Abstract
This paper is concerned with the simulation of periodic traveling deep-water free-surface water waves under the influence of gravity and surface tension in two and three dimensions. A variety of techniques is utilized, including the numerical simulation of a weakly nonlinear model, explicit solutions of low-order perturbation theories, and the direct numerical simulation of the full water wave equations. The weakly nonlinear models which we present are new and extend the work of Akers and Milewski [SIAM J. Appl. Math., 70 (2010), pp. 2390–2408] to arbitrary Bond number and fluid depth. The numerical scheme for the full water wave problem features a novel extension of the “Transformed Field Expansions” method of Nicholls and Reitich [Euro. J. Mech. B Fluids, 25 (2006), pp. 406–424] to accommodate capillary effects in a stable and rapid fashion. The purpose of this paper is apply the new numerical method, then compare small amplitude solutions of potential flow with those of the approximate model. Particular attention is paid to the behavior near quadratic resonances, an example of which is the Wilton ripple.
Highlights
Traveling water waves have been studied for over a century, most famously by Stokes, for whom weakly nonlinear periodic waves are named [1, 2]
In this paper we investigate periodic traveling waves in the potential flow equations (2.2), and a weakly nonlinear model which we derive below
Solutions to the potential flow equations are computed using an extension of the method of Transformed Field Expansions [14]
Summary
Traveling water waves have been studied for over a century, most famously by Stokes, for whom weakly nonlinear periodic waves are named [1, 2] In his 1847 paper, Stokes expanded the wave profile as a power series in a small wave slope parameter, a technique which has since become commonplace. This classic perturbation expansion, which we will refer to as the Stokes expansion, has since been applied to the water wave problem numerous times [3, 4, 5, 6, 7, 8]. The goal is to compare solutions of the potential flow equations to those of a quadratic nonlinear model, in the neighborhood of resonances which occur at quadratic order in a perturbation series expansion. Perturbation series expansions about a Stokes wave will not be valid; the quadratic model may provide a better approximation
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