DU Rui

Department of Mathematics, South East University, Nanjing 210096, China, E-mail: rdu@seu.edu.com SHI Bao-chang

Department of Mathematics, Huazhong University of Science and Technology, Wuhan 430074, China

INCOMPRESSIBLE MULTI-RELAXATION-TIME LATTICE BOLTZMANN MODEL IN 3-D SPACE*

DU Rui

Department of Mathematics, South East University, Nanjing 210096, China, E-mail: rdu@seu.edu.com SHI Bao-chang

Department of Mathematics, Huazhong University of Science and Technology, Wuhan 430074, China

(Received March 23, 2010, Revised October 19, 2010)

Multi-Relaxation-Time Lattice Boltzmann Method (MRT LBM) is of better numerical stability and has attracted more and more research interests. The previous MRT LBM included artificial compressible effects. To overcome the disadvantage, an incompressible MRT LBM has been proposed in two dimensions recently. In this article, we present incompressible MRT LBMs in 3-D space, with example of nineteen-velocity. The equilibria in momentum space are derived from an earlier incompressible Lattice Bhatnagar-Gross-Krook (LBGK) model proposed by Guo et al.. Through the Chapman-Enskog (C-E) expansion, the incompressible Navier-Stokes (N-S) equations can be recovered without artificial compressible effects. Simulations of a lid-driven cavity flow in three dimensions with =Re1 000, 2 000 and 3 200 are performed. The simulation results agree with the existing data and clearly demonstrate better numerical stability of the presented model over the incompressible LBGK model.

incompressible flow, Multi-Relaxation-Time (MRT), Lattice Boltzmann Method (LBM)

1. Introduction

The Lattice Boltzmann Method (LBM) is an innovative numerical method based on the kinetic theory to simulate various hydrodynamic problems[1-3]with its notable advantages of the LBM, such as its intrinsic parallelism of algorithm, simplicity of programming, and ease of incorporating microscopic interactions. The simplest lattice Boltzmann model is the Lattice Bhatnagar- Gross-Krook (LBGK) model, also called Single-Relaxation-Time Lattice Boltzmann Method (SRT LBM) model and has been successfully used in variety of complex fluid systems, such as multiphase fluids, suspensions in fluid and other problems[4-7]. However, some shortcomings of the LBGK model are apparent. For instance, the method may lead to numerical instability when the dimensionless relaxation time τ is close to 0.5. One way to overcome these shortcomings of the LBGK model is to use its Multi-Relaxation-Time (MRT) version[8-19]which nevertheless retains the simplicity and computational efficiency of the LBGK model.

The MRT LBM is of better numerical stability and has more degrees of freedom than the commonly-used SRT LBM. The main idea of the MRT LBM is that the collision is mapped onto the momentum space by a linear transformation and the advection is still finished in the velocity space. Most of the existing MRT LBMs have been constructed for the compressible Navier-Stokes (N-S) equations in the low Mach number limit, and it is well

understood that “compressible” error exists for simulating incompressible fluid flows. The recovered macroscopic equations from the existing MRT LBM are the approximate incompressible N-S equations through the Chapman-Enskog (C-E) expansion. Recently, an incompressible MRT LBM in 2-D space has been proposed by Du[15]based on the equilibrium distribution functions in the velocity space by Guo et al.[20]. Through the C-E expansion the incompressible N-S equations can be recovered exactly without artificial compressible effects.

In this article, we propose an incompressible MRT LBM in 3-D space, with example of nineteen-velocity. The equilibria in the momentumspace are derived from the incompressible LBGK model[20]. Through the C-E expansion, the incompressible N-S equations in 3-D space can be recovered without artificial compressible effects. Simulations of a lid-driven cavity flow in three dimensions with =Re1 000, 2 000 and 3 200 are performed. The simulation results agree well with benchmark data. Furthermore, the numerical results indicate that the presented incompressible MRT LBM of D3Q19 exhibits better numerical stability compared to the incompressible LBGK model.

2. Incompressible MRT LBM in 3-D space

For an MRT LBM with Q velocities, a set of velocity distribution functions{fα|α=0,1,…,Q?1} is defined at each lattice node. The generalized lattice Boltzmann equation reads

where feqis equalibrium function and S is

i collision matrix. The eigenvalues of S are all between 0 and 2 so as to maintain linear stability and the separation of scales, which means that the relaxation times of non-conserved quantities are much shorter than the hydrodynamic time scales. The LBGK model is the special case in which the relaxation times are all equal, and the collision matrix S=τ?1I , where I is the identity matrix.

The evolutionary progress includes two steps, collision and advection. The collision is executed in the momentum space M=RQ, while the advection is performed in the velocity space V=RQ . The evolution equation for the MRT LBM on a D-dimensional lattice reads

01Q?1in the momentum space is a diagonal matrix.

Here we take D3Q19 model as the example for its better numerical stability and accuracy. For other models, such as D3Q15 or D3Q13, their corresponding incompressible MRT LBM versions can be obtained as same as the D3Q19 model. In the D3Q19 model, space is discretized into a cubic lattice, and there are nineteen discrete velocities given by

where c=δx/δtis the particle velocity and δxand δtare the lattice grid spacing and time step respectively. From here on we shall use the units of δx=δt=1such that all the relevant quantities are dimensionless.

The equilibrium distribution function fαeqin Eq.(1) is defined as[20]

The macroscopic flow velocity and pressure are computed from the distribution functions,

It is noted that the average density ρ0is not used in the computations of the macroscopic quantities.

The transformation matrix M is given by

Through the C-E expansion, the incompressible Navier-Stokes equation can be recovered and the kinematic viscosity is ν=cs2(1/s9?1/2)δt.

3. Simulation of lid-driven cavity flow in 3-D space

Numerical simulation for the lid-driven cavity flow in 3-D space was carried out for different Reynolds numbers, which is depicted in this section. The configuration of driven cavity flow considered here consists of a 3-D square cavity whose top wall moves from left to right with constant velocity U=0.1, while the other five walls are fixed (see Fig.1) . The Reynolds number is definedRe=LU/ν, where L is the width of the top wall andν is the kinematic viscosity.

The 3-D flow structure in a cubic cavity exhibitsthe same qualitative eddy structures as its 2-D counterpart in the interior z=0-1 planes. First, we conducted the simulations withRe=1 000 and 2 000, and velocity profiles along the vertical and horizontal lines through z=0.5 plane are plotted in Fig.2.

For Re=3200, velocity vector plots in the xz-plane at y=0.50, 0.75 and the yz-plane at x=0.50, 0.75 at 26 000 time step are shown in Fig.3. From Fig.3 we can see that the flow structures in the specified planes agree well with those in Refs.[22,23], and two pairs of the Taylor-G?rtler-like vorticies are clearly visible along the span. In Table 1, the locations of vortices in the flow in the yz-plane at x=0.50, 0.75 at 26 000 time step are given.

4. Conclusions

Acknowledgments

The authors would like to thank Prof. Guo Zhao-li for helpful discussions.

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10.1016/S1001-6058(09)60116-5

* Project supported by the National Natural Science Foundation of China (Grant Nos. 70271609, 11026181).

Biography: DU Rui (1980-), Female, Ph. D., Lecturer