多层速度格子Boltzmann方法进展及展望

Progress and prospects of multi-speed lattice Boltzmann method

格子Boltzmann方法(LBM)自20世纪90年代问世以来,由于计算高效、实施简捷,在多种复杂流动的数值模拟中得到了广泛应用。传统以平衡态分布函数泰勒展开结合半经验理论推导出的LBM模型需要使用低马赫数假设,一度被认为只能适用于等温弱可压流动的计算。近年来将LBM拓展到可压缩和热流计算的模型日益增多,其中在每个离散速度方向有多个速度模态的多层速度模型,因只使用单一分布函数,物理描述上更接近事实而受到了广泛关注。我们简述了几类典型的多层速度模型的构造思路,包括早期的多层速度模型、Watari-Tsutahara模型、比热比可变多层速度模型和Hermite多项式模型。由于Hermite多项式展开法构造的多层速度模型在数学解释上较为自洽,且其低阶形态与传统等温弱可压LBM模型一致,我们着重梳理和归纳了Hermite多项式模型的构造原理与离散速度模型的求解过程,以及时间和空间离散方法。最后对LBM与传统计算流体力学方法的结合进行了简要介绍,例如LBM有限差分、LBM有限体积和LBM有限元方法,并对LBM多层速度模型目前存在的问题和未来发展方向进行了总结。

Due to high computational efficiency and implementation simplicity,lattice Boltzmann method(LBM) has been widely used to simulate a variety of complex flows since its invention in the 1990s.Traditional LBM,derived from Taylor expansion of equilibrium distribution function and semi-empirical theory,employing a low Mach-number assumption,was once considered only capable for isothermal weakly compressible flows.In recent years,several models extending LBM to compressible and thermal flow have been proposed,among those the multi-speed model with multiple speed magnitude in each directions,has received widespread attention since its single distribution function is more physically realistic.We described the ideas of several typical multi-speed models,including multi-speed models of early ages,Watari-Tsutahara model,flexible specific-heat ratio and Hermite polynomial model.Since the multi-speed model constructed by Hermite polynomial expansion method is more mathematically self-consistent,and its low-order form is consistent with the traditional isothermal weakly compressible LBM model,we focus on interpretation of the multi-speed models constructed by Hermite polynomial expansion method,including the construction principle,procedure of discrete velocity model solution and time-space discrete method.Several combination of LBM and traditional computational fluid dynamics method,including finite difference LBM,finite volume LBM,and finite element LBM are briefed.The remained issues and future directions of LBM multi-speed models are summarized in the end.