A method to find optimal 2nd-order perturbations is presented, and applied to
find the optimal spanwise-wavy surface for suppression of cylinder wake
instability. Second-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 linearised
Navier-Stokes, and (ii) the resulting method is applicable to a 2D global
instability problem. This makes it possible to formulate 2nd-order sensitivity
to shape modifications. Using this formulation, we find the optimal shape to
suppress the a 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. Furthermore, it is shown
that none of the 10 random wavy shapes alone stabilize the wake flow at Re=50,
while the optimal shape does. At Re=100, surface waviness of maximum height 1%
of the cylinder diameter is sufficient to stabilize the flow. The optimal
surface creates streaks by passively extracting energy from the base flow
derivatives and effectively altering the tangential velocity component at the
wall, as opposed to spanwise-wavy suction which inputs energy to the normal
velocity component at the wall. This paper presents a fully two-dimensional and
computationally affordable method to find optimal 2nd-order perturbations of
generic flow instability problems and any boundary control (such as boundary
forcing, shape modulation or suction).Comment: 19 pages, 6 figure