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Lid driven cavity flow

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Essam Moustafa Wahba
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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.
Numerical simulations for incompressible flow in two-sided and four-sided lid driven cavities are reported in the present study. For the two-sided driven cavity, the upper wall is moved to the right and the left wall to the bottom with equal speeds. For the four-sided driven cavity, the upper wall is moved to the right, the lower wall to the left, while the left wall is moved downwards and the right wall upwards, with all four walls moving with equal speeds. At low Reynolds numbers, the resulting flow field is symmetric with respect to one of the cavity diagonals for the two-sided driven cavity, while it is symmetric with respect to both cavity diagonals for the four-sided driven cavity. At a critical Reynolds number of 1073 for the two-sided driven cavity and 129 for the four-sided driven cavity, the flow field bifurcates from a stable symmetric state to a stable asymmetric state. Three possible flow solutions exist above the critical Reynolds number, an unstable symmetric solution and two stable asymmetric solutions. All three possible solutions are recovered in the present study and flow bifurcation diagrams are constructed. Moreover, it is shown that the marching direction of the iterative solver determines which of the two asymmetric solutions is recovered.