Optimum control of cavity flow

Abstract

For high Reynolds numbers Re = UL/nu, with an order of magnitude of a few thousands, a flow over a square cavity becomes unsteady with the growth of two-dimensional instabilities. This phenomenon is studied by computing : 1/ the branch of steady solutions with respects to the Reynolds number, using a branch tracking method ; 2/ the eigenvalues and eigenvectors of the global linearized operator with respects to Re. We thus show that the cavity is subject to a Hopf bifurcation at a critical Reynolds number denoted by Rec. After setting the computations in a supercritical case for which Re > Rec, we use an optimum control algorithm to minimize the energy of the perturbations at various terminal times T. The control will consist in unsteady blowing and succion on the cavity wall. We will analyze the phenomenology of the control law with a description of the influence of the target time T and the cost of the control which will be denoted by m.