  Recent simulations indicate that streamwise-preferential porous materials
have the potential to reduce drag in wall-bounded turbulent
flows(Gomez-de-Segura & Garcia-Mayoral 2019). This paper extends the resolvent
formulation to study the effect of such anisotropic permeable substrates on
turbulent channel flow. Under the resolvent formulation, the
Fourier-transformed Navier-Stokes equations are interpreted as a linear
forcing-response system. The nonlinear terms are considered the endogenous
forcing in the system that gives rise to a velocity and pressure response. A
gain-based decomposition of the forcing-response transfer function---the
resolvent operator---identifies response modes (resolvent modes) that are known
to reproduce important structural and statistical features of wall-bounded
turbulent flows. The effect of permeable substrates is introduced in this
framework using the Volume-Averaged Navier-Stokes equations and a generalized
form of Darcy's law. Substrates with high streamwise permeability and low
spanwise permeability are found to suppress the forcing-response gain for the
resolvent 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 reduction
trends predicted by theory and observed in numerical simulations. Simulation
results indicate that drag reduction is limited by the emergence of spanwise
rollers resembling Kelvin-Helmholtz vortices beyond a threshold value of
wall-normal permeability. The resolvent framework also predicts the conditions
in which such energetic spanwise-coherent rollers emerge. These findings
suggest that a limited set of resolvent modes can serve as the building blocks
for computationally-efficient models that enable the design and optimization of
permeable substrates for passive turbulence control.
