Wave drift enhancement effects in multi column structures

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Wave drift enhancement effects in

-F 3115781835

structures

R. EATOCK TAYLOR and S. M. HUNG

London Centre for Marine Technology, eparrment 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 squate 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 frequen-cies 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. lt 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 hydro-dynamic 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 forces47 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 float.ng 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 N2 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

s

Figure 1. Co-ordinate systems for two cylinders

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. O and 02, lie on Oy. The depth of water is d. Long crested sinusoidal waves of frequency w and amplitude A propagate in the

direction Ox.

The combined incident and scattered velocity potential may be written = Re [Oexp(iwt)] 0(kz) (1) where

igA coshk(z+d)

(2) w coshkd

and k is the wavenumber. The potential Ø is conveniently

decomposed into contributions Øj and 0D from incident

and scattered waves respectively.

In terms of cylindrical co-ordinates r1, O, we may

express ç- relative to the centre of cylinder 1 as

(i)'V(kr1)exp(in01) (3) n

The scattered potential due to cylinder 1 may be written ODdrl.O)=

a(i)2H2)(kr1)exp(in01)

(4) whereH2> _{is a Harikel function of the second kind}

satis-fying the radiation condition, and the coefficients a 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, 62 in the same form as (4), with

Wave drift enhancement effects in multi column structures: R. Eatock Taylor and S. M. Hung

y identical coettìcients u:

A

9D2(r2, 02) = I a(_j)nH2)(kr2)exp(jflO2) (5) lt is more convenient, however, to transform this into the co-ordinates r1 and 0, 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. I we have

H2(kr2)

exp[in(7r/2 - 02)]

= Y

H2

_{nl n}(ks)

m=

-jm(ki)exPHih12(7d20i)]

(6)

Equation (5) is therefore transformed into

0D2(rl,01) = a (_:\mu(2)¡)

'+n

(ks)

n=

m=-X jm(1i)ecP(1m0i) (7)

The boundary condition at cylinder i specifies that the normal velocity at the surface due to the scattered potential OD! is equal and opposite to die combined fluid velocity

due to the incident wave and scattering from cylinder 2. Hence

aoDl

= -

(a, 0)

aOD2(a,

0f)]

(a, O

8r1 Lar1 ar

Substitution of (3), (4) and (7) into (8) leads to: an(_iylH,2)'(ka)exp (mû1) n Y

(i)'1J(ka)exp(inO1)

n

-

' _a,

_{(_\mu(2) (ks)J,,(ka)}

J

n--°

m--x exp (imO1)

which holds for ail 01 in the range (0, ir). Rearranging this

equation and interchanging the order of the two convergent summations we obtain equations for a:

H,2)(ka)

2

+

_{amHn(ks)=-1, =o<n<oo}

=

(10)

The combined velocity potential expressed

in the

co-ordinates (r1, O) is then given by (1), with

O (r

= (m)n

_{exp (mû1) ni) + aH2)(k7j)}

+ m_nlfl'01

(il)

This corresponds to the incident waves plus the total effect of scattering by the two cylinders. But with the aid of (10)

Wave drift enhancement effects in multi column structures: R. Eatock Thy/or and S. M. Hung

The first and second order wave forces are obtained from Ø.

Provided therefore that we may obtain the coefficients a

from (10), we may then use these in (12) or (13) as

required. Spring and Monkrneyer4 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

i(2\2

i

+-7r\a

g2,,(a)= i

i --n!(n

l)!(\_)

ir

a

where a= ka. Furthermore

¿(2)2

h,, g2flifl

j

(2\

(n l)!-)

ir a

and in each case the relative error in the approximation is no greater than a2. The terms with negative n are obtained

from

g_2,, =g2,, (19a)

h_,, -g_2,,j-, = (- 1)'tQ,,

-g2,,j,,)

(l9b)

Equations (10) and (13) also lead immediately to the

well known results for a single cylinder. In this case

a,,= A,, _=/,, li,, (20)

g2,, g2,,

130 Applied Ocean Research, 1985, Vol. 7, No. 3

n1

n=O

nl

(18)

n2

+

Hp,,t1

= m

where Hm+n

= H.,,(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 cy1inder

-be sufficiently separated, so that interaction -

betwe-Fourier harmonics in O, in equation (12), is restricted

lowest few terms. In other words, equation (22) is assumed to be diagonally dominant, with

Hm nam «g2,,a,, m > N

for a series truncated at the term ±N; and N is taken to be n =O _{small. This may also be expressed as}

(17)

(22)

(ka

\

a>irI1,) Hm+nam

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

Subtracting (23b) from (23c) we obtain

(g2H0+H2)(a1a_1)=_H1a0

(24)

hence from (23a):

[g0+H0-2H(g2H0+H2)'ja0= 1

(25)

In line with our earlier assumptions we now suppose g0,g2 H0, H1, H2

this may he written in the simpler form:

O)

= (_j)fl_{exp (mû1)}

where . denotes the single cylinder approximation the velocity potential

(r, O)

(i) exp(inû)

//ç,\2 /ka\2 =

l(--j+(--j

result. In the long is obtained from

[Jnkr

H2)(kr)]

[

/kr\

21n(j+27+iir

wave H2)'(ka)

J

\21

/

a2\

\2J

[

L

'2J

/ka\21 x an[Hc2(

kr1)

J'(ka)

n(k7i)] (12) ¡cosO

En what follows we require this to be evaluated on the

surface of cylinder I:

r

\2JJ

k2(

a4\ (ka\41

__(c2+_ I

1miri J Jcos2O

Øa, O)=

_(-

irAn exp (mO1) (13) 4

r!

\2Ji

(21) where we have defined _{where we have used (r, O) instead of (r1, Os). This is}

equiva-=

(lin g2j)a

(14) lent to the approximation given by Lighthill (ref. (10)).

To obtain the corresponding result for the two cylinder lin =H,2)(ka); j,, =jn(ka) (15) problem we must obtain the coefficients a,, for equation

g2,,=H,2)'(ka)/Jn'(ka) (16) (10). which we write as

g0a0+110a0+H1a1H1a_1 1

(23a)

g2a_1H1a0+H0a1+112a_1= i

(23b)

Equations (26)-(28) provide relations for the coefficients a which are in convenient form for evaluation of the first

and second order wave forces.

FIRST ORDER WAVE FORCES

The first order force on cylinder i is

F' = Re [f'exp(iwt)]

(29)

where the components of f(

are obtained from the

linearised Bernoulli equation as

O 2r

ji)

_cosO

)}iwPJJ'{}adOo(kz)dz (30)

By substituting equations (2) and (13) we readily derive

the exact results

irpaA w2 =

k2

rrpaAw2

f3S°= (A1+A_1)

The single cylinder exact results, from (20), are obtained

as

e'

iirpaAw2

/

h1

= 2 (

ii

k

\

g2

(32b) The long wave length approximation for the single cylinder

's

- 27ri

ka2

f(1)__Pa2Aw2[l _iir()]

(33)

It is convenient to express the influence of the second cylinder in termsof dimensionless magnification factors,

defined as

ji) A,A_1 /

(j) =-

h,\1

g2

(ai+a_,)

2 g2 2 (34a) c(1)

_{Ai+A_i(}

hi)i

_g2 (1) _Jy Ji

-/13/

-= -

2i g2 (34b) ix

In these exact expressions we have used equation (14) to

(3 lb)

SECOND ORDER WAVE FORCES

¡ir

/1)... 1 ---(ka)(a/s)H,2(ks)

2

(i)

_{_(/)2 H12(ks) [i +}

_{(ka)2 (2H,2(ks)}

_{_H2)(ks))]}

Values of these magnification factors, f41) and /41), 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,8 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). lt may be concluded that the influence of interaction effects on the first order wave forces is well

approxiamted by the sinople approximation, over the range of parameters specified. In particular terms, however, the

effects are rather small.

(36a)

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 (32a) the fluic pressure

on the submerged surfaces of the

cylinders. Using the latter approach, it has been shown by Pinkster,2 Standing, Dacunha and Matten3 and others, that the force on a fixed body is the sum of a waterline force

f)

and a pressure drop force f,2'. These quantities may be

expressed in terms of the

velocity potential given in

equation (1), and the corresponding free surface elevation = Re [AØ exp (iwt)] (37) Thus for cylinder i we have components of mean waterline

force 2ir

f}

1

J__

(cosB

=jg {l72]r=a

adO (38) sin O O

and components of mean pressure drop force

fgì

cosOs)

»2)}P J J ivøi(

sin O adO dz (39)

d O

Applied Ocean Research, 1985. Vol. 7, No. 3 131 Wave drift enhancement effects in multi column structures: R. Eatock Taylor and S. M. Hung

Hence we deduce that express the factors in terms of a, and u_j. Making use of

the approximations given in equations (27) and (28) we

ao---('

-)

(26)

g0\

g0 obtain the results

Furthermore,

from (24) 41)

_{1 ---(H0+H2)}

(35a)

g2

H0\'

H 2J1

(i--

_{+_-_f2)}

¡HJ[ / i I

1 +H0(----

(35b)

a1a_1

(i

g0g2

g0J\

g2

g2i

g0L \g2 go) g2]

fi

I 2H1[ H21

(27) A final reduction is obtained by substituting the

approxi-g0g1L 2

g0j

g2

and by adding (23b) and (23c) we obtain

mations for g0,g2 in (17), and using the identity

2[

i

T'

a1±a_,--1 i---(Ho+H2)I

(28)

H2(ks) + H2(ks) = - H2)(ks)

ks g2L g2 J _{Therefore we have}

(A1A_,)/i

(3 la)

Wave drift enhancement effects in multi column structures: R. Eatock Taylor and S. M. Hung

Here 2denotes the mean component of x2.

By substituting equations (1) and (2), and performing

the integration with respect to z in (39) we may write these

in the form

where * denotes complex conjugate; also 0, VO are

evalua-ted 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) =f42) + jf12) ₍₄₂₎

where

2ir

f(2) =f(2) +f) =.pgA2a Re[J C(0)exp(iO)dO]

_{X Wx}

o (43a)

2n

f(2) =f(2) +f

= _{pgA2a Re[T C(0)exp (IO) do]}

Y Wy

o (43b)

The integrals with respect to O are evaluated by substituting the Fourier representation of _{given in equation (13). The}

132 Applied Ocean Research, 1985, VoL 7, No. 3

X'

'fl(

\\ _k2a2

Equation (45) gives the general expression for mean hon.

zontal drift force on a circular cylinder. If we substitute

A from equation (20) we will obtain the drift force on an

isolated cylinder.

It may be noted that in

fact A =

(- l)'A_,,

so that - Ir kd

-

n(n+l)

f42)=_PgA2a(i

5flh2kì0(

-

k2a2

)

x 1m [ÂÂ11

(46a) f(2) =o _(46b)

These results are exact to second order in wave steepness.

The long water approximation for the drift force on ail isolated cylinder is obtained by means of equations (17).

(18) and (20), using &=ka:

expression for C(0) is found to be

(45)

Table J. First ordermagnification factors

s/a 5 ka 10 25 50 100 0.05 1.044616 1.048832 1.05 1551 1.050828 1.011716 (1.0424 80) (1.046260) (1.049020) (1.049450) (1.011559) 0.10 1.0124 18 1.009562 1.00 26 54 1.00 1839 0.999050 (1.0 12289) (1.009800) (1.0035 19) (1 .00 1840) (0 .999088) 0.15 0. 99605 8 0.99825 3 0. 999 768 0.999541 1.001219 (0.99608 1) (0.998150) (0.999771) (0.9995 36) (1.001220) 0.20 0.9984 17 0. 99985 5 0.999606 0.99994 2 1.0005 18 (0.998435) (0.999884) (0.999609) (0.999950) (1.000520)

Table 2. Firstordermagnification factors /.Ly(1)

ka 5 10 25 50 lOo 0.05 0.005 294 0.011568 0.019016 0.02 7768 0.0029 16 (0.005512) (0 .0 12 13 2) (0.020 160) (0.029812) (0.002954) 0.10 0.006992 0.0 12224 0.0185 29 0.001520 0.004054 (0.007 144) (0.0126 38) (0.0 19376) (0.0015 29) (0.004113) 0.15 0.007 183 0.0 10529 0.001015 0.00276 3 0.005074 (0.0073 79) (0 .0 1094 3) (0.00 1020) (0 .00 280 1) (0.005 204) 0.20 0. 00766 3 0.00070 1 0.00196 1 0.0035 92 0.005431 (0.007 97 3) (0 .000704) (0.00198 7) (0.0036 83) (0.005644) 2ir i 'cosO

-

pgA2a )do (40) 2kd

1+

xexp{i(m_n)O](

sinh2kd)

(2) f _\SiflO Wy o 2ir

/ mn -i)

₍₄₄₎ X k2a2 2kd

1+ sinh2kd)

Only terms for which (m-n + i)

=O contribute to the

integral, and we have

}=pgA2aJ[-(

o

li'

2kd

_\

1/cosO\

+- 1-

dO (41) iT

(

2kd

2\

sinh2kdl

_]\sinoJ

f(2) =-pgA2a1

i +

n-1 n

)

¡A

n-16 2kd \)

f

-irpgA2a(

f

2 j

4\2

(2)

-

₊ sjnh 2k-d

r3

(1 + ₎ ir - /

.

/

2\

16 2)

(i4)(

_i32

c2a3 1

1+

₍

ir4)

I 9ir2 / 2kd

\

pgA2a 1 + 1(ka)3 32 \.

sinh2kd)

1f we consider the case of deep water, and neglect all terms except the first in the above series, we recover the result

f$)

_{ir2 pgA2a (ka)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: (2) =f,42)

Im(A_1A+A0A)

PX _j(2) 2 lm(À0Âr) (2) _'Y

Re(A_1.4+A0A')

113) (2)

2Irn(À0Â')

For consistency in these approximations, we evaluate the denominator also using the terms n = O, ± 1: it is therefore

given by

i /4'\

A_1+A1=--)

(a_1+a1)

ir \a

i /4'\

A_1A1 r--(J(a1+ai)

ir \&J

We may therefore proceed from the approximations given in equations (26H28). obtaining

=-

lm-1l

1 32 1 1 1

H][

ira6

gg2L

g

__(H0+H2)]}

g2 32 1 H1 [ H / 1

lH2]

n3 tggog2L

gl[

g0 g2

j}

(2)____

_Re

_{Il -}

_{i +H0(--_}

Substituting equation (17), we finally obtain the consistent lowest order approximations (i.e. independent of ka) as:

bL2,' 1 +J0(ks)+J2(ks)

(53 a)

(2)_1

_{2Y1(ks) (53b)}

Wave driftenhancement _effectsin multi columnstructures:R. Eatock Taylor and S. M. ¡Jung

(47)

'

S'a 2.@

'

S'a 2.5ø Sz= 5ØØ S'a 1.ØØ : S'o= 2a.a@ø Eqn. 53

Figure 2. Mean _drift magnifications for two bottom

supportedcylinders

The results forf42) are plotted in Fig. 2 as functions of ks

over the range O 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 42) given by equation (53a)

is surprisingly accurate.

A remarkable feature of the results for (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. j42) 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 malt-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 /42), we find that the

approximation of equation (53b) is hopelessly at variance

with the accurate solution. lt appears that at least two Fourier harmonics are essential to represent with any

degree ofverisimilitude the second order pressure gradient in

line with the two cylinders. Indeed this is hardly surprising. However, the parameter (2) is much less significant than (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 forme r.

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 plat-forms. It may be anticipated that the behaviour described

Applied Ocean Research, 1985, Vol. 7, No. 3 133

2 m(Â0A)ir(kfl)

(49)

The numerator in each case is evaluated by means of the

following readily established identify:

A_1A+ A0A= Re[A'(A_1+A1)J+i Im[A'(À_1A1)]

(50) From equations (14) and (18) we have

i12\

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

distribu-tion of horizontal drift force decreases very rapidly with distance below the free surface - indeed a major

compo-ment 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 finite 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 defmed

by a set of vertical and horizontal orthogonal planes,

thereby ensuring a highly efficient boundary element

approximation. Quadratic isoparametric elements are used in the Imite element region, and the fictitious outer surface is represented by quadratic boundary elements. A descrip-tion of the method and validadescrip-tion 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 450

_{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

9m

//////7/////////////

a) DmeflStOns _b) El emenrnesli

Figure 3. Floating cylinder geometry: (a) dimensions; (b) element mesh

134 Applied Ocean Research. 1985, Vol. 7, No. 3

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 I ) 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 1842 m down to 146 ni). 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 the total vertical forces in Cases 1, 2

and 4 are in the approximate ratios I : 2 : 4. The

correspond-ing ratios for the horizontal forces are approximately I: 4 :I 6 . This suggests that the total vertical force is simply

the sum of the forces on each cylinder considered in isola. tion; 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. lt 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 monotonicaily to a value of about I .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, ofcourse, to relate these results, based on an ideal flow calculation, to the expected behaviour of

the real fluid. Chakrabarti'2 and Standing et al.,3 have

com-pared 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

R=

Cd A

2ir(ka)2 a (54)

(in terms of the parameters defined above, with drag

coefficient Cd). If we take Cd = 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 w> 0.49 rad s' (ka> 0.22). Under related conditions, the

drift enhancement effects for the two

3

Wave drift enhancement effects in multi column structures: R. Eatock Taylor and S. M. Hung

'.0 - 1 'Q r-C r- r-C r- '.0 r-C

r-Cr- r-Cr- r- to r- to

CCC- -ri

rr-C rr-C rr-C rr-C rr-C rr-C c-i 'II 'I 'l VI C C C C C C C V C C C C C C ri r-i r-1 c ri ri c' 'o rI f) C r- r--1 'O '/-, rr- 't- 't- 't ri i I I ,/(rad s'I

Figure 4. Mean drift enhancement effects for groups of iV floating cylinders

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 w = 0.49

rad s1, 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 viscous effects

pre-dominate under these particular circumstances. On the

other hand, at the highest frequency plotted in Fig. 4, the

value of R ven 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

fre-quencies. 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

iiieiia foi rnulti-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.

CONCLUSIONS

The preceding analysis and results enable certain con-clusions 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 r-1

00

N N

z

C C C C C C C r 't r C rc' -'Q C rl r-C c' r-4 't r-C'I

-r-lrlr-l--I I I I 1-Q Cr-Cr- r--i C r-i C C N

z

- Cr- Cr- c' r- '0 't-r-i - C O 'O

_{- I}

I Cr-Cr- r-- C C C 't-o t N

z

CI ri C to r 'Q Cr- '.0 to to r-1 C to 1 TTT Q 'O C_{C C CI} r-1

z

-'ri- to to o',t- ri '1 C 'Q r- C r- Cr- 'Q 't-- 't. r't--4 o N

z

ri

-- Q - r-r- ri 'Q 'Q rn 't- 't 'l r- r-i I I I I I e s e o o e

.

Q _{t 'o e} c' r--O r-i r-r-1

fl 0 -

'0 f)2) s e Cr- r-- rl Nf,21

z

C.i 0 . A A

z

'D O 5 Q o

z

z

C., Q o

Ot

z

c r-- r- C r- C 'o ri c' c' 'Q r-r- C

-Cr- r-i C 'o C C C C C C ri 0, rl t- '.0 r-i c' r- 'I'1- 0 C', C', ,-n ri r-I I I I I I r- ri r- . -t c r-i_{,t-, r- Q',} ri C' rfl O',

Wave drift enhancement effects in multi column structures: R. Eatock Taylor and S. M. Hung

dynamic interactions are substantially more pro-nounced 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.

(fi) The long wave analysis shows that at

asymptoti-cally low frequencies (and a sufficiently large spacing between the cylinders) the total mean

horizontal force on the pair of 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

pene-trating 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

fre-quency asymptotic behaviour appears to be

independent of the precise arrangement of the two

cylinders relative to the direction of wave

inci-dence.

(y) 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 N2 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.

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.

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

contribu-tions may be misleading, unless the effects of

hydrodynamic interaction are accounted for. (vili) There is a need for some experiments to test the

aforementioned findings. To arrive at a method

of hydrodynamic synthesis which accommodates drift response criteria, it will also be necessary to perform further systematic studies of the influence

of interaction effects in a variety of multi-body

configurations.

*The theoretical basis for this rule is discussed in Mclvcr's

discus-sion accompanying this paper.

136 Applied Ocean Research, 1985, Vol. 7, No. 3

REFERENCES

I Newman, J. N. Second order slowly varying forces on vessels in irregular waves. Proc. mt. Symp. Dynamics of Marine

Vehicles and Structures in Waves, 1.Mech.E., London, 1974,

182

2 Pinkster, J. A. Mean and low frequency drift forces on floating structures, Ocean Engineering 1979.6, 593

3 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

4 Spring, B. H. and Monkmeyer, P. L. Interactions of plane

waves with vertical cylinders, Proc. /4th mt. conf, on Coastal

Engineering, Copenhagen, Denmark. ASCE 1974. 1828

5 Ohkusu, M. Ilydrodynamic forces on multiple cylinders in

waves. Proc. mt. Svmp. Dynamics of Marine Vehicles and

Structures in Waves, 1.Mech.E., London, 1974. 107

6 Matsui, T. and Tamaki. T. Hydrodynamical interaction between groups of vertical axisymmetric bodies floating in waves, Proc. Jot. Svmp. Hydrodyn. Ocean Eng. Trondheim,

1981, 817

7 Mclver, P. and Evans, D.V. Approximation of wave forces on

cylinder arrays, Applied Ocean Reasearch 1984,6, 101

8 Eatock Taylor, R. and Hung, S. M. Mean drift forces on an

articulated column in a wave tank. Applied Ocean Research 1985, 7,61

9 Abramovitz, M. and Stegun, I. A. Handbook of Mathematical

Functions, Dover Publications, New York, 1965

10 Lighthill. M. J. Waves and hydrodynamic loading, Proc. 2nd

lot.Conf.on Behaviour of Offshore Structures 1979. 1. 1

11 Eatock Taylor. R. and Zietsman, J. Hydrodynamic loading on

multicomponent bodies, Proc. 3rd mt. Conf. on Behaviour

of Offshore Structures 1982, 1,424

12 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:

I a° In=jn(a)

-(-)

n! 2

2i[

a

h0=H2)(a)a l--I

irL

\2)

hi=H2)(a)es[l

(31

hnH2)(a)(n_ l)!()

n 2

where 'y= 0.5772. Furthermore, the derivatives of the Bessel and Hankel ftnctions satisfy the following

recur-rence relations (where for our purposes C,

J or H,2):

C(a)

= -

Cj(a)

n n

C,(a)=C_1(ce)-- C0(cx)=00+1(a)+ C0(a)

a

a

We therefore obtain

H2)'(a) _H12)(a) i

2\

g0=

-J(a) J1(a)

H2) (a) + (n/a)H2(a)_ni-1

J01(a)+ (n/a)J0(a)

g20=

l+n

xl (nl)!(

ri /2)0

--

a i

\a) ]

From Dr P. McI ver, Department of Mathematics, University of Bristol, UK

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

vlinders tended to four times the force on an isolated

ylinder. Furthermore, numerical results for four truncated

cylinders were found suggesting a general rule for the

calcu-lation 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 toN times the force on an isolated cylinder, while the _{mean vertical force} on theNcylinders tends toNtimes 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 ofwater waves.

Some discussion of the effects of viscosity is given by

Eatock Taylor and Hung.

Consider a group of N identical, fixed. axisymmetric

bodies in water of uniform depth h. A co-ordinate system

(r, O, z) is chosen so that (r, O) 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 (r1, O) are centred on on the axis of the jth body which is situated at r = s.

A umform non-linear wave train is incident upon the

uodies from the direction O = ir. The resulting wave field,

including incident and scattered waves, will be represented by a velocity potential CV(, O. z, t) which may be written

as a series in the form

= _{i) + 24(2) ±}

(1)

where _{is a small parameter related to the wave steepness.}

The linear component of 4 may be writtenas (J)(')

_{= Re [f(z)(ø' +}

1))

_{exp (iwr)j}

(2) where

f(z)=

igA coshk(z+h)

ew coshkz (3) and

ø'(r, O) = exp (ikr cos 0) (4)

represents the first harmonic of the incident wave train

with frequency w, wavenumber k, and amplitude A: g is e acceleration due to gravity. The potential

_{'(r, 0) is}

Wave drift enhancement effects in multi column structures. R. Eatock Taylor and S. M. hung

ACKNOWLEDGMENTS

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 tor the jth body may also be written as a series expansion in e. so that

F1 = eFJ') + e2F2) +

...

(5)

The mean (time-averaged) second order force on a fixed

body 1

1f

(2) _{pg j T72n dl +} P IV»)!2n dS L1 Sj

+(O.O, 1)

(see Pinkster)' where L is the mean water line (with normal

ne), S1 the mean wetted surface (with normal n), r the

displacement of the free-surface from the mean level (due to the linear component of the solution) and W1 the

water-plane 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 fora

single body can be written as

ctf'(rj, 9) = VAH(kr)

cos n0 (7)

where H is the Hankel function of the first kind of order

n. The complex scattering coefficients (Aa, /1 = 0, 1,...) are, in general, functions of the wave frequency and the

body geometry. For the case of a vertical cylinder

extend-ing throughout the depth of the fluid they may be

de-termined analytically (see, for example, Mel, p. 3l2) and

it is found that A0 is O((ka)2) and A is O«ka)2'). n O 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 ln(ks)), where s i a typical spacing and the small argument asymptotic forms for the Hankel

functions have been used (Abramoxitz and Stegun,

p. 360). 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

consider-ing 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),

(6)

x [cosh2 k(z + h) + sinh2k(z +h)]

Applied Ocean Research, 1985. Vol. 7, No. 3 137

and

= A2

j (8) (9)

Ir

= j 2Lw cosh khi i -=

1 --n! (n - l)!( -

_n I Ir

or the terms corresponding to negative n. we use the

identities

Wai'e drift enhancement effects in multi column structures: R. Eatock Taylor and S. M. Hung As 2 is a constant for the incident wave there is no

contri-bution to any component of the second order force from

the integral over the waterline in equation (6). The

remain-ing terms have no dependence upon r and O and so

contribute an equal amount to the mean vertical force on

each body. That is, the total force on the iV bodies is N

times the force on a single body. In particular. the ratio of

the force on N bodies to the force on an isolated body

tends to N as ka tends to zero, in agreement with Eatock

Taylor and I-lung.

For low frequencies when the scattered field is neglected

there is no contribution to the horizontal components of the mean second order force from any of the terms in equation (6). Therefore it is necessary to include the

scattered wave field when considering the limiting

behaviour. This is perhaps most easily done by considering

the 'far-field' expressions for the second order force. As

kr1 tends to infinity (i.e. in the far field) the expression (7)

for the wave field scattered by a single body may be

approximated using the large argument asymptotic forms of the Hankel functions (Abramowitz and Stegun, p. 364) as

-) 1/2

Ji)

i(r) exp [irj

_{/4)] (10)}

where,

fl(0)=

A(if1 cos nO1

n=o

By considering a momentum balance Maruot and Newman6 derived the expression

- cos O) Ill/(0)12 dO (12)

for the component of the mean second order force on an

isolated body in the direction of wave advance. Here c is

the phase velocity and Cg 5 the group velocity of the lineansed incident wave. The formula (12) is equally

applicable to a group of bodies provided 11(0) is replaced

by a function 11(0) such that the total far field scattered

potential is

/ 2

1/2

exp [i(kr - ir/4)J (14)

The relationship between 11(0) and Hi(0) is sought in the limit of low frequency. Note that the array of bodies must

be finite, that is

it cannot extend to infmity in any

direction.

As a first step consider the scattering of the incident wave within the array. As pointed out previously, the

largest scattering coefficients (A0 and A1) are each O((ka)2)

Thus if the wave field scattered from the incident wave by

one body is further scattered by a second body this will produce a modification to the leading order scattering

coefficients of O((ka)4). Hence, as a first approximation multiple scatterings within the array may be neglected and

only the scattering of the incident was considered. The

contribution to the far field potential due to scattering by the jth body is therefore given by equation (10) with

fl(0) = A0 - ¡A1 cos O + O((ka)4) (14)

- pgA2

Fi Cg

kc7T

138 Applied Ocean Research, 1985, VoL 7, No. 3

Expanding in powers of s1Jr it is easily shown that

()»

1/2

_{/ 2}

s1/2 (15)

expkrj) =Ç) exp(ikr)

and si cos 0 = cos 0 + O -r and therefore

A0iA1cosû+O

(ka)2S/(ka)4 ç5l) r x

(-,

»1/2 exp [i(kr - J4)]

i + o(

(17)

\kr)

rj

and so by linear superposition and comparison with (13)

11(û)=Nfl1(0)+O (ka)2(ka)4

(18) The ratio of the total mean horizontal second order force on the N identical bodies to the force on an isolated body is

therefore

F

N2J(1_coso)

J(i

cosO) 11(0) dO

(19) which tends to N2 as ka tends to zero, again in agreement with Eatock Taylor and Hung. This result is easily extended

to the component of the drift force perpendicular to the

direction of wave advance.

REFERENCES

i Pinkster, J. A. Mean and low frequency wave drifting forces on

floating structures.Ocean Engineering 1979,6, 593

2 _{Longuet-Higgins, M. A. and Stewart, R. W. Radiation stress and}

mass transport in gravity waves, with application to surf beats.

Journal of Fluid Mechanics 1962, 13, 481

3 Mei. C. C. The Applied Dynamics of Ocean Surface Waves,

Wlley-lnterscience, New York, 1983

4 Abramowitz, M. J. and Stegun, L A.Handbook_ofMathematical Functions, Dover, New York,1972

4 _{Maruo, Fi. The drift of a body floating in waves.}Journal of Ship Research 1960,4,1

5 Newman. J. N. The drift force and moment on ships in waves.

JournalofShip Research 1967, 11,51

(16)

COMMENTS ON NOTE BY P. Mc [VER

We are most grateful to Dr Mclver for his interest in our

work. E-lis analysis investigates the theoretical basis for our

conjecture regarding drift enhancement effects, for groups of general axisymmetric bodies. It is always satisfactory if

a simple explanation can be found for some heuristic results obtained by numerical analysis.

The verification of our conjecture' regarding vertical forces seems to be based on an approach in which the

scattered wave field is neglected. One must be careful in

2

1ean vertical drift force on a cylinder (radius 9 m, draught 32 m,

water depth 148 ?n)

Wave drift enhancement effects in multi column structures: R. Eatock Taylor and S. M. Hung

the interpretation of this assumption. One may deduce

immediately from Mclver's equations (6) and (9) the exact

component of vertical drift force due to the first order

incident potential alone. For a single cylinder as shown in

our Fig. 3a this leads directly to the results given in the second column of the following Table. The third column gives corresponding results based on incident plus scattered

potentials (as in Table 3 of the original paper).' These

suggest that, even at low frequencies. the scattered wave contribution is as important as that from the incident wave.

But this need not affect the conclusions regarding the

influence of interactions, for cylinders in a group.

REFERENCE

I Eatock Taylor. R. and Hung, S. M. Wave drift enhancement

cffects in multi column structures. Applied Ocean Research

1985. 7

Applied Ocean Research, 1985, Vol. 7, No. 3 139

!rad s'

f51/Nm2 fz/Nm2 0.125

4862

9393

0.15 0 5 116

9627

0.250

6280

10810

0.350

7260

11800

0.450

7033

11380

0.5 50

5478

9071

0.650

3497

5 864