Numerical study of lattice Boltzmann methods for a convection–diffusion equation coupled with Navier–Stokes equations
Numerical study of lattice Boltzmann methods for a convection–diffusion equation coupled with Navier–Stokes equations
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DOI：
10.1088/1751-8113/44/5/055001
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发表时间：
2011-02
期刊：
影响因子：
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通讯作者：
Haibo Huang;X.-Y. Lu;M. Sukop
中科院分区：
文献类型：
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作者：
Haibo Huang;X.-Y. Lu;M. Sukop
许多格子Boltzmann（LB）方法已被提出用于求解对流扩散方程（CDE）。对于二维问题，通常使用D2 Q9、D2 Q5或D2 Q4速度模型。当LB对流扩散模型被用来解决耦合的Navier-Stokes方程的CDE，边界条件被发现是至关重要的精确求解耦合模拟。遵循正则化方案的想法（Latt等人2008 Phys. Rev. E 77 056703），提出了求解CDE的正则化边界条件。一个简单的外推方案也提出了诺依曼边界条件。详细讨论了三种现有边界条件和建议边界条件的空间精度。数值评估的基础上，在一个空腔和非定常泰勒-库埃特流的稳定和非定常自然对流的模拟。我们的研究表明，最简单的D2 Q4模型与O（u）的平衡分布函数的条款是能够获得同等精度的CDE的D2 Q5或D2 Q9模型的结果。一个稍微修改的LB方程求解CDE，用来取消一些不必要的条款似乎是不必要的不可压缩流。求解CDE的正则化边界条件具有二阶空间精度，是目前空间精度最好的边界条件。正则化格式和非平衡外推格式适用于处理Dirichlet和Neumann边界条件。对于零通量的Neumann边界条件，这五种边界条件都适用于给出精确的结果，其中反弹格式是最简单的一种。
Numerous lattice Boltzmann (LB) methods have been proposed for solution of the convection–diffusion equations (CDE). For the 2D problem, D2Q9, D2Q5 or D2Q4 velocity models are usually used. When LB convection–diffusion models are used to solve a CDE coupled with Navier–Stokes equations, boundary conditions are found to be critically important for accurately solving the coupled simulations. Following the idea of a regularized scheme (Latt et al 2008 Phys. Rev. E 77 056703), a regularized boundary condition for solving a CDE is proposed. A simple extrapolation scheme is also proposed for the Neumann boundary condition. Spatial accuracies of three existing and the proposed boundary conditions are discussed in details. The numerical evaluations are based on simulations of steady and unsteady natural convection flows in a cavity and an unsteady Taylor–Couette flow. Our studies show that the simplest D2Q4 model with terms of O(u) in the equilibrium distribution function is capable of obtaining results of equal accuracy as D2Q5 or D2Q9 models for the CDE. A slightly revised LB equation for solving a CDE that is used to cancel some unwanted terms does not seem to be necessary for incompressible flows. The regularized boundary condition for solving the CDE has second-order spatial accuracy and it is the best one in terms of the spatial accuracy. The regularized scheme and non-equilibrium extrapolation scheme are applicable to handle both the Dirichlet and Neumann boundary conditions. For the Neumann boundary condition with zero flux, all the five boundary conditions are applicable to give accurate results and the bounce-back scheme is the simplest one.