Lattice Boltzmann method for moving boundaries

Abstract

We propose a lattice Boltzmann method to treat moving boundary problems for solid objects moving in a fluid. The method is based on the simple bounce-back boundary scheme and interpolations. The proposed method is tested in two flows past an impulsively started cylinder moving in a channel in two dimensions: (a) the flow past an impulsively started cylinder moving in a transient Couette flow; and (b) the flow past an impulsively started cylinder moving in a channel flow at rest. We obtain satisfactory results and also verify the Galilean invariance of the lattice Boltzmann method.

Introduction

The lattice Boltzmann equation (LBE) is an explicit time marching finite difference scheme of the continuous Boltzmann equation in phase space and time [1], [2], [3]. The LBE method has a underlying Cartesian lattice grid in space as a consequence of the symmetry of the discrete velocity set and the fact that the lattice spacing δ_x is related to time step size δ_t by δ_x=cδ_t, where c is the basic unit of the discrete velocity set. This makes the LBE method a very simple scheme consisting of two essential steps: collision and advection. The collision models various interactions among fluid particles and the advection simply moves particles from one grid point to the other according to their velocities. The simplicity and kinetic nature of the LBE method are among its appealing features.

One area in the LBE method which attracts much attention is the boundary conditions in the LBE method. In particular, a much studied and often used boundary condition is the bounce-back boundary condition which mimics the particle–boundary interaction for no-slip boundary condition by reversing the momentum of the particle colliding with an impenetrable and rigid wall. The bounce-back boundary condition is most easy to implement and thus most often used in the LBE simulations. It is well understood now that the bounce-back boundary condition is indeed second-order accurate when the actual boundary position is considered to be off the grid point where the bounce-back collision takes place [4], [5], [6], [7], [8]. The bounce-back boundary condition is accurate and thus appropriate for very simple boundary geometries made of straight lines. In dealing with complex geometry of arbitrary curvatures, there are two strategies in the LBE method. One strategy is to use the body-fitted mesh and employs interpolations throughout the entire mesh [9], [10] in addition to the advection process because the computational mesh does not overlap with the underlying Cartesian lattice [11], [12], bounce-back boundary conditions are applied to the boundary nodes. The other strategy is to maintain the regular Cartesian mesh and apply interpolations to track the position of the boundary and the bounce-back boundary conditions are executed at the boundary locations which may be off the grid cartesian points [13], [14]. These methods are mostly applied to simulations of stationary objects in fluids. We shall follow the latter approach in the present study.

The lattice Boltzmann method has also been successfully applied to simulations of particulate suspensions in fluids [15], [16], [17]. In the LBE simulations of the particulate suspensions in a flow, the curved boundaries of the particles are usually approximated by zig-zag staircase thus bounce-back boundary condition can be directly applied. Despite the success of the LBE method in this area, there is no rigorous theory on the treatment of moving boundaries. In this work, we present our first attempt to systematically study the boundary boundary problem within the framework of the LBE. The method studied in this work is an extension of the method based on bounce-back and interpolations for curved boundaries proposed by Bouzidi et al. [13].

This paper is organized as follows. Section 2 describes the generalized lattice Boltzmann equation (GLBE) of d’Humières [18], [19], [20], [21]. Among its features, the GLBE has superior stability over the popular lattice BGK equation [22], [23]. Section 3 discusses the boundary conditions for the moving boundary problem, preluded by a brief introduction of the method to treat curved boundaries proposed by Bouzidi et al. [13]. Section 4 presents the numerical results. We simulate a cylinder asymmetrically placed in a channel in two dimensions with two types of initial conditions. The first is an impulsively started cylinder with a constant velocity moving in a transient Couette flow. And the second is an impulsively started cylinder with a constant velocity in the channel flow at rest. We compare the results of moving boundary simulations with that of fixed boundary. We therefore verify the Galilean invariance of the LBE method. Finally, Section 5 discusses possible directions to improve the method for moving boundary proposed in this work and concludes the paper.