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

The classical rectangular lid-driven-cavity problem is considered in which the motion of an incompressible fluid is induced by a single lid moving tan gentially to itself with constant velocity. In a system infinitely extended in the spanwise direction the flow is two-dimensional for small Reynolds n umbers. By a linear stability analysis it is shown that this basic flow bec omes unstable at higher Reynolds numbers to four different three-dimensiona l modes depending on the aspect ratio of the cavity's cross section. For sh allow cavities the most dangerous modes are a pair of three-dimensional sho rt waves propagating spanwise in the direction perpendicular to the basic f low. The mode is localized on the strong basic-state eddy that is created a t the downstream end of the moving lid when the Reynolds number is increase d. In the limit of a vanishing layer depth the critical Reynolds number app roaches a finite asymptotic value. When the depth of the cavity is comparab le to its width, two different centrifugal-instability modes can appear dep ending on the exact value of the aspect ratio. One of these modes is statio nary, the other one is oscillatory. For unit aspect ratio (square cavity), the critical mode is stationary and has a very short wavelength. Experiment s for the square cavity with a large span confirm this instability. It is a rgued that this three-dimensional mode has not been observed in all previou s experiments, because the instability is suppressed by side-wall effects i n small-span cavities. For large aspect ratios, i.e., for deep cavities, th e 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. (C) 2001 American Institute of Physics.