Analysis of Mixing of Pollutants in Water Waves and Currents

China Ocean Engineering , Vol.21 ，No.1 ,pp.115- 124◎2007 China Ocean Press , ISSN 0890-5487Analysis of Mixing of Pollutants in Water Waves and CurrentsYUAN Li-rong(袁丽蓉),b,1 , SHEN Yong-ming(沈永明力and TANG Jun(唐军)“Guangdong Province Key Laboratory of Coastal Ocean Engineering , Ocean Engineering Research Center，Sun Yat-sen University , Guangzhou 510275 ,Chinab State Key Laboratory of Coastal and Ofshore Enginering , Dalian University of Technology ，Dalian 116023 ,China( Received 6 May 2006 ; accepted 10 August 2006 )ABSTRACTA vertical two- dimensional turbulence numerical model for the interaction of waves and curents is developed in thepaper based on the nonlinear two- equation h - ε model with the VOF method . The one- dimensional equivalent advectionvelocity and equivalent mixing coeffcient are defined and the solving process is introduced : The pollutant concentrationfield , generated by an instant source in waves and currents , is caleulated with the model , and then the equivalent advc-tion velocity and equivalent mixing cofficient are obtained by calculating the time derivative of the mean and variance ofpollutant concentration probability distribution. The ffects of wave period and wave height on the equivalent mixing coef-ficient for waves and wave- currents are also investigated.Key words : mixing ; wave-current ; turbulence model ; VOF method ; pollutant concentration1. IntroductionRecently , more and more researches are focused on unstable flow fields , such as osillating flowfield , and tidal and wave-current field( Wang and Wang ,2005 ;Lu et al. ,2005 ). Based on experi-mental study and results from field observation , Li et al. ( 1994 ) regarded the fluctuations betweensectional averaged values and tidal averaged value of velocity and concentration as the main reason forequivalent dispersion . Experiments of pollutant transport for diverse wave cases in surf zones were con-ducted by Tang et al. ( 2004 ), and the results show that the movement state of pollutants in a surfzone is affected by incident waves , is quite complicated in diverse wave cases , and different from it inpure currents. But the study on the mixing in a non-breaking wave field is less studied and the conclu-sions on the relationship between wave parameters and pollutant mixing are not identical yet. Fischer etal. ( 1979 ) considered that the wave effect on dispersion can be ignored if the wavy period is farsmaller than the time scale of the pollutant diffusion. Law ( 2000 ) studied the longitudinal dispersiveeffect due to mass transport induced by progressiv中国煤化工d that the dispersive ef-fect is more significant if the wave height is mucl:MYHC N M H Ghe difusivity; T is thewave period ). An advection-diffusion model in sigma coordinates has been developed to study the mix-* This Project was supported by the National Natural Science Foundation of China( Grant No. 50579005 ) , and the Na-tional Basic Research ( 973 ) Program of China( Grant No. 2005CB724202 )1 Correspo牙有 数据. E- mail : yuanlr @ mail . sysu. edu. cn116YUAN Li-rong et al./ China Oean Engineering ,21( 1 ) , 2007 ,115- 124ing and dispersion of pollutants under the action of water waves by Yuan et al. ( 2004 ). Law andYuan' s methods are not convenient in practice because their methods completely depend on the diffu-sivity D but the approach to obtaining D has not been given .A vertical two-dimensional turbulence numerical model for the interaction of waves and currents isdeveloped in the paper based on the nonlinear two-equation h- ε model and VOF method . The veloci-ty，diffusivity and pollutant concentration distributions in waves , currents and wave-curents are ob-tained from the turbulence model. Wave effects on pollutant mixing are numerically investigatedthrough analysis of pollutant distribution in waves and wave-currents with different wave parameters.2. Mathematical Model2.1 Governing EquationsA vertical 2D turbulence numerical model is developed in the paper based on Zhang' s two-equa-tion k- ε nonlinear turbulence model( Zhang , 1994 ) with the VOF method ,and it can be written asfollows :x= 0,(1)(2 )Dt=Gi-pax+x;+ax;Tj= D(3)=- u'u',(4)- u'u’=- 3δjp +子(u.习山-Ou9δij，(5)(6)12/3Lr= ClE，(7 )Dk__Di =是((2+2)x)ul'ui'gr-E,(8)Dε刁(2) deεdu;dx;v+σ。]ax;Ce 1hu'u'Jx;(9)r\ Jc( 10)Di= Jx;\中国煤化工DF( 11)MHCNM HGwhere( u )= u;=( u 10 ) are the velocity components in horizontal and vertical directions , respective-ly ; u;'=( u' ,v') are the turbulent fluctuation velocity components in horizontal and vertical direc-tions , respectively ; G; is the mass force iρ is the fluid density ; P is the pressure ; τ and t' are theReynolds tB数插turbulent stress , respectively ;v and , are the fluid kinematic viscosity cofficientYUAN Li-rong et al./ China Oean Engineering ,21( 1 ) , 2007 ,115- 124117and turbulent viscosity coefficient , respectively ; k and ε are the turbulent kinetic energy and turbulentkinetic dissipation rate , respectively ; c is the pollutant concentration value ; F is the volume fractionfunction ; Cp ,σμ ,σ。,σ。, Ce , Ce2 and C, are constant parameters in the above equations and de-fined in the paper as( Zhang ,1994 ; Chen and Jaw ,1998 ):C, = 0.09,σ = 1.0,σ。= 1.3 ,σ。= 1.0, Ce = 1.44，Ce2 = 1.92 ,Cl = 0.1643，and these parameters are obtained from basic experiments and used universally.2.2 Calculation of the Mixing CoefficientThe pollutants will be passively advected and dispersed after the initial active entrainment phasewhen they are discharged into water , and then the dispersion mechanisms are mainly through turbulentdiffusion and longitudinal dispersion with tide , wind- induced waves and shear currents. For engineer-ing practice，the macroscopic advection velocity and the mixing cofficient of pollutants are needed ，and the equivalent advection velocity U。and the equivalent mixing coefficient K。are defined. The one-dimensional equivalent advection and mixing equation is proposed in the paper as follows :JC。.dC。2C。+ U。-=K( 12)At0xwhere C。is the pollutant concentration averaged along the vertical direction. Generally ,it is difficultto calculate U。 and K。 by analysis. In this paper , the pollutant concentration field , generated by aninstant source in waves and currents , is calculated by the model , and then the equivalent advection ve-locity U。 and the equivalent mixing coefficient K。are obtained by calculating the time derivative of themean and variance of c, , the probability distribution of pollutant concentration , when c, changes stablywith time. U。and K。are defined as follows :U。=JE(13)K。= 0.5( 14)Jt 'where , E=「t∞c,xdx ,is the meanof c。;σ2=」+o( x- E )c。dx ,is the variance of c。. ( Thesketch of the caleulation and the profile of c。are shown in Figs. 1 and 2 )xdllxid↑HTη圣L中国煤化工um.MYHCNMHGy十Fig. 1. Figure for calculating wave- current field..118YUAN Li-rong et al./ China Oean Engineering ,21( 1 ) , 2007 ,115- 124Whatever the instant source ( point , line or random distribution ) along x-direction , it is discov-ered that the time derivative of the mean and variance of pollutant concentration probability can reach astable value. In order to save computer time , the instant source is assumed to be a normal distributionfunction along x-direction.t=00.09t=10 T0.06 It=20 Tt=30 T、 t=40 T0.03Fig.2. Pollutant concentration probabilitydistribution.300600900x ()2.3 Boundary Conditions2.3.1Inlet BoundaryLi( 1989 ) has pointed out that there are three stages in the course of interaction between gravitywave and turbulent current : wave and current exist independently ，wave and current interact gradual-ly ,and then the stable wave-current is formed. So the velocities at the inlet boundary are the sum ofthe velocities of the wave and current. The velocities , turbulent kinetic energy , dissipation rate of tur-bulent kinetic energy and pollutant concentration at the inlet boundary are :u=uim+u;v=v;k= 0.005(u2+ 2)iε = C,k32/0.04h. ac=0where , the profile of uin is the logarithmic function of vertical average velocity U ; u ,and U are hori-zontal and vertical velocity produced by wave , and can be obtained from the potential flow theory. Atthe inlet boundary , F is the function of free surface elevation η which can also be obtained from the .potential flow theory .( 1 ) Wall BoundaryThe wall-function approach is employed for wall boundaries in this paper.(2 ) Outlet Boundaryu 10 and η . are handled with a sponge layer at the outlet boundary. For pure wave ，the spongelayer is treated with Larsen and Dancy' s( 1983 ) method :(x ry)0