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

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.