Linear stability analysis of the flow in an oblique lid-driven cavity

In this presentation, we analyse the linear stability of the flow in a in cavity, driven by the motion of a lid tangential to itself but not aligned with the cross section. The angle between the direction of the lid and the cross section α is varied from 0° to 90°.

At α= 0° and a cavity of square cross section, the classical Taylor-Goertler modes with high wavenumbers are recovered at a critical Reynolds number about Re_c=800, while at α = 90°, the laterally bounded Couette flow is linearly stable. Upon increasing α from 0°, the flow becomes more unstable and Re_c decreases down to Re_c = 620 at α = 22.5°. As α is further increased towards 75°, the critical modes become more elongated in the spanwise direction. Similar transitions are obtained for shallow and deep cavities.

For all angles, the lift-up effect is the main instability mechanism, extracting kinetic energy from streamwise vortices feeding streaky structures. As long as the lid velocity drives the in-plane velocity components a feedback mechanism promoting the streamwise vortices is present. However, this feedback mechanism progressively vanishes as α tend towards 90°, and the flow strongly stabilizes.