Wall-Roughness Eddy Viscosity for RANS Closures

In turbulent boundary layers over rough walls, a triple decomposition of variables into mean, spatially varying, and fluctuating components, together with a double-averaging procedure, can be used to identify extra terms in the $x$-momentum equation which describe the pressure and viscous drag forces per unit mass \textit{on account of roughness}. These terms, which are significant only in the roughness sublayer, have been measured in the DNS of Yuan \& Piomelli (2014). In this talk, we describe a model for the effects of these terms as if they make an additive contribution to the eddy viscosity within the roughness sublayer, by recasting them as a simple algebraic function of the dimensionless roughness scale $k_s^+$, for which a theoretical rationale is proposed. With this modification, $k$--$\epsilon$ and $k$--$\epsilon$--$v^\prime v^\prime$--$f$ closures for turbulent channel flow yield mean velocity profiles and friction factors which are in good agreement with measured data in rough-wall duct flows over a wide range of values of $k_s^+$ and $\hbox{Re}$.