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Hopf bifurcation of the unsteady regularized driven cavity flow. (English) Zbl 0725.76059

Summary: A numerical simulation of the unsteady incompressible flow in the unit cavity is performed by using a Chebyshev-Tau approximation for the space variables. The high accuracy of the spectral methods and the condensed distribution of the Chebyshev-collocation points near the boundary enable us to obtain reliable results for high Reynolds numbers with a moderate number of modes. It is found that the flow converges to a stationary state for Reynolds numbers (Re) up to 10,000; for Reynolds numbers larger than a critical value \(10,000<Re 1\leq 10,500\) and less than another critical value \(15,000<Re 2\leq 15,500\), the flow becomes periodic in time which indicates a Hopf bifurcation; the flow loses time periodicity for Re\(\geq Re 2\).

MSC:

76M20 Finite difference methods applied to problems in fluid mechanics
76D10 Boundary-layer theory, separation and reattachment, higher-order effects
76D05 Navier-Stokes equations for incompressible viscous fluids
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