Fourier-based sparse H₂ design of riblet geometry

The drag-reducing properties of streamwise-elongated surface corrugations, commonly referred to as riblets, have been extensively studied through experiments, simulations, and reduced-order modeling. Notably, the linearized Navier–Stokes framework introduced by Ran, Zare, and Jovanovic (J. Fluid Mech., vol. 906, 2021) employs volume penalization to model riblet-induced boundary conditions and predict drag reduction. These approaches typically quantify drag reduction as a function of a single geometric parameter, such as spanwise spacing or groove cross-sectional area, evaluated over fixed riblet shapes. As a result, identifying optimal riblet geometries across a broader design space often necessitates exhaustive parametric studies. To address this problem, we propose an optimization framework for the systematic design of optimal riblet shapes. Our approach reformulates the volume penalization term in the linearized dynamics as a structured state-feedback operator, expanded in a basis of Fourier modes representing the surface roughness profile. Targeting maximal suppression of turbulence, we pose a composite optimization problem that minimizes the steady-state variance of velocity fluctuations while promoting geometric compactness through sparsity in the Fourier basis. For a fixed spanwise wavelength (riblet spacing), the framework yields optimal wall-normal profiles that achieve maximal turbulence attenuation.