csnak, 2011 Iter J Nav Archit Oc Egg (2011) 3:201~207 http://dx.doi.org/10.3744/jnaoe.2011.3.3.201 Computatio of the iviscid drift force caused by oliear waves o a submerged circular cylider Hyeok-Ju Koh 1 ad Il-Hyoug 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 higher-harmoic 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 higher-harmoic compoets of the trasmitted wave starts to icrease. A icrease of the higher-harmoic compoets of the trasmitted wave makes the horizotal drift force be egative. It is also foud that the higher-harmoic 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: Higher-harmoic compoet; Numerical wave tak; Drift force; Fully oliear potetial flow; Trasmitted wave. INTRODUCTION Accordig to potetial flow theory ad the so-called 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 wave-iduced drift force is due to higher-order 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 secod-order effects from liear results, Ogilvie (1963) showed the existece of a mea secod-order vertical force, but foud that the horizotal mea force vaished to secod-order. Usig a Stokes expasio, Vada (1987) solved the secod-order diffractio problem i the frequecy-domai, but could ot calculate all the terms of the mea horizotal force. Loguet-Higgis (1977) observed i experimet that a freely movig, eutrally buoyat, submerged cylider experieced a egative drift force, Correspodig author: Il-Hyoug Cho e-mail: cho0904@jejuu.ac.kr causig it to move towards the wavemaker. He attributed this force mostly to wave breakig ad, to a lesser degree, to the secod-harmoic 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 two-dimesioal, fully-oliear, iviscid time-domai 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 higher-order harmoic forces ad wave trasmissio coefficiets o a submerged cylider, i a fully-oliear 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 space-periodic versio of Grilli et al. (1989) fullyoliear potetial flow model. Liu et al. (1992) applied the Higher-Order Spectral Method (HOS) to this problem ad compared computatios with aalytical results ad experimetal observatios. They also used exact deep-water Stokes waves as iitial coditios ad specified periodic coditios for upstream ad dowstream boudaries,
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 two-dimesioal (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 2D-FNPF 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 o-breakig waves, we will show that the higher-harmoic 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 higher-harmoic 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 higher-harmoic 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 2 2 1 1 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 Loguet-Higgis (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 higher-harmoic geeratio. g 1 Dx b (3) 4 2 2 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 2D-FNPF 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 o-flow 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.
Iter J Nav Archit Oc Egg (2011) 3:201~207 203 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 higher-order 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, quasi-sigular) 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 secod-order Taylor series expasios expressed i terms of a time step Δt ad of the Lagragia time derivative, D/Dt, for ϕ ad r. First-order 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. Secod-order 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 0.40. 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 zeromass-flux 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 ramp-up over a short distace i frot of the AB. Grilli ad Horrillo (1997) showed that the AB is efficiet i absorbig higher-frequecy 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 mid-legth 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,
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 time-depedet 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 quasi-steady 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 higher-harmoics are geerated i trasmitted waves. Usig the preset NWT, the time-averaged 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. Spatially-averaged 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 higher-harmoic 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(10-9 )), 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(10-6 )). 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 ramp-up time, typical computatios i the NWT quickly reach a quasi-steady 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 up-dow asymmetry idicative of oliear effects.
Iter J Nav Archit Oc Egg (2011) 3:201~207 205 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 ) 50 10-0.0851 1.4318 0.8130 100 20-0.0927 1.4911 0.8935 150 30-0.1027 1.5055 0.9437 200 40-0.1052 1.5026 0.9386 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 4th-order 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 4th-order HOS method, or the few available 2d-order results of Vada (1987). More specifically, i Fig. 5, for a shallow cylider submergece H/R=1.5, as should be expected, 4th-order 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
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 secod-harmoic 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. Spatially-averaged 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 ) 0.01-0.0071 1.6048 1.1802 0.02-0.0128 1.6000 1.1713 0.03-0.0226 1.5920 1.1552 0.04-0.0366 1.5808 1.1294 0.05-0.0518 1.5638 1.0881 0.06-0.0709 1.5477 1.0442 0.07-0.0867 1.5262 0.9879 0.08-0.1026 1.5052 0.9462 0.09-0.1116 1.4742 0.8576 0.10-0.1138 1.4502 0.7916 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 ) 0.30-0.1382 1.4254 0.8049 0.35-0.1250 1.4263 0.9963 0.40-0.1027 1.5055 0.9437 0.45-0.0989 1.4501 0.9401 0.50-0.0811 1.4905 0.8639 0.55-0.0635 1.4228 0.7968 0.60-0.0620 1.4112 0.7531 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 3 0.01 0.9933 0.0822 7.6470 10-3 0.02 0.9857 0.1572 0.0199 0.03 0.9737 0.2216 0.0294 0.04 0.9579 0.2721 0.0360 0.05 0.9354 0.3066 0.0423 0.06 0.9102 0.3281 0.0488 0.07 0.8841 0.3374 0.0547 0.08 0.8491 0.3381 0.0603 0.09 0.8102 0.3315 0.0652 0.10 0.7703 0.3274 0.0674 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 3 0.30 0.7955 0.3336 0.0733 0.35 0.8466 0.3605 0.0643 0.40 0.8485 0.3390 0.0603 0.45 0.8632 0.3162 0.0576 0.50 0.8980 0.2686 0.0515 0.55 0.9031 0.2248 0.0437 0.60 0.9145 0.1957 0.0387 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. Loguet-Higgis (1977) suggested that the egative drift force ca be attributed mostly to wave breakig, ad partly to the presece of higher-harmoic compoets of the trasmitted wave. Although our model ca simulate overturig waves, we stated with the viewpoit that the higher-harmoic 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 higher-harmoic 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.
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