Turbulent drag reduction by anisotropic permeable substrates $-$ analysis and direct numerical simulations

We explore the ability of anisotropic permeable substrates to reduce turbulent skin-friction, studying the influence that these substrates have on the overlying turbulence. For this, we perform DNSs of channel flows bounded by permeable substrates. The results confirm theoretical predictions, and the resulting drag curves are similar to those of riblets. For small permeabilities, the drag reduction is proportional to the difference between the streamwise and spanwise permeabilities. This linear regime breaks down for a critical value of the wall-normal permeability, beyond which the performance begins to degrade. We observe that the degradation is associated with the appearance of spanwise-coherent structures, attributed to a Kelvin-Helmholtz-like instability of the mean flow. This feature is common to a variety of obstructed flows, and linear stability analysis can be used to predict it. For large permeabilities, these structures become prevalent in the flow, outweighing the drag-reducing effect of slip and eventually leading to an increase of drag. For the substrate configurations considered, the largest drag reduction observed is $\approx 20-25\%$ at a friction Reynolds number $\delta^+ = 180$.