Numerical simulations of flow bifurcations inside a driven cavity

Time-dependent numerical simulations for incompressible flow in a four-sided lid driven cavity are reported in the present study. The flow in the cavity is generated by moving the upper wall to the right and the lower wall to the left, while moving the left wall downwards and the right wall upwards. All four walls are moved with equal speeds. The numerical simulations are performed by solving the unsteady two-dimensional Navier-Stokes equations, which are rewritten in terms of stream function-vorticity formulation. A compact fourth-order accurate central difference scheme is used for spatial discretization, while the implicit second-order accurate Crank-Nicolson scheme is used for discretization of the time dependent terms. Numerical test cases show that the flow inside the cavity remains steady up till a critical Reynolds number of 735. At this critical value, the flow undergoes a supercritical Hopf bifurcation which gives rise to a perfectly periodic state. The periodicity of the flow is verified through time history plots for the stream function and vorticity, Fourier power spectrum plots and phase-space trajectories. Reported streamline plots at different time instants clearly demonstrate the change in flow pattern during a single period and the merging and unmerging of the different vortices. Phase-space trajectories, on the other hand, clearly show the transition from a fixed point attractor and a steady flow regime to a limit cycle attractor and a periodic flow regime.