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

José Simão Antunes Do Carmo ,José Simão

doi:10.3390/modelling2040033

Abstract

Numerical models are useful tools for studying complex wave–wave and wave–current interactions in coastal areas. They are also very useful for assessing the potential risks of flooding, hydrodynamic actions on coastal protection structures, bathymetric changes along the coast, and scour phenomena on structures’ foundations. In the coastal zone, there are shallow-water conditions where several nonlinear processes occur. These processes change the flow patterns and interact with the moving bottom. Only fully nonlinear models with the addition of dispersive terms have the potential to reproduce all phenomena with sufficient accuracy. The Boussinesq and Serre models have such characteristics. However, both standard versions of these models are weakly dispersive, being restricted to shallow-water conditions. The need to extend them to deeper waters has given rise to several works that, essentially, add more or fewer terms of dispersive origin. This approach is followed here, giving rise to a set of extended Serre equations up to kh ≈ π. Based on the wavemaker theory, it is also shown that for kh > π/10, the input boundary condition obtained for shallow-waters within the Airy wave theory for 2D waves is not valid. A better estimate for the input wave that satisfies a desired value of kh can be obtained considering a geometrical modification of the conventional shape of the classic piston wavemaker by a limited depth θh, with θ≤ 1.0.

Highlights

Numerical models are useful tools for studying complex wave–wave and wave–current interactions in coastal areas
Serre’s standard equations are only valid for shallow-water conditions; it becomes necessary to develop an extension of these equations for applications at intermediatedepths and quasi-deep-waters
A new computational structure is proposed in this work, composed of a set of Serre equations enhanced with additional terms of dispersive origin

Summary

Introduction

Numerical models are useful tools for studying complex wave–wave and wave–current interactions in coastal areas. As noted in [6], at that time, the numerical simulation of coastal processes was carried out using models that solved Saint-Venant-type equations [7]. Such models are unable to simulate phenomena of dispersive origin, which are typical of shallow-waters and for certain types of waves. In addition to the common processes of refraction and diffraction, the phenomena of the swelling, reflection, and breaking of waves are typical of shallow-waters Such phenomena overlap, so it is the combined action of all of them that produces the flow patterns typical of coastal areas. Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations

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