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Numerical study on the effects of uneven bottom topography on freak wavesCheng Cuin,Ning Chuan Zhang,Yu Xiu Yu,Jing Bo LiState Key Laboratory of Coastal and Offshore Engineering,Dalian University of Technology,Dalian,Liaoning,Chinaa r t i c l e i n f oArticle history:Received 8 May 2011Accepted 30 June 2012Available online 5 August 2012Keywords:Freak waveVOF methodUneven bottom topographyWavelet analysisa b s t r a c tA numerical model is built by using an improved VOF method coupled with an incompressible NavierStokes solver.Exploiting the model,the freak wave formation due to the dispersive focusing mechanismis investigated numerically without uneven bottoms and in presence of uneven bottoms.During the freakwave transformation over an uneven bottom in finite water,combined effects of shoaling,refraction andreflection can modify the external characteristics of freak waves,and also can complicate the energytransfers.Furthermore,wavelet analysis method is adopted to analyze the behavior of the instantaneousenergy structure of freak waves.It is found that when the bottoms vary in height,the externalcharacteristic parameters and high frequency energy show a similar trend,but the value may be quitedifferent due to the difference in local characteristic of the bottom.&2012 Elsevier Ltd.All rights reserved.1.IntroductionFreak wave is a type of extremely large transient water wave,being close to breaking and of asymmetry in both vertical andhorizontal direction.In the time-frequency spectrum of freakwaves,strong energy density is instantly surged and seeminglycarried over to the high frequency components at the instant thefreak wave occurs(Liu and Mori,2000),therefore severe damagesto vessels,maritime structures and other facilities in the ocean areusually caused.Such waves have been observed in a large numberof basins around the world,in deep or shallow waters,with orwithout currents(Kharif and Pelinovsky,2003;Lopatoukhin andBoukhanovsky,2004;Monbaliu and Toffoli,2003),and the popularRayleigh distribution cannot precisely predict the probability ofoccurrence of freak waves due to the non-linear effects(Stansell,2004,2005;Chien et al.,2002).The previous studies have shown that several mechanismshave been suggested to explain the formation of freak waves invarious environments.Among them one can mention the dispersivefocusing,wavecurrent interaction,atmospheric forcing,spatial(geometrical)focusing,non-linear self-focusing of wave energy,and non-linear wavewave interaction(Chien et al.,2002;Kharifand Pelinovsky,2003;Lopatoukhin and Boukhanovsky,2004).Wu and Yao(2004)reported the results of laboratory mea-surements on limiting freak waves in the presence of currents.Itis found that strong opposing currents inducing partial waveblocking significantly elevate the limiting steepness and asym-metry of freak waves.Touboul et al.(2006)experimentally and numerically investi-gated the direct effect of the wind on a freak wave eventgenerated by means of a dispersive focusing mechanism.Theresults suggest that the duration of the freak wave event increaseswith the wind velocity,and the point where the waves merge hasa shift in the downstream direction,which is due to the action ofthe current induced by the wind.Peterson et al.(2003)presented that non-linear interactions ofsolitonic waves in the framework of the KadomtsevPetviashviliequation may result in particularly high and steep waves resem-bling the freak waves,and it may be a generic source of freakwaves in areas of moderate depth.Overall,these studies have provided us a good understandingto the influences on the external characteristics of freak waves.However,insight into the internal structure of freak waves playsan important role in interpreting the physical mechanism.Walkeret al.(2004)investigated the non-linear characteristics of freakwaves based on Fourier power spectrum.It is found that thefield data exhibits an anomalous set-up for the New Yearwave,whereas all the other large waves show a local set-down.The conventional Fourier method has provided substantial insightinto freak wave phenomena,but the Fourier power spectrum,as atime-averaged description of wave energy,is inappropriate forcharacterizing non-stationary signals.Wavelet method has beenproven to be a powerful tool for analyzing localized variations ofpower within a time series.By decomposing a time series intotime-frequency space,one is able to determine both the dominantmodes of variability and how those modes vary in time.By usingthis method,the analysis on time-frequency energy spectrum ofsimulated and field observed freak waves has been presented bythe authors(Cherneva and Soares,2008;Chien et al.,2002;Cuiand Zhang,2011;Mori et al.,2002).It is found that a well-definedContents lists available at SciVerse ScienceDirectjournal homepage: Engineering0029-8018/$-see front matter&2012 Elsevier Ltd.All rights reserved.http:/dx.doi.org/10.1016/j.oceaneng.2012.06.021nCorresponding author.Tel.:[phone];fax:[phone].E-mail address:(C.Cui).Ocean Engineering 54(2012)132141freak wave can be readily identified from the wavelet spectrumwhere strong energy density in the spectrum is instantly surgedand seemingly carried over to the high frequency components atthe instant the freak wave occurs.Wavelet analysis method hasbetter performance than the popular Fourier technique.One believes that the bottom topography plays important rolein modifying wave form and propagation.During the wave trans-formation from the deep-water to shallow-water over unevenbottoms,combined effects of shoaling,refraction,diffraction,andreflection can result in bending,overturning and breaking waves(Biausser et al.,2003;Choi and Wu,2006;Grilli et al.,2001).In addition,spatial(geometrical)focusing is one of the possiblephysical mechanisms of freak waves.To our knowledge,theinternal features and evolution behavior of freak waves in pre-sence of uneven bottoms need further examination.In this paper,a numerical model is built by using an improvedVOF method(Ren and Wang,2004)and the governing equationsare the Reynolds-averaged NS(RANS)equations,closed bythe two-equation keturbulence model.The component wavesfocusing method is adopted for freak wave formation,which isachieved through using a wave-maker to generate waves at oneextremity of the numerical tank and the motion of the wave-maker is prescribed according to an improved superpositionmodel(Kriebel,2000).The uneven bottoms can be characterizedby including a partial bottom-cell treatment.By using the currentmodel,freak waves without uneven bottoms and in presence ofslope and curved topography have been simulated to analyzethe effects of the uneven bottoms on the external features andinternal energy structure of freak waves.2.Mathematical model2.1.Governing equationsWhen freak waves propagate over uneven bottoms,they maybreak with turbulent fluctuation of the water particles due to thewavebottom interaction,which must be accounted for by properturbulence model.Therefore,the two-dimensional continuityequation and the Reynolds-averaged NS equations are usedas the governing equations,closed by the two-equation keturbulence model.Continuity equation:xyuyyv 01Reynolds time-averaged equations:utuuxvuy?1rpxgxn2ux22uy2!nt2ux22uy2!2ntxuxntyuyvx?23kx2vtuvxvvy?1rpygyn2vx22vy2!nt2vx22vy2!2ntyvyntxuyvx?23ky3Two-equation k eturbulence model:ktukxvkynntsk?2kx22ky2!1skntxkxntyky?2ntux?2vy?2#ntuyvx?2?e4etuexveynntse?2ex22ey2!1sentxexntyey?2Ce1ekntux?2vy?2#Ce1ekntuyvx?2?Ce2e2k5where,u and u are velocity components in x-and y-directions,respectively;yis the partial-cell parameter,which is independentof time and has a value between 0 and 1 depending on whetherthe point is inside an obstacle or in the fluid;p is the pressure;ris the fluid density;g is the gravitational acceleration;v is thecoefficient of kinematic viscosity;vtCu(k2/e)is the coefficient ofturbulent viscosity;k is the turbulent kinetic energy;eis theturbulent kinetic energy dissipation rate;Cu,sk,se,Ce1and Ce2arethe empirical constants recommended in the literature(Rodi,1993).In this work,the following standard values are used:Cu0.09,sk1.0,se1.3,Ce11.43,Ce21.92.2.2.Numerical methodThe VOF method is known for its capacity to simulate freesurface flow.This is made possible by means of a fluid fractionF(x,y,t),which has a value between zero and unity,representingthe volume fraction of a cell occupied by fluid.Thus,a cell full offluid is reflected by F1,while an empty cell will have F0.A cellthat is either intersected by a free surface or contains voids will bepartially filled with fluid and has a value of F between zero andunity.Furthermore,a free surface cell can be identified being acell with a non-zero F and having at least one neighboring cellwhere F0.The time variation of this function is governed byyFtyuFxyvFy 06The variables are solved for from a finite-difference approx-imation of the governing equations.On the discretization ofthe advection items,a third-order upwind scheme is used whendealing with inner points,and a hybrid scheme combining first-order upwind and second-order central differences is used whendealing with boundary points.The second-order central differ-ence scheme is used for the viscous terms(Ren and Wang,2004).The solution algorithm as detailed in the original VOF method isemployed(Hirt and Nichols,1981).This method allows for simulation of breaking and post-breaking waves(Biausser et al.,2003)as well as steep waves thathave high velocity near the surface(freak waves).2.3.Boundary conditions2.3.1.Boundary conditions for the freak wave-makerThe improved superposition model is used to generate freakwaves.In the model,an extreme transient wave is embedded intoa random wave train,based on a partitioning of the total waveenergy with one part of the energy going into the underlyingrandom sea and the other into the focused transient wave(Kriebel,2000).The model can be expressed asZx,t Z1x,tZ2x,t XMi 1a1icoskix?oi*teiXMi 1a2icoskix?xc?oi*t?tc?7whereZis the surface elevation at a distance x from the wavegenerator in the wave tank;M is the number of component wave;C.Cui et al./Ocean Engineering 54(2012)132141133kiando*iare the wave number and frequency of the ith compo-nent wave,respectively.Supposing the energy of the target spec-trum S(o)is distributed mainly in the frequency rangeoLoHandis divided into M bands,oinis distributed randomly in the range(oioi1);eiis the random wave phase distributed uniformly in therange(02p);a1iand a2iare the amplitudes of the random andtransient waves;the remaining parameters,xcand tc,represent thedistance and time at which the transient waves will converge:DooH?oLM8oi1oiDo9a1iffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2p1SoiDoq10a2iffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2p2SoiDoq11where,p1is equal to 80%(80%of the energy goes into the randomsea)and p2is equal to 20%(20%of the energy goes into the transientwave).This condition can produce a realistic sea containing a largewave satisfying the adopted definitions of freak wave(Kriebel,2000).The surface elevation at the wave generator(x0)isexpressed asZt Z1tZ2tXMi 1a1icos?oi*teiXMi 1a2icoski?xc?oi*t?tc?12From wave-making theory,the horizontal velocity of thewave-maker for freak waves U0(t)can be written asU0tXMi 1oi*ZtWi13where,Wiis the transfer function associated with the ithcomponent of propagating wave:Wi4sinh2kidsinh 2kid2kid14where,d is the water depth.2.3.2.Wall boundaryThe momentum equation,kinetic energy equation and dis-sipation rate equation require different grid sizes.The grid thatthe dissipation rate equation requires is much smaller than theone that the momentum equation and kinetic energy equation do(Ma,1992).The wall function method is used for the boundarycell to keep proper cell dimensions and satisfy a certain accuracycondition(Zhang,1989):lmb0L expkuun?1?15k u2n=ffiffiffiffiffiffiCup16e u3n=lm17vt Cuk2e18whereb0is a constant coefficient that is related to the roughnessof the wall and L is the characteristic length.In our numericalmodel,L is chosen as the distance from the center of the boundarycell to the wall andb0is equal to 0.0005.2.3.3.Other boundary conditionsTo limit the computational domain,an absorbing damping zoneis applied at the right end of the flume to absorb the outgoingwaves.In the absorbing damping zone,an attenuation coefficient,m(x),is introduced for velocity and wave surface elevation,which isdefined asmx e2?10l0=l?2?10lna?19where,lis the length of the absorbing damping zone,lis thedistance between the point in the absorbing damping zone and rightend of the channel;ais the damping coefficient.On the bottom boundary,a no-flow condition is prescribed asvx,0 0203.Performance proof of numerical modelIn order to demonstrate the performance of the proposednumerical model,the model results of the freak wave over a flatbottom and the fission phenomena of a solitary wave over a slopeare compared with the experimental results by one of the authors(Cui et al.,in press),the experimental results by Madsen and Mei(1969)and the numerical results by Choi and Wu(2006).3.1.Generation of freak wave over a flat bottomThe efficiency to simulate freak waves over a flat bottomhas been verified by comparing the model results with theexperimental results by one of the authors.The computation domain is shown in Fig.1,which is dis-cretized by 1600 cells in the x-direction and 30 cells in they-direction.The grid interval in the horizontal direction,dx is0.05 m in the non-absorbing zone and is 0.1 m in the absorbingdamping zone;the length of absorbing damping zone,lis 30 m;the grid interval in the vertical direction,dy is 0.025 m;thecomputing time is 220 s;the time step is 0.01 s;the water densityris 1000 kg/m3;the kinematic viscosity u is 1.002?10?6m2/s;the modified PM spectrum is chosen as the incident wavespectrum,expressed asSo Ao?5exp?Bo?421A 173Hs2T?422B 691T?423where Hsis the significant wave height;T is the average period.A wave gauge is installed at the focusing point(xc),which is2100 cm from the wave maker(x2100 cm).Wave gauge2100cm75cmWave maker50cmAbsorbing damping zone6500cm3000cmFig.1.Sketch of the computational domain for simulation of freak wave over a flat bottom.C.Cui et al./Ocean Engineering 54(2012)132141134Fig.2 shows the comparison between a numerical freak waveand the experimental results.(The experiments were carried outby one of the authors in a wave tank 56.0 m long and 1.0 m widein the State Key Laboratory of Coastal and Offshore Engineering atthe Dalian University of Technology.)The freak wave character-istic parameters,a1Hmax/Hs,a2Hmax/Hmax?,a3Hmax/Hmaxanda4Zmaxc/Hmaxare indicated in Table 1.Hmaxis the maximumwave height;Zmaxcis crest height of the maximum wave;theHmax?and Hmaxare the wave height before and after themaximum wave.The model results are in excellent agreementwith the experimental data(Cui et al.,in press),indicating the