Computation of the inviscid drift force caused by nonlinear waves on a submerged circular cylinder


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1 csnak, 2011 Iter J Nav Archit Oc Egg (2011) 3:201~207 Computatio of the iviscid drift force caused by oliear waves o a submerged circular cylider HyeokJu Koh 1 ad IlHyoug Cho 2 1 Multidiscipliary Graduate School for Wid Eergy, Jeju Natioal Uiversity, Jeju, Korea 2 Departmet of Ocea System Egieerig, Jeju Natioal Uiversity, Jeju, Korea ABSTRACT: I this paper, we focused o computig the higherharmoic compoets of the trasmitted wave passig over a submerged circular cylider to show that it is causig a horizotal egative drift force. As umerical models, a circular cylider held fixed uder free surface i deep water is adopted. As the submergece of a circular cylider decreases ad the icidet wavelegth becomes loger, the higherharmoic compoets of the trasmitted wave starts to icrease. A icrease of the higherharmoic compoets of the trasmitted wave makes the horizotal drift force be egative. It is also foud that the higherharmoic amplitudes averaged over the trasmitted wave regio become larger with the icrease of wave steepess ad wavelegth as well as the decrease of submergece depth. KEY WORDS: Higherharmoic compoet; Numerical wave tak; Drift force; Fully oliear potetial flow; Trasmitted wave. INTRODUCTION Accordig to potetial flow theory ad the socalled d Alembert paradox, there is o force actig o a submerged body i a steady state irrotatioal flow of a iviscid icompressible fluid. I such a case, oly a viscous drag force ca occur. For usteady potetial flows, such as caused by waves, however, a mea drift force ca be iduced o a submerged body. I particular, it has bee log kow that a egative drift force may be caused o submerged bodies by surface gravity waves of sufficiet steepess, i.e., oliearity; hece, this waveiduced drift force is due to higherorder effects. Usig coformal mappig, Dea (1948) thus foud that a submerged circular cylider does ot reflect waves to leadig order of steepess. Ursell (1950) cofirmed this result by derivig the complete liear solutio usig a multipole expasio method. Followig Ursell s approach ad estimatig secodorder effects from liear results, Ogilvie (1963) showed the existece of a mea secodorder vertical force, but foud that the horizotal mea force vaished to secodorder. Usig a Stokes expasio, Vada (1987) solved the secodorder diffractio problem i the frequecydomai, but could ot calculate all the terms of the mea horizotal force. LoguetHiggis (1977) observed i experimet that a freely movig, eutrally buoyat, submerged cylider experieced a egative drift force, Correspodig author: IlHyoug Cho causig it to move towards the wavemaker. He attributed this force mostly to wave breakig ad, to a lesser degree, to the secodharmoic compoet of the trasmitted wave. This coclusio, however, is ot corroborated by Miyata et al. (1988) ad Ioue ad Kyozuka (1984) measuremets, who both foud that, as the cylider was moved closer to the free surface, causig more itese wave breakig, the egative horizotal drift force was actually reduced ad ultimately eve chaged sig. A umber of twodimesioal, fullyoliear, iviscid timedomai computatios have bee proposed, to estimate strogly oliear effects caused by waves passig over submerged bodies of small equivalet diameter but large dimesio i the trasverse directio, with respect to wavelegth, such as pipelies. Usig the mixed Euleria Lagragia method, Coite (1989) calculated higherorder harmoic forces ad wave trasmissio coefficiets o a submerged cylider, i a fullyoliear potetial flow model, but did ot calculate the horizotal drift force. Torum ad Gudmestad (1990) computed particle trajectories ad Lagragia trasport caused by steep waves, represeted by exact streamfuctio Stokes waves, over a submerged cylider, i a spaceperiodic versio of Grilli et al. (1989) fullyoliear potetial flow model. Liu et al. (1992) applied the HigherOrder Spectral Method (HOS) to this problem ad compared computatios with aalytical results ad experimetal observatios. They also used exact deepwater Stokes waves as iitial coditios ad specified periodic coditios for upstream ad dowstream boudaries,
2 202 Iter J Nav Archit Oc Egg (2011) 3:201~207 a requiremet of the HOS method. I this paper, we establish the origi of the egative drift force caused by steep waves o a submerged cylider, by similarly performig twodimesioal (2D) Fully Noliear Potetial Flow (FNPF) simulatios i the time domai. I the simulatios, we use the most recet versio of the model origially developed by Grilli et al. (1989), with improvemets ad additios by Grilli ad Subramaya (1996) ad Grilli ad Horrillo (1997) (hereafter referred to as 2DFNPF model). Ulike earlier studies, our computatios are ot spaceperiodic but feature the geeratio of exact fully oliear periodic icidet waves at oe extremity of a Numerical Wave Tak (NWT), as well as wave absorptio/radiatio at the other extremity. Although our model ca simulate overturig waves, we did ot cosider wave breakig effects i this paper. For obreakig waves, we will show that the higherharmoic compoets of the trasmitted wave are the mai cause for the egative horizotal drift force o a submerged body. Numerical results will also show that the magitude of this higherharmoic compoets icreases as the body submergece decreases, ad icidet wavelegth ad steepess icrease. NEGATIVE DRIFT FORCE To establish the relatioship betwee horizotal drift force ad higherharmoic compoets of the trasmitted waves passig over a submerged body, it is useful to first obtai a simple estimate of the solutio based o the coservatio of eergy ad liear horizotal mometum. Assumig wave reflectio by a submerged circular to be egligible ad cosiderig the icidet ad trasmitted wave amplitudes o the dowstream up/dow sides of a submerged cylider, let a, b be the th harmoics of the icidet ad trasmitted wave amplitudes, respectively. With this assumptio, applicatio of the coservatio of horizotal mometum gives a expressio for the horizotal drift force to leadig order, as, g D a b (1) 2 2 x 4 1 From coservatio of eergy, a ad b are related by, a b (2) Sice, for periodic icidet waves, the amplitude of the first harmoic a 1 is much greater tha all other harmoic amplitudes, we ca eglect all a, > 1 term i Eqs. 1 ad 2. From Eqs. 1 ad 2, we ca obtai Eq. 3 provides a way to estimate D x for give trasmitted wave harmoic amplitudes, ad is a geeralizatio of the result of LoguetHiggis (1977), who oly cosidered the first (b 2 ) term oly. Although it oly represets a approximatio valid for small icidet wave steepess, Eq. 3 idicates that the horizotal force is always egative, with a magitude that icreases with the degree of higherharmoic geeratio. g 1 Dx b (3) Fig. 1 Computatioal model for the oliear wave diffractio by a fixed submerged cylider (AB: absorbig beach, AP: absorbig pisto). OVERVIEW OF NUMERICAL MODEL Goverig equatios ad umerical algorithms Equatios for the 2DFNPF wave model are briefly preseted i the followig. The velocity potetial ϕ (x, t) is used to describe iviscid irrotatioal flows i the vertical plae (x, z) ad the velocity is defied by, u=ϕ =(u, w). Cotiuity equatio i the fluid domai Ω(t) with boudary Γ(t) is a Laplace s equatio for the potetial (Fig. 1), 2 0 i t (4) O the free surface Γ f (t), ϕ satisfies the kiematic ad dyamic boudary coditios, Dr u r u o f t (5) Dt t p a D 1 gz Dt 2 o t (6) respectively, with r, the positio vector o the free surface, g the gravitatioal acceleratio, z the vertical coordiate, P a the pressure at the free surface, ad ρ the fluid desity. Alog the statioary bottom Γ b ad cylider boudary Γ c, the oflow coditio is prescribed as, 0 o b ad f c (7) where =( x, z ) is the outwards ormal vector defied o the boudary. Boudary coditios for wave geeratio o boudary Γ w ad wave absorptio o boudary Γ a are preseted i the ext sectios.
3 Iter J Nav Archit Oc Egg (2011) 3:201~ Eq. 4 is trasformed ito a Boudary Itegral Equatio (BIE), usig Gree s 2d idetity, ad solved by a BEM. The BIE is evaluated at N discretizatio odes o the boudary ad M higherorder elemets are defied to iterpolate i betwee discretizatio odes. I the preset applicatios, quadratic isoparametric elemets are used o lateral ad bottom boudaries, ad cubic elemets esurig cotiuity of the boudary slope are used o the free surface. Expressios of BEM itegrals (regular, sigular, quasisigular) for these elemets are give i Grilli et al. (1989) ad Grilli ad Subramaya (1996). Free surface boudary coditios (5) ad (6) are time itegrated based o secodorder Taylor series expasios expressed i terms of a time step Δt ad of the Lagragia time derivative, D/Dt, for ϕ ad r. Firstorder coefficiets i the series correspod to free surface coditios (5) ad (6), i which ϕ ad ϕ/ are obtaied from the BEM solutio of the BIE at time t. Secodorder coefficiets are expressed as D/Dt of Eqs. 5 ad 6, ad are calculated usig the solutio of a secod BIE for ( ϕ/ t, 2 ϕ/ t t), for which boudary coditios are obtaied from the solutio of the first problem. Detailed expressios for the Taylor series are give i Grilli et al. (1989). At each time step, global accuracy of computatios is verified by computig errors i total volume ad eergy for the geerated wave trai. Earlier work showed that these errors are fuctio of both the size (i.e., distace betwee odes) ad the degree (i.e., quadratic, cubic,...) of boudary elemets used i the spatial discretizatio, ad of the size of the selected time step. This led to adaptively selectig the optimal time step, based o a mesh Courat umber C 0 (t), which for cubic elemets has a optimum value of This value is used i the preset applicatios. Exact periodic wave geeratio i the model We use the method developed by Grilli ad Horrillo (1997) to geerate umerically exact periodic wave solutios of the FNPF problem i the model (i.e., which will propagate over costat depth without chage of form). Streamfuctio Wave Theory (SFW) is first used to calculate wave shape ad kiematics, for give height H ad period T, i water of depth h, ad the particle velocity ad acceleratio of these SFWs is specified alog a vertical wavemakig boudary (Γ w ). Sice free surface discretizatio odes represet fluid particles, durig computatios, they gradually drift away i the directio of the mea mass trasport, evetually leadig to a poor resolutio close to Γ w. This drift is cacelled by horizotally movig Γ w with the Lagragia motio of the first ode/particle o the free surface. Similarly, to prevet fluid from accumulatig i the computatioal domai, because of the mea wave mass trasport, icidet SFWs are geerated over a uiform curret U, equal ad opposite to their mea mass trasport velocity. Such waves are referred to as zeromassflux SFWs. Sice a curret slightly modifies wave characteristics due to Doppler effect, U is iteratively calculated, for specified wave characteristics as part of the SFW solutio. See details i Grilli ad Horrillo (1997). Wave eergy absorptio i the model To prevet spurious wave reflectio i the model, followig Grilli ad Horrillo (1997), a absorbig beach regio (AB) is specified at the far ed extremity of the computatioal domai (Fig. 1). The AB combies a absorbig pressure term P a =P i the dyamic free surface coditio (6), ad a actively absorbig pisto coditio (AP) o vertical boudary Γ a. To always iduce eergy dissipatio, the absorbig pressure P is specified as opposite ad proportioal to the ormal particle velocity o the free surface, with a getle rampup over a short distace i frot of the AB. Grilli ad Horrillo (1997) showed that the AB is efficiet i absorbig higherfrequecy waves if its legth is at least twice the domiat wavelegth. To better radiate lower frequecy waves out of the computatioal domai, a AP is specified, that, followig Grilli ad Horrillo (1997), moves proportioally to the mea istataeous dyamic wave force. Details of the AB ad AP implemetatio ad validatio ca be foud i Grilli ad Horrillo (1997), ad their applicatio to oliear wave shoalig, e.g., i Grilli (1998) ad Grilli ad Horillo (1999). NUMERICAL APPLICATIONS Fig. 1 shows the NWT computatioal domai used i this applicatio, with a submerged circular cylider located at midlegth of the tak. Exact periodic SFWs (with zero mea mass flux) are geerated o the wavemakig boudary, as detailed before, ad waves are dissipated both i the absorbig beach AB (of fixed legth 5m) ad usig the absorbig pisto AP. The computatioal domai is 20m log, with costat water depth h=3m. The boudary is discretized with N=310 odes M=227 elemets, with N b =60 odes discretizig the cylider boudary, ad N f =150 odes o the free surface. Iitial spacig betwee odes o the free surface is thus Δx 0 = 0.134m. The odal poits o the free surface are regridded every 20 time steps. We deote by R(=0.25m) the radius of the circular cylider, ad by H its submergece depth, measured from z=0 to the cylider axis. Wave period is specified such as to achieve deep water coditios i the tak. The trasmissio coefficiet for the th harmoic is defied as T =b /A, where A is the icidet wave amplitude ad b the amplitude of the th trasmitted wave harmoic, defied at the locatio x=15m, which marks the startig poit of the absorbig beach. Computatios are performed for icidet waves of steepess varyig i the rage 0.01kA0.1 ad legth withi 0.3kR0.6. This yields a maximum wavelegth L=5.24m, such that L/h 0 =1.75, which is clearly a deep water wave. The cylider submergece depth varies withi 1.5 H/R 2.2. Results will show the depedece of the wave trasmissio coefficiet, drift force, ad oscillatig force o parameters (ka, kr, H/R). I each simulatio, the horizotal drift forces, averaged over time τ, are calculated as,
4 204 Iter J Nav Archit Oc Egg (2011) 3:201~207 1 t0 Dx Fx t dt (8) t0 where F x D x c p d, is the timedepedet horizotal dyamic wave force o the submerged cylider of boudary Γ c, with pd p gz / t1/2, the dyamic pressure (readily available o the cylider boudary Γ c for each time step of NWT computatios), ad D x is positive i the directio of the icidet wave propagatio. The horizotal forces correspodig to the th harmoic ca be obtaied from the Fourier trasform (=1,2, ), 1 t0 it Fx Fx t e dt (9) t0 where T=2π/ω is the icidet wave period (or first harmoic period for =1). The time iterval τ for the averagig is set to 10T, ad t 0 is selected large eough so that quasisteady state is reached i the computatios. I each simulatio, harmoic aalyses are also made, for the surface elevatios computed at may successive umerical wave gauges equally spaced aroud the submerged cylider. Results will show that free higherharmoics are geerated i trasmitted waves. Usig the preset NWT, the timeaveraged spatial variatio of amplitudes of the first three harmoic of trasmitted waves, are calculated from Fourier trasforms as (for =1,2,3), 1 t0 it b x x, te dt (10) t0 Fig. 2 Volume chage error ε v =(V(t)V 0 )/V 0 ad cotiuity error ε r =( Γ ϕ/ dγ)δt/v durig computatios for ka=0.08, kr=0.4, H/R=1.5, N F /N w =30. Fig. 3 Wave elevatio computed at x=5m for ka=0.08, kr=0.4, H/R=2.0. Spatiallyaveraged values of these harmoic amplitudes are the obtaied as, 1 l g b b d 0 x x l (11) g where l g is the legth of the higherharmoic wave geeratio regio, defied i the NWT betwee 12 ad 15m (the begiig of the AB; see Fig. 1). As idicated before, this NWT has bee validated, both umerically ad experimetally, for may differet types of icidet waves ad their iteractios with obstacles or the bottom. Here, we verify the accuracy of computatios by calculatig umerical errors o volume coservatio ad cotiuity (boudary fluxes) i Fig. 2, for a typical case with ka=0.08, kr=0.4, H/R=1.5. We see, both of these errors oscillate i time, but remai very small durig a typical computatio. While the cotiuity error ε r is istataeous ad thus idicates the relative accuracy of the BEM solutio i the NWT at a give time (a very small O(109 )), the volume error ε r itegrates over time errors due to both spatial discretizatio ad time steppig, as well as effects of the AB (ad is thus larger at O(106 )). Fig. 4 Horizotal force computed o the submerged cylider, for ka=0.08, kr=0.4, H/R=2.0. We the verify that, after a short rampup time, typical computatios i the NWT quickly reach a quasisteady state: Fig. 3, thus shows wave elevatio computed at x=5m for ka=0.08, kr=0.4, H/R=2.0, ad Fig. 4 shows the correspodig horizotal force computed o the cylider. We see both of these become very closely periodic i time, ad show updow asymmetry idicative of oliear effects.
5 Iter J Nav Archit Oc Egg (2011) 3:201~ Fially, the covergece of NWT computatios with free surface discretizatio is demostrated i Table 1, for ka=0.08, kr=0.4, H/R=1.5. [Note, i Table 1, N w is the umber of odes per wavelegth.] As the umber of odes icreases (correspodig to a doublig, triplig ad quadruplig of umber of odes per wavelegth), the time averaged horizotal drift force ad 1st ad 2d harmoic forces clearly coverge. Due to the very small chages betwee the last two values, N f =150(N w =30) is selected for all computatios ad deemed to provide sufficiet accuracy ad resolutio of computatios. Table 1 Covergece of horizotal drift force o cylider, ad first ad secod harmoic forces, with the umber of free surface BEM odes N f, for ka=0.08, kr=0.4, H/R=1.5. [Note, N w deotes the umber of odes per icidet wavelegth.] N f N w D x /(ρga 2 ) F 1 x /( ρgra) F 2 x /( ρga 2 ) Fig. 7 First ad secod harmoic horizotal forces as a fuctio of the body submergece, for kr=0.4, ka=0.08. Fig. 8 Horizotal drift force as a fuctio of the body submergece H/R for kr=0.4, ka=0.12. Fig. 5 Computed first(t 1 ) ad secod(t 2 ) harmoic wave trasmissio coefficiets as a fuctio of icidet wave steepess ( ), for kr=0.4, H/R=1.5. Experimetal results of Grue (1991) are deoted by symbols ad 4thorder umerical HOS results of Liu et al. (1992) by. Fig. 6 Same results as i Fig. 5, as a fuctio of cylider submergece ( ), for kr=0.4, ka=0.08. Four sets of umerical results were computed, for the trasmissio coefficiet over the cylider (Figs. 5 ad 6), ad the horizotal force actig o the cylider (harmoics 1 ad 2 i Fig. 7; drift force i Fig. 8), as a fuctio of wave steepess ka or cylider submergece H/R. Each set of results is compared with idepedet experimetal ad/or umerical results i the figures. I the figures, we see a overall good agreemet of the preset results with earlier experimetal results by Grue (1991) ad umerical results by Liu et al. (1992), usig the 4thorder HOS method, or the few available 2dorder results of Vada (1987). More specifically, i Fig. 5, for a shallow cylider submergece H/R=1.5, as should be expected, 4thorder HOS results agree better with the preset fully oliear results for the smaller wave steepess, although relative errors o the secod harmoic trasmitted waves are a little large (HOS results were oly provided up to ka=0.08). Both umerical models slightly overpredict the measured first harmoic trasmitted wave, likely because of eergy dissipatio (ot icluded i the modelig) caused by the shallow cylider. FNPF results, however, predict the measured secod harmoic trasmitted wave quite well. I Fig. 6, for a high steepess of 0.08 but for icreasigly deeper submergece from H/R=1.5 to 2.2, HOS
6 206 Iter J Nav Archit Oc Egg (2011) 3:201~207 results stay quite close to the fully oliear results. For larger H/R, T 1 approaches 1, while T 2 decreases mootoically. As the cylider approached the free surface, stroger oliearity makes the secodharmoic term much more. I Fig. 7, as cylider submergece icreases, both harmoics of the horizotal force rapidly decrease i magitude; this agai is fully expected from the expoetial decrease with depth of dyamic pressure, for deep water waves. Fially, i Fig. 8, the measured horizotal drift force is uderpredicted by both models, ad more so the shallower the submergece, which could be due to viscous drag effects iduced by the mea drift curret (stroger o the cylider ear the free surface), ot icluded i the iviscid models. Additioally, Tables 2 ad 3 provide umerical results for the drift, ad 1st ad 2d harmoic forces, as a fuctio of wave steepess ka or waveumber kr. From these Tables, the horizotal drift forces are egative regardless of wave steepess ad icidet wavelegth ad that the magitude of this egative drift forces icreases as icidet wavelegth ad steepess icrease. Spatiallyaveraged first three harmoic amplitudes are show at Table 4 ad 5 as a fuctio of wave steepess ka or waveumber kr. The 2 d ad 3 rd harmoic amplitude icrease whe the icidet wave steepess is large ad the wavelegth is sufficietly log. By cotrast, the first harmoic amplitude shows the coutertred. Table 2 Horizotal drift force o cylider, first ad secod harmoic forces as a fuctio of wave steepess, for kr=0.4, H/R=1.5. ka D x /(ρga 2 ) F x 1 /( ρgra) F x 2 /( ρga 2 ) Table 3 Horizotal drift force o cylider, first ad secod harmoic forces as a fuctio of waveumber, for ka=0.08, H/R=1.5. kr D x /(ρga 2 ) F x 1 /( ρgra) F x 2 /( ρga 2 ) Table 4 Spatial mea first three harmoic amplitudes as a fuctio of wave steepess for kr=0.4, H/R=1.5. ka b 1 b 2 b Table 5 Spatial mea first three harmoic amplitudes as a fuctio of wave frequecies for ka=0.08, H/R=1.5. kr b 1 b 2 b CONCLUSIONS Fully Noliear Potetial Flow (FNPF) for the oliear diffractio of a submerged circular cylider are preseted ad compared to measuremets ad aother umerical predictio (HOS), with a special emphasis o the egative drift force o the cylider. LoguetHiggis (1977) suggested that the egative drift force ca be attributed mostly to wave breakig, ad partly to the presece of higherharmoic compoets of the trasmitted wave. Although our model ca simulate overturig waves, we stated with the viewpoit that the higherharmoic compoets of the trasmitted waves are the mai cause for the egative horizotal drift force o the cylider. Our umerical results are give for the horizotal drift force, harmoic amplitudes of the trasmitted waves ad oscillatory forces. It is foud that the magitude of egative drift force becomes larger with loger wavelegth, larger wave steepess ad shallower submergece. Also, the higherharmoic amplitudes averaged over the trasmitted wave regio show the same tred as the drift force. It is cocluded that the egative drift force o the cylider is caused by the higherharmoic compoets of the trasmitted wave, which arise from the oliear iteractios of waves with a submerged cylider. Our umerical model showed a overall good agreemet with earlier experimetal results by Grue (1991) ad umerical results by Liu et al.
7 Iter J Nav Archit Oc Egg (2011) 3:201~ (1992), usig the 4thorder HOS method ad simulated the higherharmoic geeratio due to oliear iteractios of streamfuctio wave by a submerged cylider through a systematic parametric ivestigatio with varyig variables such as wave steepess, wavelegth, ad submergece. ACKNOWLEDGEMENTS This work was supported by the New & Reewable Eergy of the Korea Istitute of Eergy Techology Evaluatio ad Plaig (KETEP) grat fuded by the Korea govermet Miistry of Kowledge Ecoomy. ( ). REFERENCES Coite, R., Noliear simulatio of trasiet free surface flows. I Proc. 5th Itl Cof. Num. Ship Hydro. Hiroshima. pp Dea, W.R., O the reflectio of surface waves by a submerged circular cylider. Proc. Camb. Phil. Soc. 44, pp Grilli, S.T. Skourup, J. ad Svedse, I.A., A Efficiet Boudary Elemet Method for Noliear Water Waves. Egieerig Aalysis with Boudary Elemets, 6(2), pp Grilli, S.T. ad Subramaya, R., Numerical Modelig of Wave Breakig Iduced by Fixed or Movig Boudaries. Computatioal Mechaics, 17(6), pp Grilli, S.T. ad Horrillo, J., Numerical Geeratio ad Absorptio of Fully Noliear Periodic Waves. J. Egieerig Mechaics, 123(10), pp Grilli, S.T., Depth Iversio i Shallow Water Based o Noliear Properties of Shoalig Periodic Waves. Coastal Egieerig, 35(3), pp Grilli, S.T. ad Horrillo, J., Shoalig of periodic waves over barredbeaches i a fully oliear umerical wave tak. Itl. J. Offshore ad Polar Egg, 9(4), pp Grue, J., Noliear water waves at a submerged obstacle or bottom topography. Preprit Series of Istitute of Mathmatics, Uiversity of Oslo. 2, pp Ioue, R. ad Kyozuka, Y., O the oliear wave force actig o submerged cyliders. J. Soc. Nav. Arch. Japa, 156, pp Liu, Y. Dommermuth, D.G. ad Yue, D.K.P., A higherorder spectral method for oliear wavebody iteractios. J. Fluid Mech. 245, pp LoguetHiggis, M.S., The mea forces exerted by waves o floatig or submerged bodies with applicatios to sad bars ad power machies. Proc. R. Soc. Lod. 352, pp Miyata, H. Khalil, G. Lee, Y.G. ad Kaai, M., A experimetal study of the oliear forces o horizotal cyliders. J. Kasai Soc. N.A. Japa, 209, pp Ogilvie, T.F., First ad secod order forces o a cylider submerged uder the free surface. J. Fluid Mech. 16, pp Torum, A. ad Gudmestad, O.T., Water Wave Kiematics. Kluwer Academic Publishers: Bosto. pp Ursell, F., Surface waves o deep water i the presece of a submerged circular cylider. Proc. Camb. Phil. Soc., 46, I: pp , II: pp Vada, T., A umerical solutio of the secodorder wave diffractio problem for a submerged cylider of arbitrary shape. J. Fluid Mech. 174, pp
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