Resolving turbulence and drag over textured surfaces using texture-less simulations: the case of slip/no-slip textures

Abstract

We study the effect of surface texture on an overlying turbulent flow for textures made of an alternating slip/no-slip pattern, a common model for superhydrophobic surfaces, but also a particularly simple form of texture. For texture sizes $L^+ \gtrsim 25$, the texture effectively imposes homogeneous slip boundary conditions on the overlying, background turbulence, but this is not its sole effect. The effective conditions only produce an origin offset on the background turbulence, which remains otherwise smooth-wall-like. For actual textures, however, for $L^+ \gtrsim 25$ the flow progressively departs from this smooth-wall-like regime, resulting in additional shear Reynolds stress and increased drag, in a non-homogeneous fashion not reproduced by the effective boundary conditions. We focus on the underlying physical mechanism of this phenomenon. We argue that it is caused by the non-linear interaction of the texture-coherent flow, directly induced by the surface topology, and the background turbulence, as it acts directly on the latter and alters it, not at the boundary where effective conditions are imposed, but within the flow itself. The interaction acts as a forcing on the governing equations of the background turbulence, and takes the form of cross-advective terms between the latter and the texture-coherent flow. We show this with simulations with the texture removed and additional, forcing terms in the Navier-Stokes equations, in addition to the effective boundary conditions. The forcing captures the effect of the non-linear interaction on the background turbulence without the need to resolve the texture. When the forcing is derived accounting for the background turbulence amplitude-modulating the texture-coherent flow, it captures the changes in the flow up to $L^+ \approx 70$--$100$, including the roughness function and also the changes in the flow statistics and structure.