Lattice Boltzmann method for moving boundaries
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 δx is related to time step size δt by δx=cδt, 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.
Section snippets
Description of the model
We consider a two-dimensional LBE model with nine discrete velocities (the D2Q9 model) on a square grid with grid spacing δx. In the advection step of the LBE, particles move from one node of the grid to one of its neighbors as illustrated in Fig. 1. The discrete velocities are given bywhere c=δx/δt, and the duration of the time step δt is assumed to be unity. Therefore c=1 in the units of δx=1 and δt=1.
Boundary condition for moving boundary
Our treatment for a moving boundary is a simple extension of the treatment for a curved boundary proposed by Bouzidi et al. [13]. This treatment for a curved boundary is a combination of the bounce-back scheme and interpolations. For the sake of simplicity, this boundary condition is illustrated in Fig. 2 for an idealized situation in two dimensions. Consider a wall located at an arbitrary position rw between two grid sites rj and rs, and rs is situated inside the non-fluid region – the shaded
Simulations
We conduct numerical simulations to investigate the accuracy of the proposed boundary conditions for a moving boundary. We use a cylinder asymmetrically placed in a channel in two dimensions as the basic configuration, as shown in Fig. 4. With different boundary conditions and initial conditions, we can simulate different flow situations in the channel. The flow simulations can be performed in two frames of reference. First, the position of the cylinder is fixed on the computational mesh, such
Conclusion and discussion
In this work we have proposed a lattice Boltzmann scheme to handle moving boundary problems. The proposed scheme is robust, stable, and easy to implement. The proposed scheme is tested in the simulations of two-dimensional flows past an impulsively started cylinder asymmetrically placed in a channel with different initial conditions. The results obtained with either moving boundary or fixed boundary agree well with each other. The simulations also verify the Galilean invariance of the lattice
Acknowledgements
P.L. would like to acknowledge the support from ICASE for his visit to ICASE in 2001, during which part of this work was performed. L.S.L. would like to acknowledge the partial support from CNRS for his visit to ASCI Laboratory in 2001 during which part of this work was performed, and partial support from NASA Langley Research Center under the program of Innovative Algorithms for Aerospace Engineering Analysis and Optimization and the United States Air Force Office for Scientific Research under
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