Particle Trajectories in Nonlinear Water Wave on Uniform Current

China Ocan Eninering, Vol.22, No.4, pp.611- 621◎2008 China Ocean Press, ISSN 0890-5487Particle Trajectories in Nonlinear Water Wave on Uniform Current'Hung-Chu HSU°'I, Yang Yih CHEN', Chia Yan CHENGE and Wen-Jer TSENGd●Tainan Hydrulics Laboratory ，National Cheng Kung Unitersity, Tainan 701 , Chinab Department of Marine Eruironment and Enginering, National Sun Yat Sen University,Kaohsiung 804, Chinae Coasal and 0fshore Resources Research Center ，Fisheries Research Instiue ,d Department of Ciril Enginering, Cheng Shiu Uniersiy, Kaohsiung 804, China(Received 10 October 2007; received revised formn 11 July 2008; accepted 30 August 2008)ABSTRACTThe study in this paper is focusing on trajectores of particles in the imotational progreeive waler waves coexistingwith uniform current. The perametric equations of particle trajectories over a range of levels in a Lagrangian type of de-scription are developed analyically via the EulerLagrange transformation. The Lagrangian wave period of particle motiondifering from the Eulerian wave period and the mass transport can also be obained dire:tly. The third-order solution ofparticle trajectory exhibits that they do not move in closed orbital motion but represent a net movement that decreases ex-ponentially with the water depth. Uniforn curent is found to have signifcant elfect on the trajectories and drift velocity ofgravity waves. Overall, the infuence of increased uniform curent is to increase the relative horizontal distance traveledby a particle, as well邮the magnitude of the time avernged drift velocity on the free surface. For adverse curent cases,a revere behavior is found. The obtained thind-ordler solutions satisfy the irotational condition contrasted to the Gerstnerwaves and are venified by reducing to those of two-dimensional gravity waves in Lagrangian coordinates.Key words: wue; current; Lagrangian; Euler-Lagrange transformation; partice trajetory1. IntroductionWave -curent interaction has been studied extensively in the last decades. A lange number of the-oretical solutions for wave and current within uniformn or sheared profile are well documented in the re-view articles by Peregnine (1976) , Jonsson (1990) and Thomas and Klopman (1997). There are alsoseveral experimental studies of combined wave and current which have looked at particular aspects ofthe problem (Brevik, 1980; Kemp and Simons, 1982, 1988; Thomas, 1981, 1990; Swan, 1990).In the case involving a uniformn current, the wave-curent interaction may be described by the Doppler-shift solution. The pervious theories dealing with the problems of wave-curent interactions have mostlybeen perfomed by the Eulerian approach. Publications on this issue with Lagrangian apprach are sel-dom rare. In general, Lagrangian solutions present the great advantage that the deformation of the'medium may be described explicily and the effects cannot be captured by Eulenian solutions ( Naciniand Mei, 1992; Chen et al., 2006).中国煤化工HCNMHG# This study W8s financially supported by science council of Taiwan wun gant mo. NU-91-2221-2-230-023.1 Coresponding author. E mail: bchsu@ thl. ncku. edu.tw612Hhung-Chu HSU a al./China Oeun Enginering, 22(4), 2008. 611- 621The exact solution obtained by Gerstner ( 1802) in Lagrangian coordinates possesses finite vortici-ty. Miche ( 1944 ) proposed perturbation Lagrangian solutions to the second order for gravity waterwaves. Pierson (1962) applied perturbation expansions to water wave problems with the Lagrangianformulae and obtained the first-order Lagrangian solution. Sanderson ( 1985) obtained second- order s0-lutions for smal]- amplitude intemal waves in Lagrangian system. However, the above mentioned theo-ries are highly rotational and have correspondingly litle physical interest. Recently, Chen (1994) de-rived the third order Lagrangian solutions ，which are inmotational, by incorporating the perturbation onthe water particle frequency depending on water elevation.In principle, if the flow field is completely defined in one reference frame, it can be transformedaccording to the fundamental principle of kinematics to obtain a flow description in the other frame-work. The velocity at a given position and time ( the Eulerian velocity) is identically the velocity of theparticle ( the Lagrangian velocity) that occupies the position at that time. However, this Euler-La-grange transformation is a highly nonlinear process. Wiegel (1964) and Chen ( 1996) follow themethod of Longuet-Higgins (1953) to derive the horizontal and vertical displacements of water parti-cles. The third- order expressions of the particle displacements contain a resonant tem that is non-uni-formly valid for large times and is physically unreasonable. Longuet-Higgins ( 1979) presented a simplephysical model to obtain the Lagrangian characteritics of surface waves from the Eulerian solution whileit cannot obtain the particle motion and the mass transport of the entire flow field directly. Chen andHsu (2005) showed a new method of finding an appropriate transformation from the description for awater wave motion in Eulerian coordinates to a Lagrangian type of descriptions. The generalized solu-tions of water particle trajectory and the Lagrangian wave period for all particles over a range of levelscan be derived.The purpose of this paper is to describe the trajectories of water particles in a combined wave-cur-rent motion by using an appropriate Euler-Lagrange transformation found by Chen and Hsu (2005).The third-order perturbation solution of water particles in the Lagrangian type of description was calcu-lated from the description of a wave-curent interaction field in Eulerian coordinates. The explicit para-metric fomulas highlight the trnjetories of water particles, the mass transport, the Lagrangian wavefrequency and the wave kinematics above mean water level, which is tough to describe by means of Eu-lerian method. The change of particle motion due to the coplanar and opposing uniform curtent is alsodiscussed.2. Solution of Wave Co Exist with Uniform Current in the Eulerian SystemConsider two-dimensional monochromatic wave with a steady uniform current propagaing over auniform horizontal impermeable bed. It is assumed that the fluid is irotational and inviscid so that avelocity potential exists in the fluid domain. The domain and Cartesian coordinate system are shown inFig. 1. The position of x-axis directed to the direction中国煤化工y-axis is posi-tive vertically upward from the still water level. The third-MHC N M H Gves propagatingover a uniform current in the Eulerian approach were obtained by Chen and Juang ( 1990). A summaryHung Chu HSU a al./ China Oeran Enginering, 22(4), 2008, 611- 621613of the third order velocity potential and the fre surface elevation are given as follows:yA/nFg.1. Definition sketch for a wave on a一.xuriform curent.虫=Ux+(C. =- o P[λp Icosh(y + d)sinS + 2Fzcosh2k(y + d)sin2S+ xPFsecsh3k(z+ d)sins} -Hx?(C.- U)2_sink? kd' '(1)where,3λ=ka,c. =P=%，S=kx-o。l， F= sinhkd'， F2= 8sinhtkd."F3 =一(9tanh~7hd + Stanh-hd - 53tanh~3kd + 39tanh-'I kd)cosh3hd’η= (a+1k2a33 + 14sinh? kd + 2inh'kd )cossinh4 kd.1 a2 (3土2sinh? kd)coshkdos 2ssinh3 kd+ ka2; 27 + 72sinh?kd + 72sinht kd + 24sinh° kd,cos 3S;(2)sinh' kd9 + 8sinh-kd + 8sinh4 kd,w= wo+ Hk2a2(wo- kU)(3)sinh'kdw- kU = ow，(wo- kU)2 = gktanhkd.(4)where, φ =中(x,y,t) and n(x,t) are the velocity potential and the free surface profile in the Eu-lerian coordinate. U denotes the steady uniform current; a, the wave amplitude function; C*，thewave phase velocity; d, the water depth; g, the gravitational constant and w, the relative angularfrequency. The wave number k and wave frequency σw are respectively defined by k=2x/L andσw=2x/Tw，L being the wave length and Tw the wave perigd Fa (4) ie mmmonlv refered to the中国煤化工Doppler- shifted solution.1YHCNMHG614Hung-Chu HSU a al./China Ooxan Enginering, 22(4), 2008. 611- 6213. The Euler-Lagrange TransformationLet a particle released from the initial position at initial time t= to arrive at (x, y) at the timet=t, its Lagrangian velocity of the particle marked by Lagrangian horizontal and vertical labels (a,b), W(a,b,t)= iUl(a, b,t)+jV(a,b,t) must be the same as its Eulenan velocity, R(x，y,t)= iu(x,y,t)+ jv(x,y,t) that is,R(x,y,t)= iu(x,y,t)+ jo(x,y,t) = W(a,b,t)= iUl(a,b,t) + jv(a,b,t)(5)where u(x,y,t) and v(x,y,t) denote the horizontal and vertical particle velocity components in theEulerian system，UL(a,b,t) and V[(a,b,t) represent the horizontal and vertical particle velocitycomponents in the Lagrangian system.For an incompressible fluid the continuity equation sets the invariant condition on the wave length-averaged volume integrated from water depth to free surface mapping from the Eulerian system to La-grangian system, that isa(x,y)a(a,D)=J=1.(6)Eq. (6) is the Jacobian of x,y with respect to a,b that is independent of time and is the massconservation or the continuity equation in the Lagrangian system. For solution of Eqs. (5) and (6), itis assumed that relevant physical quantities can be expanded as a power series of the perturbation pa-rameterε. This E is a small order that was inserted for identifcation of the order of the associatedtemm. The velocity and wave frequency Ow in the Eulerian system can be obtained from Eq. (1), andwe havea中u(x,y,t) =Zee'"un(x,y,t) ==ni= U+(C. - U){ AFrcoshk(y + d)cosS + 2x2 F2cosh2h(y + d)co2S+ 3λ3 F3cosh3k(y + d)cos3S};(7)ago(x,y,t) =2ee'",(x,y,t)=O=Q,n=1=(C.-v){aFhF|sinhk(y + d)sinS + 22 F2sinh2h(y + d)sin2S+ 323 F3sinh3k(y + d)sin3S }(8)σw=σ+2e'on= e'on;σo=σ中国煤化工(9)4二We account for the fact that the Lagrangian wave freYHC N M H Gnonlinearity andthe initial position of each paticle and use the Lindstedt-Poincare technique that yields unifom expan-Hung-Chu HSU e al./China Oean Engineering. 22(4), 2008, 611- 621615sions to uncover the solutions in the Lagrangian system. The particle trajectories. X= {x(a,b,t),y(a,b,t)}, the water partiele velocity W(a,b,t)= iUL(a,b,t)+ jVL(a,b,t) and the angu .lar frequency of particle motion σq(a,b) in the Lagrangian system could be assumed in the formx(a,b,I) = a+ 2e"[f(a,b,oLt) + jf.(a,b,oxo)]+ M(l);(10)y(a,b,1)= b+ 2e"[g,(a,b,oLt) + g.(a,b,ouot)];(11)σc(a,b) =2e"ou(a,b)= o(a,b)=σ=oL.(12)Eqs. (10)~(12) are the completely unknown Lagrangian solutions that can be solved sequen-tially for each order of ε.Upon substituting Eqs. (10) ~(12) into Eqs. (5) and (6), using the Taylor series expansion offunctions around the label position and the period of particle motion, and collecting terms of ε ordersyields the necessary equations to each order of approximation. The perturbed equations can be writtenas:xdYb-xbYa-1=2e"(Um +Jm +gn + gw) + [ous(rm,.+ Jow) + ouo(EwL .+6.my)]1+ iyee't*'[(fm+f)+(gmb+gn)-(fn +fa)(gm + gm)]n=1 m=l+ [na(rng+Jw.)(gms+ gm) + ouo(Umu +fm)(gmp+gmwn)- 0uao(fm +fnms)(Bmy+gww) - ono(gm + gm)(rwt+fga)- (m + fm)gm,Jt + ouo[(fm +fm)gmy,-(gm + gm)fm,]t+ [owJup(m go.s_gmwr.) + Cuwampo(gm frmq.-frw &8mq)]2} =0; (13)u(x,y,t) = u[a+e*(%+f+ M),b+ 2e*(gn+ gn),1]= u+(C. - 0\{rocord +b+ Ze(B. + 80[a, See"(% +f)+M] -olt+ 2e(ou-om)} + 2Fzcsh2k[d+br 2e"(g.+gi)] xcos2{k[a+ Ee(f.+f)+M]σLt +Ze(oun-om)}n=l .■I中国煤化工+ 3x3 Fzcosh3hb+ Se(gs + goco{YHCNMHGM]n=l616Hung-Chu HSU a al./ China Oeean Enginering , 22(4), 2008, 611- 621-0[i+. e(um-om)}} =M" + 2Eeonf:+) .+ e'oufio, + e'(oufzo, + orf1oq)+ 0(e4) =?x(a,b.t)= UE(a,b,t); (14)Hto(x,y,t)=o[a+ Ze(h +f)+M,b+ Ze(gn +g),1= (C.- Dr{rinih[d+bo+ Ze(gn +)ain{k[a+ $e(G.+f)+M]-0ut+2e(ou -om)} + 2xPrsin2[d+b+ Ze*(gn+go)]xsin2{k[a+ Ze°(f +fn)+M] -0Lt+ Ze"(own-om)}+ 3Fysinh3h{[d+c+2e(g. + 8)sin{k[a+ Eee"(fn +f)+ M]n=l-σLt +(e(aom-nm)}}= Zeeou(gmq,+ Bnmno) + e'ouB1ou .n=0+ e(ou82o +012810)+ 0(e*)= dy(a,b,u = V.(a,b,t).(15)JtEqs. (13) ~(15) can be sorted out in order of e, resuling in a cascade of ordinary diferentialproblems at the diferent orders of approximation and can be solved sucessively in the fllowing sec-tions.4. The Third-Order Approximations4.1 The Zero- and First-Order ApproximationsWith Eqs.(13)~ (15), the goveming equatins up to 0(e") and 0(e) read as:M'(t) = U;(16)fia+fia+ g16+ g1b + [oa(fion+foj) + oo(8Im+ g1o.)]t =0;(17)σLn(iq++foq) = (C - U)AFycoshk(b + d)cosS-(C. - U)AFycoshk(b + d)[ hM(t) + (σn - owo)t]sinS;(18)on(g1o;+gilon;) = (C. - U)入Fisinhk(b + d)sinS+(C. - U)AFisinhk(b+ d)[kM(t) + (σw - owo)t ]cosS;(19)where S= ka - oLt. Avoiding the occurence of tems that increase linearly with time being set to e-ro, the solution of Eqs. (16) ~ (19) can be easily obtained as:M(t) = Ut;中国煤化工σL0+ kU = 0wo= wo; :(21)MHCNMHGfi =-C.-Uc λF 1coshk(b + d)sin(ka - oul);(22)Hung Chu HSU a al./China Ceoan Engineeing , 22(4), 2008, 611 - 621617C。-Ug1 =OL0. λFrsinhk(b + d)cos(ka- σLt).(23)Eq. (20) is the basice solution, describing a steady uniform current. Eqs. (21) ~ (23) satisfyexactly all the hydrodynamics equations formulated in Lagrangian coordinates including the irmotationalcondition and differ from the Gerstner' s wave possessing finite vorticity. σL0 is the essential Lagrangianwave frequency of water particles relative to the uniform current.4.2 The Second- and Third-Order ApproxinationsTo the next order 0(e2) and to avoid the ocrence of the time growing secular termns which isnon-uniformly valid for large times, the goveming equations are given byfza +fza+ 826 + g26 =-fiaB1b + f{bgla .(C. - U)2-k2x2F}cosh2k(b+ d) +方h2λ2 F}cos2S; (24)2ooσn(fop+20) = (C. - U)XF:{k(g1 + gi)sinhk(b + d)cosS- k(fi + f)cosh(b + d)sinS} +(C。 - U)22 F2cosbh2k(b + d)cos2S1(C.-U)2，=2σL0u Ficosh2h(b+ d)-- k2 Frcos2S+ (C. - U)2x2 F2cosh2k(b + d)cos2S;(25)σuo(82of+ B20g) = (C. - U)XF;{lk(g1 + g)coshk(b + d)sinS+ k(f; + f)sinhk(b + d)cosS| + (C。- U)2λ2 F2sinh2k(b + d)sin2S= (C. - U)222 F2sinh2k(b + d)sin2S.(26)The solutions are obained by solving Eqs. (24) ~ (26) and the reults areσu=σwl=g2=0;(27)f= 41(C.-U)2. k2 Fisin2S . .(C。- U)2λ2 Fzcosh2k(b + d)sin2S;(28)σ子(C.- U)2后= 2o- h从2 Ficosh2k(b + d)anot;(29)g2 :(C.一λ2 F2sinh2k(b + d)cos2S0101(C.一U)2+4- k2 FPsinh2k(b + d).(30)The horizontal Lagrangian paricle trajectory in the second order approximation includes a periodiccomponent f2 and non-periodic function f2 increasing linearly in time and independent of lagrangianhorizontal label a which presents the mass transport. The中国煤化工_ second hamoniccomponent and a tem that is the function of wave steepnesLlagrangian verti-cal label b, independent of time and a second-order verticCNMHGeph.From Eq. (29) the mass transport velocity is explicitly involved in the second-order parametricHung-Cu HSU a al./ China Ocean Enginering, 22(4), 2008, 611- 621solution and can be obtained as follows:起55a+2=长= (1- (2xF0coh2k(b+d)，c.=o，(31)C.ro'where the over-bar denotes time average taken for one complete orbit, i.e. the period of particle mo-tion and C. is the linear phase speed. In Fig. 2 the mass transport velocity u * /C. is plotted againstthe water depth and the unifom current in the wave-curent interaction field. The second-order masstransport velocity increases when the wave train encounters an opposing current, while decreasing whenthe curent is in the same direction. For this, owing to the coplanar current following the waves makesthe wave amplitude smaller, while opposing current amplifies with the wave amplitude. From Fig. 3,we can clearly see that uniform curent has a significant efect on the drift velocity . The increase of fol-lowing curent is to enhance the magnitude of the drift velocity over the whole range of depths and asignificant amount of fluid that has been transported forward, on the contrary, the drift velocity de-creases by the adverse current. Eq. (31), a second-order quantity, is the same as that obtained byLonguet-Higgins ( 1953) by reducing to the pure progressive water waves.0.0 |0.0-0.2-0.4-0.6. UIC.=0-U/C.-0U/C.=0.1UIC.-0.1----- -U/C.-=0.2-0.8. U/C.=0.1. U/C.-0.2UIC.=0.2-1.00.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14-1:20.3-0.2-0.10.00.10.2030.40.50.6Ku'/0。xFig. 2. The dimensionless drift velocity profile at the Fig. 3. The dimensionless uriformn current plus massrelative water depth d/L=0.5 and the relativetransport velocity profile at the relative waterwave height H/L= 0.1 under various currentdeph d/L = 0.5 and the relaive wave heightconditions.H/L=0.1 under various curent conditions.The third-order olutions can be obtained from the nonhomogeneous boundary value problems de-rived from the terns of the corresponding orders. The procedures are sinilar to the above denivationsand the details can be omitted here. After a lengthy but staightforward manipulation, the third-ordersolutions in Lagrangian form are given by[ou=on-C. - U)2. k2λ2 FPcosh2k(b + d)σn(32)lon2= iTk2a'(σo - hU)(9ranh*中国煤化工YHCNMHGf= g3=0;(33)Hung-Chu HSU t al./China Oean Enginering, 22(4), 2008, 611- 621.619fs = M333cosh3hk(b + d)sin3S + M31Ccosh3k(b + d)sinS+ M3scoshk(b + d)sin3S + M311coshk(b + d)sinS;(34)g3 = N333sinh3h(b + d)cos3S + N331sinh3k(b + d)cosS+ N3nsinhk(b + d)cos3S + N3usinhk(b + d)cosS,(35)where cofficients M333, M331，M313, M311 , N33, N31, N3u3, and N311 are listed as fllows:M33= -C。-U"x3F3,M331= -[g(C.-UYwFFz+1(C-)323月].M33=-5(C.- U)2k3FF2-1C. ;-k2>3所,612 叫M311=σnλF,otoN33=Cs -一x3F3,σL03(C.- U)21(C.-U)3h2λ3F3,N31= 2o似3F1Fz+ 4-N1s=-之一1(C.一U)2ka3F\F2,N311= -σmnC.-UAF.Eq. (32) is the second- order angular frequency corretion of a particle motion. The first term isthe second-order Eulerian wave frequency and the second temm varies with the vertical elevation andcurrent. The third-order solutions for Eqs. (34) and (35) are periodic functions that are combinationsof boh the first and the third harmonic components. It is also noted that the new nonlinear third-orderasymptotic solutions can be reduced to those by Chen (1994) for pure two-dimensional gravity waterwaves.In Fig. 4 by plotting the trajectories at the free surface the influence of uniform curent on theparticle trajectories is also presented when the uniform curent is from weak to strong. It can be seenthat in each complete orbit a particle advances horizontally through a distance by the existence of amean horizontal drift or mass transport in the direction of wave propagation. It is also evident that thevariations of water particle trajectory in wave-current field are due to the nonlinear interaction of theuniform current and particle velocity or mass transport velocity. The larger the component of particlevelocity and unifom flow, the stronger nonlinear interaction will be. For the cases of waves with copla-nar current, the effect of increasing current speed is generally to increase the magnitude and extent ofthe time averaged drift velocity since the uniformcurent is中国煤化工3gation and pro-duces an increase in a relatively horizontal distance travelevith the case ofwaves without curent. Also, a reverse behavior is noticedTCH.CNMH. Gaen an opoigcurrent. The adverse curent blocks the advance of the particles and decreases the relatively horizontal620Hung- Chu HSU a al./China 0ean Enginering, 22(4), 2008, 611- 621distance traveled by water particles that the surface takes less time to complete one orbit as does theparticle in the case of following current .0.08(@)0.06-0.040.04-0.02会0.00-0.02-0.02|00M-0.04-0.04-F;=0-- F,-0.01----F,= -0.01-0.06F,=0 --.F,=0.03 ---F,=-0.03-0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6x/c)Fig. 4. The thind-order trajectories of pearticles atthe free surface for relative water depths“0.00d/L = 0.2 and the relative wave height-0.02-7H/L = 0. 04 under various uniform flow一-F,0-- F,=0.05---F,- -0.05conditions .-080.0010203040.50.65. Concluding RemarksIn the present study, it is shown that for the particle trajectories of nonlinear irrotational waterwaves on uniform curent the explicitly third-order Lagrangian paranetric solutions are obtained by theEuler-Lagrange transformnation and this problem has not yet been solved in the previous study. This so-lution satisfies the original nonlinear goveming equations in Lagrangian coordinates and the irotationalcondition. The mass transport velocity and the angular frequency of the particle motion differ from theEulerian wave frequency that is a function of the wave steepness , uniformn curent and the vertical labelof each individual particle can also be obtained directly . The third- order trajectory solution of particlesexhibits that they do not move as a closed orbital motion but represent a net movement. For adversecurrent cases, the particle displacement decreases due to the forward drift velocity resisted by the cur-rent, as the current speed increases. For following current cases, a reverse behavior is found. In thenon-uniform current case, the new nonlinear parametric solution is reduced to those by Chen ( 1994)for two dimensional gravity waves. Finally, this paper provides a new method of obtaining the La-grangian solutions from the given Eulerian solutions for the wave-current interaction flow.中国煤化工ReferencesMHCNMHGBrevik, I, 1980. Flume experiment on waves and current I . smooth bed, Coast. Eng., 4, 89~ 110. .Chen, Y. Y. and Juang, W. J，1990. Primary analysis on wave current interaction, Proc. 12h Conf. on OceanHung Chu HSU a al./China Oean Enginering, 22(4), 2008, 611-621621Eng., 248 ~ 265. (in Chinee)Chen, Y. Y., 1994. 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Mech., 16, 9~ 117.Pierson, W. J., 1962. Perturbation analysis of the Navier- Stokes equations in Lagrangian form with selected linear solu-tion, J. Geophy, Res., 67(8): 3151 ~ 3160.Sanderson, B., 1985. A Lagrangian solution for internal waves, J. Fluid Mech., 152, 191 ~ 202.Swan, C., 1990. An experimental study of waves on a strongly sheared curent profile, Prceeding of 22nd Coastal En-gineering Conference, ASCE, 1, 489~ 502.Thomas, G. P., 1981. Wave-curent interactins: an experimental and mumerical study, part1. linear waves, J. Flu-id Mech., 110, 457~474.Thomas, G. P., 1990. Wave-cument interactions: an experimental and mmerical study, part 2. nonlinear waves, J.Fuid Mech., 216, 505 ~ 536.Thonas, G. P. and Klopman, G., 1997. Wave curent interacions in the near-shore region, Aduxances in Fluid Me-chanics, Computational Mechanics Publications, 66~84.Wiegel, R. L, 1964. OCeangraphical Enginering, PrenticeHall, New Jensey, 37~40.中国煤化工MYHCNMHG