Numerical Simulations Of Compressible Mixing Layers With A Discontinuous Galerkin Method And Structural Ensemble Study

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.