Numerical and experimental study of the transformation of a solitary wave over a shelf or isolated obstacle

Abstract

In order to model the evolution of a solitary wave near an obstacle or over an uneven bottom, the long-wave equations including curvature effects are introduced to describe the deformation and fission of a barotropic solitary wave passing over a shelf or an obstacle. The numerical results obtained from these equations are shown to be in good agreement with an analytical model derived by Germain (1984) in the framework of a generalized shallow-water theory, and with experimental results collected in a large channel equipped with a wave generator. Given the initial conditions, i.e. amplitude of the incident solitary wave, water depth in the deep region, and height of the shelf or the barrier, it is possible to predict the amplitude and number of the transmitted solitary waves as well as the amplitude of the reflected wave, and to describe the shape of the free surface at any time.

A Solitary Wave Propagates through a Gap into a Channel with an Upward Step

“…The numbers of wave fission are related to the step height and incident wave height. Readers can refer detail discussions in Seabra-Santoes et al (1987) [10] and experimental work of Losada et al (1989) [12]. As L 2 =60, it represents a channel with a sudden expansion of the width.…”

“…The related problems of solitary wave fission on a shelf have been investigated by many researchers in last two decades. The fission process that separates an incident solitary wave into a sequence of solitary waves with amplitude in decreasing order after climbing upon a two-dimensional (2D) shelf was presented in numerical solutions and laboratory measurements by [10]. The wave fission in a three-dimensional canal with an uneven bottom and a wider channel width has been noticed to be very different from that in 2D case.…”

Transformation of surface and internal waves at the bottom step

“…C учетом формул, выведенных в работе [27], коэффициенты трансформации можно уточ-нить, учитывая конечность амплитуды набегающего солитона. Сохранение формы импульсов и их длительностей при трансформации солитонов на резком скачке глубины у края уступа нашло подтверждение в последующих численных расчетах [29] и лабораторных экспериментах [30,31]. После трансформации в прошедшем и отраженном импульсах нарушается баланс между амплиту-дой и длительностью, характерный для стационарного солитона.…”

“…В последующих лабораторных экспериментах изучалась лишь трансформация длинных (по сравнению с глубиной бас-сейна) уединенных волн. Так, в работе [30] авторы приводят результаты большой серии экспериментов, в которых менялась высота подводного уступа и амплитуда набегающей уединенной волны. При этом амплитуды волн в ряде случаев были не ма-лыми, так что наблюдались нелинейные эффекты, приводящие даже к обрушению прошедших за уступ волн.…”

Finite volume and pseudo-spectral schemes for the fully nonlinear 1D Serre equations

“…Later, these equations were independently rediscovered by Su and Gardner [64] and by Green, Laws and Naghdi [38]. The extension of Serre equations for general uneven bathymetries was derived by Seabra-Santos et al [58]. In the Soviet literature these equations were known as the Zheleznyak-Pelinovsky model [75].…”