Modeling and large-eddy simulation (LES) of a turbulent boundary layer over linearly-varying surface roughness

An empirical model is presented, after Rotta (1962), that describes the development of a fully-developed turbulent boundary layer in the presence of surface roughness with nominal roughness length-scale $k_s$ that varies with stream-wise distance $x$. For $Re_x = U_e(x)\,x/\nu$ large, use is made of the log-wake model of the local turbulent mean-velocity profile that contains the Hama roughness correction $\Delta U^+(k_s^+)$ for the asymptotic, fully rough regime. It is shown that the skin friction coefficient $C_f$ is constant in $x$ only for $k_s = \alpha x$, where $\alpha$ is a dimensionless number. For $U_e(x)=A\,x^m$, this then gives a two-parameter $(\alpha,m)$ family of solutions for boundary-layer flows that are self similar in the variable $z/(\alpha\,x)$ where $z$ is the wall-normal co-ordinate. Trends observed in this model are supported by wall-modeled LES of the zero-pressure-gradient turbulent boundary layer ($m=0$) at very large $Re_x$. It is argued that the present results suggest that, in the sense that $C_f$ is spatially constant and independent of $Re_x$, this class of flows can be interpreted as providing the asymptotically-rough equivalent of Moody-like diagrams for boundary layers in the presence of small-scale roughness.