Optimal wavy surface to suppress vortex shedding using second-order sensitivity to shape changes

A method to find optimal 2nd-order perturbations is presented, and applied to find the optimal spanwise-wavy surface for the suppression of cylinder wake instability. As shown in recent studies (Hwang et al., 2013, Tammisola et al., 2014, Del Guercio et al., 2014), 2nd-order perturbations are required to capture the stabilizing effect of spanwise waviness, which is ignored by standard adjoint-based sensitivity analyses. Here, previous methods are extended so that (i) 2nd-order sensitivity is formulated for base flow changes satisfying the linearized Navier-Stokes, and (ii) the resulting method is applicable to a 2D global instability problem. This makes it possible to formulate the 2nd-order sensitivity to shape modifications. This formulation is used to find the optimal shape to suppress the cylinder wake instability. The optimal shape is then perturbed by random distributions in full 3D stability analysis to confirm that it is a local optimal at the given amplitude and wavelength. At Re=100, surface waviness of maximum height 1% of the cylinder diameter is sufficient to stabilize the flow. The optimal surface creates streaks passively by extracting energy from the base flow derivatives and altering the tangential velocity component at the wall. This paper extends previous techniques to a fully two-dimensional method to find boundary perturbations which optimize the 2nd-order drift. The method should be applicable to generic flow instability problems, and to different types of control, such as boundary forcing, shape modulation or suction.