Elsevier

Journal of Computational Physics

Volume 227, Issue 18, 10 September 2008, Pages 8472-8487
Journal of Computational Physics

A mass conserving boundary condition for the lattice Boltzmann equation method

https://doi.org/10.1016/j.jcp.2008.06.003Get rights and content

Abstract

In lattice Boltzmann (LB) simulations, the widely used wall boundary conditions (BCs) proposed by Filipova and Hänel (FH) and Mei, Luo and Shyy (MLS) result in constant mass leakage in certain circumstances. In this paper, we have analyzed the source of the leakage. Based on this analysis, we propose a second-order accurate mass conserving wall BC. In our BC, the distribution function at a wall node is decomposed into its equilibrium and non-equilibrium parts. The mass conservation is guaranteed by enforcing a mass conserving rule in the construction of the fictitious equilibrium distribution part. We have shown through several benchmark test problems involving steady and unsteady flows that our new BC not only eliminates the constant mass leakage, but also has many other advantages over the FH and MLS BCs.

Introduction

In LBE simulations [1], [2], [3], [4], to some extent, developing accurate and efficient BCs is as important as developing an accurate computation scheme itself, since they will influence the stability of the computation. The most common and simplest solid wall BC is the bounce-back boundary condition. In this BC, when a particle distribution streams to a wall node, it scatters back to the fluid node along its incoming link. However, the bounce-back BC only gives first-order numerical accuracy. To improve it, many BCs have been proposed in the past [5], [6], [7], [8], such as the halfway bounce-back scheme [9], [10], extrapolation scheme [11] and non-equilibrium bounce-back scheme [12]. However, most of these BCs are only suitable for flat walls. When applied to curved walls, these BCs will result in jagged boundaries, and additional errors will therefore be introduced. Recently, Filipova and Hänel (FH) proposed a curved wall BC [13], which later was improved by Mei, Luo and Shyy (MLS) [14], [15]. Both the FH and MLS BCs are second-order accurate and have the ability to model curved geometries, and therefore are widely used in the literature. However, these models do not consider body forces, like gravity or a magnetic field, and we have found that when a body force is applied on the fluids, the requirement of mass conservation cannot be exactly met. Hence, based on the FH and MLS BC, we developed a mass conserving solid wall BC for the cases with body force. While, Chen’s volumetric (as opposed to pointwise) lattice Boltzmann theory [16], [17], [18] can ensure mass conservation in the whole simulation domain with a body force, the true strengths of LBE methods lie in their ability to simulate multi-phase fluids for both single and multi-component fluids. The most widely used multi-phase, multi-component or thermal LBE models are based on the pointwise LBE theory [19], [20], [21], [22], [23], [24], so it is still necessary to develop a mass conserving solid curved wall BC for the pointwise LBE model. Doing so extends the applicability of these multi-component, multi-phase and thermal models to many more practical problems.

Section snippets

Formulation of the FH and MLS BCs

Our mass conserving solid wall BC is derived from the FH and MLS BC, so we will briefly discuss both of them. As shown in Fig. 1, eα and eα¯ denote directions opposite to each other, xb is a boundary node, and xf is a fluid node. The curved wall is located between a boundary node and fluid node, with Δ=|xf-xw||xf-xb| denoting the fraction of an intersected link in the fluid region. Obviously, 0  Δ  1. In order to finish the streaming step, we need to know f˜α¯(xb,t) at boundary node xb, where f˜α¯

Benchmark tests for the new BC

The new mass conserving BC is as simple to implement as the FH and MLS BCs yet demonstrates a better performance, as in the previously presented example. In this subsection, we will use several benchmark problems to test and illustrate this new BC further.

Discussion

In summary, we propose a second-order accurate mass conserving boundary condition for the LBE method. Several benchmark test problems were used to validate the accuracy and examine the robustness of the proposed BC. Compared with the FH and MLS BCs, our new BC has numerous advantages. First, the mass leakage is uniformly smaller than in the other schemes, and will not result in the constant mass leakage that occurs for the other BCs in some cases. These cases include larger magnitudes of

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