Diffraction Computation In Fhase-Average Wave Model.doc

Diffraction Computation In Phase-Averaged Wave ModelJia Xiao1 Pan Jun-ning21 Estuarine&Coastal Science Research Center,201201,Shang Hai,C2 Nanjing Hydraulic Research Institute,210029,Nan Jing,ChinaAbstract:Comparing to phase-resolving numerical wave model,phase-averaged wave model have lots of advantages,especially on time effency and detailed source terms.But diffraction can not be considered in these kind models.Since SWAN(Simulating WAve Nearshore) Version 40.51,diffraction is introduced to the model with phase-decoupled method.In this paper,the applicability of the diffraction calculation in the SWAN model is tested by calculation with half-infinite breakwater and harbour bathymetry. Then,improved eikonal equation in mild-slope equation is applied to the SWAN model to improve the diffraction term, accuracy and convergence are validated.Keywords:numerical wave model,phase-averaged model,diffraction,SWAN model,numerical convergence1.IntroduceAmong many kinds of numerical wave models,phase-averaged wave models(such as WAM model,SWAN model)have many advantages,such as its state-of-the-art source terms,its computational efficiency,coupled calculation with other hydrodynamic factors and so on.But,diffraction calculation can not be simulated in all phase-averaged models.Diffraction simulation should use other phase-resolving models(such as Mild Slope Equation,Boussinesq Equation).Because advantages in the phase-averaged models,many researchers tried a lot to introduce diffraction to phase-averaged models1.New version of SWAN model,since version 40.512 uses phase-decoupled method to calculate diffraction.After that, feasibility and applicability are analysed.Enet3 calculates wave field behind semi-infinite breakwater,and compare the results with analytical solutions.He also gives requirements that make diffraction calculation convergence. Nelson Violante-Carvalho4studies irregular waves diffraction and reflection in harbour basin,and compares the results with physical models results and MIKE21 BWs results.In this paper,improvement of the diffraction calculation is introduced in the SWAN model.Then,semi-infinite breakwater case and harbour case are calculated to find the feasibility of the improvement.Also,to find the applicability of the diffraction calculation in the SWAN model.2.The SWAN modelThe SWAN model uses the same equation and source terms with the WAM model,and additionally,some numerical calculation improvement and shallow terms are added to the SWAN model.The model uses directional spectrum to express real ocean waves which have randomicity characters,the simulations can be closer to the real waves.The SWAN model uses the action balance equation to compute the evolution of the wave field in time and special space.In Cartesian coordinates,the equation is given by:Where action density is defined as with relative frequency and E energy density.The velocity and are the propagation speed in frequency space and and directional space.The term S is the source term.The SWAN model uses fully implicit finite difference method,which can makes calculation unconditionally stable.This method allows larger time step in calculation,So the SWAN model is quite time saving.3.Phase-decoupled method in diffraction calculation of the SWAN modelTo accommodate diffraction in action balance equation, at 2003, Holthuijsen used phase-decoupled method to calculate diffraction in the SWAN model.It is expressed in terms of the directional turning rate of the individual wave components in the 2D wave spectrum. Exactly,this method is basd on the mild-slope equationWhen current does not exist and diffraction is ignored, =0.Then the propagation speed in space can be expressed: (1) eikonal equation can be expressed: (2) So,we can get the diffraction parameter as below: (3) In order to make calculation more convergent,some smoothing also has been done4.Improvement in diffraction calculationAccording to PH.D Pan5s study about elliptical mild-slope equation,improves diffraction term by adding high order bathymetry variation term and nonlinear dispersion term.Datailed information can be refered to the thesis of Jia6.In Pans study,mild-slope equation was separated to real part and imaginary part,two equations can be got.Wave action balance equation, (4) And, eikonal equation, (5)Where m stands for frequency bin number and n stands for wave direction bin number. ； ， ，with wave amplitude.So,if currents are ignored,the diffraction term in the SWAN can be described, Where (6) (7) (8)Datailed expressions of fromula(6),(7),(8) are given in the SWAN technical manual7 and research work of Zhang & Edge8,Kirby & Dalrymple9 and so on.5.ValidationTwo bathymetrys has been set to validate the diffraction calculation in the SWAN model.5.1 Semi-infinite BreakwaterEnet has calculated the diffraction propagation of the narrow spectrum behind semi-infinite breakwater,and summarized some convergence requirements.In this paper,similar experiment has been done to test if the dispersion in the diffraction term can improve the convergence.There exists a semi-infinite breakwater in 10 meters deep water.Reflection is ignored.Incident wave parameters are choosen as:Hs=1.0m,Tm=6.0s,and short-crestedness .Default diffraction No diffraction请将图片中的中文改成英文。 Modified diffraction 图片中的中文改成英文。Fig 1 Wave field behind the semi-infinite breakwater with short-crestedness =1.5No diffraction Default diffraction 请将图片中的中文改成英文。 请将图片中的中文改成英文。Modified diffraction 请将图片中的中文改为英文。Fig 2 Wave field behind the semi-infinite breakwater with short-crestedness =10Default diffraction No diffraction 请将图片中的中文改成英文。请将图片中的中文改成英文。请将图片中的中文改成英文。Modified diffraction Fig 3 Wave field behind the semi-infinite breakwater with short-crestedness =30Analyse figures above,it is obvious that adding dispersion can improve convergence in the calculation,but,adding dispersion also decrease the wave height behind the simi-infinite breakwater a little.And diffractions effect gets smaller when the short-crestedness gets larger.The effect of directional spreading in the SWAN model reduces the importance of diffraction.5.2 Harbour BasinIn SWAN Mannual,it is not recommended to calculate the wave field in harbour basin or in front of reflecting breakwaters.In this paper,a harbour basin physical model is used,because inside the basin,along the dock line, carpolite slopes are set,so reflection is quite small.When reflection is ignored,the wave field in the basin can be calculated by the SWAN model.The bathymetry shows below.Fig 4 Physical model of a harbour At south side of the model,there is a wave maker.Incedent wave parameter is given in table 1.Table 1 Design wave parameters used in the physical modelIncedent conditionReturn Period: 50 aWave parameterWave directionH4（m）H13（m）T（s）ESESE5.04.28.4In SWAN models setup,Jonswap spectrum is used with .Water level is 6.48m.In physical model,wave height scale is choosed as 1:110 and wave period scale is 1:. Amended SWAN model is used to calculate the wave field in the basin.The calculated results are transformed to actual scale.请将图片中的中文改成英文。 Fig 5 Results comparison (Physical model left,SWAN model right) ,ESE wave direction请将图片中的中文改成英文。 Fig 6 Results comparison (Physical model left,SWAN model right) ,SE wave directionIn this simulation,mesh numbers are strictly choosen,.Short-crestedness should not less than 10.When incident wave parameter is correctly choosen, reasonable wave fields calculated are quite close to the observed wave field.Wave height near the dock is smaller than observed data because reflection is ignored totally.6.Discussion and conclusionsThe present study revealed that the SWAN model can do diffraction calculation when phase-decoupled method is used,but quite lots of practical limitations exists.Especially narrow banded frequency waves and narrow banded directional waves.Grid resolution is quite important to stabilities of the calculation.The tests of the semi-infinite breakwater showed that instabilities occur easily when diffraction term is turned on.Adding nonlinear dispersion to the diffraction term makes the calculation more stable,but the shortcoming is that the adding the dispersion also decrease the wave height a little.Group study has not been done to find if the dipersion can work well at various water depth.The tests for harbour basin showed that stable results can get when SWAN model is well set.Reflections adding causes instabilities easily.Waves in basin area can be simulated quite well,but wave height near dock is smaller than physical models results because of reflection is ignored.The influence of high order bathymetry term has not been validated because of lack of proper experiment data.7.References1 Rivero, F.J., A.S. Arcilla, and E. Carci, Analysis of diffraction in spectral wave models. Proc. 3rd Int. Symp. Ocean Wave Measurement and Analysis, WAVES 97, ASCE, New York, 1997,pp.431-445.2 Holthuijsen, L.H., A. Herman, and N. Booij, Phase-decoupled refraction-diffraction for spectral wave models. Coastal Engineering, 49, 2003,pp. 291-