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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\(\geq Re 2\).

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
Full Text: DOI
[1] Bruneau, C.H.; Jouron, C., C. R. acad. sci. Paris, 307, 359, (1988)
[2] Canuto, C.; Hussaini, M.Y.; Quarteroni, A.; Zang, T.A., Spectral methods in fluid dynamics, (1987), Springer-Verlag New York/Berlin · Zbl 0636.76009
[3] Fortin, A.; Fortin, M.; Gervais, J.J., J. comput. phys., 70, 295, (1987)
[4] Ghia, U.; Ghia, K.N.; Shin, C.T., J. comput. phys., 48, 387, (1982)
[5] {\scJ. W. Goodrich, K. Gustafson, and K. Halasi}, Hopf bifurcation in the driven cavity, J. Comput. Phys., to appear. · Zbl 0702.76052
[6] Gustafson, K.; Halasi, K., J. comput. phys., 70, 271, (1987)
[7] Haidvogel, D.B.; Zang, T.A., J. comput. phys., 30, 167, (1979)
[8] Kim, J.; Moin, J. comput. phys., 59, 308, (1985)
[9] Quéré, Le; De Roquefort, T.Aliziary, J. comput. phys., 57, 210, (1985)
[10] Orszag, S.A., J. comput. phys., 37, 70, (1980)
[11] Shen, J., (), (unpublished)
[12] Shen, J., Model. math. anal. num., 22, 677, (1988)
[13] Shen, J., Comput. methods appl. mech. eng., 80, 273, (1990)
[14] {\scJ. Shen}, “Projection Methods for Time Dependent Navier-Stokes Equations,” Appl. Math. Let., to appear. · Zbl 0754.35111
[15] Temam, R., Navier-Stokes equations. theory and numerical analysis, (1979), North-Holland Amsterdam · Zbl 0426.35003
[16] Temam, R., Infinite dimensional dynamcal systems in mechanics and physics, (1988), Springer-Verlag New York
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