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Wave drift enhancement effects in multi column structures
Wave drift enhancement effects in multi column structures R. E A T O C K T A Y L O R a n d S. M. H U N G
London Centre for Marine Technology, Department of Mechanical Engineering, University College London, UK
A theoretical assessment is made of mean wave drift forces on groups of vertical circular cylinders, such as the columns of a floating offshore platform. A complete analytical solution is obtained for two cylinders extending from seabed to free surface, and a long wave approximation is found to provide reliable predictions of the drift force in line with the waves at low frequencies. For moderate separation between the two cylinders, this force is found to tend at low frequencies to a value four times the force on an isolated cylinder. A numerical method is employed to study two surface piercing cylinders truncated below the free surface, and an arrangement of four vertical cylinders characteristic of a floating offshore platform. The mean vertical drift force is found to be reasonably well approximated, over the frequency range of practical interest, by the force on an individual cylinder considered in isolation multiplied by the number of cylinders in the group. Interaction effects, however, have a profound influence on the total horizontal drift force. At low frequencies this force is found to tend to the force on an isolated cylinder multiplied by the square of the number of cylinders in the group.
INTRODUCTION
The design of floating and compliant offshore structures such as semisubmersibles and tension leg platforms has stimulated interest in the wave drift responses of such systems. Lightly damped resonances can be excited in irregular seas by non-linear wave effects, which yield forces at sum and difference components of the basic wave frequency spectrum. Because of the low natural frequencies of certain rigid body modes of floating and compliant structures, the difference frequency components of wave force can have profound implications for the platform motions. It has been found that observed behaviour may be successfully explained on the basis of a wave force model containing terms linear and quadratic in the body motions and wave kinematics. It is the quadratic components which are commonly assumed to be primarily responsible for low frequency drift responses of many such structures in irregular seas (e.g. refs. 2-3). A closely related problem is that of mean drift in regular waves. This is not only easier to model, but also sheds considerable light on the behaviour of compliant systems in irregular seas. It is useful therefore to examine the occurrence of the mean wave drift forces themselves. Considerable success has been achieved in the theoretical prediction of these forces, at least for relatively simple body geometries. But at least as important, from the view point of hydrodynamic synthesis, is development of an understanding of their dependence on underwater geometry. The latter has not hitherto received much attention in the published literature. This paper is motivated by the discovery that hydrodynamic interactions within a group of surface piercing bodies can profoundly influence the wave drift forces: the drift force on one body in a group can be increased several Accepted January 1985. Discussion closes September 1985.
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Applied Ocean Research, 1985. Vol. 7, No. 3
fold over the force on the same body in isolation. Previous work on first order wave forces 4-7 has illustrated some of the phenomena associated with interactions and multiple wave scattering, but it has been found that significant enhancement of these first order forces generally only arises when the bodies are relatively closely spaced (i.e. less than two diameters for vertical circular cylinders). Preliminary results discussed in ref. 8, however, suggest that drift forces may be much more strongly influenced by multi-body interactions. To shed light on this behaviour, we consider here groups of vertical surface piercing cylinders. The first order problem has been considered by several authors (e.g. refs. 4-7). Here we tackle the problem by means of three complementary approaches. We use a complete analytical solution to make the calculations for two fixed vertical circular cylinders, stretching from the seabed through the free surface; and we also develop a highly simplified version of this analytical solution, obtained with the aid of a long wavelength approximation. Finally we consider some numerical results for floating vertical cylinders, truncated at some distance below the free surface. These last have geometries characterised by typical semisubmersible or tension leg platforms. The drift forces on these more complex systems are evaluated by a combined finite element boundary integral procedure. The results suggest that, under certain realistic conditions, the total horizontal mean drift force on a group of N vertical cylinders may be in the region of N 2 times the force on a single cylinder in isolation. The corresponding total vertical drift force, however, is of the order of N times the isolated cylinder force. ANALYSIS FOR TWO FIXED VERTICAL CYLINDERS
We consider two cylinders of radius a separated by a distance s, extending from the seabed to the free surface, 0141-1187/85/030128-12 $2.00 © 1985 CML Publications
Wave drift enhancement effects in multi column structures: R. Eatock Taylor and S. M. Hung identical coefficients an:
*m(r2, 02) = ~
a.(--O'H(~2)(~':)exp(in02)
(5)
i,!=--oo
It is more convenient, however, to transform this into the co-ordinates rl and 01, using the Graf addition theorem for Bessel functions (equation 9.1.79 given by Abramovitz and Stegun).9 In terms of the geometry shown in Fig. 1 we have
92
H(2)(kr2) exp[--in(rrl2 -- 02)] =
r2
~
H(2m)+n(ks)
X Jm(krl) exp[--im(n/2 -- 001
(6)
Equation (5) is therefore transformed into
q~D2(rl,01) -- ~
an
n=--o,
~ (--i)mU(2m)+n(kS) m=--o,
x Jm(krl) exp(im01) =--X
Figure 1.
Co-ordinate systems for two cylinders
The boundary condition at cylinder 1 specifies that the normal velocity at the surface due to the scattered potential ~ol is equal and opposite to the combined fluid velocity due to the incident wave and scattering from cylinder 2. Hence
aOOl
_[aoi
Or1 (a, 0,) = as shown in Fig. 1. A fixed co-ordinate system Oxyz is located half way between the cylinders, with the z axis pointing positive upwards and Oxy in the mean free surface. The centres of the two cylinders, O1 and 02, lie on Oy. The depth of water is d. Long crested sinusoidal waves of frequency co and amplitude A propagate in the direction Ox. The combined incident and scattered velocity potential may be written ¢p = Re [~ exp(icot)] ~o(kz)
~o(gZ) -
co
cosh kd
aom
Lar--~(a, 0,) + - -
] (a, 0,)
an(--i)nu(z)'(ka) exp (inO,) n=--oo
=--
(1)
~
(--i)nJn(ka)exp(in01)
-- ~ an Z (--i)mH(m2)+n(kS)Jm(ka) n =--** m =--*~
x exp(imO 0 (2)
and k is the wavenumber. The potential ~ is conveniently decomposed into contributions ~/ and 0O from incident and scattered waves respectively. In terms of cylindrical co-ordinates r~, 01, we may express ~I relative to the centre of cylinder 1 as
(8)
Substitution of (3), (4) and (7) into (8) leads to:
where
igA cosh k(z + d)
(7)
(9)
which holds for all 01 in the range (0, 21r). Rearranging this equation and interchanging the order of the two convergent summations we obtain equations for an :
an--+ Jn(ka)
~ araI-I(m2)+n(kS)------1,--00<.<00 m :-** (10)
~z(rl, O1) = ~
(--i)nJn(krl) exp(inO0
(3)
n:--oo
The combined velocity potential expressed in the coordinates (rl, 01) is then given by (1), with
The scattered potential due to cylinder 1 may be written
e~(rl, 01) q~Dl(rl, 01) :
~ an(--i)nH(2)(kr,)exp(in01)
(4)
n =--c:c
where H (2) is a Hankel function of the second kind satisfying the radiation condition, and the coefficients an are to be determined from the boundary condition on the cylinder. In view of the symmetry about Ox, the scattered potential due to cylinder 2 must be expressible in terms of cylindrical co-ordinates r2, 02 in the same form as (4), with
(-i) n
exp(inO0 [Jn(~l)+ anH(n2)(krl)
+ m=-** ~ amH(m2)+n(ks)Jn(krt)]
(11)
This corresponds to the incident waves plus the total effect of scattering by the two cylinders. But with the aid of (10)
Applied Ocean Research, 1985, Vol. 7, No. 3
129
Wave drift enhancement effects in multi column structures: R. Eatock Taylor and S. M. Hung
this may be written in the simpler form:
where " denotes the single cylinder result. In the long wave approximation the velocity potential is obtained from
~(rl-, 01) ~b(y, 0 ) = =
~
X
(--i)nexpqn01)
IIU(ka)
a n IHn(2)(kr,)
jn (ka)
Jn(kr,)]
~ (--i)nexp(inO)IJn(kr) - - - ! H(2)(kriJ n=-~ k g2n
\ 2/
(121
\2//
In what follows we require this to be evaluated on the surface of cylinder 1 : ~(a, 01) =
L
(-i)nAnexp(in01)
(13)
?./:--~
4
r 2+
1 -- in
cos 20
+...
where we have defined An = (hn --g2n]n) an
(14)
hn = H(2)(ka);
(15)
/n = J,(ka)
gzn = H(2)'(ka)/Jn(ka)
(16)
The first and second order wave forces are obtained from ~. Provided therefore that we may obtain the coefficients an from (10), we may then use these in (12) or (13) as required. Spring and Monkmeyer4 used this approach to investigate first order forces. Results for second order drift forces are given below.
THE LONG WAVE APPROXIMATION In the case of small ka it is convenient, as well as being instructive, to develop an approximate analysis. This may be used to obtain results valid in the long wavelength limit (relative to cylinder radius). We employ the small argument expansions for the Bessel functions, given in the Appendix, to derive
1 +/(2) 2
(21)
where we have used,, (r, 0) instead of (rl, 01). This is equivalent to the approximation given by Lighthill (ref. (10)). To obtain the corresponding result for the two cylinder problem we must obtain the coefficients a n for equation (10), which we write as g2nan +
~
Hm+na m = -
I
(22)
m =--e~
where Hm+n=H~)+n(ks ). One set of coefficients may thereby be obtained for each dimensionless wavenumber ka and cylinder spacing ks, as in the method of Spring and Monkmeyer4 and others. Here, however, we invoke the long wave approximation, in combination with a further assumption, to derive results in a very simple form. Our further assumption is to consider the cylinders to be sufficiently separated, so that interaction between the Fourier harmonics in 0, in equation (12), is restricted to the lowest few terms. In other words, equation (22) is assumed to be diagonally dominant, with Hm +na m "~ g2nan
m > N
for a series truncated at the term +-N; and N is taken to be small. This may also be expressed as
n=0
nk@/
gzn(~) =
i {27 n 1----n!(n--1)!t; )
(17)
/kaY'
n>~l
n
for small ka. The success of the assumption can then be assessed by comparison of results with those based on solution of equation (22) with a large number of terms. Our simple solution is based on using only the terms n = 1, + 1. Equations (22) become
where a =/ca. Furthermore
-;,L
n=0
i4 n= 1
hn--g2njn~-
(18) goao + tloao + H l a l - - H a a - i
7T
i(n
1),(~3n
n/>2
and in each case the relative error in the approximation is no greater than ~ . The terms with negative n are obtained from g - 2 n = g2n
(19a)
h - n - - g - 2 n ] - n = (-- 1)n(hn --g2nJn)
(19b)
Equations (10) and (13) also lead immediately to the well known results for a single cylinder. In this case an . . . .
1
g2n
130
;
hn d, = ln--g2n
Applied Ocean Research, 1985, Vol. 7, No. 3
= -- 1
g2a_ 1 - - H l a o + Hoal + 112a_ 1 = -- 1
(23b)
g2al + Hlao + n2al + Hoa_ l = -- 1
(23c)
Subtracting (23b) from (23c) we obtain (g2--Ho+H2)(al--a_l)
= -- 2Hxa o
(24)
hence from (23a): [go + Ho-- 2H?(g2 -- Ho +/-/2) -1 ] ao = -- 1 In line with our earlier assumptions we now suppose
(20)
(23a)
go,g2>> Ho,H1,H2
(25)
Wave drift enhancement effects in multi column structures: R. Eatock Taylor and S. M. Hung Hence we deduce that ao ~ - -
1(T0) 1
He
(26)
go
1 #0) ~ 1 - - - (He+ H2) g2
Furthermore, from (24) a 1 -- a_ 1 ~
2 H , ( _HO~(lq 1-1o 1-I2 1 gog2 go/\ g2
[
-'~2H1 l + H o
(gl
express the factors in terms of al and a_l. Making use of the approximations given in equations (27) and (28) we obtain the results (35a) (35b)
1) H2] --go --~-2
(27)
gogl
A f'mal reduction is obtained by substituting the approximations for go,g2 in (17), and using the identity
and by adding (23b) and (23c) we obtain
~2)(ks) + ~2)(ks) = 2~(2)(ks)
al 4- a_l ~ - - 1 - - - (He + Hz) (28) g2 g2 Equations (26)-(28) provide relations for the coefficients an which are in convenient form for evaluation of the first and second order wave forces.
ks
Therefore we have iTr
iz0) ~ 1 -- 2 (ka)(a/s) Ha(2)(ks)
(36a)
#(1' ~--4(ka)2Hl2)(ks)[ 1+ 4 (ka)2 (2n~2'(ks)--H~i)(ks))] FIRST ORDER WAVE FORCES
The first order force on cylinder 1 is F O) = Re [fO) exp(i~t)]
(29)
where the components of fO) are obtained from the linearised Bernoulli equation as
0 27r
fY), -
i f ~lsino~adO~o(kz)dz ccos0
(30)
-d 0 By substituting equations (2) and (13) we readily derive the exact results fxO) = 7rPaA~2
k2
(A~--A_,)/i
(31a)
f(1) = 7rpaA~_~2 (A1 + A_I)
Values of these magnification factors, #xO) and #yO), are given in Tables 1 and 2 respectively, as functions of the spacing parameter s/a. Results are given for values of dimensionless frequency ka = 0.05, 0.10, 0.15, 0.20. For each ka and s/a combination in the tables, two figures are given. The upper is obtained from the more complete solution, a based on truncation of equation (22) at the terms m =-+ 10 (i.e. equivalent in accuracy to the results of Spring and Monkmeyer). 4 The figure placed lower in the table, in parentheses, is obtained from the approximate solution (equations (36a) and (26b) for Tables 1 and 2 respectively). It may be concluded that the influence of interaction effects on the first order wave forces is well approxiamted by the simple approximation, over the range of parameters specified. In particular terms, however, the effects are rather small.
(31b) SECOND ORDER WAVE FORCES
The single cylinder exact results, from (20), are obtained
as fix(')-
2iTrpaA ~o2 [ h1 F ~ ]'--Z]
(32a)
L (1) = 0
(32b)
The long wave length approximation for the single cylinder is
fx(l)~--~pa2Aw2[1-iTr(-~y]
(33)
It is convenient to express the influence of the second cylinder in termsof dimensionless magnification factors, defined as ,tt(1) _ f(x1) A I - - A - I { .
hl'~ 1
2
g.._22
= - 2 (a,+a_,)
r/= Re [A¢ exp(i~t)]
~
2ff f(~)
dO
(38)
d0 dz
(39)
iT/ ]r,=a ~ Sin 0
Pg
0 and components of mean pressure drop force
hl)l g2
\--~2/
(37)
Thus for cylinder 1 we have components of mean waterline force
(34a)
f(1) _ Al+A_l(jl .(1) _JY
t,'-y --?~,)
We are concerned here with mean horizontal drift forces on fixed cylinders, which may be obtained by means of a far field momentum formulation, or direct integration of the fluic pressure on the submerged surfaces of the cylinders. Using the latter approach, it has been shown by Pinkster, 2 Standing, Dacunha and Matten 3 and others, that the force on a fixed body is the sum of a waterline force f(~) and a pressure drop force fA2). These quantities may be expressed in terms of the velocity potential given in equation (1), and the corresponding free surface elevation
---~(al--a_l) (34b)
In these exact expressions we have used equation (14) to
x~= 1 -d 0
\sin0
Applied Ocean Research, 1985, Vol. 7, No. 3
131
Wave drift enhancement effects in multi column structures." R. Eatock Taylor and S. M. Hung Table 1. Firstorder magnificationfactors ~ ) 5
10
25
50
100
1.044616 (1.042480) 1.012418 (1.012289) 0.996058 (0.996081) 0.998417 (0.998435)
1.048832 (1.046260) 1.009562 (1.009800) 0.998253 (0.998150) 0.999855 (0.999884)
1.051551 (1.049020) 1.002654 (1.003519) 0.999768 (0.999771) 0.999606 (0.999609)
1.050828 (1.049450) 1.001839 (1.001840) 0.999541 (0.999536) 0.999942 (0.999950)
1.011716 (1.011559) 0.999050 (0.999088) t.001219 (1.001220) 1.000518 ( 1.000520)
5
10
25
50
100
0.005294 (0.005512) 0.006992 (0.007144) 0.007183 (0.007379) 0.007663 (0.007973)
0.011568 (0.012132) 0.012224 (0.012638) 0.010529 (0.010943) 0.000701 (0.000704)
0.019016 (0.020160) 0.018529 (0.019376) 0.001015 (0.001020) 0.001961 (0.001987)
0.027768 (0.029812) 0.001520 (0.001529) 0.002763 (0.002801) 0.003592 (0.003683)
0.002916 (0.002954) 0.004054 (0.004113) 0.005074 (0.005204) 0.005431 (0.005644)
0.05 0.10 0.15 0.20
Table 2. Firstorder magnificationfactors ~(~) k a ~ 0.05 0.10 0.15 0.20
Here .~a denotes the mean component o f x 2. By substituting equations (1) and (2), and performing the integration with respect to z in (39) we may write these in the form f(~)f~)} = _ 1
y
_ 21"7[ /cos 0 )dO
expression for C(O) is found to be c(o)
=
(--i)m-nAmA*n
x exp[i(m--n)O]
(40)
4 pgA~a J q~*~ sin O o
1m ~ ~
(
l +sinh2kd /
mn
27[
fA~
lpgA 2a
1 1
2kd
o 1(
s 2kd
\
"~
•
Only terms for which (m -- n + 1) = 0 contribute to the integral, and we have
kdJV¢ q/cos2)
+~-1 sin]a--~kd)~O*J~sin
dO
(41)
where * denotes complex conjugate; also ~, V¢ are evaluated at r = a, and V in equation (41) is the two dimensional gradient operator. The total mean horizontal drift force vector may then conveniently be written f(2) =f(2) + if~2)
(42)
where
sinh 2kd ] n=_~ X( n (kn~- i 1)
1)
(45)
Equation (45) gives the general expression for mean horizontal drift force on a circular cylinder. If we substitute An from equation (20) we will obtain the drift force on an isolated cylinder. It may be noted that in fact An = (-- 1)nj - n, so that
27r
f(x2'=f(~)x+e(jl~)x:=lpgA2aRe[f C(O)exp(iO)dO] o
(43a) x Im [in J*+l]
27[
f(2)=f(~)+f~=~pgA2aRe[f
C(O)exp(iO)dO] o
(43b)
The integrals with respect to 0 are evaluated by substituting the Fourier representation of ¢ given in equation (13). The
132
Applied Ocean Research, 1985, Vol. 7, No. 3
fy(2) = 0
(46a) (46b)
These results are exact to second order in wave steepness. The long water approximation for the drift force on an isolated cylinder is obtained by means of equations (17), (18) and (20), using a = ka:
Wave drift enhancement effects in multi column structures: R. Eatock Taylor and S. M. Hung 2kd
).
--16
~
f(2) ,~ ~-2pgA2a( 1 + sinh 2kd/Im ff ot2 ]
16 +(19rr232
;).2a3
}
3
To obtain the appropriate drift forces on the two cylinder configuration, we revert to equation (45), and as for the first order forces we truncate the series so that only the terms n = 0, -+ 1 are included. Again it is convenient to define dimensionless magnification factors: f(2)
Im(A_,A~+AoA ~
=/(2) =
2 Im(.,ioA*)
f(2)
(48a)
Re(A_,A*+AoA*)
= fx(2) -
2 Im (A o.Zi*)
(48b)
For consistency in these approximations, we evaluate the denominator also using the terms n = 0, + 1: it is therefore given by 2 Im (AoA*) ~ 7r(/ca) a
(49)
The numerator in each case is evaluated by means of the following readily established identify:
A_IA* + AoA*= Re[A*(A_, + A,)]+ i Im[A*(A_,--A,)] (5o) From equations (14) and (18) we have i 2 A o : - - ~ r (a-S)ao
A_i+AI=~
i(4)=
(51a)
(--a_,+a,)
,(4)
A_x--A, =-- ~" ~ (a-1 + a,)
(51b)
(51c)
We may therefore proceed from the approximations given in equations (26)-(28), obtaining
=--r2a----~ Im ~og2 ]
g, j 1 -
S/a=
5.000
x
S/e=
18.008
[]
S/o=
20.000
Eqn.
53a
(47)
/(2) ~. I n2pgA2a(ka) a
#(2)
2.500
+
++÷.+v++ +
If we consider the case of deep water, and neglect all terms except the first in the above series, we recover the result
"x(2)
2.881
S/a=
- -
( 1 __i4~__~)(1 + ~-'~-~4)i32'~+ " "
,( 2kd \ ogA2a 1 + s . ~ - ~ ) ( k a )
S/a= v
(Ho+H2) (52a)
t I"
2
4
6
8
10
kS
Figure 2. Mean drift magnifications for two bottom supported cylinders
The results for/ax(2) are plotted in Fig. 2 as functions of ks over the range 0 to 10. These are compared with results from the more complete solution, 8 for the spacings (s/a) = 2.5, 5, 10 and 20. Some results from the more complete solution are also given for the cylinders touching (s/a = 2.001). It is seen that at large spacings the low frequencies, the approximate solution for/~x(2) given by equation (53a) is surprisingly accurate. A remarkable feature of the results for /ax(2) is the behaviour at low values of ks. Both the accurate numerical results and the simple approximate analysis suggest that as ks tends asymptotically to zero, /ax(2) tends to 2. The physical interpretation is that the drift force on one of the pair of the cylinders tends to twice the force which would act on the cylinder in isolation: indeed the total horizontal drift force on the two cylinder group, in line with the direction of wave propagation, tends to four times the force on a single cylinder in isolation. Results described below for other mult-cylinder geometries suggest that this is a special case of a more general law governing the horizontal drift force on a group of vertical cylinders. Most importantly, the force cannot be approximated by the sum of the forces on the individual cylinders taken in isolation. Such a simple superposition would underestimate the resulting drift force by at least 37% over the range ks < 2. The error could exceed 100% for close spacings and low frequencies. Turning now to the results for/ay(2) , we fimd that the approximation of equation (53b) is hopelessly at variance with the accurate solution. It appears that at least two Fourier harmonics are essential to represent with any degree of verisimilitude the second order pressure gradient in line with the two cylinders. Indeed this is hardly surprising. However, the parameter lay(2) is much less significant than /ax(2), since it corresponds to the mean drift splitting force on the cylinder group. Unless the two cylinders are free to move independently, both first and second order splitting forces are resisted by structural stiffness. The effects of the latter forces are then insignificant compared with the former.
(52b) Substituting equation (17), we finally obtain the consistent lowest order approximations (i.e. independent of ka) as:
#(2) ~_, 1 + Jo(ks) + 12J2(ks)
(53a)
/@(2)~ _ 12}'1 (ks)
(53b)
NUMERICAL RESULTS FOR OTHER MULTICYLINDER GEOMETRIES
We consider now drift enhancement effects for cylinder arrangements which are more representative of floating structures such as semisubmercibles and tension leg platforms. It may be anticipated that the behaviour described
Applied Ocean Research, 1985, Vol. 7, No. 3
133
Wave drift enhancement effects in multi column structures: R. Eatock Taylor and S. M. Hung above, for two cylinders extending to the seabed in deep water, would also be manifested for two surface piercing cylinders truncated somewhat below the free surface (i.e. two columns of a TLP). This is because the vertical distribution of horizontal drift force decreases very rapidly with distance below the free s u r f a c e - indeed a major compoment arises only in the vicinity of the waterline. Here we verify this conjecture and extend the previous results, by examining horizontal and vertical drift force enhancement effects for groups of two and four equal vertical cylinders, truncated below the free surface. We show that interactions between the cylinders do not lead to significant enhancement of mean vertical drift effects, but mean horizontal drift forces can be strongly influenced when the wavelength is sufficiently large compared with cylinder diameter. To obtain these results, a well proven numerical method was employed, based on a combined t'mite element and boundary integral procedure. Finite elements are used to represent the flow field in the vicinity of the submerged body surface (in this case the cylinder group), and a boundary element idealisation is adopted on a fictitious surface surrounding the body. The latter surface is defined by a set of vertical and horizontal orthogonal planes, thereby ensuring a highly efficient boundary element approximation. Quadratic isoparametric elements are used in the f'mite element region, and the fictitious outer surface is represented by quadratic boundary elements. A description of the method and validation of the computer program (DYHANA) is given in Ref. 11. The cases considered here are based on groups of cylinders of radius 9 m and draught 32 m, in water depth 148 m. The geometry for the single cylinder (Case 1) is shown in Fig. 3a, with the associated element mesh in Fig. 3b. In Case 2, two such cylinders are separated by 54 m (centre to centre), in a line perpendicular to the direction of the incident wave. This is closely related to the two cylinder case considered above, except that now the cylinders do not extend to the seabed. Case 3 is similar to Case 2, with the line joining the cylinders making an angle of 45 ° to the direction of the waves. In Case 4, four cylinders are placed at the corners of a square, of side 54 m, and the incident wave propagates in a direction parallel to two sides of the square. Case 5 is similar to Case
4, but the incident wave is directed parallel to a diagonal of the square. The arrangements for the five cases are summarised in Table 3. The finite element meshes for Cases 2-5 are based on the single cylinder mesh (Case 1) shown in Fig. 3b. This is facilitated by exploiting one or two planes of symmetry, as appropriate: these are specified in the input to the computer program DYHANA. Some results for the total mean horizontal and vertical drift forces on the cylinder groups are given in Table 3, for frequencies within the range 0.125-0.65 radians/s (i.e. wavelengths from 1842m down to 146m). For long wavelengths the results for Cases 2 and 3 are seen to be very similar, as are those for Cases 4 and 5. In the limit of zero frequency the horizontal drift force always tends to be zero. But more significant is the ratio between the forces at a given frequency or wavelength, for the difference cases. At low frequencies ~ahe total vertical forces in Cases 1, 2 and 4 are in the approximate ratios 1 : 2 : 4. The corresponding ratios for the horizontal forces are approximately 1 : 4 : 16. This suggests that the total vertical force is simply the sum of the forces on each cylinder considered in isolation; but at low frequencies the total horizontal force is the corresponding single cylinder force multiplied by the square of the number of cylinders in the group. This appears to be a generalisation of the two cylinder result obtained from the analytical formulation of the previous section. The enhancement of the horizontal drift forces, arising from the interaction effects in the two and four cylinder arrangements, is clearly illustrated in Fig. 4. This shows the ratio of the total mean horizontal force on the group on N cylinders (N = 2 or 4 here) to the mean force on N isolated cylinders. At low frequencies this ratio appears to tend to the value N, rather than to the value 1 given by the sum of forces on the cylinders neglecting interaction effects. The same low frequency limit is reached regardless of the precise plan arrangement of the cylinders. Of considerable practical interest is the rate at which the drift enhancement effect decreases with increase in wave frequency (or ka), for a given geometry. It may be noticed in Fig. 4 that for Case 2 (two cylinders in a line parallel to the wave crests) the decay is much slower than for the other cases. Up to the highest frequency plotted (corresponding to a period of approximately 10 s), the drift enhancement decays monotonically to a value of about 1.4. In Cases 3-5, however, the ratio drops lower than one within the range plotted, although for Case 4 it again exceeds one at the highest frequency point. It is important, of course, to relate these results, based on an ideal flow calculation, to the expected behaviour of the real fluid. Chakrabarti 12 and Standing et al.,3have compared the mean drift force associated with diffraction at a single cylinder (i.e. Case 1 considered above)with the steady force arising from viscous effects. They show at low frequencies the theoretical ratio of viscous to potential drift force in deep water is given approximately by
ca
I///////////////////// a)
Oin,ension8
b)
E l e m e n t mesh
Figure3. Floating cylinder geometry: (a) dimensions; (b) element mesh
134
Applied Ocean Research, 1985, Vol. 7, No. 3
A
R - - 2rr (ka) 2 a
(54)
(in terms of the parameters defined above, with drag coefficient Ca). If we take Ca = 0.7 and A = 4 m (i.e. a wave height of 8 m), this suggests that potential effects exceed viscous effects on the single 9 m radius cylinder provided that co > 0.49 rad s-1 (ka > 0.22). Under related conditions, the drift enhancement effects for the two
Wave drift enhancement effects in multi column structures: R. Eatock Taylor and S. M. Hung
-,
N-2
?
illl
I
I
]
l
[
l
l
©
t
V
N-2
x
N-4
+
A
A
•
•
N-~
A
A V
A ×
E
V
l i l l l l l
0
÷ I o.1
, N
I 0.2
I 0.3
I
I
0.~
0.2
a¢
I 0.6
I --
o 7
~/(rad s-l)
~ Q Q Q Q
Figure 4. Mean drift enhancement effects for groups of N floating cylinders
0
0
0
0
0
0
0
0
0
IIitl]] 0
0
0
0
0
0
IIIIIII ©
0
0 0 0 0 0 0 0
IITTTII
cylinder configuration (Case 2) appear to be of practical significance, although the influence of interactions on the viscous contribution is at present unknown. The design implications of the four cylinder results in Table 3 and Fig. 4 are also still speculative. At ¢o = 0.49 rad s -l, corresponding to R = 1, the total horizontal drift force for Case 4 may be estimated as some 50% of the force on four isolated cylinders. Hence the ~iscous effects predominate under these particular circumstances. On the other hand, at the highest frequency plotted in Fig. 4, the value of R given by equation (54) is 0.33 and the drift enhancement for Case 4 is 1.21. For these latter conditions, therefore, potential effects appear to predominate, and interaction effects between the four cylinders may be significant. Overall, the results suggest that it may be unwise to disregard the enhancement of mean horizontal drift forces, due to wave scattering by multiple cylinders at low frequencies. Expressions such as equation (54), intended to provide an estimate of the relative magnitude of viscous and potential drift effects, are unlikely to provide reliable criteria for multi-colunm structures. Further work, both theoretical and experimental, is required before general rules become available for making simple estimates of the drift behaviour of such structures.
0
t
CONCLUSIONS
E ~ d d d ~ d
C ~4
~ 0 0 0 0 0 0
The preceding analysis and results enable certain conclusions to be drawn regarding prediction of mean drift forces on multicolumn structures. The major finding is that where wave diffraction is primarily responsible for mean drift forces, hydrodynamic interactions can in principle lead to substantial increases in drift force, as compared with results obtained by simple superposition without regard to interaction. The following points may also be noted: (i) An analytical solution has been obtained for the mean horizontal drift force on two circular cylinders placed in a line parallel to the incident wave crests. The cylinders extend from the seabed through the free surface. The effects of hydro-
Applied Ocean Research, 1985, Vol. 7, No. 3
135
Wave drift enhancement effects in multi column structures: R. Eatock Taylor and S. M. Hung
dynamic interactions are substantially more pronounced for mean drift forces than for first order wave frequency forces. (ii) A simple long wave approximation has been developed for the two cylinder problem. This gives good agreement with the more accurate solution for first order forces as long as ka <<,0.2 and s/a/> 5. The agreement for the mean drift force in line with the direction of wave propagation is also satisfactory over this range. (iii) The long wave analysis shows that at asymptotically low frequencies (and a sufficiently large spacing between the cylinders) the total mean horizontal force on the pair o f cylinders tends to exactly twice the total force which would be predicted by summing the forces on each cylinder considered in isolation. (iv) Numerical results from a finite element - boundary integral approximation confirm this behaviour for horizontal force on two circular cylinders penetrating the free surface but truncated well above the seabed. However, the total mean vertical force on such a pair of truncated cylinders tends to be the same value as would be obtained by adding the forces on each isolated cylinder. This low frequency asymptotic behaviour appears to be independent of the precise arrangement of the two cylinders relative to the direction of wave incidence. (v) Further numerical results, for four truncated cylinders, suggest the existence of a general rule:* the total mean horizontal drift force on a group of N cylinders tends to N 2 times the force on one cylinder in isolation; but the total vertical force tends to N times the equivalent single cylinder force. Hence interaction effects are shown to be potentially important for horizontal drift forces, but not for vertical drift forces. (vi) The enhancement or magnification of horizontal drift forces due to hydrodynamic interactions has been evaluated over a range of frequencies, for various two and four cylinder configurations. In the case of two cylinders aligned parallel to the wave crests, the enhancement effects decay with increasing frequency at a much more gradual rate than for the other cases examined. Substantial effects are predicted over a realistic range of physical parameters. (vii) At the frequencies where these enhancement effects are pronounced the possible contribution of viscous effects to mean drift forces must also be assessed. Expressions indicating the relative importance of viscous versus diffraction contributions may be misleading, unless the effects of hydrodynamic interaction are accounted for. (viii) There is a need for some experiments to test the aforementioned findings. To arrive at a method o f hydrodynamic synthesis which accommodates drift response criteria, it ~ also be necessary to perform further systematic studies of the influence of interaction effects in a variety of multi-body configurations.
REFERENCES 1
2 3 4 5 6
7 8 9 10 11 12
Newman, J. N. Second order slowly varying forces on vessels
in irregular waves. Proc. Int. Symp. Dynamics of Marine Vehicles and Structures in Waves, I.Mech.E., London, 1974, 182 Pinkster, J. A. Mean and low frequency drift forces on floating structures, Ocean Engineering 1979, 6,593 Standing, R. G., Dacunha, N. M. C. and Matten, R. B. Slowly varying second order wave forces: theory and experiment. National Maritime Institute, Report R 138, December 1981 Spring, B. H. and Monkmeyer, P. L. Interactions of plane waves with vertical cylinders, Proc. 14th Int. Conf. on Coastal Engineering, Copenhagen, Denmark, ASCE 1974, 1828 Ohkusu, M. Hydrodynamic forces on multiple cylinders in waves. Proc. Int. Symp. Dynamics of Marine Vehicles and Structures in Waves, I.Mech.E., London, 1974, 107 Matsui, T. and Tamaki, T. Hydrodynamical interaction between groups of vertical axisymmetric bodies floating in waves, Proc. lnt. Symp. Hydrodyn. Ocean Eng. Trondheim, 1981, 817 Mclver, P. and Evans, D.V. Approximation of wave forces on cylinder arrays, Applied Ocean Reasearch 1984, 6,101 Eatock Taylor, R. and Hung, S. M. Mean drift forces on an articulated column in a wave tank. Applied Ocean Research 1985, 7,61 Abramovitz, M. and Stegun, I. A. Handbook of Mathematical Functions, Dover Publications, New York, 1965 Lighthill, M. J. Waves and hydrodynamic loading, Proc. 2nd Int. Conf. on Behaviour of Offshore Structures 1979, 1, 1 Eatock Taylor, R. and Zietsman, J. Hydrodynamic loading on multicomponent bodies, Proc. 3rd lnt. Conf. on Behaviour of Offshore Structures 1982, 1,424 Chakrabarti, S. K. Steady drift force on vertical cylinder viscous vs potential, Applied Ocean Research 1984, 6, 73
APPENDIX - EXPANSIONS FOR SMALL ARGUMENTS
The following expansions are given by Abramovitz and Stegun, 9 for Bessel functions of a small argument:
jn =
n!\2/
136
Applied Ocean Research, 1985, Vol. 7, No. 3
2i[in(. + ~-k \ 2 1
ho=B~:)(a)~l
hl=Ht2)(a)~-~[2 1+; ;i(27]
hn = H(2)(a) ~ - (n -- 1)!
n >/2
7r
where 7 = 0.5772. Furthermore, the derivatives of the Bessel and Hankel f~nctions satisfy the following recurrence relations (where for our purposes Cn = Jn or//(2)): ~(a)
= - - C 1( a ) n
C~,(a) = C n _ l ( a ) - -
a
n
Cn(a ) = - Cn+ l(a ) + - Cn(a ) ot
We therefore obtain H~2)' (a)
Hl2)(a) ~---
gO= J~(a)
1 + /(_2"~
Jx(,~)
rr\a/
-H(~2~(a) + (n/a)H(~2)(a) g2n =
*The theoretical basis for this rule is discussed in Mclver's discussion accompanying this paper.
"
1 + n!
--Jn+ 1(a) + (n/a) Jn(a)
x
(n-l)!
~
n
~n!
~
Wave drift enhancement effects in multi column structures: R. Eatock Taylor and S. M. Hung _in. =1
ACKNOWLEDGMENTS
"2 "2n
n>~l
~r ' ( n - - l ) ! ( ~ )
For the terms corresponding to negative n, we use the identities
Jn = (-- 1)nJ - n ;
//(2) = ( - - 1)nil(-2)
From Dr P. Mclver, Department of Mathematics, University of Bristol, UK
This work was supported by the Science and Engineering Research Council and the Cohesive Programme on the Dynamics of Compliant Structures.
the first order component of the scattered wave field. The force vector for the ]th body may also be written as a series expansion in e, so that
DISCUSSION OF:
=
Wave drift enhancement effects in multicolumn structures
by R. Eatock Taylor and S. M. Hung In their paper Eatock Taylor and Hung have examined the mean wave drift forces on groups of vertical circular cylinders. Using a long wave approximation for the case of two cylinders extending throughout the fluid depth they found that, at low frequencies, the inline drift force on two cylinders tended to four times the force on an isolated cylinder. Furthermore, numerical results for four truncated cylinders were found suggesting a general rule for the calculation of wave drift forces on groups of cylinders in the low frequency limit. The authors speculated that the mean horizontal drift force on N cylinders tends to N times the force on an isolated cylinder, while the mean vertical force on the N cylinders tends to N times the force on an isolated cylinder. Here these rules are examined from an analytical point of view and confirmed. The results are derived under the usual assumptions of the inviscid theory of water waves. Some discussion of the effects of viscosity is given by Eatock Taylor and Hung. Consider a group of N identical, fLxed, axisymmetric bodies in water of uniform depth h. A co-ordinate system (r, 0, z) is chosen so that (r, 0) are polar co-ordinates in a horizontal plane and the vertical z-axis is directed vertically upwards with the zero at the level of the mean free surface. The origin of co-ordinates is chosen to be 'within' the group of bodies. Local polar co-ordinates (rl, 0]) are centred on on the axis of the jth body which is situated at r = sj. A uniform non-linear wave train is incident upon the bodies from the direction 0 = rr. The resulting wave field, including incident and scattered waves, will be represented by a velocity potential ap(r, 0, z, t) which may be written as a series in the form (I) = e ( I ) ( i ) "~- e2~ (2) + . . .
(1)
where e is a small parameter related to the wave steepness. The linear component of ~ may be written as b(i) = Re [f(z)(~O) + ~s(i)) exp (--/cot)]
(2)
where
igA cosh k (z + h) f(z)
-
eco
cosh kz
(3)
+ erp
+...
(s)
The mean (time-averaged) second order force on a fixed body ~(2)=
i f 2pg
rl2nidl+21 p f iVdPO)12ndS zi
si
+/3I¢/(0, 0, 1)
(6)
(see Pinkster) 1 where 1,] is the mean water line (with normal nt), S~ the mean wetted surface (with normal n), 77 the displacement of the free-surface from the mean level (due to the linear component of the solution) and I~ the waterplane area. The mean set down pressure P depends upon the second order component of the incident wave and is readily calculated (Longuet-Higgins and Stewart). 2 An overbar is used to denote a time averaged quantity. If the local terms that decay exponentially away from the body are neglected, then the scattered wave field for a single body can be written as
¢~1)(rj,Oi)= ~. AnHn(Icri) cosnOj
(7)
n=O
where Hn is the Hankel function of the first kind of order n. The complex scattering coefficients (An, n = O, 1 .... ) are, in general, functions of the wave frequency and the body geometry. For the case of a vertical cylinder extending throughout the depth of the fluid they may be determined analytically (see, for example, Mel, p. 312) 3 and it is found that A o is O((ka) 2) and A n is O((ka)2n), n 4=0 where a is the cylinder radius. For a circular cylinder that does not reach the bed or similar axisymmetric bodies it is reasonable to expect the scattering coefficients to behave similarly. Thus for long waves the scattered wavefield (equation (7)) within the array of bodies can be expected to be at most O((ka) 2 In(ks)), where s is a typical spacing and the small argument asymptotic forms for the Hankel functions have been used (Abramoxitz and Stegun, p. 360). 4 Hence, for long waves it is reasonable to neglect multiple scattering events within the array and a first approximation to the forces may be obtained by considering the incident wave field only. Consider first the mean vertical second order forces on the bodies. For long waves the scattered field may be neglected, then from equations (3) and (4), n = ½A 2
and $(/i)(r, 0) = exp(ikr cos 0)
(4)
represents the first harmonic of the incident wave train with frequency w, wavenumber k, and amplitude A: g is the acceleration due to gravity. The potential SsO)(r, 0) is
(8)
and q" 2
1[
gkA
]2
17" '1 = 2[e~-c-ososhk/~J
(9)
x [cosh2/~(z+ h) + s~/c(z +h)] Applied Ocean Research, 1985, Vol. 7, No. 3
137
London Centre for Marine Technology, Department of Mechanical Engineering, University College London, UK
A theoretical assessment is made of mean wave drift forces on groups of vertical circular cylinders, such as the columns of a floating offshore platform. A complete analytical solution is obtained for two cylinders extending from seabed to free surface, and a long wave approximation is found to provide reliable predictions of the drift force in line with the waves at low frequencies. For moderate separation between the two cylinders, this force is found to tend at low frequencies to a value four times the force on an isolated cylinder. A numerical method is employed to study two surface piercing cylinders truncated below the free surface, and an arrangement of four vertical cylinders characteristic of a floating offshore platform. The mean vertical drift force is found to be reasonably well approximated, over the frequency range of practical interest, by the force on an individual cylinder considered in isolation multiplied by the number of cylinders in the group. Interaction effects, however, have a profound influence on the total horizontal drift force. At low frequencies this force is found to tend to the force on an isolated cylinder multiplied by the square of the number of cylinders in the group.
INTRODUCTION
The design of floating and compliant offshore structures such as semisubmersibles and tension leg platforms has stimulated interest in the wave drift responses of such systems. Lightly damped resonances can be excited in irregular seas by non-linear wave effects, which yield forces at sum and difference components of the basic wave frequency spectrum. Because of the low natural frequencies of certain rigid body modes of floating and compliant structures, the difference frequency components of wave force can have profound implications for the platform motions. It has been found that observed behaviour may be successfully explained on the basis of a wave force model containing terms linear and quadratic in the body motions and wave kinematics. It is the quadratic components which are commonly assumed to be primarily responsible for low frequency drift responses of many such structures in irregular seas (e.g. refs. 2-3). A closely related problem is that of mean drift in regular waves. This is not only easier to model, but also sheds considerable light on the behaviour of compliant systems in irregular seas. It is useful therefore to examine the occurrence of the mean wave drift forces themselves. Considerable success has been achieved in the theoretical prediction of these forces, at least for relatively simple body geometries. But at least as important, from the view point of hydrodynamic synthesis, is development of an understanding of their dependence on underwater geometry. The latter has not hitherto received much attention in the published literature. This paper is motivated by the discovery that hydrodynamic interactions within a group of surface piercing bodies can profoundly influence the wave drift forces: the drift force on one body in a group can be increased several Accepted January 1985. Discussion closes September 1985.
128
Applied Ocean Research, 1985. Vol. 7, No. 3
fold over the force on the same body in isolation. Previous work on first order wave forces 4-7 has illustrated some of the phenomena associated with interactions and multiple wave scattering, but it has been found that significant enhancement of these first order forces generally only arises when the bodies are relatively closely spaced (i.e. less than two diameters for vertical circular cylinders). Preliminary results discussed in ref. 8, however, suggest that drift forces may be much more strongly influenced by multi-body interactions. To shed light on this behaviour, we consider here groups of vertical surface piercing cylinders. The first order problem has been considered by several authors (e.g. refs. 4-7). Here we tackle the problem by means of three complementary approaches. We use a complete analytical solution to make the calculations for two fixed vertical circular cylinders, stretching from the seabed through the free surface; and we also develop a highly simplified version of this analytical solution, obtained with the aid of a long wavelength approximation. Finally we consider some numerical results for floating vertical cylinders, truncated at some distance below the free surface. These last have geometries characterised by typical semisubmersible or tension leg platforms. The drift forces on these more complex systems are evaluated by a combined finite element boundary integral procedure. The results suggest that, under certain realistic conditions, the total horizontal mean drift force on a group of N vertical cylinders may be in the region of N 2 times the force on a single cylinder in isolation. The corresponding total vertical drift force, however, is of the order of N times the isolated cylinder force. ANALYSIS FOR TWO FIXED VERTICAL CYLINDERS
We consider two cylinders of radius a separated by a distance s, extending from the seabed to the free surface, 0141-1187/85/030128-12 $2.00 © 1985 CML Publications
Wave drift enhancement effects in multi column structures: R. Eatock Taylor and S. M. Hung identical coefficients an:
*m(r2, 02) = ~
a.(--O'H(~2)(~':)exp(in02)
(5)
i,!=--oo
It is more convenient, however, to transform this into the co-ordinates rl and 01, using the Graf addition theorem for Bessel functions (equation 9.1.79 given by Abramovitz and Stegun).9 In terms of the geometry shown in Fig. 1 we have
92
H(2)(kr2) exp[--in(rrl2 -- 02)] =
r2
~
H(2m)+n(ks)
X Jm(krl) exp[--im(n/2 -- 001
(6)
Equation (5) is therefore transformed into
q~D2(rl,01) -- ~
an
n=--o,
~ (--i)mU(2m)+n(kS) m=--o,
x Jm(krl) exp(im01) =--X
Figure 1.
Co-ordinate systems for two cylinders
The boundary condition at cylinder 1 specifies that the normal velocity at the surface due to the scattered potential ~ol is equal and opposite to the combined fluid velocity due to the incident wave and scattering from cylinder 2. Hence
aOOl
_[aoi
Or1 (a, 0,) = as shown in Fig. 1. A fixed co-ordinate system Oxyz is located half way between the cylinders, with the z axis pointing positive upwards and Oxy in the mean free surface. The centres of the two cylinders, O1 and 02, lie on Oy. The depth of water is d. Long crested sinusoidal waves of frequency co and amplitude A propagate in the direction Ox. The combined incident and scattered velocity potential may be written ¢p = Re [~ exp(icot)] ~o(kz)
~o(gZ) -
co
cosh kd
aom
Lar--~(a, 0,) + - -
] (a, 0,)
an(--i)nu(z)'(ka) exp (inO,) n=--oo
=--
(1)
~
(--i)nJn(ka)exp(in01)
-- ~ an Z (--i)mH(m2)+n(kS)Jm(ka) n =--** m =--*~
x exp(imO 0 (2)
and k is the wavenumber. The potential ~ is conveniently decomposed into contributions ~/ and 0O from incident and scattered waves respectively. In terms of cylindrical co-ordinates r~, 01, we may express ~I relative to the centre of cylinder 1 as
(8)
Substitution of (3), (4) and (7) into (8) leads to:
where
igA cosh k(z + d)
(7)
(9)
which holds for all 01 in the range (0, 21r). Rearranging this equation and interchanging the order of the two convergent summations we obtain equations for an :
an--+ Jn(ka)
~ araI-I(m2)+n(kS)------1,--00<.<00 m :-** (10)
~z(rl, O1) = ~
(--i)nJn(krl) exp(inO0
(3)
n:--oo
The combined velocity potential expressed in the coordinates (rl, 01) is then given by (1), with
The scattered potential due to cylinder 1 may be written
e~(rl, 01) q~Dl(rl, 01) :
~ an(--i)nH(2)(kr,)exp(in01)
(4)
n =--c:c
where H (2) is a Hankel function of the second kind satisfying the radiation condition, and the coefficients an are to be determined from the boundary condition on the cylinder. In view of the symmetry about Ox, the scattered potential due to cylinder 2 must be expressible in terms of cylindrical co-ordinates r2, 02 in the same form as (4), with
(-i) n
exp(inO0 [Jn(~l)+ anH(n2)(krl)
+ m=-** ~ amH(m2)+n(ks)Jn(krt)]
(11)
This corresponds to the incident waves plus the total effect of scattering by the two cylinders. But with the aid of (10)
Applied Ocean Research, 1985, Vol. 7, No. 3
129
Wave drift enhancement effects in multi column structures: R. Eatock Taylor and S. M. Hung
this may be written in the simpler form:
where " denotes the single cylinder result. In the long wave approximation the velocity potential is obtained from
~(rl-, 01) ~b(y, 0 ) = =
~
X
(--i)nexpqn01)
IIU(ka)
a n IHn(2)(kr,)
jn (ka)
Jn(kr,)]
~ (--i)nexp(inO)IJn(kr) - - - ! H(2)(kriJ n=-~ k g2n
\ 2/
(121
\2//
In what follows we require this to be evaluated on the surface of cylinder 1 : ~(a, 01) =
L
(-i)nAnexp(in01)
(13)
?./:--~
4
r 2+
1 -- in
cos 20
+...
where we have defined An = (hn --g2n]n) an
(14)
hn = H(2)(ka);
(15)
/n = J,(ka)
gzn = H(2)'(ka)/Jn(ka)
(16)
The first and second order wave forces are obtained from ~. Provided therefore that we may obtain the coefficients an from (10), we may then use these in (12) or (13) as required. Spring and Monkmeyer4 used this approach to investigate first order forces. Results for second order drift forces are given below.
THE LONG WAVE APPROXIMATION In the case of small ka it is convenient, as well as being instructive, to develop an approximate analysis. This may be used to obtain results valid in the long wavelength limit (relative to cylinder radius). We employ the small argument expansions for the Bessel functions, given in the Appendix, to derive
1 +/(2) 2
(21)
where we have used,, (r, 0) instead of (rl, 01). This is equivalent to the approximation given by Lighthill (ref. (10)). To obtain the corresponding result for the two cylinder problem we must obtain the coefficients a n for equation (10), which we write as g2nan +
~
Hm+na m = -
I
(22)
m =--e~
where Hm+n=H~)+n(ks ). One set of coefficients may thereby be obtained for each dimensionless wavenumber ka and cylinder spacing ks, as in the method of Spring and Monkmeyer4 and others. Here, however, we invoke the long wave approximation, in combination with a further assumption, to derive results in a very simple form. Our further assumption is to consider the cylinders to be sufficiently separated, so that interaction between the Fourier harmonics in 0, in equation (12), is restricted to the lowest few terms. In other words, equation (22) is assumed to be diagonally dominant, with Hm +na m "~ g2nan
m > N
for a series truncated at the term +-N; and N is taken to be small. This may also be expressed as
n=0
nk@/
gzn(~) =
i {27 n 1----n!(n--1)!t; )
(17)
/kaY'
n>~l
n
for small ka. The success of the assumption can then be assessed by comparison of results with those based on solution of equation (22) with a large number of terms. Our simple solution is based on using only the terms n = 1, + 1. Equations (22) become
where a =/ca. Furthermore
-;,L
n=0
i4 n= 1
hn--g2njn~-
(18) goao + tloao + H l a l - - H a a - i
7T
i(n
1),(~3n
n/>2
and in each case the relative error in the approximation is no greater than ~ . The terms with negative n are obtained from g - 2 n = g2n
(19a)
h - n - - g - 2 n ] - n = (-- 1)n(hn --g2nJn)
(19b)
Equations (10) and (13) also lead immediately to the well known results for a single cylinder. In this case an . . . .
1
g2n
130
;
hn d, = ln--g2n
Applied Ocean Research, 1985, Vol. 7, No. 3
= -- 1
g2a_ 1 - - H l a o + Hoal + 112a_ 1 = -- 1
(23b)
g2al + Hlao + n2al + Hoa_ l = -- 1
(23c)
Subtracting (23b) from (23c) we obtain (g2--Ho+H2)(al--a_l)
= -- 2Hxa o
(24)
hence from (23a): [go + Ho-- 2H?(g2 -- Ho +/-/2) -1 ] ao = -- 1 In line with our earlier assumptions we now suppose
(20)
(23a)
go,g2>> Ho,H1,H2
(25)
Wave drift enhancement effects in multi column structures: R. Eatock Taylor and S. M. Hung Hence we deduce that ao ~ - -
1(T0) 1
He
(26)
go
1 #0) ~ 1 - - - (He+ H2) g2
Furthermore, from (24) a 1 -- a_ 1 ~
2 H , ( _HO~(lq 1-1o 1-I2 1 gog2 go/\ g2
[
-'~2H1 l + H o
(gl
express the factors in terms of al and a_l. Making use of the approximations given in equations (27) and (28) we obtain the results (35a) (35b)
1) H2] --go --~-2
(27)
gogl
A f'mal reduction is obtained by substituting the approximations for go,g2 in (17), and using the identity
and by adding (23b) and (23c) we obtain
~2)(ks) + ~2)(ks) = 2~(2)(ks)
al 4- a_l ~ - - 1 - - - (He + Hz) (28) g2 g2 Equations (26)-(28) provide relations for the coefficients an which are in convenient form for evaluation of the first and second order wave forces.
ks
Therefore we have iTr
iz0) ~ 1 -- 2 (ka)(a/s) Ha(2)(ks)
(36a)
#(1' ~--4(ka)2Hl2)(ks)[ 1+ 4 (ka)2 (2n~2'(ks)--H~i)(ks))] FIRST ORDER WAVE FORCES
The first order force on cylinder 1 is F O) = Re [fO) exp(i~t)]
(29)
where the components of fO) are obtained from the linearised Bernoulli equation as
0 27r
fY), -
i f ~lsino~adO~o(kz)dz ccos0
(30)
-d 0 By substituting equations (2) and (13) we readily derive the exact results fxO) = 7rPaA~2
k2
(A~--A_,)/i
(31a)
f(1) = 7rpaA~_~2 (A1 + A_I)
Values of these magnification factors, #xO) and #yO), are given in Tables 1 and 2 respectively, as functions of the spacing parameter s/a. Results are given for values of dimensionless frequency ka = 0.05, 0.10, 0.15, 0.20. For each ka and s/a combination in the tables, two figures are given. The upper is obtained from the more complete solution, a based on truncation of equation (22) at the terms m =-+ 10 (i.e. equivalent in accuracy to the results of Spring and Monkmeyer). 4 The figure placed lower in the table, in parentheses, is obtained from the approximate solution (equations (36a) and (26b) for Tables 1 and 2 respectively). It may be concluded that the influence of interaction effects on the first order wave forces is well approxiamted by the simple approximation, over the range of parameters specified. In particular terms, however, the effects are rather small.
(31b) SECOND ORDER WAVE FORCES
The single cylinder exact results, from (20), are obtained
as fix(')-
2iTrpaA ~o2 [ h1 F ~ ]'--Z]
(32a)
L (1) = 0
(32b)
The long wave length approximation for the single cylinder is
fx(l)~--~pa2Aw2[1-iTr(-~y]
(33)
It is convenient to express the influence of the second cylinder in termsof dimensionless magnification factors, defined as ,tt(1) _ f(x1) A I - - A - I { .
hl'~ 1
2
g.._22
= - 2 (a,+a_,)
r/= Re [A¢ exp(i~t)]
~
2ff f(~)
dO
(38)
d0 dz
(39)
iT/ ]r,=a ~ Sin 0
Pg
0 and components of mean pressure drop force
hl)l g2
\--~2/
(37)
Thus for cylinder 1 we have components of mean waterline force
(34a)
f(1) _ Al+A_l(jl .(1) _JY
t,'-y --?~,)
We are concerned here with mean horizontal drift forces on fixed cylinders, which may be obtained by means of a far field momentum formulation, or direct integration of the fluic pressure on the submerged surfaces of the cylinders. Using the latter approach, it has been shown by Pinkster, 2 Standing, Dacunha and Matten 3 and others, that the force on a fixed body is the sum of a waterline force f(~) and a pressure drop force fA2). These quantities may be expressed in terms of the velocity potential given in equation (1), and the corresponding free surface elevation
---~(al--a_l) (34b)
In these exact expressions we have used equation (14) to
x~= 1 -d 0
\sin0
Applied Ocean Research, 1985, Vol. 7, No. 3
131
Wave drift enhancement effects in multi column structures." R. Eatock Taylor and S. M. Hung Table 1. Firstorder magnificationfactors ~ ) 5
10
25
50
100
1.044616 (1.042480) 1.012418 (1.012289) 0.996058 (0.996081) 0.998417 (0.998435)
1.048832 (1.046260) 1.009562 (1.009800) 0.998253 (0.998150) 0.999855 (0.999884)
1.051551 (1.049020) 1.002654 (1.003519) 0.999768 (0.999771) 0.999606 (0.999609)
1.050828 (1.049450) 1.001839 (1.001840) 0.999541 (0.999536) 0.999942 (0.999950)
1.011716 (1.011559) 0.999050 (0.999088) t.001219 (1.001220) 1.000518 ( 1.000520)
5
10
25
50
100
0.005294 (0.005512) 0.006992 (0.007144) 0.007183 (0.007379) 0.007663 (0.007973)
0.011568 (0.012132) 0.012224 (0.012638) 0.010529 (0.010943) 0.000701 (0.000704)
0.019016 (0.020160) 0.018529 (0.019376) 0.001015 (0.001020) 0.001961 (0.001987)
0.027768 (0.029812) 0.001520 (0.001529) 0.002763 (0.002801) 0.003592 (0.003683)
0.002916 (0.002954) 0.004054 (0.004113) 0.005074 (0.005204) 0.005431 (0.005644)
0.05 0.10 0.15 0.20
Table 2. Firstorder magnificationfactors ~(~) k a ~ 0.05 0.10 0.15 0.20
Here .~a denotes the mean component o f x 2. By substituting equations (1) and (2), and performing the integration with respect to z in (39) we may write these in the form f(~)f~)} = _ 1
y
_ 21"7[ /cos 0 )dO
expression for C(O) is found to be c(o)
=
(--i)m-nAmA*n
x exp[i(m--n)O]
(40)
4 pgA~a J q~*~ sin O o
1m ~ ~
(
l +sinh2kd /
mn
27[
fA~
lpgA 2a
1 1
2kd
o 1(
s 2kd
\
"~
•
Only terms for which (m -- n + 1) = 0 contribute to the integral, and we have
kdJV¢ q/cos2)
+~-1 sin]a--~kd)~O*J~sin
dO
(41)
where * denotes complex conjugate; also ~, V¢ are evaluated at r = a, and V in equation (41) is the two dimensional gradient operator. The total mean horizontal drift force vector may then conveniently be written f(2) =f(2) + if~2)
(42)
where
sinh 2kd ] n=_~ X( n (kn~- i 1)
1)
(45)
Equation (45) gives the general expression for mean horizontal drift force on a circular cylinder. If we substitute An from equation (20) we will obtain the drift force on an isolated cylinder. It may be noted that in fact An = (-- 1)nj - n, so that
27r
f(x2'=f(~)x+e(jl~)x:=lpgA2aRe[f C(O)exp(iO)dO] o
(43a) x Im [in J*+l]
27[
f(2)=f(~)+f~=~pgA2aRe[f
C(O)exp(iO)dO] o
(43b)
The integrals with respect to 0 are evaluated by substituting the Fourier representation of ¢ given in equation (13). The
132
Applied Ocean Research, 1985, Vol. 7, No. 3
fy(2) = 0
(46a) (46b)
These results are exact to second order in wave steepness. The long water approximation for the drift force on an isolated cylinder is obtained by means of equations (17), (18) and (20), using a = ka:
Wave drift enhancement effects in multi column structures: R. Eatock Taylor and S. M. Hung 2kd
).
--16
~
f(2) ,~ ~-2pgA2a( 1 + sinh 2kd/Im ff ot2 ]
16 +(19rr232
;).2a3
}
3
To obtain the appropriate drift forces on the two cylinder configuration, we revert to equation (45), and as for the first order forces we truncate the series so that only the terms n = 0, -+ 1 are included. Again it is convenient to define dimensionless magnification factors: f(2)
Im(A_,A~+AoA ~
=/(2) =
2 Im(.,ioA*)
f(2)
(48a)
Re(A_,A*+AoA*)
= fx(2) -
2 Im (A o.Zi*)
(48b)
For consistency in these approximations, we evaluate the denominator also using the terms n = 0, + 1: it is therefore given by 2 Im (AoA*) ~ 7r(/ca) a
(49)
The numerator in each case is evaluated by means of the following readily established identify:
A_IA* + AoA*= Re[A*(A_, + A,)]+ i Im[A*(A_,--A,)] (5o) From equations (14) and (18) we have i 2 A o : - - ~ r (a-S)ao
A_i+AI=~
i(4)=
(51a)
(--a_,+a,)
,(4)
A_x--A, =-- ~" ~ (a-1 + a,)
(51b)
(51c)
We may therefore proceed from the approximations given in equations (26)-(28), obtaining
=--r2a----~ Im ~og2 ]
g, j 1 -
S/a=
5.000
x
S/e=
18.008
[]
S/o=
20.000
Eqn.
53a
(47)
/(2) ~. I n2pgA2a(ka) a
#(2)
2.500
+
++÷.+v++ +
If we consider the case of deep water, and neglect all terms except the first in the above series, we recover the result
"x(2)
2.881
S/a=
- -
( 1 __i4~__~)(1 + ~-'~-~4)i32'~+ " "
,( 2kd \ ogA2a 1 + s . ~ - ~ ) ( k a )
S/a= v
(Ho+H2) (52a)
t I"
2
4
6
8
10
kS
Figure 2. Mean drift magnifications for two bottom supported cylinders
The results for/ax(2) are plotted in Fig. 2 as functions of ks over the range 0 to 10. These are compared with results from the more complete solution, 8 for the spacings (s/a) = 2.5, 5, 10 and 20. Some results from the more complete solution are also given for the cylinders touching (s/a = 2.001). It is seen that at large spacings the low frequencies, the approximate solution for/~x(2) given by equation (53a) is surprisingly accurate. A remarkable feature of the results for /ax(2) is the behaviour at low values of ks. Both the accurate numerical results and the simple approximate analysis suggest that as ks tends asymptotically to zero, /ax(2) tends to 2. The physical interpretation is that the drift force on one of the pair of the cylinders tends to twice the force which would act on the cylinder in isolation: indeed the total horizontal drift force on the two cylinder group, in line with the direction of wave propagation, tends to four times the force on a single cylinder in isolation. Results described below for other mult-cylinder geometries suggest that this is a special case of a more general law governing the horizontal drift force on a group of vertical cylinders. Most importantly, the force cannot be approximated by the sum of the forces on the individual cylinders taken in isolation. Such a simple superposition would underestimate the resulting drift force by at least 37% over the range ks < 2. The error could exceed 100% for close spacings and low frequencies. Turning now to the results for/ay(2) , we fimd that the approximation of equation (53b) is hopelessly at variance with the accurate solution. It appears that at least two Fourier harmonics are essential to represent with any degree of verisimilitude the second order pressure gradient in line with the two cylinders. Indeed this is hardly surprising. However, the parameter lay(2) is much less significant than /ax(2), since it corresponds to the mean drift splitting force on the cylinder group. Unless the two cylinders are free to move independently, both first and second order splitting forces are resisted by structural stiffness. The effects of the latter forces are then insignificant compared with the former.
(52b) Substituting equation (17), we finally obtain the consistent lowest order approximations (i.e. independent of ka) as:
#(2) ~_, 1 + Jo(ks) + 12J2(ks)
(53a)
/@(2)~ _ 12}'1 (ks)
(53b)
NUMERICAL RESULTS FOR OTHER MULTICYLINDER GEOMETRIES
We consider now drift enhancement effects for cylinder arrangements which are more representative of floating structures such as semisubmercibles and tension leg platforms. It may be anticipated that the behaviour described
Applied Ocean Research, 1985, Vol. 7, No. 3
133
Wave drift enhancement effects in multi column structures: R. Eatock Taylor and S. M. Hung above, for two cylinders extending to the seabed in deep water, would also be manifested for two surface piercing cylinders truncated somewhat below the free surface (i.e. two columns of a TLP). This is because the vertical distribution of horizontal drift force decreases very rapidly with distance below the free s u r f a c e - indeed a major compoment arises only in the vicinity of the waterline. Here we verify this conjecture and extend the previous results, by examining horizontal and vertical drift force enhancement effects for groups of two and four equal vertical cylinders, truncated below the free surface. We show that interactions between the cylinders do not lead to significant enhancement of mean vertical drift effects, but mean horizontal drift forces can be strongly influenced when the wavelength is sufficiently large compared with cylinder diameter. To obtain these results, a well proven numerical method was employed, based on a combined t'mite element and boundary integral procedure. Finite elements are used to represent the flow field in the vicinity of the submerged body surface (in this case the cylinder group), and a boundary element idealisation is adopted on a fictitious surface surrounding the body. The latter surface is defined by a set of vertical and horizontal orthogonal planes, thereby ensuring a highly efficient boundary element approximation. Quadratic isoparametric elements are used in the f'mite element region, and the fictitious outer surface is represented by quadratic boundary elements. A description of the method and validation of the computer program (DYHANA) is given in Ref. 11. The cases considered here are based on groups of cylinders of radius 9 m and draught 32 m, in water depth 148 m. The geometry for the single cylinder (Case 1) is shown in Fig. 3a, with the associated element mesh in Fig. 3b. In Case 2, two such cylinders are separated by 54 m (centre to centre), in a line perpendicular to the direction of the incident wave. This is closely related to the two cylinder case considered above, except that now the cylinders do not extend to the seabed. Case 3 is similar to Case 2, with the line joining the cylinders making an angle of 45 ° to the direction of the waves. In Case 4, four cylinders are placed at the corners of a square, of side 54 m, and the incident wave propagates in a direction parallel to two sides of the square. Case 5 is similar to Case
4, but the incident wave is directed parallel to a diagonal of the square. The arrangements for the five cases are summarised in Table 3. The finite element meshes for Cases 2-5 are based on the single cylinder mesh (Case 1) shown in Fig. 3b. This is facilitated by exploiting one or two planes of symmetry, as appropriate: these are specified in the input to the computer program DYHANA. Some results for the total mean horizontal and vertical drift forces on the cylinder groups are given in Table 3, for frequencies within the range 0.125-0.65 radians/s (i.e. wavelengths from 1842m down to 146m). For long wavelengths the results for Cases 2 and 3 are seen to be very similar, as are those for Cases 4 and 5. In the limit of zero frequency the horizontal drift force always tends to be zero. But more significant is the ratio between the forces at a given frequency or wavelength, for the difference cases. At low frequencies ~ahe total vertical forces in Cases 1, 2 and 4 are in the approximate ratios 1 : 2 : 4. The corresponding ratios for the horizontal forces are approximately 1 : 4 : 16. This suggests that the total vertical force is simply the sum of the forces on each cylinder considered in isolation; but at low frequencies the total horizontal force is the corresponding single cylinder force multiplied by the square of the number of cylinders in the group. This appears to be a generalisation of the two cylinder result obtained from the analytical formulation of the previous section. The enhancement of the horizontal drift forces, arising from the interaction effects in the two and four cylinder arrangements, is clearly illustrated in Fig. 4. This shows the ratio of the total mean horizontal force on the group on N cylinders (N = 2 or 4 here) to the mean force on N isolated cylinders. At low frequencies this ratio appears to tend to the value N, rather than to the value 1 given by the sum of forces on the cylinders neglecting interaction effects. The same low frequency limit is reached regardless of the precise plan arrangement of the cylinders. Of considerable practical interest is the rate at which the drift enhancement effect decreases with increase in wave frequency (or ka), for a given geometry. It may be noticed in Fig. 4 that for Case 2 (two cylinders in a line parallel to the wave crests) the decay is much slower than for the other cases. Up to the highest frequency plotted (corresponding to a period of approximately 10 s), the drift enhancement decays monotonically to a value of about 1.4. In Cases 3-5, however, the ratio drops lower than one within the range plotted, although for Case 4 it again exceeds one at the highest frequency point. It is important, of course, to relate these results, based on an ideal flow calculation, to the expected behaviour of the real fluid. Chakrabarti 12 and Standing et al.,3have compared the mean drift force associated with diffraction at a single cylinder (i.e. Case 1 considered above)with the steady force arising from viscous effects. They show at low frequencies the theoretical ratio of viscous to potential drift force in deep water is given approximately by
ca
I///////////////////// a)
Oin,ension8
b)
E l e m e n t mesh
Figure3. Floating cylinder geometry: (a) dimensions; (b) element mesh
134
Applied Ocean Research, 1985, Vol. 7, No. 3
A
R - - 2rr (ka) 2 a
(54)
(in terms of the parameters defined above, with drag coefficient Ca). If we take Ca = 0.7 and A = 4 m (i.e. a wave height of 8 m), this suggests that potential effects exceed viscous effects on the single 9 m radius cylinder provided that co > 0.49 rad s-1 (ka > 0.22). Under related conditions, the drift enhancement effects for the two
Wave drift enhancement effects in multi column structures: R. Eatock Taylor and S. M. Hung
-,
N-2
?
illl
I
I
]
l
[
l
l
©
t
V
N-2
x
N-4
+
A
A
•
•
N-~
A
A V
A ×
E
V
l i l l l l l
0
÷ I o.1
, N
I 0.2
I 0.3
I
I
0.~
0.2
a¢
I 0.6
I --
o 7
~/(rad s-l)
~ Q Q Q Q
Figure 4. Mean drift enhancement effects for groups of N floating cylinders
0
0
0
0
0
0
0
0
0
IIitl]] 0
0
0
0
0
0
IIIIIII ©
0
0 0 0 0 0 0 0
IITTTII
cylinder configuration (Case 2) appear to be of practical significance, although the influence of interactions on the viscous contribution is at present unknown. The design implications of the four cylinder results in Table 3 and Fig. 4 are also still speculative. At ¢o = 0.49 rad s -l, corresponding to R = 1, the total horizontal drift force for Case 4 may be estimated as some 50% of the force on four isolated cylinders. Hence the ~iscous effects predominate under these particular circumstances. On the other hand, at the highest frequency plotted in Fig. 4, the value of R given by equation (54) is 0.33 and the drift enhancement for Case 4 is 1.21. For these latter conditions, therefore, potential effects appear to predominate, and interaction effects between the four cylinders may be significant. Overall, the results suggest that it may be unwise to disregard the enhancement of mean horizontal drift forces, due to wave scattering by multiple cylinders at low frequencies. Expressions such as equation (54), intended to provide an estimate of the relative magnitude of viscous and potential drift effects, are unlikely to provide reliable criteria for multi-colunm structures. Further work, both theoretical and experimental, is required before general rules become available for making simple estimates of the drift behaviour of such structures.
0
t
CONCLUSIONS
E ~ d d d ~ d
C ~4
~ 0 0 0 0 0 0
The preceding analysis and results enable certain conclusions to be drawn regarding prediction of mean drift forces on multicolumn structures. The major finding is that where wave diffraction is primarily responsible for mean drift forces, hydrodynamic interactions can in principle lead to substantial increases in drift force, as compared with results obtained by simple superposition without regard to interaction. The following points may also be noted: (i) An analytical solution has been obtained for the mean horizontal drift force on two circular cylinders placed in a line parallel to the incident wave crests. The cylinders extend from the seabed through the free surface. The effects of hydro-
Applied Ocean Research, 1985, Vol. 7, No. 3
135
Wave drift enhancement effects in multi column structures: R. Eatock Taylor and S. M. Hung
dynamic interactions are substantially more pronounced for mean drift forces than for first order wave frequency forces. (ii) A simple long wave approximation has been developed for the two cylinder problem. This gives good agreement with the more accurate solution for first order forces as long as ka <<,0.2 and s/a/> 5. The agreement for the mean drift force in line with the direction of wave propagation is also satisfactory over this range. (iii) The long wave analysis shows that at asymptotically low frequencies (and a sufficiently large spacing between the cylinders) the total mean horizontal force on the pair o f cylinders tends to exactly twice the total force which would be predicted by summing the forces on each cylinder considered in isolation. (iv) Numerical results from a finite element - boundary integral approximation confirm this behaviour for horizontal force on two circular cylinders penetrating the free surface but truncated well above the seabed. However, the total mean vertical force on such a pair of truncated cylinders tends to be the same value as would be obtained by adding the forces on each isolated cylinder. This low frequency asymptotic behaviour appears to be independent of the precise arrangement of the two cylinders relative to the direction of wave incidence. (v) Further numerical results, for four truncated cylinders, suggest the existence of a general rule:* the total mean horizontal drift force on a group of N cylinders tends to N 2 times the force on one cylinder in isolation; but the total vertical force tends to N times the equivalent single cylinder force. Hence interaction effects are shown to be potentially important for horizontal drift forces, but not for vertical drift forces. (vi) The enhancement or magnification of horizontal drift forces due to hydrodynamic interactions has been evaluated over a range of frequencies, for various two and four cylinder configurations. In the case of two cylinders aligned parallel to the wave crests, the enhancement effects decay with increasing frequency at a much more gradual rate than for the other cases examined. Substantial effects are predicted over a realistic range of physical parameters. (vii) At the frequencies where these enhancement effects are pronounced the possible contribution of viscous effects to mean drift forces must also be assessed. Expressions indicating the relative importance of viscous versus diffraction contributions may be misleading, unless the effects of hydrodynamic interaction are accounted for. (viii) There is a need for some experiments to test the aforementioned findings. To arrive at a method o f hydrodynamic synthesis which accommodates drift response criteria, it ~ also be necessary to perform further systematic studies of the influence of interaction effects in a variety of multi-body configurations.
REFERENCES 1
2 3 4 5 6
7 8 9 10 11 12
Newman, J. N. Second order slowly varying forces on vessels
in irregular waves. Proc. Int. Symp. Dynamics of Marine Vehicles and Structures in Waves, I.Mech.E., London, 1974, 182 Pinkster, J. A. Mean and low frequency drift forces on floating structures, Ocean Engineering 1979, 6,593 Standing, R. G., Dacunha, N. M. C. and Matten, R. B. Slowly varying second order wave forces: theory and experiment. National Maritime Institute, Report R 138, December 1981 Spring, B. H. and Monkmeyer, P. L. Interactions of plane waves with vertical cylinders, Proc. 14th Int. Conf. on Coastal Engineering, Copenhagen, Denmark, ASCE 1974, 1828 Ohkusu, M. Hydrodynamic forces on multiple cylinders in waves. Proc. Int. Symp. Dynamics of Marine Vehicles and Structures in Waves, I.Mech.E., London, 1974, 107 Matsui, T. and Tamaki, T. Hydrodynamical interaction between groups of vertical axisymmetric bodies floating in waves, Proc. lnt. Symp. Hydrodyn. Ocean Eng. Trondheim, 1981, 817 Mclver, P. and Evans, D.V. Approximation of wave forces on cylinder arrays, Applied Ocean Reasearch 1984, 6,101 Eatock Taylor, R. and Hung, S. M. Mean drift forces on an articulated column in a wave tank. Applied Ocean Research 1985, 7,61 Abramovitz, M. and Stegun, I. A. Handbook of Mathematical Functions, Dover Publications, New York, 1965 Lighthill, M. J. Waves and hydrodynamic loading, Proc. 2nd Int. Conf. on Behaviour of Offshore Structures 1979, 1, 1 Eatock Taylor, R. and Zietsman, J. Hydrodynamic loading on multicomponent bodies, Proc. 3rd lnt. Conf. on Behaviour of Offshore Structures 1982, 1,424 Chakrabarti, S. K. Steady drift force on vertical cylinder viscous vs potential, Applied Ocean Research 1984, 6, 73
APPENDIX - EXPANSIONS FOR SMALL ARGUMENTS
The following expansions are given by Abramovitz and Stegun, 9 for Bessel functions of a small argument:
jn =
n!\2/
136
Applied Ocean Research, 1985, Vol. 7, No. 3
2i[in(. + ~-k \ 2 1
ho=B~:)(a)~l
hl=Ht2)(a)~-~[2 1+; ;i(27]
hn = H(2)(a) ~ - (n -- 1)!
n >/2
7r
where 7 = 0.5772. Furthermore, the derivatives of the Bessel and Hankel f~nctions satisfy the following recurrence relations (where for our purposes Cn = Jn or//(2)): ~(a)
= - - C 1( a ) n
C~,(a) = C n _ l ( a ) - -
a
n
Cn(a ) = - Cn+ l(a ) + - Cn(a ) ot
We therefore obtain H~2)' (a)
Hl2)(a) ~---
gO= J~(a)
1 + /(_2"~
Jx(,~)
rr\a/
-H(~2~(a) + (n/a)H(~2)(a) g2n =
*The theoretical basis for this rule is discussed in Mclver's discussion accompanying this paper.
"
1 + n!
--Jn+ 1(a) + (n/a) Jn(a)
x
(n-l)!
~
n
~n!
~
Wave drift enhancement effects in multi column structures: R. Eatock Taylor and S. M. Hung _in. =1
ACKNOWLEDGMENTS
"2 "2n
n>~l
~r ' ( n - - l ) ! ( ~ )
For the terms corresponding to negative n, we use the identities
Jn = (-- 1)nJ - n ;
//(2) = ( - - 1)nil(-2)
From Dr P. Mclver, Department of Mathematics, University of Bristol, UK
This work was supported by the Science and Engineering Research Council and the Cohesive Programme on the Dynamics of Compliant Structures.
the first order component of the scattered wave field. The force vector for the ]th body may also be written as a series expansion in e, so that
DISCUSSION OF:
=
Wave drift enhancement effects in multicolumn structures
by R. Eatock Taylor and S. M. Hung In their paper Eatock Taylor and Hung have examined the mean wave drift forces on groups of vertical circular cylinders. Using a long wave approximation for the case of two cylinders extending throughout the fluid depth they found that, at low frequencies, the inline drift force on two cylinders tended to four times the force on an isolated cylinder. Furthermore, numerical results for four truncated cylinders were found suggesting a general rule for the calculation of wave drift forces on groups of cylinders in the low frequency limit. The authors speculated that the mean horizontal drift force on N cylinders tends to N times the force on an isolated cylinder, while the mean vertical force on the N cylinders tends to N times the force on an isolated cylinder. Here these rules are examined from an analytical point of view and confirmed. The results are derived under the usual assumptions of the inviscid theory of water waves. Some discussion of the effects of viscosity is given by Eatock Taylor and Hung. Consider a group of N identical, fLxed, axisymmetric bodies in water of uniform depth h. A co-ordinate system (r, 0, z) is chosen so that (r, 0) are polar co-ordinates in a horizontal plane and the vertical z-axis is directed vertically upwards with the zero at the level of the mean free surface. The origin of co-ordinates is chosen to be 'within' the group of bodies. Local polar co-ordinates (rl, 0]) are centred on on the axis of the jth body which is situated at r = sj. A uniform non-linear wave train is incident upon the bodies from the direction 0 = rr. The resulting wave field, including incident and scattered waves, will be represented by a velocity potential ap(r, 0, z, t) which may be written as a series in the form (I) = e ( I ) ( i ) "~- e2~ (2) + . . .
(1)
where e is a small parameter related to the wave steepness. The linear component of ~ may be written as b(i) = Re [f(z)(~O) + ~s(i)) exp (--/cot)]
(2)
where
igA cosh k (z + h) f(z)
-
eco
cosh kz
(3)
+ erp
+...
(s)
The mean (time-averaged) second order force on a fixed body ~(2)=
i f 2pg
rl2nidl+21 p f iVdPO)12ndS zi
si
+/3I¢/(0, 0, 1)
(6)
(see Pinkster) 1 where 1,] is the mean water line (with normal nt), S~ the mean wetted surface (with normal n), 77 the displacement of the free-surface from the mean level (due to the linear component of the solution) and I~ the waterplane area. The mean set down pressure P depends upon the second order component of the incident wave and is readily calculated (Longuet-Higgins and Stewart). 2 An overbar is used to denote a time averaged quantity. If the local terms that decay exponentially away from the body are neglected, then the scattered wave field for a single body can be written as
¢~1)(rj,Oi)= ~. AnHn(Icri) cosnOj
(7)
n=O
where Hn is the Hankel function of the first kind of order n. The complex scattering coefficients (An, n = O, 1 .... ) are, in general, functions of the wave frequency and the body geometry. For the case of a vertical cylinder extending throughout the depth of the fluid they may be determined analytically (see, for example, Mel, p. 312) 3 and it is found that A o is O((ka) 2) and A n is O((ka)2n), n 4=0 where a is the cylinder radius. For a circular cylinder that does not reach the bed or similar axisymmetric bodies it is reasonable to expect the scattering coefficients to behave similarly. Thus for long waves the scattered wavefield (equation (7)) within the array of bodies can be expected to be at most O((ka) 2 In(ks)), where s is a typical spacing and the small argument asymptotic forms for the Hankel functions have been used (Abramoxitz and Stegun, p. 360). 4 Hence, for long waves it is reasonable to neglect multiple scattering events within the array and a first approximation to the forces may be obtained by considering the incident wave field only. Consider first the mean vertical second order forces on the bodies. For long waves the scattered field may be neglected, then from equations (3) and (4), n = ½A 2
and $(/i)(r, 0) = exp(ikr cos 0)
(4)
represents the first harmonic of the incident wave train with frequency w, wavenumber k, and amplitude A: g is the acceleration due to gravity. The potential SsO)(r, 0) is
(8)
and q" 2
1[
gkA
]2
17" '1 = 2[e~-c-ososhk/~J
(9)
x [cosh2/~(z+ h) + s~/c(z +h)] Applied Ocean Research, 1985, Vol. 7, No. 3
137