Flow Properties of Fully Nonlinear Model Equations for Surface Waves

Abstract

The focus of this thesis is wave motion in shallow water. In particular, we investigate some properties of flows underneath long waves in shallow water and present the results in two parts. The first part contains a systematic derivation of four balance equations, namely mass, momentum, energy and tangent velocity at the free surface. The asymptotic derivation of the conservation laws is obtained due to the surface motion of long, fully nonlinear water waves. We use the Serre-Green- Naghdi system, which is an asymptotic, fully nonlinear, weakly dispersive wave model to describe the considered waves. It is found that the derived conservation equations are satisfied exactly by the solution of the Serre-Green-Naghdi system when the bottom is flat. In the case of varying depth, mass and momentum conservation equations are satisfied exactly and the energy conservation is satisfied in an approximate sense. Moreover, they all reduce correctly to the equivalent derivations in both the Boussinesq and the shallow water scalings. In the case of flat bottom, we find what appears to be a new conservation law in the full Euler system. This conservation laws involves the tangential velocity, and reduces to the well known fourth conservation law in the Serre-Green-Naghdi system. We also describe particle trajectories in the Serre-Green-Naghdi approximation, and we find that the particles associated with the Serre-Green-Naghdi equations experience a backward drift which is in conflict with the Stokes drift. In the second part, we apply balance laws associated with the Korteweg-de Vries equation to study the evolution of a shoaling wave. The employed nonlinear expression for energy flux eliminates the discontinuity of wave height which normally appears in such studies. The results show an increase in wave height due to the decrease in water depth and they are in good agreement with the numerical results based on full Euler computations.