Quasi-Three-Dimensional Simulation of Crescent-Shaped Waves
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i-Three-Dimensional Simulation of Crescent-Shaped Waves A. S. Dosaev* Institute of Applied Physics, Russian Academy of Sciences, Nizhny Novgorod, 603950 Russia *e-mail: [email protected] Received March 26, 2019; revised March 23, 2020; accepted June 3, 2020
Abstract—Properties and limitations of a quasi-three-dimensional water-wave model put forward by V.P. Ruban and based on the assumption of narrow directional distribution of the wave spectrum are studied. Within the approximate equations of motion, a stability problem for a finite amplitude Stokes wave to threedimensional perturbations is considered. A zone of instability corresponding to five-wave interactions is examined. It is shown that, despite the fact that the corresponding modes consist of harmonics that are relatively far from the main direction, the increment values are close to those given by the exact equations of motion. The subsequent stages of the three-dimensional instability growth exhibit a plausible dynamics, leading to formation of crescent-shaped waves. A modification to cubic components of the Hamiltonian functional of the model is suggested that eliminates a spurious zone of instability for perturbations propagating almost perpendicularly to the main direction. Keywords: surface water waves, nonlinear waves, potential flow, conformal variables DOI: 10.1134/S0001433820050035
INTRODUCTION Many characteristic features of the dynamics of water waves can be explained in terms of the model of the potential flow of an incompressible fluid with a free surface. The potential flow is an object convenient for both analytical study and numerical simulation, because its motion can be described by integro-differential equations at the fluid boundary, which reduces the dimension of the problem. For the numerical simulation of two-dimensional potential waves, the conformal transform method is widely used. In conformal coordinates, the equations of motion are written in such a way (see, for example, [1]) that the computational complexity of evaluating the corresponding right-hand sides on a spatial grid consisting of N nodes is determined by the complexity of the fast Fourier transform O ( NlogN ). The advantages of the approach also include the ability to describe configurations with overhanging free surface (when elevation is not a single-valued function of the horizontal coordinate); beside the obvious example of breaking waves, this situation is also possible for nonbreaking waves of the gravity-capillary range [2]. A review of the history of the method and a detailed bibliography can be found in [3]. Three-dimensional modeling of potential waves within the exact equations of motion is possible via boundary-integral method [4, 5], which, however, has a rather high computational cost. Another approach to exact modeling of potential waves involves solving the
Laplace equation for the potential on a three-dimensional grid [6], while requiring that the free surface does not overhang. The quasi-three-dimensional wave model proposed by V.P. Ruban [7] is a ge
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