THE CROSS-FLOW DRAG ON A MANOEUVRING SHIP. J. P. HOOFf. MARIN, Wageningen, The Netherlands

Transcription

1 f!j Prgamon Ocan Engng, Vol. 21, No. 3, pp , 1994 Elsvir Scinc Ltd Printd in Grat Britain $ THE CROSS-FLOW DRAG ON A MANOEUVRNG SHP J. P. HOOFf MARN, Wagningn, Th Nthrlands Abstract-n this papr, an analysis is givn of th xprimntally drivd local latral forc on a manouvring ship as a raction to th ship's latral vlocity. Th tsts wr prformd with modls consisting of svral sgmnts. Spcial attntion is paid to th longitudinal distribution of th non-linar componnt of th latral forc, th so-calld cross-flow drag. This aspct is of utmost importanc whn th non-linar contribution bcoms dominant as will occur in a tight turn during which th ship's drift vlocity bcoms rlativly larg compard to th ahad spd. NOMENCLATURE AP aft prpndicular lift cofficint Cd drag cofficint C cd corrctd drag cofficint Fn Froud numbr = U(g*Lpp )0.5 FP forward prpndicular G ship's cntr of gravity g acclration du to gravity L lift on th ship = forc prpndicular to th rsultant spd U of th ship Lpp ship's lngth n lngth of th nth sgmnt my addd mass in latral dirction pr unit lngth r ship's rat of turning T ship's draught 'n draught of th nth sgmnt u ship's spd componnt in th longitudinal dirction U rsultant spd of ship =-{u 2 + V 2)0.5 v ship's spd componnt in th latral dirction x longitudinal dirction in th ship fixd systm of coordinats x longitudinal dirction in th Earth fixd systm of coordinats y latral dirction in th Earth fixd systm of coordinats Y latral forc to th ship N yawing momnt around G No yawing momnt around th midship sction drift angl = atn(v/u) P mass dnsity of th watr. 1. NTRODUCTON FOR THE prdiction of th ship's manouvrability, us can b mad of computr simulations. For this approach mathmatical modls must b availabl by which th rlvant hydrodynamic forcs on th manouvring ship ar dscribd. Oftn ths mathmatical dscriptions of th hydrodynamic phnomna hav bn drivd from a rgrssion analysis of th rsults of captiv modl tsts for th ship considrd [s Hooft (1986)]. Howvr, such mathmatical dscriptions ar not always 329

2 330 J. P. HOOFT satisfactory bcaus of th lack of a physical maning of th dscriptions. This imprfction may lad to th following problms: Th impossibility of comparing th hydrodynamic cofficints of two diffrnt ships. This problm may b srious whn dsigning th ship for optimum manouvrability. A limitd accuracy of th manouvring prdictions bcaus of th fact that th simulation rsults will dpnd on th accuracy by which th cofficints hav bn drivd by th rgrssion analysis. For th prdiction of th ship's manouvrability in th initial dsign stag, no modl tst rsults ar usually availabl for th dtrmination of th hydrodynamic cofficints. On can thn us computr simulations only if th hydrodynamic cofficints can b stablishd othrwis. Nowadays it is not yt possibl to dtrmin th hydrodynamic manouvring cofficints by mans of thortical mthods. Thrfor, mpirical mthods hav bn dvlopd by mans of which, in th initial dsign stag, th hydrodynamic cofficints can b stimatd as a function of th ship's main dimnsions; s, for xampl, nou t al. (1981) and Kijima t al. (1990). n mpirical mthods th linar hydrodynamic cofficints ar dscribd rathr accuratly as a function of only a fw aspcts of th ship's dimnsions. Two rasons can b givn for th achivd accuracy: 1. Th linar hydrodynamic cofficints ar most likly rathr indpndnt of local paramtrs of th ship's hull form (at th bow and/or th strn). 2. For a wid rang of ships th linar hydrodynamic cofficints hav bn dtrmind xprimntally. This mans that an accptabl lvl of confidnc has bn achivd in th dscription of th cofficints as a function of th ship's main paramtrs. t appars that th non-linar contribution of th hydrodynamic charactristics can b stimatd only roughly by mpirical mthods. This is vry inconvnint whn tight turns hav to b prdictd by mans of computr simulations. n such manouvrs th non-linar contributions play a significant rol. Thr rasons ar mostly givn for this unfavourabl aspct: 1. Th non-linar hydrodynamic cofficints ar rathr snsitiv to th local form paramtrs of th ship. This mans that much mor information (data) is rquird as a function of th largr numbr of paramtrs. 2. Only for a limitd numbr of ships hav non-linar hydrodynamic cofficints bn dtrmind xprimntally. This mans that only a limitd lvl of confidnc has bn achivd in th dscription of th cofficints as a.function of th ship's main paramtrs. 3. Oftn only th hydrodynamic cofficints hav bn publishd without th actual modl tst rsults. Som of th authors us quadratic non-linar cofficints, whil othrs apply trtiary non-linar cofficints. n this way th validity of a prsntd non-linar cofficint is limitd and cannot b compard with th corrsponding cofficint for anothr hull form. t was thought that a bttr dscription of th non-linar componnt of th latral forc could b achivd if on could stablish th local non-linar forc componnt

3 Cross-flow drag on a manouvring ship 331 instad of th total forc on th ship. This local componnt was dfind by th local cross-flow drag cofficint; s, for xampl, Hooft (1987). Burchr (1972), ark (1972), Matsumoto (1983) and Buklman (1988) hav prsntd th rsults of thir xprimnts with sgmntd modls. From ths tsts th latral forcs on ach of th sgmnts can b drivd. n th prsnt papr a dscription is givn of th analysis of tst rsults with sgmntd modls. Basd on this analysis an mpirical mthod can b drivd to stimat th distribution of th non-linar latral forc componnt ovr th ship's lngth. Th intntion of this mthod S: 1. To achiv a mor accurat prdiction of th non-linar componnt of th latral forc. 2. To raliz a mor accurat mathmatical dscription of th hydrodynamic forcs from masurmnts on th total modl. 3. To stablish vntually a simulation program that is capabl of prdicting th ship's manouvrability with sufficint accuracy. Such a manouvring prdiction mthod would provid a mans to invstigat th ffct of th variations of th ship's hull form on its manouvrability. 2. THE VARATON OF THE LATERAL VELOCTY OVER THE LENGTH OF THE SHP Whil considring th ship's manouvring prformanc, thr vlocity componnts can b discrnd at ach instant (s Fig. 1): U, th ahad vlocity of th cntr of gravity G v, th latral vlocity of th cntr of gravity r, th rat of turning around th vrtical axis. Th combination of drift vlocity v and yaw rat r lads to a local latral vlocity v(x) that varis ovr th lngth of th ship: v(x) = v + X*r. (1),! Whn considring a turn by a port (positiv) ruddr angl thn it is sn that th ship will turn to port (ngativ r) at a positiv drift vlocity v. n this turn th local x - y u FG. 1. Dfinition of th ship-fixd systm of coordinats.

4 332 J. P. HOOFT drift vlocity bcoms zro (th turning point) somwhr nar th bow of th ship (x>o), whil at th strn th latral vlocity will bcom quit larg rlativ to th ship's ahad spd. As a consqunc of this rsult it is found that, in gnral, th latral hydrodynamic raction forc at th bow will b small, whil at th strn quit a larg forc will b xprincd. 3. THE LATERAL FORCE AS A REACTON TO THE DRFT VELOCTY Whn considring first th latral hydrodynamic forc on th ship as a raction to th ship's drift vlocity v only (at zro rat of turning), th following dscription is oftn applid: Y(v) = Yuv*u*v + Yvv*v*lvl (2) or, in a non-dimnsional form: Y(3)' = Y*cos(3)*sin(l3) + Y*sin(3)*/sin(3)1 (3) in which th latral forc has bn mad non-dimnsional by dividing by 0.5p Lpp 2U2 or 0.5p LppTUZ with th total vlocity U dfind by: U (u 2 + v 2)o.s. (4) Th drift angl 13 is, dfind by: f3 = arctan(vlu). (5) Th total latral forc Y(v) is th rsultant of all latral forcs, Y n ( v) on ach of th sgmnts in which th ship modl has bn subdividd: whil th total yawing momnt around th cntr of gravity amounts to: (6) N(v) = 2: (x n* YnCv)). (7) For furthr analysis th local latral forc on ach sgmnt is writtn in a nondimnsional form according to: with n th lngth of th sgmnt and tn its avrag draught; s, for xampl, Fig. 2. According to Equation (3) on now dscribs th non-dimnsional local latralforc Yn( l3 )' by mans of a linar cofficint CYn and a non-linar cofficint Cd.: in which th linar cofficint Cyn corrsponds to th drivativ Y and th drag cofficint Cd; corrsponds to - Y. t can b shown that in Equation (2) th linar trm is linarly dpndnt on th longitudinal vlocity componnt u and not on th total vlocity U. This mans that for zro forward spd (u = 0) no linar contribution in th latral forc xists. This (8) (9)

5 ;f..,' Cross-flow drag on a manouvring ship 333 Todd 70 modl; Lpp/T = 17.5; zro trim. 5th sgmnt from th bow; Fn = ' FG. 2. Exampl of th masurd latral forc on a sgmnt as a function of th drift angl 13; obtaind from Buklman (1988). holds not only for th latral forc on th total ship but also for th local latral forc on ach of th sgmnts of a ship modl.'. f on considrs th lift L to b a latral forc that is prpndicular to th total undisturbd incoming flow U, thn th cofficint Cyn is rlatd to th local lift cofficint 13 (n) as follows:. nsrting this rsult into Equation (9) lads to: n th application of Equation (11), us is mad of th assumption that th lift cofficint 13 (n) is indpndnt of th drift angl {3. This mans that it is accptd that th local cross-flow drag cofficint is a function of th drift angl (3. Various physical argumnts can b givn to prov this assumption to b corrct. t should b born in mind that both th lift cofficint and th cross-flow drag cofficint vary ovr th lngth of th ship. From th tst rsults on ach sgmnt as prsntd in Fig. 2 on dtrmins th local lift cofficint 13 (n) at th rang of small drift angls (3. With this valu of 13 ( n) on dtrmins for ach sgmnt th cross-flow drag cofficint from th masurmnts at highr drift angls: (10) (11) (12) From this quation on finds, in Tabl 1, th valus of th drag cofficints Cd; for th tst rsults in Fig. 2, which wr prsntd by Buklman (1988). 2

6 334 J. P. ROOFT TABLE 1. EXAMPLE OF THE DERVATON OF THE CROSS-FLOW DRAG COEFFCENTS Cd; FROM THE MEASUREMENTS Drift angl 13 (in dgrs) Masurd dimnsionlss latitudinal forc y Drag cofficint Cd; with Cf>, (5) = Todd 70 modl (7 sgmnts); Lpp/T = 17.5; zro trim; 5th sgmnt from th bow; Fn = THE LOCAL LFT COEFFCENT (x) According to Jons (1946), on finds that th local lift pr unit lngth is dtrmind by th instantanous apparnt acclration of th local latral addd mass of watr alongsid th ship: L/; = v*u*my/; with my bing th latral addd mass of watr pr unit lngth of th ship and bing th distanc of th cross-sction from th forward prpndicular. n Equation (13) th drivativs L/; and my/; ar dfind by: TABLE 2. CoMPARSON BETWEEN MEASURED AND CALCULATED LNEAR LATERAL FORCE COMPONENT (N A NON DMENSONAL FORM) ON EACH OF THE SEVEN SEGMENTS OF A MODEL OF THE TODD 70 SERES, NO TRM, Fn = 0.15; SEE BEUKELMAN (1988) Lpp/T = Lpp/T = Lpp/T = (13) L/; = dl/d; my/; = dmy/d. (14) From intgration of L/; in Equation (13) on finds that th lift forc L ovr a distanc btwn f and (with f bing th closst to th FP) amounts to: L = v*u * i/;a /;/ my/; d = v *u * (my{) - my{f)) from which it is sn that in thory th total lift on. th ship will b zro (paradox of d'almbrt) bcaus of th fact that in an idal fluid th addd mass pr unit lngth my is zro at th bow (f = 0) as wll as at th stm ( = Lpp). n Tabls 2 and 3 th local lift cofficint 13 (n) on ach of th svn sgmnts of (15) CLf>, C L f3 CLf>, CLf>, C L f3 C L f3 Sgmnt No. (masurd) (calculatd) (masurd) (calculatd) (masurd) (calculatd)

7 1 4 * Cross-flow drag on a manouvring ship 335 TABLE 3. COMPARSON BETWEEN MEASURED AND CALCULATED LNEAR LATERAL FORCE COMPONENT (N A NON DMENSONAL FORM) ON EACH OF THE SEVEN SEGMENTS OF A MODEL OF THE TODD 70 SERES, Lpp/T = 17.5, Fn = 0.15; SEE BEUKELMAN (1988) Trim = Zro trim Trim = CL!3 CL!3 CL!3 CL!3 CL!3 CL!3 Sgmnt No. (masurd) (calculatd) (masurd) (calculatd) (masurd) (calculatd) th ship modl is givn as drivd from th xprimnts dscribd by Buklman (1988). n ths tabls th local lift cofficints ar also prsntd, drivd from Equation (15). From th comparison btwn th xprimntal and thortical valus in Tabls 2 and 3 it is sn that th thortical valus for th hull form considrd agr wll with th masurmnts for th fiv most forward sgmnts. Th thory by Jons (1946) for dtrmining th local lift forc can also b chckd in anothr way whil applying Equation (15). For this purpos on rducs from th masurmnts th distribution of my() by a summation of th lift drivativ L u v ( n) ovr th sgmnts aftr Equation (15) has bn rwrittn by: with my bing zro at th forward prpndicular. n Figs 3 and 4 th xprimntal valus of my ar prsntd as drivd by mans of Equation (16) using L uv from th modl tst rsults prsntd by Buklman (1988). n Figs 3 and 4 th thortical valus of my ar also plottd. From th comparison of th thortical and xprimntal valus in ths figurs it is sn that th thory should b corrctd for thr-dimnsional ffcts on th valu of my lading to myc: in which rn() is th corrction trm as a function of th longitudinal location. Having dtrmind th corrctd latral "addd mass" distribution myc() it will b possibl to dtrmin th corrctd drivativ myt;c: (16) (17) myt;c() = myct;() = dmyc()/d. (18) ; Applying th valu myt;c in Equation (15) on now finds that th total lift on th ship amounts to: Whn applying th thory of Jons (1946), whil using th corrctd valus of my, (19)

8 " 336 J. P. HooF calculatd LPP/T = masurd LPP/T = ,-,,-., LPP/T = ", ',\ /' " // --.., "'-,,','/ " / \-. J7 " \ /i \\ / / \ ',, ' i '<,,/ \'\ \ j,,-_. \\ 1'., <,/ \,/. / i 1/._..:.:::-..:.._ \ "".....,._. ', / /.'7. 1///../ " "\,. "'" "\., / \c ' <, <, 1/ -i t"-_ ''-. / // "'-. " / ' / <, V t o FG. 3. Comparison of th calculatd latral mass my pr unit lngth with th valus drivd from th xprimnts with sgmntd modls for a Todd 70 hull form without trim at Fn = 0.15; s Buklman (1988). ; on thn finds for th total yawing momnt du to th lift forc distribution ovr th ship's lngth: f;,a PP N(around th midship sction) = v*u* (O,5Lpp -) *mycd

9 ' Cross-flow drag on a manouvring ship 337 calculatd masurd LPP/T = \ \ \ ",- / \ _r--+--" \ / "'-,,/ V/, / \ / C U,,- / ' ">1 " / ", 1'-",', V'\ y', \ \ ' / '1"= \' +3.4' \ / ", ',_/,' / /, \\ // / "'\ \., 'f \' \\ // \\,r/ /=A: \, '\. \, \\ '\, " -, -,,' , f t---- FG. 4. Comparison of th calculatd latral mass my pr unit lngth with th valus drivd from th xprimnts with sgmntd modls for a Todd 70 hull form with LpplT = 17.5 at Fn = 0.15; s Buklman (1988). 0.4 o or No = v*u* [-myc(pp)*lppl2 + l A PP ; myc() d]. (20)

10 J. P. HOOFf if 5. THE LOCAL CROSS-FLOW DRAG COEFFCENT Cd(13J Aftr having dtrmind th lift cofficint 13 (n) on ach sgmnt, on thn stablishs th cross-flow drag cofficint Cd n(r3) on ach sgmnt by mans of Equation (12). Som of th rsults drivd in this way ar prsntd in Fig. 5. n this figur th distribution of Cd; ovr th ship lngth has bn plottd for various drift angls J3. Th following commnts on th rsults in Fig. 5 can b givn: Th cross-flow drag cofficint on th first sgmnt from th bow has most probably bn causd by th bow wav. Thrfor it is assumd that this valu dpnds on th forward spd of th ship; s also th findings by Matsumoto (1983). Asid of th cross-flow drag cofficint on th most forward sgmnt(s) it is sn from th rsults in Fig. 5 that for small drift angls J3 th valu of Cd will incras at largr distancs from th bow until som maximum valu is attaind aftr which Cd will dcras a littl. At incrasing drift angls J3 this curv of Cd ovr th ship's lngth will mov forward. For larg drift angls of approximatly 90 th ship sails narly abam. n this condition th distribution of th cross-flow drag cofficint ovr th ship's lngth mainly dpnds on th form of th ship's hull and to a lssr dgr on th local Rynolds numbr. n Fig. 6 this distribution of Cd is prsntd for a containrship and a tankr; th rsults in this figur follow from Matsumoto (1983). Th valu of Cd() at a givn cross-sction will vary at incrasing drift angls from th valus shown in Fig. 5 to th valus shown in Fig. 6. On now considrs th rlation btwn Cd n(j3 =1= 90 ) at an arbitrary drift angl and CdnCJ3 = 90 ) at a drift angl of 90 for th various ship typs and ship conditions. For this purpos us is mad of th corrctd drag cofficint Ccd n(j3) which is dfind by:. Cd(J3 =1= 90) C c i J3) = Cd(J3 = 90) (21) lading to th fact that at 90 drift angl th cofficint Ccd quals unity ovr th whol lngth of th ship. Combining th rsults in Figs 5 and 6 according to Equation (21) will yild th rsults prsntd in Fig. 7. n Fig. 8 th cours of Ccd ovr th ship lngth is schmatically indicatd for incrasing drift angls. From th rsults of th corrctd drag cofficint CcdnCJ3) in Fig. 7 it is concludd that this cofficint is still dpndnt on th longitudinal location and th drift angl, but is no longr dpndnt on th ship's hull form. 6. CONCLUSONS Exprimnts hav bn carrid out with sgmntd modls during which.th latral forcs on ach of th sgmnts wr masurd. Ths tsts wr prformd as a function of th ship's latral vlocity (in a towing tank) and as a function of th ship's rat of turning (undr a rotating arm facility). n this papr th rsults ar prsntd of th analysis of th xprimnts with sgmntd modls as a function of th ship's latral vlocity. Th sam kind of rsults wr found by analysing th rotating arm tsts as a function of th ship's rat of turning.

11 .' Cross-flow drag on a manouvring ship 339 o LPPT LPP/T = LPPT fj = fj = ẹ ().2-0.5!l FPF L: fj = fj = 20 o() 0.6 n o A 1! 0 1!,, 1!, ẹ APF 0.5 FG. 5. Distribution of th cross-flow drag cofficint ovr th ship's lngth, drivd from th xprimnts with a sgmntd modl for a Todd 70 hull form without trim at Fn = 0.15; s Buklman (1988)., S

12 J. P. HOOFT 1: jj h CONTANER TANKER '. 0.4, O FG. 6. Distribution of th cross-flow drag cofficint ovr th ship's lngth, drivd from th captiv drift tsts with two sgmntd modl at 13 = 90 ; s Matsumoto (1983). Th conclusions obtaind in th prsnt study can b summarizd as follows: Basd on Jons' thory (1946), th distribution of th linar latral forc ovr th ship lngth can b prdictd as a function of th ship's form by mans of mpirical formulations.

13 Cross-flow drag on a manouvring ship 341 LPPT = LPPT = LPPT {3 = 12 1:1 (5 0.6 i.... 1:1 <3 0.6 m (!).. ( a -.. '" m 0.5 t - : (!) 1.4 {3 = 16 1:1 <3 0.6 i '" m.. 1l A 1:1 o 0.6 o m.. - i ẹ. (!) t - 0.& t FG. 7. Distribution of th corrctd cross-flow drag cofficint Ccd ovr th ship's lngth, drivd from th xprimnts with a sgmntd modl for a Todd 70 hull form without trim at Fn = 0.15; s Buklman (1988). mr iliiiiiiliiiiiiiiiiiiiiiiiiiiiiiiil

14 _._ E 342 J. P. Hoosr ""---/ <, / '- /--.., <:«'. \ \ \ / '" \ f3: 90 13: 10 f3:30 \ 13: 5 0; \ 13: 70,. \, \ \ \ -. " -. "- "- "- Ccd FG. 8. Schmatic indication of th forward shift of th distribution of Ccd() at incrasing. Th distribution of th cross-flow drag cofficint can b prdictd for a latral drifting ship (B = 90 ) as a function of th ship's form paramtrs by mans of mpirical formulations. With th knowldg of th drag cofficint for 90 drifting it is possibl to prdict th cross-flow drag cofficint for any drift angl as a function of th ship's form paramtrs by mans of mpirical formulations. Th rsults prsntd in this papr show that it is possibl to dvlop a manouvring prdiction program that is basd on a physical thory rathr than on rgrssion. Such a program lads to mor accurat prdictions of.th ship's manouvrability bcaus th ffcts of local hull form paramtrs ar takn into account. REFERENCES BEUKELMAN, W Longitudinal distribution of drift forcs for a ship modl. Tchnical Univrsity of Dlft, Dpartmnt of Hydronautica, Rport No BURCHER, R. K Dvlopmnts in ship manouvrability. J. R. nst. nav. Archit. 114, CLARKE, D A two-dimnsional strip mthod for surfac ship hull drivativs: comparison of thory with xprimnts on a sgmntd tankr modl. J. mch. Engng Sci. 14, HOOF!, J. P Computr simulations of th bhavior of maritim structurs. Mar. Tchnol. 23, HOOF!, J. P Furthr considrations on mathmatical manouvring modls. RNA ntrnational Confrnc 'On Ship Manouvrability-Prdiction and Achivmnt', London, V.K. NouE, S., HRANO, M., KuMA, K. and TAKASHNA, J A practical calculation mthod of ship manouvring motion. nt. Shipbuilding Prog. 28, JONES, R. T Proprtis of low-aspct ratio pointd wings at spds blow and abov th spd sound, NACA Rport No KuMA, K., NAKR, Y., TANAKA, S. and FURUKAWA, Y On a numrical simulation for prdicting ship manouvring prformanc. 19th TTC, Madrid, Spain. MATSUMOTO, N. and SUEMTSU, K Hydrodynamic forc acting on a hull in manouvring motion. J. Kansai Soc. nav. Archit. 190,