The Wave Excitation Forces on a Floating Vertical Cylinder in Water of Infinite Depth

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1 Th Wv Exctton Forcs on Flotng Vrtcl Cylnr n Wtr of Infnt Dpth Wll Fnngn, Mrtn Mr, J Goggns 3,* Collg of Engnrng n Infortcs, Ntonl Unvrsty of Irln, Glwy, Irln chool of Mthtcs, ttstcs n Appl Mthtcs, Ntonl Unvrsty of Irln, Glwy, Irln 3 Ryn Insttut for Envronntl, Mrn n Enrgy Rsrch, Ntonl Unvrsty of Irln, Glwy, Irln * Tl: , E-l: j.goggns@nuglwy. Abstrct: Whn crryng out ny nurcl olng t s vtl to hv n nlytcl pproxton to nsur tht rlstc rsults r obtn. Th nurcl olng of wv nrgy convrtrs s n ffcnt n nxpnsv tho of unrtng ntl optston n xprntton. Thrfor, th n objctv of ths ppr s to trn n nlytcl soluton for th hv, surg n ptch wv xctton forcs on flotng cylnr n wtr of nfnt pth. Th bounry vlu probl tchnqu, usng th tho of sprton of vrbls, s ploy to rv th vlocty potntls throughout th flu on. A Fourr trnsfor s us to rprsnt nfnt pth. Atonlly, vloc s xpnson thor s us to nvrt th coplct cobn Fourr sn/cosn trnsfor. An syptotc pproxton s tn for low frquncy ncnt wvs n orr to crt n nlytcl soluton to th probl. Grphcl rprsnttons of th wv xctton forcs wth rspct to ncnt wv frquncs for vrous rft to rus rtos r prsnt, whch cn sly b us n th sgn of wv nrgy convrtrs. Kywors: Infnt pth, Wv nrgy, Wv structur ntrcton, Wv wtr probl Noncltur rus of cylnr... A pltu of ncnt wv... b rft of cylnr... F forc... N F c Fourr cosn trnsfor... F,xt...surg xctton forc N F 3,xt... hv xctton forc N G grvty... s - wvnubr... - ntgr... n j j-coponnt of th norl... p ()coffcnt... q coffcnt... q ()coffcnt... r rus... B wtt surfc... t t.. v flow vlocty... s - x horzontl coornt... z vrtcl coornt... ε Nunn sybol... θ polr coornt... rs sprton constnt... ρ nsty... g 3 frquncy on vlocty potntl... s - I ncnt wv vlocty potntl... s - s ntror scttrng vlocty potntl... s - xtror ffrcton vlocty potntl.. s - s xtror scttrng vlocty potntl... s - Φ t on vlocty potntl... s - ω wv ngulr frquncy... s -. Introucton On of th n stgs n th sgn of wv nrgy convrtrs (WECs) s th nurcl ollng of gvn convrtr. In ths ppr, n nlytcl soluton for th wv xctton forcs on flotng cylnr n wtr of nfnt pth s prov. Th soluton wll ct s tho of vltng th rsults fro nurcl ols of WECs, s t provs n stton of forcs on cylnr rprsntton of n rbtrry shp xsytrc WEC. 75

2 Th soluton of th scttrng n rton probl for flotng bos, n fnt or nfnt pth wtr, hs bng xplor for cs for vrous shps of bos. In 948, Frtz Ursll[] xplor th forcs on n nfntly long horzontl flotng cylnr n nfntly p wtr n, n 955, r Thos vloc[] solv th rton probl for flotng hlf-rs sphr n nfntly p wtr. In 97, G rrtt[3] solv th scttrng probl by trnng th vrtcl forc, horzontl forc n torqu for crculr oc n wtr of fnt pth. In 975, B lc[4] loo t th wv forcs on bos whch r vrtclly xsytrc usng n ntgrl quton forulton n wtr of fnt pth. In 98, Yung[5] prsnt st of thortcl ss n png coffcnt for flotng cylnr n fnt pth, whch h lso trunct for th nfnt pth probl. In 3, Bhtt n Rhn[6] us slr tchnqu s vloc to solv, lthough usng s-nlytc soluton, th scttrng n rton probl for flotng vrtcl cylnr n wtr of fnt pth. Prvously, n nlytcl soluton for wv xctton forcs on flotng cylnr n wtr of nfnt pth hs not bn rv. Thrfor, th soluton rv n ths ppr s for s-subrg vrtcl cylnr n nfnt pth wtr. A bounry vlu probl s us to rv n nlytc soluton, fro th scttrng probl, for th hv, surg n ptch xctton forcs.. Mthoology Th probl consrs vrtcl cylnr, of rus,, n wth rft, b, whch cn ov n surg, hv or ptch oton, n n ncnt wv of pltu, A, n ngulr frquncy, ω, s pct n Fg.. Th wv progrsss n th postv x-rcton wth th orgn t th stll wtr lvl (WL) n th postv z-rcton s vrtclly ownwrs. In th forulton of th soluton, nubr of ssuptons r us: Th wtr s both ncoprssbl, s frquncs r low, n ffctvly vsc. As th r hs such sll nsty, prssur chng s nglgbl n, thus, s t constnt prssur. Th surfc tnson t r-wtr ntrfc s nglgbl. Th wtr s t constnt nsty n tprtur. Th Rynols nubr for th flow s suffcntly sll for th flow to rn lnr. Th wvs r progrssv n only trvl n on rcton n th wv oton s rrottonl. Fg. Grphcl st-up of th Bounry Vlu Probl for Vrtcl Cylnr 76

3 Yung[5] n Bhtt n Rhn[6] ploy th tchnqu of vng th on nto two rgons, whch s us n ths ppr. Th two rgons r th ntror rgon, whch s th r unrnth th cylnr, n th xtror rgon, whch s th rnng r of th flu (Fg. ). Th probl s solv n th frquncy on. Thrfor, th vlocty potntl,, to b solv s trnsfor to th frquncy on, s follows: { -ωt } Φ r,θ, z, t = R r,z,θ whr Φ s th t on vlocty potntl, r s rus, θ s th ngl, s th stnr gnry unt, ω s th wv ngulr frquncy of th wv, n s th frquncy on vlocty potntl. Th forc s thn clcult by ntgrtng th vlocty potntl ovr th wtt surfc r of th cylnr, B, usng th followng quton: B Fˆ = ρω n j () whr ρ s wtr nsty, n j s th j-coponnt of th norl, s surfc n F s th forc, whr F = R{ Fˆ } -ωt. Th qutons n bounry contons tht n to b stsf throughout th probl r: th Lplc s quton, th p wtr conton, th fr surfc quton n th rton conton, rspctvly[7]: = + + r r r r + θ z = (3) s z (4) ω - g = on z =, r (5) z l r( - ) = (6) r r whr s th wvnubr ( = ω /g). nc th oton s rrottonl n ncoprssbl, th Lplc s quton ws rrv t by substtutng v = nto. v =, whr v s th flow vlocty. Th soluton bng vlop s for nfntly p wtr. Thus, th p wtr conton fns th flow vlocty nr th s b. Th fr surfc quton fns th vlocty potntl t th fr surfc wy fro th flotng boy. Th rton conton fns th vlocty potntl of th wv t th stnc fro th boy whn th ffct of th boy on th wv hs sspt. Th scttrng probl ls wth th xctton forc on fx boy n, thrfor, th followng structurl bounry contons ust b pos: z = on z =, whr z = z - b (7) r = t r = (8) 77

4 whr n r th ntror n xtror scttrng vlocty potntls, rspctvly. nc w r lng wth nfnt pth, Fourr sn/cosn trnsfor s ploy whn lng wth th vrtcl or z-coponnt. For th ntror rgon, n orr to stsfy th structurl quton (Eq. (7)) Fourr cosn trnsfor s rqur. Thrfor, ntroucng constnt,, yls: ( ) F C r,θ, z = r,θ, z cosz z (9) ( ) r,θ, z = F c r,θ, z cosz whr F c s th Fourr cosn trnsfor. Th tho of sprton of vrbls s us to solv th Lplc s quton (Eq. (3)) n orr to forult n xprsson for th ntror scttrng vlocty potntl, s follows: ( r,θ, z) = p ( ) ( r) I cosz cos θ () I = whr I s th of frst Bssl functon of orr n p () s n unnown coffcnt. K[8] gvs th ncnt wv vlocty potntl, I, n th frquncy on for p wtr n oblqu s s: ga ω - z + ( r,θ, z) = - ε J ( r) cos θ I = whr J s th frst Bssl functon of orr, ε s th Nunn sybol, fn by ε = n ε = for. lrly, for th xtror rgon, whn lng wth nfnt pth n th tho of sprton of vrbls, Fourr sn/cosn trnsfor s us. In orr to stsfy th fr surfc quton (Eq. (5)), cobnton of th Fourr sn n Fourr cosn trnsfor s rqur. Agn, ntroucng constnt, th followng s obtn: ( ( r,θ, z) ) = ( r,θ, z)[ cosz - sn z] F cosz z (3) whr s th xtror ffrcton vlocty potntl. Th vloc s xpnson thor [9] s us to obtn th nvrs Fourr trnsfor. lrly, th tho of sprton of vrbls s us to solv th Lplc s quton (Eq. (3)) n orr to forult n xprsson for th xtror ffrcton vlocty potntl, whch s gvn s: r,θ, z = [q, + = q + r - z ( ) K ( r) [ cosz - sn z ] ]cos θ K (4) 78

5 Thrfor, snc th scttrng vlocty potntl s th su of th ncnt n ffrcton vlocty potntls (.. = I + ) n ncorportng -gaω - ε + nto th ( r,θ, z) tr n Eq. (4), th scttrng vlocty potntl for th xtror probl s gvn s: ga ( r) + -z ( r,θ, z) = - ε [{J r + q, } ( ) = ω q ( ) K ( r) + [ cosz - sn z ] ]cos θ + K whr s th frst nl functon of orr n K s th of scon Bssl functon of orr. Th unnown coffcnts of p () n Eq. (), n q, n q () n Eq. (5), r foun by tchng th vlocty potntls cross th bounry t r =. Th contons whch r to b stsf t th bounry r: ( r,θ, z) = ( r,θ, z), f b z (5) (6) r,θ, z r = r,θ, z, f r b z (7) r,θ, z r =, f z b (8) 3. Rsults In orr to crt n nlytcl soluton, syptotc pproxtons for th xctton forcs r rv for low frquncy wvs or, n othr trs, whn th wvnubr,, tns towrs zro. Thrfor, n ton to Eq. (6)-(8), th pproxton tht tns towrs zro s pos whn tchng th ntror scttrng vlocty potntl, gvn n Eq. (), n th xtror scttrng vlocty potntl, gvn n Eq. (5), cross th bounry r = n orr to solv for th unnown coffcnts p (), q, n q (),. Usng ths tonl pproxton, t ws foun tht q () tns to zro n th coffcnt, q,, s pproxt s: q, = -J ' (9) '( ) n th coffcnt, p (), s gvn s: p + ga = - ε ω {J + J ' -b ( ) } ' + () whr pr s th rvtv. Thrfor, n nlytcl pproxton s crt n shown grphclly for vrous rft, b, to rus,, rtos n Fg. -4. Whn clcultng th surg, or horzontl, xctton forc th only non-zro soluton s whn =, s ths s th only non-zro soluton to th ntgrl cosθ cosθ θ, whch rss n th forc clculton. Furthror, whn ntgrtng th vlocty potntl ovr th surfc r, 79

6 th ntgrton s prfor only ovr th curv surfc of th cylnr n, hnc, th xtror vlocty potntl t r = s us. Thrfor, th surg xctton forc, Fˆ, xt, s gvn s: b Fˆ = ρω ( r,θ,z) n = ρω (,θ, z) n z θ,xt = - B ρga ε ρga = - {J ( - ) b {J - J ' }- ' ( - b ) - J' }- = ' cos θ cosθ θ whr n = -cos θ. Grphcl rprsnttons of surg xctton forcs wth rspct to ncnt wv frquncs for vrous rft to rus rtos of vcs r shown n Fg.. Whn clcultng th hv, or vrtcl, xctton forc fro th vlocty potntl, th only non-zro soluton s whn s qul to zro u to th ntgrl cosθ θ. Furthror, whn ntgrtng th vlocty potntl ovr th surfc r, th ntgrton s prfor only ovr th bs of th cylnr n, hnc, th ntror vlocty potntl, t z =, s us. Thrfor, th hv xctton forc, Fˆ 3, xt, s gvn s: Fˆ 3,xt = ρω = -ρω = -ρω r,θ, z n 3 = ρω r,θ, B I r p I I r p I n r r cos θ θ 3 r r θ () whr n 3 = -. Grphcl rprsnttons of hv xctton forcs wth rspct to ncnt wv frquncs for vrous rft to rus rtos of vcs r shown n Fg. 3.,6 F,xt /ρga,4,,8,6,4 b/=. b/=.5 b/= b/=.5 b/=, 3 4 Fg. Th norls surg (or horzontl) xctton forc, n th frquncy on, s functon of for vrous rus to rft rtos. 8

7 F 3,xt /ρga,8,6,4, b= b/=. b/=.5 b/=.5 b/= b/= 3 4 Fg. 3 Th norls hv (or vrtcl) xctton forc, n th frquncy on, s functon of for vrous rft to rus rtos. Th ptch, or torqu, xctton forc rss fro th surg n hv forcs on t h wtt surfc of th cylnr. Th ptch s tn bout th xs whch s trnsvrs to th ncnt wv t th cntr of th bs, s shown by T n Fg.. Whn clcultng th ptch th only non-zro soluton, slr to surg, s whn =. Thrfor, th ptch xctton forc, Fˆ 5, xt, s gvn s: Fˆ 5,xt = ρω = -ρω B + ρω b = -ρga{j r,θ, z n p 5,θ, z - J I I z - b ( r) ' cosθ z θ + ρω ( ) } ' b ( z - b) -z r,θ, r z cosθ r θ (8) Grphcl rprsnttons of ptch xctton forcs wth rspct to ncnt wv frquncs for vrous rft to rus rtos of vcs r shown n Fg Dscusson n Conclusons An nlytcl soluton to trn th hv, surg n ptch wv xctton forcs on flotng cylnr n wtr of nfnt pth hs bn prsnt n ths ppr. For s of us n th sgn of wv nrgy convrtrs, grphcl rprsntton of th wv xctton forcs wth rspct to th ncnt wv frquncs for vrous rft to rus rtos of vcs r gvn. In prtculr, th hv, surg n ptch xctton forcs, whch r th only thr forcs on n xsytrc vc, wr rv. Th nlytcl solutons wr obtn usng n syptotc pproxton for low frquncy ncnt wvs. 8

8 F 5,xt /ρga 3,8,6,4,,8,6,4, b/=. b/=.5 b/=.5 b/= b/= 3 4 Fg. 4 Th norls ptch (or torqu) xctton forc, n th frquncy on, s functon of for vrous rft to rus rtos. Acnowlgnts Th frst uthor woul l to cnowlg th fnncl support fro th Ntonl Unvrsty of Irln unr th Collg of Engnrng & Infortcs Postgrut Fllowshp. Rfrncs [] F. Ursll, On Th vng Moton of Crculr Cylnr on th urfc of Flu, Th Qurtrly Journl of Mchncs n Appl Mthtcs, (), 949, pp [] T. vloc, Wvs u to F lotng phr Mng Proc vng Osclltons. Procngs of th Royl octy of Lonon, rs A, Mthtcl n Physcl cncs, 3(84), 955, pp. -7. [3] C.J.R. Grrtt, Wv forcs on crculr oc, Journl of Flu Mchncs, 46, 97, pp [4] J.L. Blc, Wv forcs on vrtcl xsytrc bos, Journl of Flu Mchncs, 67(), 975, pp [5] R.W. Yung, A ss n png of vrtcl cylnr n fnt-pth wtrs, Appl Ocn Rsrch, 3(3), 98, pp [6] D.D. Bhtt n M. Rhn, On scttrng n rton probl for cylnr n wtr of fnt pth, Intrntonl Journl of Engnrng cnc, 4(9), 3, pp [7] C.M. Lnton n P.McIvr, nboo of Mthtcl Tchnqus for Wv/tructur Intrctons, Chpn & ll/crc,. [8] C.. K, Nonlnr Wvs n Offshor tructurs, Avnc rs on Ocn Engnrng, Worl cntfc Publshng Co. Pt. Lt, 8. [9] A. Chrbrt, On th oluton of th Probl of cttrng of urfc-wtr Wvs by th Eg of n Ic Covr, Procngs: Mthtcl, Physcl n Engnrng cncs, 456(997), pp

(A) the function is an eigenfunction with eigenvalue Physical Chemistry (I) First Quiz

(A) the function is an eigenfunction with eigenvalue Physical Chemistry (I) First Quiz 96- Physcl Chmstry (I) Frst Quz lctron rst mss m 9.9 - klogrm, Plnck constnt h 6.66-4 oul scon Sp of lght c. 8 m/s, lctron volt V.6-9 oul. Th functon F() C[cos()+sn()] s n gnfuncton of /. Th gnvlu s (A)