A Comparative Study on Generation and Propagation of Nonlinear Waves in Shallow Waters

Abstract

This study is concerned with the generation and propagation of strongly nonlinear waves in shallow water. A numerical wave flume is developed where nonlinear waves of solitary and cnoidal types are generated by use of the Level I Green-Naghdi (GN) equations by a piston-type wavemaker. Waves generated by the GN theory enter the domain where the fluid motion is governed by the Navier–Stokes equations to achieve the highest accuracy for wave propagation. The computations are performed in two dimensions, and by an open source computational fluid dynamics package, namely OpenFoam. Comparisons are made between the characteristics of the waves generated in this wave tank and by use of the GN equations and the waves generated by Boussinesq equations, Laitone’s 1st and 2nd order equations, and KdV equations. We also consider a numerical wave tank where waves generated by the GN equations enter a domain in which the fluid motion is governed by the GN equations. Discussion is provided on the limitations and applicability of the GN equations in generating accurate, nonlinear, shallow-water waves. The results, including surface elevation, velocity field, and wave celerity, are compared with laboratory experiments and other theories. It is found that the nonlinear waves generated by the GN equations are highly stable and in close agreement with laboratory measurements.

Keywords

• Green–Naghdi equations
• Navier–Stokes equations
• Boussinesq equations
• KdV equation
• solitary wave
• cnoidal wave