Approximate velocity formula over mobile sediment bed induced by velocity-skewed waves and current

A velocity formula is proposed for flow over a mobile sediment bed induced by velocity-skewed waves and current. The formula is obtained by a separation of waves and current velocities and requires seven free variables related to free stream velocity and sediment characteristics. The formula includes two parts: (1) a wave part consisting of the free stream velocity and defect function, which considers phase lead, wave boundary layer thickness, and mobile bed level, and (2) a current part, which changes the wave part through the boundary layer thickness and mobile bed level, influenced by wave eddy viscosity. The wave part is superior to the current part such that the boundary layer characteristics are mainly determined by the waves. The wave process is divided into four quarter stages to model the asymmetries of parameters in velocity and acceleration. Under velocity-skewed waves and current conditions, velocity asymmetry and residual velocity are accounted for in the formula, especially the offshore residual velocity with a large phase lag and mobile bed effect with a small phase lag. Phase lead is shown to be non-constant in velocity-skewed oscillatory sheet flow and corresponds to an extra offshore residual velocity. •Velocity formula is proposed for sheet flow transport beneath wave and weak current.•Mobile bed, phase lag, boundary layer thickness and flow asymmetry are presented.•Wave-current interactions are separated.•Residual velocity and mobile sediment bed effect are performed.•Inconstant phase lead in velocity-skewed flow corresponds to extra residual velocity.