Diffusion on unstructured triangular grids using Lattice Boltzmann

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Abstract

In this paper, we present a Lattice Boltzmann scheme for diffusion on unstructured triangular grids. In this formulation there is no need for interpolation, as is required in other LB schemes on irregular grids. At the end of the propagation step, the lattice gas particles arrive exactly at neighbouring lattice sites, as is the case in LB schemes on Bravais lattices. The scheme is constructed using the constraints that the moments of the equilibrium distribution equals that of the Maxwell–Boltzmann distribution. For a special choice of the relaxation parameter (ω=1) we show that our LB scheme is identical to a cell-centred Finite Volume scheme on an unstructured triangular grid. Consequences of the use of the unstructured grid on the required data structures is discussed in detail. Subsequent simulation show that diffusion on this unstructured grid is isotropic.

Introduction

Lattice Boltzmann (LB) has become a powerful numerical technique for solving complex fluid phenomena, such as multiphase flow and flow in porous media [1]. However, most LB schemes are implemented on uniform structured grids, i.e. Bravais lattices. For many other types of applications this restriction to uniform grids is quite disadvantageous. Several formulations of LB on irregular grids have been developed [2], [3], [4], but most of these formulations involve an extra interpolation step, which imposes undesired numerical diffusion.

In a previous paper [5], we have presented an LB scheme for convection–diffusion on a rectangular grid with non-uniform lattice spacings, i.e. an irregular, but still structured grid. This scheme is without an extra interpolation step, but adheres the original conceptual framework of the LB schemes on Bravais lattice, where the lattice gas particles always propagate to adjacent lattice sites. In this paper, we take a step further in complexity of the grid, and present an LB scheme for diffusion on unstructured triangular grids. The scheme is constructed using the requirements that the moments of the equilibrium distribution are equal that those of the classical Maxwell–Boltzmann distribution. In principle, these requirements are not restricted to Bravais lattices, and hence the particle velocities may be different for different lattice sites. Consequently, one can construct Lattice Boltzmann schemes for unstructured grids.

In this paper, we construct the equilibrium distribution for diffusion on a 2D unstructured triangular grid. For the special case of ω=1, we show the equivalence between the LB scheme and Finite Volume schemes, like in our previous papers [5], [6] This equivalence with Finite Volume/Finite Difference has also been noted for LB schemes modelling fluid dynamics on Bravais lattices [7] and for diffusion [8]. In the case of ω=1, the LB scheme is more or less equivalent with the artificial compressibility scheme [7], [9]. These insights show that Lattice Boltzmann and Finite Volume can benefit from each other by transfer of concepts. Finite Volume schemes for irregular unstructured grids are well developed, and can provide good directions how to develop LB schemes for such grids.

Because unstructured grids are more complex than Bravais lattices, we will discuss in detail the datastructures and algorithm, as used in our implementation. Subsequently, we show some results of trial simulations. From those results, we conclude that the diffusion is isotropic, as demanded by the requirements on the equilibrium distribution.

Section snippets

Triangular Bravais lattice

Before developing the scheme for unstructured triangular grids, it is instructive to develop first the Lattice Boltzmann scheme on a triangular Bravais lattice. For this lattice we assume that the Wigner–Seitz cell is a triangle, with possible unequal sides. The lattice site is located at the intersection of the orthogonal bisectors of the triangle. The lattice site is inside the Wigner–Seitz cell if all angles of the triangle are less than π/2. This triangular lattice is shown in Fig. 1.

The

Equivalence with finite volume

In a previous paper [5], we have shown that for convection–diffusion on rectangular Bravais lattices the Lattice Boltzmann scheme is equivalent with a Finite Volume schemes (Lax–Wendroff), if the relaxation parameter ω=1. For this special case the Lattice Boltzmann equation reads fi∗(x+ei,t+Δt)=fieq(x,t).Here we investigate whether this equivalence still holds for diffusion on (irregular) triangular grids. Herbin and Labergerie [10] have presented a Finite Volume scheme for convection–diffusion

Data structure and algorithm

Because of the complexity of the unstructured triangular grid, compared to the triangular Bravais lattice, the required data structures will also be more complex. In this section, we will describe these structures and the associated algorithm as used in our implementation.

We distinguish three data types, related to the grid: (1) Vertices, (2) Cells, and (3) BoundaryCells. The vertices structure is an array of points, containing the coordinates of the vertices of the lattice cells (control

Diffusion on unstructured grid

We have implemented the Lattice Boltzmann scheme for diffusion on unstructured grids for the special case of ω=1. The grid is generated with a commercial Finite Element Package and is imported into our Lattice Boltzmann implementation. The grid is generated with Delauny triangulation of a circle with radius 1, and is shown in Fig. 2.

We have tested our implementation for diffusion into a cylinder, having Dirichlett boundary conditions. Initially the density at all cells is ρ(x,y)=ρ0=1. At the

Conclusions

In this paper, we have presented a Lattice Boltzmann scheme for diffusion on unstructured triangular grids. This new scheme follows the same concepts as the classical LB schemes for Bravais lattices. During propagation the discrete velocity set of the lattice gas particles move them directly to adjacent lattice sites. Collision is modelling by a relaxation towards an equilibrium distribution. As for all LB schemes on Bravais lattices, the equilibrium distribution can be derived from the

R.G.M. van der Sman received his MS degree in Applied Physics from the Technical University of Delft in 1987. From 1989 to 1992 he was an industrial researcher at the Dutch Post and Telecom, and since 1992 he has been a research employee at the Agrotechnological Research Institute (ATO) in Wageningen, specialising in heat and mass transfer in processing and storage of food. In 1999 he received his PhD degree from University of Wageningen, based on the research at ATO. In 2001 he joined the Food

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R.G.M. van der Sman received his MS degree in Applied Physics from the Technical University of Delft in 1987. From 1989 to 1992 he was an industrial researcher at the Dutch Post and Telecom, and since 1992 he has been a research employee at the Agrotechnological Research Institute (ATO) in Wageningen, specialising in heat and mass transfer in processing and storage of food. In 1999 he received his PhD degree from University of Wageningen, based on the research at ATO. In 2001 he joined the Food Process Engineering group of Prof. Remko Boom at the University of Wageningen, working in parttime next to the job at ATO. His scientific interests are (Lattice Boltzmann) simulations of heat and mass transfer in foods, porous media flow, and flows of dispersed foods as emulsions and suspensions.

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