Spatially localized convolution kernels for feedback control of transitional flows

Optimal (ℋ₂) linear feedback controllers are computed for the Orr-Sommerfeld/Squire equations for an array of wavenumber pairs {k_x, k_z} and then inverse-transformed to the physical domain. The feedback kernels minimize both transient energy growth and input-output transfer function norms in the controlled linear system representing small perturbations to a laminar channel flow. It is shown that this calculation yields feedback convolution kernels with localized physical-domain support. These kernels decay exponentially with distance from the actuator, so they can be truncated a finite distance from each actuator while retaining any desired degree of accuracy. The truncated kernels may then be used in decentralized control of distributed flow. Spatial localization provides the critical link which connects controllers designed for the spatially periodic model system to application on spatially evolving physical systems. Not all formulations of the present problem lead to physical-space controllers with localized spatial support. The feedback convolution kernels so determined are then implemented in direct numerical simulations of transitional flows with both random and oblique-wave finite magnitude initial flow perturbations. The ability of the linear control feedback to stabilize the nonlinear flow system is demonstrated for finite initial flow perturbations with magnitudes well beyond the threshold which induces transition to turbulence in the uncontrolled system