A priori roughness function from texture-less simulations: the case of slip/no-slip textures

We study the effect of surface texture on an overlying turbulent flow. For very small textures, we have previously discussed that turbulence remains smooth-wall-like, other than experiencing an apparent origin offset for different flow components. For slip/no-slip textures, a common model for superhydrophobic surfaces, this effect reduces to the flow experiencing slip conditions in the streamwise and spanwise directions and zero transpiration at the surface. We have previously reported that these effective boundary conditions for the overlying turbulence persist at least up to texture sizes L⁺ ≈ 50. However, beyond L⁺ ≈ 20 the texture interacts with the overlying turbulence in a non-homogenous fashion, and turbulence is no longer smooth-wall-like. This is the typical effect of surface texture observed in rough surfaces, and results in a degradation of drag. We argue that the texture modifies the overlying turbulence through cross-advective terms between the background turbulence and the flow directly induced by the presence of the texture. To verify this, we introduce additional, forcing terms in the Navier-Stokes equations in homogenous-slip-length simulations that capture the effect of the non-linear interaction on the background turbulence. The interaction can then be accounted for without the need to resolve the surface texture. We show that this captures the changes both in drag and turbulent statistics and spectra up to texture sizes L⁺ ≈ 70.