Evolution of wide-spectrum unidirectional wave groups in a tank: an experimental and numerical study

Evolution of unidirectional nonlinear wave groups with wide spectra is studied experimentally and numerically. As an example of such an evolution, focusing of an initially wide wave train that is modulated both in amplitude and in frequency, to a single steep wave at a prescribed location along the laboratory wave tank is investigated. When numerous frequency harmonics arrive at the focusing location in phase, a very wave steep single emerges. The experimental study was carried out in two wave flumes that differ in size by an order of magnitude: a 330 m long Large Wave Channel in Hanover, and in 18 m long Tel-Aviv University wave tank. The spatial version of the Zakharov equation was applied in the numerical simulations. Detailed quantitative comparison is carried out between the experimental results and the numerical simulations. Spectra of the 2nd order bound waves are calculated using the theoretical model adopted. It is demonstrated that with the contribution of bound waves accounted for, a very good agreement between experiments and simulations is achieved.     

Freak waves under the action of wind: experiments and simulations

The freak wave formation due to the dispersive focusing mechanism is investigated experimentally without wind and in presence of wind. An asymmetric behaviour between the focusing and defocusing stages is found when the wind is blowing over the mechanically generated gravity wave group. This feature corresponds physically to the sustain of the freak wave mechanism on longer periods of time. Furthermore, a weak amplification of the freak wave and a shift in the downstream direction of the point where the waves merge are observed. The experimental results suggest that the Jeffreys' sheltering mechanism could play a key role in the coherence of the group of the freak wave. Hence, the Jeffreys' sheltering theory is introduced in a fully nonlinear model. The results of the numerical simulations confirm that the duration of the freak wave event increases with the wind velocity.     

Shallow-water equations with dispersion. Hyperbolic model

A hyperbolic shallow-water model is constructed with allowance for nonlinear and dispersive effects. The model describes solitonlike solutions in a range of wave velocities and predicts the breaking of smooth waves when the limiting amplitude is attained. The model is found to be adequate by comparison with experimental data on the evolution of a wave packet generated by the moving lateral wall of a channel. Lavrent'ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 2, pp. 40–46, March–April, 1998.     

Numerical study of breaking waves by a two‐phase flow model

A two‐phase flow model, which solves the flow in the air and water simultaneously, is presented for modelling breaking waves in deep and shallow water, including wave pre‐breaking, overturning and post‐breaking processes. The model is based on the Reynolds‐averaged Navier–Stokes equations with the k ?ε turbulence model. The governing equations are solved by the finite volume method in a Cartesian staggered grid and the partial cell treatment is implemented to deal with complex geometries. The SIMPLE algorithm is utilised for the pressure‐velocity coupling and the air‐water interface is modelled by the interface capturing method via a high resolution volume of fluid scheme. The numerical model is validated by simulating overturning waves on a sloping beach and over a reef, and deep‐water breaking waves in a periodic domain, in which good agreement between numerical results and available experimental measurements for the water surface profiles during wave overturning is obtained. The overturning jet, air entrainment and splash‐up during wave breaking have been captured by the two‐phase flow model, which demonstrates the capability of the model to simulate free surface flow and wave breaking problems.Copyright © 2011 John Wiley & Sons, Ltd.     

Large-scale vorticity generation by breakers in shallow and deep water

Water wave breaking is of considerable importance in the transfer of momentum, and in other transfers, between the atmosphere and oceans. Typically breaking occurs on deep water as events that have finite duration and finite spatial extent. Near shore lines most of the water motions are dominated by breaking waves. Recent work on the generation of vorticity by breaking waves and bores in the surf zone on beaches is considered and typical vortical structures are briefly discussed. Consideration of deep water breaking leads to the proposal that the end result of a breaking event in deep water may be a coherent structure within the resulting current field. Such a structure is topologically equivalent to half a vortex ring.     

Observations on waveforms of capillary and gravity-capillary waves

Due to extreme conditions in the field, there has not been any observational report on three-dimensional waveforms of short ocean surface waves. Three-dimensional waveforms of short wind waves can be found from integrating surface gradient image data (Zhang 1996a). Ocean surface gradient images are captured by an optical surface gradient detector mounted on a raft operating in the water offshore California (Cox and Zhang 1997). Waveforms and spatial structures of short wind waves are compared with early laboratory wind wave data (Zhang 1994, 1995). Although the large-scale wind and wave conditions are quite different, the waveforms are resoundingly similar at the small scale. It is very common, among steep short wind waves, that waves in the capillary range feature sharp troughs and flat crests. The observations show that most short waves are far less steep than the limiting waveform under weak wind conditions. Waveforms that resemble capillary-gravity solitons are observed with a close match to the form theoretically predicted for potential flows (Longuet-Higgins 1989, Vanden-Broeck and Dias 1992). Capillaries are mainly found as parasitic capillaries on the forward face of short gravity waves. The maximum wavelength in a parasitic wave train is less than a centimeter. The profiles of parasitic wave trains and longitudinal variations are shown. The phenomenon of capillary blockage (Phillips 1981) on dispersive freely traveling short waves is observed in the tank but not at sea. The short waves seen at sea propagate in all directions while waves in the tank are much more unidirectional.     

Instability of nonlinear standing waves in front of a vertical wall

The standing waves formed in front of a vertical breakwater in Gdańsk North Port Harbour are examined. To simplify the wave-structure interaction problem, a laboratory experiment and mathematical model were designed. Standing waves in a limited space were generated under resonance conditions between a vertical wall and wave generator. Such slowly growing standing waves eventually become unstable and, consequently, create an impact impulse on the vertical wall. The dynamics of the vertical wall and hydrodynamics of the standing wave were measured and compared with a numerical model derived by variational calculus. The phenomenon of standing wave instability observed in nature was reproduced by laboratory experimentation and numerical simulation. The presented mechanism of breaking standing waves is more complicated in reality due to wave randomness and the multidirectional wave field.     

Laboratory study on the evolution of waves parameters due to wave breaking in deep water

Understanding of the occurrence of the wave breaking, the process of the wave breaking and evolution of waves after they break in deep water is crucial to simulate the growth of wind wave in ocean. In this study, deep-water breaking waves with various spectral types, center frequencies and frequency bandwidths are generated in a wave flume based on energy focusing theory. The time series of the wave surface elevation along the flume are obtained by 22 wave probes mounted along the central line of the flume. The characteristics of deep-water wave breaking are analyzed using the spectrum analysis based on the Fast Fourier Transform (FFT). For small center frequency the maximum height of wave surface generated using the Pierson–Moskowitz (P–M) spectrum is produced and the impact of the frequency width is small in wave breaking zone. While the spectral type has a significant impact on the local wave steepness during breaking, the influence of center frequency and frequency width on the local wave steepness is very weak. The significant wave steepness changes significantly after wave breaking, but it remains stable in the upstream or the downstream of wave breaking zone. After wave breaking, the peak frequency remains stable, but the spectrally weighted wave frequency changes significantly. The relationship between the level of downshift and the incident wave steepness is approximately linear. By analyzing the energy spectra, it is found that the energy loses near high frequency of controlling frequencies range and increases near peak frequency during the wave breaking. After wave breaking, the total energy dissipates remarkably with increasing breaking intensity.     

Numerical simulation of interaction between wind and 2D freak waves

This paper presents a newly developed approach for the numerical modelling of wind effects on the generation and dynamics of freak waves. In this approach, the quasi arbitrary Lagrangian–Eulerian finite element method (QALE-FEM) developed by the authors of this paper is combined with a commercial software (StarCD). The former is based on the fully nonlinear potential model, in which the wind-excited pressure is modelled using a modified Jeffreys' model [9]. The latter has a volume of fluid (VOF) solver which can handle violent air–wave interaction problems. The combination can simulate the interaction between freak waves and winds with an improved computational efficiency. The numerical approach is validated by comparing its predictions with experimental data. Satisfactory agreements are achieved. Detailed numerical investigations of the interaction between winds and 2D freak waves are carried out, which not only explore different air flow states but also reveal the wind effects on the change of freak wave profiles. Both breaking and non-breaking freak waves are considered.     

Pressure Beneath a Solitary Water Wave: Mathematical Theory and Experiments

We study the pressure beneath a solitary water wave propagating in an irrotational flow at the free surface of water with a flat bed. The investigation is divided into two parts. The first part concerns a rigorous nonlinear study of the governing equations for water waves. We prove that the pressure in the fluid beneath a solitary wave strictly increases with depth and strictly decreases horizontally away from the vertical line beneath the crest. The second part of the paper describes an experimental study. Excellent agreement is found to exist between the theoretical predictions and the measurements. Our conclusions are also supported by numerical simulations.     

Long time interaction of envelope solitons and freak wave formations

This paper concerns long time interaction of envelope solitary gravity waves propagating at the surface of a two-dimensional deep fluid in potential flow. Fully nonlinear numerical simulations show how an initially long wave group slowly splits into a number of solitary wave groups. In the example presented, three large wave events are formed during the evolution. They occur during a time scale that is beyond the time range of validity of simplified equations like the nonlinear Schrödinger (NLS) equation or modifications of this equation. A Fourier analysis shows that these large wave events are caused by significant transfer to side-band modes of the carrier waves. Temporary downshiftings of the dominant wavenumber of the spectrum coincide with the formation large wave events. The wave slope at maximal amplifications is about three times higher than the initial wave slope. The results show how interacting solitary wave groups that emerge from a long wave packet can produce freak wave events.Our reference numerical simulation are performed with the fully nonlinear model of Clamond and Grue [D. Clamond, J. Grue, A fast method for fully nonlinear water wave computations, J. Fluid Mech. 447 (2001) 337–355]. The results of this model are compared with that of two weakly nonlinear models, the NLS equation and its higher-order extension derived by Trulsen et al. [K. Trulsen, I. Kliakhandler, K.B. Dysthe, M.G. Velarde, On weakly nonlinear modulation of waves on deep water, Phys. Fluids 12 (10) (2000) 2432–2437]. They are also compared with the results obtained with a high-order spectral method (HOSM) based on the formulation of West et al. [B.J. West, K.A. Brueckner, R.S. Janda, A method of studying nonlinear random field of surface gravity waves by direct numerical simulation, J. Geophys. Res. 92 (C11) (1987) 11 803–11 824]. An important issue concerning the representation and the treatment of the vertical velocity in the HOSM formulation is highlighted here for the study of long-time evolutions.     

On the nonlinear dynamics of the traveling-wave solutions of the Serre system

We numerically study nonlinear phenomena related to the dynamics of traveling wave solutions of the Serre equations including the stability, the persistence, the interactions and the breaking of solitary waves. The numerical method utilizes a high-order finite-element method with smooth, periodic splines in space and explicit Runge–Kutta methods in time. Other forms of solutions such as cnoidal waves and dispersive shock waves are also considered. The differences between solutions of the Serre equations and the Euler equations are also studied.     

A coupled level set and volume-of-fluid method for sharp interface simulation of plunging breaking waves

A coupled level set and volume-of-fluid (CLSVOF) method is implemented for the numerical simulations of interfacial flows in ship hydrodynamics. The interface is reconstructed via a piecewise linear interface construction scheme and is advected using a Lagrangian method with a second-order Runge–Kutta scheme for time integration. The level set function is re-distanced based on the reconstructed interface with an efficient re-distance algorithm. This level set re-distance algorithm significantly simplifies the complicated geometric procedure and is especially efficient for three-dimensional (3D) cases. The CLSVOF scheme is incorporated into CFDShip-Iowa version 6, a sharp interface Cartesian grid solver for two-phase incompressible flows with the interface represented by the level set method and the interface jump conditions handled using a ghost fluid methodology. The performance of the CLSVOF method is first evaluated through the numerical benchmark tests with prescribed velocity fields, which shows superior mass conservation property over the level set method. With combination of the flow solver, a gas bubble rising in a viscous liquid and a water drop impact onto a deep water pool are modeled. The computed results are compared with the available numerical and experimental results, and good agreement is obtained. Wave breaking of a steep Stokes wave is also modeled and the results are very close to the available numerical results. Finally, plunging wave breaking over a submerged bump is simulated. The overall wave breaking process and major events are identified from the wave profiles of the simulations, which are qualitatively validated by the complementary experimental data. The flow structures are also compared with the experimental data, and similar flow trends have been observed.     

Boundary‐fitted non‐linear dispersive wave model for regions of arbitrary geometry

A vertically integrated non‐linear dispersive wave model is expressed in non‐orthogonal curvilinear co‐ordinate system for simulating shallow or deep water wave motions in regions of arbitrary geometry. Both dependent and independent variables are transformed so that an irregular physical domain is converted into a rectangular computational domain with contravariant velocities. Thus, the wall condition for enclosures surrounding a typical physical domain, such as a channel, port or harbor, is satisfied accurately and easily. The numerical scheme is based on staggered grid finite‐difference approximations, which result in implicit formulations for the momentum equations and semi‐explicit formulation for the continuity equation. Test cases of linear wave propagation in converging, diverging and circular channels are performed to check the reliability of model simulations against the analytical solutions. Cnoidal waves of different steepness values in a circular channel are also considered as examples to non‐linear wave propagation within curved walls. In closing, remarks concerning versatility and practical uses of the numerical model are made. Copyright © 2004 John Wiley & Sons, Ltd.     

Damping of large-amplitude solitary waves

We consider the damping of large-amplitude solitary waves in the framework of the extended Korteweg-de Vries equation (that is, the usual Korteweg-de Vries equation supplemented with a cubic nonlinear term) modified by the inclusion of a small damping term. The damping of a solitary wave is studied for several different forms of friction, using both the analytical adiabatic asymptotic theory and numerical simulations. When the coefficient of the cubic nonlinear term has the opposite sign to the coefficient of the linear dispersive term, the extended Kortweg-de Vries equation can support large-amplitude “thick” solitary waves. Under the influence of friction, these “thick” solitary waves decay and may produce one or more secondary solitary waves in this process. On the other hand, when the coefficient of the cubic nonlinear term has the same sign as the coefficient of the linear dispersive term, but the opposite sign to the coefficient of the quadratic nonlinear term, the action of friction may cause a solitary wave to decay into a wave packet.     

Physical mechanisms of the rogue wave phenomenon

A review of physical mechanisms of the rogue wave phenomenon is given. The data of marine observations as well as laboratory experiments are briefly discussed. They demonstrate that freak waves may appear in deep and shallow waters. Simple statistical analysis of the rogue wave probability based on the assumption of a Gaussian wave field is reproduced. In the context of water wave theories the probabilistic approach shows that numerical simulations of freak waves should be made for very long times on large spatial domains and large number of realizations. As linear models of freak waves the following mechanisms are considered: dispersion enhancement of transient wave groups, geometrical focusing in basins of variable depth, and wave-current interaction. Taking into account nonlinearity of the water waves, these mechanisms remain valid but should be modified. Also, the influence of the nonlinear modulational instability (Benjamin–Feir instability) on the rogue wave occurence is discussed. Specific numerical simulations were performed in the framework of classical nonlinear evolution equations: the nonlinear Schrödinger equation, the Davey–Stewartson system, the Korteweg–de Vries equation, the Kadomtsev–Petviashvili equation, the Zakharov equation, and the fully nonlinear potential equations. Their results show the main features of the physical mechanisms of rogue wave phenomenon.     

Non-reflecting boundary conditions applicable to general purpose CFD simulators

In simulations of propagating blast waves the effects of artificial reflections at open boundaries can seriously degrade the accuracy of the computations. In this paper, a boundary condition based on a local approximation by a plane traveling wave is presented. The method yields small artificial reflections at open boundaries. The derivation and the theory behind these so-called plane-wave boundary conditions are presented. The method is conceptually simple and is easy to implement in two and three dimensions. These non-reflecting boundary conditions are employed in the three-dimensional computational fluid dynamics (CFD) solver FLACS, capable of simulating gas explosions and blast-wave propagation in complex geometries. Several examples involving propagating waves in one and two dimensions, shock tube and an example of a simulation of a propagating blast wave generated by an explosion in a compressor module are shown. The numerical simulations show that artificial reflections due to the boundary conditions employed are negligible. © 1998 John Wiley & Sons, Ltd.     

A depth‐integrated non‐hydrostatic finite element model for wave propagation

This paper presents a two‐dimensional finite element model for simulating dynamic propagation of weakly dispersive waves. Shallow water equations including extra non‐hydrostatic pressure terms and a depth‐integrated vertical momentum equation are solved with linear distributions assumed in the vertical direction for the non‐hydrostatic pressure and the vertical velocity. The model is developed based on the platform of a finite element model, CCHE2D. A physically bounded upwind scheme for the advection term discretization is developed, and the quasi second‐order differential operators of this scheme result in no oscillation and little numerical diffusion. The depth‐integrated non‐hydrostatic wave model is solved semi‐implicitly: the provisional flow velocity is first implicitly solved using the shallow water equations; the non‐hydrostatic pressure, which is implicitly obtained by ensuring a divergence‐free velocity field, is used to correct the provisional velocity, and finally the depth‐integrated continuity equation is explicitly solved to satisfy global mass conservation. The developed wave model is verified by an analytical solution and validated by laboratory experiments, and the computed results show that the wave model can properly handle linear and nonlinear dispersive waves, wave shoaling, diffraction, refraction and focusing. Copyright © 2013 John Wiley & Sons, Ltd.