Wave evolution over submerged sills: tests of a high-order Boussinesq model

A Boussinesq model accurate to O(μ)⁴, μ=k₀h₀ in dispersion and retaining all nonlinear effects is derived for the case of variable water depth. A numerical implementation of the model in one horizontal direction is described. An algorithm for wave generation using a grid-interior source function is derived. The model is tested in its complete form, in a weakly nonlinear form corresponding to the approximation δ=O(μ²), δ=a/h₀, and in a fully nonlinear form accurate to O(μ²) in dispersion Wei, G., Kirby, J.T., Grilli, S.T., Subramanya R. (1995). A fully nonlinear Boussinesq model for surface waves: Part 1. Highly nonlinear unsteady waves. J. Fluid Mech., 294, 71–92]. Test cases are taken from the experiments described by Dingemans Dingemans, M.W. (1994). Comparison of computations with Boussinesq-like models and laboratory measurements. Report H-1684.12, Delft Hydraulics, 32 pp.] and Ohyama et al. Ohyama, T., Kiota, W., Tada, A. (1994). Applicability of numerical models to nonlinear dispersive waves. Coastal Engineering, 24, 297–313.] and consider the shoaling and disintegration of monochromatic wave trains propagating over an elevated bar feature in an otherwise constant depth tank. Results clearly demonstrate the importance of the retention of fully-nonlinear effects in correct prediction of the evolved wave fields.