LES of channel flow with wall roughness

We describe LES, using a spectral-element method, of turbulent channel flow with sub-grid wall roughness at moderate to large Reynolds numbers. The wall-model of Chung and Pullin ({\it JFM, 2009}) is extended to account for local subgrid roughness by the addition of a single empirical roughness function $u_\tau\,\Delta^+(k_s^+)$, where $k_s^+ = k_s\,u_\tau/\nu$ and $k_s$ is the equivalent sand roughness. This term is used in both the inner-scaling ansatz used to reduce the unsteady term of the wall-normal integration of the stream-wise momentum equation and also in the derivation of the ``log-like'' profile used to provide the outer slip velocity, as a boundary condition for the LES, at a raised virtual wall. In these LES, standard expressions for $\Delta^+(k_s^+)$ of the Colebrook type are used. The LES produce results that include the mean velocity profile and some turbulence statistics as a function of Reynolds number for fixed ratio of $k_s$ to the channel half height.