Numerical Simulation on Nonlinear Evolution of Rogue Waves on Currents Based on the NLS Equation

2011 ◽  
Author(s):  
Hanhong Hu ◽  
Ning Ma
Keyword(s):  
Numerical Simulation ◽  
Deep Water ◽  
Water Waves ◽  
Rogue Waves ◽  
Domain Model ◽  
Energy Concentration ◽  
Current Field ◽  
Nls Equation ◽  
Steady Current ◽  
Deep Water Wave

In this paper, nonlinear instability and evolution of deep-water rogue waves on following and opposing currents were described by numerical simulation for laboratory investigation. The generation of rogue waves in a numerical tank by means of wave focusing technique had been studied. Here a spatial domain model of current modified nonlinear Schro¨dinger (NLSC) equations in one horizontal dimension (1D) was established for describing the deep-water wave trains in a prescribed stationary current field. The transient water waves (TWW) was adopted as the initial condition of the NLSC equation. The steady current was added to see the effect of wave-current interaction on the energy concentration of gravity waves. The influence of current as well as other terms in the NLSC equations on wave height, inclination, particle velocity and acceleration are shown. Meanwhile, the focusing time/position of TWW influenced by the current field is investigated, which is of course a very important factor in experimental research when we generate rogue waves in the laboratory.

Numerical simulation of viscous, nonlinear and progressive water waves

2009 ◽  
Vol 637 ◽  
pp. 443-473 ◽  
Author(s):  
ASHISH RAVAL ◽  
XIANYUN WEN ◽  
MICHAEL H. SMITH
Keyword(s):  
Numerical Simulation ◽  
Shear Stress ◽  
Deep Water ◽  
Water Waves ◽  
Water Wave ◽  
Shear Stresses ◽  
Stress Distributions ◽  
Energy Decay Rate ◽  
Average Wind ◽  
Anticlockwise Rotation

A numerical simulation is performed to study the velocity, streamlines, vorticity and shear stress distributions in viscous water waves with different wave steepness in intermediate and deep water depth when the average wind velocity is zero. The numerical results present evidence of ‘clockwise’ and ‘anticlockwise’ rotation of the fluid at the trough and crest of the water waves. These results show thicker vorticity layers near the surface of water wave than that predicted by the theories of inviscid rotational flow and the low Reynolds number viscous flow. Moreover, the magnitude of vorticity near the free surface is much larger than that predicted by these theories. The analysis of the shear stress under water waves show a thick shear layer near the water surface where large shear stress exists. Negative and positive shear stresses are observed near the surface below the crest and trough of the waves, while the maximum positive shear stress is inside the water and below the crest of the water wave. Comparison of wave energy decay rate in intermediate depth and deep water waves with laboratory and theoretical results are also presented.

Numerical Simulation of Rogue Waves Based on the Fourth-Order NLS Equation for Laboratory Experimental Investigation

2010 ◽  
Author(s):  
Hanhong Hu ◽  
Ning Ma ◽  
Xuefeng Wang ◽  
Xiechong Gu
Keyword(s):  
Experimental Investigation ◽  
Fourth Order ◽  
Rogue Waves ◽  
Structure Design ◽  
Domain Model ◽  
Random Wave ◽  
Offshore Structure ◽  
Nls Equation ◽  
Transient Wave ◽  
Wave Trains

The main purposes of investigating the generation of the rogue waves in offshore engineering include: 1) prediction of its occurrence to protect the offshore structure from attacking; 2) the experimental investigation of rogue waves/structure interaction for the structure design. The latter one calls high requirement of wave generation and calculation. In this paper, we establish a spatial domain model of fourth order nonlinear Schro¨dinger (NLS) equation for describing deep-water wave trains in moving coordinate system. For the first purpose mentioned above, this paper presents the evolution of random wave trains in real sea state described by the Joint North Sea Wave Project (JONSWAP) power spectrum numerically, which is governed by the NLS equation. The parameters of the spectrum are evaluated to discuss their effect on the occurrence of rogue waves. For the second purpose to generate rogue waves in experimental tank efficiently, the transient wave is focused for its allowance of precise determination of concentration place/time. First we simulate the three-dimensional transient waves in the numerical tank modeling the deepwater basin with double-side multi-segmented wave-maker in Shanghai Jiao Tong University (SJTU) with linear superposing theory. To discuss its nonlinearity for the guidance of experiment, the transient wave is set as the initial condition of the NLS equation and the difference from the linear simulation is presented, which could be given as the suggestion to the preparation of experiment.

The effect of the induced mean flow on solitary waves in deep water

1998 ◽  
Vol 355 ◽  
pp. 317-328 ◽  
Author(s):  
T. R. AKYLAS ◽  
F. DIAS ◽  
R. H. J. GRIMSHAW
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Deep Water ◽  
Water Waves ◽  
Wave Equations ◽  
Soliton Solutions ◽  
Phase Speed ◽  
Mean Flow ◽  
Nls Equation ◽  
Envelope Soliton

Two branches of gravity–capillary solitary water waves are known to bifurcate from a train of infinitesimal periodic waves at the minimum value of the phase speed. In general, these solitary waves feature oscillatory tails with exponentially decaying amplitude and, in the small-amplitude limit, they may be interpreted as envelope-soliton solutions of the nonlinear Schrödinger (NLS) equation such that the envelope travels at the same speed as the carrier oscillations. On water of infinite depth, however, based on the fourth-order envelope equation derived by Hogan (1985), it is shown that the profile of these gravity–capillary solitary waves actually decays algebraically (like 1/x2) at infinity owing to the induced mean flow that is not accounted for in the NLS equation. The algebraic decay of the solitary-wave tails in deep water is confirmed by numerical computations based on the full water-wave equations. Moreover, the same behaviour is found at the tails of solitary-wave solutions of the model equation proposed by Benjamin (1992) for interfacial waves in a two-fluid system.

Phenomena on Rogue Waves and Rational Solution of Korteweg-de Vries Equation

2013 ◽  
Author(s):  
Ni Song ◽  
Wei Zhang ◽  
Qian Wang
Keyword(s):  
Shallow Water ◽  
Water Waves ◽  
Kdv Equation ◽  
Rational Solution ◽  
Rogue Waves ◽  
Nls Equation ◽  
Engineering Structures ◽  
Weakly Nonlinear ◽  
Nonlinear Mechanism ◽  
De Vries

An appropriate nonlinear mechanism may create the rogue waves. Perhaps the simplest mechanism, which is able to create considerate changes in the wave amplitude, is the nonlinear interaction of shallow-water solitons. The most well-known examples of such structure are Korteweg-de Vries (KdV) solitons. The Korteweg-de Vries (KdV) equation, which describes the shallow water waves, is a basic weakly dispersive and weakly nonlinear model. Basing on the homogeneous balanced method, we achieve the general rational solution of a classical KdV equation. Numerical simulations of the solution allow us to explain rare and unexpected appearance of the rogue waves. We compare the rogue waves with the ones generated by the nonlinear Schrödinger (NLS) equation which can describe deep water wave trains. The numerical results illustrate that the amplitude of the KdV equation is higher than the one of the NLS equation, which may causes more serious damage of engineering structures in the ocean. This nonlinear mechanism will provide a theoretical guidance in the ocean and physics.

scholarly journals Theoretical and experimental evidence of non-symmetric doubly localized rogue waves

2014 ◽  
Vol 470 (2171) ◽  
pp. 20140318 ◽  
Author(s):  
Jingsong He ◽  
Lijuan Guo ◽  
Yongshuai Zhang ◽  
Amin Chabchoub
Keyword(s):  
Deep Water ◽  
Rogue Waves ◽  
Nls Equation ◽  
Third Order ◽  
Weakly Nonlinear ◽  
Wave Flume ◽  
Symmetric Pattern ◽  
System A ◽  
Special Case ◽  
Good Agreement

We present determinant expressions for vector rogue wave (RW) solutions of the Manakov system, a two-component coupled nonlinear Schrödinger (NLS) equation. As a special case, we generate a family of exact and non-symmetric RW solutions of the NLS equation up to third order, localized in both space and time. The derived non-symmetric doubly localized second-order solution is generated experimentally in a water wave flume for deep-water conditions. Experimental results, confirming the characteristic non-symmetric pattern of the solution, are in very good agreement with theory as well as with numerical simulations, based on the modified NLS equation, known to model accurately the dynamics of weakly nonlinear wave packets in deep water.

Current Effects on the Generation and Evolution of the Peregrine Breather-Type Rogue Wave

2015 ◽  
Author(s):  
Wenyue Lu ◽  
Jianmin Yang ◽  
Longbin Tao ◽  
Haining Lu ◽  
Xinliang Tian ◽  
...  
Keyword(s):  
Rogue Wave ◽  
Evolutionary Process ◽  
Rogue Waves ◽  
Nls Equation ◽  
Sea State ◽  
Steady Current ◽  
Wave Heights ◽  
Current Interaction ◽  
Physical Mechanics ◽  
Peregrine Breather

Rogue wave is a kind of surface gravity waves with much larger wave heights than expected in normal sea state. Since this extreme sea event always occurred in the areas with strong currents, the wave-current interaction was considered to be one of the physical mechanics of the formation of the rogue wave. Some breather type solutions of the NLS equation have been considered as prototypes of rogue waves in ocean which usually appears from smooth initial condition only with a certain disturbance. In this paper, we have numerically studied evolutionary process of the Peregrine breather rogue wave based on the current modified fourth order nonlinear Schrödinger equation (the CmNLS equation). During the generation and evolution of the Peregrine breather rogue wave, the effects of the steady current was investigated by comparing with the results without current. The differences of the focusing position/time were observed due to the current influence.

scholarly journals Multi-rogue waves solutions to the focusing NLS equation and the KP-I equation

2011 ◽  
Vol 11 (3) ◽  
pp. 667-672 ◽  
Author(s):  
P. Dubard ◽  
V. B. Matveev
Keyword(s):  
Fiber Optics ◽  
Deep Water ◽  
Acoustic Waves ◽  
Liquid Surface ◽  
Large Family ◽  
Rogue Waves ◽  
Point Of View ◽  
Rational Solutions ◽  
Nls Equation ◽  
Physical Importance

Abstract. We construct a multi-parametric family of quasi-rational solutions to the focusing NLS equation, presenting a profile of multiple rogue waves. These solutions have also been used by us to construct a large family of smooth, real localized rational solutions of the KP-I equation quite different from the multi-lumps solutions first constructed in Bordag et al. (1977). The physical relevance of both equations is very large. From the point of view of geosciences,the focusing NLS equation is relevant to the description of surface waves in deep water, and the KP-I equation occurs in the description of capillary gravitational waves on a liquid surface, but also when one considers magneto-acoustic waves in plasma (Zhdanov, 1984) etc. In addition, there are plenty of equations of physical importance, having their origin in fiber optics, hydrodynamics, plasma physics and many other areas, which are gauge equivalent to the NLS equation or to the KP-I equation. Therefore our results can be easily extended to a large number of systems of physical interest to be discussed in separate publications.

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