum of velocity fluctuations in the center of mixing layers has an inertial region with the slope k=-4 in log-log frame. To visualize the compressibility effect on large scale structures in scompressible mixing layers, a hybrid visualization technique that is composed of vorticity, divergence of velocity and pressure gradient is utilized, with which expansion and compressed zones around large scale structures are shown clearly, as Mc increasing the compressed regions shrink till a shocklet is formed. In three dimensions case, Q2 is used instead of vorticity to identify structures. It is found that span-wise roller structures dominate the field in the initial region of mixing layer, rib structures, stream-wise vorticities and the forest of hairpin structures are resulted down-stream for vortex stretch mechanism and complex interaction among structures. The self-similarity in fully developed mixing layers is excellent both in our two and three dimensional numerical results, even some high order statistical variables such as skewness and flatness in three dimensional cases, which verify the credibility of current simulations. To analyze the function of different physical process in the flow, the control equations of momentum and turbulence Reynolds stress in Favre averaged form are used. It is found that, the function of pressure gradient is to transport momentum from stream direction to vertical direction; while for the case of turbulence Reynolds stress equation, the production gains energy from averaged field and feeds the turbulence diffusion, pressure diffusion, viscosity diffusion and viscosity dissipation.
There is a long time dream to predict mean flow field with structures in turbulence study, the well organized coherent structures in mixing layers make it is possible here. Recently, structural ensemble dynamic (SED) theory introduced by 2009 She et al. sheds light on the closure problem based on structures in turbulence study. In this theory, the averaged control equations are decomposed into state functions, differential calculator and unknown terms, after introduce order functions, which is the bridge that connect state functions and unknown terms, the control equations is closed. Here the meaning of order functions is to describe the statistical information of structures, which is from traditional flow structures but goes further. In this thesis, the frame of turbulence model based on SED is configured. With the study of numerical results, we analyze the spatial variation of order functions emerged in current control equations, obtain the empirical laws for some order functions in temporal mixing layers, and find the self-similarity of order functions in spatially developing mixing layers, which together makes it is possible to construct the next generate turbulence models based on SED for compressible mixing layers in the near future.
引文
[1]佘振苏,什么是湍流世纪难题.教育部,科学技术部,中国科学院,国家自然科学基金委员会. 10000个科学难题:物理学卷.北京:科学出版社, 2009.
[2] Brown, G.,A. Roshko, On density effects and large structure in turbulent mixing layers. Journal of Fluid Mechanics, 1974, 64(4): 775-816.
[3] Chinzei, N., G. Masuya, T. Komuro, A. Murakami, K. Kudou, Spreading of two-stream supersonic turbulent mixing layers. Physics of Fluids, 1986, 29: 1345-1347.
[4] Papamoschou, D.,A. Roshko, The compressible turbulent shear layer: an experimental study. J. Fluid Mech, 1988, 197: 453-477.
[5] Clemens, N.T.,M.G. Mungal, Large-scale structure and entrainment in the supersonic mixing layer. Journal of Fluid Mechanics, 1995(284): 171-216.
[6] Elliott, G.,M. Samimy, Compressibility effects in free shear layers. Physics of Fluids A: Fluid Dynamics, 1990, 2: 1231-1240.
[7] Goebel, S.,J. Dutton, Experimental study of compressible turbulent mixing layers. AIAA journal, 1991, 29(4): 538-546.
[8] Rossmann, T., M. Mungal, R. Hanson, Evolution and growth of large-scale structures in high compressibility mixing layers*. Journal of Turbulence, 2002, 3(1): 9-9.
[9] Bernal, L.,A. Roshko, Streamwise vortex structure in plane mixing layers. Journal of Fluid Mechanics, 1986, 170: 499-525.
[10] Michalke, A., On the inviscid instability of the hyperbolic- tangent velocity profile(Inviscid instability, streamlines and vorticity distribution of disturbed hyperbolic tangent velocity profile). Journal of Fluid Mechanics, 1964, 19: 543-556.
[11] Ragab, S.A.,J.L. Wu, Linear instabilities in two-dimensional compressible mixing layers. Physics of Fluids A: Fluid Dynamics, 1989, 1: 957-966.
[12] Jackson, T.,C. Grosch, Inviscid spatial stability of a compressible mixing layer. Journal of Fluid Mechanics, 1989, 208: 609-637.
[13] Zhuang, M., P. Dimotakis, T. Kubota, The effect of walls on a spatially growingsupersonic shear layer. Physics of Fluids A: Fluid Dynamics, 1990, 2(4): 599-604.
[14] Lu, G.,S. Lele, On the density ratio effect on the growth rate of a compressible mixing layer. Physics of Fluids, 1994, 6(2): 1073-1075.
[15] Day, M.J., W.C. Reynolds, N.N. Mansour, The structure of the compressible reacting mixing layer: Insights from linear stability analysis. Physics of Fluids, 1998, 10(4): 993-1007.
[16] Michalke, A., On spatially growing disturbances in an inviscid shear layer. J. Fluid Mech, 1965, 23(3): 521-544.
[17] Monkewitz, P.,P. Huerre, Influence of the velocity ratio on the spatial instability of mixing layers. Physics of Fluids, 1982, 25(7): 1137-1143.
[18] Corcos, G.M.,F.S. Sherman, Vorticity Concentration and the Dynamics of Unstable Free Shear Layers. J. Fluid Mech., 1976, 73: 241-264.
[19] Lasheras, J.,H. Choi, Three-dimensional instability of a plane free shear layer: an experimental study of the formation and evolution of streamwise vortices. Journal of Fluid Mechanics, 1988, 189: 53-86.
[20] Jimenez, J., A Spanwise Structure in the Plane Mixing Layer. J. Fluid Mech., 1983., 132: 319-345.
[21] Rogers, M.,R. Moser, Spanwise scale selection in plane mixing layers. Journal of Fluid Mechanics, 1993, 247: 321-337.
[22] Moser, R.,M. Rogers, The three-dimensional evolution of a plane mixing layer: pairing and transition to turbulence. Journal of Fluid Mechanics, 1993, 247: 275-320.
[23] Moser, R., M. Rogers, D. Ewing, Self-similarity of time-evolving plane wakes. Journal of Fluid Mechanics, 1998, 367: 255-289.
[24] Bradshaw, P., The Effect of Initial Conditions on the Development of a Free Shear layer. J.Fluid Mech., 1966, 26: 225-236.
[25] Konrad, J.H., An Experimental Investigation of mixing in Two-Dimensional Turbulent Shear Flows with Application to Diffusion-Limited Chemical Reactions Ph.D. thesis California Institute of Technology, 1977.
[26] Breidenthal, R.E., Structure in Turbulent Mixing Layers and Wakes Using a Chemical Reaction. J.Fluid Mech., 1981, 125: 397-409.
[27] Rudy, D.,S. Birch, Mean flow development and surface heating for an attaching compressible free shear layer. Journal of Spacecraft and Rockets, 1974, 11: 348-351.
[28] Slessor, M.D., M. Zhuang, P.E. Dimotakis, Turbulent shear-layer mixing: growth-rate compressibility scaling. Journal of Fluid Mechanics, 2000, 414: 35-45.
[29] Bogdanoff, D., Compressibility effects in turbulent shear layers. AIAA journal, 1983, 21(6): 926-927.
[30] Sarkar, S., The stabilizing effect of compressibility in turbulent shear flow. Journal of Fluid Mechanics, 1995, 282: 163-186.
[31] Tumin, A., Optimal streamwise vortices intended for supersonic mixing enhancement. AIAA journal, 2003, 41(8): 1542-1546.
[32] Clemens, N.,M. Mungal, Two- and three-dimensional effects in the supersonic mixing layer. AIAA journal, 1992, 30(4): 973-981.
[33] Clemens, N.,M. Mungal, A planar Mie scattering technique for visualizing supersonic mixing flows. Experiments in fluids, 1991, 11(2): 175-185.
[34] Rossmann, T., M. Mungal, R. Hanson, Acetone PLIF and schlieren imaging of high compressibility mixing layers. AIAA Paper, 2001, 2001-0290.
[35] Passot, T.,A. Pouquet, Numerical simulation of compressible homogeneous flows in the turbulent regime. J. Fluid Mech, 1987, 181: 441-466.
[36] Papamoschou, D., Evidence of shocklets in a counterflow supersonic shear layer. Physics of Fluids, 1995, 7(2): 233-235.
[37] Lee, S., S. Lele, P. Moin, Eddy shocklets in decaying compressible turbulence. Physics of Fluids A, 1991, 3(4): 657-664.
[38] Lele, S.K., Direct numerical simulation of compressible free shear flows. AIAA-89-0374, 1989
[39] Sandham, N.,H. Yee, A numerical study of a class of TVD schemes for compressible mixing layers. NASA TM 102194, 1989.
[40] Vreman, B., H. Kuerten, B. Geurts, Shocks in direct numerical simulation of the confined three-dimensional mixing layer. Physics of Fluids, 1995, 7(9): 2105-2107.
[41] Vreman, A., N. Sandham, K. Luo, Compressible mixing layer growth rate and turbulence characteristics. Journal of Fluid Mechanics, 1996, 320: 235-258.
[42]傅德薰,马延文,时间发展平面混合流的三维演化.力学学报, 1998, 30(2): 130-137.
[43] Sabin, C., An analytical and experimental study of the plane, incompressible, turbulent free shear layer with arbitrary velocity ratio and pressure gradient ASME, TRANSACTIONS, SERIES D-JOURNAL OF BASIC ENGINEERING 1965, 87: 421-428.
[44] Abramovich, G., T. Girshovich, S. Krasheninnikov, A. Sekundov, I. Smirnova, The theory of turbulent jets. Moscow Izdatel Nauka, 1984.
[45] Dimotakis, P., Two-dimensional shear-layer entrainment. AIAA journal, 1986, 24(11): 1791-1796.
[46] Barone, M., W. Oberkampf, F. Blottner, Validation case study: Prediction of compressible turbulent mixing layer growth rate. AIAA journal, 2006, 44(7): 1488.
[47] Bradshaw, P., Compressibility effects on free shear layers. In Conference on Complex Turbulent Flows. Stanford University, 1981, 1: 364–368.
[48] Goebel, S.G.,J.C. Dutton, Velocity measurements of compressible, turbulent mixing layers AIAA P.-1990-0709, 1990
[49] Sarkar, S., G. Erlebacher, M. Hussaini, H. Kreiss, The analysis and modelling of dilatational terms in compressible turbulence. Journal of Fluid Mechanics, 1991, 227: 473-493.
[50] Fujiwara, H., Y. Matsuo, C. Arakawa, A turbulence model for the pressure-strain correlation term accounting for compressibility effects. International Journal of Heat and Fluid Flow, 2000, 21(3): 354-358.
[51] Pantano, C.,S. Sarkar, A study of compressibility effects in the high-speed turbulent shear layer using direct simulation. Journal of Fluid Mechanics, 2002, 451: 329-371.
[52] Huang, S.,S. Fu, Modelling of pressure-strain correlation in compressible turbulent flow. Acta Mechanica Sinica, 2008, 24(1): 37-43.
[53] Gruber, M., N. Messersmith, J. Dutton, Three-dimensional velocity field in a compressible mixing layer. AIAA journal, 1993, 31(11): 2061-2067.
[54] Chang, C.T., C.J. Marek, C. Wey, R.A. Jones, M.J. Smith, Turbulence Measurement in a Reacting and Non-reacting Shear layer at High Subsonic Mach Number 9thSymposium of Turbulent Shear Flows, Kyoto, Japan, 1993.
[55] Urban, W.,M. Mungal. A PIV study of compressible shear layers. 1998.
[56] Olsen, M.,J. Dutton, Planar velocity measurements in a weakly compressible mixing layer. Journal of Fluid Mechanics, 2003, 486: 51-77.
[57] Barre, S., C. Quine, J. Dussauge, Compressibility effects on the structure of supersonic mixing layers: experimental results. Journal of Fluid Mechanics, 1994, 259: 47-78.
[58] Elliott, G., M. Samimy, S. Arnette, The characteristics and evolution of large scale structures in compressible mixing layers. Physics of Fluids, 1995, 7(4): 864-876.
[59] Messersmith, N.,J. Dutton, Characteristic features of large structures in compressible mixing layers. AIAA journal, 1996, 34(9): 1814-1821.
[60] Freund, J., S. Lele, P. Moin, Compressibility effects in a turbulent annular mixing layer. Part 1. Turbulence and growth rate. Journal of Fluid Mechanics, 2000, 421: 229-267.
[61] Freund, J., P. Moin, S. Lele, Compressibility effects in a turbulent annular mixing layer. Part 2. Mixing of a passive scalar. Journal of Fluid Mechanics, 2000, 421: 269-292.
[62] Simone, A., G. Coleman, C. Cambon, The effect of compressibility on turbulent shear flow: a rapid-distortion-theory and direct-numerical-simulation study. Journal of Fluid Mechanics, 1997, 330: 307-338.
[63] Sandham, N.D.,H.C. Yee, A numerical study of a class of TVD schemes for compressible mixing layers. NASA Thechnical Memorandum 102194, 1989.
[64] Breidenthal, R., The sonic eddy-a model for compressible turbulence. AIAA P90-0495, 1990
[65] Rogers, M.M.,R.D. Moser, Direct simulation of a self-smilar turbulent mixing layer. Phys. Fluids, 1994, 6(2): 903-923.
[66] Miller, J.,J. Wang, Passive scalars, random flux, and chiral phase fluids. Physical review letters 1996, 76(9): 1461-1464.
[67] Dutton, J.C., R.F. Burr, S.G. Goebel, N.L. Messersmith, Compressibility and Mixing in Turbulent Free Shear Layers. 12th Symposium on Turbulence, Rolla, Mo., 1990
[68] Naughton, J., L. CATTAFESTA III, G. Settles, An experimental study of compressible turbulent mixing enhancement in swirling jets. Journal of Fluid Mechanics, 1997, 330: 271-305.
[69] Dowling, D.R.,P.E. Dimotakis, Similarity of the concentration field of gas-phase turbulent jets. J. Fluid Mech, 1990, 218: 109-142.
[70]刘君,高树椿,超声速自由剪切流动的数值模拟和理论分析.空气动力学学报, 1995, 13(2): 152-158.
[71]沈利平,林建忠,傅斌,随时间发展的三维混合层流场的研究.力学学报, 1996, 28(1): 1-7.
[72] Fu, D.,Y. Ma, A high order accurate difference scheme for complex flow fields. Journal of Computational Physics, 1997, 134(1): 1-15.
[73]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), 1996 26(7): 657-664.
[74]邵雪明,林建忠,余钊圣,混合层中四涡合并过程的数值研究.科技通报, 2000 16(3): 169-180.
[75]李沁,张涵信,高树椿,关于超声速剪切流动的数值模拟.空气动力学学报, 2000,增刊(18): 67-77.
[76] Li, Q.,S. Fu, Numerical simulation of high-speed planar mixing layer. Computers & Fluids, 2003, 32(10): 1357-1377.
[77]沈清,王强,庄逢甘,超声速平面剪切层声辐射涡模态数值分析.力学学报, 2007, 39(1): 7-14.
[78]苗文博,程晓丽,王强,可压缩燃烧反应转捩混合层直接数值模拟.力学学报, 2008, 40(1): 114-120.
[79]潘宏禄,史可天,马汉东, DNS/LES方法在剪切湍流模拟中的应用.空气动力学学报, 2009, 27(4): 444-450.
[80]罗纪生,吕祥翠,超音速混合层稳定性分析及增强混合的研究.力学学报, 2004 36(2): 202-207.
[81]王强,傅德薰,马延文,可压缩平面混合层稳定性数值计算.计算物理, 1997 14((4,5)): 413-416.
[82]熊红亮,崔尔杰,可压缩混合层流动特性与增混措施研究. 2003空气动力学前沿研究学术研讨会, 2003.
[83]甘才俊,何枫,杨京龙,王金勇,熊红亮,李烺,可压缩混合层中大尺度结构的演化.第六届全国流动显示学术交流会, 2006.
[84]易仕和,赵玉新,何霖,超声速混合层转捩过程时空特性的实验研究第十二届全国激波与激波管学术会议论文集, 2006.
[85]赵玉新,超声速混合层时空结构的实验研究.国防科学技术大学博士毕业论文, 2008.
[86]陈军,王铁进,时晓天,王克,佘振苏,超声速混合层大尺度流动结构显示实验研究.第六届全国流动显示学术交流会论文集.福建武夷山;, 2006.
[87]时晓天,陈军,王铁进,王克,佘振苏,可压缩平面混合层中准周期条带结构的提取气体物理-理论与应用, 2006, 1(1): 44-48.
[88] Chen, J., T.-J. Wang, X.-T. Shi, Wavy Structures in Compressible Planar Mixing Layer. The 9th Asian Symposium on Visualization. Hong Kong, China;, 2007.
[89]王铁进,平面超声速混合层实验技术研究.北京大学博士毕业论文, 2009.
[90]林建忠,余钊圣,沈利平,混合层流场的特性及其大涡结构的研究.力学进展, 1997 27(2): 217-231.
[91]杨武兵,庄逢甘,沈清,可压缩混合层流动近十年研究进展.力学进展, 2008, 38(1): 62-76.
[92] Sandham, N.D.,W.C. Reynolds, Some inlet-plane effect on the numerically simulated spatially-developing mixing layer. Turbulent shear Flows 6. Berlin: Springer- Verlag, 1989 441-454.
[93] Comte, P., M. Lesieur, E. Lamballais, Large and small scale stirring of vorticity and a passive scalar in a 3D temporal mixing layer. Physics of Fluids A, 1992, 4: 2761.
[94] McCracken, M.E.,J. Abraham, Simulations of gas-gas mixing layers with a lattice Boltzmann binary fluid model. International Journal of Modern Physics C, 2005, 16(4): 533-547.
[95] Bernard, P., Grid-free simulation of the spatially growing turbulent mixing layer. AIAA journal, 2008, 46(7): 1725-1737.
[96] Knaepen, B., O. Debliquy, D. Carati, Direct numerical simulation and large-eddysimulation of a shear-free mixing layer. Journal of Fluid Mechanics, 2004, 514: 153-172.
[97] Lee, S., S. Lele, P. Moin, Eddy shocklets in decaying compressible turbulence. Physics of Fluids A: Fluid Dynamics, 1991, 3: 657.
[98] Lu, P.J.,K.C. Wu, Numerical investigation on the structure of a confined supersonic mixing layer. Physics of Fluids A, 1991, 3: 3063-3079.
[99] Yee, H.C., N.D. Sandham, M.J. Djomehri, Low-dissipative high-order shock-capturing methods using characteristic-based filters. Journal of Computational Physics, 1999, 150(1): 199-238.
[100] Yee, H.C., B. Sjogreen, N.D. Sandham, A. Hadjadj, Progress in the development of a class of efficient low dissipative high order shock-capturing methods Proceeding of the Symposium in Computational Fluid Dynamics for the 21st Century, 2000
[101] Kourta, A.,R. Sauvage, Computation of supersonic mixing layers. Physics of Fluids, 2002, 14(11): 3790-3797.
[102] Baumann, C.,J. Oden, A discontinuous hp finite element method for convection-diffusion problems. Computer Methods in Applied Mechanics and Engineering, 1999, 175(3-4): 311-341.
[103] Bassi, F.,S. Rebay, Numerical evaluation of two discontinuous Galerkin methods for the compressible Navier-Stokes equations. International Journal for Numerical Methods in Fluids, 2002, 40(1-2): 197-207.
[104] Van der Vegt, J.,H. Van der Ven, Space-time discontinuous Galerkin finite element method with dynamic grid motion for inviscid compressible flows. J. Comput. Phys, 2002, 182: 546-585.
[105] Hovorka, R., V. Canonico, L. Chassin, U. Haueter, M. Massi-Benedetti, M. Federici, T. Pieber, H. Schaller, L. Schaupp, T. Vering, Nonlinear model predictive control of glucose concentration in subjects with type 1 diabetes. PHYSIOLOGICAL MEASUREMENT 2010, 7: 49.
[106] Wang, T., X. Shi, J. Chen, Z. She, Multi-sclae structures in compressible turbulent mixing layers. Modern Physics Letters B, 2010, 24(13): 1429-1432.
[107] Reed W.D.,Hill T.R., Triangular mesh methods for the neutron transport equationLos Alamos Scientific Laboratory Report LA-UR-73-479, 1973.
[108] Lesaint, P.,P. Raviart, On a finite element method for solving the neutron transport equation. Mathematical aspects of finite elements in partial differential equations, 1974: 89-123.
[109] Johnson, C.,J. Pitk ranta, Convergence of a fully discrete scheme for two-dimensional neutron transport. SIAM Journal on Numerical Analysis, 1983: 951-966.
[110] Peterson, T., A note on the convergence of the discontinuous Galerkin method for a scalar hyperbolic equation. SIAM Journal on Numerical Analysis, 1991, 28(1): 133-140.
[111] Qun, L.,Z. Aihui, Convergence of the discontinuous Galerkin method for a scalar hyperbolic equation. Acta Mathematica Scientia, 1993, 13(2): 207-210.
[112] Cockburn B.,Shu C.-W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for scalar conservation laws II: General framework Math. Comp. , 1989, 52: 411—435.
[113] Cockburn B., Lin S.Y., Shu C.-W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: One dimensional systems. J. Comp. Phys. , 1989, 84: 90-113.
[114] Yan, J.,C. Shu, Local discontinuous Galerkin methods for partial differential equations with higher order derivatives. Journal of Scientific Computing, 2002, 17(1): 27-47.
[115] Spencer, B.W.,B.G. Jones, Statistical investigation of pressure and velocity fields in the turbulent two-stream mixing layer. AIAA Paper 71-613, 1971.
[116] Shu, C.-W.,S. Osher, Efficient implementation of essentially non-oscillatory shock-capturing schemes, II. Journal of Computational Physics, 1989, 83(1): 32-78.
[117] Cockburn, B.,C. Shu, Runge¨CKutta discontinuous Galerkin methods for convection-dominated problems. Journal of Scientific Computing, 2001, 16(3): 173-261.
[118] Yee, H.C., M. Vinokur, M.J. Djomehri, Entropy splitting and numerical dissipation. Journal of Computational Physics, 2000, 162(1): 33-81.
[119] Xu, Z.,C. Shu, Anti-diffusive finite difference weno methods for shallow water with transport of pollutant. Journal of Computational Mathematics, 2006, 24(3): 239-251.
[120] Thompson, K., Time-dependent boundary conditions for hyperbolic systems, II. Journal of Computational Physics, 1990, 89(2): 439-461.
[121] Poinsot, T.,S. Lelef, Boundary conditions for direct simulations of compressible viscous flows. Journal of Computational Physics, 1992, 101(1): 104-129.
[122]李启兵,应用BGK格式对可压缩混合层的数值研究.清华大学博士毕业论文, 2003.
[123] Stanley, S.,S. Sarkar, Simulations of spatially developing two-dimensional shear layers and jets. Theoretical and Computational Fluid Dynamics, 1997, 9(2): 121-147.
[124] Sandham, N.D., The effect of compressibility on vortex pairing. Physics of Fluids, 1994, 6(2): 1063-1072.
[125] Papamoschou, D.,S. Lele, Vortex induced disturbance field in a compressible shear layer. Physics of Fluids A: Fluid Dynamics, 1993, 5(6): 1412-1419.
[126] Shi, X., J. Chen, Z. She, Visualization of Compressibility Effect on Large Scale Structures in Spatially Developing Compressible Mixing Layers. J. Visualization (accepted), 2010.
[127] Hall, J., P. Dimotakis, H. Rosemann, Experiments in nonreacting compressible shear layers. AIAA journal, 1993, 31(12): 2247-2254.
[128] Wagner, R.D., Mean flow and turbulence measurements in a Mach 5 free shear layer NASA TN D7366, 1973
[129] Urban, W.D.,M.G. Mungal, Planar velocity measurements in compressible mixing layers. Journal of Fluid Mechanics, 2001, 431: 189-222.
[130] Hunt, J., A. Wray, P. Moin. Eddies, streams, and convergence zones in turbulent flows. 1988.
[131] Jeong, J.,F. Hussain, On the identification of a vortex. Journal of Fluid Mechanics, 1995, 285: 69-94.
[132] Barre, S., P. Braud, O. Chambres, J. Bonnet, Influence of inlet pressure conditions on supersonic turbulent mixing layers. Experimental thermal and fluid science, 1997, 14(1): 68-74.
[133] She, Z., N. Hu, Y. Wu, Structural ensemble dynamics based closure model for wall-bounded turbulent flow Acta Mechanica Sinica, 2009, 25(5): 731-736.
[134]佘振苏,面向工程应用的湍流基础研究-湍流世纪难题的突破与新一代湍流工程湍流模型探索.中科院力学所, 2009年8月19日[C].北京, 2009.
[135]张志雄,基于结构系综的湍流复杂系统研究.北京大学博士毕业论文, 2009.
[136] Kline, S., W. Reynolds, F. Schraub, P. Runstadler, The structure of turbulent boundary layers. Journal of Fluid Mechanics, 1967, 30(04): 741-773.
[137] She, Z.S., E. Jackson, S.A. Orszag, Intermittent votex structures in homogeneous isotropic turbulence Nature, 1990, 344, 15 March.
[138] Wu, X.,P. Moin, Direct numerical simulation of turbulence in a nominally zero-pressure-gradient flat-plate boundary layer. Journal of Fluid Mechanics, 2009, 630: 5-41.
[139] Kolmogrov, A.N., The local structure of turbulence in incompressible viscous ?uid for very large Reynolds. Dokl Akad Nauk SSSR, 1941, 30: 9-13.
[140] She, Z.S.,E. Leveque, Universal scaling laws in fully-developed turbulence. Phys Rev Lett 1994 72(3): 336-339.
[141]时晓天,裴杰,陈军,佘振苏,可压缩混合层流动的结构系综研究.中国力学学术大会.郑州, 2009.
[2] Brown, G.,A. Roshko, On density effects and large structure in turbulent mixing layers. Journal of Fluid Mechanics, 1974, 64(4): 775-816.
[3] Chinzei, N., G. Masuya, T. Komuro, A. Murakami, K. Kudou, Spreading of two-stream supersonic turbulent mixing layers. Physics of Fluids, 1986, 29: 1345-1347.
[4] Papamoschou, D.,A. Roshko, The compressible turbulent shear layer: an experimental study. J. Fluid Mech, 1988, 197: 453-477.
[5] Clemens, N.T.,M.G. Mungal, Large-scale structure and entrainment in the supersonic mixing layer. Journal of Fluid Mechanics, 1995(284): 171-216.
[6] Elliott, G.,M. Samimy, Compressibility effects in free shear layers. Physics of Fluids A: Fluid Dynamics, 1990, 2: 1231-1240.
[7] Goebel, S.,J. Dutton, Experimental study of compressible turbulent mixing layers. AIAA journal, 1991, 29(4): 538-546.
[8] Rossmann, T., M. Mungal, R. Hanson, Evolution and growth of large-scale structures in high compressibility mixing layers*. Journal of Turbulence, 2002, 3(1): 9-9.
[9] Bernal, L.,A. Roshko, Streamwise vortex structure in plane mixing layers. Journal of Fluid Mechanics, 1986, 170: 499-525.
[10] Michalke, A., On the inviscid instability of the hyperbolic- tangent velocity profile(Inviscid instability, streamlines and vorticity distribution of disturbed hyperbolic tangent velocity profile). Journal of Fluid Mechanics, 1964, 19: 543-556.
[11] Ragab, S.A.,J.L. Wu, Linear instabilities in two-dimensional compressible mixing layers. Physics of Fluids A: Fluid Dynamics, 1989, 1: 957-966.
[12] Jackson, T.,C. Grosch, Inviscid spatial stability of a compressible mixing layer. Journal of Fluid Mechanics, 1989, 208: 609-637.
[13] Zhuang, M., P. Dimotakis, T. Kubota, The effect of walls on a spatially growingsupersonic shear layer. Physics of Fluids A: Fluid Dynamics, 1990, 2(4): 599-604.
[14] Lu, G.,S. Lele, On the density ratio effect on the growth rate of a compressible mixing layer. Physics of Fluids, 1994, 6(2): 1073-1075.
[15] Day, M.J., W.C. Reynolds, N.N. Mansour, The structure of the compressible reacting mixing layer: Insights from linear stability analysis. Physics of Fluids, 1998, 10(4): 993-1007.
[16] Michalke, A., On spatially growing disturbances in an inviscid shear layer. J. Fluid Mech, 1965, 23(3): 521-544.
[17] Monkewitz, P.,P. Huerre, Influence of the velocity ratio on the spatial instability of mixing layers. Physics of Fluids, 1982, 25(7): 1137-1143.
[18] Corcos, G.M.,F.S. Sherman, Vorticity Concentration and the Dynamics of Unstable Free Shear Layers. J. Fluid Mech., 1976, 73: 241-264.
[19] Lasheras, J.,H. Choi, Three-dimensional instability of a plane free shear layer: an experimental study of the formation and evolution of streamwise vortices. Journal of Fluid Mechanics, 1988, 189: 53-86.
[20] Jimenez, J., A Spanwise Structure in the Plane Mixing Layer. J. Fluid Mech., 1983., 132: 319-345.
[21] Rogers, M.,R. Moser, Spanwise scale selection in plane mixing layers. Journal of Fluid Mechanics, 1993, 247: 321-337.
[22] Moser, R.,M. Rogers, The three-dimensional evolution of a plane mixing layer: pairing and transition to turbulence. Journal of Fluid Mechanics, 1993, 247: 275-320.
[23] Moser, R., M. Rogers, D. Ewing, Self-similarity of time-evolving plane wakes. Journal of Fluid Mechanics, 1998, 367: 255-289.
[24] Bradshaw, P., The Effect of Initial Conditions on the Development of a Free Shear layer. J.Fluid Mech., 1966, 26: 225-236.
[25] Konrad, J.H., An Experimental Investigation of mixing in Two-Dimensional Turbulent Shear Flows with Application to Diffusion-Limited Chemical Reactions Ph.D. thesis California Institute of Technology, 1977.
[26] Breidenthal, R.E., Structure in Turbulent Mixing Layers and Wakes Using a Chemical Reaction. J.Fluid Mech., 1981, 125: 397-409.
[27] Rudy, D.,S. Birch, Mean flow development and surface heating for an attaching compressible free shear layer. Journal of Spacecraft and Rockets, 1974, 11: 348-351.
[28] Slessor, M.D., M. Zhuang, P.E. Dimotakis, Turbulent shear-layer mixing: growth-rate compressibility scaling. Journal of Fluid Mechanics, 2000, 414: 35-45.
[29] Bogdanoff, D., Compressibility effects in turbulent shear layers. AIAA journal, 1983, 21(6): 926-927.
[30] Sarkar, S., The stabilizing effect of compressibility in turbulent shear flow. Journal of Fluid Mechanics, 1995, 282: 163-186.
[31] Tumin, A., Optimal streamwise vortices intended for supersonic mixing enhancement. AIAA journal, 2003, 41(8): 1542-1546.
[32] Clemens, N.,M. Mungal, Two- and three-dimensional effects in the supersonic mixing layer. AIAA journal, 1992, 30(4): 973-981.
[33] Clemens, N.,M. Mungal, A planar Mie scattering technique for visualizing supersonic mixing flows. Experiments in fluids, 1991, 11(2): 175-185.
[34] Rossmann, T., M. Mungal, R. Hanson, Acetone PLIF and schlieren imaging of high compressibility mixing layers. AIAA Paper, 2001, 2001-0290.
[35] Passot, T.,A. Pouquet, Numerical simulation of compressible homogeneous flows in the turbulent regime. J. Fluid Mech, 1987, 181: 441-466.
[36] Papamoschou, D., Evidence of shocklets in a counterflow supersonic shear layer. Physics of Fluids, 1995, 7(2): 233-235.
[37] Lee, S., S. Lele, P. Moin, Eddy shocklets in decaying compressible turbulence. Physics of Fluids A, 1991, 3(4): 657-664.
[38] Lele, S.K., Direct numerical simulation of compressible free shear flows. AIAA-89-0374, 1989
[39] Sandham, N.,H. Yee, A numerical study of a class of TVD schemes for compressible mixing layers. NASA TM 102194, 1989.
[40] Vreman, B., H. Kuerten, B. Geurts, Shocks in direct numerical simulation of the confined three-dimensional mixing layer. Physics of Fluids, 1995, 7(9): 2105-2107.
[41] Vreman, A., N. Sandham, K. Luo, Compressible mixing layer growth rate and turbulence characteristics. Journal of Fluid Mechanics, 1996, 320: 235-258.
[42]傅德薰,马延文,时间发展平面混合流的三维演化.力学学报, 1998, 30(2): 130-137.
[43] Sabin, C., An analytical and experimental study of the plane, incompressible, turbulent free shear layer with arbitrary velocity ratio and pressure gradient ASME, TRANSACTIONS, SERIES D-JOURNAL OF BASIC ENGINEERING 1965, 87: 421-428.
[44] Abramovich, G., T. Girshovich, S. Krasheninnikov, A. Sekundov, I. Smirnova, The theory of turbulent jets. Moscow Izdatel Nauka, 1984.
[45] Dimotakis, P., Two-dimensional shear-layer entrainment. AIAA journal, 1986, 24(11): 1791-1796.
[46] Barone, M., W. Oberkampf, F. Blottner, Validation case study: Prediction of compressible turbulent mixing layer growth rate. AIAA journal, 2006, 44(7): 1488.
[47] Bradshaw, P., Compressibility effects on free shear layers. In Conference on Complex Turbulent Flows. Stanford University, 1981, 1: 364–368.
[48] Goebel, S.G.,J.C. Dutton, Velocity measurements of compressible, turbulent mixing layers AIAA P.-1990-0709, 1990
[49] Sarkar, S., G. Erlebacher, M. Hussaini, H. Kreiss, The analysis and modelling of dilatational terms in compressible turbulence. Journal of Fluid Mechanics, 1991, 227: 473-493.
[50] Fujiwara, H., Y. Matsuo, C. Arakawa, A turbulence model for the pressure-strain correlation term accounting for compressibility effects. International Journal of Heat and Fluid Flow, 2000, 21(3): 354-358.
[51] Pantano, C.,S. Sarkar, A study of compressibility effects in the high-speed turbulent shear layer using direct simulation. Journal of Fluid Mechanics, 2002, 451: 329-371.
[52] Huang, S.,S. Fu, Modelling of pressure-strain correlation in compressible turbulent flow. Acta Mechanica Sinica, 2008, 24(1): 37-43.
[53] Gruber, M., N. Messersmith, J. Dutton, Three-dimensional velocity field in a compressible mixing layer. AIAA journal, 1993, 31(11): 2061-2067.
[54] Chang, C.T., C.J. Marek, C. Wey, R.A. Jones, M.J. Smith, Turbulence Measurement in a Reacting and Non-reacting Shear layer at High Subsonic Mach Number 9thSymposium of Turbulent Shear Flows, Kyoto, Japan, 1993.
[55] Urban, W.,M. Mungal. A PIV study of compressible shear layers. 1998.
[56] Olsen, M.,J. Dutton, Planar velocity measurements in a weakly compressible mixing layer. Journal of Fluid Mechanics, 2003, 486: 51-77.
[57] Barre, S., C. Quine, J. Dussauge, Compressibility effects on the structure of supersonic mixing layers: experimental results. Journal of Fluid Mechanics, 1994, 259: 47-78.
[58] Elliott, G., M. Samimy, S. Arnette, The characteristics and evolution of large scale structures in compressible mixing layers. Physics of Fluids, 1995, 7(4): 864-876.
[59] Messersmith, N.,J. Dutton, Characteristic features of large structures in compressible mixing layers. AIAA journal, 1996, 34(9): 1814-1821.
[60] Freund, J., S. Lele, P. Moin, Compressibility effects in a turbulent annular mixing layer. Part 1. Turbulence and growth rate. Journal of Fluid Mechanics, 2000, 421: 229-267.
[61] Freund, J., P. Moin, S. Lele, Compressibility effects in a turbulent annular mixing layer. Part 2. Mixing of a passive scalar. Journal of Fluid Mechanics, 2000, 421: 269-292.
[62] Simone, A., G. Coleman, C. Cambon, The effect of compressibility on turbulent shear flow: a rapid-distortion-theory and direct-numerical-simulation study. Journal of Fluid Mechanics, 1997, 330: 307-338.
[63] Sandham, N.D.,H.C. Yee, A numerical study of a class of TVD schemes for compressible mixing layers. NASA Thechnical Memorandum 102194, 1989.
[64] Breidenthal, R., The sonic eddy-a model for compressible turbulence. AIAA P90-0495, 1990
[65] Rogers, M.M.,R.D. Moser, Direct simulation of a self-smilar turbulent mixing layer. Phys. Fluids, 1994, 6(2): 903-923.
[66] Miller, J.,J. Wang, Passive scalars, random flux, and chiral phase fluids. Physical review letters 1996, 76(9): 1461-1464.
[67] Dutton, J.C., R.F. Burr, S.G. Goebel, N.L. Messersmith, Compressibility and Mixing in Turbulent Free Shear Layers. 12th Symposium on Turbulence, Rolla, Mo., 1990
[68] Naughton, J., L. CATTAFESTA III, G. Settles, An experimental study of compressible turbulent mixing enhancement in swirling jets. Journal of Fluid Mechanics, 1997, 330: 271-305.
[69] Dowling, D.R.,P.E. Dimotakis, Similarity of the concentration field of gas-phase turbulent jets. J. Fluid Mech, 1990, 218: 109-142.
[70]刘君,高树椿,超声速自由剪切流动的数值模拟和理论分析.空气动力学学报, 1995, 13(2): 152-158.
[71]沈利平,林建忠,傅斌,随时间发展的三维混合层流场的研究.力学学报, 1996, 28(1): 1-7.
[72] Fu, D.,Y. Ma, A high order accurate difference scheme for complex flow fields. Journal of Computational Physics, 1997, 134(1): 1-15.
[73]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), 1996 26(7): 657-664.
[74]邵雪明,林建忠,余钊圣,混合层中四涡合并过程的数值研究.科技通报, 2000 16(3): 169-180.
[75]李沁,张涵信,高树椿,关于超声速剪切流动的数值模拟.空气动力学学报, 2000,增刊(18): 67-77.
[76] Li, Q.,S. Fu, Numerical simulation of high-speed planar mixing layer. Computers & Fluids, 2003, 32(10): 1357-1377.
[77]沈清,王强,庄逢甘,超声速平面剪切层声辐射涡模态数值分析.力学学报, 2007, 39(1): 7-14.
[78]苗文博,程晓丽,王强,可压缩燃烧反应转捩混合层直接数值模拟.力学学报, 2008, 40(1): 114-120.
[79]潘宏禄,史可天,马汉东, DNS/LES方法在剪切湍流模拟中的应用.空气动力学学报, 2009, 27(4): 444-450.
[80]罗纪生,吕祥翠,超音速混合层稳定性分析及增强混合的研究.力学学报, 2004 36(2): 202-207.
[81]王强,傅德薰,马延文,可压缩平面混合层稳定性数值计算.计算物理, 1997 14((4,5)): 413-416.
[82]熊红亮,崔尔杰,可压缩混合层流动特性与增混措施研究. 2003空气动力学前沿研究学术研讨会, 2003.
[83]甘才俊,何枫,杨京龙,王金勇,熊红亮,李烺,可压缩混合层中大尺度结构的演化.第六届全国流动显示学术交流会, 2006.
[84]易仕和,赵玉新,何霖,超声速混合层转捩过程时空特性的实验研究第十二届全国激波与激波管学术会议论文集, 2006.
[85]赵玉新,超声速混合层时空结构的实验研究.国防科学技术大学博士毕业论文, 2008.
[86]陈军,王铁进,时晓天,王克,佘振苏,超声速混合层大尺度流动结构显示实验研究.第六届全国流动显示学术交流会论文集.福建武夷山;, 2006.
[87]时晓天,陈军,王铁进,王克,佘振苏,可压缩平面混合层中准周期条带结构的提取气体物理-理论与应用, 2006, 1(1): 44-48.
[88] Chen, J., T.-J. Wang, X.-T. Shi, Wavy Structures in Compressible Planar Mixing Layer. The 9th Asian Symposium on Visualization. Hong Kong, China;, 2007.
[89]王铁进,平面超声速混合层实验技术研究.北京大学博士毕业论文, 2009.
[90]林建忠,余钊圣,沈利平,混合层流场的特性及其大涡结构的研究.力学进展, 1997 27(2): 217-231.
[91]杨武兵,庄逢甘,沈清,可压缩混合层流动近十年研究进展.力学进展, 2008, 38(1): 62-76.
[92] Sandham, N.D.,W.C. Reynolds, Some inlet-plane effect on the numerically simulated spatially-developing mixing layer. Turbulent shear Flows 6. Berlin: Springer- Verlag, 1989 441-454.
[93] Comte, P., M. Lesieur, E. Lamballais, Large and small scale stirring of vorticity and a passive scalar in a 3D temporal mixing layer. Physics of Fluids A, 1992, 4: 2761.
[94] McCracken, M.E.,J. Abraham, Simulations of gas-gas mixing layers with a lattice Boltzmann binary fluid model. International Journal of Modern Physics C, 2005, 16(4): 533-547.
[95] Bernard, P., Grid-free simulation of the spatially growing turbulent mixing layer. AIAA journal, 2008, 46(7): 1725-1737.
[96] Knaepen, B., O. Debliquy, D. Carati, Direct numerical simulation and large-eddysimulation of a shear-free mixing layer. Journal of Fluid Mechanics, 2004, 514: 153-172.
[97] Lee, S., S. Lele, P. Moin, Eddy shocklets in decaying compressible turbulence. Physics of Fluids A: Fluid Dynamics, 1991, 3: 657.
[98] Lu, P.J.,K.C. Wu, Numerical investigation on the structure of a confined supersonic mixing layer. Physics of Fluids A, 1991, 3: 3063-3079.
[99] Yee, H.C., N.D. Sandham, M.J. Djomehri, Low-dissipative high-order shock-capturing methods using characteristic-based filters. Journal of Computational Physics, 1999, 150(1): 199-238.
[100] Yee, H.C., B. Sjogreen, N.D. Sandham, A. Hadjadj, Progress in the development of a class of efficient low dissipative high order shock-capturing methods Proceeding of the Symposium in Computational Fluid Dynamics for the 21st Century, 2000
[101] Kourta, A.,R. Sauvage, Computation of supersonic mixing layers. Physics of Fluids, 2002, 14(11): 3790-3797.
[102] Baumann, C.,J. Oden, A discontinuous hp finite element method for convection-diffusion problems. Computer Methods in Applied Mechanics and Engineering, 1999, 175(3-4): 311-341.
[103] Bassi, F.,S. Rebay, Numerical evaluation of two discontinuous Galerkin methods for the compressible Navier-Stokes equations. International Journal for Numerical Methods in Fluids, 2002, 40(1-2): 197-207.
[104] Van der Vegt, J.,H. Van der Ven, Space-time discontinuous Galerkin finite element method with dynamic grid motion for inviscid compressible flows. J. Comput. Phys, 2002, 182: 546-585.
[105] Hovorka, R., V. Canonico, L. Chassin, U. Haueter, M. Massi-Benedetti, M. Federici, T. Pieber, H. Schaller, L. Schaupp, T. Vering, Nonlinear model predictive control of glucose concentration in subjects with type 1 diabetes. PHYSIOLOGICAL MEASUREMENT 2010, 7: 49.
[106] Wang, T., X. Shi, J. Chen, Z. She, Multi-sclae structures in compressible turbulent mixing layers. Modern Physics Letters B, 2010, 24(13): 1429-1432.
[107] Reed W.D.,Hill T.R., Triangular mesh methods for the neutron transport equationLos Alamos Scientific Laboratory Report LA-UR-73-479, 1973.
[108] Lesaint, P.,P. Raviart, On a finite element method for solving the neutron transport equation. Mathematical aspects of finite elements in partial differential equations, 1974: 89-123.
[109] Johnson, C.,J. Pitk ranta, Convergence of a fully discrete scheme for two-dimensional neutron transport. SIAM Journal on Numerical Analysis, 1983: 951-966.
[110] Peterson, T., A note on the convergence of the discontinuous Galerkin method for a scalar hyperbolic equation. SIAM Journal on Numerical Analysis, 1991, 28(1): 133-140.
[111] Qun, L.,Z. Aihui, Convergence of the discontinuous Galerkin method for a scalar hyperbolic equation. Acta Mathematica Scientia, 1993, 13(2): 207-210.
[112] Cockburn B.,Shu C.-W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for scalar conservation laws II: General framework Math. Comp. , 1989, 52: 411—435.
[113] Cockburn B., Lin S.Y., Shu C.-W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: One dimensional systems. J. Comp. Phys. , 1989, 84: 90-113.
[114] Yan, J.,C. Shu, Local discontinuous Galerkin methods for partial differential equations with higher order derivatives. Journal of Scientific Computing, 2002, 17(1): 27-47.
[115] Spencer, B.W.,B.G. Jones, Statistical investigation of pressure and velocity fields in the turbulent two-stream mixing layer. AIAA Paper 71-613, 1971.
[116] Shu, C.-W.,S. Osher, Efficient implementation of essentially non-oscillatory shock-capturing schemes, II. Journal of Computational Physics, 1989, 83(1): 32-78.
[117] Cockburn, B.,C. Shu, Runge¨CKutta discontinuous Galerkin methods for convection-dominated problems. Journal of Scientific Computing, 2001, 16(3): 173-261.
[118] Yee, H.C., M. Vinokur, M.J. Djomehri, Entropy splitting and numerical dissipation. Journal of Computational Physics, 2000, 162(1): 33-81.
[119] Xu, Z.,C. Shu, Anti-diffusive finite difference weno methods for shallow water with transport of pollutant. Journal of Computational Mathematics, 2006, 24(3): 239-251.
[120] Thompson, K., Time-dependent boundary conditions for hyperbolic systems, II. Journal of Computational Physics, 1990, 89(2): 439-461.
[121] Poinsot, T.,S. Lelef, Boundary conditions for direct simulations of compressible viscous flows. Journal of Computational Physics, 1992, 101(1): 104-129.
[122]李启兵,应用BGK格式对可压缩混合层的数值研究.清华大学博士毕业论文, 2003.
[123] Stanley, S.,S. Sarkar, Simulations of spatially developing two-dimensional shear layers and jets. Theoretical and Computational Fluid Dynamics, 1997, 9(2): 121-147.
[124] Sandham, N.D., The effect of compressibility on vortex pairing. Physics of Fluids, 1994, 6(2): 1063-1072.
[125] Papamoschou, D.,S. Lele, Vortex induced disturbance field in a compressible shear layer. Physics of Fluids A: Fluid Dynamics, 1993, 5(6): 1412-1419.
[126] Shi, X., J. Chen, Z. She, Visualization of Compressibility Effect on Large Scale Structures in Spatially Developing Compressible Mixing Layers. J. Visualization (accepted), 2010.
[127] Hall, J., P. Dimotakis, H. Rosemann, Experiments in nonreacting compressible shear layers. AIAA journal, 1993, 31(12): 2247-2254.
[128] Wagner, R.D., Mean flow and turbulence measurements in a Mach 5 free shear layer NASA TN D7366, 1973
[129] Urban, W.D.,M.G. Mungal, Planar velocity measurements in compressible mixing layers. Journal of Fluid Mechanics, 2001, 431: 189-222.
[130] Hunt, J., A. Wray, P. Moin. Eddies, streams, and convergence zones in turbulent flows. 1988.
[131] Jeong, J.,F. Hussain, On the identification of a vortex. Journal of Fluid Mechanics, 1995, 285: 69-94.
[132] Barre, S., P. Braud, O. Chambres, J. Bonnet, Influence of inlet pressure conditions on supersonic turbulent mixing layers. Experimental thermal and fluid science, 1997, 14(1): 68-74.
[133] She, Z., N. Hu, Y. Wu, Structural ensemble dynamics based closure model for wall-bounded turbulent flow Acta Mechanica Sinica, 2009, 25(5): 731-736.
[134]佘振苏,面向工程应用的湍流基础研究-湍流世纪难题的突破与新一代湍流工程湍流模型探索.中科院力学所, 2009年8月19日[C].北京, 2009.
[135]张志雄,基于结构系综的湍流复杂系统研究.北京大学博士毕业论文, 2009.
[136] Kline, S., W. Reynolds, F. Schraub, P. Runstadler, The structure of turbulent boundary layers. Journal of Fluid Mechanics, 1967, 30(04): 741-773.
[137] She, Z.S., E. Jackson, S.A. Orszag, Intermittent votex structures in homogeneous isotropic turbulence Nature, 1990, 344, 15 March.
[138] Wu, X.,P. Moin, Direct numerical simulation of turbulence in a nominally zero-pressure-gradient flat-plate boundary layer. Journal of Fluid Mechanics, 2009, 630: 5-41.
[139] Kolmogrov, A.N., The local structure of turbulence in incompressible viscous ?uid for very large Reynolds. Dokl Akad Nauk SSSR, 1941, 30: 9-13.
[140] She, Z.S.,E. Leveque, Universal scaling laws in fully-developed turbulence. Phys Rev Lett 1994 72(3): 336-339.
[141]时晓天,裴杰,陈军,佘振苏,可压缩混合层流动的结构系综研究.中国力学学术大会.郑州, 2009.