Bounds on the energy stability limit of plane parallel shear flows

Abstract

The variational problem determining the energy stability limit ReE of the Reynolds number Re is analyzed for plane parallel shear flows of viscous incompressible fluids. The perturbations are assumed to be periodic in the horizontal directions and are two-dimensional, varying either streamwise or spanwise, or three-dimensional. We derive for these 2D and 3D limits various estimates valid for arbitrary (normalized) shear profiles: The 3D limit is estimated from below in terms of the 2D limits, the streamwise limit is shown to be always greater than more than half of the spanwise limit, and the 2D limits are estimated from below by explicit numbers. To test the quality of the estimates they are applied to Couette- and Poiseuille flow, where the exact values of the limits are known.