Resolvent-informed design of anisotropic permeable substrates for turbulence control

We utilize an extended version of the resolvent formulation to design anisotropic permeable substrates for passive turbulence control. The resolvent formulation interprets the Fourier transformed Navier-Stokes equations as a forcing-response system: the linear terms map the action of the nonlinear terms to a velocity and pressure response. A gain-based decomposition of the forcing-response transfer function (the resolvent operator) identifies flow features (resolvent modes) that reproduce important structural and statistical features of wall-bounded turbulent flows. One particular resolvent mode serves as a useful surrogate for the dynamically important near-wall cycle. The effect of permeable substrates is introduced in this framework using the Volume-Averaged Navier-Stokes equations. Substrates with high streamwise permeability and low wall-normal permeability are found to suppress the near-wall resolvent mode, which is consistent with conditions in which drag reduction has been observed in recent numerical simulations. Performance deteriorates when high-gain spanwise constant rollers resembling Kelvin-Helmholtz vortices emerge over the porous medium. A parametric study is used to identify permeability combinations that have drag reduction potential.