基于多松弛时间格子Boltzmann方法的浅水大涡模拟方法研究

Due to the inherent parallelism, the lattice Boltzmann method (LBM) is very suitable to the numerical simulations of the turbulent flows. Compared to the traditional large eddy simulation (LES) based on the macroscopic kinetic equations, the LES based on the LBM, especially on the multiple relaxation time (MRT) model, could preserve more spatio-temperal details of the turbulence. However, the calculation of the strain rate tensor for the current LES of the shallow water turbulence based on the MRT-LBM introduces spurious oscillations, which interfere to the turbulent fluctuations and result in the loss of the accuracy and stability. In the proposal, it is intended to reconstruct the equilibrium moment functions in the moment space and develop a frame work based on the MRT-LBM to solve Shallow Water Equations (SWEs) for LES of turbulent flows, referred to as MRT-LBM-SWEs-LES. The subgrid scale (SGS) turbulence effects are represented by the standard Smagorinsky model. The strain rate tensor for SGS models are expressed in terms of nonequilibrium moments. It takes the advantages of the separable modes, avoid the incorrect assumption, eliminate the spurious oscillations interfering with turbulent fluctuations and improve the methodological consistency, stability and accuracy of the MRT-LBM to simulate the shallow water turbulence. The framework would preserve more spatio-temperal information. Subsequently, the MRT-LBM-SWEs-LES will be applied to simulate the coastal current in the North of South China Sea. This would further verify the accuracy and viability of the present model. All these would further promote the development of the shallow water turbulence and LBM.

由于天生的并行性，格子Boltzmann方法（LBM）非常适合紊流数值模拟。相比较传统的基于宏观方程的大涡模拟方法（LES），基于LBM的LES能够保留更多的湍流脉动信息，尤其是多松弛时间（MRT）的LBM。然而，现行的MRT-LBM的浅水紊流LES计算应变率张量的方法会引入“伪振荡”干扰紊流结构，影响方法的数值稳定性和计算准确性。本项目拟通过构建合适的矩空间的平衡态矩函数，建立基于MRT-LBM的LES用于浅水紊流模拟，即（MRT-LBM-SWEs-LES）。MRT-LBM-SWEs-LES采用标准Smagorinsky模型考虑湍动能耗散效应。通过矩空间的一阶非平衡矩计算应变率张量获得涡粘系数，可充分发挥MRT-LBM各模态矩分离的优势，消除应变率张量计算的不合理假定，去除“伪振荡”对紊流结构的干扰，提高MRT-LBM模拟浅水紊流的方法一致性、计算稳定性和准确性。

浅水流动是自然界和工程问题中广泛存在的自由表面流动，包括了河道、湖泊、河口潮汐、近岸和浅海环流等，是泥沙输运、河口冲淤、污染物对流扩散等问题的水动力基础。格子Boltzmann方法（LBM）具有显式求解、边界条件处理简单，程序易于实现以及天生的并行性，适合于大规模流场计算，尤其是其计算局部性和良好的并行性对紊流的直接数值模拟（DNS）和大涡模拟（LES）具有重要意义。LES通过数值方法直接求解大尺度运动，并建立模型考虑小尺度脉动的耗散效应。湍动能耗散效应通过亚格子（SGS）模型表示。LES兼顾了雷诺平均方法（RANS）和DNS的特点，缓解了对计算资源的苛刻要求，减少计算量，适用于高雷诺数紊流模拟，又能提供比雷诺平均方法更多的紊流信息。目前流行的基于浅水多松弛时间（multi-relaxation time，MRT）LBM的紊流模拟方法，未利用MRT-LBM各模态矩分离的优势，仍然采用非EDFs的二阶矩计算应变率张量获得涡粘系数，且假定 与SWEs不相符，这些都将引入额外的数值误差。. 本项目建立基于MRT-LBM的浅水紊流大涡模拟方法（简记为MRT-LBM-SWEs-LES），实现利用一阶非平衡矩计算应变率张量 以获得涡粘系数 ，从而在计算方法一致性、计算准确性等方面提高MRT-LBM模拟浅水紊流的能力。在此基础上开展模型数值模拟与验证和紊流模型选取方面的工作。主要研究内容如下：1）基于现行的浅水格子Boltzmann模型框架，探讨不同作用力模型对底坡项（静水压力项）的表达能力，为合适的作用力模型选取提供支撑；2）通过理论分析推导出新的矩空间的平衡矩函数和一阶非平衡矩计算应变率张量的计算表达式，建立基于MRT-LBM的浅水紊流大涡模拟方法，实现利用一阶非平衡矩计算应变率张量 以获得涡粘系数 。.项目提出利用一阶非平衡距实现MRT-LBM的LES模拟紊流，消除不合理的假定引起的数值“伪振荡”，提高MRT-LBM模拟浅水紊流的准确性与稳定性，进一步推动MRT-LBM紊流LES的发展。