Forcing scheme analysis for the axisymmetric lattice Boltzmann method under incompressible limit
Forcing scheme analysis for the axisymmetric lattice Boltzmann method under incompressible limit
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不可压缩极限下轴对称格子Boltzmann法的受力方案分析
DOI：
10.1103/physreve.95.043311
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发表时间：
2017
影响因子：
2.4
通讯作者：
Chew Jia Wei
中科院分区：
文献类型：
--
作者：
Zhang Liangqi;Yang Shiliang;Zeng Zhong;Chen Jie;Yin Linmao;Chew Jia Wei
由于标准晶格玻尔兹曼（LB）方法是针对笛卡尔Navier-Stokes （NS）方程提出的，因此在轴对称LB方法中需要额外的源项来表示轴对称效应。因此，轴对称LB模型的准确性和适用性取决于源项离散化所采用的强迫格式。本文通过在扩散尺度下推导宏观方程，对基于梯形规则的三种强制方案，即直接强制方案和半隐式中心强制方案进行了理论分析。特别地，将标准LB法的有限差分解释推广到带源项的LB方程，并对轴对称LB法不同强迫格式的精度进行了评价。理论分析表明，直接强迫方案产生的离散点阵效应是截断误差项的一部分，因此不会影响具有一般力项的标准LB方法的整体精度（即只考虑动量方程中的源项），但会导致轴对称LB模型的宏观方程不正确。另一方面，基于梯形规则的格式和半隐式中心格式都具有避免离散点阵效应和恢复正确宏观方程的优点。数值试验验证了理论分析的正确性，结果表明，直接强制方案对轴对称LB模拟的数值稳定性和精度都有影响，这表明不受离散点阵效应影响的强制方案是轴对称LB方法所必需的。
Because the standard lattice Boltzmann (LB) method is proposed for Cartesian Navier-Stokes (NS) equations, additional source terms are necessary in the axisymmetric LB method for representing the axisymmetric effects. Therefore, the accuracy and applicability of the axisymmetric LB models depend on the forcing schemes adopted for discretization of the source terms. In this study, three forcing schemes, namely, the trapezium rule based scheme, the direct forcing scheme, and the semi-implicit centered scheme, are analyzed theoretically by investigating their derived macroscopic equations in the diffusive scale. Particularly, the finite difference interpretation of the standard LB method is extended to the LB equations with source terms, and then the accuracy of different forcing schemes is evaluated for the axisymmetric LB method. Theoretical analysis indicates that the discrete lattice effects arising from the direct forcing scheme are part of the truncation error terms and thus would not affect the overall accuracy of the standard LB method with general force term (i.e., only the source terms in the momentum equation are considered), but lead to incorrect macroscopic equations for the axisymmetric LB models. On the other hand, the trapezium rule based scheme and the semi-implicit centered scheme both have the advantage of avoiding the discrete lattice effects and recovering the correct macroscopic equations. Numerical tests applied for validating the theoretical analysis show that both the numerical stability and the accuracy of the axisymmetric LB simulations are affected by the direct forcing scheme, which indicate that forcing schemes free of the discrete lattice effects are necessary for the axisymmetric LB method.