A mass conserving boundary condition for the lattice Boltzmann equation method
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 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 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 at boundary node xb, where
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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