Modeling of Wave Propagation from Arbitrary Depths to Shallow Waters – A Review

Abstract

Numerical models are a useful instrument for studying complex superposition of wave-wave and wave-current interactions in coastal and estuarine regions and to investigate the interaction of waves with complex bathymetries or structures built in nearshore areas. Moreover, since their applications are significantly less expensive and more flexible than the construction of physical models, they are a convenient tool to support design. The ability of the standard "oussinesq and Serre or Green and Naghdi equations to reproduce these nonlinear processes is well known. However, these models are restricted to shallow water conditions, and addition of other terms of dispersive origin has been considered since the s, particularly for approximations of the "oussinesq-type. To allow applications in a greater range of ν / , other than shallow waters, where ν is the water depth at rest and λ is the wavelength, a new set of Serre-type equations, with additional terms of dispersive origin, is developed and tested with the available data and with a numerical solution of a "oussinesq-type model, also improved with dispersive characteristics. Δx-plicit and implicit methods of finite-difference are implemented to solve both approximations of "oussinesq and Serre types with improved dispersive characteristics. " finite element method is also implemented to solve an extension of a "oussinesq-type model that takes into account wave-current interactions. "pplication examples to solve real-world problems are shown and discussed. The performances of both HΓ models are compared with experimental data of very demanding applications, namely ξ a highly nonlinear solitary wave propagating up a slope and reflecting from a vertical wall and ξξ a periodic wave propagating in intermediatedepth waters upstream a trapezoidal bar, followed by very shallow waters over the bar, and again in intermediate-depth water conditions downstream.

Remarks on the Boundary Conditions for a Serre-Type Model Extended to Intermediate-Waters

“…where η is the free surface elevation; ξ represents bathymetry; u, v, and w are velocity components; and p is the pressure. After some mathematical developments, the following dimensionless equations of motion are obtained in the second approach (order 2 in σ, σ 2 ) [33,34]:…”