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Recent simulations indicate that streamwise-preferential porous materialshave the potential to reduce drag in wall-bounded turbulentflows(Gomez-de-Segura & Garcia-Mayoral 2019). This paper extends the resolventformulation to study the effect of such anisotropic permeable substrates onturbulent channel flow. Under the resolvent formulation, theFourier-transformed Navier-Stokes equations are interpreted as a linearforcing-response system. The nonlinear terms are considered the endogenousforcing in the system that gives rise to a velocity and pressure response. Again-based decomposition of the forcing-response transfer function---theresolvent operator---identifies response modes (resolvent modes) that are knownto reproduce important structural and statistical features of wall-boundedturbulent flows. The effect of permeable substrates is introduced in thisframework using the Volume-Averaged Navier-Stokes equations and a generalizedform of Darcy's law. Substrates with high streamwise permeability and lowspanwise permeability are found to suppress the forcing-response gain for theresolvent mode that serves as a surrogate for the energetic near-wall cycle.This reduction in mode gain is shown to be consistent with the drag reductiontrends predicted by theory and observed in numerical simulations. Simulationresults indicate that drag reduction is limited by the emergence of spanwiserollers resembling Kelvin-Helmholtz vortices beyond a threshold value ofwall-normal permeability. The resolvent framework also predicts the conditionsin which such energetic spanwise-coherent rollers emerge. These findingssuggest that a limited set of resolvent modes can serve as the building blocksfor computationally-efficient models that enable the design and optimization ofpermeable substrates for passive turbulence control.

Resolvent-based predictions for turbulent flow over anisotropic permeable substrates