Linear stability of rectangular cavity flows driven by anti-parallel motion of two facing walls

The flow in an infinite slab of rectangular cross-section is investigated numerically by a finite volume method. Two facing walls which move parallel to each other with the same velocity, but in opposite directions, drive a plane flow in the cross-section of the slab. A linear stability analysis shows that the two-dimensional flow becomes unstable to different modes, depending on the cross-sectional aspect ratio, when the Reynolds number is increased. The critical mode is found to be stationary for all aspect ratios. When the separation of the moving walls is larger than approximately twice the height of the cavity, the basic flow forms two vortices, each close to one of the moving walls. The instability of this flow is of centrifugal type and similar to that in the classical lid-driven cavity problem with a single moving wall. When the moving walls are sufficiently close to each other (aspect ratio less than 2) the two vortices merge and form an elliptically strained vortex. Owing to the dipolar strain this flow becomes unstable through the elliptic instability. When both moving walls are very close, the finite-length plane-Couette flow becomes unstable by a similar elliptic mechanism near both turning zones. The critical mode produces wide streaks reaching far into the cavity. For a small range of aspect ratios near unity the flow consists of a single vortex. Here, the strain field is dominated by a four-fold symmetry. As a result the instability process is analogous to the instability of a Rankine vortex in an quadripolar strain field, resulting from vortex stretching into the four corners of the cavity.