2001-2002 > Academic research: refereed > Publication in refereed journalVersion of RecordPublishe

Li, YS

Zhan, JM

Journal of Hydrodynamics

2001-2002 > Academic research: refereed > Publication in refereed journalVersion of RecordPublishedVoR allowe

PolyU Institutional Repository

Journal of Hydrodynamics , Ser.B ,4 (2001),24-27China Ocean Press , Beijing -Printed in ChinaA LARGE EDDY SIMULATION TURBULENCE MODEL FOR COASTAL SEASAND SHALLOW WATER PROBLEMS Zhan Jie-min ,Department of Applied Mechanics and Engineering , Zhong shan University , Guang zhou 510275 , ChinaLi Yok-sheungDepartment of Civil &Structural Engineering , The Hong Kong Poly technic University , Hong Kong , China(Received Nov.1 , 2000)ABSTRACT:　 In large scale motions of circulations in coastalseas and shallow-water problems , different characteristics of flowin the horizontal plane and in the ver tical direction are expected.In this paper , a new lar ge eddy simulation model was propo sed.There are some differences be tw een the present method and theo ther LES models.The philosophy of the large eddy simulationand the directional eddy viscosity method were applied in the ho r-izontal plane and in the vertical direction , respectively.Compar-ied with the o ther LES models in which there is no difference be-tw een ho rizontal viscosity and ver tical visco sity , the proposedmethod is resonable.KEY WORDS:　Large Eddy Simulation (LES), directional ed-dy viscosity method , coastal sea , shallow wa ter1.　INTRODUCTIONThe LES approach originated f rom the needs ofmeteorologists to predict or simulate the global weath-er .Generally Smagorinsky (1963)was credi ted wi ththe initial development of LES.The fi rst application ofthe LES approach to problems of eng ineerig interestw as made by Deardorff(1970), who treated the chan-nel f low wi th reasonable success.The development andapplications of LES have also been made by many otherresearchers , fo r example , Aldama (1982), Ferziger(1977)and Su(1990).Many applicat ions of large ed-dy simulation have succeeded in studies of the dynamicsof the atmosphereic ci rculation , but there is a lack ofthose in hydraulics.However , the LES as a viable toolfor oceanic turbulence research has been recognized on-ly recently and its oceanic applications are just begin-ning (McWilliams et al.1993 , Skylingstad and Den-bo , 1995).Because of the use of the Smagorinsky-Lil-ly approach , the viscosi ty in the Large Eddy Simula-tion(LES)no t only varies w ith grid sizes , but alsov aries w ith the trace of the st rain rate tensor S ij , Be-cause the value of the nondimensional constant Cs inthe Smagorinsky-Lilly viscosity formula can be predict-ed (Lilly 1967)using the Kolmogo rov - 53powerlaw , the variable viscosi ty is more reasonable than aconstant one.In a few previous models in hydraulics(Bedfo rd and Babajimopoulos 1980 , Babajimopoulosand Bedford 1980 , and Denbo and Sky lingstad 1996),there is no dif ference between horizontal viscosity andvertical viscosi ty.In the present study , the philosophyof the large eddy simulation and the directional eddyviscosity method are applied in the horizontal plane andin the vertical direction , respectively.2.　LARGE EDDY SIMULATION AND PARAME-TERIZATION OF VERTICAL VISCOSITYUsing the Gaussian filter and filtering hydrody-namics equations , we obtain  u x +  v y +  w z =0 (1)  u t + (uu) x + (vu) y + (wu) z -f v =-1ρ  p x(2)  v t + (uv) x + (vv) y + (wv) z +f u =-1ρ  p y(3)  w t + (uw) x + (vw) y + (ww) z =-1ρ  p z -g(4)In shallow w ater problems , because of the differ-ent characteristics of f low , and g rid sizes , in the hori-24 Project supported by the National Natural Science Foundation o f China.(No:19802023)zontal plane and in the vert ical direction , dif ferent val-ues of viscosity in the horizontal plane and in the verti-cal direction are expected.Using the Tay lor expan-sion , the f iltered value of the nonlinear terms in thehorizontal momentum equation can be approximated byu iu j =u i ·u j +L(u i , u j)+u′iu′j (5)in w hich i , j =1 ,2 andL(ui , u j)= Δ212γ u i x u j x +Δ222γ u i y u j y +Δ232γ u i z u j zwhere γis called the filter coefficient and equals6.0 and Δi is the filter size in the i th direction.If the scale of the g rid cell is small enough , Eq.(5)is av ailable for i , j =1 ,2 , 3 .Generally , limiltedby the capability of computers , the scale of the g ridcell(especially on the ho rizontal plane of the shallow-water flow field)canno t be small enough.So Eq.(5)is only used for the terms uu , uv vv .In the horizon-tal plane , using the smagorinsky-Lilly approach , theterm u′iu′j can be modeled asu′iu′j -23 δij E =-12 μ(  u i xj +  u j xi) (6)in w hichμ=(Cs Δ)2[(  u x)2 +(  v y)2 +12(  u y +  v x)2] 1/ 2 ,E = 12u′iu′j (7)and δij is the Kronecker delta;Δis a leng th related tothe g rid cell by Δ=(Δ1 Δ2)1/2;Cs is a dimensionlessconstg ant(=0.2 in this paper), which can be predict-ed using the Kolmogorov -53power law ; E is theSGS turbulent kinet ic energy per unit mass.2 E actslike pressure forces and can be absorbed by the pres-sure-g radient term so that in fact the pressure can bereplaced as an unknown quant ity by the modified pres-sure P = p + 23 E (8)Because of the dif ferent characteristics of flow inthe ho rizontal plane and in the vertical direction , theautho rs suggest that the nonlinear terms uw and vware t reated using the directional eddy viscosity method ,e.g .,ρuw =ρ u ·  w -τxz =ρ( u ·  w -ε u z) (9)ρvw =ρ v · w -τx z =ρ( v ·  w -ε v z) (10)For different problems , the formula of εhas dif ferentforms.For the simulation of the South China Sea , thefollow ing vertical viscosity has been used , which isbased upon Irish Sea observations (Bowden et al.1959 , Davies and Law rence 1995).ε=0.0025( u 2 + v 2)12 h (11)where  u ,  v =vertically-averaged velocity componentsand h is w ater depth.By assuming hydrostatic pressure dist ribution inthe vertical direct ion , the equations of u , v ,  w 　and ηcan be obtatined , where ηis the w ater surface eleva-tion.3.　NUMERICAL METHODTo reduce the ef fects of numerical errors on thephy sical process , we need very stable and high o rdernumerical schemes.Therefo re , based on the stag geredmesh , the spline interpolat ion and Lag rang rian inter-polation are used in the spatial discretization in the hor-izontal plane and in the vertical direct ion , respectively.In the time discretization , the following fourth-o rderaccurate scheme is used:(f n+12)k = f n +(f n+1)k2-Δt8{[( f t)n+1] k-( f t)n}(12)(f n+1)k+1 = f n +2Δt3[( f t)n+12 ] k +Δt6·{[( f t)n+1] k +( f t)n} (13)and(f n+1)0 = f n , [( f t)n+1] 0 =( f t)n when k =0wi th f representing  u , v o r η.Know ing (f n+12)k ,[( f t)n+12] k can be obtained from the equat ions of  u , v25o r η.The iterative procedure is continued until theabsolute values of the differences in  u ,  v 　and ηbe-tween two i terations are less than 10-5.When the it-erat ive procedure at each time step is completed ,(f n+1)k+1is denoted by f n+1.Numerical experiments show that , the proposedmodel has good numerical stabili ty and high-order ac-curacy.The ef fect of numerical errors on the physicalprog ress is very small.4.　EXAMPLE AND DISCUSSIONA second-order accurate version of the proposedLES model has been verified (Zhan and Li , 1993 ,1998).In this paper , the high order accurate LESmodel is applied to simulate the f low pattern of theSouth China Sea(SCS)driven by seasonal wind st ress-es and the Kuroshio.The SCS is one of the largest marginal seas of thew estern Pacific Ocean (Xu , Su and Qiu 1995 , Wyrki1961 , Metzger and Hurlburt 1996).The computa-tional domain is between 98°E and 126°E and betw een3°S and 26°N.The grid Size is 0.25°×0.25°.Fromsurface to bot tom , the thicknesses of the five w aterlayers used in the simulation are 60m , 150m , 300m ,500m and h1 , whereh 1 = 0 , 　　　h ≤1010mh -1010 , 　h >1010m (14)For h1 >0 , the bot tom stress components are used.When h1 =0 ,  u1 = v 1 =0 .The time step size is150s.The division of w ater body in the vertical driec-tion is based on the hydrog raphic analysis of the SCSby Xu , Su and Qiu(1995).To simulate correct ly the flow pat tern of the SCSdriven by seasonal wind st resses and the Kuroshio ,Computation was run for two years using data interpo-lated f rom monthly mean w ind st ress f ields.The openboundary conditions on 126°E and 26°N are interpolat-ed from the results obtained by Cai and Li (1997),and Li and Cai (1999).The first annual cycle w asused to w arm up the model f rom a mot ionless state.The simulated f low patterns in the surface layer in Jan-uary is show n in Fig.1.It can be seen that there arecyclonic eddies near the Natuna Island.In the gulf ofThailand , eddy circulation can be observed.The samephenomena are also reported in Naga report 2(Wyrki1961).In the Beibu Gulf , there is also eddy circula-tion , because of the geometric shape of the Gulf.When the Kuroshio f low s near the Bashi Channel , themain part of the Kuroshio w ill turn right and flownortheastw ard back into the Pacif ic.In the northwestof the Luzon and the southeast of the Taiw an , thereare st rong eddies.It seems that the eddies mix the w a-ter f rom the SCS and the Kuroshio.In the SCS , manymeso-scale eddies can be observed.This phenomenon isa common feature of oceanic circulations.Fig.1　The simulated flow pattern in the surface layer in Jan-uary using the LES model　ACKNOWLEDGMENTThe Scientif ic Research Foundation for the Re-turned Oversea Chinese Scholars , State Educat ionMinist ry .REFERENCES1. 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A large eddy simulation turbulence model for coastal seas and shallow water problems

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