On the limiting Stokes wave of extreme height in arbitrary water depth

Xiaoxu Zhong,Shijun Liao

doi:10.1017/jfm.2018.171

Abstract

Both Schwartz (J. Fluid Mech., vol. 62 (3), 1974, pp. 553–578) and Cokelet (Phil. Trans. R. Soc. Lond., vol. 286 (1335), 1977, pp. 183–230) failed to gain convergent results for limiting Stokes waves in extremely shallow water by means of perturbation methods, even with the aid of extrapolation techniques such as the Padé approximant. In particular, it is extremely difficult for traditional analytic/numerical approaches to present the wave profile of limiting waves with a sharp crest of$120^{\circ }$included angle first mentioned by Stokes in the 1880s. Thus, traditionally, different wave models are used for waves in different water depths. In this paper, by means of the homotopy analysis method (HAM), an analytic approximation method for highly nonlinear equations, we successfully gain convergent results (and especially the wave profiles) of the limiting Stokes waves with this kind of sharp crest in arbitrary water depth, even including solitary waves of extreme form in extremely shallow water, without using any extrapolation techniques. Therefore, in the frame of the HAM, the Stokes wave can be used as a unified theory for all kinds of waves, including periodic waves in deep and intermediate depths, cnoidal waves in shallow water and solitary waves in extremely shallow water.

Highlights

The two-dimensional steady progressive gravity wave is one of the most classic problems in fluid mechanics, which can be tracked back to Stokes (1847, 1880), and has been widely studied by a lot of researchers
It is found that the convergence of all Fourier coefficients of the solutions can be guaranteed by choosing a proper convergence-control parameter hin the framework of the homotopy analysis method (HAM), as shown in figure 2
Since the Fourier series is used to represent the free surface with a sharp pointed crest, using a large number of Fourier coefficients is inevitable

Summary

Introduction

The two-dimensional steady progressive gravity wave is one of the most classic problems in fluid mechanics, which can be tracked back to Stokes (1847, 1880), and has been widely studied by a lot of researchers Vanden-Broeck & Miloh (1995) employed series truncation methods, which use a refinement of Davies–Tulin’s approximation (Davies 1951; Tulin 1983), to solve the steep gravity waves By means of these methods, accurate numerical results can be obtained in the cases of d/λ 0.0168. An approach that can yield accurate results for the two-dimensional limiting (extreme) progressive gravity wave in arbitrary water depth without using any kind of extrapolation technique is of great value This is the motivation of this paper. There exist two challenges for the traditional perturbation methods: (i) the series solutions diverge either when the water depth is rather small or when the wave height approaches the peak value; (ii) the computational efficiency is rather low when the terms of the Fourier coefficients are large.

Mathematical description of the limiting Stokes wave

The mathematical approach based on the HAM

The limiting Stokes wave in infinite depth

The limiting Stokes wave in finite depth

Findings

Concluding remarks