Numerical simulation of mean cross-shore currents: A stream function approach

A numerical model applying a stream function formulation is developed and tested to simulate the two-dimensional, wave-induced return flow in the nearshore region, commonly referred to as the undertow. The model is developed using the stream function, ψ, to represent the wave-induced return flow and tests are conducted of the effects of various wave theories and boundary conditions on the flow. Solution of ψ produces the mean velocity in the horizontal and vertical directions, providing a description of the complete flow field in the two-dimensional vertical plane perpendicular to the shoreline.; The governing equation, a fourth-order partial differential equation in ψ, is not specific to wave-induced flows, therefore, the details of the wave forcing are introduced entirely through the boundary conditions. In this way the model provides a convenient platform to evaluate different combinations of applied boundary conditions. The model includes the ability to study spatial variations in the turbulent eddy viscosity. The present model also accommodates monochromatic or random wave fields and arbitrary bottom profiles, allowing the modeling of flows over bar-trough beach profiles or any other irregular profile geometry.; Comparison to measurements of wave-induced return flows demonstrates that the model is able to predict the vertical structure and magnitude of the return flow to within 25% to 30% (typical) for most applications. If the return flow rate is properly predicted (or fit), the model can predict the vertical structure of the horizontal velocity to within 10% (typical). Simulation of six laboratory and field experiments demonstrates the capability of the model and highlights the challenges of return flow modeling. The simulations indicate that the most important parameter in return flow modeling is the proper prediction of the wave-induced volumetric transport. Further improvement of the predictive capability of the model inside the surf zone stems from the application of a vertically varying eddy viscosity field based on the turbulence generated by breaking waves. Comparison of the vertical structure of the return flow seaward and landward of the break point illustrates differences in the appropriate boundary conditions and eddy viscosity distributions that should be applied in return flow modeling.