Laminar-Turbulent Transition in Stratified Wakes

Abstract

This chapter presents experimental and theoretical results on the transition from a laminar to a turbulent wake in a stratified fluid. The case of a cylinder is analysed in detail at low Reynolds number since it gives rise to the famous von Karman vortex street when the Reynolds number exceeds a critical value. This value highly depends on the stratification and on the tilt angle of the cylinder. A moderate stratification tends to suppress the von Karman vortex street, in agreement with the stabilisation of shear flows at high Richardson numbers. However, it is surprising to see that a strong stratification destabilises the flow when the cylinder is tilted. This new von Karman vortex street is allowed because the vortices exhibit horizontal streamlines although the vortices are tilted. The experimental stability diagram obtained by dye visualisations are compared to numerical results. At larger Reynolds numbers, the 2D von Karman vortex street leads to a 3D instability. Shadowgraph visualisations clearly reveal that the unstable mode is similar to the mode A well known in homogeneous cylinder wakes if the cylinder is vertical. This mode seems to be more unstable for moderate stratifications and more stable for strong stratifications. When the cylinder is tilted a new unstable mode appears at moderate Froude numbers, which exhibits thin undulated dark lines. This mode is due to a Kelvin-Helmholtz instability of the critical layer which appears in each tilted vortex of the von Karman street. Finally, at high Reynolds numbers, the wake becomes turbulent in the early stages for the case of a sphere. However, the late stages of the wake exhibit once again a von Karman street of flat horizontal vortices. The size and the velocity of the wake vary algebraically with time. These scaling laws can be predicted by a simple model of turbulent diffusion in the horizontal direction and of viscous diffusion in the vertical direction.