Two-Dimensional Modeling of Three-Dimensional Waves

An approximate method of direct modeling of three-dimensional surface waves based on the complete equations of the potential motion of a liquid with a free surface in a curved non-stationary coordinate system is proposed. The separation of the velocity potential into nonlinear and linear components is used. The two-dimensional equations of the model are derived on the basis of an exact three-dimensional equation for the nonlinear component of the velocity potential written on the surface. The equation contains the first and second vertical derivatives of the potential on the surface; thus, the system of equations turns out to be unclosed. The analysis of the results of accurate three-dimensional modeling allowed us to establish that the first and second derivatives are linearly related to each other. This connection allows a closed two-dimensional (surface) formulation of the problem of three-dimensional waves. The first derivative of the potential (i.e., the vertical velocity on the surface) is calculated from the equation for the velocity potential on the surface using iterations. The relationship between the derivatives of the potential is not completely accurate, so in general, the method is approximate. Nevertheless, the model accurately reproduces the evolution of the wave field and its main statistical characteristics. The most obvious advantage of the model is its high efficiency: the speed of integration of a two-dimensional model is about two orders of magnitude higher than the speed of an equivalent three-dimensional model. The model is designed to quickly reproduce the dynamics of a two-dimensional wave field based on information about the wave spectrum.