Modelling of undertow by a one-equation turbulence model

Abstract A model for the circulation current in the vertical plane in the surf zone, the so-called undertow, is formulated. The model describes the time-averaged shear stresses caused by the wave breaking, which drives the current, and the resulting velocity distribution. The time-averaged shear stress is composed of the following contributions: (a) the radiation stress gradient, including the vertical momentum flux of the nonuniform wave motion; (b) the momentum associated with the surface rollers; (c) the streaming in the wave boundary layer, calculated from the actual simulated velocities in the wave boundary layer; (d) the contribution due to the slope of the mean water surface, the set-up. The set-up is a free parameter, which is determined so that the resulting mean current velocity profile fulfills the continuity equation. The time-averaged velocity distribution is determined by a flow model based on a one-equation turbulence model to calculate the eddy viscosity. Two contributions are included in the description of the production of turbulence, the normal shear production, which is dominant in the wave boundary layer, and the production from the broken waves, which dominates away from the bed. The model simulates the unsteady flow during a wave cycle. The model results have been compared with laboratory measurements from the literature of turbulence in the surf zone and mean current velocity profiles with and without a net cross-shore discharge.