E Lab.. Sh,essbu&kunde Technic. r. WAVE RESISTANCE AND LIFT ON CYLINDERS BY A COUPLED ELEMENT TECHNIQUE R. Eatck Taylr and G.X. Wu Lndn Centre fr Marine Technlgy Department f Mechanical Engineering University Cllege Lndn Trringtn Place Lndn WC1E 7JE July 1985
CONTENTS Intrductin i Gverning Equatins 4 Numerical Prcedure 6 3.1 Galerkin Methd in the Near Field Regin 6 3.2 Green Functin 8 3.3 Discretisatin f the far field integrai equatin I Discussin f Results... 13 Cnclusin........ References...16
-1- ABS TRACT This paper presents a new applicatin f ne f the mst prmising methds in ship hydrdynamics t free surface flw prblems. It cncerns a bdy having unifrm frward speed in an incmpressible and inviscid fluid. The methd cmbines lcalised finite elements in the near field with representatin by a bundary integral equatin in the far field. The free surface cnditin is linearised but the bdy surface bundary cnditin is satisfied exactly. Calculatins fr different submerged cylinders are made, as a means f assessing the effectiveness f the new apprach. Agreement with existing results is excellent, and it is cncluded that the methd is suited t further develpment fr the case f an arbitrary bdy prgressing in waves. 1. Intrductin Accurate estimatin f the resistance f a ship, prir t cnstructin, is f vital imprtance in design. As part f this prcess, based n the classical separatin f the ttal frce int viscus drag effects and wave resistance, it is desirable t have efficient numerical techniques fr predicting the flw caused by a ship mving in an inviscid and incmpressible fluid. The prblem may cnveniently be psed in terms f a velcity ptential functin satisfying Laplace's equatin and apprpriate bundary cnditins. Unfrtunately, hwever, the resulting mathematical prblem is nt easy t slve, as a result f the awkward nn-linear free surface cnditin at an unknwn bundary and the cnditins at infinity. Sme f the difficulties may be avided by satisfying a linearised bundary cnditin at the mean psitin f the free surface, and using an artificial radiatin cnditin. The resulting s-called Neumann-Kelvin (N-K) prblem, with which this paper is cncerned, has been shwn <17> t be a useful representatin fr certain cases, such
-2- as the thin surface ship and deeply submerged bdies. While the physical prblem is three dimensinal, it is valuable t cnsider first the tw dimensinal case. This enables theretical techniques t be assessed and numerical methds investigated at a simpler level than fr the fully three dimensinal case. The first wrk n the tw dimensinal N-K prblem was cmpleted by Havelck in 1936 <10>. In that paper, cited by almst every fllwing authr, Havelck expanded the ptential int a series satisfying the Laplace equatin, the free surface cnditin and the radiatin cnditin. The unknwn cefficients in the series were fund by impsing the bdy surface cnditin. Althugh the methd was limited t the case f a submerged circular cylinder, its great cntributin t the N-K prblem is undeniable. - The prblem f a cylinder f arbitrary shape mving belw the free surface remained unslved until abut thirty years later. With the help f a cmputer, Giesing and Smith <8> were able t btain numerical slutins based n the traditinal representatin by a surce distributin ver the bdy. The surface f the bdy was divided int many segments and n each segment the strength f the surce was assumed cnstant. By invking the bdy surface cnditin the surce strength culd be fund. In this way slutins were btained fr the N-K prblem f an arbitrary cylinder, bth with and withut circulatin. Applicatin f this methd t the multiple-bdy prblem was als made. Quite similarly, Chang and Pien <3> used diple instead f surce distributins ver the bdy. The advantage claimed fr this methd is that the ptential can be fund frm the diple strength withut any further cmplicated integratin. Results were btained fr submerged bdies, and srne attempt was made t attack the prblem f a flating cylinder. Abut a decade ag, anther pwerful numerical methd in engineering, namely the finite element methd, was intrduced t ship hydrdynamics. This was first applied t the harmnic mtin prblem <2>. T avid the difficulty f the infinite dmain in this methd,
3 the fluid was divided int tw regins: the near field in which finite elements were used; and the far field in which series r integral equatin representatins were adpted. This methd was later applied t the N-K prblem by Bai <1>, Mei and Chan <13>, and (in a smewhat different frm) by Yeung and Buger <21>. In these latter papers a series slutin was always used t represent the far field, althugh the details f the methds differ. While there is n dubt abut the effectiveness f the series slutin, ne ntes hwever that it is difficult t find a series fr the three dimensinal N-K prblem in the general case f a bdy mving in waves. Furthermre, even fr the tw dimensinal N-K prblem, an integral equatin representatin in the far field may be mre effective fr deep water-and fr multiple-bdy prblems, as has been shwn in the case f harmnic mtin by Eatck Taylr and Zietsman <6>. This is cnfirmed in the present investigatin. An imprtant purpse f this wrk is als t establish a sund mathematical structure fr the prblem f a bdy prgressing in waves. This is usually slved by simplified tw dimensinal strip theries, which have many inherent limitatins. In mre recent years, linearised three dimensinal thery has been used t slve this prblem, as shwn by Chang <4>, Guevel and Bugis <9>, Inglis and Price <11>, Kbayashi <12>. These all used cnventinal methds f singularity distributins ver the bdy surface and waterline. It has been fund, hwever, that the numerical calculatins required by these methds are nt easy, and the cmputatins are very expensive. In the present wrk we investigate the ptential advantages f an alternative methd by first cnsidering the N-K prblem. The fllwing sectins suimnarise the theretical apprach. Results are then given fr a circular cylinder in infinite and finite water depths, fr an elliptical cylinder at an angle f attack, and fr a multiple-bdy prblem. It is cncluded that the mathematical prcedures are very simple while the
4 results are excellent. 2. Gverning Equatins We cnsider the prblem f a submerged bdy mving thrugh a fluid with a free surface at cnstant speed U. As shwn in figure 1, a crdinate system Oxy is chsen with x-axis pinting in the directin f frward speed, y-axis pinting vertically upwards and the rigin lcated n the undisturbed free surface. Based n the assumptins made fr N-K prblem, the velcity ptential satisfies Laplace's equatin \L4j C) - (t) z7 - in the whle fluid dmain; the kinematic cnditin n the bdy surface S0 =UR where nx is the cmpnent in the x directin f the unit inward nrmal t the bdy; the bttm cnditin fr finite water f depth d; r '/=-cl (3-Q) (3-b) fr infinite water depth; and the free surface cnditin where y- (4) _)= /'U2.
-5- and g is the acceleratin due t gravity. T cmplete the specificatin f this prblem, an apprpriate radiatin cnditin has t be impsed. A generally accepted cnditin fllws frm the assumptin that there is n wave ahead f the bdy, r the fluid far ahead the bdy is undisturbed. But the cnditin at x = - depends n whether the flw is subcritical r supercritical, defined by Frude number F =U/Jcd<(and Fl respectively. In the subcritical case it is assumed that there is a wave behind the bdy, whereas in supercritical flw there is nt. In mathematical frm, the cnditins upstream may be written as - (S - cz) Dwnstream the cnditin is X> - - b) fr F,<l, where w(x,y) crrespnds t a wave scillating with X; and =:.Ç C. 5-c) fr Fl. As already nted, hwever, this radiatin cnditin is artificial. Indeed we bserve that based n this cnditin the resistance n the bdy is zer when Fi. This is nt the case in practice, althugh experiments d cnfirm that there is a large drp f resistance at the critical pint F=l. Once a slutin has been fund fr the velcity ptential, the pressure distributin n the bdy (r anywhere else) can be btained frm the well knwn Bernulli equatin P=- ) (6)
-6 after neglecting the static buyancy cntributin. (It shuld be emphasised here that althugh higher rder terms n the free surface can be neglected when the bdy is deeply submerged, they must be retained in this equatin in the regin near t the bdy.) The resistance R and lift L n the bdy can then be fund by integratin ver its surface. Thus R = ds L = p ds (Thy) 3. Numerical Prcedure 3.1 Galerkin Methd in the Near Field Regin It seems that rigrus thery ften lags behind its practical applicatin. Fr this prblem, uniqueness and existence have nt yet been prved in a general sense <5>, <14>, <15>. Fr a given prblem, hwever, if ne and nly ne slutin can be fund, the result itself suggests heuristic evidence fr that particular case; but ne may nt generalise frm this cnclusin t ther prblems. Based n this philsphy, we nly try t find here the slutin fr specific prblems, rather than attempting t establish frmally their existance and uniqueness. T suit ur numerical prcedure we express the gverning equatins f the previus sectin in a mre cnvenient frni. We make use f ideas which have been successfully implemented fr the harmnic mtin prblem <6>. The fluid dmain is divided int tw parts, the near field Rl and the far field R2, and the crrespnding ptential functins are taken t be and 4'. satisfies equatins (1), (2), (3) and (4)
-7- and satisfies equatins (1), (3), (4) and (5). T ensure cntinuity f the pential and velcity thrughut the fluid dmain, the fllwing cnditins are impsed n the bundary S. separating Rl frm R2: c 4 -- - b) where n pints ut f the apprpriate dmain. In the near field, a finite element representatin is used, and the ptential is expressed by means f shape functins Nt, r ç) where is the value f the ptential n the element ndes and is the number f ndes. Unlike the harmnic mtin prblem, the differential equatin here is nt btainable frm a variatinal statement. Instead we use the weak frm r Galerkin methd t btain ur apprximatin: fç 2Z =0 (1.0) Frm Green's first identity, and the bundary cnditin n S, we btain - Ñds CtI) prvided that the bundary 1S1 is fully submerged. We determine n S frm equatin (8), and an integral equatin in R2 relates t its gradient.
-8 3.2 Green Functin The Green functin G(x,y,a,b) emplyed in the integral equatin n S can be defined as the ptential due t a surce mving at cnstant speed. It is the slutin f the fllwing equatins: + 4. =0 =0, <#QO - where is the Dirac delta functin and the pint (a,b) is assumed t lie belw the free surface. Frm these equatins ne may find the slutin (except fr an arbitrary additive cnstant) <19> as G=+'.)+ PvJJ - csk(b+d) E4 1.) h 0cb-c.) i) - + -ticc.d) Sntç(.x-.) where r= [)2+ = ( 4 b)1] (t C) (1)
-9- k0 is the slutin f ) sr d - = and is nly nn-zer if F<l. The last term in equatin (16) is assciated with the singularity at k = k0 in the integrand, which shuld be deleted when Fril. Numerically it is nt dificult t evaluate this Green functin fr any given pints P(x,y) and Q(a,b), apart frm sme tedius effrt in dealing with the singularity. Here we adpta similar prcedure t that used in the harmnic mtin prblem <6>. We write G =(--)* & + 2&- ZIt 1) d) i+ct). (t ) where G - Çcp (Z-c) (L)= J COS ca-4) -cu (}A) jo rì4jaípi (ZO-c) L=1 J (ZU - ci.) çze)
- l - In the case f infinite water depth, z. vç C(1 ) we have Sicl'.)(XcA) (.20 = And in a manner similar t equatin (19), G* may be written as * ç 2 2..) where v0 ç2q k- = (L$-k) Q- )-r(k- çv) u) *) k cl\< (23-O.) (23) (23-c) (23-d) 3.3 Discretisatin f the far field integral equatin We use the integral equatin r GJds e (24) where S=S_+S +S +S. This is btained frm Green's secnd identity. Using the cnditins fr G and c n 2 these bundaries, we derive the fllwing frm:
ì cp«, ZL'X 2 'x 2S) As discussed by Ursell <18>, the cntributin f the last tw terms in this equatin is at mst an additive cnstant, which can be neglected withut lsing generality. Thus z ç z When P is n the bundary, written as equatin (26) may be (21) where is the subtended angle cnditin (8) we have at pint P. Using (2g) The minus sign here arises frm the change in directin f the nrmal. We nw apprximateby means f shape functins M.: c2)
- 12 - where is the number f ndes n the bundary Equatin (28) may then be written ac-..= c. 3) "J where.- is the Krnecker delta functin. In matrix frm, we have (3 i) Thus, we btain = (32) with 3) Cmbining equatins (il) and (32) and fllwing the prcedure in <6>, we finally btain W 34-) where ALt is a symmetric matrix with cefficients - = * d cl )
- 13 - is the square matrix with cefficients ç1 = K % (3 ) is the clumn matrix cmpsed f the unknwn cefficients.; and P is the clumn matrix with cefficients L p.= nds. *% J? A particular advantage f this methd is that the bundary S. can be chsen arbitrarily. This makes it pssible t evaluate the surface integral explicitly. It is cnvenient t specify a rectangular bundary fr S., and then the main task in evaluating the surface integrals f equatin (30) is the calculatin f XYGcb t- ç3-c) n a hrizntal part f bundary S._; and n a vertical part f bundary Sa.. Here p = 0,1,2 fr the quadratic shape functins used t btain the results belw. Evaluatin f the varius matrices is, therefre, straightfrward. 4. Discussin f Results T shw the applicability f this methd t a variety f prblems, results fr several cases are given belw. The first is a mving circular cylinder f radius a submerged at a depth h = 2a belw the free surface in water f infinite depth. This prblem has been slved
- 14 - by Havelck analytically. T demnstrate the cnvergence f the present methd, a typical mesh f 12 elements is shwn in figure 2, and table 1 gives results fr several meshes frm carse (8 elements) t fine (24 elements). The resistance R and lift L have been nn-dimensinalised by dividing the crrespnding quantities by vga, where A is the area f the circle. Frm this table it is clear that the results cnverge very rapidly. A cmparisn f these results with Havelck's analytical slutin <10> is given in figure 3, and they are seen t be indistinguishable at the scale pltted. The influence f water depth d n the resistance and lift f a circular cylinder is shwn in figure 4, based n results f the present methd. It can be seen that this influence starts t becme imprtant when d<la. The resistance is then significantly increased by the reductin f water depth. The physical explanatin fr this is that when the gap between the cylinder and bttm becmes smaller, the flw in the vertical directin is prgressively blcked. Therefre the nly way fr fluid t pass by the bdy is fr its hrizntal speed t increase. As a cnsequence, the wave amplitude will be increased, as given by the fllwing equatin: It is bvius that the bdy will suffer larger resistance when the wave amplitude is larger. Results fr resistance and lift n tw elliptical cylinders are cnsidered next. The ratis f minr axis b t majr axis a were 0.5 and 0.125 respectively. In each case the centre f the cylinder is lcated ne bdy length belw the free surface (i.e. h = 2a), with the majr axis tilted 100 upwards frm the hcrizntal. The results shwn in figure 5 are in very gd agreement with thse f Yeung and Buger <21>, bth in subcritical and supercritical ranges f Frude number.
- 15 - T illustrate the flexibility f this methd, we have als calculated results fr a multiple-bdy prblem. The gemetry fr the tw submerged circular cylinders is shwn in figure 6, while the results are given in figure 7. These are based n a mesh cmprising tw grups f elements similar t the grup shwn in figure 2. The results are nn-dimensinalised in terms f the area A f ne cylinder. It can be seen that the effect f the secnd bdy n the first is t reduce bth resistance and lift n the upstream bdy. There is a regin f high pressure in frnt f a mving bdy and a regin f lw pressure behind it. The resistance is due t their difference. Therefre when the tw bdies mve tgether, the lw pressure behind the upstream bdy is increased by the high pressure in frnt f the dwnstream bdy, thus causing a reductin f the resistance n the upstream bdy. It is fund that the resistance and lift n the dwnstream bdy scillates with the Frude Number; the resistance can even be negative. This can be understd if ne realises that there is always a wave behind a mving bdy, s that the dwnstream bdy is mving in the wave generated by the upstream bdy. On the ther hand, it is interesting t nte that the ttal frce n tw cylinders is never negative. 5. Cnclusin In this paper, it has been shwn hw the Neumann-Kelvin prblem may be slved by a very effective methd. Results frm several test cases are in excellent agreement with previusly published results btained by a variety f ther means. The present wrk als prvides a mathematical basis fr mre general cases. The extensin t the three dimensinal prblem is in principle straightfrward, if the three dimensinal Green functin and a three dimensinal mesh are used. Furthermre this methd appears suited t develpment fr the case f a mving bdy in waves. Indeed the advantages f this methd may then becme mre bvius, since it shuld nt be necessary t integrate the secnd derivatives f the Green functin ver the bdy
- 16 - surface, and the line integral can be calculated explicitly. These tw aspects have hithert been difficult t deal with. References Bai, K.J. A lcalised finite element methd fr steady tw-dimensinal free-surface flw prblems. First mt. Cnf. Nun. Ship Hydrdyn. 209-230, (1975). Bai, K.J. and Yeung, R. W. Numerical slutins t free-surface flw prblems. Tenth Syrup. Naval. Hydrdyn. 609-633, (1974). Chang, M. S. and Pien, P. C. Hydrdynamic frce n a bdy mving beneath a free surface. First mt. Cnf. Nun. Ship Hydrdyn. 539-559, (1975). Chang, M. S. Cmputatins f three-dimensinal ship-mtins with frward speed. Secnd mt. Cnf. Nun. Hydrdyn. 124-135, (1977). Dem, J. C. Existence, uniqueness and regularity f the slutin f the Neumann-Kelvin prblem fr tw r three dimensinal submerged bdies. Secnd. mt. Cnf. Nun. Ship Hydrdyn. 57-77, (1977). Eatck Taylr, R. and Zietsman, J. A cmparisn f lcalised finite element frmulatins fr the tw dimensinal wave diffractin and radiatin prblem. mt. J. Num. Meth. Eng. 19, 1355-1384, (1981). Euvrard, D., Jami, A., Lenir, M. and Martin, D. Recent prgress twards an ptimal cupling f finite elements and surce distributin prcedures. Third mt. Cnf. Num. Ship Hydrdyn. 193-219, (1981). Giesing, J. P. and Smith, A. M. O. Ptential flw abut tw-dimensinal hydrfils. J. Fluid Mech. 28, 113-129, (1967) Guevel, P. and Bugis, J. Ship-mtins with frward speed in infinite depth. mt. Shipbuilding Prg. 29, 103-117, (1982)
- 17 - Havelck, T. H. The frce n a circular cylinder submerged in a unifrm stream. Prc. Ry. Sc. Lndn, A 137, 526-534, (1936). Inglis, R. B. and Price, W. G. A three dimensinal ship mtin thery - cmparisn between theretical predictins and experimental data f the hydrdynamic cefficients with frward speed. Trans. R.I.N.A. 124, 141-157, (1981) Kbayashi, M. On the hydrdynamic frces and mments acting n an arbitrary flating bdy with a cnstant frward speed. J.S.N.A. Japan 150, 61-72, (1981) Nei, C. C. and Chan, H. S. A hybrid methd fr steady linearised free-surface flw. mt. J. Num. Meth. in Eng. 10, 1153-1175, (1976) Newman, J. N. The thery f ship mtins. Adv. Appl. Mech. 18, 221-283, (1978). Simn M. J. and tfrsell, F. Uniqueness in the linearised tw-dimensinal water-wave-prblem. J. Fluid. Mech. 148, 137-154, (1984) Susuki, K. Numerical studies f the Neumann- Kelvin prblem fr a tw dimensinal semi-submerged bdy. Third Int. Cnf. Num. Ship Hydrdyn. 193-219, (1981) Tuck, E. O. The effect f nn-linearity at the free surface n flw past a submerged cylinder. J. Fluid Mech. 22, 401-414, (1965) tjrsell, F. Mathematical ntes n the twdimensinal Kelvin-Neumann prblem. Thirteenth Symp. Naval. Hydrdyn. Vl. 2, (1980). Wehausen, J. V. and Laitne, E. V. Surface waves in Handbuch der Physik, Vl. 9, 446-778. Springer, Berlin, (1960) Wehausen, J. V. The wave resistance f ships.
- 18 - Adv. Appi. Mech. 13, 93-245, (1973). 21. Yeung, R. W. and Buger, Y. C. Hybrid integral equatin methd fr steady ship-wave prblem. Secnd mt. Cnf. Num. Hydrdyn., 160-175, (1977).
List f figures and tables. Figure 1. Definitin f gemetry and fluid regins. Figure 2. 12 element mesh fr submerged circular cylinder. Figure 3. Cmparisn f resistance and lift with Havelck's analytical slutin fr a circular cylinder < 10 >. Figure 4. Frces n a circular cylinder in water f different depths: Infinite water depth; --s-- d = loa --V-- d = 7a; --+-- d = 4.Sa;. --E-- d = 4a. (a) Resistance; (b) Lift. Figure 5. Cmparisn f frces n submerged elliptical cylinders at 100 angle f attack. Yeung and Buger < 21 >, b/a = 0.5; Yeurig and Buger < 21 >, b/a = 0.125; Present methd. (a) Resistance; (b) Lift. Figure 6. Gemetry fr multi-bdy prblem. Figure 7. Cmparisn f frces n single and multiple bdies: single bdy; /1 upstream bdy; dwnstream bdy. (a) Resistance; (b) Lift. Table 1. Cnvergence f results fr a submerged circular cylinder (h = 2a).
S F X SJ R1 R2 SB \'\'ÇY / //)\\\\ Figure 1. Definitin f gemetry and fluid regins.
Figure 2. 12 element mesh r submerged circular cylinder.
4 O) 0 5 8 9 FROUDE NUMBER FnU/(gh)5 RESULTS BY PRESENT METHOD RESISTANCE BY HAVELOCK'S METHOD LIFT BY HAVELOCK'S METHOD Figure 3. Cmparisn f resistance and lift with Havelck's analytical slutin fr a circular cylinder < 10 >.
H 12 18 XI0 3 X / X X I' i /,/' E9 a. ///,,, / ---i----- -3 ii 5 G 7 8 x10-i FROUDE NUMBER Fn=U/(gh)e5 FROUDE NUMBER Fn=U/(gh)'5 6 7 8910 (a) Resistance (b) Lift. Figure 4. Frces n a circular cylinder in water f different depths. Infinite water depth; --A--- d loa --Y--- d = 7a; --+-- d 4.5a; ---{-- d 4a. X10
x1@-1 5 X10 2 1. / 'S 'S / -J -2 FROUDE NUMBER FriU/(gd)05-5 8 10 0 5 IO IS X IO-i FROUDE NUMBER Fn=U/(gd)B5 (a) Resistance (b) Lift. Figure 5. Cmparisn f frces n submerged elliptical cylinders at 100 angle f attack. Yeung and Buger < 21 >, b/a = 0.5; Yeung and Buger < 21 >, b/a 0.125; Present methd. X I-1 20
h 2a k 4a Figure 6. Gemetry fr multi-bdy prblem.
-2 X10 0 0 I A 0/ A A AA J A A U 000 C) A A 5 7 8 9 10 FROUDE NUMBER Fn=U/(gh)05 xi0-i -2 5 6 7 8 9 XIOH FROUDE NUMBER Ffl=U/(9h)e5 (a) Resistance, (b) Lift. Figure 7. Cmparisn f frces n single and multiple bdies: single bdy; A upstream bdy dwnstream bdy.
NUMBER OF ELEMENTS 8 12 16 20 24 0.5 0.02507 0.02719 0.02739 O.027'44 0.027)45 g 0.6 O.14296 0.15049 0.15108 0.15118 0.15121 0.7 0.30639 0.31989 0.32085 0.32102 0.32113 0.8 0.40383 0.41939 0.42034 0.42045 0.42077 0.9 0.42308 0.43737 0.43853 0.43849 0.43737 1.0 0.39846 0.41730 0.41121 0.41130 0.41129 1.1. 0.35673 0.36653 0.36712 0.36715 0.36722 0.5 0.11729 0.11607 0.11595 0.11614 0.11615 0.6 0.20365 0.20087 0.20103 0.20209 0.20112 0.7 0.17889 0.17842 0.17795 0.17781 0.17758 0.8 0.05664 0.05993 0.05925 0.05906 0.05854 0.9-0.08010-0.07416-0.07553-0.07541-0.07389 1.0-0.19378-0.18669-0.18738-0.18777-0.18790 1.1-0.28109-0.27325-0.27383-0.27465-0.27480 Table 1 Cnvergence f results fr a submerged circular cylinder (h2a)