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NumericalstudyontheeffectsofunevenbottomtopographyonfreakwavesChengCuin,NingChuanZhang,YuXiuYu,JingBoLiStateKeyLaboratoryofCoastalandOffshoreEngineering,DalianUniversityofTechnology,Dalian,Liaoning,ChinaarticleinfoArticlehistory:Received8May2011Accepted30June2012Availableonline5August2012Keywords:FreakwaveVOFmethodUnevenbottomtopographyWaveletanalysisabstractAnumericalmodelisbuiltbyusinganimprovedVOFmethodcoupledwithanincompressibleNavier–Stokessolver.Exploitingthemodel,thefreakwaveformationduetothedispersivefocusingmechanismisinvestigatednumericallywithoutunevenbottomsandinpresenceofunevenbottoms.Duringthefreakwavetransformationoveranunevenbottominfinitewater,combinedeffectsofshoaling,refractionandreflectioncanmodifytheexternalcharacteristicsoffreakwaves,andalsocancomplicatetheenergytransfers.Furthermore,waveletanalysismethodisadoptedtoanalyzethebehavioroftheinstantaneousenergystructureoffreakwaves.Itisfoundthatwhenthebottomsvaryinheight,theexternalcharacteristicparametersandhighfrequencyenergyshowasimilartrend,butthevaluemaybequitedifferentduetothedifferenceinlocalcharacteristicofthebottom.&2012ElsevierLtd.Allrightsreserved.1.IntroductionFreakwaveisatypeofextremelylargetransientwaterwave,beingclosetobreakingandofasymmetryinbothverticalandhorizontaldirection.Inthetime-frequencyspectrumoffreakwaves,strongenergydensityisinstantlysurgedandseeminglycarriedovertothehighfrequencycomponentsattheinstantthefreakwaveoccurs(LiuandMori,2000),thereforeseveredamagestovessels,maritimestructuresandotherfacilitiesintheoceanareusuallycaused.Suchwaveshavebeenobservedinalargenumberofbasinsaroundtheworld,indeeporshallowwaters,withorwithoutcurrents(KharifandPelinovsky,2003;LopatoukhinandBoukhanovsky,2004;MonbaliuandToffoli,2003),andthepopularRayleighdistributioncannotpreciselypredicttheprobabilityofoccurrenceoffreakwavesduetothenon-lineareffects(Stansell,2004,2005;Chienetal.,2002).Thepreviousstudieshaveshownthatseveralmechanismshavebeensuggestedtoexplaintheformationoffreakwavesinvariousenvironments.Amongthemonecanmentionthedispersivefocusing,wave–currentinteraction,atmosphericforcing,spatial(geometrical)focusing,non-linearself-focusingofwaveenergy,andnon-linearwave–waveinteraction(Chienetal.,2002;KharifandPelinovsky,2003;LopatoukhinandBoukhanovsky,2004).WuandYao(2004)reportedtheresultsoflaboratorymea-surementsonlimitingfreakwavesinthepresenceofcurrents.Itisfoundthatstrongopposingcurrentsinducingpartialwaveblockingsignificantlyelevatethelimitingsteepnessandasym-metryoffreakwaves.Toubouletal.(2006)experimentallyandnumericallyinvesti-gatedthedirecteffectofthewindonafreakwaveeventgeneratedbymeansofadispersivefocusingmechanism.Theresultssuggestthatthedurationofthefreakwaveeventincreaseswiththewindvelocity,andthepointwherethewavesmergehasashiftinthedownstreamdirection,whichisduetotheactionofthecurrentinducedbythewind.Petersonetal.(2003)presentedthatnon-linearinteractionsofsolitonicwavesintheframeworkoftheKadomtsev–Petviashviliequationmayresultinparticularlyhighandsteepwavesresem-blingthefreakwaves,anditmaybeagenericsourceoffreakwavesinareasofmoderatedepth.Overall,thesestudieshaveprovidedusagoodunderstandingtotheinfluencesontheexternalcharacteristicsoffreakwaves.However,insightintotheinternalstructureoffreakwavesplaysanimportantroleininterpretingthephysicalmechanism.Walkeretal.(2004)investigat...