Numerical investigation of four-lid-driven cavity flow bifurcation using the multiple-relaxation-time lattice Boltzmann method

As a fundamental subject in fluid mechanics, sophisticated cavity flow patterns due to the movement of multi-lids have been routinely analyzed by the Computational Fluid Dynamics community. This paper seeks to make a systematic study over the complex four-lid-driven cavity flows using the multiple-relaxation-time (MRT) lattice Boltzmann method (LBM). The flow is generated by moving the top wall to the right and the bottom wall to the left, while moving the left wall downwards and the right wall upwards, with an identical moving speed. The present MRT-LBM results reveal a lot of important features of bifurcated flow, such as the symmetry and steady characteristics of cavity flows at low Reynolds numbers, the two-stage multiplicity of stable asymmetric and unstable symmetric cavity flow patterns when the Reynolds number exceeds its first and second critical values (corresponding to the first and second steady bifurcation stages), respectively, as well as the flow periodicity after a further critical Reynolds number is reached (referred to as Hopf bifurcation point). For the steady flow regions, the detailed characteristics are reported that include the locations of the vortex centers, the values of stream function at the vortex centers. For the first and second steady bifurcations as well as the Hopf bifurcation phenomena, in the present MRT simulations, the critical Reynolds numbers are predicted at 132.5 ± 0.5, 359 ± 1, and 720 ± 7, respectively. For the numerically observed periodic flows, the history plots for the stream function and vorticity and the corresponding phase-space trajectories, as well as the merging and unmerging details of the different vortices during a single period of the change in flow pattern are all examined. Through comparison against the stability analysis and numerical results reported elsewhere, not only does the MRT-LBM approach exhibit its fairly satisfactory accuracy, but also its remarkable capability for investigating the multiplicity of complex flow patterns.