Hopf bifurcation in the driven cavity

Goodrich, John W.; Gustafson, Karl; Halas, Kadosa

doi:10.1016/0021-9991(90)90204-e

Cited by 73 publications

(36 citation statements)

References 15 publications

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“…Indeed, it is observed that the presence of singular boundary conditions degrades convergence from spectral to algebraic. A time-marching procedure was employed to generate the steady state; in the light of the work of Goodrich et al (1990) and Shen (1991), such a technique is likely to be successful for Reynolds numbers below about 10 4 , well beyond our envelope of investigation. Algorithmic details for the computation of the twodimensional steady state may be found elsewhere (Theofilis 2003).…”

Section: Pressure-gradient-driven Flow Through a Rectangular Ductmentioning

confidence: 99%

“…Unsteady approaches to the problem yield a yet further twist to the overall picture. The work of Goodrich, Gustafson & Halasi (1990) and Shen (1991) indicates that the flow converges to a (two-dimensional) steady state for Reynolds numbers up to about 10 4 . The former work suggests that for Reynolds numbers in the range 10 000 < Re < 10 500 the flow becomes temporally periodic with a Hopf bifurcation, whilst Shen (1991) shows the flow to become quasi-periodic in the range 15 000 < Re < 15 500 in the regularized case, with a non-uniform sliding of the lid (in order to avoid difficulties with flow singularities at the top corners of the cavity).…”

Section: Introductionmentioning

confidence: 99%

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Viscous linear stability analysis of rectangular duct and cavity flows

2004

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The viscous linear stability of four classes of incompressible flows inside rectangular containers is studied numerically. In the first class the instability of flow through a rectangular duct, driven by a constant pressure gradient along the axis of the duct (essentially a two-dimensional counterpart to plane Poiseuille flow – PPF), is addressed. The other classes of flow examined are generated by tangential motion of one wall, in one case in the axial direction of the duct, in another perpendicular to this direction, corresponding respectively to the two-dimensional counterpart to plane Couette flow (PCF) and the classic lid-driven cavity (LDC) flow, and in the fourth case a combination of both the previous tangential wall motions. The partial-derivative eigenvalue problem which in each case governs the temporal development of global three-dimensional small-amplitude disturbances is solved numerically. The results of Tatsumi & Yoshimura (1990) for pressure-gradient-driven flow in a rectangular duct have been confirmed; the relationship between the eigenvalue spectrum of PPF and that of the rectangular duct has been investigated. Despite extensive numerical experimentation no unstable modes have been found in the wall-bounded Couette flow, this configuration found here to be more stable than its one-dimensional limit. In the square LDC flow results obtained are in line with the predictions of Ding & Kawahara (1998b), Theofilis (2000) and Albensoeder et al. (2001b) as far as one travelling unstable mode is concerned. However, in line with the predictions of the latter two works and contrary to all previously published results it is found that this mode is the third in significance from an instability analysis point of view. In a parameter range unexplored by Ding & Kawahara (1998b) and all prior investigations two additional eigenmodes exist, which are both more unstable than the mode that these authors discovered. The first of the new modes is stationary (and would consequently be impossible to detect using power-series analysis of experimental data), whilst the second is travelling, and has a critical Reynolds number and frequency well inside the experimentally observed bracket. The effect of variable aspect ratio $A\in[0.5,4]$ of the cavity on the most unstable eigenmodes is also considered, and it is found that an increase in aspect ratio results in general destabilization of the flow. Finally, a combination of wall-bounded Couette and LDC flow, generated in a square duct by lid motion at an angle $\phi\in(0,{\pi}/{2})$ with the homogeneous duct direction, is shown to be linearly unstable above a Reynolds number $\Rey\,{=}\,800$ (based on the lid velocity and the duct length/height) at all $\phi$ parameter values examined. The excellent agreement with experiment in LDC flow and the alleviation of the erroneous prediction of stability of wall-bounded Couette flow is thus attributed to the presence of in-plane basic flow velocity components.

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Section: Pressure-gradient-driven Flow Through a Rectangular Ductmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Viscous linear stability analysis of rectangular duct and cavity flows

2004

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“…It is suspected that the slow convergence at this Reynolds number is due to the strong transient nature of the flow where several significant secondary flows appear and interact with the main circulating vortex. Goodrich et al [67] has found the flow to be unsteady at Re ~ 5000.…”

Section: Present Results Hwang and Fanmentioning

confidence: 99%

A primitive variable, strongly implicit calculation procedure for two and three-dimensional unsteady viscous flows: applications to compressible and incompressible flows including flows with free surfaces

Chen

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“…The driven cavity does not show stability for a Reynolds number larger than 7500, instead there is bounded oscillations of the energy even for very small time-steps. We have not analysed the nature of this oscillation, although the reasons may be associated with the dynamical features of the physical problem as reported in the literature for the driven cavity in References [27][28][29], where other numerical schemes were used. The reference value given by Botella and Peyret [30] is !…”

Section: Stabilitymentioning

confidence: 99%

Effect of boundary vorticity discretization on explicit stream-function vorticity calculations

Sousa

Sobey

2005

Int. J. Numer. Meth. Fluids

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SUMMARYThe numerical solution of the time-dependent Navier-Stokes equations in terms of the vorticity and a stream function is a well tested process to describe two-dimensional incompressible ows, both for uid mixing applications and for studies in theoretical uid mechanics. In this paper, we consider the interaction between the unsteady advection-di usion equation for the vorticity, the Poisson equation linking vorticity and stream function and the approximation of the boundary vorticity, examining from a practical viewpoint, global iteration stability and error. Our results show that most schemes have very similar global stability constraints although there may be small stability gains from the choice of method to determine boundary vorticity. Concerning accuracy, for one model problem we observe that there were cases where the boundary vorticity discretization did not propagate to the interior, but for the usual cavity ow all the schemes tested had error close to second order.

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Hopf bifurcation in the driven cavity

Cited by 73 publications

References 15 publications

Viscous linear stability analysis of rectangular duct and cavity flows

Viscous linear stability analysis of rectangular duct and cavity flows

A primitive variable, strongly implicit calculation procedure for two and three-dimensional unsteady viscous flows: applications to compressible and incompressible flows including flows with free surfaces

Effect of boundary vorticity discretization on explicit stream-function vorticity calculations

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