Conservation equations for open-channel flow: effects of bed roughness and secondary currents

Time-averaged velocity fields in uniform open-channel flows over rough beds may exhibit spatial heterogeneities due to the effects of bed roughness and secondary currents (SCs). The latter typically originate from the turbulence anisotropy and spatial heterogeneity introduced by the solid and mixed corners (i.e., between sidewalls and water surface), but may also appear due to roughness spanwise heterogeneities, e.g., associated with patchy vegetation distributions or streamwise sediment ridges on the channel bed. In this paper, we propose rigorous conservation equations for momentum, kinetic energy and fluid stresses accounting for the contributions of bed roughness and SCs, separately. Particular attention is given to the terms regulating the energy exchanges between roughness-induced and SC-related motions, which are expected to provide information on the physical mechanisms leading to the generation of roughness-induced SCs. The proposed approach is illustrated using a large-eddy simulation of a rough-bed open-channel flow.