High-Order Models of Nonlinear and Dispersive Wave in Water of Varying Depth with Arbitrary Sloping Bottom

High-order models with a dissipative term for nonlinear and dispersive wave in water of varying depth with an arbitrary sloping bottom are presented in this article. First, the formal derivations to any high order of mu(= h/lambda, depth to deep-water wave length ratio) and epsilon(= a/h, wave amplitude to depth ratio) for velocity potential, particle velocity vector, pressure and the Boussinesq-type equations for surface elevation eta and horizontal velocity vector (U) over right arrow at any given level in water are given. Then, the exact explicit expressions to the fourth order of mu are derived. Finally, the linear solutions of eta, (U) over right arrow, C (phase-celerity) and C-g (group velocity) for a constant water depth are obtained. Compared with the Airy theory, excellent results can be found even for a water depth as large as the wave legnth. The present high-order models are applicable to nonlinear regular and irregular waves in water of any varying depth (from shallow to deep) and bottom slope (from mild to steep).

Numerical Calculation for Nonlinear Waves in Water of Arbitrarily Varying Depth with Boussinesq Equations

Based on the high order nonlinear and dispersive wave equation with a dissipative term, a numerical model for nonlinear waves is developed, It is suitable to calculate wave propagation in water areas with an arbitrarily varying bottom slope and a relative depth h/L(0)less than or equal to1. By the application of the completely implicit stagger grid and central difference algorithm, discrete governing equations are obtained. Although the central difference algorithm of second-order accuracy both in time and space domains is used to yield the difference equations, the order of truncation error in the difference equation is the same as that of the third-order derivatives of the Boussinesq equation. In this paper, the correction to the first-order derivative is made, and the accuracy of the difference equation is improved. The verifications of accuracy show that the results of the numerical model are in good agreement with those of analytical Solutions and physical models.

QUASI-PERIODIC WAVES AND QUASI-SOLITARY WAVES IN STRATIFIED FLUID OF SLOWLY VARYING DEPTH

The nonlinear waves in a stratified fluid of slowly varying depth are investigated in this paper.The model considered here consists of a two-layer incompressible constant-density inviscid fluid confined by a slightly uneven bottom and a horizontal rigid wall.The Korteweg-de Vries(KdV)equation with varying coefficients is derived with the aid of the reductive perturbation method.By using the method of multiple scales,the approximate solutions of this equation are obtained.It is found that the unevenness of bottom may lead to the generation of socalled quasi-periodic waves and quasi-solitary waves,whose periods,propagation velocities and wave profiles vary slowly.The relations of the period of quasi-periodic waves and of the amplitude,propagation velocity of quasi-solitary waves varying with the depth of fluid are also presented.The models with two horizontal rigid walls or single-layer fluid can be regarded as particular cases of those in this paper.

A Mooring System Deployment Design Methodology for Vessels at Varying Water Depths

In this paper,a methodology for designing mooring system deployment for vessels at varying water depths is proposed.The Non-dominated Sorting Genetic Algorithm-II(NSGA-II)is combined with a self-dependently developed vessel-mooring coupled program to find the optimal mooring system deployment considering both station-keeping requirements and the safety of the mooring system.Two case studies are presented to demonstrate the methodology by designing the mooring system deployments for a very large floating structure(VLFS)module and a semi-submersible platform respectively at three different water depths.It can be concluded from the obtained results that the mooring system can achieve a better station-keeping ability with relatively shorter mooring line when deployed in the shallow water.The safety factor of mooring line is mainly dominated by the maximum instantaneous tension increment in the shallow water,while the pre-tension has a decisive influence on the safety factor of the mooring line in the deep water.