Eddy genesis and transformation of Stokes flow in a double-lid driven cavity

Abstract

Stokes flow is considered in a rectangular driven cavity of depth 2 H and width 2 L, with two stationary side walls and two lids moving in opposite directions with speeds U1 and U2. The flow is governed by two control parameters: the cavity aspect ratio, A = H/L, and the speed ratio, S = U1/ U2. The solution for the streamfuntion is expressed as an infinite series of Papkovich-Faddle eigenfunctions, which is then expanded about any stagnation point to reveal changes in the local flow structure as A and S are varied. An (S, A) control space diagram is constructed, which exhibits an intricate structure due to the intersection and confluence of several critical curves representing flow bifurcations at degenerate critical points. There are eight points where two critical curves intersect and the flow bifurcations are described and interpreted with reference to the theoretical work of Bakker ( Bifurcations in Flow Patterns, Kluwer Academic, 1991) and Brøns and Hartnack ( Phys. Fluids, 1999, 11, 314). For a speed ratio in the range -1 ≤ S < O the various flow trnsformations are tracked as A increases in the range O < A < 3.2, and hence the means is identified by which new eddies appear and become fully developed. It is shown that for S ≠ O, the number of eddies increases from 1 to 3 via several key flow transformations, which become more complicated as | S| is reduced.