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Added mass and damping of a vertical cylinder in finite-depth waters
Added mass and damping of a vertical cylinder in finite-depth waters R O N A L D W. Y E U N G Department of Ocean Enyineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA (Received 1 May 1980, revised 23 May 1980)
A comprehensive set of theoretical added masses and wave damping data for a floating circular cylinder in finite-depth water is presented. The hydrodynamic problem is solved by matching eigen functions of the interior and exterior problems. The resulting infinite system is solved directly and found to have excellent truncation characteristics. Added mass and damping are given for heave, sway, and roll motion, as well as coupling coefficients for sway and roll. It is shown that the heave added mass is logarithmic singular and the damping approaches a constant in the low-frequency limit. Transition of the behaviour in finite-depth water to deep water is also discussed.
INTRODUCTION Added mass and damping of bodies oscillating in a free surface has been a subject of considerable research interest. Their calculations have traditionally been indispensable in the prediction of ship motions by the naval architects. In more recent years, the mushrooming of various nearshore and offshore projects has generated a substantial demand on the knowledge of hydrodynamics of floating structures operating in finite-depth water. Examples of these are the development of a floating airport in Japan and of an offshore floating nuclear power station in the USA. There exists a variety of elaborate numerical techniques for tackling the case by case situation. For example, Faltinsen and Michelsen 6 and Garrison 8 and Fenton 13 used the traditional wave source distribution method, Yeung 23 a hybrid integral-equation formulation based on the infinite-fluid source, Bai and Yeung I and Chen and Mei 5 finite-element variational formulations, and even more recently Berhault" an integro-variational method. f h e s e elaborate procedures are normally too expensive to allow one to investigate efficiently the effects of various parameters on the hydrodynamics of simple structures. Notably absent in the published literature is a comprehensive set of hydrodynamic inertia and damping coefficients for the relatively simple case of a vertical circular cylinder oscillating in a finite-depth fluid. Admittedly, this is a problem that can be solved, in fact if its not already solved in part, by any of the referenced procedures. As examples, Garrison 8 presented some results for a specific cyclinder in varying depths, Isshiki and Hwang 11 for another cTlinder at one specific depth but with incorrect results, and Maeda and Eguchi 15 for circular disks in varying depths. It is the intent of this paper to provide the ocean engineers with this missing gap of information. It is noteworthy that Garrett 7 and Black et al. 4 studied extensively the wave exciting forces and scattering characteristics of vertical cylinders. The present results should complement theirs. In a strict sense, by the use of 0141 1187/81/030119 1552.00 ©1981 CML Publications
Haskind's relation iv, one could deduce the damping coefficient from the published exciting forces, and with the use of the Kramers-Kronig relation (39 below) further calculate the added mass, provided that the infinitefrequency added mass were known. This is generally too much of a handicap for the practitioners. Because of the axi-symmetry of the geometry, the problem can be treated simply by the use of eigen functions alone. Extensive use of this technique has already been made by Hilaly 1°, Miles and Gilbert 16, Garrett 7 and Black et al. 4. The formulation described below follows closely that of Garrett, but the presentation, viewpoint, and solution are different. Garrett developed the expressions of the interior and exterior problems in terms of the potential at the common boundary, and match the normal derivatives together. In this work, the interior problem and exterior problem was treated as a Dirichlet and Neumann type, respectively, as if the conditions on the common boundary were known. By introducing expressions of the complementary region in the boundary condition, the matching is automatically completed. In this manner, the coupling of the coefficients of the two sets of eigen functions can also be seen very clearly. This approach apparently has not been taken by previous workers. All three types of radiation problems, heave, sway, and roll can be treated on a common platform. It should be pointed out that in Black's work 4 the farfield solutions of the radiation problems, which were solved by a variational formulation, were actually used to obtain the exciting forces by Haskind's relation, but no direct results on the radiation forces themselves were presented. Radiation forces for submerged circular cylinders were obtained recently by Tung 18, who solved for the flux at the common boundary before computing the coefficients of the eigen functions. The present procedure yields these coefficients directly. Of particular interest in the results shown is that the heave added mass is logarithmic singular, the nondimensional damping coefficient (non-dimensionalized in the standard way with the frequency in the denominator)
Applied Ocean Research, 1981, Vol. 3, No. 3
119
/tdded mass amt dampinq ol a vertical cylinder in finite-depth water. R. W. Yeunq Y
cylinder's bottom surface. We will proceed to obtain the solution(0 "~in the interior region (r ~i a).
x
Interior solution z
The interior solution satisfying equations (2), (4), and (5) can be written as follows:
z=l
(7)
.~(i) q o-(- i'¢"..(i,)h _~_ ~ "V p
where the homogeneous solution @h° and particular solution ~o - r p~ satisfy the following boundary conditions:
h(=l)
d
t_- I: ~l)
"~'///////////
Figure A.
= o,
= ~
(8)
Cz [z=d
/~/// O C,,--n- = = O,d = 0 , (?q)!i)
Coordinate system and notations
a constant as the wave frequency approaches zero. Separate low-frequency asymptotic analysis confirms this behaviour, which can also be verified by the Kramers Kronig relation. Aside from the diagonal values of the added mass and damping matrix, also presented are the coupling coefficients between sway and roll.
~O(hi) r= a _- - q) (e) - -
(9)
It is noteworthy that the matching of the potential between the interior and exterior regions is effectively achieved by equation (9). After some calculations, we can arrive at the following results: (~) = ~d(Z 2 --
MATHEMATICAL
q~(i) p
r2/2)
(10)
FORMULATION
j
We consider a vertical cylinder of finite draft heaving time-harmonically in water of depth/~. Let the Oxy plane be the bottom surface of the fluid, (r,O) be polar coordinates in the horizontal plane, and the z-axis point upwards (Fig. A). At the outset, we will assume all space variables are non-dimensionalized by water depth. Thus, for example, a represents the radius to water-depth ratio; d the bottom clearance to water-depth ratio. If inviscidfluid flow is assumed, we can decompose the velocity potential O(r,O.z.tl as follows:
2
.
.
,, = 1 i,)(/.,y)
z,, = d
(11)
where I o is the modified Bessel function of the first kind order zero and % are Fourier coefficients, which are given by: d ~n = dr2 [(~(e) -- (~)]r
----oc°s(2.z) dz,
n =0,1...
(12)
0
O(r,O,z,t) = ] i R e [ - iaq)lr,z)~ ,e i,,]
(1)
where a is the angular frequency, ~3 the heave amplitude, and i = v / ~ l . In writing down equation (1) the axisymmetric behaviour of the fluid motion has been observed. The non-dimensional time-complex spatial potential q~(r,z) can be easily shown to satisfy the following equations: V2qJ(r,z) =0
/)~o _ ?z
v(p : =
1
=0,
(2)
~'
3z I:~o
=0
( ~ r _= 0 _
Applied Ocean Research, 1981, Vol. 3, No. 3
03 (a,,)c - " 0 s(z,,_Idz " " q)l,
,~
(6)
03)
L
O
which are immediately calculatable, we thus have the following relation:
(4)
where g is the acceleration due to gravity. Evidently, the only inhomogeneous boundary condition comes from the
120
~,,* __ -
2/ ~ , = d q¢~(a,z)cos (51
d<<,z<~l
d
d
.q 0~
Although the exterior potential is not known at the moment, the second term of equation (12) can be evaluated using equation (10). Defining the real coefficients
7z
.
dz-c(,
(14)
0
If% were known, then equations (10) and (11) describe the potential ~o(1) completely.
Exterior solution
The exterior problem can be thought of as one driven by a flux emitted by the interior region. Its solution can be written in terms of the following well-known eigen expansion22:
Added mass and damping of a vertical cylinder in finite-depth water: R. W. Yeun9 q~t~)(r,z)= L AkRk(mkr)Zk(mkZ)' 0 <~Zr>~a <~1
(15)
where the coupling integrals C,k are defined by:
k=0
2f C.k--~ COS -~z
where A k a r e unknown coefficients. The non-dimensional eigen values are solutions of the equations m tanh m=v
,
(Htol}(mor)
ktmkr~= )Ko(mkr )
for k=O for k ~> 1
. . . .
Zk(mkZ)dz
0
(16)
where m represents mo o r im k for k 7> 1. The functions R k are given by R"
n__0,1 k = O, 1....
d
2i - )"sinh(m°d)/m°d =
(17)
for k=O \moaJ A (24) for k = l , 2. . . . 2sin(mkd-- n~) mkd (mkd - mz) (mkd + mz)
The quantity S, occurring in equation (23) is defined by: where H o is the Hankel function of order zero, chosen to satisfy the radiation condition, and K o the Macdonald function of order zero. The functions Zk(mk z) form a normalized orthogonal set in [0,I] and are defined by:
(cosh(m,,z)//N~; '2, N o---~(1 + sinh(2mo)/(2mo)) Zk(m~z) = e ~"'C~O'Sm (2,kZN/) S t = ~(1 + sin(2mk)/(2m~))
(18) The inner product (Zk, Z j ) ~ 6 k j , where 6kj is the Kronecker delta, using equation (6) and matching the derivative of tpt~ with that of the interior solution at r = a we have:
d<~z<~h k=OmkAkR'k(mka)Zk(mkz)= q~i)(a,z) O<~z
{0
ng , / n x a \
/nxa\
which vanishes at n = 0 in this instance, but not the other modes of motion. It is understood that in equations (22} and (23) the R~,s are evaluated at toga. The coupled infinite system for e, and Ak is in an ideal form to eliminate either set of unknowns. One has the option of solving for either 5. or Ak first. It can be shown to be slightly more stable to obtain the former. For the heave problem, this also has the added convenience of allowing one to know ~0~i~with little further calculations. Hence, ct,- L e,j~j = g,,
(19)
for n = 0, 1, ...
(25)
j=O
where where the prime indicates differentiation with respect to the argument. By making use of the orthogona[ properties of the Z~,s in equation (19), and substituting equation (7) into the right-hand side, we obtain: d
gn =
mkAkR'k(mka) = f ~i)(a,Z)Zk(mkz~z + A~
where d
A *= f ~--~q~( a,Z)Zk(mkZ)dz
(21)
0
Equations (14) and (20) are the key results of our present analysis. In its present form, it is easy to see that an inhomogeneous boundary condition on the vertical surface of the vertical cylinder merely modifies the definition of A~ by contributing an integral from d to h(=l). Equations (14) and (20) are coupled because q ~ in equation (14) depend on A k, whereas q~0 in equation (20) depend on ~,. If equations (11) and (15) are now introduced, the following equations result:
~. = L RkC,kAk + e*
n = 0, 1, 2
(22)
k--0, 1, 2,...
(23)
k=0
*)
S n C n k & n -[- A k
n=
mkRk
CnkCjkfRk'~le
kk=0
"'k
/,
--[WYp'~k
(26)
\''k/_J
fR
- - ~n
(27)
(20)
0
A k=
~~
e
The infinite system (25) was observed to have excellent truncation characteristics. For a 1~ accuracy, it was rarely necessary to go beyond 20 equations. To the author's knowledge no other existing method allows one to obtain the solution so efficiently and conveniently. It is perhaps of interest to note that for the heave problem the coefficients e,, n = 1, 2 . . . . is decoupled from s o since S o vanishes; but the above structure is the most general, allowing immediate extension to sway and roll motions later on. To complete this analysis, we recall that the dimensional heave added mass P3~ and damping 3.33 can be obtained as follows: 2~
a
(28) 0
0
After performing the necessary integration and nondimensionalization, we obtain: /~3a + i23a - - J ] 3 3
"-[- i~33/ff
pr~fi3
Applied Ocean Research, 1981, Vol. 3, No. 3 121
Added mass .nd damping of a vertical cylinder m finite-depth water: R. 14/.. Yeung (l/d\
I [a\
Ire%
/d\ &
=21.4ka)-16~")+"[ ~4 + 2ta),,~'(-)
,7,S, ]) (;ini~
) ;' + log2 - log(m0a)] ~ + O(mgaqog(moa))
t~3.~ 2)4,/+ a[ -
l~
)
(37) (29)
I
/,s3-~4a l+(m~¢0-' n2 - ; ' + l o g 2 - log(re,a)
where we recall a is the dimensional radius,
O(m~a"~ Iog(moa)l
LOW-FREQUENCY ASYMPTOTIC BEHAVIOUR It is possible to obtain an expression for the low-frequency behaviour of the heave added mass and damping in the following simple manner. Since ~o~1 satisfies equations (2), (3) and (5), integration in the z-direction yields: 1
1 ? 80"') + mZq¢"~(r,0)= 0, r&.r-?r
q)i~)(r,z)dz
where (P(")tr)= (I
(30) and mo is the non-dimensional low frequency wave number, viz. m 0 =,,/v. Similarly, for the interior region, a
dl ? r-a E@i, + 1 =0, I" ( T
where
(oli)(rJ=d1 ( qo.~lr.z)dz
(31)
L} 1
+ (38)
where 7 is Euler's constant. Hence it can be seen that "~33 approaches a constant of value n&4h, whereas IL<~ is logarithmic singular. The constant value 0f233 can also be deduced by the use of Haskind's relation if one noted that in the long-wave limit, the exciting force per unit wave amplitude is simply p# multiplied by the water-plane area. The behaviour of #33 is less obvious. In fact, in this limit, since the heave added-mass in two dimensions reduces from a logarithmic singular behaviour in deep water ~9 to a finite-behaviour in shallow water z2-'4, and since the three-dimensional added mass in deep water is already known to be finite ~, one would be tempted to conclude otherwise. Mathematically, we may explain this by observing that the behaviour of the added-mass is related to the behaviour of damping by the Kramers-Kronig relations. In non-dimensional form, one of them ~2 can be written as:
~1"
j,
o
p(va)-p(~c)=~ ~(e) and ~(0 are the depth-averaged values, To leading order in mo, we may assume in equation (30) that ~01el(r,0) ~ - ~ , which is basically the shallow-water approximation 5. This is also equivalent to assuming that the terms corresponding to k > 0 in equation (15) are small: thus for a given mo the larger a is, the better the approximation. Equations (30) and (31) have the following solution form:
~o~el(r)= AoH~oll(mor ) a <~r <<,ec
2(z)d= =__ va o
(39) t
0
where ~ denotes that Cauchy's principal value is to be taken. Since I~33(~c) is a finite constant and ,~33(0) = g~l/4h, it follows that:
(32) lim p33(va)= K -4,~log(va)~ 2,~ l°g(m°a)
(40)
va~O
~o")(r)=%-r2/4d
O<~r<<,a
(33)
To determine A o and % we match ~(~) with ~9lil and ~¢fl with d ~ ~, the latter being the consequence of the fact that ~") and ~,1 each applies to domain of different vertical dimensions. The results are: ~le~(r)= a H~ol~(mor)
2moH]l)(moa) ~o(i)[r]_ a H(o1)(mOa) ' '-2mo~)
+
(34)
a 2 - - 1.2 4j
(35)
Integrating over the cylinder's bottom, we obtain the following closed-form solution
l[a
U33_1_i233=~ 4~+af(moa)
]
. f(moa) =
.,,,. o ~moa) rnoa HllI )(moa ) (36)
If we assume further that a/h = 0(1), the following limiting behaviour can be obtained as moa-*O.
122
Applied Ocean Research, 1981, Vol. 3, No. 3
which is precisely the singular behaviour arrived at earlier, Physically, we can explain this by imagining that the fluid bottom and rigid free surface together trap the cylinder as if it were a cascade in the vertical direction. This effectively forces the fluid particles to move twodimensionally in the horizontal plane. Viewing from the top. we now have essentially a pulsating two-dimensional source which is known to have a singular added mass. It should be pointed out that because of the neglect of the evanescent modes we cannot expect the simple matching conducted here to yield an exact value of the constant K. The error of this matching is of 0(e-~""~). This is consistent with the fact that equation (36) was observed to yield reasonably accurate prediction when 6//~ is > 1. To determine K exactly, the rigid free-surface problem must be solved, this can be done using the numerical procedure in this paper but is not pursued here.
SWAY M O T I O N It is a simple task to generalize the ideas presented in the first section for the case of sway motion. Because of the boundary condition
Added mass and dampin9 of a vertical cylinder in finite-depth water." R. W. Yeun 9 __ ~, fcosO~(t) ~ = nx¢ ~t) = ~ c~n 0
d<~z<~h, z=d
ct~*=0,
r=a O<~r<.a
n = 0, 1, ...
(50)
(41) J'sinh mo -sinh(mod)'~/tN1/2 m A*=~sin mk--sin (mkd) (/~ k k,
where n~ is the x-component of the interior normal ~and ~(t) the sway velocity of the cylinder, we define: ~(r,O,z,t) = hRe[q~(r,z)~ ]cos0
~H' t' >(m~,a)/H~' 'i Rk/R'k=(Kl(mka)/K ],
(42)
Hence, the boundary-value problem for the axisymmetric function ~0(r,z) is now as follows:
Sk =
(51)
k= 0 k >~l
(52)
I d/4a
k=0
(53)
~kTz
k >11
{ ~ll(mka)/l I ,
V2tp(r,z)-~=0
(43)
c~c~z v~p~= ~=0
(44)
If equation (25) is now solved as before, equation (23) allows the determination of the values of Ak'S for ~o(~) in equation (15). Whence, added mass and damping can be calculated and they are given by the following formula:
(45)
~11 +i211 "-'- ~p~2(~__d ) --a~l__U)k=O_,;-~, ~ AkA~R k (54)
(46)
When viewed from the formulation as presented, it is now evident that the sway problem has an identical structure to that of the heave. The roll-motion problem can be treated essentially in an identical manner.
_~11+i~al/a
3~z ~ ==a=0
O<<.r<~a
~z~ =o=0 =1
d<~z<~l
-1
~'
(47)
r--a
ROLL MOTION
Since equation (45) is homogeneous, it is clear that
q~.1 p =0
We assume that the roll motion occurs about the point (0,0,11. The dependence of the boundary condition on 0 is the same as the case of sway. It is appropriate to introduce the following definition of ~o(r,z)
(48)
Also, because of equation (43) the interior homogeneous solution is now given by:
(55)
O(r,O,z,t) = h2 Re[ ~o(r,z)~5]cosO --Y~a/
(49)
)~nz
-t- 2., ~n - c O S n=l Ia(2,a)
where (5 is the roll velocity. Accordingly, one may show in equations (43-47), only equations (45) and (46) have to be modified; viz:
while the exterior solution is still given by equation (15) with the order of the Bessel functions defined by equation (17) being 1 instead of 0. The inhomogeneity now arises from A*, which consists of merely an integral from d to h. Therefore, the entire solution procedure described earlier is completely applicable. In particular, only the following trivial modifications of the definitions of the quantities used (25) are necessary:
C3~z z z=d= --r
O<~r<~a
3~r rr=a=-(1-z)
(56)
I<<.z<~l
(57)
Table 1. Behaviour o["ct, and A k for varying number ~]' equations N=3
N = 10
N=20
Roll: d/h=0.5, h/d=2.5, t o o , = 1.0
~ k
Re
0 1 2 3
AkR'k lm
Re
~k lm
11,83429 2.61280 -2.43323 0.64106 0,22991 -0.02399 -0.28583 0.06968 -0,07093 0.00776 0.06081 -0.00804 0.03547 -0.00396 0.06216 -0.01022 a qPss, 2551=(0"37270, 0"11968)
k
Re
AkR'k Im
Re
~k Im
k
Re
AkR'k lm
Re
lm
0 1 2 3
11.82702 2.61625 -2.43558 0.64197 0.23270 -0.02433 -0.28172 0.06928 -0.07273 0.00798 0.06182 -0.00817 0.03678 -0.00412 0.05861 -0.00983 a i Pss,)-5,~)={ 0.37193, 0"119881
0 tl.82547 2.61695 - 2 A 3 6 0 6 0.64215 I 0.23338 -0.02441 -0,28097 0.06922 2 -0.07321 0.00803 0.06206 -0.00819 3 0.03716 -0.00416 0.05806 -0.00977 a t(p55, ),551={0.37179, 0.119921
0 1 2 3
63.42547 0.53782 -0.12836 0.05614
0 63.42550 5.37668 -28,67539 5.37678 1 0.53782 -0.00298 0.32239 -0.00180 2 -0.12836 0.00071 -0.08344 0.00047 3 0.05614 -0.00031 0.03751 -0.00021 a ~1P55, )-ss)=t 1.18235, 0.13446)
Roll: d/'h = 1.0, a//7 = 5.0, moa =0,5 0 I 2 3
63.42517 0.53784 -0.12836 0.05614
5.37663 -28.67510 5.37673 -0.00298 0.32237 -0.00180 0.00071 -0.08343 0.00047 -0.00031 0.03750 -0.00021 a-J(p55, 25~)={I.18258, 0.13446)
.
.
.
.
.
5.37667 -28.67536 5.37678 -0.00298 0.32239 -0,00180 0.00071 -0.08344 0.00047 -0.00031 0.03751 -0.00021 a I{it55, ,;.5~)=(I.18237, 0.13446)
.
.
.
.
L
Applied Ocean Research, 1981, Vol. 3, No. 3
123
Added mass and damping ola vertical cylinder in finite-depth water: R. W. Yeung
1.~ao b~
5,0
a=5.0
4.8--
I3.88
3.0m
0.60~
p.aa/~
~
}~33/a ~
-
2.0~
d=O.i0
8.28--
1.0--
~ 0.0
9
I 11
0
0 1.0/
/i.0
~ _
~"~--~.._
/o.9o
0.25,0.i0
I [ l , l l [ I 1,~ 20
r-~-r ]- ~--rm-q-3,~
--I-q~-~-VTqTT~-]T[-.T~-I-T-[-TTq--
0.00--
4 rfloa
moa
Fig. 1. 5 0
l
I. 00-~ a
a=l.0
o8o_ {3.60--
~33/a
d=O.lO
2.0
o.4
_
I I
1
0.25
04
I
0.75 0.9( /
I~.~
I .~
I 2.0 rlloa
T~--[--I-[--FT--] 3.0
4.0
1 .0
r,d.0
2.El
3.121 rnoa
4.~
5.1~
6
Fig. 2. Thus, the roll motion has the combined properties of both heave and sway: inhomogeneous conditions occur in both the interior and exterior regions. The particular solution satisfying equation (56) can be easily derived, and is given by:
9.(,~= - 1 F 2 r- ar3qj
158)
which contributes to both the calculation of ~* and Ak* Performing the necessary elementary integrals due to
124
Applied Ocean Research, 1981, Vol, 3, No. 3
equations (571 and (58), we get
t
,,
(
%* = i l
_ad
1
liar-']
[3-4,dJ ,
2ad( - ) " - i
tn~:)2
o d
(59)
,> 1
t
A*=J I ~r(" q#)"I.:. Zkdz + f(z - 1)Zkdz 0
,,:0
d
(60)
Added mass and damping of a vertical cylinder in finite-depth water: R. W. Yeung 5.0
1.00--.]- b a=O. 5
0.
4.0--
d=o: ~O
3. o - [ ~ _ _ _
0.60. -i.0
X33/a 0.40.
_
~.~
1 .0--
0)75
~
I I I
0.0
~
I [ I 1 .0
0.0
0.90
I ~
0.
[ 1" I 2.0
0,00-~-~
I [ J I I I I 3.0
4.0
0.0
1.0
2.0
moa
3.0
4.0
5.0
6.0
moa
Fig. 3. 2.00
8.0
-b
a=O. 2
I . 80--
7.0-~
eel 40-S.O--
d=O •I0
1 20--
la3s/a
X33/a.~ 1 00 _
4.0----
0.25 3.0-~
e Be-
0.75
Z
e 60-0.90
-4
0 40-0 20--
0 00--
e. e~-T--T.-0.0
I .0
2.0 Ill a 0
3.0
,.o
4,0
2.0
3.0
,.o
5.0
6.,,
rnoa
Fig. 4. Figures l 4. Added mass (a) and damping (b)Jor heaving motion: a/'h = 5.0; fi/h= 1.0: fi/'h =0.5; a/fi =0.2
sinh(mod)F3 2 3. 2 =Nk
I/2
L
-] m~I +"Jc°shm°
sinh(m°d) 2cosh (mod)] (tood)
smm,dlq
(61)
sin(mkd)F3'rnkd---LSa-- ~d32 + d]-..!2 F2COSmk L (mkd)--( c°s r a g + - mkd )J As far as solution of equation (25) is concerned these are the only changes when compared to the sway problem. If the following definition of the integrals:
Applied Ocean Research, 1981, Vol. 3, No. 3 125
Added mass and damping o/'a vertical cylinder in finite-depth water: R. W. Yeung 1 . 28
0.80~ d/h=l.
-
0
O. 70--
1.10--
-
I
4
a=l.
0
.3333
0.90
O. 20
IX33
_~ O •70--
$0
68--
kca=l. 0
X33 _
~
0.
\
~0.50
88--,
8. 4 0 ~ 1o
o.
"~
0 . 20
i
a
-'
moa
moa
Figure 5. Low-frequency behaviour o./heare added mass and damping as ai'h~O. ((l/]]=1.0)
2
.
5
0
8.
~
.......... Eqn (36)
--
'exact'
2. 0. a=2.0 -AI
X33/a
1'58--'11"~I -
""" .
.
a=5.0 . . ~
.
~a3/"
.
ii
.
.
.
.
.
.
0.40~
.
.......................
!
1 .00---4
I
~ d=O.l
:']=i. 0
.a=2.0 0.58 _
d=O.l/•
=
I'
"-'
"
.
d=l.
-"
0
a
o.~
b
''''
ri r-7--r- .
'l''-v-1
0.00-
2
l l,i
l,i
v-l-l-7--r--l--r- I i
~.~
,.,~
~.o
~.<~
,.,~
,~.o
moa
Figure 6.
Ek = F( - z)Zkdz = N~
d
1/2
~.<~
i
~.o
moa
Comparison of exact solution with asymptotic jormula
1l
,.~
( (d -- 1)sinh(mkd)mo(mod )4
cosh mo-cosh (rood) m °2
cos m k --cos (mkd)
(d-- 1)sin
(eq. 36)jot heave added mass and damping
mk
t~k 2
are introduced, the added moment of inertia P55 and roll hydrodynamic damping 255 can be written as: 126 Applied Ocean Research, 1981, Vol. 3, No. 3
(62j
I
I
''1
4..0
Added mass and damping of a vertical cylinder infinite-depth
water: R. W Yeung
0.80
b
a
d= 0.001
0.60 0.80
0.60+
0.40 \I
0.40
0.50
0.20
0.00 2:0
010
4:0
610
8 ,.I
0
mea
Fig. 7.
’.20/
a=l.O
a
II ;i
0.001
0.600.80
0.20
x11
0.60
4
0.40
0.401
n “.-” 5”
\
0.75 0.20
0.90
:
II
0 0:0
210
410
610
8 .0
0.0
2.0
mea
4.0
6.0
8
sa
Fig. 8.
0
i ’ f k=O
A,R,(m,a)E,
(63)
where the first two terms are contributions resulting from the bottom surface and the last from the side of the
cylinder. It is worthwhile to point out that unlike the case of sway motion, these hydrodynamic coefficients do not reduce to the case of a full cylinder touching the bottom as the bottom clearance reduces to zero. This is because the boundary condition on the cylinder’s bottom does not vanish in this limit and the interior particular solution still contributes to the calculation of tl,* and At. The coupling coefficients between sway and roll can be obtained from either the sway or roll solution. The expression from the roll motion is simpler because of the absence of the contribution from the cylinder’s bottom and is given by:
Applied Ocean Research,
1981, Vol. 3, No. 3
127
Added mass and dampi,w o/ a vertical cylinder infinite-depth water: R. W. Yeung
G 8~3~
2g . . . . . . . . . . a=O. 5 -4
'
a
00-~ /
i
b
d=
~- o.ool o
I
r/1
o.oo
8 @
2.8
6 @
4 0
8.8
2.~3
0.0
4.0
moa
moa
Fig. 9. 20
I .
8.0
610
0.80 a=O. 2
d= 0 0 ---//i/~--\ O. 001 I
b
a
4
@. 8.88--
60t
0.75
XH
laH 0.20 e. ee---
8,48--
-1
0.50
, ---0.90
0.90
0.40-~ 0.2{3--
7 O.S2.L__-------~ 0.
0.
@@---~--T
I
r
T
]
I
[
,
00
moa
moa
Fig. 10. Figures 7-10.
Added mass (a) and damping ( b ) . / b r s w a y motion: fi,,'h = 5.0: ~,"h = 1.0: ~/'h = 0 . 5 : fi.,"h = 0.2
"
-
+i)~5|/a pTza2h 2
-
1 a k: o
Ak k~k
(64)
where A *(~) is the expression given by equation (51). In the actual computations,/~15 + i)~5 obtained from the sway solution agree within 1% with/~51 + i25x, which provided a confidence check on the results.
128
Applied Ocean Research, 1981, Vol. 3, No. 3
N U M E R I C A L RESULTS A N D D I S C U S S I O N The numerical work involved in the solution of equation (25) is reasonably straightforward. Care, however, must be exercised in the summing of the series (26), whose major contribution comes mainly from values of k when mkd mz orj~. It is possible to convert the complex system to a real one but this option was not taken as it was found
Added mass and dampin9 of a vertical cylinder in finite-depth water." R. W. Yeun9 2.58
0.88. a
a=5.0
2. 8 8--: / ~ 0
,
~.e°I
d= .3333
~
b
0.50
8.48 1.0
0"3333~
8.88
' 8.8
I 2.~
'
I 4.8
'
I 6.8
' ~.8
Fig. 11.
~.8
~.8
~.~
nloa
moa
8.68
0. a
150
a=2.5
8.58--
8.48--
L_050
8.3B--
8.28--
8.10--
8.88--
' 8.8
Fig. 12.
J
2.~
'
[ 4.8
'
I 6.0
'
8.0
mo a
that most of the computation efforts are in the generation of the matrix elements rather than the solution of the system. The infinite system has excellent truncation characteristics. Table 1 shows typically how the first four coefficients of~k and A k, and the resultant added mass and damping behave as one increases the number of equations, N. Generally, convergence is slower for smaller a//~, and larger d//~. Numerical checks were conducted on the hydrodynamic damping by comparing the exciting force and moment with those given by Garrett v and Black and Mei 3, on the added mass by comparing results for the case of a circular disk ( d = l ) with Maeda and Eguchi 15. Excellent agreement was observed. Other checks based on
moa
asymptotic formulae discussed above as well as symmetric properties of the coupling coefficients were also applied.
Heave motion Figures 1~, show the non-dimensional added mass and damping for four different 6/h ratios, in descending order, with d/~ as a parameter. The coefficients are weighted by the factor/~/& The horizontal axis is chosen as moa instead of the more conventional parameter va since most of the variations concentrate near the zero-frequency end. The low-frequency behaviour of these curves is in agreement with that based on the asymptotic analysis. For 6//~ bigger
Applied Ocean Research, 1981, Vol. 3, No. 3
129
Added mass and dampinq of'a vertical cylinder in finite-depth water. R. W. Yeung
0 100
0,0,40-a
b
a=l.O
d=O.3333
"] X55/a
080
0
1%5h
0.020--!
0.50
0 0701 0.75
0.010-0 060-0.90 I
0 050-00
I
2,0
'
4.0
0.000
I
6.0
8.0
,0
2.0
rnoa
Fig. 13.
4.0
-
6.0
8
3.0
4.0
mo a
0.2S0
o.
"
0.
oo
100
°°°.5
0. 080 t
0. 0 6 0 - ~55/a --
~
~
Xss/a -
0. 1 0 0 - -
0. 040--
0.0S0--
0. 020-
0.90 ~0.75
0. 000 t 0.0
I
I
|.0
Fig. 14. Figures 11-14.
I
[
I
2.0
]
0. 000 :
i
3.0
4.0
! .0
2.0
Added moment o[ inertia (a) and damping (b)./br roll motion: fi..'h= 5.0: a.'h = 2.5: a'h = 1.0: fi..'h=0.5
than 0.5, and the added mass and damping have features similar to those of two-dimensional cylinders in deep water: viz. the damping decays monotonically from a finite value as the frequency increases, whereas the added mass starts from a logarithmic behaviour and attains a minimum before approaching its asymptotic infinitefrequency limit. The occurrence of a minimum is a necessary condition for satisfying the following equality21:
XJ
,J(ff(va)-I~(O°)d(va) ~ =0
0 130
0.0
moa
moa
Applied Ocean Research, 1981, Vol. 3, No. 3
(65)
which is valid for all modes of motion. This minimum is observed to be located at higher moa values for cylinders of shallower drafts. This is also the case for sway motion. As to be expected, shallow-draft cylinders have more damping than deeper ones of the same radius. It is interesting to note how the finite-depth behaviour transforms to its deep-water counterpart. Naturally, the transition occurs sooner for shallower cylinders than deep ones. Figure 3 shows the initial and Fig. 4 the full development of a maximum in the damping curves. The slight undulation in the added mass is related to the occurrence of these maximum (via K r a m e r s - K r o n i g ' s relation). Note that as 6 / / ~ 0 , equation (38) or equivalent
Added mass and damping of a vertical cylinder in finite-depth water." R. W. Yeung 3 . gO
3.0
~t51 0.50
2.0
75
0.75 0.
0 00
-I .0d 0.0
I
'
2.0
I
'
4 0
I
'
8.0
1
I
~.~
8.0
'
I
2.~
'
moa
t
~.~
'
I
~.~
' 8.0
moa
Fig. 15. 0.88----,
0. GO
a
a=2.5
0.
0, $ 0 - -
0.3333
0.20
0.40--
0.
)~51
O. 4 0
-
3333
50
}asl
0.30--
0.30
~ - - - O. 50
75
0.20--
0.90
0
0.
0.20,
90
-o.~o
_• 0.0
0.50 /
'
I 2.0
'
I 4.0
'
0.3333~
1 6.0
' 8.0
m° a
0.0
2,0
4,0
6.0
8.0
moa
Fig. 16. analysis yields 233 (0)=0, thus removing the logarithmic behaviour of the added mass in equation (39), and resulting in a finite value of zero-frequency heave added mass in deep water, which is well known from Havelock's results 9. We should note in passing that Wang 2°, who solved the problem of the heaving motion of a sphere in finite-depth water, has damping curves that behave precisely in this manner. Unfortunately, his added-mass low-frequency behaviour is in error. Figure 5 shows in a slightly clearer fashion how the transition we just discussed occurs. Figure 6 shows the effectiveness of the asymptotic formula (36) for four different geometric configurations.
This is representative of the type of agreement one gets when {l/h > 1 regardless of the value of d/h, For smaller h//~, the inclusion of the evanescent modes in equation (15) is necessary in order to obtain more accurate predictions (not shown).
Sway motion The sway added mass and damping are shown in Figs. 7 10. Regardless of the value of h/h the damping attains a maximum at mo a,.~ 1.2, then decays gradually. This is essentially the same behaviour as the exciting force of Garrett 7. The added mass curves each has now a
Applied Ocean Research, 1981, Vol. 3, No. 3
131
,4dded mass and dampin.q o/a certical cylimler infinite-depth water: R. W. Yeung 0.0S0-~
0
................................. a:l
. 040
0
~ d = 0 . 9 0
I
-0
0S0
~--
\
-//
-0 10 ~~0.20
-0. 060--
/
b
a -o.
\
150--
'
l
0.0
I
'
4.0
2.E~
I
'
6.0
8.0
0.0
2.0
4.0
6.0
8.0
ilia
moa
Fig. 17. 8
@. 0 5 8
82S
a=0.5
.90
d=O. 75 @
888-
0. 000-75 .50
-e. ese-_
_
//~..~
-8
82S-
-0
8S0-
~.0.3333
Ia51 -0. I00--
3333
)'51 -0 075-
.20
0.20
-8. 158--
-0.
-8
1 @e-
-0
125-
-0
150
200--
b
a
' 0
0
[ 2.8
I
1 4.0
I
I
6.8
' 8
0
i 0.0
I 2.0
I
~
4.B
T
]
6.0
r 8
Illoa
Fig. 18. Figures 15- 18.
Coupling added mass (a) aml dumping (b) between sway amt rolh fi.h = 5.0: fi.'h = 2.5: fi.'h = 1.0: fi..'h = 0 . 5
maximum and a minimum. We might add that while it may appear in Fig. 7 that the added mass decays gradually as moa increase, thus seemingly violating equation (65), in actuality computations showed that the curve eventually rises again after some large value of moa. The damping curves corresponding to d//~=0.001 can be compared directly with MacCamy and Fuch's classical result 14 of the horizontal exciting force on a full cylinder, The added mass results approaches the two-dimensional
132
Applied Ocean Research, 1981, Vol. 3, No. 3
limit of an infinitely long cylinder as moa-*O. In this limit, we note that a bottom gap clearance of 20% reduces a strip-theory estimate by 10 to 25% as •/h ranges from 0.2 to 5.0. It is possible to show that the limiting value of the sway added mass is given by:
l -d) /ql =
l+ii
for ~//~>>1
(66)
Added mass and damping o f a vertical cylinder in finite-depth water." R. W. Yeun9 Roll motion
ACKNOWLEDGEMENTS
T h e roll a d d e d m o m e n t of inertia and h y d r o d y n a m i c d a m p i n g are shown in Figs. 11-14. Since both the b o t t o m a n d the side c o n t r i b u t e to the h y d r o d y n a m i c m o m e n t , the b e h a v i o u r of the curves in terms of d/h is m o r e complex. N o longer w o u l d a shallow-draft cylinder necessarily have m o r e wave d a m p i n g that a d e e p - d r a f t one. The results d e p e n d s t r o n g l y on the ~/h ratio. F u r t h e r m o r e , with certain c o m b i n a t i o n of g e o m e t r i c p a r a m e t e r s , e.g. a / h = 1.0, d/h~-0.5, or ~//~=0.5, d / ~ = 0 . 7 5 , the roll d a m p i n g a b o u t the centre of r o t a t i o n b e c o m e s vanishingly small, resulting in an a l m o s t c o n s t a n t value of the a d d e d m o m e n t of inertia as a function of frequency. G a r r i s o n s has n o t e d this relatively c o n s t a n t b e h a v i o u r of a specific cylinder for which he was c o m p u t i n g m o t i o n response in a seaway. This p h e n o m e n o n is a p p a r e n t l y related to the possibility that the phasing a n d m a g n i t u d e of the wave systems due to the side a n d the b o t t o m are such t h a t they become cancelative. It occurs when the draft to r a d i u s r a t i o is a p p r o x i m a t e l y 0.5. Associated with this situation, is also a relatively small exciting roll m o m e n t . In fact, at certain specific frequencies, the roll m o m e n t practically vanishes altogether, This b e h a v i o u r is o b s e r v a b l e in the u n p u b l i s h e d d a t a of Black a n d Mei 3. The roll-sway coupling coefficients are shown in Figs. 15-18. T h e sign c o n v e n t i o n is t h a t a negative value of p51 or 251 will generate a clockwise m o m e n t a b o u t the axis of r o t a t i o n when the cylinder is given a positive sway a c c e l e r a t i o n of velocity. G e n e r a l l y speaking, the beh a v i o u r of b o t h a d d e d mass and d a m p i n g is very similar to the c o r r e s p o n d i n g b e h a v i o u r of sway except, of course, with the possible sign change. It is of interest to note t h a t cylinders of shallow drafts generate c o u n t e r - c l o c k w i s e m o m e n t s whereas the d e e p - d r a f t e d ones generate clockwise m o m e n t s . T h e t r a n s i t i o n occurs when the cylinder draft is a p p r o x i m a t e l y o n e - h a l f of its radius. This beh a v i o u r is consistent with the fact that in sway m o t i o n the pressure v a r i a t i o n on the surface of the cylinder is directly related to cos 0. Exciting forces T h e d a m p i n g d a t a presented in this p a p e r can be used to predict the exciting force a n d m o m e n t . F o r c o m p l e t e ness, we p r o v i d e the f o r m u l a e below. If the incident wave of unit a m p l i t u d e is a s s u m e d to be of the form R e [ e x p ( i m o x - i a t ) ] , a n d if X 1, X3, (Xs) designates the c o m p l e x a m p l i t u d e of the wave exciting force (or m o ment), then
IXal/rtPga2
=~4(1-d)D L lrmo a2
2 , , ] '12
(67)
The a u t h o r t h a n k s Mr. Sea H. K i m for his valuable assistance in the graphics of the results. This w o r k is s u p p o r t e d p r i m a r i l y by the Office of N a v a l Research, u n d e r T a s k N R 062-611. P a r t i a l s u p p o r t by the N a t i o n a l Science F o u n d a t i o n u n d e r c o n t r a c t E N G 7 7 17187 is also gratefully acknowleged.
REFERENCES l 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
iX31/7~p,Qa2= [
n~oa2D ,~33] 1/2
(68) 21
4 D )~5511/2 IXsI/'nPaa3=[ ~ooJ
(69)
where D(mo) = t a n h m o + mo(l - tanh2mo). T h e phase angles of the force a n d m o m e n t , t h o u g h not given here, are a v a i l a b l e from G a r r e t t 7 a n d has the very f o r t u n a t e feature of being practically i n d e p e n d e n t of the g e o m e t r i c p a r a m e t e r s a / h a n d d/X!
22 23 24
Bai, K. J. and Yeung, R. W. Numerical solution to free-surface flow problems, lOth Syrup. Naval Hydrodynamics, Cambridge, Mass, 1974 Berhault, C. An integro-variational method for interior and exterior flee-surface flow problems, Appl. Ocean Res. 1980, 2, 33 Black, J. L. and Mei, C. C. Scattering and radiation of water waves, Water Resources and Hydrodynamics Lab. Rep. 121 (MIT), April 1970, 248 pp. Black, J. L., Mei, C. C. and Bray, C. G. Radiation and scattering of water waves by rigid bodies, J. Fluid Mech. 1971, 46, (1), 151 Chen, H. S. and Mei, C. C. Oscillations and wave forces in a man.made harbor in the open sea, lOth Syrup. Naval Hydrodynamics, Cambridge, Mass, 1974 Faltinsen, O. M, and Michelsen, F. C. Motions of large structures in waves at zero Froude number, Proc. Marine Vehiclesin Waces Symp., london, 1974, pp. 91- 106 Garrett, C. J. R. Wave forces on a circular dock, J. Fluid Mech. 1971, 46, (1), 129 Garrison, C. J. Hydrodynamics of large objects in the sea, Part II: Motion of free-floating bodies, J. Hydronautics 1975, 9, (2), 58 Havelock, T. Waves due to a floating sphere making periodic heaving oscillations, Proc. R. Soc. (A) 1955, 231, 1 Hilaly, N. Diffraction of water waves over bottom discontinuities, Univ. of California, Berkeley, Rept HEL 1 7, Sept. 1967, 142 pp. Isshiki, H. and Hwang, J. H. An axis-symmetrical dock in waves, Seoul National University, Ship Hydrodynamics Laboratory Rept. No. 73--1, January 1973, 38 pp. Kotik, J. and Mangulis, V. On the Kramers-Kronig relations for ship motions, Int. Shipbuild. Progr. 1962, 9, (97), 361 Fenton, J. D. Wave forces on vertical bodies of revolution, J. Fluid Mech. 1978.85, [2), 241 MacCamy, R. and Fuchs, R. Wave forces on a pile - - a diffraction theory, Beach Erosion Board, Corps of Engineers, Tech Memo 69, 1954 Maeda, H. and Eguchi, S. On the hydrodynamic forces for shallow-draft ships in shallow water-on the heave hydrodynamic forces, Japan. Soc. Naval Archit., Spring Meet. 1976, pp. 45-52 Miles, J. and Gilbert, F. Scattering of gravity waves by a circular dock, J. Fluid Mech. 1968, 34, (4), 783 Newman. J. N. Exciting forces on fixed bodies in waves, J. Ship Res. 1962, 6, (3), 10 Tung, C. C. Hydrodynamic forces on submerged vertical circular cylindrical tanks under ground excitation, Appl. Ocean Res. 1979, 1, 75 Ursell, F. On the heaving motion of a circular cylinder on the free surface of a fluid, Q. J. Mech. Appl. Math. 1949, If (Pt. 2) Wang, S. The hydrodynamic forces and pressure distribution for an oscillating sphere in a fluid of finite depth, MI TDept. of Naval Architecture and Marine Engineering Report, June 1966 Wehausen, J. V. The motion of floating bodies, A. Rev. Fluid Mech. 1971, 3, 237 Ursell, F. On the virtual-mass and damping coefficients for long waves in water of finite depth, J. Fluid Mech. 1976, 76, 17 Yeung,R. W. A singularity distribution method for free-surface flow problems with an oscillating body, Report NA 7~6, Univ. of Calif., Berkeley, 1973 Yeung,R. W. and Newman, J. N. Discussion paper on Paper by Sayer, P. and Ursell, F. On the virtual mass, at long wavelengths, of a half-immersed circular cylinder heating on water of finite depth, l lth Symp. Naval Hydrodynamics, London, 1976, pp. 56(~ 561
Applied Ocean Research, 1981, Vol. 3, No. 3
133
A comprehensive set of theoretical added masses and wave damping data for a floating circular cylinder in finite-depth water is presented. The hydrodynamic problem is solved by matching eigen functions of the interior and exterior problems. The resulting infinite system is solved directly and found to have excellent truncation characteristics. Added mass and damping are given for heave, sway, and roll motion, as well as coupling coefficients for sway and roll. It is shown that the heave added mass is logarithmic singular and the damping approaches a constant in the low-frequency limit. Transition of the behaviour in finite-depth water to deep water is also discussed.
INTRODUCTION Added mass and damping of bodies oscillating in a free surface has been a subject of considerable research interest. Their calculations have traditionally been indispensable in the prediction of ship motions by the naval architects. In more recent years, the mushrooming of various nearshore and offshore projects has generated a substantial demand on the knowledge of hydrodynamics of floating structures operating in finite-depth water. Examples of these are the development of a floating airport in Japan and of an offshore floating nuclear power station in the USA. There exists a variety of elaborate numerical techniques for tackling the case by case situation. For example, Faltinsen and Michelsen 6 and Garrison 8 and Fenton 13 used the traditional wave source distribution method, Yeung 23 a hybrid integral-equation formulation based on the infinite-fluid source, Bai and Yeung I and Chen and Mei 5 finite-element variational formulations, and even more recently Berhault" an integro-variational method. f h e s e elaborate procedures are normally too expensive to allow one to investigate efficiently the effects of various parameters on the hydrodynamics of simple structures. Notably absent in the published literature is a comprehensive set of hydrodynamic inertia and damping coefficients for the relatively simple case of a vertical circular cylinder oscillating in a finite-depth fluid. Admittedly, this is a problem that can be solved, in fact if its not already solved in part, by any of the referenced procedures. As examples, Garrison 8 presented some results for a specific cyclinder in varying depths, Isshiki and Hwang 11 for another cTlinder at one specific depth but with incorrect results, and Maeda and Eguchi 15 for circular disks in varying depths. It is the intent of this paper to provide the ocean engineers with this missing gap of information. It is noteworthy that Garrett 7 and Black et al. 4 studied extensively the wave exciting forces and scattering characteristics of vertical cylinders. The present results should complement theirs. In a strict sense, by the use of 0141 1187/81/030119 1552.00 ©1981 CML Publications
Haskind's relation iv, one could deduce the damping coefficient from the published exciting forces, and with the use of the Kramers-Kronig relation (39 below) further calculate the added mass, provided that the infinitefrequency added mass were known. This is generally too much of a handicap for the practitioners. Because of the axi-symmetry of the geometry, the problem can be treated simply by the use of eigen functions alone. Extensive use of this technique has already been made by Hilaly 1°, Miles and Gilbert 16, Garrett 7 and Black et al. 4. The formulation described below follows closely that of Garrett, but the presentation, viewpoint, and solution are different. Garrett developed the expressions of the interior and exterior problems in terms of the potential at the common boundary, and match the normal derivatives together. In this work, the interior problem and exterior problem was treated as a Dirichlet and Neumann type, respectively, as if the conditions on the common boundary were known. By introducing expressions of the complementary region in the boundary condition, the matching is automatically completed. In this manner, the coupling of the coefficients of the two sets of eigen functions can also be seen very clearly. This approach apparently has not been taken by previous workers. All three types of radiation problems, heave, sway, and roll can be treated on a common platform. It should be pointed out that in Black's work 4 the farfield solutions of the radiation problems, which were solved by a variational formulation, were actually used to obtain the exciting forces by Haskind's relation, but no direct results on the radiation forces themselves were presented. Radiation forces for submerged circular cylinders were obtained recently by Tung 18, who solved for the flux at the common boundary before computing the coefficients of the eigen functions. The present procedure yields these coefficients directly. Of particular interest in the results shown is that the heave added mass is logarithmic singular, the nondimensional damping coefficient (non-dimensionalized in the standard way with the frequency in the denominator)
Applied Ocean Research, 1981, Vol. 3, No. 3
119
/tdded mass amt dampinq ol a vertical cylinder in finite-depth water. R. W. Yeunq Y
cylinder's bottom surface. We will proceed to obtain the solution(0 "~in the interior region (r ~
x
Interior solution z
The interior solution satisfying equations (2), (4), and (5) can be written as follows:
z=l
(7)
.~(i) q o-(- i'¢"..(i,)h _~_ ~ "V p
where the homogeneous solution @h° and particular solution ~o - r p~ satisfy the following boundary conditions:
h(=l)
d
t_- I: ~l)
"~'///////////
Figure A.
= o,
= ~
(8)
Cz [z=d
/~/// O C,,--n- = = O,d = 0 , (?q)!i)
Coordinate system and notations
a constant as the wave frequency approaches zero. Separate low-frequency asymptotic analysis confirms this behaviour, which can also be verified by the Kramers Kronig relation. Aside from the diagonal values of the added mass and damping matrix, also presented are the coupling coefficients between sway and roll.
~O(hi) r= a _- - q) (e) - -
(9)
It is noteworthy that the matching of the potential between the interior and exterior regions is effectively achieved by equation (9). After some calculations, we can arrive at the following results: (~) = ~d(Z 2 --
MATHEMATICAL
q~(i) p
r2/2)
(10)
FORMULATION
j
We consider a vertical cylinder of finite draft heaving time-harmonically in water of depth/~. Let the Oxy plane be the bottom surface of the fluid, (r,O) be polar coordinates in the horizontal plane, and the z-axis point upwards (Fig. A). At the outset, we will assume all space variables are non-dimensionalized by water depth. Thus, for example, a represents the radius to water-depth ratio; d the bottom clearance to water-depth ratio. If inviscidfluid flow is assumed, we can decompose the velocity potential O(r,O.z.tl as follows:
2
.
.
,, = 1 i,)(/.,y)
z,, = d
(11)
where I o is the modified Bessel function of the first kind order zero and % are Fourier coefficients, which are given by: d ~n = dr2 [(~(e) -- (~)]r
----oc°s(2.z) dz,
n =0,1...
(12)
0
O(r,O,z,t) = ] i R e [ - iaq)lr,z)~ ,e i,,]
(1)
where a is the angular frequency, ~3 the heave amplitude, and i = v / ~ l . In writing down equation (1) the axisymmetric behaviour of the fluid motion has been observed. The non-dimensional time-complex spatial potential q~(r,z) can be easily shown to satisfy the following equations: V2qJ(r,z) =0
/)~o _ ?z
v(p : =
1
=0,
(2)
~'
3z I:~o
=0
( ~ r _= 0 _
Applied Ocean Research, 1981, Vol. 3, No. 3
03 (a,,)c - " 0 s(z,,_Idz " " q)l,
,~
(6)
03)
L
O
which are immediately calculatable, we thus have the following relation:
(4)
where g is the acceleration due to gravity. Evidently, the only inhomogeneous boundary condition comes from the
120
~,,* __ -
2/ ~ , = d q¢~(a,z)cos (51
d<<,z<~l
d
d
.q 0~
Although the exterior potential is not known at the moment, the second term of equation (12) can be evaluated using equation (10). Defining the real coefficients
7z
.
dz-c(,
(14)
0
If% were known, then equations (10) and (11) describe the potential ~o(1) completely.
Exterior solution
The exterior problem can be thought of as one driven by a flux emitted by the interior region. Its solution can be written in terms of the following well-known eigen expansion22:
Added mass and damping of a vertical cylinder in finite-depth water: R. W. Yeun9 q~t~)(r,z)= L AkRk(mkr)Zk(mkZ)' 0 <~Zr>~a <~1
(15)
where the coupling integrals C,k are defined by:
k=0
2f C.k--~ COS -~z
where A k a r e unknown coefficients. The non-dimensional eigen values are solutions of the equations m tanh m=v
,
(Htol}(mor)
ktmkr~= )Ko(mkr )
for k=O for k ~> 1
. . . .
Zk(mkZ)dz
0
(16)
where m represents mo o r im k for k 7> 1. The functions R k are given by R"
n__0,1 k = O, 1....
d
2i - )"sinh(m°d)/m°d =
(17)
for k=O \moaJ A (24) for k = l , 2. . . . 2sin(mkd-- n~) mkd (mkd - mz) (mkd + mz)
The quantity S, occurring in equation (23) is defined by: where H o is the Hankel function of order zero, chosen to satisfy the radiation condition, and K o the Macdonald function of order zero. The functions Zk(mk z) form a normalized orthogonal set in [0,I] and are defined by:
(cosh(m,,z)//N~; '2, N o---~(1 + sinh(2mo)/(2mo)) Zk(m~z) = e ~"'C~O'Sm (2,kZN/) S t = ~(1 + sin(2mk)/(2m~))
(18) The inner product (Zk, Z j ) ~ 6 k j , where 6kj is the Kronecker delta, using equation (6) and matching the derivative of tpt~ with that of the interior solution at r = a we have:
d<~z<~h k=OmkAkR'k(mka)Zk(mkz)= q~i)(a,z) O<~z
{0
ng , / n x a \
/nxa\
which vanishes at n = 0 in this instance, but not the other modes of motion. It is understood that in equations (22} and (23) the R~,s are evaluated at toga. The coupled infinite system for e, and Ak is in an ideal form to eliminate either set of unknowns. One has the option of solving for either 5. or Ak first. It can be shown to be slightly more stable to obtain the former. For the heave problem, this also has the added convenience of allowing one to know ~0~i~with little further calculations. Hence, ct,- L e,j~j = g,,
(19)
for n = 0, 1, ...
(25)
j=O
where where the prime indicates differentiation with respect to the argument. By making use of the orthogona[ properties of the Z~,s in equation (19), and substituting equation (7) into the right-hand side, we obtain: d
gn =
mkAkR'k(mka) = f ~i)(a,Z)Zk(mkz~z + A~
where d
A *= f ~--~q~( a,Z)Zk(mkZ)dz
(21)
0
Equations (14) and (20) are the key results of our present analysis. In its present form, it is easy to see that an inhomogeneous boundary condition on the vertical surface of the vertical cylinder merely modifies the definition of A~ by contributing an integral from d to h(=l). Equations (14) and (20) are coupled because q ~ in equation (14) depend on A k, whereas q~0 in equation (20) depend on ~,. If equations (11) and (15) are now introduced, the following equations result:
~. = L RkC,kAk + e*
n = 0, 1, 2
(22)
k--0, 1, 2,...
(23)
k=0
*)
S n C n k & n -[- A k
n=
mkRk
CnkCjkfRk'~le
kk=0
"'k
/,
--[WYp'~k
(26)
\''k/_J
fR
- - ~n
(27)
(20)
0
A k=
~~
e
The infinite system (25) was observed to have excellent truncation characteristics. For a 1~ accuracy, it was rarely necessary to go beyond 20 equations. To the author's knowledge no other existing method allows one to obtain the solution so efficiently and conveniently. It is perhaps of interest to note that for the heave problem the coefficients e,, n = 1, 2 . . . . is decoupled from s o since S o vanishes; but the above structure is the most general, allowing immediate extension to sway and roll motions later on. To complete this analysis, we recall that the dimensional heave added mass P3~ and damping 3.33 can be obtained as follows: 2~
a
(28) 0
0
After performing the necessary integration and nondimensionalization, we obtain: /~3a + i23a - - J ] 3 3
"-[- i~33/ff
pr~fi3
Applied Ocean Research, 1981, Vol. 3, No. 3 121
Added mass .nd damping of a vertical cylinder m finite-depth water: R. 14/.. Yeung (l/d\
I [a\
Ire%
/d\ &
=21.4ka)-16~")+"[ ~4 + 2ta),,~'(-)
,7,S, ]) (;ini~
) ;' + log2 - log(m0a)] ~ + O(mgaqog(moa))
t~3.~ 2)4,/+ a[ -
l~
)
(37) (29)
I
/,s3-~4a l+(m~¢0-' n2 - ; ' + l o g 2 - log(re,a)
where we recall a is the dimensional radius,
O(m~a"~ Iog(moa)l
LOW-FREQUENCY ASYMPTOTIC BEHAVIOUR It is possible to obtain an expression for the low-frequency behaviour of the heave added mass and damping in the following simple manner. Since ~o~1 satisfies equations (2), (3) and (5), integration in the z-direction yields: 1
1 ? 80"') + mZq¢"~(r,0)= 0, r&.r-?r
q)i~)(r,z)dz
where (P(")tr)= (I
(30) and mo is the non-dimensional low frequency wave number, viz. m 0 =,,/v. Similarly, for the interior region, a
dl ? r-a E@i, + 1 =0, I" ( T
where
(oli)(rJ=d1 ( qo.~lr.z)dz
(31)
L} 1
+ (38)
where 7 is Euler's constant. Hence it can be seen that "~33 approaches a constant of value n&4h, whereas IL<~ is logarithmic singular. The constant value 0f233 can also be deduced by the use of Haskind's relation if one noted that in the long-wave limit, the exciting force per unit wave amplitude is simply p# multiplied by the water-plane area. The behaviour of #33 is less obvious. In fact, in this limit, since the heave added-mass in two dimensions reduces from a logarithmic singular behaviour in deep water ~9 to a finite-behaviour in shallow water z2-'4, and since the three-dimensional added mass in deep water is already known to be finite ~, one would be tempted to conclude otherwise. Mathematically, we may explain this by observing that the behaviour of the added-mass is related to the behaviour of damping by the Kramers-Kronig relations. In non-dimensional form, one of them ~2 can be written as:
~1"
j,
o
p(va)-p(~c)=~ ~(e) and ~(0 are the depth-averaged values, To leading order in mo, we may assume in equation (30) that ~01el(r,0) ~ - ~ , which is basically the shallow-water approximation 5. This is also equivalent to assuming that the terms corresponding to k > 0 in equation (15) are small: thus for a given mo the larger a is, the better the approximation. Equations (30) and (31) have the following solution form:
~o~el(r)= AoH~oll(mor ) a <~r <<,ec
2(z)d= =__ va o
(39) t
0
where ~ denotes that Cauchy's principal value is to be taken. Since I~33(~c) is a finite constant and ,~33(0) = g~l/4h, it follows that:
(32) lim p33(va)= K -4,~log(va)~ 2,~ l°g(m°a)
(40)
va~O
~o")(r)=%-r2/4d
O<~r<<,a
(33)
To determine A o and % we match ~(~) with ~9lil and ~¢fl with d ~ ~, the latter being the consequence of the fact that ~") and ~,1 each applies to domain of different vertical dimensions. The results are: ~le~(r)= a H~ol~(mor)
2moH]l)(moa) ~o(i)[r]_ a H(o1)(mOa) ' '-2mo~)
+
(34)
a 2 - - 1.2 4j
(35)
Integrating over the cylinder's bottom, we obtain the following closed-form solution
l[a
U33_1_i233=~ 4~+af(moa)
]
. f(moa) =
.,,,. o ~moa) rnoa HllI )(moa ) (36)
If we assume further that a/h = 0(1), the following limiting behaviour can be obtained as moa-*O.
122
Applied Ocean Research, 1981, Vol. 3, No. 3
which is precisely the singular behaviour arrived at earlier, Physically, we can explain this by imagining that the fluid bottom and rigid free surface together trap the cylinder as if it were a cascade in the vertical direction. This effectively forces the fluid particles to move twodimensionally in the horizontal plane. Viewing from the top. we now have essentially a pulsating two-dimensional source which is known to have a singular added mass. It should be pointed out that because of the neglect of the evanescent modes we cannot expect the simple matching conducted here to yield an exact value of the constant K. The error of this matching is of 0(e-~""~). This is consistent with the fact that equation (36) was observed to yield reasonably accurate prediction when 6//~ is > 1. To determine K exactly, the rigid free-surface problem must be solved, this can be done using the numerical procedure in this paper but is not pursued here.
SWAY M O T I O N It is a simple task to generalize the ideas presented in the first section for the case of sway motion. Because of the boundary condition
Added mass and dampin9 of a vertical cylinder in finite-depth water." R. W. Yeun 9 __ ~, fcosO~(t) ~ = nx¢ ~t) = ~ c~n 0
d<~z<~h, z=d
ct~*=0,
r=a O<~r<.a
n = 0, 1, ...
(50)
(41) J'sinh mo -sinh(mod)'~/tN1/2 m A*=~sin mk--sin (mkd) (/~ k k,
where n~ is the x-component of the interior normal ~and ~(t) the sway velocity of the cylinder, we define: ~(r,O,z,t) = hRe[q~(r,z)~ ]cos0
~H' t' >(m~,a)/H~' 'i Rk/R'k=(Kl(mka)/K ],
(42)
Hence, the boundary-value problem for the axisymmetric function ~0(r,z) is now as follows:
Sk =
(51)
k= 0 k >~l
(52)
I d/4a
k=0
(53)
~kTz
k >11
{ ~ll(mka)/l I ,
V2tp(r,z)-~=0
(43)
c~c~z v~p~= ~=0
(44)
If equation (25) is now solved as before, equation (23) allows the determination of the values of Ak'S for ~o(~) in equation (15). Whence, added mass and damping can be calculated and they are given by the following formula:
(45)
~11 +i211 "-'- ~p~2(~__d ) --a~l__U)k=O_,;-~, ~ AkA~R k (54)
(46)
When viewed from the formulation as presented, it is now evident that the sway problem has an identical structure to that of the heave. The roll-motion problem can be treated essentially in an identical manner.
_~11+i~al/a
3~z ~ ==a=0
O<<.r<~a
~z~ =o=0 =1
d<~z<~l
-1
~'
(47)
r--a
ROLL MOTION
Since equation (45) is homogeneous, it is clear that
q~.1 p =0
We assume that the roll motion occurs about the point (0,0,11. The dependence of the boundary condition on 0 is the same as the case of sway. It is appropriate to introduce the following definition of ~o(r,z)
(48)
Also, because of equation (43) the interior homogeneous solution is now given by:
(55)
O(r,O,z,t) = h2 Re[ ~o(r,z)~5]cosO --Y~a/
(49)
)~nz
-t- 2., ~n - c O S n=l Ia(2,a)
where (5 is the roll velocity. Accordingly, one may show in equations (43-47), only equations (45) and (46) have to be modified; viz:
while the exterior solution is still given by equation (15) with the order of the Bessel functions defined by equation (17) being 1 instead of 0. The inhomogeneity now arises from A*, which consists of merely an integral from d to h. Therefore, the entire solution procedure described earlier is completely applicable. In particular, only the following trivial modifications of the definitions of the quantities used (25) are necessary:
C3~z z z=d= --r
O<~r<~a
3~r rr=a=-(1-z)
(56)
I<<.z<~l
(57)
Table 1. Behaviour o["ct, and A k for varying number ~]' equations N=3
N = 10
N=20
Roll: d/h=0.5, h/d=2.5, t o o , = 1.0
~ k
Re
0 1 2 3
AkR'k lm
Re
~k lm
11,83429 2.61280 -2.43323 0.64106 0,22991 -0.02399 -0.28583 0.06968 -0,07093 0.00776 0.06081 -0.00804 0.03547 -0.00396 0.06216 -0.01022 a qPss, 2551=(0"37270, 0"11968)
k
Re
AkR'k Im
Re
~k Im
k
Re
AkR'k lm
Re
lm
0 1 2 3
11.82702 2.61625 -2.43558 0.64197 0.23270 -0.02433 -0.28172 0.06928 -0.07273 0.00798 0.06182 -0.00817 0.03678 -0.00412 0.05861 -0.00983 a i Pss,)-5,~)={ 0.37193, 0"119881
0 tl.82547 2.61695 - 2 A 3 6 0 6 0.64215 I 0.23338 -0.02441 -0,28097 0.06922 2 -0.07321 0.00803 0.06206 -0.00819 3 0.03716 -0.00416 0.05806 -0.00977 a t(p55, ),551={0.37179, 0.119921
0 1 2 3
63.42547 0.53782 -0.12836 0.05614
0 63.42550 5.37668 -28,67539 5.37678 1 0.53782 -0.00298 0.32239 -0.00180 2 -0.12836 0.00071 -0.08344 0.00047 3 0.05614 -0.00031 0.03751 -0.00021 a ~1P55, )-ss)=t 1.18235, 0.13446)
Roll: d/'h = 1.0, a//7 = 5.0, moa =0,5 0 I 2 3
63.42517 0.53784 -0.12836 0.05614
5.37663 -28.67510 5.37673 -0.00298 0.32237 -0.00180 0.00071 -0.08343 0.00047 -0.00031 0.03750 -0.00021 a-J(p55, 25~)={I.18258, 0.13446)
.
.
.
.
.
5.37667 -28.67536 5.37678 -0.00298 0.32239 -0,00180 0.00071 -0.08344 0.00047 -0.00031 0.03751 -0.00021 a I{it55, ,;.5~)=(I.18237, 0.13446)
.
.
.
.
L
Applied Ocean Research, 1981, Vol. 3, No. 3
123
Added mass and damping ola vertical cylinder in finite-depth water: R. W. Yeung
1.~ao b~
5,0
a=5.0
4.8--
I3.88
3.0m
0.60~
p.aa/~
~
}~33/a ~
-
2.0~
d=O.i0
8.28--
1.0--
~ 0.0
9
I 11
0
0 1.0/
/i.0
~ _
~"~--~.._
/o.9o
0.25,0.i0
I [ l , l l [ I 1,~ 20
r-~-r ]- ~--rm-q-3,~
--I-q~-~-VTqTT~-]T[-.T~-I-T-[-TTq--
0.00--
4 rfloa
moa
Fig. 1. 5 0
l
I. 00-~ a
a=l.0
o8o_ {3.60--
~33/a
d=O.lO
2.0
o.4
_
I I
1
0.25
04
I
0.75 0.9( /
I~.~
I .~
I 2.0 rlloa
T~--[--I-[--FT--] 3.0
4.0
1 .0
r,d.0
2.El
3.121 rnoa
4.~
5.1~
6
Fig. 2. Thus, the roll motion has the combined properties of both heave and sway: inhomogeneous conditions occur in both the interior and exterior regions. The particular solution satisfying equation (56) can be easily derived, and is given by:
9.(,~= - 1 F 2 r- ar3qj
158)
which contributes to both the calculation of ~* and Ak* Performing the necessary elementary integrals due to
124
Applied Ocean Research, 1981, Vol, 3, No. 3
equations (571 and (58), we get
t
,,
(
%* = i l
_ad
1
liar-']
[3-4,dJ ,
2ad( - ) " - i
tn~:)2
o d
(59)
,> 1
t
A*=J I ~r(" q#)"I.:. Zkdz + f(z - 1)Zkdz 0
,,:0
d
(60)
Added mass and damping of a vertical cylinder in finite-depth water: R. W. Yeung 5.0
1.00--.]- b a=O. 5
0.
4.0--
d=o: ~O
3. o - [ ~ _ _ _
0.60. -i.0
X33/a 0.40.
_
~.~
1 .0--
0)75
~
I I I
0.0
~
I [ I 1 .0
0.0
0.90
I ~
0.
[ 1" I 2.0
0,00-~-~
I [ J I I I I 3.0
4.0
0.0
1.0
2.0
moa
3.0
4.0
5.0
6.0
moa
Fig. 3. 2.00
8.0
-b
a=O. 2
I . 80--
7.0-~
eel 40-S.O--
d=O •I0
1 20--
la3s/a
X33/a.~ 1 00 _
4.0----
0.25 3.0-~
e Be-
0.75
Z
e 60-0.90
-4
0 40-0 20--
0 00--
e. e~-T--T.-0.0
I .0
2.0 Ill a 0
3.0
,.o
4,0
2.0
3.0
,.o
5.0
6.,,
rnoa
Fig. 4. Figures l 4. Added mass (a) and damping (b)Jor heaving motion: a/'h = 5.0; fi/h= 1.0: fi/'h =0.5; a/fi =0.2
sinh(mod)F3 2 3. 2 =Nk
I/2
L
-] m~I +"Jc°shm°
sinh(m°d) 2cosh (mod)] (tood)
smm,dlq
(61)
sin(mkd)F3'rnkd---LSa-- ~d32 + d]-..!2 F2COSmk L (mkd)--( c°s r a g + - mkd )J As far as solution of equation (25) is concerned these are the only changes when compared to the sway problem. If the following definition of the integrals:
Applied Ocean Research, 1981, Vol. 3, No. 3 125
Added mass and damping o/'a vertical cylinder in finite-depth water: R. W. Yeung 1 . 28
0.80~ d/h=l.
-
0
O. 70--
1.10--
-
I
4
a=l.
0
.3333
0.90
O. 20
IX33
_~ O •70--
$0
68--
kca=l. 0
X33 _
~
0.
\
~0.50
88--,
8. 4 0 ~ 1o
o.
"~
0 . 20
i
a
-'
moa
moa
Figure 5. Low-frequency behaviour o./heare added mass and damping as ai'h~O. ((l/]]=1.0)
2
.
5
0
8.
~
.......... Eqn (36)
--
'exact'
2. 0. a=2.0 -AI
X33/a
1'58--'11"~I -
""" .
.
a=5.0 . . ~
.
~a3/"
.
ii
.
.
.
.
.
.
0.40~
.
.......................
!
1 .00---4
I
~ d=O.l
:']=i. 0
.a=2.0 0.58 _
d=O.l/•
=
I'
"-'
"
.
d=l.
-"
0
a
o.~
b
''''
ri r-7--r- .
'l''-v-1
0.00-
2
l l,i
l,i
v-l-l-7--r--l--r- I i
~.~
,.,~
~.o
~.<~
,.,~
,~.o
moa
Figure 6.
Ek = F( - z)Zkdz = N~
d
1/2
~.<~
i
~.o
moa
Comparison of exact solution with asymptotic jormula
1l
,.~
( (d -- 1)sinh(mkd)mo(mod )4
cosh mo-cosh (rood) m °2
cos m k --cos (mkd)
(d-- 1)sin
(eq. 36)jot heave added mass and damping
mk
t~k 2
are introduced, the added moment of inertia P55 and roll hydrodynamic damping 255 can be written as: 126 Applied Ocean Research, 1981, Vol. 3, No. 3
(62j
I
I
''1
4..0
Added mass and damping of a vertical cylinder infinite-depth
water: R. W Yeung
0.80
b
a
d= 0.001
0.60 0.80
0.60+
0.40 \I
0.40
0.50
0.20
0.00 2:0
010
4:0
610
8 ,.I
0
mea
Fig. 7.
’.20/
a=l.O
a
II ;i
0.001
0.600.80
0.20
x11
0.60
4
0.40
0.401
n “.-” 5”
\
0.75 0.20
0.90
:
II
0 0:0
210
410
610
8 .0
0.0
2.0
mea
4.0
6.0
8
sa
Fig. 8.
0
i ’ f k=O
A,R,(m,a)E,
(63)
where the first two terms are contributions resulting from the bottom surface and the last from the side of the
cylinder. It is worthwhile to point out that unlike the case of sway motion, these hydrodynamic coefficients do not reduce to the case of a full cylinder touching the bottom as the bottom clearance reduces to zero. This is because the boundary condition on the cylinder’s bottom does not vanish in this limit and the interior particular solution still contributes to the calculation of tl,* and At. The coupling coefficients between sway and roll can be obtained from either the sway or roll solution. The expression from the roll motion is simpler because of the absence of the contribution from the cylinder’s bottom and is given by:
Applied Ocean Research,
1981, Vol. 3, No. 3
127
Added mass and dampi,w o/ a vertical cylinder infinite-depth water: R. W. Yeung
G 8~3~
2g . . . . . . . . . . a=O. 5 -4
'
a
00-~ /
i
b
d=
~- o.ool o
I
r/1
o.oo
8 @
2.8
6 @
4 0
8.8
2.~3
0.0
4.0
moa
moa
Fig. 9. 20
I .
8.0
610
0.80 a=O. 2
d= 0 0 ---//i/~--\ O. 001 I
b
a
4
@. 8.88--
60t
0.75
XH
laH 0.20 e. ee---
8,48--
-1
0.50
, ---0.90
0.90
0.40-~ 0.2{3--
7 O.S2.L__-------~ 0.
0.
@@---~--T
I
r
T
]
I
[
,
00
moa
moa
Fig. 10. Figures 7-10.
Added mass (a) and damping ( b ) . / b r s w a y motion: fi,,'h = 5.0: ~,"h = 1.0: ~/'h = 0 . 5 : fi.,"h = 0.2
"
-
+i)~5|/a pTza2h 2
-
1 a k: o
Ak k~k
(64)
where A *(~) is the expression given by equation (51). In the actual computations,/~15 + i)~5 obtained from the sway solution agree within 1% with/~51 + i25x, which provided a confidence check on the results.
128
Applied Ocean Research, 1981, Vol. 3, No. 3
N U M E R I C A L RESULTS A N D D I S C U S S I O N The numerical work involved in the solution of equation (25) is reasonably straightforward. Care, however, must be exercised in the summing of the series (26), whose major contribution comes mainly from values of k when mkd mz orj~. It is possible to convert the complex system to a real one but this option was not taken as it was found
Added mass and dampin9 of a vertical cylinder in finite-depth water." R. W. Yeun9 2.58
0.88. a
a=5.0
2. 8 8--: / ~ 0
,
~.e°I
d= .3333
~
b
0.50
8.48 1.0
0"3333~
8.88
' 8.8
I 2.~
'
I 4.8
'
I 6.8
' ~.8
Fig. 11.
~.8
~.8
~.~
nloa
moa
8.68
0. a
150
a=2.5
8.58--
8.48--
L_050
8.3B--
8.28--
8.10--
8.88--
' 8.8
Fig. 12.
J
2.~
'
[ 4.8
'
I 6.0
'
8.0
mo a
that most of the computation efforts are in the generation of the matrix elements rather than the solution of the system. The infinite system has excellent truncation characteristics. Table 1 shows typically how the first four coefficients of~k and A k, and the resultant added mass and damping behave as one increases the number of equations, N. Generally, convergence is slower for smaller a//~, and larger d//~. Numerical checks were conducted on the hydrodynamic damping by comparing the exciting force and moment with those given by Garrett v and Black and Mei 3, on the added mass by comparing results for the case of a circular disk ( d = l ) with Maeda and Eguchi 15. Excellent agreement was observed. Other checks based on
moa
asymptotic formulae discussed above as well as symmetric properties of the coupling coefficients were also applied.
Heave motion Figures 1~, show the non-dimensional added mass and damping for four different 6/h ratios, in descending order, with d/~ as a parameter. The coefficients are weighted by the factor/~/& The horizontal axis is chosen as moa instead of the more conventional parameter va since most of the variations concentrate near the zero-frequency end. The low-frequency behaviour of these curves is in agreement with that based on the asymptotic analysis. For 6//~ bigger
Applied Ocean Research, 1981, Vol. 3, No. 3
129
Added mass and dampinq of'a vertical cylinder in finite-depth water. R. W. Yeung
0 100
0,0,40-a
b
a=l.O
d=O.3333
"] X55/a
080
0
1%5h
0.020--!
0.50
0 0701 0.75
0.010-0 060-0.90 I
0 050-00
I
2,0
'
4.0
0.000
I
6.0
8.0
,0
2.0
rnoa
Fig. 13.
4.0
-
6.0
8
3.0
4.0
mo a
0.2S0
o.
"
0.
oo
100
°°°.5
0. 080 t
0. 0 6 0 - ~55/a --
~
~
Xss/a -
0. 1 0 0 - -
0. 040--
0.0S0--
0. 020-
0.90 ~0.75
0. 000 t 0.0
I
I
|.0
Fig. 14. Figures 11-14.
I
[
I
2.0
]
0. 000 :
i
3.0
4.0
! .0
2.0
Added moment o[ inertia (a) and damping (b)./br roll motion: fi..'h= 5.0: a.'h = 2.5: a'h = 1.0: fi..'h=0.5
than 0.5, and the added mass and damping have features similar to those of two-dimensional cylinders in deep water: viz. the damping decays monotonically from a finite value as the frequency increases, whereas the added mass starts from a logarithmic behaviour and attains a minimum before approaching its asymptotic infinitefrequency limit. The occurrence of a minimum is a necessary condition for satisfying the following equality21:
XJ
,J(ff(va)-I~(O°)d(va) ~ =0
0 130
0.0
moa
moa
Applied Ocean Research, 1981, Vol. 3, No. 3
(65)
which is valid for all modes of motion. This minimum is observed to be located at higher moa values for cylinders of shallower drafts. This is also the case for sway motion. As to be expected, shallow-draft cylinders have more damping than deeper ones of the same radius. It is interesting to note how the finite-depth behaviour transforms to its deep-water counterpart. Naturally, the transition occurs sooner for shallower cylinders than deep ones. Figure 3 shows the initial and Fig. 4 the full development of a maximum in the damping curves. The slight undulation in the added mass is related to the occurrence of these maximum (via K r a m e r s - K r o n i g ' s relation). Note that as 6 / / ~ 0 , equation (38) or equivalent
Added mass and damping of a vertical cylinder in finite-depth water." R. W. Yeung 3 . gO
3.0
~t51 0.50
2.0
75
0.75 0.
0 00
-I .0d 0.0
I
'
2.0
I
'
4 0
I
'
8.0
1
I
~.~
8.0
'
I
2.~
'
moa
t
~.~
'
I
~.~
' 8.0
moa
Fig. 15. 0.88----,
0. GO
a
a=2.5
0.
0, $ 0 - -
0.3333
0.20
0.40--
0.
)~51
O. 4 0
-
3333
50
}asl
0.30--
0.30
~ - - - O. 50
75
0.20--
0.90
0
0.
0.20,
90
-o.~o
_• 0.0
0.50 /
'
I 2.0
'
I 4.0
'
0.3333~
1 6.0
' 8.0
m° a
0.0
2,0
4,0
6.0
8.0
moa
Fig. 16. analysis yields 233 (0)=0, thus removing the logarithmic behaviour of the added mass in equation (39), and resulting in a finite value of zero-frequency heave added mass in deep water, which is well known from Havelock's results 9. We should note in passing that Wang 2°, who solved the problem of the heaving motion of a sphere in finite-depth water, has damping curves that behave precisely in this manner. Unfortunately, his added-mass low-frequency behaviour is in error. Figure 5 shows in a slightly clearer fashion how the transition we just discussed occurs. Figure 6 shows the effectiveness of the asymptotic formula (36) for four different geometric configurations.
This is representative of the type of agreement one gets when {l/h > 1 regardless of the value of d/h, For smaller h//~, the inclusion of the evanescent modes in equation (15) is necessary in order to obtain more accurate predictions (not shown).
Sway motion The sway added mass and damping are shown in Figs. 7 10. Regardless of the value of h/h the damping attains a maximum at mo a,.~ 1.2, then decays gradually. This is essentially the same behaviour as the exciting force of Garrett 7. The added mass curves each has now a
Applied Ocean Research, 1981, Vol. 3, No. 3
131
,4dded mass and dampin.q o/a certical cylimler infinite-depth water: R. W. Yeung 0.0S0-~
0
................................. a:l
. 040
0
~ d = 0 . 9 0
I
-0
0S0
~--
\
-//
-0 10 ~~0.20
-0. 060--
/
b
a -o.
\
150--
'
l
0.0
I
'
4.0
2.E~
I
'
6.0
8.0
0.0
2.0
4.0
6.0
8.0
ilia
moa
Fig. 17. 8
@. 0 5 8
82S
a=0.5
.90
d=O. 75 @
888-
0. 000-75 .50
-e. ese-_
_
//~..~
-8
82S-
-0
8S0-
~.0.3333
Ia51 -0. I00--
3333
)'51 -0 075-
.20
0.20
-8. 158--
-0.
-8
1 @e-
-0
125-
-0
150
200--
b
a
' 0
0
[ 2.8
I
1 4.0
I
I
6.8
' 8
0
i 0.0
I 2.0
I
~
4.B
T
]
6.0
r 8
Illoa
Fig. 18. Figures 15- 18.
Coupling added mass (a) aml dumping (b) between sway amt rolh fi.h = 5.0: fi.'h = 2.5: fi.'h = 1.0: fi..'h = 0 . 5
maximum and a minimum. We might add that while it may appear in Fig. 7 that the added mass decays gradually as moa increase, thus seemingly violating equation (65), in actuality computations showed that the curve eventually rises again after some large value of moa. The damping curves corresponding to d//~=0.001 can be compared directly with MacCamy and Fuch's classical result 14 of the horizontal exciting force on a full cylinder, The added mass results approaches the two-dimensional
132
Applied Ocean Research, 1981, Vol. 3, No. 3
limit of an infinitely long cylinder as moa-*O. In this limit, we note that a bottom gap clearance of 20% reduces a strip-theory estimate by 10 to 25% as •/h ranges from 0.2 to 5.0. It is possible to show that the limiting value of the sway added mass is given by:
l -d) /ql =
l+ii
for ~//~>>1
(66)
Added mass and damping o f a vertical cylinder in finite-depth water." R. W. Yeun9 Roll motion
ACKNOWLEDGEMENTS
T h e roll a d d e d m o m e n t of inertia and h y d r o d y n a m i c d a m p i n g are shown in Figs. 11-14. Since both the b o t t o m a n d the side c o n t r i b u t e to the h y d r o d y n a m i c m o m e n t , the b e h a v i o u r of the curves in terms of d/h is m o r e complex. N o longer w o u l d a shallow-draft cylinder necessarily have m o r e wave d a m p i n g that a d e e p - d r a f t one. The results d e p e n d s t r o n g l y on the ~/h ratio. F u r t h e r m o r e , with certain c o m b i n a t i o n of g e o m e t r i c p a r a m e t e r s , e.g. a / h = 1.0, d/h~-0.5, or ~//~=0.5, d / ~ = 0 . 7 5 , the roll d a m p i n g a b o u t the centre of r o t a t i o n b e c o m e s vanishingly small, resulting in an a l m o s t c o n s t a n t value of the a d d e d m o m e n t of inertia as a function of frequency. G a r r i s o n s has n o t e d this relatively c o n s t a n t b e h a v i o u r of a specific cylinder for which he was c o m p u t i n g m o t i o n response in a seaway. This p h e n o m e n o n is a p p a r e n t l y related to the possibility that the phasing a n d m a g n i t u d e of the wave systems due to the side a n d the b o t t o m are such t h a t they become cancelative. It occurs when the draft to r a d i u s r a t i o is a p p r o x i m a t e l y 0.5. Associated with this situation, is also a relatively small exciting roll m o m e n t . In fact, at certain specific frequencies, the roll m o m e n t practically vanishes altogether, This b e h a v i o u r is o b s e r v a b l e in the u n p u b l i s h e d d a t a of Black a n d Mei 3. The roll-sway coupling coefficients are shown in Figs. 15-18. T h e sign c o n v e n t i o n is t h a t a negative value of p51 or 251 will generate a clockwise m o m e n t a b o u t the axis of r o t a t i o n when the cylinder is given a positive sway a c c e l e r a t i o n of velocity. G e n e r a l l y speaking, the beh a v i o u r of b o t h a d d e d mass and d a m p i n g is very similar to the c o r r e s p o n d i n g b e h a v i o u r of sway except, of course, with the possible sign change. It is of interest to note t h a t cylinders of shallow drafts generate c o u n t e r - c l o c k w i s e m o m e n t s whereas the d e e p - d r a f t e d ones generate clockwise m o m e n t s . T h e t r a n s i t i o n occurs when the cylinder draft is a p p r o x i m a t e l y o n e - h a l f of its radius. This beh a v i o u r is consistent with the fact that in sway m o t i o n the pressure v a r i a t i o n on the surface of the cylinder is directly related to cos 0. Exciting forces T h e d a m p i n g d a t a presented in this p a p e r can be used to predict the exciting force a n d m o m e n t . F o r c o m p l e t e ness, we p r o v i d e the f o r m u l a e below. If the incident wave of unit a m p l i t u d e is a s s u m e d to be of the form R e [ e x p ( i m o x - i a t ) ] , a n d if X 1, X3, (Xs) designates the c o m p l e x a m p l i t u d e of the wave exciting force (or m o ment), then
IXal/rtPga2
=~4(1-d)D L lrmo a2
2 , , ] '12
(67)
The a u t h o r t h a n k s Mr. Sea H. K i m for his valuable assistance in the graphics of the results. This w o r k is s u p p o r t e d p r i m a r i l y by the Office of N a v a l Research, u n d e r T a s k N R 062-611. P a r t i a l s u p p o r t by the N a t i o n a l Science F o u n d a t i o n u n d e r c o n t r a c t E N G 7 7 17187 is also gratefully acknowleged.
REFERENCES l 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
iX31/7~p,Qa2= [
n~oa2D ,~33] 1/2
(68) 21
4 D )~5511/2 IXsI/'nPaa3=[ ~ooJ
(69)
where D(mo) = t a n h m o + mo(l - tanh2mo). T h e phase angles of the force a n d m o m e n t , t h o u g h not given here, are a v a i l a b l e from G a r r e t t 7 a n d has the very f o r t u n a t e feature of being practically i n d e p e n d e n t of the g e o m e t r i c p a r a m e t e r s a / h a n d d/X!
22 23 24
Bai, K. J. and Yeung, R. W. Numerical solution to free-surface flow problems, lOth Syrup. Naval Hydrodynamics, Cambridge, Mass, 1974 Berhault, C. An integro-variational method for interior and exterior flee-surface flow problems, Appl. Ocean Res. 1980, 2, 33 Black, J. L. and Mei, C. C. Scattering and radiation of water waves, Water Resources and Hydrodynamics Lab. Rep. 121 (MIT), April 1970, 248 pp. Black, J. L., Mei, C. C. and Bray, C. G. Radiation and scattering of water waves by rigid bodies, J. Fluid Mech. 1971, 46, (1), 151 Chen, H. S. and Mei, C. C. Oscillations and wave forces in a man.made harbor in the open sea, lOth Syrup. Naval Hydrodynamics, Cambridge, Mass, 1974 Faltinsen, O. M, and Michelsen, F. C. Motions of large structures in waves at zero Froude number, Proc. Marine Vehiclesin Waces Symp., london, 1974, pp. 91- 106 Garrett, C. J. R. Wave forces on a circular dock, J. Fluid Mech. 1971, 46, (1), 129 Garrison, C. J. Hydrodynamics of large objects in the sea, Part II: Motion of free-floating bodies, J. Hydronautics 1975, 9, (2), 58 Havelock, T. Waves due to a floating sphere making periodic heaving oscillations, Proc. R. Soc. (A) 1955, 231, 1 Hilaly, N. Diffraction of water waves over bottom discontinuities, Univ. of California, Berkeley, Rept HEL 1 7, Sept. 1967, 142 pp. Isshiki, H. and Hwang, J. H. An axis-symmetrical dock in waves, Seoul National University, Ship Hydrodynamics Laboratory Rept. No. 73--1, January 1973, 38 pp. Kotik, J. and Mangulis, V. On the Kramers-Kronig relations for ship motions, Int. Shipbuild. Progr. 1962, 9, (97), 361 Fenton, J. D. Wave forces on vertical bodies of revolution, J. Fluid Mech. 1978.85, [2), 241 MacCamy, R. and Fuchs, R. Wave forces on a pile - - a diffraction theory, Beach Erosion Board, Corps of Engineers, Tech Memo 69, 1954 Maeda, H. and Eguchi, S. On the hydrodynamic forces for shallow-draft ships in shallow water-on the heave hydrodynamic forces, Japan. Soc. Naval Archit., Spring Meet. 1976, pp. 45-52 Miles, J. and Gilbert, F. Scattering of gravity waves by a circular dock, J. Fluid Mech. 1968, 34, (4), 783 Newman. J. N. Exciting forces on fixed bodies in waves, J. Ship Res. 1962, 6, (3), 10 Tung, C. C. Hydrodynamic forces on submerged vertical circular cylindrical tanks under ground excitation, Appl. Ocean Res. 1979, 1, 75 Ursell, F. On the heaving motion of a circular cylinder on the free surface of a fluid, Q. J. Mech. Appl. Math. 1949, If (Pt. 2) Wang, S. The hydrodynamic forces and pressure distribution for an oscillating sphere in a fluid of finite depth, MI TDept. of Naval Architecture and Marine Engineering Report, June 1966 Wehausen, J. V. The motion of floating bodies, A. Rev. Fluid Mech. 1971, 3, 237 Ursell, F. On the virtual-mass and damping coefficients for long waves in water of finite depth, J. Fluid Mech. 1976, 76, 17 Yeung,R. W. A singularity distribution method for free-surface flow problems with an oscillating body, Report NA 7~6, Univ. of Calif., Berkeley, 1973 Yeung,R. W. and Newman, J. N. Discussion paper on Paper by Sayer, P. and Ursell, F. On the virtual mass, at long wavelengths, of a half-immersed circular cylinder heating on water of finite depth, l lth Symp. Naval Hydrodynamics, London, 1976, pp. 56(~ 561
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133