Numerical Simulations of Flow Bifurcations Inside a Driven Cavity

Abstract

Time-dependent numerical simulations for incompressible flow in a four-sided lid driven cavity are reported in the present study. The flow 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. Numerical simulations are performed by solving the unsteady two-dimensional Navier-Stokes equations in stream function-vorticity form. A compact fourth-order accurate central difference scheme is used for spatial discretization, while the second-order accurate Crank-Nicolson scheme is used for discretization of the time dependent terms. Numerical test cases show that the cavity flow remains steady up till a critical Reynolds number of 735. At this critical value, the flow undergoes a supercritical Hopf bifurcation, giving rise to a perfectly periodic state. Flow periodicity 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. Moreover, phase-space trajectories show the transition from a fixed point attractor and a steady flow regime to a limit cycle attractor and a periodic flow regime.