E L S E V I E R
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Wave and current forces on a vertical cylinder free to
surge and sway
S. Malenica
Institut Frangais du Pétrole, BP 311, 92506 Rueil-Malmaison, France
P. J . Clark
Heriot-Watt University, Department of Civil and Offshore Engineering, Riccarton, Edinburgh EH14 4AS, UK êl
B . Molin
Ecole Supérieure d'Ingénieurs de Marseille, Technopdle de Chateau Gombert, 13451 Marseille Cedex 20, France
(Received 14 November 1994; accepted 20 February 1995)
The problem of a vertical circular cylinder submitted to both regular waves and moderate current in water of finite depth is considered. The cylinder is free to move in surge and sway at the frequency of the incoming waves. First-order and steady second-order forces are calculated. In the calculation of mean drift forces and wave drift damping coefficients, both near-field and far-field methods are used and compared with numerical results provided by Grue [Personal communication (1993)], and with those obtained by a simple formula for wave drift damping [Aranha, J. A. P., A formula for wave damping in the drift of a floating body,
/ Fluid Mech, lis (1994) 147-155; Clark, P I , Malenica, S., & Molin, B., An
heuristic approach to wave drift damping, Appl. Ocean Res., 15 (1993) 53-55. The method used for the calculation of the potentials is semi-analytical and based on eigenfunction expansions.
1 I N T R O D U C T I O N
The response of floating bodies to the action of waves and current has been considered i n many works during the past 10 years. There are two main reasons for the i m -portance o f this problem. The first is the need to calcu-late the influence of a moderate current on the radiation-diffraction response of a body; and the second is the pos-sibiHty of calculating the slowly varying drift forces and wave drift damping coefficients which are important in the study of slow drift motion.
A t the onset of these studies, attention was first given to the developement o f general numerical codes for ar-bitrary bodies, and different methods were proposed.'"^ The most difficult quantity to calculate is the so-called wave drift damping coefficient which is defined'' as the derivative of the steady second-order force F with respect to the forward velocity C/ (B = -dfjdU). A l l these meth-ods solve the complete problem for small positive and negative forward velocities, and the wave drift damping is obtained finally by numerical differentiation.
Recently,^'^ a new approach has been proposed. The idea is to introduce, i n the original boundary value prob-lem for a small forward speed, a suppprob-lementary pertur-bation with respect to the forward speed coefficient T ( =
Ucoo/g), which permits the calculation o f the wave drift
damping directly f r o m the solution of the T-order prob-lem. I n Refs [5] and [6], the complete semi-analytical so-lution for the problem o f a fixed vertical cylinder i n finite^ or infinite' water depth has been given.
A surprisingly simple formula for the calculation of wave drift damping''"' was found to give exactly the same results as the complicated theory developed i n Refs [5] and [6]. Since, so far, there is no convincing theoretical explanation for the validity of this formula, i n this pa-per we extend the semi-analytical solution to the more complicated problem o f a cylinder no longer linked to the sea bottom, but free to respond to the incident waves. For the sake of simplicity, only two degrees of freedom (surge and sway) are considered. The wave d r i f t damping calculations are performed by direct pressure integration on the body and also by the far-field method. Results are
also compared with those obtained by a general numeri-cal method.'"
Since the problem of a body advancing with a forward speed i n waves is completely equivalent to the problem of a body in waves and current these two notions will sometimes be interchanged.
2 T H E O R Y
We make use o f the standard assumptions of idealised potential flow f o r wliich the following boundary value problem (B.V.R) can be formulated.
I n the fluid:
A4> = 0 (1) On the free surface:
9 $ .,^^„d^
(2) where H is the exact position of the free surface defined b y :
1 /94> 1
g\dt 2 (3)
On the material boundaries we write the no-flow condi-tion:
( V $ - v ) n = 0 (4)
where v is the velocity o f the boundary.
Finally, one must satisfy an appropriate radiation con-dition at infinity. We recall here the convention utilised throughout this paper, that is that the normal vector n is pointing out of the fluid.
I n this paper, the assumption of small current (forward speed) is adopted and, i n the expressions to follow, only terms of order 0(C/) are retained. The linearisation of the problem is performed by introduction o f the following perturbation series for the potential:
^ iP, t) = (P) + £0"' (P, t) + a'(f)^^> (P) (5)
where E is the perturbation coefficient which corresponds to the wave slope E = koA (ko is the wave number, A the wave amplitude), ï>^'^'> is the steady potential due to the presence of the current, is its first-order unsteady perturbation and 0(2) is the steady second-order potential which must be considered in the calculation of wave d r i f t damping.
Similarly, we obtain quantities at the exact position of the free surface by Taylor series expansion about the mean position z = 0:
/ ( H ) = / ( 0 ) ^ - H ^
z=0 (6)
This leads to the following free surface conditions and free surface elevations. For 0(1)
d<j) (0) = 0
and for 0{E)
dz S'"' = 0 , ( 1 ) (7) (8) dt dz^ 1 (dcf>w dt (9) (10) where V o ( = d/dx, d/dy, 0) is the horizontal gradient and the expressions are to be applied on z = 0.
As we w i l l see later, i n the calculation o f wave drift damping, we need only the value o f the steady second-order potential for the zero-forward-speed case and f r o m now on the notation (^'^^ is used for this potential. So, by repeating the same procedure for t / = 0 we can obtain the following free surface condition for this potential:
3<^
(2)
dz
1 a<^(io) 92^(10)
g dt dz^ (11)
where 0'"" represents the first-order potential without forward speed and the overbar denotes time averaging over one wave period.
We proceed in a similar way with the hnearisation of the body condition:
/ ( P ) = / ( P o ) + ( P ^ - v ) / ( P o ) (12)
where P is the instantaneous position of the point on the body, Po is its mean position and / ( P ) represents either scalar or vector quantities.
Since we only consider translatory motions in the hor-izontal plane, the displacement vector P Q P is the same everywhere:
PoP = § (13)
Similarly, the normal vector at point Po does not change:
np = n (14) where n denotes the normal vector at point Po.
W i t h this in mind, the foUowing conditions on the body can be obtained." For 0 ( 1 )
V<^t'»n= 0 and for 0 ( f ) dt - U (PoP •
v )
(0) (15) (16) these conditions being apphed on the mean surface of the body ,SBO.I n the same way, for the steady second-order potential, the condition b e c o m e s ;
Wave and current forces on a vertical cylinder In
(17) 2.1 Forces
The most obvious way to calculate the forces is by inte-gration of the pressure over the wetted body surface:
pndS SB
(18)
where the pressure p follows f r o m the Bemoulh equation 9$ 1
(19) 2.1.1 First-order forces
First order forces are obtained by keeping only terms of order 0 ( f ) in eqn (18): f f /
F
= - e J J
( 1 ) + C / V ^ ' o ' V ^ ^ M n d ^ (20) dtwhere SBO is the mean wetted surface. 2.1.2 Steady second-order forces
We are interested here only ui the horizontal steady second-order forces. They are calculated by both the near-field and far-field methods.
Direct pressure integration (near-field). By
introduc-ing the pressure expression in eqn (18), and takintroduc-ing ac-count of the fact that the cylinder only performs hori-zontal motions, we obtain:
' = - e J J | ^ ( v 4 , ' " ) ' + c/v0(°'v0">
SBO V ^ + C/V(V<#.'°'V<^(") | n o d 5 2g CBO (I) dt + 2 C / ^ V ^ < ° > V < ^ < " no dC (21)Momentum equation (far-field). We recall here the
classical expression (see Ref. [13], for example), the inte-grals being performed on a vertical cyhnder at infinity:
F =
/ / | [ i K
' ' ) ' " o - V o 0 < > ' ^ '
0 -ƒƒ
rdzde
2g di nor do (22)
I n both expressions, only terms of order 0(.U) in the forward velocity are retained and no = («i, «2, 0) is the horizontal normal vector.
3 P O S I T I O N O F T H E P R O B L E M I N T H E F R E Q U E N C Y D O M A I N
Excitation of the body is caused by regular incident waves with the well-known expression for the potential:
fc^l" (P,/) = !R
I
igA cosh ko (z + H)Wo coshAro^T
^ glto(^cosP-l-^sin^)
(23) with P = (x, y, z) denoting the coordinates o f the point in the body-fixed coordmate system, and p the angle be-tween the wave direction and the x-axis.
So, the body will experience excitation at the frequency
(jde = COo - koU cos/3, which is called the encounter
fre-quency. COo is the fundamental frequency of the incom-ing wave and it satisfies the dispersion relation cog = ^^0 tmhkoH. This permits us to suppose that all quan-tities of order 0 ( f ) are periodic at frequency coe, for ex-ample:
f<^<" (P,t) = %[c]}{F)e-'""''],
§ = l R { ( ? i , 5 2 , 0 ) e - - = ' ' (24) This leads to the following condition on the free surface (z = 0):
-Co2<^-2itt)eC/Vo0""Vo</)
On the cyhnder (r = a) we obtain:
^ = [ - i a ) e § - C / ( § - V ) V 0 « " l n (26)
on <- J
which can be written as''
2
9 ^ dn= -ia)eXly(«,+ ; ^ m ; ) (27)
y=l
V / with («,, «2. 0) = no, ( w i , mj. 0) = - (no • V ) V<^^"' (28) The other parts are not interestmg because we consider only surge and sway first-order motions.S. Malenica et al.
The standard decomposition of the potential is then mtroduced:
(29)
with 01 the incident potential, 4>d the diffracted potential, and 0Ry the radiation potentials corresponding to the first-order motions in surge (gi), and in sway
(^2)-The 01 + 0 d potential and radiation potentials, 0Ry, satisfy the free surface condition, eqn (25), and the f o l -lowing conditions on the body:
90d dn 901^ 90Ry dn ' dn nj + iU -mi We (30) A n d finally we recall here the radiation condition for 0ci and 0Ry which can be found i n Ref. [10]:
^ h (0) cosh^i (9) (z + H) ^,^,f,ui^n(n^^)
Vr coshki{e)H
+ 0 { l / r ) (31)
with kl (9) = kAl + lUw^cosOlgitsinhkeH + k^H
Icosh^keH)]} and wl = gk^tanhk^H.
A similar procedure is followed for the steady
second-order potential 0<2) and the following free surface and body boundary conditions are obtained. O n the free sur-face (z = 0):
90 (2)
dz - 2g^
and on the cylinder (r = a):
, 9 V 9^2 9 0 < 2 ) _ 1 - : 4 - § o | : ) V ( p * „ : 0 A 9 x •dy (32) (33) where (p is the potential of order 0(s) in the case o f zero forward speed, eqn (41), §1 and §2 are the first-order amphtudes of motions in surge and sway for the zero-forward-speed case, eqn (53) and * denotes the complex conjugate.
3.1 First-order forces
I n the frequency domain, eqn (20) becomes:
¥=-Q ( - i 6 ü e 0 + ^ 7 V 0 " " V 0 ) n d 6 • (34)
By introducing the notations:
^ • = -eJ I [ - icOe ( 0 1 + 0d) + U V 0 » ' V ( 0 i + 0 d ) ] « , d 5 (35) and wloij + iw^bij = q {wl4> SBO + itOef7V0<'''V0Ry) Kids' (36) with Fi the excitation force, ay the added mass matrix and bij the damping matrix, and summation by the i n -dex j understood, we can write the equation of dynamic equihbrium o f the body:
-wl [Mij + aij) - iw,bij\ §y = Fi (37)
where Mij is the surge sway mass matrix, f r o m which the amphtudes of motions §y can easily be deduced.
3.2 Steady second-order forces 3.2.1 Near-field method { V 0 V 0 * + 4C/V0^'"V0<2' 2 § [ i a ) e V 0 * + C/V ( V 0 « » V 0 * ) ] } n o d s c o i f g . CBO (^eg no dC (38) 3.2.2 Far-field method V 0 V 0 * n o - 2 V o 0 9 0 * dn dS (39) 4 T H E P E R T U R B A T I O N B Y T = Uwo/g
The idea of introducing a new perturbation about the for-ward speed coefficient, T , which has been used i n Refs [5] and [6], is followed here:
01 = (Ph 0 d = <Pd + T 0 d .
0Ry = CPRy + T(//Ry, gy = + T § j (40) where T = Uwo/g is the ratio between the forward ve-locity and the phase veve-locity for infinite depth.
This leads to the following free surface conditions. For 0(1) - v c p + ^ = 0 az (41) and for 0 ( T ) -v(// + ^ = 2iVo0""Vo(p oz
- 2ko cos fiqp - iqf
9z2 with V = wl/g.
We note here that we can express the stationary poten-tial 0 " " as a sum of the potenpoten-tial o f the u n i f o r m current
-X and its steady perturbation due to the body which, in
the case of a complete vertical cyUnder, is a dipole: (42)
0 ( 0 ) = c f , - x = - ( y + r ) c o s ö
and so the last term i n eqn (42) disappears. On the cylinder we have, f o r 0 ( 1 )
(43) dqp . I V t O - = -ia;o I § > , (44) 7 = 1 and f o r 0 ( T ) dip dn = - i c o o i l U } - ï ' j ^ COS p ) n j 1 (45) The perturbation by T is also introduced i n eqn (29), which gives the foUowing decomposition f o r the poten-tials:
<P = cpi + cpd - io^o X §°<PRy 7 = 1
(/^ = (//d - iwo X (?) - §yT7Cos/3) (pRy (46)
§ > R 7 - (47)
with the following free surface and body boundary con-ditions. For z = 0 -v(//d + ^ = 2 i V o 0 Vo (cpd + <Pi) - 2 i ^ ^ - Ikocospcpd dx -vipRj + = 2 i V o 0 VocpRy oz and f o r r = a 9(Pd dn 9tpRi - 2 i ^ - 2 A : o c o s ^ ( P R y (48) d<Pi ' dn = Kl = - cos 0 9<PR2 an = n2 = - sin 9 a(//d = 0 Sn = 0 dtpR\ dn i = —nil V 2i va di{jR2 dn i = —nj2 V 2i va c o s 2 ö (49) Similarly the free surface elevation, eqn (10), becomes:
S ' " = ! R { r j ( ' > e - ' " ' ' } = lR{(r7<"» + T f j " " ) e - ' " ' = ' } (50) with: , , ( 1 0 ) = H^<p ^(U) = i^^^o r _ ^ ^ ^ ^ ^ ^ _^ i ^ ^ ^ ( o ) ^ ^ ^ g I V V 4.1 First-order problem
The dynamic equation, eqn (37) becomes, f o r 0 ( 1 ) -cog (M,y + 4 ) - ia)o*?y]
= gicoo (cpi + ( p D ) « , d 5 ' and f o r 0 ( T ) = gicoo '/'d + ;^ ( i V 0 ' " ' V - ko cos (51) (52) (53) •SBO X (cpi + <pd) n , d S 0 R
i ( i V 0 < o . V - 2A:ocos^ )<PRy tti dS
2 col^^ cosPMij (54)
where the matrices ajj and b'^j are the added mass and damping matrices for the zero-forward-speed problem and are defined by:
tola^j + itüobl = Qwl qPRj-nidS (55)
•SBO
The solution of these two sets of equations gives the am-plitudes of motion of the cylinder and this terminates the calculation of the potential 0 = qp + Tip.
84
4.2 Steady second-order forces 4.2.2 Far-field method For the steady second-order forces the followmg notation
is introduced:
F = D - C/B (56)
with D denoting the drift force, D = (Du Di), and B the wave d r i f t damping matrix:
B =
Bix B22 (57)
Since we are studying the case of a body advancing with a forward speed in the x-direction, only the value of 5 =
(Bn, B]2) can be calculated. For the other two elements,
we must consider the problem of the body advancing in the ^i-direction. However, i n the axisymmetric single cyhnder case these two problems are very similar and the values of B21 and B22 can be deduced f r o m B\] and B\2. 4.2.1 Near-field method ( Vcp Vcp* + 2itoo§° Vcp*) no dS V <p<p*nodC CDO (58) 2g V ( p V ( / / * - t - 2 - ^ V 0 « * ) v 0 ( 2 ) too
+ icüog** V(//* + icoo ( g ' - ^ cos ^ g " ) V(p^ + ^ g » v ( V 0 ( ' " V ( p * ) l n o d S
too ^ •'J
- vcpip* - ko COS pqpcp*
CBO
i ( p V 0 < ' » V ( p * ] n o d C (59)
w i t h , by the convention used here, no = ( - c o s ö , - s i n Ö, 0) and d/dn = ~d/dr.
The contribution f r o m the steady second-order poten-dal 0<2) is calculated in the way proposed i n Ref. [13]:
V0«')V0'2)no dS
90(2) [.SBO
dn V o 0 d S (60)
[.SBO+SFS
which avoids the explicit calculation o f the potential 0(2)
D = -^% Vogs dcp* dn SBO i v ( p V < p * n o j dS-F ^ qp(p*nodC CBO (61) B 2g f f / „ 9 0 * 9 ( p „ J ( V o c p - ^ + ^ V o 0 * dn dn SBO - V ( p V 0 * n o j dS
+ {vq)ip* - ko cos Pcpqp*) node (62)
with no = (cos 6, sin 6, 0) and 9/9« = 9/9r.
As was pointed out m Ref. [5], this expression has a conservative f o r m f o r B\ \, i.e. i t does not depend on the position of the control surface as long as only the far-field values of the potentials are considered. Therefore, we can apply this expression directly to the cylinder. However,''' for B12 we must add the following mtegral to expression (62). QIÜO 3-^ dcp dcp* dy dx dS 2g ' CBO (63)
where Sp is the mean free surface (z = 0,r> a) and d/ds represents the contour derivative which i n the case of one cyhnder is ( 1 / 0 ( 9 / 9 0 ) .
5 S O L U T I O N F O R T H E P O T E N T I A L S 5.1 Potentials qp
The solutions f o r these potentials are well known and we just recall the expressions here.
cpi = - ï ^ / o ( z ) too X X e'"'('^'2-'')/,„(A:or)e' W10 (64) CPD = ^ / 0 ( Z ) too X X e™(-/2-?)z„,oi/,„(A:or)e'''"' (65)
<PRI <PR2 /o ( z ) BoHx ihr) 00 + Xf"(^)BnKi{k„r) n=l /o ( Z ) ^o^^l ( ^ 0 ^ ) CO + X ƒ « ( ^ ) ^ « ^ 1 ( ^ " ' • ) n=l C O S Ö (66) sin0
with Z„o = j',„{kQa)/H'„,{koa), and:
2Cb V „ 2C„
5 „ =
koH[ (koa) k'o' " k„K[{k„a)kl coshkoiz + H) (67) (68) ƒ « ( z ) = cosh/co/T cosA:,, ( z + / / ) cos k„H (69) Co = u ( z ) d z - 1 C„ = ( z ) d z (70) -H 0 2 ƒ ƒ , ?
and V = ko tanhkoH = -k„ tank„H. 5.2 Potentials i/y
Each potential (ƒ/ satisfies the same kind of B.V.P. and the solution is given for the general case. Consider the foUowing B.YR A(/y = 0, 0 > z > - 7 7 düj - dcp -vqj + ^ = 2iVo(#>Vog9 - 2 i - ^ - Ikocosficp, oz ox 0 dip dn = 0, r = a '^ = 0,z=-H dz (71)
and a radiation condition at infinity to be defined. We note here that the potential cp, in the case of one cyUnder, either for the diffraction or radiation problem, can be written in the form:
00
cp ir, e, z) = fo ( z ) go ir. 0)
+ X
ƒ« ( z ) g„ (^ 6) (72)n=l
Following the method given in Ref. [5], the potential ip is divided into three parts which satisfy the following con-ditions on the free surface and on the body:
-Vlp2 + - V ( / / 3 + dip\ dz djh dr dipi dz dip2 dr dip3 dz dip3 dr dcp
•• -2i^ -Iko cos Pep
V ( z , Ö ) (73)
0
- v ( z , ö ) (74)
2iVo0Vo97
0 (75) As pointed out i n Ref. [5], for the first potential ip\, an
explicit particular solution can be found i n the following form.
( d \
= - 2 1—-f^rocosjS X
\dx J (76)
where x must satisfy the Laplace equation in the fluid, zero normal velocity on the sea bottom, and the following free surface condition:
9X
The solution proposed i n Ref. [5] is as foUows.
dcp (77)
= X
9 ^ 9v dfn iz) ,1=0 + fn ( Z ) ^k„ dg„ (r, e) dk„ gn {r. e) (78) Since the potential i n the case of one cyUnder can be written i n the f o r m of eqn (72), we can exhibit a simpler solution: /i=0 dK dv ( z + / / ) ^ g „ ( r , 0 ) + -rfn (2) dz dgn jr. 6) dr (79)This solution ehminates some homogeneous solutions present i n eqn (78). I t can be written as follows.
.//, = X - 2
n = 0 dk, dv dz i c o s 0 dgn_ dr .sm 6 dgn 1 — + koCOsPgn de^ / i r i c o s e f l
^ - f r ^
k„ I \ dr 9r2 - i s i n e l ^ + r ^ o c o s ^ ^ 9r dO dr (80)We can see that this solution has a secular term which causes the solution to increase as r goes to infinity. This
S. Malenica et al.
happens because the T perturbation introduces a free sur-face condition in which the forcing term on the right-hand side contains terms which are solutions of the homoge-neous problem. However we can show that the solution to eqn (80) satisfies the radiation condirion, eqn (31), at order T , that is:
( / / - 2i/coCosÖAi (0)
As we can see, this condition is secular, which means that our perturbation by T is non-uniform and valid only m the vicinity of the body However, this will cause no problem i n the calculation of the wave drift damping co-efficient by the far-field method because, as we have ex-plained, the expression f o r wave drift damping has a con-servative f o r m and can be applied on the cylinder where the solution f o r (//] is valid.
The ipz potential cancels the normal velocity on the cyhnder induced by the potential (//i, and it is the stan-dard hnear scattering potential with a homogeneous free surface condition, a standard radiation condition at infin-ity and known normal velocinfin-ity on the body I t can easily be found by eigenfunction expansion as follows.
fo iz) P„^H,„ ihr)
«=i
Jm9
(82)
where we distinguish the propagative part associated with the Hankel functions H„{kr) = Jmihr) + lYmihr) and the local evanescent part associated with the modified Bessel functions K„,{k„r).
The P,„„ coefficients are found by enforcing the follow-mg condition on the cylinder.
PmO = -2Co koH' {koa) u /o/c -H dz f [A + H M d z dz (83) -2C„ k„K;„ (k„a) vf... fnfn<iz + f df
Xvf,„ J
f.iz + H)^dz
(84)in which the coefficients vfm and vf,„, follow f r o m the ex-pression for viz, 9) after the Fourier decomposition:
V ( z , 9) = dipi dr
= 1 1
CO 0 0 n=Ora=-oo + fn iz) vf. dz Jm9 (85) The last part of the potential, ip^, also satisfies a non-homogeneous free surface condition, eqn (75). Because of the rapid decay o f the forcing term on the free surface, this potential satisfies a standard radiation condition and the ring-source method^''' is used for its calculation. Only the final solution is recalled here:00
ip3(r,z.9)= X '/^3m(r,z)e™^
0 0
= X • ""iQ/o iz) H,„ (kor) m = - o o r
< J
[Jm (kop) a - Z,„oHm (kop)] Qm (p) pdp oo + 2Xc„f„iz)K,Aknr) n=\ X [/,„ {k„p) a - Z„,„K,„ {k„p)] Q,„ {p)pdp+ TTiCb/o iz) [J,„ ikor)
- Z^oJfm ikor)] 0 0 X H,„{kop)Qm{p)pdp r oo + 2 X C„f„ iz) [ƒ,„ ik„r) «=1 - Z,„„K,„ik„r)] 0 0 X ƒ A;,, {k„p)Q,„{p)pdp e'"'^
where z„„, = l'jk„a)/K'„,ik„a), and: CO
Ö(/-,0) = 2 i V o 0 V o ( p = X Qmir)e""^
(86)
(87)
Since we calculate the wave d r i f t damping by both the near-field and far-field methods, we need to know the near-field and far-field expressions for the (//3 potential. By puttmg r = Ö in eqn (86) we obtain:
OJ, <a,^- 2Co\:H,Akop)Qm{p)pdp
Wave and current forces on a vertical cylinder
_ ~ 2C„i: K,Ak„p)Q,n{p)pép
^ k„aK:„{k„a) -^"^'^
(88) and by taking the hmit ïor R = r ^ oo;
(//3„, (7?, z) = TTiCo/o (z) (koR) CO
X [Jm (kop) - Zy^Hm (kop) ] Qm {p)pdp (89) This hmit f o r r ^ oo can be obtained because of the rapid decay of the free surface forcing term Q{r. 9). I t would not be the case in the problem of, for example, second-order diflfraction, where the same method can be used, because then the forcing term decays very slowly.
Finally, i n the case of the radiation problem, we must add to the solution one more component because of the non-homogenous condition on the cyhnder. I t is clear that this potential, which we denote as 1//4, can be found in the same way as the (//2 potential because of the same f o r m o f the B.V.P. Only, f o r the sake of clarity, the two problems are treated separately.
6 R E S U L T S AND D I S C U S S I O N S
Since one o f the reasons o f this study was to give refer-ence results f o r the vahdation of numerical methods we first present some intermediate results f o r the first-order quantities. A l l results are f o r a cyhnder of radius equal to the water depth {H/a= 1) and the mass of the cylinder is taken to be equal to its displacement (Ma = ga'^nH, i =
1,2).
Figures 1-3 represent the first-order exciting force, eqn (35), on the complete vertical circular cyhnder in the cases of small positive and negative forward speeds, and f o r diflferent incident angles p. Fn is the Froude number de-fined by Fn = UI Figures 4 and 5 show the free sur-face elevation, eqn (50), around the cyhnder for the same cases. Figures 6 and 7 represent the first-order added mass and damping coeflicients, eqn (36), and finally. Figs 8-10 show the corresponding first-order amphtudes of motion in surge and sway, eqn (37). We can see the hnportant mfluence of the current on the first-order quantities, es-pecially on the excitmg forces and free surface elevation. Figure 11 represents the wave d r i f t forces, eqns (58) and (61), f o r the fixed cyhnder and Fig. 12 the corresponding values o f the wave drift damping coeflUcients, eqns (59) and (62).
We can observe a good agreement with the results ob-tained by a purely numerical method.'" I n addition, a complete agreement is obtained between these results and the simple formula,' which we recall here:
5 i (too) = ] — A (0)0) too
-I-0! Z)i (COo) cos^
4.00-Qga?A 3.00H 2.50-2.00-^ 1.50- 1.00-Fn = -0.05 f n = 0. Fn = +0.05 0.00 0.50 1.00 1.50 2.00 Fig. 1. First-order exciting force in the ;c-direction for
/? = 0. 3.00-Qga?A 2.00H 1.50-i.ooH 0.50-0.00 0.50 1.00 1.50 2.00
Fig. 2. First-order exciting force in ttie A:-direction for ^ = 7T/4. 3.00-( ' ) l Qga^A \ 2.00-H 1.50-1 I.OOH 0.50-0.00 0.50 1.00 1.50 A-na 2.00 Fig. 3. First-order exciting force in the ^-direction for
S. Malenica et al.
Fig. 6. First-order added mass coefficient.
1.80
Fig. 11. Drift force for the fixed cylinder. ^'^^ ' ° ""'^ ^"'ay and for ^ = 0.
0.00 0.50 1.00 1.50 f.^^ 2.00 0.00 0.50 1.00 1.50 j.^^ 2.00
Fig. 12. Wave drift damping coefficient for the fixed Fig. 15. Wave drift damping coefficient for the cylinder cylinder. free to surge and sway and for ^ = TT/4.
S. Malenica et al.
0.700
Fig. 16. Difference Ixtween the results for wave drift damping obtained by the semi-analytical method (Bu)
and those obtained by a simple formula (^f,).
1 a A iCCo)
a dp Wo
where a is the ratio between the group and phase velocity « = 1/2 + koH/ smhlkoH. This agreement holds to 4 decimal places, and is also vahd for an array of fixed vertical cylinders.'
Figure 13 shows the results for the wave d r i f t forces on the cylmder free to surge and sway at the frequency of the incoming waves, for the incidences ^ = 0 and P = T r / 4 , and Figs 14 and 15 represent the wave d r i f t damping coef-ficient for each case. The results obtained by the far-field and near-field methods are identical, and the agreement with the numerical results provided by Grue'* is also very good.
I n contrast, the results obtained by the simple formula do not agree with the setni-analytical results, except i n the case of a wave incidence p = r r / 4 . Careful investigation of the results has shown that, for a given frequency and water depth, the difference between the semi-analytical results and those obtained by the simple formula is equal to some factor tunes cos 2^, and this factor varies as shown in Fig. 16. However, we did not succeed in ehciting this difference in a simple manner f r o m the solution at zero forward speed only
A C K N O W L E D G E M E N T S
The authors thank D r Grue f r o m Oslo University for exchange of results and useful discussions.
R E F E R E N C E S
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