HYDRODYNAMIC ANALYSES OF HIGH SPEED TRANSOM STERN HULL FORMS IN WAVES BY 3-D GREEN FUNCTION METHOD

Transcription

1 HYDRODYNAMIC ANALYSES OF HIGH SPEED TRANSOM STERN HULL FORMS IN WAVES BY 3-D GREEN FUNCTION METHOD Yoshiyuki Inou, Yokohaa National Univrsity, Japan Md. Karuzzaan, Yokohaa National Univrsity, Japan SUMMERY A coputr cod YNU-SEA of 3-D Grn function thod with forward spd has bn dvlopd and usd to analyz th hydrodynaic radiation and diffraction forcs and otion rsponss for high spd crafts in wavs. Th 3-D Grn function thod can b applid to hydrodynaic analyss of arbitrary hull for, howvr it can b hardly applid to high spd transo strn hull fors. Th stra lin of flow fild of high spd craft of transo strn is sparatd fro transo strn. So th boundary conditions of wttd hull surfac in high spd should b diffrnt fro thos in low spd. In this papr, th authors show so application th coputr cod YNU-SEA to th hydrodynaic analyss of high spd transo strn hull fors in wavs. Th rsults of prsnt calculations ar copard to th xprintal ons by prvious rsarchrs. Th nurical rsults for singl hull craft and cataaran hulls rval that th prsnt calculation thod givs th accurat nurical coputations of sakping probls for high spd crafts of transo strns advancing in wavs. Finally, so discussions ar ad basd on ths nurical rsults which ay b hlpful for th accurat prdiction of sakping prforanc of high spd crafts in wavs.. INTRODUCTION Th corcial application of high spd vssls has incrasd significantly ovr th past yars. Rcnt yars hav sn any high spd vssls, carrying passngrs and vhicls, xposd to svr sa conditions. A fatur of ths high spd vssls that ar ostly in cataaran configurations is that thir roll and pitch natural priods ar uch or siilar than is th cas for ono hulls. This ay lad to a corkscrw typ otion, which is not only uncofortabl for passngrs but can ipos gratr loads on th structur of th vssls. In ordr to assss th cofort and safty of ths vssls, it is ncssary to undrstand thir bhavior in a sa way and this ust includ thir sakping prforanc in obliqu as wll as had wavs. At th sa ti thr has bn a nd to dvlop tchniqus for th prdiction of th sakping prforancs of such vssls, particularly at highr oprational spds. Such a nd ariss fro th iportant rol that sakping can play whn assssing th oprational viability, availability and safty of high spd vssls in particular sa conditions. Thr hav bn so systatic invstigations of th sakping of high spd ono hulls. Blok and Buklan (984), Grigoropoulos and Loukakais (998) hav invstigatd th sakping charactristics for sris hull fors. Boovic and Sahoo (998) has also studid th otion analysis of high spd displacnt hull fors. Many rsarchs hav also bn carrid out to study th prforancs of high spd cataarans. Inccik t al (99), Matsui t al (993), Hudson t al () and Molland t al () hav invstigatd sakping charactristics for high spd cataarans. It is donstratd that strip thory can prdict rspons otions of th convntional ono hull ship in wavs at low to odrat Froud nubrs. Howvr at high Froud nubrs, th thr-dinsional ffcts bco doinant and strip thory fails to prdict th hydrodynaic prforanc of high spd crafts. Though th 3-D Grn function thod with forward spd nds long coputational ti, it should b th or accurat assuption of th flow fild probl on an advancing ship in wavs. Nowadays th availability of uch fastr coputr aks th 3-D Grn function thod with forward spd or practical than bfor. An advantag of th 3-D Grn function thod is its wid rang of applicability with rspct to hull for, forward spd and frquncy. In this papr, sakping analyss for ono and ulti hull vssls with transo strn hulls ar considrd. Th 3-D Grn function thod with translating and pulsating sourc has bn adoptd to calculat th hydrodynaic forcs for advancing vssls. Th 3-D Grn function thod is on of th ost vrsatil tchniqus for calculating th potntial flow fild in haronic oscillation for a singl hull vssl with arbitrary shap and in th prsnt analyss th 3-D Grn function thod is also usd for th ulti hull bodis without any chang to th nurical approach. Th solutions obtaind by th 3-D Grn function thod copltly dtrin th potntial flow fild, so that all quantitis rlvant to th syst can b drivd fro th vlocity potntial. Th YNU-SEA has bn usd to solv such a probl. Coputations hav bn carrid out for odls with transo strn hulls configurd in ono and ulti hull configurations.th nurical coputations ar validatd by coparing th calculatd rsults with th xprintal ons rportd by Molland t al (). Th nurical rsults obtaind by th YNU-SEA ar found in fairly good agrnt with th xprintal rsults. Sssion A 75

2 HYDRODYNAMIC ANALYSIS Lt (x, y, z) b th right-hand Cartsian coordinat syst with xy plan on th an fr surfac and th z axis dirctd vrtically upwards through th cntr of a ship as shown in Fig.. To siplify th analysis it is assud that th body aintains its initial cours at constant forward spd U and oscillats haronically with th ncountr frquncy ω ( = ω ku cos χ ) in wav with frquncy of ω and is aking wav attack angl of χ. Hr k ( = ω g ) is th wav nubr. Th coordinat syst is translating at th sa forward spd U as that of th body. Fig. Coordinat syst. BOUNDARY CONDITIONS In prsnt analysis, it is assud that th fluid is inviscid and incoprssibl, th flow is irrotational and both incoing wav lvation and body oscillations ar sall. Th radiation and diffraction vlocity potntials φ (=~7) ust satisfy th Laplac quation and th corrsponding boundary condition is iposd on th an position of th fluid boundary as follows: φ = and φ = in th fluid With th body hull boundary conditions, φ U = n + iω (for =-6) and φ 7 φ = (for =7) and th fr surfac conditions at z = φ iω + U φ + g = x z whr th gnralizd noral n is dfind by ( n, n, n 3 ) = n and ( n 4, n5, n6 ) = r n () () (3) (4) (5) with n is th outward unit noral vctor on th hull surfac and r is th position vctor with rspct to th cntr of th gravity of a ship. Nglcting th prturbation of th stady flow du to th prsnc of th body, th coponnts of Eq. can b siplifid to = = 3 = 4 =, 5 = n3 and 6 = n (6). VELOCITY POTENTIAL Th total potntial fild φ T is th su of two vlocity potntial filds: a ti indpndnt potntial fild φ du s to ship stady forward otion, and a priodic potntial fild φ du to th ship s oscillatory otions. Th total potntial can b writtn as φ iωt T = Ux + φs + φ (7) Th priodic potntial φ can b sparatd into contributions of incidnt wav potntial φ, radiation wav potntial φ du to six ods of otion and diffraction wav potntial φ in th fild: 7 6 (8) φ = φ + φ + iω φ 7 = X whr, X (=~ 6) is th coplx otion rspons aplitud in -th od. Hr =,,3,4,5,6 rfrs to surg, sway, hav, roll, pitch and yaw otions, rspctivly. Ifζ A b th aplitud of incidnt wav thn th incidnt wav potntial φ can b xprssd as, (9) φ = i ζ Aω kz ik ( x cos χ + y sin χ ) k Th potntial function φ (=~ 7) can b obtaind by introducing a singularity distribution ovr th hull boundary surfac. If σ ( is considrd as th strngth of a sourc distributd ovr th hull boundary surfac at point Q thn th potntial φ (P) at any point P insid th fluid can b xprssd by using Grn function, G(P, as: U φ( P) = σ( QG ) ( P, Qds ) + G( P, σ( Qn ) dy 4π SH CH g () whr contour intgral C H is ovr th intrsction to th hull surfac S H and th fr surfac. Th lin intgral of scond tr in Eq. is firstly introducd by Brad (97). It is drivd fro th fr surfac boundary condition of Eq. 4 which includs coupling of forward spd U and x dirctional coponnt of inducd vlocity by th advancing sourc. iw t Grn function G( x, y, z; x, y, z ) is dfind as vlocity potntial for a fild point P(x,y,z) du to a unit Sssion A 76

3 pulsating sourc locatd at a point Q ( x, y, z). It satisfis th linarizd fr surfac boundary condition xprssd in Eq. 4 and Laplac s Eq. outward sourc point (singularity). According to th Whausn and Laiton (96), and othrs as Inglis and Pric (98), th Grn function of th translating and pulsating sourc for watr of infinit dpth can b writtn as: = + g G( P, R R π γ f ( θ, k) dθdk+ g π π γ L g + π π π L f ( θ, k) dθdk whr k k f ( θ, k) = gk ( ω + ku cosθ ) ( x, y, z) is th position of singularity and β = Uω / g ; if β <. 5 thn γ = and if β. 5 thn γ = arccos( / 4β ) ; R = ( x x) + ( y y) + ( z z) R = ( x x ) + ( y y ) + ( z+ z f ( θ, k) dθdk [( z+ z ) + i( x x )cosθ ] cos[ k( y y )sinθ ] ) () () (3) (4) Th contours L and L ar illustratd in Fig. with singularitis atgk, 4β cosθ gk3 = ω β cosθ (5) gk, + 4β cosθ gk4 = ω β cosθ (6) Fig. Contours L and L usd in intgrations of quation in Eq..3 NUMERICAL CALCULATION OF VELOCITY POTENTIAL Taking th noral drivativ of Eq. and lt P approach Q, thn th Eq. bcos φ ( P) = σ( P) SH G( P, U σ( ds+ g G( P, σ( ndy (7) For th nurical calculation, th hull surfac is discrt by N nubr of panls. Th discrt for of Eq.7 for th point P l (l=~n) and Q ( = ~N) can b rwrittn as CH N GPQ ( l, ) U σ ( Pl ) σ( Q ) ds+ S = n g C ;=~6 (8) ;=7 To carry out th surfac intgration of Grn function in Eq. 8, Hss and Sith (964) thod is applid for ach quadrilatral hull surfac panl S. So ach panl can b xprssd by singl point sourc σ ( Q ) (=~7) of which strngth ar quivalnt to kp th hull surfac at Q (=~N) in -th od. To carry out th intgration of Grn function on ach lnt lin C, it is approxiatd as a straight lin. Whn th sourc dnsity σ ( Q ) on ach panl and lin lnt at Q ( = ~N) is obtaind by solving th Eq. 8, th vlocity potntial φ ( Pl ) (l=~n, and =~7) can b calculatd by using Eq. and xprssd as (9) whr, th scond tr suation is only for th intrsction to th hull surfac panls..4 HYDRODYNAMIC COEFFICIENTS AND EXCITING FORCES Aftr dtrining th vlocity potntial φ ( P l ) on ach lnt by th body surfac boundary conditions, by nglcting sall ffct fro a ti indpndnt potntial fild φ du to stady forward otion of ship, th s radiation forcs (i=,,3) and onts (i=4,5,6) in th -th od can b obtaind by: φ F i = ρ ω X φ + iωux nids () S x Fro th radiation forcs and onts xprssd in Eq., addd ass cofficints a i and daping cofficints b ar obtaind as i a b i i GPQ ( n Pl l, ) ( ) ndy = φ N U φ ( Pl ) = σ ( Q ) G( Pl, Q ) ds+ 4π S C = g U φ = ρ R φ + i nids () s ω x U φ = ρω I φ + i nids () s ω x Wav xciting forcs and onts F can b obtaind by i t U ( φ + φ7 ) F = i iρω ω ( φ + φ7 ) + i ni ds S ω x i G ( P l, Q ) ndy (3) Sssion A 77

4 .5 EQUATION OF MOTION In ordr to prdict th ship otions in wavs, th ship is rgardd as a rigid floating body having six dgrs of frdo with coplx aplituds X ( =~ 6) and th otion quation at th cntr of th gravity of a ship can b xprssd in frquncy doain as 6 (4) [ ω ( M i + ai ) + iωbi + ci ] X = Fi = (i=~6) whr, M is th inrtia atrix and c i i is th hydrostatic stiffnss atrix in i-th od du to otion in -th od. 3 NUMERICAL CALCULATIONS A coputr cod YNU-SEA has bn dvlopd according to th athatical forulations dscribd in prvious sction. In ordr to vrify th validity of th dvlopd coputr cod, nurical coputations for ship with transo strn hull shown in Fig. 3 has bn calculatd by using th YNU-SEA cod. Mono hull vssls usd for th prsnt analyss ar of 4b and 5b typ as dscribd by Molland t al (). Nurical calculations hav bn prford to prdict otion rsponss for transo strn hull ships advancing in had wavs at diffrnt Froud nubrs. Th principal particulars for odls 4b and 5s ar shown in Tabl. Th nurical rsults ar copard with xprintal rsults invstigatd by Molland t al. (). Modl tsts hav bn carrid out in th Southapton Institut Tst Tank of Univrsity of Southapton, England. Tabl. Principal Particulars of Transo Hull Spcification Modl 4b Modl 5s Lngth,L ().6.6 L/B () 9. B/T ().. C B C P C M Th prsnt nurical approach has bn xtndd for th ulti hull vssls. Nurical rsults for th transo strn hull cataaran ar also copard with xprintal rsults invstigatd by Molland t al. (). Singl hull of th cataaran is considrd as 4b typ hull. Twin hulls ar sparatd at a distanc of S/L= and S/L=.4. Th shing and arrangnt of th odl ship usd for th prsnt nurical calculations ar shown in Fig.4. Fig. 4 Msh arrangnts for Transo strn cataaran 4. RESULTS AND DISCUSSIONS Nurical coputation for ships with transo strn hull has bn calculatd by using th YNU-SEA cod. Th nurical rsults ar copard with xprintal rsults invstigatd by Molland t l. (). Figs. 5~ 8 show hav and pitch otion rsponss for 4b typ ono hull odl as dscribd in Tabl for Froud nubrs Fn=. and Fn=.. =. Fig. 3 Msh arrangnts for Transo strn Hull (L/g) / Fig. 5 Hav transfr function for 4b ono hull at Fn=. Sssion A 78

5 . =.. =. Exp.(Southapton) (L/g) / Fig. 6 Pitch transfr function for 4b ono hull at Fn= (L/g) / Fig. 9 Hav transfr function for 5s ono hull at Fn= Fn= (opn) Exp. (Southapton ) (closd). =. Exp.(Southapton) (L/g) / (L/g) / Fig. 7 Hav transfr function for 4b ono hull at Fn= Fig. Pitch transfr function for 5s ono hull at Fn=.. = (opn) Exp.(Southapton) (closd).5. = (L/g) / (L/g) / Fig. 8 Pitch transfr function for 4b ono hull at Fn= Fig. Hav transfr function for 5s ono hull at Fn= For Fn=. transo strn shs ar not includd in th nurical calculations. Th prsnt nurical calculations show ovrall good agrnt with th xprintal rsults. Though transo strn shs ar includd for th nurical calculation for 4b ono hull at Fn= to invstigat th rsults for with and without transo strn rsults, it is obsrvd fro th Fig. 7 and Fig. 8 that oitting th transo strn shs hav a vry ngligibl influnc on th prdictd otion rsponss. Hav and pith otion rsponss for Fn=. ar shown in Fig.9 and Fig. for 5s typ ono hull odl. Hav otion rspons is found highr than th xprintal ons though pith otion rspons shows good agrnt with th xprintal ons. Nurical rsults for Fn= ar shown in Fig. and Fig.. Hav otion rspons is also found highr than xprintal ons though th trnd of pitch otion rspons shows vry good agrnt with th xprintal rsults. Sssion A 79

6 .5. =. = (L/g) / Fig. Pitch transfr function for 5s ono hull at Fn= Th coputr cod of YNU-SEA has furthr bn applid to calculat th otion rsponss for cataaran vssls. For this purpos cataaran with 4b typ hull for is invstigatd. Twin hulls ar sparatd at a distanc of D/L= and D/L=.4. Fig.3 and Fig. 4 show th hav and pitch otion rsponss for D/L= at Fn=. and Fig.5 and Fig. 6 show thos at Fn=. In ths nurical calculations th transo strn shs ar not includd. Fro ths figurs it can b sn that th prsnt nurical calculations show ovrall good agrnt with th xprintal rsults. Fig. 7 and Fig. 8 show hav and pitch rsponss for D/L=.4 at Fn=.. Hav rspons shows a diffrnt charactristic than that of xprintal ons but Pitch otion rspons shows a vry good agrnt with xprintal rsults. Fig. 9 and Fig. show hav and pitch rsponss for D/L=.4 at Fn= (L/g) / Fig. 4. Pitch transfr function for 4b cataaran at Fn=. and D/L=.. = Exp.(Southapton) (L/g) / Fig. 5 Hav transfr function for 4b cataaran at Fn= and D/L=.. =.. = Exp.(Southapton) (L/g) / (L/g) / Fig. 3 Hav transfr function for 4b cataaran hull at Fn=. and D/L=. Fig. 6 Pitch transfr function for 4b cataaran at Fn= and D/L=. Sssion A 8

7 at Fn= and D/L=.4. =..5. = (L/g) / (L/g) / Fig. 7. Hav transfr function for 4b cataaran at Fn=. and D/L=.4 Fig. Pitch transfr function for 4b cataaran at Fn= and D/L=.. = (L/g) / Fig. 8 Pitch transfr function for 4b cataaran at Fn=. and D/L=.4.5. = (L/g) / Fig. 9 Hav transfr function for 4b cataaran 5. CONCLUSIONS In this study, th coputations of th hydrodynaic forcs and otion rsponss for high spd crafts in wavs hav bn prsntd. A coputr cod of YNU- SEA has bn dvlopd by 3-D Grn function with forward spd. Th nurical calculations for otion rsponss of transo strn hull vssls in ono and twin hull configurations advancing in wavs hav bn carrid out by using th YNU-SEA. Th rsults of prsnt calculations ar copard with th xprintal ons. Th ovrall agrnt btwn xprintal rsults with prsnt nurical rsults by 3-D Grn function thod was favorabl. Nurical rsults rval that th prsnt 3- D Grn function thod of classic forulations can b usd as a practical tool to prdict th radiation and diffraction forcs and th wav inducd otions for th high spd ulti hull floating bodis advancing in rgular wavs. REFERENCES. BLOK, J. and BEUKELMAN, W., Th High Spd Displacnt Ship Systatic Sris Hull Fors- Sakping Charactristics, Trans. Of SNAME, Vol. 9, GRIGOROPOULOS, G. J. and LOUKAKIS, T. A., Sakping Prforanc of a Systatic Sris of Fast Monohulls, RINA Intrnational Confrnc on High Spd Craft Motions and Manuvrability, London, UK, BOJOVIC, P. and SAHOO, P., A Study of Motion Analysis of High Spd Displacnt Hull Fors, Procdings of 7 th Intrnational Syposiu on Practical Dsign of Ships and Mobil Units (PRADS 98), Th Hagu, Holland, INCECIK, A., MORRISON, B.F. and RODGERS, A. J. Exprintal Invstigation Of Rsistanc and Sakping Charactristics of a Cataaran Dsign, Sssion A 8

8 Proc. of th st Intrnational confrnc on Fast Sa Transportation (FAST 9), Trondhi, Norway, Matsui, S., SHAO, S.M., WANG, Y.C., and TANAKA, K., Th Exprintal Invstigations on Rsistanc and Sakping Qualitis of High Spd Cataarans, Proc. of th nd Intrnational confrnc on Fast Sa Transportation (FAST 93), Yokohaa, Japan, HUDSON, D., MOLLAND, A., PRICE, W.G., and TEMARAL, P., Sakping Prforanc of High Spd Cataaran Vssls in Had and Obliqu Wavs, Procdings of th 6th Intrnational confrnc on Fast Sa Transportation (FAST ), Southapton,UK,. 7. MOLLAND, A., WELLICOME, J.F.,TEMAREL, Cic J. and TAUNTON, D. J., Exprintal Invstigation Of th Sakping Charactristics of Fast Displacnt Cataarans in Had and Obliqu Wavs, Transactions of RINA,. 8. BRAD, R., Th Rprsntation of a Givn Ship For by Singularity Distribution Whn th Boundary Condition of th Fr Surfac is Linarizd, Journal of Ship Rsarch, WEHAUSEN, J.V. and LAITONE, E.V., Surfac Wavs, Hundbuck dr Physic, Vol. 9, Springr, 96.. INGLIS, R.B. and PRICE, W.G., Calculation of th Vlocity Potntial of a Translating and Pulsating Sourc Transactions of.rina, 98. Sssion A 8