Optimization of two dimensional riblet geometries using the Resolvent analysis.*

We utilize an extended version of the resolvent formulation proposed by McKeon and Sharma (2010, J. Fluid Mech.) for the optimization of patterned walls for passive turbulence control. Under the resolvent analysis, the Navier-Stokes equations are interpreted as a forcing-response system: the nonlinear convective terms are considered to be a forcing to the linear system, generating a velocity and pressure response. A gain-based decomposition of the forcing-response transfer function---the resolvent operator---yields a set of highly amplified velocity and pressure modes. Previous work has shown that these high-gain modes reproduce statistical and structural features for smooth-wall flows. In the extended formulation, the effect of complex walls is introduced into the governing equations using a volume penalization technique. A gain-based decomposition of this modified system reproduces the deterioration in performance observed in previous direct numerical simulations for channel flow over rectangular riblets of increasing size. Building on these tests, we parametrically study the effect of triangular riblets of varying size and shape to identify optimal geometries.