基于非平衡气体动理学的时间松弛湍流建模

This project proposes new time-relaxation turbulent models based on the gas-kinetic theory, namely the Relaxation Turbulence Gas-Kinetic Scheme (TRTGKS) and the multi-scale Time-Relaxation Turbulence Unified Gas-Kinetic Scheme (TRTUGKS) to improve the traditional eddy-viscosity model. In order to capture the multi-scale turbulent relaxation characteristics on mesh size scale, a duel-time relaxation model will be developed to account for the energy exchanges among different scales through the gas kinetic model (BGK-Turbulence). LES-like macroscopic equations with six conserved variables can be derived from the BGK-Turbulence model. It provides more degrees of freedom than that in the Navier-Stokes equations, which may not have sufficient variables to resolve unresolved turbulent flow with respect to the grid resolution. The new added variable, i.e., "non-trivial invariant", is used to account for the unresolved turbulent vortex flow, which is defined as “unresolved vortex rotational energy K”. Based on the macroscopic formulation, the physical model for the grid size dependent "turbulent source term" is constructed in the closed time-relaxation turbulence model TRTGKS. With the high-order spatial reconstruction and the LUSGS acceleration algorithm, high-order and efficient implicit TGTGKS method will be obtained. To overcome the near equilibrium assumption used in the Chapman-Enskog expansion in TRTGKS, an integral analytic solution of BGK-Turbulence at a cell interface will be used to update the highly non-equilibrium turbulent distribution function, along with the update of conserved variables through TRTGKS. As a result, a complete multi-scale time-relaxation non-equilibrium turbulence model TRTUGKS will be constructed on the mesh size scale. In summary, the development of the time-relaxation TRTGKS and TRTUGKS turbulence models will enrich the theory for the multi-scale turbulence modeling and construct the numerical algorithms. It will be of great help in providing alternative ways for the turbulent research and engineering applications.

本课题基于非平衡气体动理学理论，发展突破涡粘模式的时间松弛非平衡湍流模式(TRTGKS)和多尺度时间松弛非平衡湍流模式(TRTUGKS)。针对湍流多尺度涡结构的松弛特性，构造粗网格上求解湍流的双松弛气体动理学模型(BGK-Turbulence)。基于BGK-Turbulence，推导粗网格上“不可分辨涡转动能”的演化方程，以克服粗网格上求解湍流时NS方程提供的宏观自由变量不够的问题。对“不可分辨涡转动能”的演化方程中“湍流源项”进行直接物理建模，发展出封闭的时间松弛非平衡湍流模式TRTGKS，并发展高效高精度隐式TGTGKS。通过BGK-Turbulence更新湍流场速度分布函数并由TRTGKS更新宏观守恒量，发展出全新的多尺度时间松弛非平衡湍流模式TRTUGKS。该项目发展的时间松弛湍流模式将提供湍流多尺度直接建模的新理论和数值方法，为重大湍流问题研究提供新的探索方向。

本课题组完成了高阶气体动理学格式（HGKS）的雷诺平均NS方程（RANS）湍流模拟，隐式大涡模拟（iLES）和直接数值模拟（DNS）。提出了非平衡时间松弛湍流模型，完成了非平衡时间松弛湍流模型的典型可压缩湍流模拟。对于高雷诺数光滑湍流模拟，基于高阶格式的RANS模拟优于二阶格式；对于高雷诺数间断湍流模拟，湍流模型误差占误差主导地位。在有限体积框架HGKS的低雷诺数可压缩湍流模拟中，iLES模拟准确度优于显式LES。对于直接数值模拟，HGKS的模拟精度和效率与高阶有限差分格式相当，且对于强间断可压缩湍流模拟具有更强的鲁棒性。我们提出了非平衡时间松弛湍流模型，构建了粗网格上湍流源项，完成了非平衡时间松弛湍流模型的可压缩各向同性衰减湍流和平面混合层的湍流模拟，加深了对动理学框架下时间松弛湍流模型的理解。该项目证实了HGKS在可压缩流模拟领域的巨大优势，发展的时间松弛湍流模式为可压缩湍流建模提供了可行的探索方向。