The present study employs the multi relaxation time (MRT) lattice Boltzmann equation (LBE) to study the flow instability within lid driven cavity at different depth-width aspect ratios. The focuses are to examine the correlation between the depth-width aspect ratio and the transition Reynolds number, and to identify the oscillatory instability mechanism. Simulations were applied to two and three dimensional cavity flows at different depth aspect rations. For square and cubic cavity flows, the predicted results were validated with available benchmark solutions [48, 62, 63].
For two-dimensional flows, MRT LBE was used to compute flow fields at different Reynolds numbers (100 to 7500) and cavity depth aspect ratios (K) (1 to 4). It is found that the merger of the bottom corner vortices into a primary vortex (PV) and the reemergence of the corner vortices as the Reynolds number increases, are more evident for the deep cavity flows. When the depth aspect ratio is 4, four PVs were predicted by MRT model for Reynolds number beyond 1000, which were not captured by previous single relaxation time (SRT) BGK model. Present study also shows that with the increase of the cavity depths, the first Hopf bifurcation Reynolds number decreases.
In addition, since MRT LBM is explicit, it is suitable for parallel computing. Graphical Processing Unit (GPU) is used to speed up the simulation for 2D flows, and the computing platform is NVIDIA TeslaTM C2050 GPU. In the present study, CPU and GPU comparisons performed have shown that the maximum GPU speedup is 20.4 times faster than its Intel CoreTM i7-920 CPU counterpart. MRT LBE was further adopted to simulate three dimensional cavity flows at various Reynolds numbers (100-1900) and depth aspect ratios (1-4). Compared to its 2D counterpart along the wall bisector, at high Reynolds number, the presence of secondary flows causes the dramatically different both on flow structure and velocity profile. Also, the critical Reynolds number (Recr) for the first Hopf bifurcation in 3D cavity was found to be much lower than its 2D cavity counterpart. For 3D cubic cavity, the Recr for the onset of first Hopf bifurcation is 1763.7, which is much lower than its square cavity counterpart at Recr ∼ 8300. The present result is consistent with the experimental result [63], where Recr locates in the range of (1700<Recr<1970). When Reynolds numbers increase above the critical Reynolds numbers, the flow will experience transition from periodic flow to chaotic flow with
broad frequency distribution.
Similar to 2D cavity, the Recr in 3D cavity decreases in tandem with the increase of the cavity depth, which however saturates beyond K > 2. We propose two evidences to show that this instability phenomenon is related to the stabilizing effect of the confined space from the bottom wall. Firstly, based on the predicted flow
structures, it is found the emergence of the Taylor-Goertler-like (TGL) vortices at deep cavity is more eminent. This may contribute partly to the onset of Hopf
bifurcation at lower Reynolds number. On the other hand, the development of TGL vortices in a cubic cavity is suppressed from the bottom wall. Secondly, by monitoring
the amplitudes of kinetic energy oscillation in the cavity, the mechanism of stability is found to be caused by the instability of sidewall boundary layers. The oscillatory sources are localized at the upstream of the interface between the first PV and its induced counterclockwise vortex. Meanwhile, the lower part of first PV would attach to the bottom wall at K=1 or contact to the secondary PV when K ≥ 2. Thus, the
constrained force in cubic cavity is more vigorous.

The present study employs the multi relaxation time (MRT) lattice Boltzmann equation (LBE) to study the flow instability within lid driven cavity at different depth-width aspect ratios. The focuses are to examine the correlation between the depth-width aspect ratio and the transition Reynolds number, and to identify the oscillatory instability mechanism. Simulations were applied to two and three dimensional cavity flows at different depth aspect rations. For square and cubic cavity flows, the predicted results were validated with available benchmark solutions [48, 62, 63].
For two-dimensional flows, MRT LBE was used to compute flow fields at different Reynolds numbers (100 to 7500) and cavity depth aspect ratios (K) (1 to 4). It is found that the merger of the bottom corner vortices into a primary vortex (PV) and the reemergence of the corner vortices as the Reynolds number increases, are more evident for the deep cavity flows. When the depth aspect ratio is 4, four PVs were predicted by MRT model for Reynolds number beyond 1000, which were not captured by previous single relaxation time (SRT) BGK model. Present study also shows that with the increase of the cavity depths, the first Hopf bifurcation Reynolds number decreases.
In addition, since MRT LBM is explicit, it is suitable for parallel computing. Graphical Processing Unit (GPU) is used to speed up the simulation for 2D flows, and the computing platform is NVIDIA TeslaTM C2050 GPU. In the present study, CPU and GPU comparisons performed have shown that the maximum GPU speedup is 20.4 times faster than its Intel CoreTM i7-920 CPU counterpart. MRT LBE was further adopted to simulate three dimensional cavity flows at various Reynolds numbers (100-1900) and depth aspect ratios (1-4). Compared to its 2D counterpart along the wall bisector, at high Reynolds number, the presence of secondary flows causes the dramatically different both on flow structure and velocity profile. Also, the critical Reynolds number (Recr) for the first Hopf bifurcation in 3D cavity was found to be much lower than its 2D cavity counterpart. For 3D cubic cavity, the Recr for the onset of first Hopf bifurcation is 1763.7, which is much lower than its square cavity counterpart at Recr ∼ 8300. The present result is consistent with the experimental result [63], where Recr locates in the range of (1700<Recr<1970). When Reynolds numbers increase above the critical Reynolds numbers, the flow will experience transition from periodic flow to chaotic flow with
broad frequency distribution.
Similar to 2D cavity, the Recr in 3D cavity decreases in tandem with the increase of the cavity depth, which however saturates beyond K > 2. We propose two evidences to show that this instability phenomenon is related to the stabilizing effect of the confined space from the bottom wall. Firstly, based on the predicted flow
structures, it is found the emergence of the Taylor-Goertler-like (TGL) vortices at deep cavity is more eminent. This may contribute partly to the onset of Hopf
bifurcation at lower Reynolds number. On the other hand, the development of TGL vortices in a cubic cavity is suppressed from the bottom wall. Secondly, by monitoring
the amplitudes of kinetic energy oscillation in the cavity, the mechanism of stability is found to be caused by the instability of sidewall boundary layers. The oscillatory sources are localized at the upstream of the interface between the first PV and its induced counterclockwise vortex. Meanwhile, the lower part of first PV would attach to the bottom wall at K=1 or contact to the secondary PV when K ≥ 2. Thus, the
constrained force in cubic cavity is more vigorous.

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