The objective of the present thesis is to answer the question: `can anisotropic permeable substrates reduce turbulent skin-friction drag?'

The first part of the thesis aims to extend the existing understanding on how complex surfaces of small texture size can reduce drag. We show that the effect of these surfaces can be reduced to an offset between the apparent, virtual smooth wall perceived by the mean flow and that perceived by the overlying turbulence, but turbulence remains otherwise smooth-wall-like. The drag reduction produced by these surfaces is therefore proportional to the difference between the two virtual origins.

In the second part of the thesis, we study the influence that anisotropic permeable substrates have on the overlying turbulence and show the potential of these substrates to reduce drag. For this, we conduct direct numerical simulations of channel flows bounded by permeable substrates. For small permeabilities, we observe a linear regime, where drag reduction is proportional to the aforementioned offset between the virtual origin perceived by the mean flow and that perceived by turbulence. For the substrates under study, the former is governed by the streamwise permeability and the latter by the spanwise permeability. This linear regime breaks down as spanwise-coherent structures begin to appear, which increase the turbulent mixing and consequently increase the drag. These structures are attributed to a Kelvin-Helmholtz-like instability of the mean flow, a common feature to a variety of obstructed flows, and their onset can be predicted using a linear stability analysis. This analysis shows, and the simulations corroborate, that the governing parameter for the breakdown is the wall-normal permeability. As this permeability increases, the drag-increasing, spanwise-coherent structures become prevalent in the flow, outweighing the drag-reducing effect of the virtual origins and eventually leading to an increase of drag. Based on the virtual-origin theory and the linear stability analysis, we build a predictive model for the behaviour of drag-reducing substrates, which estimates, with good accuracy, the drag reduction observed in the simulations. The present results and the models we subsequently developed provide guidelines for the design of drag-reducing permeable substrates. For the substrate configurations considered, the largest drag reduction observed is $\approx 20-25\%$ at a friction Reynolds number $\delta +=180$, which is at least twice that obtained for riblets.