Rough-wall turbulent boundary layers with constant skin friction

A semi-empirical model is presented that describes the development of a fully developed turbulent boundary layer in the presence of surface roughness with length scale k_s that varies with streamwise distance x. Interest is centred on flows for which all terms of the von Kármán integral relation, including the ratio of outer velocity to friction velocity U⁺_∞ ≡ U_∞/u_ζ, are streamwise constant. For Rex assumed large, use is made of a simple log-wake model of the local turbulent mean-velocity profile that contains a standard mean-velocity correction for the asymptotic fully rough regime and with assumed constant parameter values. It is then shown that, for a general power-law external velocity variation U_∞ ∼ x^m, all measures of the boundary-layer thickness must be proportional to x and that the surface sand-grain roughness scale variation must be the linear form k_s(x) = αx, where x is the distance from the boundary layer of zero thickness and is a dimensionless constant. This is shown to give a two-parameter (m,α) family of solutions, for which U⁺_∞ (or equivalently C_f) and boundary-layer thicknesses can be simply calculated. These correspond to perfectly self-similar boundary-layer growth in the streamwise direction with similarity variable z/(αx), where z is the wall-normal coordinate. Results from this model over a range of α are discussed for several cases, including the zero-pressure-gradient (m = 0) and sink-flow (m =-1) boundary layers. Trends observed in the model are supported by wall-modelled large-eddy simulation of the zero-pressure-gradient case for Re_x in the range 10⁸-10¹⁰ and for four values of α. Linear streamwise growth of the displacement, momentum and nominal boundary-layer thicknesses is confirmed, while, for each α, the mean-velocity profiles and streamwise turbulent variances are found to collapse reasonably well onto z/(αx). For given α, calculations of U⁺_∞ obtained from large-eddy simulations are streamwise constant and independent of Re_x when this is large. The present results suggest that, in the sense that U⁺_∞ (α, m) is constant, these flows can be interpreted as the fully rough limit for boundary layers in the presence of small-scale linear roughness.