Premnath Kannan N, Banerjee Sanjoy
Department of Chemical Engineering, University of California, Santa Barbara, California 93106, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2009 Sep;80(3 Pt 2):036702. doi: 10.1103/PhysRevE.80.036702. Epub 2009 Sep 11.
Cascaded lattice Boltzmann method (cascaded-LBM) employs a class of collision operators aiming to stabilize computations and remove certain modeling artifacts for simulation of fluid flow on lattice grids with sizes arbitrarily larger than the smallest physical dissipation length scale [Geier, Phys. Rev. E 63, 066705 (2006)]. It achieves this and distinguishes from other collision operators, such as in the standard single or multiple relaxation-time approaches, by performing relaxation process due to collisions in terms of moments shifted by the local hydrodynamic fluid velocity, i.e., central moments, in an ascending order by order at different relaxation rates. In this paper, we propose and derive source terms in the cascaded-LBM to represent the effect of external or internal forces on the dynamics of fluid motion. This is essentially achieved by matching the continuous form of the central moments of the source or forcing terms with its discrete version. Different forms of continuous central moments of sources, including one that is obtained from a local Maxwellian, are considered in this regard. As a result, the forcing terms obtained in this formulation are Galilean invariant by construction. To alleviate lattice artifacts due to forcing terms in the emergent macroscopic fluid equations, they are proposed as temporally semi-implicit and second order, and the implicitness is subsequently effectively removed by means of a transformation to facilitate computation. It is shown that the impressed force field influences the cascaded collision process in the evolution of the transformed distribution function. The method of central moments along with the associated orthogonal properties of the moment basis completely determines the analytical expressions for the source terms as a function of the force and macroscopic velocity fields. In contrast to the existing forcing schemes, it is found that they involve higher-order terms in velocity space. It is shown that the proposed approach implies "generalization" of both local equilibrium and source terms in the usual lattice frame of reference, which depend on the ratio of the relaxation times of moments of different orders. An analysis by means of the Chapman-Enskog multiscale expansion shows that the cascaded-LBM with forcing terms is consistent with the Navier-Stokes equations. Computational experiments with canonical problems involving different types of forces demonstrate its accuracy.
级联格子玻尔兹曼方法（cascaded-LBM）采用了一类碰撞算子，旨在稳定计算并消除某些建模伪影，以模拟尺寸任意大于最小物理耗散长度尺度的格子网格上的流体流动[盖尔，《物理评论E》63，066705（2006）]。它通过在由局部流体动力学流体速度（即中心矩）偏移的矩方面，以不同的松弛率按顺序递增地执行碰撞引起的松弛过程来实现这一点，并与其他碰撞算子区分开来，例如在标准的单松弛时间或多松弛时间方法中。在本文中，我们提出并推导了级联格子玻尔兹曼方法中的源项，以表示外力或内力对流体运动动力学的影响。这本质上是通过使源项或强迫项的中心矩的连续形式与其离散形式相匹配来实现的。在这方面，考虑了源的不同形式的连续中心矩，包括从局部麦克斯韦分布得到的一种。结果，通过构造，在此公式中获得的强迫项是伽利略不变的。为了减轻由于宏观流体方程中强迫项引起的格子伪影，将它们提出为时间上半隐式且二阶的，随后通过一种变换有效地消除隐式性以方便计算。结果表明，外加力场在变换后的分布函数的演化中影响级联碰撞过程。中心矩方法以及矩基的相关正交特性完全确定了作为力和宏观速度场函数的源项的解析表达式。与现有的强迫方案相比，发现它们在速度空间中涉及高阶项。结果表明，所提出的方法意味着在通常的格子参考系中局部平衡和源项的“推广”，这取决于不同阶矩的松弛时间之比。通过查普曼 - 恩斯科格多尺度展开的分析表明，带有强迫项的级联格子玻尔兹曼方法与纳维 - 斯托克斯方程一致。涉及不同类型力的典型问题的计算实验证明了其准确性。
