Three-dimensional centrifugal-flow instabilities in the lid-driven-cavity problem

Abstract

The classical rectangular lid-driven-cavity problem is considered in which the motion of an incompressible fluid is induced by a single lid moving tangentially to itself with constant velocity. In a system infinitely extended in the spanwise direction the flow is two-dimensional for small Reynolds numbers. By a linear stability analysis it is shown that this basic flow becomes unstable at higher Reynolds numbers to four different three-dimensional modes depending on the aspect ratio of the cavity’s cross section. For shallow cavities the most dangerous modes are a pair of three-dimensional short waves propagating spanwise in the direction perpendicular to the basic flow. The mode is localized on the strong basic-state eddy that is created at the downstream end of the moving lid when the Reynolds number is increased. In the limit of a vanishing layer depth the critical Reynolds number approaches a finite asymptotic value. When the depth of the cavity is comparable to its width, two different centrifugal-instability modes can appear depending on the exact value of the aspect ratio. One of these modes is stationary, the other one is oscillatory. For unit aspect ratio (square cavity), the critical mode is stationary and has a very short wavelength. Experiments for the square cavity with a large span confirm this instability. It is argued that this three-dimensional mode has not been observed in all previous experiments, because the instability is suppressed by side-wall effects in small-span cavities. For large aspect ratios, i.e., for deep cavities, the critical three-dimensional mode is stationary with a long wavelength. The critical Reynolds number approaches a finite asymptotic value in the limit of an infinitely deep cavity.