Discrete effects on boundary conditions of the lattice Boltzmann method for convection-diffusion equations with curved geometries

Abstract

In this work, we are concerned with the discrete effects of curved boundary conditions in the lattice Boltzmann method (LBM) for convection-diffusion equations (CDEs). The mathematical derivations show that for previous boundary conditions only with the distance ratio, the relaxation time to eliminate the numerical slip at curved boundary geometries cannot remain constant. Motivated by a single-node boundary scheme which contains a free parameter besides the distance ratio, a strategy is then proposed within the framework of multiple-relaxation-time (MRT) model to achieve uniform relaxation times such that the numerical slip can be removed. It is also shown that the derived relation adapted with the halfway boundary scheme to eliminate the numerical slip fails for the case of curved boundaries. Some simulations are performed for the case of inclined and curved boundaries, and the numerical results are consistent with our theoretical analysis.

Introduction

In the past three decades, the lattice Boltzmann method (LBM) been successfully advanced from simulating complex fluid flows [1] to heat and mass transfer processes, which are usually governed by convection-diffusion equations (CDEs) [2]. In addition to the lattice Boltzmann (LB) models for CDEs, the boundary treatment implemented at boundary nodes (the lattice nodes nearest to the physical boundary) is another important problem. As have been revealed in flow simulations [[3], [4], [5], [6], [7], [8]], the discrete effect on the boundary condition also must be minimized to derive reliable results for CDEs [[9], [10], [11], [12], [13]]. Since the local information at the current node is only contained, the single-node boundary technique would be more expected to match the local computation of LBM, especially for the case of complex boundaries and narrow spaces in porous media [14]. With the two-relaxation-time (TRT) model, Ginzburg [9,10] analyzed the single-node anti-bounce-back (ABB) boundary condition for CDEs. On the basis of the multiple-relaxation-time (MRT) model, Dubois et al. [11,15] used the Taylor expansion to investigate the ABB boundary scheme for the Dirichlet boundary condition. Their results both show that the accuracy of the boundary condition is affected by the combination of two relaxation rates. Zhang et al. [12] theoretically studied their halfway bounce-back boundary (HBB) scheme within the Bhatnagar-Gross-Krook (BGK) model for CDEs. For the diffusion in Couette flow with wall injection, their mathematical derivations reveal a nonzero numerical slip related with the relaxation time and the square of lattice spacing. Later, the discrete effect of the HBB condition was specially analyzed by Cui et al. [13] using the MRT model for CDEs. They further provided the formulae to eliminate the numerical slip by the free relaxation parameter s₂ in the MRT model. However, in all these works regarding the halfway boundary scheme, the derivation results are theoretically valid only for the case of aligned straight walls with respect to the underlying lattice [9,15].

For the case of curved boundaries, the location of boundary nodes shall change locally from the physical wall. One straightforward strategy for this issue is to approximate the curved surface by some halfway points between the boundary and solid nodes [12,13]. However, such implementation will lose its accuracy under coarse grid resolutions. To improve the accuracy of numerical simulations for CDEs, various LB curved boundary conditions have been proposed to account for the curved geometry [9,[16], [17], [18], [19]]. Noteworthily, it is revealed in Refs. [9, 15, 20] that besides the relaxation time and the lattice spacing, the numerical slip (or the accuracy) of curved boundary conditions is related with the location of physical wall between lattice nodes. Therefore, when one attempts to remove the discrete effects from curved boundary conditions, the corresponding relaxation parameter would be varied at different boundary nodes with different lattice directions. This will make the collision operator anisotropic in the LBM. Therefore, it is inspiring us to resolve this important problem. In the present paper, a single-node boundary condition [21], which contains a free parameter besides the distance ratio, is analyzed detailedly within the MRT model for the diffusion problems with a parabolic distribution. Based on the mathematical derivations, a strategy to obtain uniform relaxation parameters is then provided to eliminate the numerical slip. Some numerical examples are then performed with the case of inclined and curved boundaries. The numerical results demonstrate the availability of our theoretical analysis for diffusion and thermal convective flow problems.