VNUJournalofScience,EarthSciences23(2007)160‐169 160 Anumericalmodelforthesimulationofwavedynamics inthesurfzoneandnearcoastalstructures VuThanhCa* Center for Marine and Ocean-Atmosphere Interaction Research, Vietnam Institute of Meteorology, Hydrology and Environment Received07March2007 Abstract.Thispaperdescribesanumericalmodelforthesimulationofnearshorewavedynamics andbottomtopographychange.Inthispart,thenearshorewavedynamicsissimulatedbysolving the depth integrated Boussinesq approximation equations for nearshore wave transformation togetherwithcontinuityequation witha Crank‐Nicholsonscheme.Thewaverunuponbeachesis simulatedbyascheme,similartothe VolumeOfFluid(VOF)technique.Thewaveenergylossdue to wave breaking and shear generated turbulence is simulated by a ε − k model, in which the turbulence kinetic energy (TKE) generation is assumed as the sum of those respectively due to wavebreakingandhorizontalandverticalshear. Theverificationofthenumericalmodelagainstdataobtainedfromvariousindoorexperiments reveals that the model is capable of simulating the wave dynamics, turbulence and bottom topography change under wave actions. The simulation of turbulence in the surf zone and near coastalstructuresenable the model realisticallysimulatesthe contribution ofsuspendedsediment transportintothebedtopographychange. Keywords:Wavedynamics;Waverunup;Waveenergy;Surfzone;Boussinessqmodel. 1.Introduction 1 Extensive researches on the wave dynamics, sediment transport and bottom topography change in the nearshore area, especiallyinthesurfzone[1‐5,7,9,12,14‐17] have elucidated various aspects of coastal processes, such as the dynamics of wave breaking, characteristics of turbulence in the surf zone, structure of the undertow, the developmentofbottomboundarylayerunder breakingwaves,therateofbedloadtransport, uptakeofbedmaterialforsuspension,settling rateofsuspendedsedim entetc. _______ *Tel.:84‐913212455. E‐mail:vuca@vkttv.edu.vn Nadaoka [9] found by indoor experimentsthatduringwavebreaking,large vortices were formed and rapidly extended both vertically and horizontally. Ting and Kirby [15‐17] by conducting experiments withdifferentwaveconditionsfoundthatthe advective and diffusive transports of TKE play a major role in the distribution of turbulence, especially under plunging breaker. They also found that under spilling breakers (the breaking of relatively steep wavesonagentleslope),thetimevariationof TKE was relatively small, and th e time average transport of TKE was directed offshore. Under plunging breakers (the breaking of less steep waves on a gentle slope), there was a large time variation of VuThanhCa/VNUJournalofScience,EarthSciences23(2007)159‐168 160 TKE, and its time averaged transport is directedon‐shore. For situations with negligible alongshore sediment tr ansp ort, the status of a be ach depends on the cross‐shore transport of sediment,whichis closelyrelatedwithwave conditions. If the shoreward transport of sediment by incoming waves exceeds the offshore transport of sediment by ret r eating waves and the undertow, there will be a net onshore transport of sediment, resulting in beach accretion. Otherwise, the beach is in equilibriumstateoreroded. During a storm, turbulence generated by the breaking of a relatively short wind wave has not been significantly dissipated when a newwavearrivesandbreaks.Thus,the time variation of TKE is relatively small, and the combination of wave‐induced flow and undertowmaytransportTKEandsuspended sediment offshore. This results in the offshore‐directed transport of sand during storm and the associated beach erosion. On the other hand, post storms, turbulence generated by the breaking of a long period‐ small amplitude swell has significantly dissipated when the wave retreats. Thus, thereisalargetimevariationofTKE,andthe peaks in turbulence intensity and suspended sediment concentration coincide with incoming waves. Accordingly, onshore transportofTKEandsuspended sedimentby incoming waves exceeds the offshore transport by retreating waves and the undertow. This results in a net onshore transport of suspended sediments and helps explaining the onshore‐directed transport of sediment during calm weather and the consequentpoststormbeachrecovery. Schaffer [14] and Madsen [7] developed models for the simulation of the nearshore wave dynamics based on Boussinesq approximation equations. The wave energy loss due to breaking is simulated by employing a surface roller model. Due to the instability of the numerical code resulting from the tr eat m ent of the surface roller wave energy loss, Schaffer [14] had to use a smoothingtechniquetostabilizethesolution. Rakha et al [12,13] presented a quasi‐2D and a quasi‐3D phase resolving hydrodynamic and sediment transport models. In these mo dels, the horizontal transport of TKE, and the associated transport of suspended sediment are neglected.However,asdiscussedpreviously, results of Nadaoka et al [9] and Ting and Kirby[16]show thatthehorizontaltransport ofTKEinthesurfzoneisveryimportantand should not be neglected. Thus, without accounting for this, it is not easy to simulate the beach erosion during storm and the consequentrecoveryafterthe storm. NadaokaandOno[10]presentedadepth‐ integrated k‐model where the TKE production rate was evaluated with a Rankineeddy model. In this model, the TKE dissipation rate and the eddy viscosity was evaluated by employing an empirical length scale. The model had not been verified againstexperimentaldata. Also,waverunup onbeach,whichismainlyresponsibleforthe erosion of foreshore during storms, is not simulatedinthismodel. Regarding all the above mentioned facts, the purpose of this study is to develop a numerical model that can simulate the nearshore wave dynamics, including wave breaking and wave runup, the generation, transportanddissipationofTKE. 2. Governing equations of the numerical modelfornearshorewavedynamics In this study, the near‐shore wave VuThanhCa/VNUJournalofScience,EarthSciences23(2007)159‐168 161 dynamics are simulated by solution of two‐ dimensional depth integ rated Boussinesq approx im at ion equations, including bottom friction and wave energy loss due to wave breaking and shear. The main equations of thenumericalmodelarewrittenas: 0= ∂ ∂ + ∂ ∂ + ∂ ∂ ty q x q y x η (1) 0 2 6 2 3 2 3 2 3 2 33 2 =+− ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂∂∂ ∂ + ∂∂ ∂ − ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂∂∂ ∂ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂∂ ∂ + ∂ ∂ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ x c bx y x y x yx xx Qq d f M tyx q tx q h h q tyxh q tx h x gd d qq yd q xt q η (2) 0 2 6 2 3 2 3 2 3 2 33 2 =+− ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂∂∂ ∂ + ∂∂ ∂ − ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂∂∂ ∂ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂∂ ∂ + ∂ ∂ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ y c by x y x y yyxy Qq d f M tyx q ty q h h q tyxh q ty h y gd d q yd qq xt q η (3) where x q and y q are respectively the depth integrated flow discharges in x and y directions; η is the water surface elevation; d isthe instantaneous water depth; h is the still water depth; c f is the bed friction coefficient; Q is the total discharge, defined as 22 yx qqQ += ; and bx M and by M represent the wave energy loss due to breaking, evaluated by introducing an eddy viscosity andexpressedas: () () () () ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∂ ∂ ∂ ∂ + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∂ ∂ ∂ ∂ = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∂ ∂ ∂ ∂ + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∂ ∂ ∂ ∂ = y dq df yx dq df x M y dq df yx dq df x M y tD y tDby x tD x tDbx // // νν νν (4) InEq.(4), t ν istheeddyviscosity;and D f is an empirical coefficient, determined based onthecalibrationofthenumericalmodel. When waves are breaking on beach, a part of the lost wave energy is transformed into turbulence energy. At the beginning of the wave breaking process, the turbulence is confinedintoa small portion ofthebreaking wave crest, the surface roller; after that, turbulence eddies rapidly expand in vertical and horizontal dire ctions [9, 15‐17]. The turbulence under wave breaking is very complexandfullythree‐dimensional.Thus,a 3Dmodelisrequiredforapropersimulation of turbulence processes here. However, such a model would require an excessive computational time and at the moment is not suitable for a practical application. On the otherhand, based on resultsof Nadaokaetal [9],TingandKirby[15‐17],itcanbeestimated that in the surf zone, the ti me scale for turbulence energy transport in the vertical direction is much shorter than that in the horizontaldirections.Thus,thesimulationof the transport of TKE in the horizontal direction is more important than that in the vertical direction. Therefore, in the present study, the TKE is assumed un ifor mly distributedinthewholewaterdepth,andthe depth ‐integrated equations for the production, transport and dissipation of the TKEanditsdissipationrateread: () () , // ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∂ ∂ ∂ ∂ + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∂ ∂ ∂ ∂ + −= ∂ ∂ + ∂ ∂ + ∂ ∂ y dk d yx dk d x P y vk x uk t k t t t t r σ ν σ ν ε (5) () () () ε εε σ ν ε σ ν εεε εε ε ε 21 / / CPC ky d d y x d d xy v x u t r t t −+ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∂ ∂ ∂ ∂ + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∂ ∂ ∂ ∂ = ∂ ∂ + ∂ ∂ + ∂ ∂ (6) where k and ε are respectively the depth integrated TKE and its dissipation rate; u and v are respectively phase‐depth averagedflowvelocitiesinxandydirections; t σ , ε σ , ε 1 C , ε 2 C are closure coefficients. In VuThanhCa/VNUJournalofScience,EarthSciences23(2007)159‐168 162 Eq.(6), r P is theTKE productionrate,which is assumed as a summation of the TKE production due to bottom friction rb P , horizontal sh ear rs P and wave breaking rw P as: rwrsrbr PPPP ++= (7) Withknownvaluesof k and ε ,theeddy viscosityisevaluatedas: () εν ε dkC t / 2 = , (8) where )09.0( = ε C isconstant. The scheme for the simulation of wave runup and rundown on the beach is explained in the next section. By employing this scheme, the present model can simulate the wave setup, set down on the beach, and theerosionofforeshoreduringstormevents. 3. Boundary and initial conditions and numericalscheme 3.1.Boundaryandinitialconditions It is possible to use a weekly wave reflected boundary condition such as the Summerfeld radiation condition at the offshore boundary to let reflected waves freelygoingout ofthecomputationalregion. However, this linear wave theory based boundary condition, when applied in combination with a nonlinear wave model, does not ensure mass conservation and may lead to an accumulation or lost of water insidethecomputationalregion.Thus,inthis study,watersurfaceelevationunderwavesis givenattheoffshoreboundary. Wave‐absorbing zones are introduced at the lateral boundaries to minimize wave reflection. The bed friction coefficient c f in these zones is assumed constant within first five meshes from the lateral boundaries, and then increases linearly with the distances fromthe boundaries towardsthe ends ofthe waveabsorbingzones.Finally,attheendsof the wave absorbing zones, the Summerfeld radiation condition for long waves are introduced to letremaining waves going out of the computational region. A free slip boundary condition is applied at surfaces of thecoastalstructures. Zero gradients of k and ε are assumed at the offshore, lateral boundaries and at surfacesofcoastalstructures. A scheme similar to that of Hibberd and Peregrine [5] is used to compute the wave runuponthebeach.Asketchoftheschemeis shown in Fig. 1. In this scheme, when the shore is approached, all the dispersion terms in Eqs. (2) and (3) are turned off. Additionally, a cell side wetted function, defined as the wetted portion over the total length of a cell side, and a cell wetted area function, defined as the wetted portion over the total cell area are introduced to account for the fact that water flows only in wetted parts of the cells on the instantaneous shoreline. Then, the continuity equation (Eq. 1) and momentum equations (Eqs. 2 and 3) can be derived by a method similar to Vu et al[19]andbecome: 0= ∂ ∂ + ∂ ∂ + ∂ ∂ t S y qf x qf yxxy η (9) () () ,0 / 1 / 1 11 2 2 =+ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∂ ∂ ∂ ∂ − ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∂ ∂ ∂ ∂ − ∂ ∂ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ x cx t x t yx xx Qq d f y dq Sd yS x dq Sd xSx gd d qSq ySd Sq xSt q ν ν η (10) VuThanhCa/VNUJournalofScience,EarthSciences23(2007)159‐168 163 () () 0 / 1 / 1 11 2 2 =+ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∂ ∂ ∂ ∂ − ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∂ ∂ ∂ ∂ − ∂ ∂ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ y c y t y t yxyy Qq d f y dq Sd yS x dq Sd xSy gd d Sq ySd qSq xSt q ν ν η (11) where x f and y f arerespectivelythecellside wetted functions corresponding to x and y directions, and S is the cell area wetted function. Fig.1.Thecoordinatesystemandmethodforthe evaluationofawettinganddryingboundary. The procedure for determining the cell side wetted function and the cell area wetted function in the numerical scheme will be discussedinthenextsection. A still water is assumed at the beginning of the computation. With this, all variables aresetequaltozeroinitially. 3.2.Numericalscheme Equations (1‐3) and (5‐6) are integrated numerically on a spatially staggered grid system, where components of the flow discharge are evaluated at surfaces, and bed elevation, k and ε are evaluated at the centersof control volumes. The sketchof the coordinates and computational mesh is showninFig.1.Asitwillbediscussedlater, in the present scheme, the water level inside acell is evaluatedatthecenterofthewetted area inside the cell. A second order accurate Crank‐Nicholsonschemeisemployedforthe time discretization for all equations, and a central differencing scheme is employed for spatial discretization of Eqs. (1) to (3). The spatial disretization for advection terms of Eqs. (5) and (6), governing the transport, diffusion, generation and dissipation of k and ε , follows the third order accurate QUICK scheme, and that for the diffusion terms follows the central differencing scheme. As the discretization scheme is implicit, an iterative scheme similar to the SIMPLE scheme of Patankar [11] is employed. At the beginning of a new time step, the computation of the flow discharges requires the still unknown water level and eddy viscosity. Thus, at first, the water level ateachnewtimestepisassumedequaltothe valueattheprevioustimestep.Then,Eqs.(2) and (3) are solved to get the flow discharges in x and y directions, respectively. The new values of the flow discharges are substituted into the continuity equation to compute the new water le vel. Also, with the new water level, the thickness of the surface roller is evaluated. Then, Eqs. (5) and (6) are integratedtoget k and ε ,andconsequently the new coefficient of eddy viscosity. All newly obtained water level, flow discharges and coefficient of eddy viscosity are substituted back into Eqs. (2) and (3) to compute the new components of the flow discharge. The procedure is repeated until convergedsolutionsarereached. The wetted periphery inside a computational mesh at the intersection betweenthewatersurfaceandthebeach,the cell side wetted function and the cell area wetted function at each time step are VuThanhCa/VNUJournalofScience,EarthSciences23(2007)159‐168 164 evaluated explicitly based on the water level, bed elevation and the bed slope in two directions.Theprocedureforthisisshownin Fig.1.Thebedelevationsatcellcorners(such aspointsA,B,CandDinFig.1)areevaluated as the average value of the bed elevation at four adjacent points. For example, the bed elevationatpointCinthisfigureisevaluated as: 4 ,11,11,, jijijiji c bbbb b ++++ +++ = ,(12) where b c is the bed elevation at point C, and b i,j , b i,j+1 , b i+1,j+1 and b i+1,j are respectively the bed elevations at the center of cells (i,j), (i,j+1), (i+1,j+1)and(i+1,j). The water level at a cell side is averaged from the water levels at two adjacent cells. For example, the water level on the side BC ofcell i,jinFig.1isevaluatedas: 2 1,, + + = jiji bc ηη η , (13) where bc η , ji, η and 1, +ji η are respectively water levels at the cell side BC, and in the cells(i,j)and(i,j+1). If one of adjacent cells to a cell side is completely dry (with the value of the area wetted function equal to zero), the average water level at the cell side is assumed equal tothewaterlevelatthewettedcell.Basedon the bed elevation at its two ends and the average water level on a cell side, the intersected point between the water surface and the cell side, and the wetted portion of the side are determined. When the average waterlevelonthecellsideishigherthanthe bed elevation at its two ends, the side is consideredtotally submerged intothe water, and the corresponding value of the cell side wettedfunctionis1.Forothercases,valueof the cell side wetted function equals to the ratioofthe lengthofthewettedportion over the total length of the cell side. After getting allthewettedpointsonfoursidesofthecell, the wetted periphery and the wetted area inside a cell are determined by connecting two adjacent wetted points with a straight line. This wetted periphery is shown by the dottedlineinFig.1.Thewettedareaincelli,j in this figure is the portion of the cell from the dotted line to offshore. The wetted periphery and area inside the cell are kept constantforatimestep. 4.Modelverification 4.1. Wave transformation and characteristics of turbulence due to wave breaking on a natural beach To verify the accuracy of the numerical model on the simulation of the wave transformation on a natural beach, existing experimental data on the wave dynamics in the nearshore area obtained by Ting and Kirby [15 ‐17] are used. The experiments were carried out in a two‐dimensional wave flumeof40mlong,0.6mwideand1.0mdeep. A plywood false bottom was installed in the flume to create a uniform slope of 1 on 35. Regular waves with heights and periods equalto12.7cm,2s and8.7cm,5sareusedas incoming waves respectively for spilling breakerandplungingbreakerexperiments. Fig. 2 shows the sketch of the Ting and Kirby [15‐17] experiments. Computation was carried out with the same conditions of the experiments. The critical water surface slope for a broken wave to be recovered φ0 is set equalto6 0 ,accordingtoMadsenetal[7]. VuThanhCa/VNUJournalofScience,EarthSciences23(2007)159‐168 165 Wave generator 0.4m 35 1 0.38m Fig.2.ExperimentsbyTingandKirby[15‐17]. As cited by various authors [2, 4], when waves are breaking, a major part of the lost wave energy is dissipated directly in the shearlayerbeneaththesurfaceroller,andonly aminorpartofitistransformedintoturbulent energy. Thus, a turbulence model may underestimate the wave energy lost due to breaking. To account for this, an empirical coefficient D f was introduced in Eq. 4. Calibrationswere carriedouttofindthebest valueofthiscoefficient.Vuetal[18]founda constant value of 1.5 for this coefficient for theirone‐dimensionalmodel.However,their computational results show that the coefficient does not provide adequate wave energy dissipation, and the computed wave heights after breaking is significantly larger thantheobservedones. As mentioned previously, wave breaking happens with a sudden loss of wave energy. This in a numerical model can be simulated by a sudden increase in the “energy dissipation coefficient” D f . As the breaking waveprogresses onshore, the growth of TKE mayaccompanyanincreaseinthecoefficient. On the other hand, turbulence length scale, and the corresponding turbulence intensity decrease with water depth, leading to a decrease in the coefficient. Thus, in this study, the coefficient is assumed suddenly increases at the breaking point, then gradually increases towards the shore, and thendecreaseswiththe decreaseinthewater depthinthefollowingform: 2 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − += mb m mb b D h h h xx baf , (14) where a and b are constants, to be determined from calibration; x and x b are respectively the coordinates in the on‐ offshore direction at the point under considerationandthebreakingpoint; m h and mb h arethecorrespondingmeanwaterdepths attherespectivepoints. Fig.3showsthecomparisonbetweenon‐ offshoredistributionsoftimeaveragedmean water surface elevation, minimum water surface elevation, maximum water surface elevation, and wave height for the spilling breaker, computed by the model (with D f evaluated following Eq. (14), 05.0=a and 1 = b ), and observed by Ting and Kirby [15, 16]. -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0 1 2 3 4 5 6 7 8 9 10 11 12 13 Horizontal Distance (m) Height (m) Bed Comp. Etaav Comp. Etamax Comp. Etamin Comp. Waveh Obs. Wavh Obs. Etaav Obs. Etamax Obs. Etamin Fig.3.Comparisonbetweenobservedandcomputed timeaveragedwaveheight,highest,lowestand meanwatersurfaceelevationforspillingbreaker. ExperimentaldatafromTingandKirby[15,16]. ItcanbeseeninFig.3thatthemodelcan accurately predict the wave breaking point and provides adequate wave energy dissipation after breaking. The maximum, minimumandmeanwaterlevelsatallpoints in the computational region are also predicted by the model with good accuracy. The general satisfactory agreement between computed and observed data shown in the VuThanhCa/VNUJournalofScience,EarthSciences23(2007)159‐168 166 figure suggests that the model can simulate nearshore wave processes, such as wave energy loss due to breaking, wave setup, setdownetc.withacceptableaccuracy. Figures (4) to (7) respectively show the time variation of ensemble averaged (phase‐ averaged) non‐dimensional water surface elevation, depth‐averaged horizontal flow velocity,TKE,andadvectivetransportrateof TKE, computed by the model and observed by Ting and Kirby [15, 16] at () 642.7/ =− mbb hxx .Thetimetinthefiguresis non‐dimensionalized by wave period T. For convenient, the same coordinate system in Ting and Kirby [15‐17] is employed in this study. The computed time variation of ensemble‐averaged water surface elevation fluctuation, non‐dimensionalized by local mean water depth h m (equal the sumof local still water depth and mean water surface fluctuation η ), shown in Fig. 4 agrees very well with observed data. The agreement between computed and observed time variation of phase and depth‐averaged horizontal flow velocity, non‐ dimensionalized by the local long‐wave celerity c (defined as () Hhgc m += , with H as the deepwater wave height) also agrees satisfactorily with observed data. The agreement between computed and observed phase and depth‐averaged non‐dimensional TKE and its advective transport is less satisfactory than that of the water level or flow velocity. It must be noted that the computation of TKE employs a depth‐ integrated ε −k model,whichinvolvesmany approximation assumptions and may not accurately predict the TKE production, transport and dissipation under a complex situation such as wave breaking. Among all, theweakestpointofthis model might be the depth‐integrated approximation. It is commonly known that just after wave breaking, turbulence is concentrated only inside the surface roller, and flow in the region below remains irrotational. Thus, a depth‐integrated model for the generation, transport and dissipation of TKE cannot be considered as a good approximation for this situation. However, despite of all inadequate assumptions and approximations, order of TKEpredictedbythe model,showninFig.6, agreeswellwiththeobservedone.Regarding difficultiesinpredictingtheTKEunderwave breaking with a numerical model, it can be saidthatthenumericalmodelcanpredictthe TKE and its advective transport with satisfactoryaccuracy. -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 00.20.40.60.81 t/T ( ζ -< ζ >)/ h Fig.4.Computedandobservedphase‐averaged watersurfaceelevationat(x‐x b)/hb=7.462.Spilling breaker. -0.2 -0.1 0 0.1 0.2 0.3 0.4 00.20.40.60.81 t/T <u>/c VuThanhCa/VNUJournalofScience,EarthSciences23(2007)159‐168 167 Fig.5.Computedandobservedphase‐depth averagedhorizontalflowvelocity at(x-x b )/h b =7.462.Spillingbreaker. The agreement between computed and observedadvectivetransportsofTKE,shown inFig.7,isbetterthanthatfortheTKEitself. Results of Ting and Kirby [15, 16] show that there is a tendency of offshore (negative) transport of TKE. The computational results by the present model also reveals the same tendency; ho wever, as shown in Fig. 8, the residual advective offshore transport of TKE evaluated by the numerical model is significantlysmallerthantheobservedone. From the general agreement between computed and observed values of various wave characteristics, it can be remarked that the numerical model can simulate wave transformation in the nearshore region with anacceptableaccuracy. 0 0.001 0.002 0.003 0.004 0.005 0.006 00.20.40.60.811.2 t/T k /c 2 Fig.6.Computedandobservedphase‐depth averagedrelativeturbulentintensity at(x-x b )/h b =7.462.Spillingbreaker. -1 -0.5 0 0.5 1 1.5 2 00.20.40.60.81 t/T <u>k/c 3 (X10 - 3 ) Fig.7.Computedandobservedphase‐depth averagedrelativeadvectivetransportrateofTKE inthehorizontaldirectionat(x-x b )/h b =7.462. Spillingbreaker. 4.2.Waverunuponbeach To verify the accuracy of the simulation bythepresent numerical modelonthewave runup on beach, experimental data of Mase andKobayashi[8]areused.Thesketchofthe experiment is shownin Fig. 10. As shown in the figure, the experiments were carried out in a wave flume with the length of 27 m, depth of 0.75 m and width of 0.50 m. An irregular wave generator is installed at one end of the wave flume. At the other end is a model beach with a foreshore slope of 1/20. The water depth in front of the slope is set constantandequalto0.47m.Thewaverunup on the beach is recor ded by a wave meter. Wave groups used in the experiments are expressedas: () [] () [] ()() ,2cos2cos 12cos 2 1 12cos 2 1 max ftft ftft ππδ δπδπ η η = −++= (15) where max η is the amplitude of the incoming waves, f is the wave frequency, and ∆ is the variation in the relative wave frequency. During the experiments, max η was taken as 5 cm. VuThanhCa/VNUJournalofScience,EarthSciences23(2007)159‐168 168 -0.05 -0.025 0 0.025 0.05 0 5 10 15 20 25 Time (sec) Water Surface Elevation (m) Fig.8.Computedandobservedwaverunupheight. T=2.5s,∆=0.1. Fig. 8 shows an example of comparison between observed and computed wave runup for different wave periods. It can be seen in the figures that the computed wave runup heights agree very satisfactorily with theobservedvalues. The computational results (not shown) also reveal that short period waves are dissipated much more rapidly on the beach compared with long period waves. The very satisfactory agreement between computed and observed wave runup heights reveals that the numerical model can accurately simulatewaverunuponbeaches. The model is also verified for its applicability of computing waves near coastalstructures. 5.Conclusions A numerical model has been developed for the simulation of the wave dynamics in the near shore area and in the vicinity of coastal structures. It has been found that the numerical model can satisfactorily simulate the wave transformation, including wave breaking, wave runup on the beach, and turbulence generated by wave breaking and shear. As the model is a depth‐integrated, two‐dimensionalinthehorizontaldirections, the computational time is relatively short. Thus, the application of the model for simulation of wave transformation in the field, especially in the vicinity coastal structures and inside harbours is very promising. References [1] D. Cox, N. Kobayashi, Kinematic undertow model with logarithmic boundary layer, Journal ofWaterway, Port, Coastal,andOceanEngineering 123/6(1997)354. [2] W.R.Dally,C.A.Brown ,Amodelinginve stig ati on of the breaking wave roller with application to cross‐shore currents, Journal of Geophysical Research100(1995)24873. [3] A.G. Davies,J.S.Ribberink,A. Temperville,J.A. Zyserman, Comparisons between sediment transport models and observations made in wave and current flows above plane beds, CoastalEngineering31(1997)163. [4] R. Deigaard, Mathematical modelling of waves inthesurfzone,Prog.ReportISVA69(1989)47. [5] S. Hibberd, H.D. Peregrine, Surf and runup on beach:Auniformbore,JournalofFluidMechanics 95(1979)323. [6] C.W. Hirt, Nichols,Volumeof fluidmethodfor the dynamics of free boundaries, Journal of ComputationalPhysics39(1981)201. [7] P.A. Madsen, O.R. Sorensen, H.A. Schaffer, Surf zone dynamics simulated by a Boussinesq typemodel.Part1:Modeldescriptionandcross‐ shore motion of regular waves, Coastal Engineering33(1997)255. [8] H. Mase, N. Kobayashi, Low frequency swash oscillation, Journal of Japan Society of Civil EngineersII‐22/461(1993)49. [9] K. Nadaoka, M. Hino, Y. Koyano, Structure of the turbulent flow field under breaking waves in the surf zone, Journal of Fluid Mechanics 204 (1989)359. [...]... resolving cross shore sediment transport model for beach profile evolution, Coastal Engineering 31 (1997) 231. [13] K .A. Rakha, A quasi‐3D phase‐resolving hydrodynamic and sediment transport model, Coastal Engineering 34 (1998) 277. [14] H .A. Schaffer, P .A. Madsen, R. Deigaard, A Boussinesq model for waves breaking in shallow water, Coastal Engineering 20 (1993) 185. [15] F.C.K. Ting, J.T. Kirby, Observation of undertow ... F.C.K. Ting, J.T. Kirby, Observation of undertow [16] [17] [18] [19] 169 and turbulence in laboratory surf zone, Coastal Engineering 24 (1994) 51. F.C.K. Ting, J.T. Kirby, Dynamics of surf zone turbulence in a strong plunging breaker, Coastal Engineering 24 (1995) 177. F.C.K. Ting, J.T. Kirby, Dynamics of surf zone turbulence in a spilling breaker, Coastal Engineering 27 (1996) 131. Vu Thanh ... Coastal Engineering 27 (1996) 131. Vu Thanh Ca, K. Tanimoto, Y. Yamamoto, Numerical simulation of wave breaking by a k‐ε model, Proceedings of Coastal Engineering, JSCE 47 (2000) 176. Vu Thanh Ca, Y. Ashie, T. Asaeda, A k‐ ε turbulence closure model for the atmospheric boundary layer including urban canopy, Boundary‐Layer Meteorology 102 (2002) 459. ...Vu Thanh Ca / VNU Journal of Science, Earth Sciences 23 (2007) 159‐168 [10] K. Nadaoka, O. Ono, Time‐Dependent Depth‐ Integrated Turbulence Closure Modeling of Breaking Waves, Coastal Engineering ACSE (1998) 86. [11] S.V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere Publ. Co., London, 1980. [12] K .A. Rakha, R. Deigaard, I. Broker, A phase resolving cross shore sediment transport model . VNUJournal of Science,EarthSciences23(2007)160‐169 160 A numerical model for the simulation of wave dynamics in the surf zone and near coastal structures VuThanhCa* Center for Marine and. computing waves near coastal structures. 5.Conclusions A numerical model has been developed for the simulation of the wave dynamics in the near shore area and in the vicinity. of the numerical model on the simulation of the wave transformation on a natural beach, existing experimental data on the wave dynamics in the nearshore area obtained