Quantitative coarse graining of laminar fluid flow penetration in rough boundaries

The interaction between a fluid and a wall is described with a certain boundary condition for the fluid velocity at the wall. To understand how fluids behave near a rough wall in a completely laminar flow regime, the fluid velocity at every point on the rough surface may be provided. This approach requires detailed knowledge of, and likely depends strongly on the roughness. Another approach of modelling the boundary conditions of a rough wall is to coarse grain and extract a penetration depth over which on average the fluid penetrates into the roughness. In this work, we examine the impact of well-defined patterned surfaces on the fluid flow behaviour. We considered two extreme cases: one with horizontal ridges and another with vertical ridges on the wall and an intermediate case with ridges at an angle on the wall. We show that for a broad range of periodic roughness patterns and relative flow velocities, a universal penetration depth function can be obtained. We obtain these results with experiments and complementary numerical simulations. We evaluate how this penetration depth depends on the various roughness parameters such as ridge depth, ridge spacing and ridge angle. Our results present a novel approach to investigating wall roughness boundary conditions by considering the penetration depth delta that captures the spatially averaged behaviour of the decaying velocity profile between the asperities. We find that this penetration depth delta can be rescaled into a simple exponential master curve delta = delta infinity (1 - e -kD/S ) for horizontal ridges with varying depth D and spacing S. A similar variation of delta with D and S is observed for vertical ridges, but with a smaller magnitude delta infinity , while for ridges at an angle, the penetration depth lies between the two extreme cases.