Reorganization of Coherent Vortices in Shear Layers under the Action of Solid-Body Rotation

Abstract

We examine the effects of solid-body rotation, characterized by an angular velocity $\vec \Omega$ , on a mixing layer and a plane wake (in a plane perpendicular to $\vec \Omega$) upon which is superposed a small three-dimensional random perturbation. Using Kelvin’s theorem in a frame rotating with $\vec \Omega$, and with the aid of arguments based either on the straining of absolute vortex filaments by the ambiant shear, or introducing in a simplified case an exact solution whose results are close to a three-dimensional linear stability analysis, it is proposed that rotation is always stabilizing (with respect to the non-rotating situation) in the cyclonic case. In the anticyclonic case, a slight rotation is destabilizing, with an intensification of the stretching of longitudinal hairpin vortices. This effect is maximum at a local Rossby number of unity according to the nonlinear theory. From the exact solution point of view, this value R _o = 1 is the crossover between stability and instability, as already predicted by former studies. At high rotation rates (low Rossby numbers), both cyclonic and anticyclonic vortices are stabilized, in agreement with the Taylor-Proudman theorem. In order to assess these theoretical predictions, we perform three-dimensional numerical simulations. We find a critical Rossby number (based on the maximum initial vorticity of the basic flow) at which maximum three-dimensional anticyclonic destabilization is achieved. In this case, a detailed examination of the structures shows that the flow is highly anisotropic with very elongated streamwise hairpin vortices of small spanwise wavelength. Finally, the same theoretical considerations applied to rotating developed turbulence permit to infer that, for turbulent Rossby numbers of the order of one, cyclonic vortices of axis parallel to.$\vec \Omega$ are reinforced, while anticyclonic vorticity is disrupted. We present calculations which support this conjecture in the case of initially quasi two-dimensional turbulence. The applications for geophysical or engineering flows are briefly discussed.