Theoretical evaluation of the Reynolds shear stress and flow parameters in transitionally rough turbulent boundary Layers

Abstract

The theory by W.K. George and L. Castillo (Zero-pressure gradient turbulent boundary layers, Appl. Mech. Rev. 50 (1997), pp. 689–729) is extended for rough surfaces and numerically implemented on zero pressure gradient turbulent boundary layers. The method is based on the similarity transformations of the Navier–Stokes equations. From these equations, a composite profile for the Reynolds shear stress-⟨ uv⟩ is obtained over the entire boundary layer. The solution is in good agreement with the experiments in the inner and outer regions, for hydraulically smooth (k+ < 5) and transitionally rough surfaces up to roughness parameter of k+ ≈ 55. Beyond this limit, the accuracy of the solutions decreases with k+, especially in the inner region of the mean velocity. However, the accuracy always increases with the Reynolds number, Re_θ. In addition, the eddy viscosity ⟨ ν_T⟩/ν and flow parameters, including the x-dependence, have also been computed and tested with experimental data. Furthermore, the friction power law for smooth/rough surfaces has been used for all calculations and comparisons with direct methods and velocity-based methods are shown to be in good agreement with the theory.