A 2D Model for 3D Periodic Deep-Water Waves

Dmitry Chalikov

doi:10.3390/jmse10030410

Abstract

The paper is devoted to further development of an accelerated method for simulation of the two-dimensional surface waves at infinite depth with the use of a two-dimensional model derived with simplifications of the three-dimensional equations for potential periodic deep-water waves. A 3D full wave model (FWM) is based on a numerical solution of a 3D Poisson equation written in the surface-fitted coordinates for a nonlinear component of the velocity potential. For sufficient vertical resolution used for the Poisson equation, the 3D model provides very high accuracy. The simplified model is based on the 2D Poisson equation written for a free surface. This exact equation contains both the first and second derivatives of the velocity potential, i.e., it is unclosed. The analysis of the accurate solutions for the 3D velocity potential obtained with the 3D model shows that those variables are linearly connected to each other. This property allows us to obtain a 2D equation for the first derivative of the velocity potential (i.e., vertical velocity on surface), which gives the closed 2D formulation for a 3D problem of two-dimensional waves. The previously developed scheme was not universal since the parameters of the closure scheme had to be adjusted to the specific setting. The current paper offers a new formulation of a closing scheme based on the integral parameters of wave field. The method of closing the equations, as well as the numerical parameters, were chosen on the basis of the multiple numerical experiments with the full nonlinear wave model (FWM) and selection of a suitable closing scheme. That is why the given model can be called Heuristic Wave Model (HWM). The connection between the first and second variables is not precise; hence, the method as a whole cannot be exact. However, the derived 2D model is able to reproduce different statistical characteristics of the 2D wave field with good accuracy. The main advantage of the model developed is its high performance exceeding that of 3D model by about two decimal orders.

Highlights

The current paper offers a new formulation of a closing scheme based on the integral parameters of wave field
A brief review of different 3D numerical methods developed for investigation of wave processes is given in [1]
full nonlinear wave model (FWM) for infinite depth is used as a basic model for building up a simplified model that allows for obtaining similar results at a much lower cost

Summary

Introduction

A brief review of different 3D numerical methods developed for investigation of wave processes is given in [1]. The multiple numerical calculations with an accurate 3D model [9] showed that the vertical profiles of a nonlinear e have quite a universal simple structure, which allows component of the velocity potential φ e and w eζ can be closely connected It was shown in (10) that dependence the suggestion that w e on w eζ is very close to the linear one, but the coefficients in such connection depend of w on external integral parameters of the wave field. PreviThe presence of w ously, the author regarded a surface condition (17) as a means of validation of the numerical scheme for the 3D Full Wave Model (FWM) based on a numerical solution of an equation for the velocity potential (see example of such calculation in [8]).

An Example of Simulation of the Wave Field Development on the Basis of HFM

Conclusions