Two-Dimensional Wave Modeling Based on Conformal Mapping

Abstract 

High accuracy was confirmed by validation of a non-stationary model against known solutions and by comparison between the results obtained with different resolutions in the horizontal. The method developed is applied for simulation of wave evolution with different initial conditions. The numerical experiments with the initially monochromatic waves of different steepness show that the model is able to simulate the breaking conditions when the surface becomes a multi-valued function of the horizontal coordinate. An estimate of the critical initial wave height that separates non-breaking and eventually breaking waves is obtained. Simulation of nonlinear evolution of a wave field represented initially by two modes with close wave numbers (amplitude modulation) and a wave field with a phase modulation is given. Both runs result in appearance of large and very steep waves. Both of them also break if the initial amplitudes are sufficiently large. The interaction of two monochromatic waves at water surface enters a different dynamic regime if their wave numbers become very close. In the course of evolution of two waves, downshifting of the initial wave energy and growth of the first mode occur depending on wave steepness and a relative distance between modes in Fourier space.