Law of the Wall Formulation for Separating Flow Over a Rough Hill

Abstract

Simple asymptotic arguments together with the mixing length hypothesis are used to derive a local analytical solution that accounts for adverse pressure gradient and separation effects for flows over rough walls. To model the wall roughness effects, a new parametrization method is advanced for the proposed solution in terms of the friction velocity (u∗), the local pressure gradient (∂xp) and the roughness length (z0). The near-wall solution and the parametrization function are validated against some DNS (smooth wall flow) and LDA (smooth and rough wall flows) data. The experiments were carried out in a water-channel and the extent of separated flow was made to vary as a function of the roughness and the Reynolds number. Global optimization algorithms based on four different direct search methods are used to assess the relevant flow parameters. The analysis includes regions of attached as well as separated flow. Forty two velocity profiles are tested against the proposed expression for the parametrization function, including two profiles that satisfy the solution of Stratford. The present theory furnishes consistent results that might be implemented in predictive numerical models for complex flows.