Keywords: multi relaxation time (MRT), lattice Boltzmann model, graphic processing unit (GPU), three dimensional lid-driven cavity flow, high Reynolds number flows.

The lid driven cavity flow as a classical benchmark problem has been extensively studied using both numerical approaches and experimental techniques. Despite its geometric simplicity, the flow exhibits a variety of flow features and provides an ideal platform to examine vortex dynamics. Stability in such two-dimensional analysis schemes is an issue of great interest. Also, numerical results reveal that the critical Reynolds number, where the first Hopf bifurcation takes place, is around 8000 for a square cavity flow [1] while Lin et al. [2] further showed that this value decreases with the increase of the depth-width ratio.

On the other hand, three-dimensional cavity flows were investigated both by experiments and numerical simulations. Iwatsu et al. [3] adopted a different numerical scheme to simulate cubic cavity flows, where steady solution was shown to exist at Reynolds number at Re=2000.

However, the study of critical Reynolds number of three dimensional cavity flow, especially with different depth-width ratios is still not extensively explored. Previous works also indicated that aspect ratio has strong influence on the fluid stability. However, this topic has received little attention. Therefore, the present study aims to examine the range of critical Reynolds number as well as the relationship between critical Reynolds number and depth-width ratio. The D3Q19 MRT lattice Boltzmann model is adopted here due to its enhanced stability at high Reynolds number flows.

As an explicit numerical scheme with intensive local computation, the LBM algorithm is very suitable for parallelization. This can be achieved using graphical processing units (GPU) with the compute unified device architecture (CUDA). As a result of the immense computing demand for three dimensional MRT lattice Bolztamnn simulation, multi GPU computations will also be adopted. The computation platform is a single node multi-GPU system consisting of three nVIDIA M2070 devices with the OpenMP framework. Its performance relative to CPU will also be addressed.

The results show that transition takes place between 1700<Re<2000 and 1000<Re<1500, respectively for cubic cavity and for deep cavity flows with an aspect ratio of 2 and 3. This indicates that increase of the depth-width aspect ratio would induce the transition at lower Reynolds number and is consistent with the previous results for two-dimensional cavities, though the critical Reynolds number is much lower for three dimensional cavity.

1

F. Auteri, N. Parolini, L. Quartapelle, "Numerical investigation on the stability of singular driven cavity flow", J. Comput. Phys., 183, 1, 2002.

2

L.S. Lin, Y.C. Chen, C.A. Lin, "Multi relaxation time lattice Boltzmann simulations of deep lid driven cavity flows at different aspect ratios", Computers and Fluids, 45, 233, 2011.

3

R. Iwatsu, K. Ishii, T. Kawamura, K. Kuwahara, J.M. Hyun, "Numerical simulation of three-dimensional flow structure in a driven cavity", Fluid Dyn. Res., 5, 173, (1989).

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