Waves In Water Of Finite Depth Research Articles

Numerical Study on Collisions of Solitons of Surface Waves in Finite Water Depth

Head-on collisions between two solitary waves in the framework of the nonlinear Schrödinger (NLS) equation were investigated using the Fourier spectral method. When solitary waves undergo collision, the peak value of surface elevation (hereafter referred to as ζmax) exhibits fluctuations with increasing relative water depths k0h (where k0 is the wave number and h is the water depth). ζmax is approximately equal to the sum of the peak values of the two solitary waves with smaller wave steepness ε0 (ε0 = k0a0, a0 is the free background amplitude parameter), and it exhibits fluctuations for ε0 > 0.10. Similar results have been observed in the study of head-on collisions for four solitary waves. These results show that the water depth and wave steepness play important roles in the collision of solitary waves, and the effects of the interactions of intense wave groups are important in studies of the mechanisms and manifestations of freak oceanic waves.

Four-wave resonant interaction of surface gravity waves in finite water depth

We investigated the four-wave resonant and quasi-resonant interactions in a special degenerated case, wherein bichromatic mother waves are generated to give birth to a daughter wave. Through theoretical and numerical analyses, we first discover the threshold value of water depth when the four-wave resonance diminishes. Our work can shed light on the limiting shallow water depth where the energy exchange due to four wave resonance creases.

Large eddy simulation of focused breaking waves with different wave steepness

The profile of fluid field and wave geometry, kinematic and dynamic characteristics of focused breaking waves in finite water depth with different wave steepness are studied with a constant wave steepness wave focusing method in this paper. The flow field is solved by the two-phase flow model in REEF3D based on the incompressible Navier–Stokes equation and the Large Eddy Simulation (LES) model to simulate the complex breaking process. The numerical model is validated by comparing wave elevation of both focused non-breaking and breaking case with the experimental results and satisfactory agreements are achieved. A series of simulation cases with different global steepness are carried out to analyze the focused waves of different breaking types. The difference between the preset focused condition and the real ones and the area of air cavity rolled by the wave jets are both indicates that the process of wave focusing and breaking show a strong nonlinear profile. The maximum values of the horizontal velocity occur after the impingement moment and can reach 2.5 Cp for extreme steep waves. Three typical wave breaking criteria in finite water depth are calculated and discussed successively. For the geometric criterion analysis, the global wave steepness and local steepness have a good performance both for predicting wave breaking and presenting the breaking intensity. The kinematic breaking criterion ratio shows that the previous threshold 1 at the breaking incipient scenario is not sufficient to predict the wave breaking. Besides, it is found that the time interval when the kinematic breaking criterion ratio big than 1 may related to the wave breaking type. The energy changing rate for both the focusing and dissipation processes increase linearly with the global wave steepness and show a good ability as a dynamic breaking criterion and also prove the asymmetric profile of breaking waves.

Modulational instability and statistical properties of irregular waves in finite water depth

Modulational instability is one of the possible basic mechanisms for the generation of rogue waves in deep water. Under the condition of finite depth, this instability is reduced and disappears as k1h<1.363 (k1 is the wavenumber of the carrier wave and h is the water depth). In this study, we addressed this phenomenon through three approaches, theoretical analysis of the Zakharov’s equation and the nonlinear Schrödinger equation and direct numerical simulations of the Euler equation. Their results on the modulational instability over arbitrary depths, particularly the growth rates of sidebands, are analysed and compared. In addition, the consequences of the suppression of modulational instability for realistic ocean waves are illustrated by direct simulations of long-crested, irregular waves at finite water depth. The results show that many phenomena related to the third-order nonlinearity (i.e. the downshift of the spectral peak frequency, the enhancement of kurtosis of surface elevations and rouge wave occurrence probability) are suppressed with decreasing depth k1h and eventually cease to exist as k1h<1.363. Considering the dependence of wave evolution on water depth, two Benjamin–Feir indices for arbitrary depths derived from Zakharov’s equation and nonlinear Schrödinger equation are used as indicators to estimate the modulational instability effect on nonlinear wave fields. Our results confirm that both parameters provide reasonable predictions of the modulational instability and probability of rogue wave occurrence.

EVOLUTION OF UNSTABLE WAVE PACKETS OVER VARIABLE BATHYMETRY

Several field observations have reported the formation of rogue waves in coastal zones, see Chien et al. (2002) for an example in Taiwanese sea. The mechanisms that lead to the occurrence of rogue waves in finite water depth to shallow water are not well understood yet under the conjecture of modulation instability. Indeed, this theory for uni-directional waves shows that when kh is lower than a threshold of 1.363 in homogeneous water depth conditions, the wave train becomes stable to side-band perturbations. Then if the wave train is stable, the appearance of rogue waves is not possible within this linear stability framework. One explanation may come from the complex wave transformation mechanisms in variable bathymetry, especially, for cases of steep slopes or near the edge between a steep slope and a gentle slope as it is the case of the continental shelf. Very few laboratory experiments have been so far addressing the influence of the bathymetry on extreme wave occurrence (Baldock and Swan (1996), Kashima et al. (2012), Ma et al. (2015)).Recorded Presentation from the vICCE (YouTube Link): https://youtu.be/a5M4PS-Lo4Q

Phase-suppressed hydrodynamics of solitons on constant-background plane wave

The significance of local phase shifts in the experimental realization of soliton and breather waves in finite water depth as well as in deep water is explored. When suppressing the corresponding phase shift, the coherence of the wave envelope disintegrates by forming distinct soliton patterns. All experimental results are in excellent agreement with the predictions based on the nonlinear Schr\odinger equation framework, suggesting the universality of observed dip and extreme wave pattern dynamics.

A nonlinear Schrödinger equation for gravity waves slowly modulated by linear shear flow**Project supported by the National Key Research and Development Program of China (Grant Nos. 2016YFC1401404 and 2017YFA0604102) and the National Natural Science Foundation of China (Grant No. 41830533).

Assume that a fluid is inviscid, incompressible, and irrotational. A nonlinear Schrödinger equation (NLSE) describing the evolution of gravity waves in finite water depth is derived using the multiple-scale analysis method. The gravity waves are influenced by a linear shear flow, which is composed of a uniform flow and a shear flow with constant vorticity. The modulational instability (MI) of the NLSE is analyzed, and the region of the MI for gravity waves (the necessary condition for existence of freak waves) is identified. In this work, the uniform background flows along or against wave propagation are referred to as down-flow and up-flow, respectively. Uniform up-flow enhances the MI, whereas uniform down-flow reduces it. Positive vorticity enhances the MI, while negative vorticity reduces it. Hence, the influence of positive (negative) vorticity on MI can be balanced out by that of uniform down (up) flow. Furthermore, the Peregrine breather solution of the NLSE is applied to freak waves. Uniform up-flow increases the steepness of the free surface elevation, while uniform down-flow decreases it. Positive vorticity increases the steepness of the free surface elevation, whereas negative vorticity decreases it.

Two-dimensional modulation instability of wind waves

It is widely known that deep-water waves are modulationally unstable and that this can be modelled by a nonlinear Schrödinger equation. In this paper, we extend the previous studies of the effect of wind forcing on this instability to water waves in finite depth and in two horizontal space dimensions. The principal finding is that the instability is enhanced and becomes super-exponential and that the domain of instability in the modulation wavenumber space is enlarged. Since the outcome of modulation instability is expected to be the generation of rogue waves, represented within the framework of the nonlinear Schrödinger equation as a Peregrine breather, we also examine the effect of wind forcing on a Peregrine breather. We find that the breather amplitude will grow at twice the rate of a linear instability.

Numerical Study on Change of Added Resistance due to Incident Waves in Finite Water Depth

AbstractIn this study, the effect of water depth on vertical motion responses and added resistance of ships in waves was numerically examined. The applied numerical method corresponded to the time ...

Finite-amplitude steady-state wave groups with multiple near-resonances in finite water depth

Finite-amplitude wave groups with multiple near-resonances are investigated to extend the existing results due to Liu et al. (J. Fluid Mech., vol. 835, 2018, pp. 624–653) from steady-state wave groups in deep water to steady-state wave groups in finite water depth. The slow convergence rate of the series solution in the homotopy analysis method and extra unpredictable high-frequency components in finite water depth make it hard to obtain finite-amplitude wave groups accurately. To overcome these difficulties, a solution procedure that combines the homotopy analysis method-based analytical approach and Galerkin method-based numerical approaches has been used. For weakly nonlinear wave groups, the continuum of steady-state resonance from deep water to finite water depth is established. As nonlinearity increases, the frequency bands broaden and more steady-state wave groups are obtained. Finite-amplitude wave groups with steepness no less than $0.20$ are obtained and the resonant sets configuration of steady-state wave groups are analysed in different water depths. For waves in deep water, the majority of non-trivial components appear around the primary ones due to four-wave, six-wave, eight-wave or even ten-wave resonant interactions. The dominant role of four-wave resonant interactions for steady-state wave groups in deep water is demonstrated. For waves in finite water depth, additional non-trivial high-frequency components appear in the spectra due to three-wave, four-wave, five-wave or even six-wave resonant interactions with the components around the primary ones. The amplitude of these high-frequency components increases further as the water depth decreases. Resonances composed by components only around the primary ones are suppressed while resonances composed by components around the primary ones and from the high-frequency domain are enhanced. The spectrum of steady-state resonant wave groups changes with the water depth and the significant role of three-wave resonant interactions in finite water depth is demonstrated.

KAM for gravity water waves in finite depth

KAM for gravity water waves in finite depth

Hogner model of wave interferences for farfield ship waves in shallow water

The remarkably simple yet realistic Hogner model of farfield ship waves, previously considered to analyze the apparent wake angle associated with the highest waves that result from constructive interferences among the divergent waves created by a fast ship in deep water, is applied to the more general case of uniform finite water depth. This theory is used to illustrate notable features of ship waves in finite water depth. In particular, numerical applications to a monohull ship and catamarans illustrate two notable features of ship waves in shallow water: specifically, the apparent wake angle can be much smaller than Havelock's classical asymptote or cusp angles, and the apparent wake angle in finite water depth is nearly identical to the apparent wake angle in deep water if the water depth is greater than the ship length. Moreover, the numerical illustrations demonstrate the paramount importance of interferences among divergent waves for fast ships. Indeed, constructive interferences among the divergent waves created by a fast ship largely determine the variation of the amplitude of the waves across the ship wake, and therefore the apparent wake angle and the wave drag of the ship, a critical element of ship design. These numerical illustrations also corroborate main conclusions of an approximate analysis of wave interferences in shallow water for monohull ships and catamarans modeled as 2-point wavemakers.

Statistical distribution of wave-induced drift for random ocean waves in finite water depth

Based on the linear random wave solutions of water wave equations in finite water depth, a theoretical statistical distribution of the wave-induced drift is derived. The parameters involved in the distribution can be determined by the water depth and the wave spectrum of ocean waves. As an illustrative example, we consider a simple wind-generated sea and the parameters are calculated for typical wind speeds and water depths by using Phillips spectrum. The behaviours of the distribution and the effects of wind speed and water depth on the distribution are investigated.

Existence and amplitude bounds for irrotational water waves in finite depth.

We prove the existence of solutions to the irrotational water-wave problem in finite depth and derive an explicit upper bound on the amplitude of the nonlinear solutions in terms of the wavenumber, the total hydraulic head, the wave speed and the relative mass flux. Our approach relies upon a reformulation of the water-wave problem as a one-dimensional pseudo-differential equation and the Newton-Kantorovich iteration for Banach spaces.This article is part of the theme issue 'Nonlinear water waves'.

Linear-shear-current modified Schrödinger equation for gravity waves in finite water depth.

A nonlinear Schrödinger equation for the propagation of two-dimensional surface gravity waves on linear shear currents in finite water depth is derived. In the derivation, linear shear currents are assumed to be a linear combination of depth-uniform currents and constant vorticity. Therefore, the equation includes the combined effects of depth-uniform currents and constant vorticity. Next, using the equation, the properties of the modulational instability of gravity waves on linear shear currents are investigated. It is showed that shear currents significantly modify the modulational instability properties of weakly nonlinear waves. Furthermore, the influence of linear shear currents on Peregrine breather which can be seen as a prototype of freak waves is also studied. It is demonstrated that depth-uniform opposing currents can reduce the breather extension in both the time and spatial domain in intermediate water depth, but following currents has the adverse impact, indicating that a wave packets with freak waves formed on following currents contain more hazardous waves in finite water depth. However, the corresponding and coexisting vorticity can counteract the influence of currents. Additionally, if the water depth is deep enough, shear currents have negligible effect on the characteristics of Peregrine breathers.

A comparative study between the IGN-2 equations and the fully nonlinear, weakly dispersive Boussinesq equations

This work focuses on the comparison between the fully nonlinear, and weakly dispersive Boussinesq equations and the irrotational Green–Naghdi Level 2 (IGN-2) equations for incompressible and inviscid fluid and irrotational flow resulting from waves in finite water depth. We provide in this paper for the first time a comprehensive comparison of the GN/IGN and Boussinesq wave theories for strongly nonlinear and weakly dispersive waves. The Boussinesq approach results in many different theories depending on the different orders of approximation of nonlinearity and dispersion used in the corresponding perturbation series. The fully nonlinear, weakly dispersive Boussinesq equations have been widely used in problems involving coastal regions and harbors because it is more efficient than fully nonlinear, strongly dispersive Boussinesq equations. A competing theory is the strongly nonlinear IGN-2 wave theory. Since both sets of equations have comparable complexity, it is appropriate to compare the results of these equations with known numerically exact solutions to determine the advantages of each approach. In this study, we use the same iterative algorithm to obtain the steady solutions of the IGN-2 and Boussinesq equations for periodic waves. The steady solutions of the IGN-2 and Boussinesq equations are studied for four different wavelengths that cover a range of applicability of the theories. We show in this study that the Boussinesq equations give more accurate results for the velocity distribution at the wave-crest transection. The IGN-2 equations give more accurate results for the velocity distribution at the wave-trough transection. Moreover, it is shown that the IGN-2 equations give more accurate results on the wave speed and wave profile, and that they can treat accurately waves higher than the ones that the Boussinesq equations can treat. The calculations show that both of these sets of equations are very good for strongly nonlinear wave simulations and the limitations of each are different.

Finite time blow-up and breaking of solitary wind waves.

The evolution of surface water waves in finite depth under wind forcing is reduced to an antidissipative Korteweg-de Vries-Burgers equation. We exhibit its solitary wave solution. Antidissipation accelerates and increases the amplitude of the solitary wave and leads to blow-up and breaking. Blow-up occurs in finite time for infinitely large asymptotic space so it is a nonlinear, dispersive, and antidissipative equivalent of the linear instability which occurs for infinite time. Due to antidissipation two given arbitrary and adjacent planes of constant phases of the solitary wave acquire different velocities and accelerations inducing breaking. Soliton breaking occurs in finite space in a time prior to the blow-up. We show that the theoretical growth in amplitude and the time of breaking are both testable in an existing experimental facility.

On cusped solitary waves in finite water depth

It is well-known that the Camassa–Holm (CH) equation admits both of the peaked and cusped solitary waves in shallow water. However, it was an open question whether or not the exact wave equations can admit them in finite water depth. Besides, it was traditionally believed that cusped solitary waves, whose 1st-derivative tends to infinity at crest, are essentially different from peaked solitary ones with finite 1st-derivative. Currently, based on the symmetry and the exact water wave equations, Liao [1] proposed a unified wave model (UWM) for progressive gravity waves in finite water depth. The UWM admits not only all traditional smooth progressive waves but also the peaked solitary waves in finite water depth: in other words, the peaked solitary progressive waves are consistent with the traditional smooth ones. In this paper, in the frame of the linearized UWM, we give, for the first time, some explicit expressions of cusped solitary waves in finite water depth, and besides reveal a close relationship between the cusped and peaked solitary waves: a cusped solitary wave is consist of an infinite number of peaked solitary ones with the same phase speed, so that it can be regarded as a special peaked solitary wave. This also well explains why and how a cuspon has an infinite 1st-derivative at crest. Besides, it is found that, when wave height is small enough, the effect of nonlinearity is negligible for the interaction of peaked waves so that these explicit expressions are good enough approximations of peaked/cusped solitary waves in finite water depth. In addition, like peaked solitary waves, the vertical velocity of a cusped solitary wave in finite water depth is also discontinuous at crest (x=0), and especially its phase speed has nothing to do with wave height, too. All of these would deepen and enrich our understandings about the cusped solitary waves.

Experimental evidence of the modulation of a plane wave to oblique perturbations and generation of rogue waves in finite water depth

We present a laboratory experiment in a large directional wave basin to discuss the instability of a plane wave to oblique side band perturbations in finite water depth. Experimental observations, with the support of numerical simulations, confirm that a carrier wave becomes modulationally unstable even for relative water depths k0h &amp;lt; 1.36 (with k the wavenumber of the plane wave and h the water depth), when it is perturbed by appropriate oblique disturbances. Results corroborate that the underlying mechanism is still a plausible explanation for the generation of rogue waves in finite water depth.

On the steady-state fully resonant progressive waves in water of finite depth

AbstractThe steady-state fully resonant wave system, consisting of two progressive primary waves in finite water depth and all components due to nonlinear interaction, is investigated in detail by means of analytically solving the fully nonlinear wave equations as a nonlinear boundary-value problem. It is found that multiple steady-state fully resonant waves exist in some cases which have no exchange of wave energy at all, so that the energy spectrum is time-independent. Further, the steady-state resonant wave component may contain only a small proportion of the wave energy. However, even in these cases, there usually exist time-dependent periodic exchanges of wave energy around the time-independent energy spectrum corresponding to such a steady-state fully resonant wave, since it is hard to be exactly in such a balanced state in practice. This view serves to deepen and enrich our understanding of the resonance of gravity waves.