The hydraulically smooth limit of flow over surfaces with spanwise heterogeneity

We study turbulent flows over surfaces featuring alternating streamwise strips with shear-free and no-slip wall boundary conditions via direct numerical simulations. The pattern wavelength (l) varies from 200 to 2 wall units, reaching the hydraulically smooth limit. The surface slip leads to drag reduction by giving rise to a slip velocity at the wall, but the surface spanwise heterogeneity contributes to drag increase by introducing secondary flows and more energetic turbulence, which can be parametrized with the roughness function. The results show that the slip at the wall dominates over the roughness function in most cases, leading to drag reduction, but the roughness function turns more dominant when l is within a few wall units, giving rise to net drag increase in the supposed hydraulically smooth regime. The drag increase is unexpected as the flat surface is a mixture of no-slip and slip conditions. To gain an insight into the mechanism responsible for the drag increase, we derive a decomposition of the roughness function based on the Navier-Stokes equations. The decomposition shows that turbulence and secondary flows contribute to the roughness function. However, when the secondary flows are suppressed as l is reduced to a few wall units, turbulence persists and contributes to the roughness function, leading to a positive roughness function and overall drag increase in the hydraulically smooth regime.