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Mean drift forces on an articulated column oscillating in a wave tank
M e a n drift forces on an articulated c o l u m n oscillating in a w a v e t a n k 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, University College London, UK
An analytical solution is presented for calculating the regular wave induced respones of an articulated column in a wave tank. Extension of the procedure leads to calculation of second order mean drift forces and moments. Hydrodynamic interaction between a cylindrical column and the parallel walls of the tank is shown theoretically to be highly significant and experimental data are presented to corroborate this finding. Key Words: Mean drift, articulated column, wave forces, wave tank
INTRODUCTION Design of offshore structures such as semi-submersibles, tension leg platforms and articulated columns has motivated recent interest in the prediction of wave forces on large diameter cylinders. One important aspect of his problem relates to oscillatory flows at low Keulegan-Carpenter numbers, where inertial and diffraction effects are predominant and drag forces may be neglected from theoretical predictions. For such cases the use of numerical techniques is widespread and they were first developed to solve the linear potential flow problem, associated with waves being scattered by arbitrary three dimensional bodies. These techniques have then been extended so that mean and low frequency second order wave drift forces may be predicted (e.g. Pinkster, 13 Standing, Dacunha and Mattena7). Attempts have also been made to validate the theoretical drift force results by means of wave tank experiments, in some cases with mixed success. One of the difficulties experienced by some investigators, who have attempted to measure wave drift forces on models appears to be related to the problem of testing in a narrow wave tank. Mean drift forces on large bodies are associated with wave scattering. Any reflections of scattered waves from the side walls of a tank are therefore likely to have a significant influence on the magnitude of measured drift force. An associated problem is that the drift behaviour of a structure comprising multiple cylindrical legs is likely to be rather different from that of an isolated cylinder. Multiple scattering effects, which are unimportant in the context of linear wave forces, may cause major changes to second order drift forces. This paper attempts to shed some light on this phenomenon by examining the behaviour of a cylindrical articulated column. Theoretical predictions are obtained for a column in the open sea, and in a narrow wave tank. The latter results are compared with experimental data, which appear to confirm the predicted effects related to tank wall reflections.
Accepted November 1984. Discussion closes June 1985.
number of its images reflected in the parallel sides of the tank. The problem may be formulated in terms of a velocity potential in a semi-infinite region, subject to boundary conditions on the free surface, on the seabed, and on the body and its images. The linearised formulation corresponds to the multibody problem which has been studied by several investigators for simple body shapes and for restricted ranges of conditions. Spring and Monkmeyer 14 evaluated the first order wave forces on closely spaced groups of vertical cylinders. No assumption was made regarding incident wave angle or cylinder arrangement. A series solution was obtained, the coefficients of which were evaluated from the solution of a matrix equation. The size of this equation was related to the number of cylinders, hence limiting the approach to smali groups. Results were given for two cylinders in various configurations relative to the incident wave. The same method was later used by Chakrabarti 3 to approximate the first order horizontal force on a cylinder at the centre of a wave tank, by employing three pairs of image cylinders. More recently, McIver and Evans 1° formulated a new matrix equation method for the linear multi cylinder problem. In this method, efficiency is improved by approximating the effect on one cylinder of scattered waves from other cylinders as that of plane incident waves. An improved approximation was also given involving a non-plane correction term. Another recent approximate theory, based on a low scattering (small body) assumption, has been described by KyUingstad. 7 In neither of the cited works were results given for the wave tank problem. By observing symmetry around each cylinder in an equally spaced infinite array orthogonal to the direction of the incident wave, Spring and Monkmeyer is deduced a matrix equation solution such that the dimension of the matrix becomes independent of the number of cylinders considered. The condition modelled by this method is equivalent to that of a stationary vertical column situated midway between parallel tank walls. Results were obtained for first order pressures and forces. Massels considered the above problem with arbitrary incident wave direction by using an approximated incident potential in the vicinity of the cylinders. Simple formulae for first order wave forces were obtained. The approximation, however, confines the applicability of the method
Applied Ocean Research, 1985, VoL 7, No. 2
0141-1187/85/020066-13 $2.00 © 1985 CML Publications
THEORETICAL BASIS Solution of the irrotational flow problem for a body in a wave tank is equivalent to that of the body plus a large
66
Mean drift forces on an articulated column oscillating in a wave tank." R. Eatock Taylor and S. M. Hung to cases of small separation and when the ratio of cylinder size to wavelength is small (i.e. low scattering). Masumoto et al. 9 calculated first order and second order mean wave forces on multiple rows and infinite columns of spar buoy type structures. The simplicity of their formulation is due to the use of approximated incident and transmitted wave potentials. The applicability of this method also relies on the size and spacing of the bodies being small compared with wavelength and spacing. The behaviour of wave power devices in a canal was investigated by Srokosz.16 The general solution 6 for the velocity potential between parallel walls was employed, and approximations were made regarding the local disturbance due to the device. More recently, Miles u utilised avariational integral equation and Fourier representation to solve the problem of a submerged or surface piercing vertical cylinder between parallel tanks walls. The results are valid provided that the tank width is small compared with the wavelength. In the following, we use an exact analytical formulation and approximation is only introduced at the final stage of obtaining numerical results, where it is necessary to truncate the infinite series. The procedure is similar to that adopted by Spring and Monkmeyer is for the fixed cylinder in the centre of the wave tank: here, however, there is no limitation regarding the position of the cylinder in the tank, and it is free to pitch in a plane parallel to the tank walls. Both first order wave forces and responses are obtained, and mean drift forces. ANALYSIS OF HYDRODYNAMIC INTERACTION EFFECTS
For an arbitrary shaped body in an ideal fluid with a free surface and a horizontal seabed at water depth d, the first order velocity potential associated with wave scattering can be expressed (cf. Wehausen and Laitone 19) in cyfindrical coordinates. If the origin of the fixed reference coordinate system (ro, 00, z) is taken at the mean free surface, with the vertical axis 0z directed positive upwards, this velocity potential may be written as qbB = Re
[t~B
exp(iwt)]
where d)n=R(r)®(O)Z(z ) is obtained as a solution to Laplace's equation
0 2 ¢ + 1 0 ¢ + 1 02¢ + - 02¢ -=0 br 2 r ~r r 2 b02 0z 2 __
(l)
in the radial coordinates are
/~)(kr0),
/ = 0
[K,.(KFo),
/ >/1
(3)
where H(m2) and K m are respectively the Hankel function of the second kind and the modified Bessel function of the first kind of order m. The functions in the vertical coordinate z are coshk(z + d)
Z/(KF)
,
cosh kd
/=0
(4)
=,
cos K/(z + d) ,
cos rid
]>~1
The wave number k is defined by 092
k tanh(kd) = - g
(5)
and the quantities Ki are the positive real roots of 092
K/tan(Kfl) = -- ---, g
] ~> 1
(6)
It is convenient to use the alternative definition k = Ko. When there are I bodies in an otherwise unbounded sea, the scattered potential can be expressed by superposition. Local coordinate systems (ri, Oi, z) are introduced, centred on the ith body (i = 1 to I), permitting the total scattered potential to be written
1 (OB= E
~
~. [Cq'mPm(K/ri) c°smOi
i=0 m = - ~ ]=0
+ C]/mPm(r/ri) sinmOi] Zj(riz)
(7)
Through use of the Bessel addition theorem (stated in Appendix A), the terms expressed in the ith body coordinates (ri, Oi, z) may be transformed to expressions in the reference coordinates (ro, 0o, z). Thus the scattered potential of equation (7) is written:
CB(ro, 0o, z) /
subject to the boundary conditions
~-
E Ej = O
=
= -c°2 -¢ ~O Oz z=o g
l
em(K/ro) =
i=o
z:o , -O0 z=-a = 0 0z
~
~o
X
X
m=--*o
+C~m
~
Gi]m n (Kiro, nOo)
n=--oo
s Giimn(rlro, n0o)] Zi(r]z )
(8)
n=--ao
Also at infinity
with exp ( - ikr) 0 c ~ - -
c
cosm00
Gijmn s = 6mnPm(K/r°) sinm00'
i= 0
(9)
as r -+ oo. Equation (1) then gives (see Drake s) where 6rn n is the Kronecker delta, and
cPB(ro,Oo,z)= ~. ]=0
~
[CfinPm(K/ro)cosmOo
c Gijmn
m =-~
+ C;mPm(Kjro) sinm0o] Zj(r/z)
(2)
The coefficients Cfm , C]m are obtained from the boundary conditions: they correspond to the terms in ¢8 that are symmetric and antisymmetric about 0o = 0. The functions
+ --
s
= (--1)mpm+n(r/loi) Qn(r/ro)
cos(m + n) 0oi
+
¢os sin
(n0o)
+
sin ] sin(m + n) Ooi cos(n0o)/,j
i >t 1
(10)
Applied Ocean Research, 1985, VoL 7, No. 2
67
Mean drift forces on an articulated column oscillating in a wave tank•" R. Eatock Taylor and S. M. Hung Here loi coordinate relative to functions Q
and Ooi are the radial distance and angular of the origin of the local coordinate system the origin of the reference coordinates. The are Bessel functions of the first kind: jn(kro)
] = 0
Qn(K/r°) = [ In(K/ro)'
]> 1
C~.m = Cojm c $
- - C ( 2 i - - 1)]m : C z i j m
: C~/m
(13)
,I
© ©
=
4
© © 1
0
2
y
rn=-,~
i=0
2
n ....
(-1) Gqmn(K/ro, nO©) Zl(t~/z } i=0
(14)
n =-~
By taking i from 0 to infinity we provide an exact representation of the effect of the tank walls. By taking i = 0 and 1 we obtain the case of two cylinders in the open sea. We note that the number of unknown coefficients is in fact independent of the number of image cylinders employed. In cases where bl = b2 in Fig. 1, it may be shown that there is no coupling between symmetric and anti-symmetric components of velocity potential. Since the boundary condition associated with incident waves propagating in the direction normal to the line of cylinders has only symmetric terms, the antisymmetric components of the scattered potential are identically zero. Under this special condition, the series form in equation (14) may be shown to reduce, for a fixed cylinder, to the series solution obtained by Spring and Monkmeyer) s Equation (14) also applies to any body geometry that is symmetric about the 0o = 0 plane, and may be employed for both diffraction and radiation problems. Attention is restricted here to surface piercing circular cylinders of radius a. The diffraction potential 4~D satisfies 0¢D
---
Or
0~ I
Or
on r = a
(15)
where the incident velocity potential 4~z is given in the reference coordinates by
Oi =
~ n=--~
igAzo(kz)(--i)nJn(kro)cos(nOo)
(16)
(.,O
for a regular incident wave of amplitude A. The radiation potential q)R generated by column oscillation in the direction of wave propagation satisfies
0@ Or
- i ~ f ( z ) cos00
on r = a
(17)
where f(z) is the amplitude of horizontal motion of the cylinder cross section at elevation z (the cross section being assumed to be rigid). Making use of the orthogonality of the eigenfunctions Zo(kz), Zi(~/z) when integrated over the depth of the water column, and enforcing the body boundary conditions, we obtain for each value o f ] two sets of linear algebraic equations for the unknown coefficients associated with the diffraction and radiation problems respectively. From the symmetric components (terms in cosn0 in equation (14)) we obtain:
© ©
~" 5
j=O
(12)
In what follows it is convenient to omit the subscript 0. Thus making use of equations (12) and (13), we may write
© ©
4~B(ro,0o, z)
(11)
The nth term of the infinite series for Gi]mn signifies the contribution to the velocity potential of the nth harmonic (in the reference coordinate system) by the ruth harmonic solution of body i. The subscript] identifies dependence on the ]th vertical eigenfunction. The analysis leading to equation (10) is completely general as regards the shape of the individual bodies or their spatial arrangement. In this investigation, we are primarily interested in the flow past equal diameter cylinders placed in a line perpendicular to the direction of the incident waves. We shall consider specifically the cases of (a) two cylinders, and of (b) a large number tending to infinity. Case (a) is examined to provide a cross check with other published data, while case (b) is related to a cylinder placed in a wave tank. It is readily seen that the effect of the parallel tank walls may be simulated by an arrangement in the open sea consisting of such an array of cylinders. Each cylinder in this imaginary array is an image of the single real cylinder, reflected in the two parallel tank walls. The image cylinders extend to infinity on either side of the real cylinder, although for the purposes of computation it is feasible to employ a finite number of cylinders in the array. A typical arrangement is shown in Fig. 1. We take the origin of the reference coordinates (ro, 0o, z) at the centre of the cylinder numbered 0. The local axes (ri, Oi, z) for each other cylinder have the origin at the centre of cylinder i, and 0i is measured in the same direction for each cylinder• We exploit the symmetry of the odd and even (symmetric and antisymmetric) functions of 0o in equations (9) and (10). The scattered velocity potential for cylinder i within each set of zero normal flow boundaries (shown as dotted lines) is equal to the corresponding potential for cylinders i + 2, i + 4, i - 2, i - 4 etc. The scattered potentials for cylinder i + 1, i -+ 3 etc are the mirror images of the potential for cylinder i, as reflected in the boundary equidistant between the corresponding cylinders. Under these circumstances, the following symmetries are found: S
equation (8) as
3
2
i
m=-~
X [C~m cos(m + n) Ooi + (--1)iCfin sin(m + n) Ooi] i=1
× (--1)mpm +n(n/loi) Qn(gia) + CTmP'n(Kia) = Bin, Figure 1. Geometry of reference cylinder and its images in walls of wave tank
68
Applied Ocean Research, 1985, Vol. 7, No. 2
--~
(17)
Mean drift forces on an articulated column oscillating in a wave tank: R. Eatock Taylor and S. M. Hung
(-i)n+lJ'n(ka),
/=0
B/n =
(18)
~0,
Fx Fy
(¢z + ~bo)\sin 0 /
f>~l 21r
For the radiation problem the right hand side is
( My ] = icop Xmx!
0
f :(z) Z/%z) dz --el B/n =
, n=+l 0
;! z}(K/z) dz (19)
0,
n=0, lnl>l
From the antisymmetric components (terms in sinn0o in equation (14)) we obtain, for both radiation and diffraction problems:
~_
~ [CTmsin(m+n)Ooi-(-1)iC~m cos(m+n)Ooil
m=--** i=1 I
$
t
(-- 1)mpm +n(t¢]loi) Qn(gja) + C}mPn(gja) = 0 (20) If we truncate the various series, we may write these equations in matrix form. Thus we assume that we may retain only J terms in the series over ], and M terms in the series over the Fourier harmonics m and n. The number of image cylinders is taken to be I. Equations (17)-(20) are therefore expressed in the form [Arc A]c
ArS l [ C]C] : A]s [ C / sj [B/]
(21)
where 1
CC
t
A/nra = 8mnP'n(K/a) + • (--1)mPm+n(r]loi) i=1
x Q'n(r/a) cos(m + n) Ooi 1
ff o
t
ic°s°o
(q~z + q~D)(z + d) k-- sin 0 ]
dO dz (23)
It is clear that only the first Fourier harmonics (n = -+ 1) in the potentials ¢I and q~o contribute to the forces and moments. The added inertia and damping terms are obtained in a similar manner from the radiation potential q~R. These terms also involve only the first Fourier harmonics (n =-+ 1), although it should be noted that solution is required for all of the coefficients C/m: this in turn re~luires formulation of equations (21) and (22) over the full range of values of n. Finally the wave induced response of the articulated column (i.e. the tilt angle in the direction of propagation of waves down the tank) may be evaluated by direct solution of the equation of motion. MEAN DRIFT FORCES A method has been given by Pinkster 13 for evaluating mean drift forces and moments on floating bodies. This involves direct integration of the hydrodynamic pressure on the instantaneous submerged surface of the body, in conjunction with a perturbation expansion of the governing variables in terms of wave steepness. Retention of terms up to second order in the expansions, and evaluation of the time averages, leads to the mean drift forces and moments. The approach has been extended by Drake, Eatock Taylor and Matsui4 to include the second order contribution due to motion of an articulated column at the seabed. A well known alternative method for obtaining horizontal mean drift forces, and moment about a vertical axis, uses the far field momentum approach.12 Although simpler to formulate this is not suitable for the present work. We thereforefore adopt the former approach to write the mean drift forces and overturning moments on a pair of cylinders and on an articulated cylinder in a wave tank. The expressions involve the total first order potential ~, where
.
~b= ~z + q~B
i=1
(24)
and equations (16) and (14) are used to obtain ~I and ~B respectively. The latter includes contributions from both diffraction and radiation potentials. For the general case where a column is free to undergo small rotations about two orthogonal horizontal axis at the base of the tank, we may define a complex rotation vector ~ = (f2x, £2y, 0), corresponding to harmonic response at frequency co. We thus obtain
1
Ajn m = ~" (-1)m pm+n(giloi) Q'n(g/a) sin(m + n) Ooi i=1 I
A/n m = 8mnPn(gja)-- ~ (--1)i+mpra+n(rjloi) i=1
X Qn(gja) cos(m + n)Ooi
0
-a
CS A/n m = ~ (--1)i+mp m +n(t~jloi)Qn(gja) sln(m + n) Ooi
$c
dO dz
0 -el
(22)
The matrices A/, B/, and C/are of order M x M, M x 1 and M x 1 respectively, and the subscript ] runs from 0 to J. The subscripts m and n also take both positive and negative values, as seen from equation (17) and (20). For each value o f / , equation (21) may be solved for the diffraction and radiation problems, giving in each case the coefficients C/C m and Cfm. These may then be substituted into equation (14) to give the potentials q~o and q5R. Hence for a cylinder hinged at z = - d the first order complex hydrodynamic forces and moments may be obtained from
if(2)/T(" x2, = ~ Re [ ( - -Pg 2
cBJ +off so
• CW
It/+
Xxf2,--X,~2x,2(nXldl \ nyl \ ny
{½1V~12+iw(Rx~2)'.V¢*}(nx) ds ] ny
(25)
Applied Ocean Research, 1985, Vol. 7, No. 2 69
Mean drift forces on an articulated column oscillating in a wave tank." R. t:'atock Tavh)r and S. M. Hzmg NUMERICAL
jl,~x(2), = 2 Re [ - - ~ -
flrT+Xxg2y--Xy~2xIRzl \ - - n y /} dl cw
--ioop fo*(Xx~2y--Xyg2x)R z ( n ~ , ) d , cB ~r +;ll
{½lv$12+i¢°(Rx ~)'V¢*} so
x (Rznx--Rxnz) ds] \Rzny Rynz/
(26)
where r/is the first order free surface elevation, given by -
at
(27)
Oz
X = (Xx, Xy, Xz) is the position vector (relative to the origin in the mean free surface) of a point on the submerged surface S of the body; R = (Rx, Ry, Rz) is the vector from the point about which the moment is evaluated to the point of integration; n x, ny and n z are direction cosines; C w and Cs are the intersections between So and horizontal planes a't the mean free surface and the seabed respectively (it is assumed in the above that n z = 0 around C w and 6'3). In the case of the circular cylindrical column of radius a, pitching dnly in a plane parallel to the direction of wave propagation, we obtain (in scalar form) 2rr
2[Re-7Pg of [l~+mkcosOl2]acosOdO 277
-- i¢op j
[q~*ff]z = _ a a 2 cos20 dO
0 2rr
0
0
--d
+i°3(z--zA)~O¢*sinO} ] a O c°sOadOdz O
(28)
2n
m~
= 2 Re
-- 7
.
[Ir/+aqJ cos0 121==o aza cos 0 dO
o 2Tf
--icop f
[O*~]z=_da 2 cos2,0(d+zA)dO
0 2rr
0
0
In order to implement the preceding analysis it is necessary to truncate the various series at a finite number of" terms. We consider here some of the implications of" this truncation. From equation (10), it can be seen that if the Fourier series in 0 is included up to the nth harmonic, then both the Hankel function of the second kind 11(2) and the modified Bessel function K need to be calculated up to order 2n. These have as argument the term g/loi. We recall that the •i are the generalised wave numbers ('the roots of equation (6)), and loi is the distance from the reference cylinder to the ith image. From tables 9.4 and 9 . l l of Abramowitz and-Stegun, 1 it is clear that the high order ( > 5 0 ) Bessel functions ./ and K with small arguments can easily have values outside the normal numeric range of computers. Thus Jso(1) = 2.9E--80 and Ks0(1 ) = 3.4E77. This computational limitation has been found to preclude the use of more than 25 harmonics in 0 under general conditions. Fortunately, numerical experiments described below confirm that in most cases, only a small number (~<6) of harmonics is needed for first and second order forces to converge. Exceptions to this rule have only been found for cases where the cylinders are almost touching each other. Under these latter circumstances, the highly confined flow near a cylinder may require higher harmonics for convergence. Similarly, a larger number of harmonics is required for accurate calculation of the free surface elevation at large distances from the cylinder. This is associated with using the Bessel function Jn(kr) to describe a plane progressive wave, for which clearly the radial coordinate is not well suited. Let us now consider the series in ]', for the radiation problem (since / = 0 is the only term required for the diffraction problem). Due to the orthogonality of the eigenfunctions Z/ in equation (2), the coefficients of this series (for each value of m) are uncoupled. This permits the contribution of each successive term in the series to be efficiently monitored (e.g. in evaluation of the hydrodynamic coefficients). In this way it has been found that convergence of results is generally rapid. The remaining series that needs to be truncated relates to the number of image cylinders employed on each side of the reference cylinder: this is the series over the index i in equation (7) for the wave tank problem. If we consider the transformed expressions obtained after use of the Bessel addition theorem, equation (8)-(10), we see that the function Pm+n(Kjloi) is a measure of the influence of cylinder i on the reference cylinder. Except for the term corresponding to ] = 0, Prn+n is a modified Bessel function K. The rapid decay of this function with large argument implies that contributions from cylinders which are far away may safely be neglected. The term ] = 0 corresponds to a progressive wave component, and Pm+n is a Hankel function of the second kind. Ensuring convergence of these terms, for increasing i, is more troublesome. To obtain converged results efficiently, it is necessary to use the asymptotic form of the Hankel function, namely
--d
•O~* sinO }cosOa(z--,A)dO dz](29) + ico(z+za)~a30 where ~ is the complex pitch amplitude and (0, 0, ZA) denotes the centre of rotation on the seabed pivot.
70
CONSIDERATIONS
Applied Ocean Research, 1985, Vol. 7, No. 2
H(Z)(r)
•v rrr
exp i •
2
+-2
r
(30)
The convergence is then inversely proportional to the square root of the distance to the image cylinder. This
Mean drift forces on an articulated column oscillating in a wave tank: R. Eatock Taylor and S. M. Hung asymptotic approximation may therefore be used in the evaluation of the terms Ajmn in (22). We have, for example (replacing i by j as the summation index):
assuming that the series converges. This is transformed using the formula in Appendix B, with
B*
8 =-kBz
v=-+z; i Hg{,(kloi) j=1
+ ‘2’
(36)
B2
H,$,(kl,,j)
j=l
(31) The infinite sum is then evaluated by transforming it to integral, using the formula of Lipschitz given in Appendix Reference to Fig. 1 shows that there are three cases be considered, so that all image cylinders are included the series. These are given by
an B. to in
loj = B1 + jB*
(32)
Bz = 2(b* + b,)
(33)
where
and
The resulting integral may be transformed into a finite range and evaluated numerically. An exception to the general case of a cylinder placed asymmetrically in the tank arises when kBz is equal to an integer multiple of 2n. The infinite series then becomes
j;o+P2 (v
which does not converge. This arises when the tank width (b, + b2) is an integer multiple of the half wavelength h, and corresponds physically to the excitation of standing waves across the width of the tank. In reality the amplitude of these waves must be limited by non-linear and viscous effects. As far as our irrotational flow theory is concerned, the solution is singular at these values of wavelength. For the case of a cylinder placed on the centreline of the tank (b, = b2), the general solution given above is slightly modified. We take
(34) loi = jB for the three cases respectively. In each case we may write
=$$
exp[ -ikB2(;+j)]
(35) -+r+i B2
Table 1. First order wave forces on one cylinder of cz two cylinder pair (ka = 0.4), compared with reference 14
lla
2.500 3.250 4.000 5.000 6.250 7.500
7.0122 (6.7246) 5.1843 (5.1817) 4.6415 (4.6394) 4.4069 (4.4049) 4.2456 (4.244 1) 4.1497 (4.1477) 4.1141 (4.1121)
0.5542 (0.5542) 0.4753 (0.4753) 0.4181 (0.4181) 0.3794 (0.3794) 0.3371 (0.3371) 0.2887 (0.2887) 0.2459 (0.2459)
where B = 2bI is the tank width. The corresponding series is then singular when kB is an integer multiple of 2n; i.e. the tank width is a multiple of the wavelength. Physically the difference from the preceding case is that antisymmetric standing wave modes cannot be excited when the cylinder is placed symmetrically in the tank.
($+j)-“’
=~exp[-ik~2(~+~~]~~~~(-ik~2~!,*
2.000
(37)
156.8 (157.9) 163.6 (163.7) 166.0 (166.1) 167.4 (167.5) 168.9 (168.9) 170.4 (170.4) 171.6 (171.7)
78.1 (78.2) 67.2 (67.3) 50.5 (50.6) 32.6 (32.6) (77::) -23.7 (- 23.8) -53.8 (-53.8)
1.6550 (1.5880) 1.2237 (1.2236) 1.0955 (1.0955) 1.0402 (1.0402) 1.0021 (1.0021) 0.9794 (0.9794) 0.9710 (0.9711)
DISCUSSION OF RESULTS
The theory presented above has been employed to investigate the drift forces on an articulated column on the centreline, and offset from the centre, in a wave tank. First, however, we examine two closely spaced cylinders in the open sea. Our results have been obtained by using the same theoretical formulation as for the wave tank problem, and enable some comparisons to be made with predictions obtained in an alternative way?4 In Table 1 values are given for first order forces on one of the two cylinder pair, corresponding to equation (23). The tabulated results are parameters R and 6, defined according to ref. 14 by
It may be recalled that F, is the first order force in line with the direction of wave propagation, and F,, is the force along the line of the c linders. Values are given for R,, R,, 8) between F, and the corresponding S,, a,,, and the ratio ~1,
Applied Ocean Research, 1985, Vol. 7, No. 2
71
Mean drift jbrces on an articulated column oscillating in a wave tank: R. Eatock Taylor and S. M. llung Table 2. Mean drift Jbrees on one cylinder of a two cylinder pair (ka = 0.4)
2.000 2.500 3.250 4.000 5.000 6.250 7.500 10.000 100.000 1000.000
0.78440 0.50644 0.42732 0.38232 0.33196 0.27538 0.23012 0.18118 0.21407 0.20875
9.41225 0.62951 0.22961 0.12042 0.05450 0.01179 -0.01038 -0.02319 -0.00052 -0.00145
3.72875 2.40704 2.03132 1.81741 1.57803 1.30905 1.09389 0.86127 1.01762 0.99232
force on a single cylinder in the open sea. Thus /~(1) is a direct measure of the interaction effect. Results in Table 1 are limited to the single frequency considered in ref. 14, corresponding to ka = 0.4, over a range of spacings l between the two cylinders. The values in parentheses are from Spring and Monkmeyer 14 (corrected by multiplying the values in Table 1 of the cited reference by the factor 2cosh 1.6). Our results are obtained from equation (14), using Fourier harmonics up to +100 in equation (7). Except when the cylinders are touching (l/a = 2.0), the results from the two methods are seen to be virtually identical. Differences may be assumed to arise from truncation of the series at different numbers of Fourier harmonics - i t is not clear from ref. 14 how many terms were employed. In Table 2 we give the total mean drift forces fix(z) and /7(2) on one of the two cylinder pair, and the ratio #(2) between/7(2) and the corresponding drift force on a single cylinder in the open sea. Again these results are for/ca = 0.4. It is found that the influence of hydrodynamic interactions on the drift forces persists over much wider separations than is the case for first order wave forces. Indeed substantial magnification of drift forces can arise from interactions between widely separated cylinders. Additional results, to be presented in the future, suggest that for separations greater than about five radii the magnification/a(2) tends to a value close to two, as ka tends to zero: while under the same conditions/a(J ) tends to one. For smaller separations, the drift force magnification is substantially greater than 2 at low frequencies. This behaviour associated with hydrodynamic interaction effects has important implications for the mean drift of structures such as tension leg platforms involving several closely spaced columns. We turn now to results for the cylinder in a wave tank. Our calculations of first order force on a fixed cylinder at the centre line of a tank agree to within one part in the fifth significant digit with those given in Table 2 of ref. 15. The results cover the non-dimensional frequency range 0.1 < k a < 1.0 and tank width B in the range given by 0.628 < kB < 30.8: they fall within the extremes B/a = 2.5 and B / a = 3 0 0 . We employed 100 image cylinders and Fourier harmonics up to + 100 for these calculations. The same procedure was then adopted in obtaining the drift pivot moment on a model articulated column in a wave tank. The geometry is that used for the experiments by Drake, Eatock Taylor and Matsui. 4 The cylindrical column having a radius of 100 mm was tested in a tank of width 2.2 m and in water of depth 1 m. The column was initially free to articulate in a plane parallel to the length of the tank, about a simple hinge surrounded by a flexible
72
Applied Ocean Research, 1985, Vol. 7, No. 2
collar. The centre of rotation was 825 mm tinder tile mean free surface, and below this point was attached a rigid section of cylinder as shown in Fig. 2. As described in ref. 4, a vibration absorber was then incorporated into the test arrangement so that the column was prevented from responding to the dynamic force in regular waves. It could, however, adopt a steady tilt, from which the mean drift force could be obtained. Further details have been given by Drake. s Let us first consider the theoretical predictions for this column. Figure 3 provides an indication of the convergence of the numerical results for second order drift moment on the column when placed on the centreline of the tank. The mean overturning moment about the hinge has been non-dimensionalised by the quantity ogA2ad, and plotted
20
~i
i
I o
I
__t
/
/
/
/
/
/
I / I/
I
Figure 2.
o6't
/
m
I
1
I
I
/1111
1
I
Model o f articulated column
---~---N
varying)
I=15
-
vorylng,
N=15
~
- I
° 62
o
.60.
% C71 CD
If-/
^\ IS--
~ 58
ogB
J
°54
M, I Figure3. Convergence o f mean drift m o m e n t about hinge f o r f i x e d column at eentreline o f wave tank, ka = 1 (M = number o f Fourier harmonics, I = number o f image cylinders)
Mean drift forces on an articulated column oscillating in a wave tank: R. Eatock Taylor and S. M Hung .8_ +
,6_
%: UT~
/
,4_
+
\ ..]Y',if, /
.2_
+
+1
/
X10-1 k~
Figure 4. Mean drift overturning m o m e n t about hinge for fixed column: - theory for column on tank centreline; - - - theory for column in open sea; + experimental averages for column on tank centreline
1.0.
,8.
-t5
~<
.6_
+
ffL, IZ2
.4.
.2_
/
"/
/
XIO-I
ka
Figure 5. Mean drift overturning m o m e n t about hinge for fixed column: - - theory for column offset from tank centreline; - - - theory .for column in open sea; + experimental averages for offset column.
against the number of terms in the relevant series. M corresponds to the number of Fourier harmonics, and I corresponds to the number of image cylinders. Results are given for the frequency at which ka = 1. Convergence is seen to be rapid. Figure 4 shows the mean drift overturning moment about the hinge plotted against non-dimensional wave number ka. Theoretical results are given for the column fixed on the tank centreline, and for an identical column in open sea conditions. Large jumps are apparent in the theoretical predictions for the wave tank arrangement, in the region of the values ka = 0.29, 0.57, 0.86. These correspond to wavelengths of 2.2m, 1.1m, 0.73m and are associated with the frequencies of symmetric standing waves across the 2.2 m width of the tank. It will be recalled from the previous section that the theoretical solution is in fact singular at these precise frequencies.
For a cylinder placed asymmetrically in a tank, the theoretical solution is singular when the tank width is an integer multiple of half the wavelength. As an example of the theoretical behaviour, Fig. 5 compares results for the previously considered cylinder in the open sea with those for the same cylinder located 238 mm off the centreline of the tank. Additional jumps in the mean drift moment are indeed observed in the region of values ka = 0.43, 0.71, 1.0, corresponding to the frequencies of asymmetric standing waves across the tank width. Figures 4 and 5 also display the experimental values of mean drift overturning moment on the column, placed on the centreline of the tank and 238 mm offset respectively. The results plotted correspond to averages from different runs with different wave heights. In both figures the experimental points, with the exception of the two lowest frequencies, are all closer (or at least as close) to the tank theoretical results than to the open sea predictions. The wave tank confinement effects appear to provide at least a partial explanation of the apparent scatter of the experimental averages about the smooth predictions obtained for the cylinder in the open sea. It must also be noted, however, that there is considerable difficulty in obtaining experimental data of this type, particularly at low frequencies: both the small magnitude of the second order f o r c e s compared with first order wave frequency excitation - and the strong influence of any reflections from the ends of the tank, impose serious constraints on the experimental technique. Further insight into the confinement effects associated with the wave tank walls may be sought from the theoretical solutions. It is instructive to examine the amplitude of the wave surface elevation in the vicinity of a rLxed cylinder, first under open sea conditions and then in the wave tank. Figure 6 shows the open sea surface elevation within a rectangle 2.2 m wide and 2 m long, centred on the axis of the cylinder of radius 100 mm. The co-ordinate at each point is the amplitude of the total wave elevation at that point: thus the plot should not be interpreted as an instantaneous snap shot, but rather as an indication of the disturbance of the incident wave which has a constant ordinate of unity. The particular condition illustrated in Fig. 6 is for a wavelength of 1.101 m. Figure 7 shows at the same scale the corresponding surface when the same regular wave is diffracted by the same cylinder placed at the centreline of the 2.2 m wide wave tank. The striking difference between these two figures is associated with the presence of near standing waves across the tank in the latter case. The free surface is seriously disturbed, both in the region between cylinder and tank walls as well as upstream and downstream of the cylinder. The incident wave amplitude is increased by some 50% due to the near resonant condition (exact resonance occurring for an incident wave 1.1 m long: but the theory breaks down in this condition). Instantaneous cross sections of the scattered wave elevation, for a wavelength 1.101m, are shown in Figs. 8 and 9. These cross sections are taken parallel to the incident wave crests, at the centreline of the cylinder, for successive time steps of one eighth of the wave period. The conditions of Figs. 8 and 9 correspond to the open sea and wave tank conditions of Figs. 6 and 7 respectively. The sharp contrast between the standing wave behaviour of Fig. 9, and the radiating wave of Fig. 8, is readily apparent. The standing waves are characterised by antinodes at the tank walls and the cylinder surface, with nodes in between. With the cylinder placed at the centreline of the tank, this pheno-
Applied Ocean Research, 1985, Vol. 7, No. 2
73
Mean drift forces on an articulated column oscillating in a wave tank: R. Eatock Taylor and S. M. Hutl~
Figure 6.
Free surface amplitude around column m open sea ()~ = 1. l O1 m)
Figure7.
Free
74
surface amplitude
around column at c e n t r e l i n e o f t a n k ( B = 2 . 2 m , )~= l . l O l m )
Applied Ocean Research, 1985, Vol. 7, No. 2
Mean drift forces on an articulated column oscillating in a wave tank: R. Eatock Taylor and S. M. Hung X10-1 15.
--t=
0
---t-
]/8
....... t=
2/8
T T
.... t =
3/8
T
---t-
~/8
T
.... t-
5/8
T
10.
g ~
5
d
0
~
~_
.....
t=
8/8
T
....
t=
7/8
T
-5
-10
-15 -15
10
-5
0
5
10 xi0-I
15
Y
Figure 8. 1.101 m).
Scattered wave cross sections in open sea (X =
X10-I 15
t=0 t-
1/8
T
t=
2/8
T
10 .........
g ~
/.
5
~
o
-:-g,-:-:-'-.,.t-,
\
'"
. '
--
t=
3/8
T
----
L=
4/8
T
-
--t=
5/8 T
r
-1~ -15
-10
-5
0
5
10 x10 I
15
T
Figure 9. Scattered wave cross sections in tank (k = 1.1 O1 m, column central)
Figure 10.
menon will occur whenever the incident wavelength is close to an integer submultiple of the tank width (in this case a factor of 1/2). Another condition is represented by Figs. 10 and 11, showing total wave amplitude and instantaneous scattered wave cross sections at the same scales as Figs. 6 and 8 respectively, and for the same spatial arrangement of the cylinder in the tank but for a wavelength of 1.465m. Figure 11 shows that the scattered waves are reflected from the tank walls as they propagate down the length of the tank; but there is no evidence of standing waves. The behaviour illustrated by Figs. 6-11 relates to open sea and wave tank conditions giving rise to the drift moments in Fig. 4. Specifically they correspond to results at ka = 0.571 and ka = 0.429. We now consider the cylinder offset by 238 mm from the tank centreline, namely the geometry leading to the results of Fig. 5. For the wavelength 1.465 m (ka = 0.429) the resulting total wave amplitude and instantaneous scattered wave cross sections are shown in Figs. 12 and 13 respectively (again at the same scales). Once again we see the characteristic standing wave behaviour, although the only difference between Figs. 11 and 13 is that the cylinder has been offset from the tank centreline. The introduction of a small asymmetry into the geometrical arrangement of the cylinder in the tank is seen to give rise to resonant type behaviour when the half wavelength is close to an odd integer submultiple of tank width (in this case 1/3). This is in addition to the even submulfiples which alone cause standing waves for symmetric geometry. The occurrence of the submultiples is not related to the magnitude of the cylinder offset, merely to its existence.
CONCLUSIONS An analytical solution has been developed for calculating mean drift forces and moments on a circular cylindrical
Free surface amplitude around column at centreline o f tank (B = 2.2 m, X = 1.465 m).
Applied Ocean Research, 1985, Vol. 7, No. 2
75
Mean drift forces on an articulated column oscillating in a wave tank." R. l/atock Taylor and S. 34. Hung X18 1 15.
7<
t::
8
t
1~9
T
t
2,8
T
t
3/8
]
18.
.,
5i
,±j
g~
LE :> <
i
I
c]d LL'
[5~
•
t- 5-8
-
t:
-
--
- t :
T
6/8
7/8
Io
ally modified even for large separations between the t w o cylinders. In the case of the cylinder centrally placed in a wave tank, major differences from open sea behaviour can arise if the tank width is an integer multiple of incident wavelength. Equally important discrepancies also arise when the tank width is an odd multiple of half wavelength, if the cylinder is offset from the centreline of the tank. This behaviour exists irrespective of the dimensions of the cylinder relative to the tank and is not a conventional problem of 'blockage'. These findings have serious implications for the investigation of drift response by testing models in a wave tank.
-15 -15
Llg
25
g
5
lg X10-1
15
¥
X~8
1 -
Figure 11. Scattered wave cross sections in tank (X = 1.465 m, column central)
L=
/ ¢'
articulated column, confined within a parallel sided wave tank. The solution procedure has also been applied to the problem of two closely spaced circular cylinders in line with the crests of an incident wave system. In each case results have also been obtained for first order (linear) wave forces, which are found to be virtually identical to theoretical results published elsewhere} 4' 15 The emphasis of this paper is on obtaining drift force results. These illustrate a striking influence of hydrodynamic interaction between the two cylinders or the single cylinder and tank walls. Even though first order forces and moments may not be significantly affected, drift forces are substanti-
Figure 12.
76
>~:
"4 //7"
:u
// i
-" /
,x /
f
/
b
/
/ / '• ~
,
"~\
--(J ,~'.._..G'
.5
cm k±J ca/
" :,.:.,
_
'
//
,/ ,4~,
,,k " ',,\
1/8
T
t=
2/8
7
t=
3/8
7
- t~
-5/8
%=
6,'8
1
'/8
,-
/
tO
<< 15
_._ 15 ]
Figure 13. Scattered wave cross sections in tank (X = 1.465 m, offset column)
Free surface amplitude around offset column (bl = 1.338 m, b2 = 0.862 m, X = 1.465 rn)
Applied Ocean Research, 1985, Vol. 7, No. 2
-
;3
t=
Mean drift forces on an articulated column oscillating in a wave tank." R. Eatock Taylor and S. M. Hung Experimental results obtained in the 2.2 m wide tank at University College London have highlighted this phenomenon. It is vital to avoid investigations of drift force at frequencies corresponding to the occurrence of standing waves across the tank. In view of current interest in slowly varying drift responses in irregular seas, it is desirable to be able to measure quadratic transfer functions by testing in regular bichromatic waves. The avoidance of standing waves in these conditions is more demanding than in monochromatic incident waves, and adds to the importance of understanding the interaction phenomenon.
APPENDIX A Transformation o f co-ordinates by Bessel addition theorem Looking at Fig. A I , it is known that (Watson 18) exp(v/~) Cu(,o) =
~. Cv+m(Z) ~3re(z) exp(miq~) m=-**
such that/3 = J for C=Jor
Y
and/3 = I for
C=K
ACKNOWLEDGEMENTS The authors are indebted to Dr K. R. Drake for the provision of experimental data for the articulated column. The work was supported by the Science and Engineering Research Council and the Cohesive Programme on the Dynamics of Compliant Structures, through Grant GR/C/ 7137.8 to the London Centre for Marine Technology.
furthermore
p = x / Z ~ + z 2 - 2Zz cos From Fig. A2, we have __
2
2
rg - x/lp~ + rp - 2lpgrp cos(0p
-
Opg)
so that putting
lpg=Z, rp=Z,
~):Op--Opg, ~):~--Ogq-Opg
we obtain
REFERENCES 1 Abramowitz, M. and Stegun, I. A. Handbook of Mathematical Functions, Dover, 1972 2 Bateman, H. Higher TranscendentalFunctions, 1, McGraw-Hill, 1953 3 Chakrabarti, S. K. Wave forces on multiple vertical cylinders, J. Waterway, Port, Coastal and Ocean Div., ASCE May 1978, 147 4 Drake, K. R., Eatock Taylor, R. and Matsui, T. The drift of an articulated column in regular waves, Proc. The Royal Society 1984, A394, 363 5 Drake. K. R. Dvnandcs of large diameter articulated columns in waves, Thesis submitted in partial fullfillment of the requirement for the Ph.D. degree, London University, 1984 6 Havelock, T. H. Forced surface-waves on water, Philosoph. Mag. 1929, 8, 569 7 Kyllingstad, A. A low-scattering approximation for the hydrodynamic interactions of small wave-power devices, Appl. Ocean Res. 1984,6, 132 8 Massel, S. R. Interaction of water waves with cylinder barrier, J. Waterway, Port, Coastal and Ocean Div., ASCE May 1976, 165 9 Masumoto, A., Yamagami, Y. and Sakata, R. Wave forces on multiple floating bodies, Appl. Ocean Res. 1982, 4, 2 10 Mclver,P. and Evans, D. V. Approximation of wave forces on cylinder arrays, Appl. Ocean Res. 1984, 6, 101 11 Miles,J. W. Surface wave diffraction by a periodic row of submerged ducts, J. FluMMech. 1983, 128, 155 12 Newman, J. N. The drift force and moment on ships in waves, J. Ship Res. March 1967, 51 13 Pinkster, J. A. Low frequency second order wave exciting forces on floating structures, PubI. No. 650, Netherland Ship Model Basin, Wageningen, 1980 14 Spring, B. H. and Monkmeyer, P. L. Interaction of plane waves with vertical cylinders, Proc. 14th Int. Con]: on Coastal Engineering, Chapter 107, pp. 1828-1849, 1975 15 Spring, B. H. and Monkmeyer, P. L. Interaction of plane waves with a row of cylinders, Proc. 3rd Speciality Conf. on Ovil Engineering in the Oceans, ASCE, Chapter 2, pp. 979-998, 1975 16 Srokosz, M. A. Some relations for bodies in a canal, with an application to wave-power absorption, J. Fluid Mech. 1980, 99, 145 17 Standing, R. G., Dacunha, N. M. C. and Matten, R. B. Mean wave drift forces: theory and experiment, National Maritime Institute Report NMI R124, 1981 18 Watson, G. N. A Treatise on the Theory of Bessel Functions, Cambridge University Press, 1944 19 Wehausen,J. V. and Laitone, E. V. Surface Waves, Handbuch der Physik, IX, Springer-Verlag, 1960
exp [vi(rr --Og + Opg)] Cu(rg)
Cv+m(lpg) /3rn(rp) exp [mi(Op - Opg)] m=-~
Z
Figure A1.
Geometry for Bessel addition theorem
rg
rp O~
Opg
lpg
Figure A2. ~r~, o~)
Relationship between coordinates (rp, Op} and
Applied Ocean Research, 1985, VoL 7, No. 2
77
Mean drift ]brces on an articulated column oscillating in a wave tank. R. tfatock Taylor aJ~dS. M. 1tung Multiplying throughout by exp
exp(viOpg) yields
If we put z = exp(iO ), we may obtain Lipschitz's to, mula:
[ui(n - Og)] Cv(rg)
exp(ijO)
oo
=
y"
Cv+m(lpg)~m(rp)exp[imOp--i(m+v)Opg]
1 ~
m~--~
t s-1 e x p l - v t )
= r(s~- J i ~ oxpiiO) expi ~ t; d,
since
0
exp(vin) = (--1) ~, exp(-- viOg)= cos vOg--i sin vOg Hence finally
1 i
COS
Cu(rg)
sin
VOg= (--1) v
_+
~ m=--o
Cu+m(lpg)~m(rp) +
cos(m
t s-1 e x p ( - u t ) [ ( e x p ( t ) - e x p ( - i O ) ] [ ~ e x p ) = t + io)l [ e x p ( t ) ~ e x p ( ~ t ~
P(s)
~o
0
sin
1;
]
mOp + sin (m + p) Opgeos mOp ] + v) Opg cos sin
t s-1
e x p ( - v t ) [exp(t) - exp(-iO)]
p(s)
2(cosh t - cos O) 0
=
APPENDIX B
Summation of the asymptotic form of Hankel function
1 {f
ts-lexp[-(v-1)t]
2F(
cosh t -- cosO 0
From ref. 2 we use the result
~o--°° cb(z, s, u) =
78
i=
zj _ 1 f t s-lexp(-vt) (v + j) s P(s) 1 z exp(- t) dt 0
Applied Ocean Research, 1985, Vol. 7, No. 2
f l 0
-- cososh 0
dt
dt
at
London Centre for Marine Technology, University College London, UK
An analytical solution is presented for calculating the regular wave induced respones of an articulated column in a wave tank. Extension of the procedure leads to calculation of second order mean drift forces and moments. Hydrodynamic interaction between a cylindrical column and the parallel walls of the tank is shown theoretically to be highly significant and experimental data are presented to corroborate this finding. Key Words: Mean drift, articulated column, wave forces, wave tank
INTRODUCTION Design of offshore structures such as semi-submersibles, tension leg platforms and articulated columns has motivated recent interest in the prediction of wave forces on large diameter cylinders. One important aspect of his problem relates to oscillatory flows at low Keulegan-Carpenter numbers, where inertial and diffraction effects are predominant and drag forces may be neglected from theoretical predictions. For such cases the use of numerical techniques is widespread and they were first developed to solve the linear potential flow problem, associated with waves being scattered by arbitrary three dimensional bodies. These techniques have then been extended so that mean and low frequency second order wave drift forces may be predicted (e.g. Pinkster, 13 Standing, Dacunha and Mattena7). Attempts have also been made to validate the theoretical drift force results by means of wave tank experiments, in some cases with mixed success. One of the difficulties experienced by some investigators, who have attempted to measure wave drift forces on models appears to be related to the problem of testing in a narrow wave tank. Mean drift forces on large bodies are associated with wave scattering. Any reflections of scattered waves from the side walls of a tank are therefore likely to have a significant influence on the magnitude of measured drift force. An associated problem is that the drift behaviour of a structure comprising multiple cylindrical legs is likely to be rather different from that of an isolated cylinder. Multiple scattering effects, which are unimportant in the context of linear wave forces, may cause major changes to second order drift forces. This paper attempts to shed some light on this phenomenon by examining the behaviour of a cylindrical articulated column. Theoretical predictions are obtained for a column in the open sea, and in a narrow wave tank. The latter results are compared with experimental data, which appear to confirm the predicted effects related to tank wall reflections.
Accepted November 1984. Discussion closes June 1985.
number of its images reflected in the parallel sides of the tank. The problem may be formulated in terms of a velocity potential in a semi-infinite region, subject to boundary conditions on the free surface, on the seabed, and on the body and its images. The linearised formulation corresponds to the multibody problem which has been studied by several investigators for simple body shapes and for restricted ranges of conditions. Spring and Monkmeyer 14 evaluated the first order wave forces on closely spaced groups of vertical cylinders. No assumption was made regarding incident wave angle or cylinder arrangement. A series solution was obtained, the coefficients of which were evaluated from the solution of a matrix equation. The size of this equation was related to the number of cylinders, hence limiting the approach to smali groups. Results were given for two cylinders in various configurations relative to the incident wave. The same method was later used by Chakrabarti 3 to approximate the first order horizontal force on a cylinder at the centre of a wave tank, by employing three pairs of image cylinders. More recently, McIver and Evans 1° formulated a new matrix equation method for the linear multi cylinder problem. In this method, efficiency is improved by approximating the effect on one cylinder of scattered waves from other cylinders as that of plane incident waves. An improved approximation was also given involving a non-plane correction term. Another recent approximate theory, based on a low scattering (small body) assumption, has been described by KyUingstad. 7 In neither of the cited works were results given for the wave tank problem. By observing symmetry around each cylinder in an equally spaced infinite array orthogonal to the direction of the incident wave, Spring and Monkmeyer is deduced a matrix equation solution such that the dimension of the matrix becomes independent of the number of cylinders considered. The condition modelled by this method is equivalent to that of a stationary vertical column situated midway between parallel tank walls. Results were obtained for first order pressures and forces. Massels considered the above problem with arbitrary incident wave direction by using an approximated incident potential in the vicinity of the cylinders. Simple formulae for first order wave forces were obtained. The approximation, however, confines the applicability of the method
Applied Ocean Research, 1985, VoL 7, No. 2
0141-1187/85/020066-13 $2.00 © 1985 CML Publications
THEORETICAL BASIS Solution of the irrotational flow problem for a body in a wave tank is equivalent to that of the body plus a large
66
Mean drift forces on an articulated column oscillating in a wave tank." R. Eatock Taylor and S. M. Hung to cases of small separation and when the ratio of cylinder size to wavelength is small (i.e. low scattering). Masumoto et al. 9 calculated first order and second order mean wave forces on multiple rows and infinite columns of spar buoy type structures. The simplicity of their formulation is due to the use of approximated incident and transmitted wave potentials. The applicability of this method also relies on the size and spacing of the bodies being small compared with wavelength and spacing. The behaviour of wave power devices in a canal was investigated by Srokosz.16 The general solution 6 for the velocity potential between parallel walls was employed, and approximations were made regarding the local disturbance due to the device. More recently, Miles u utilised avariational integral equation and Fourier representation to solve the problem of a submerged or surface piercing vertical cylinder between parallel tanks walls. The results are valid provided that the tank width is small compared with the wavelength. In the following, we use an exact analytical formulation and approximation is only introduced at the final stage of obtaining numerical results, where it is necessary to truncate the infinite series. The procedure is similar to that adopted by Spring and Monkmeyer is for the fixed cylinder in the centre of the wave tank: here, however, there is no limitation regarding the position of the cylinder in the tank, and it is free to pitch in a plane parallel to the tank walls. Both first order wave forces and responses are obtained, and mean drift forces. ANALYSIS OF HYDRODYNAMIC INTERACTION EFFECTS
For an arbitrary shaped body in an ideal fluid with a free surface and a horizontal seabed at water depth d, the first order velocity potential associated with wave scattering can be expressed (cf. Wehausen and Laitone 19) in cyfindrical coordinates. If the origin of the fixed reference coordinate system (ro, 00, z) is taken at the mean free surface, with the vertical axis 0z directed positive upwards, this velocity potential may be written as qbB = Re
[t~B
exp(iwt)]
where d)n=R(r)®(O)Z(z ) is obtained as a solution to Laplace's equation
0 2 ¢ + 1 0 ¢ + 1 02¢ + - 02¢ -=0 br 2 r ~r r 2 b02 0z 2 __
(l)
in the radial coordinates are
/~)(kr0),
/ = 0
[K,.(KFo),
/ >/1
(3)
where H(m2) and K m are respectively the Hankel function of the second kind and the modified Bessel function of the first kind of order m. The functions in the vertical coordinate z are coshk(z + d)
Z/(KF)
,
cosh kd
/=0
(4)
=,
cos K/(z + d) ,
cos rid
]>~1
The wave number k is defined by 092
k tanh(kd) = - g
(5)
and the quantities Ki are the positive real roots of 092
K/tan(Kfl) = -- ---, g
] ~> 1
(6)
It is convenient to use the alternative definition k = Ko. When there are I bodies in an otherwise unbounded sea, the scattered potential can be expressed by superposition. Local coordinate systems (ri, Oi, z) are introduced, centred on the ith body (i = 1 to I), permitting the total scattered potential to be written
1 (OB= E
~
~. [Cq'mPm(K/ri) c°smOi
i=0 m = - ~ ]=0
+ C]/mPm(r/ri) sinmOi] Zj(riz)
(7)
Through use of the Bessel addition theorem (stated in Appendix A), the terms expressed in the ith body coordinates (ri, Oi, z) may be transformed to expressions in the reference coordinates (ro, 0o, z). Thus the scattered potential of equation (7) is written:
CB(ro, 0o, z) /
subject to the boundary conditions
~-
E Ej = O
=
= -c°2 -¢ ~O Oz z=o g
l
em(K/ro) =
i=o
z:o , -O0 z=-a = 0 0z
~
~o
X
X
m=--*o
+C~m
~
Gi]m n (Kiro, nOo)
n=--oo
s Giimn(rlro, n0o)] Zi(r]z )
(8)
n=--ao
Also at infinity
with exp ( - ikr) 0 c ~ - -
c
cosm00
Gijmn s = 6mnPm(K/r°) sinm00'
i= 0
(9)
as r -+ oo. Equation (1) then gives (see Drake s) where 6rn n is the Kronecker delta, and
cPB(ro,Oo,z)= ~. ]=0
~
[CfinPm(K/ro)cosmOo
c Gijmn
m =-~
+ C;mPm(Kjro) sinm0o] Zj(r/z)
(2)
The coefficients Cfm , C]m are obtained from the boundary conditions: they correspond to the terms in ¢8 that are symmetric and antisymmetric about 0o = 0. The functions
+ --
s
= (--1)mpm+n(r/loi) Qn(r/ro)
cos(m + n) 0oi
+
¢os sin
(n0o)
+
sin ] sin(m + n) Ooi cos(n0o)/,j
i >t 1
(10)
Applied Ocean Research, 1985, VoL 7, No. 2
67
Mean drift forces on an articulated column oscillating in a wave tank•" R. Eatock Taylor and S. M. Hung Here loi coordinate relative to functions Q
and Ooi are the radial distance and angular of the origin of the local coordinate system the origin of the reference coordinates. The are Bessel functions of the first kind: jn(kro)
] = 0
Qn(K/r°) = [ In(K/ro)'
]> 1
C~.m = Cojm c $
- - C ( 2 i - - 1)]m : C z i j m
: C~/m
(13)
,I
© ©
=
4
© © 1
0
2
y
rn=-,~
i=0
2
n ....
(-1) Gqmn(K/ro, nO©) Zl(t~/z } i=0
(14)
n =-~
By taking i from 0 to infinity we provide an exact representation of the effect of the tank walls. By taking i = 0 and 1 we obtain the case of two cylinders in the open sea. We note that the number of unknown coefficients is in fact independent of the number of image cylinders employed. In cases where bl = b2 in Fig. 1, it may be shown that there is no coupling between symmetric and anti-symmetric components of velocity potential. Since the boundary condition associated with incident waves propagating in the direction normal to the line of cylinders has only symmetric terms, the antisymmetric components of the scattered potential are identically zero. Under this special condition, the series form in equation (14) may be shown to reduce, for a fixed cylinder, to the series solution obtained by Spring and Monkmeyer) s Equation (14) also applies to any body geometry that is symmetric about the 0o = 0 plane, and may be employed for both diffraction and radiation problems. Attention is restricted here to surface piercing circular cylinders of radius a. The diffraction potential 4~D satisfies 0¢D
---
Or
0~ I
Or
on r = a
(15)
where the incident velocity potential 4~z is given in the reference coordinates by
Oi =
~ n=--~
igAzo(kz)(--i)nJn(kro)cos(nOo)
(16)
(.,O
for a regular incident wave of amplitude A. The radiation potential q)R generated by column oscillation in the direction of wave propagation satisfies
0@ Or
- i ~ f ( z ) cos00
on r = a
(17)
where f(z) is the amplitude of horizontal motion of the cylinder cross section at elevation z (the cross section being assumed to be rigid). Making use of the orthogonality of the eigenfunctions Zo(kz), Zi(~/z) when integrated over the depth of the water column, and enforcing the body boundary conditions, we obtain for each value o f ] two sets of linear algebraic equations for the unknown coefficients associated with the diffraction and radiation problems respectively. From the symmetric components (terms in cosn0 in equation (14)) we obtain:
© ©
~" 5
j=O
(12)
In what follows it is convenient to omit the subscript 0. Thus making use of equations (12) and (13), we may write
© ©
4~B(ro,0o, z)
(11)
The nth term of the infinite series for Gi]mn signifies the contribution to the velocity potential of the nth harmonic (in the reference coordinate system) by the ruth harmonic solution of body i. The subscript] identifies dependence on the ]th vertical eigenfunction. The analysis leading to equation (10) is completely general as regards the shape of the individual bodies or their spatial arrangement. In this investigation, we are primarily interested in the flow past equal diameter cylinders placed in a line perpendicular to the direction of the incident waves. We shall consider specifically the cases of (a) two cylinders, and of (b) a large number tending to infinity. Case (a) is examined to provide a cross check with other published data, while case (b) is related to a cylinder placed in a wave tank. It is readily seen that the effect of the parallel tank walls may be simulated by an arrangement in the open sea consisting of such an array of cylinders. Each cylinder in this imaginary array is an image of the single real cylinder, reflected in the two parallel tank walls. The image cylinders extend to infinity on either side of the real cylinder, although for the purposes of computation it is feasible to employ a finite number of cylinders in the array. A typical arrangement is shown in Fig. 1. We take the origin of the reference coordinates (ro, 0o, z) at the centre of the cylinder numbered 0. The local axes (ri, Oi, z) for each other cylinder have the origin at the centre of cylinder i, and 0i is measured in the same direction for each cylinder• We exploit the symmetry of the odd and even (symmetric and antisymmetric) functions of 0o in equations (9) and (10). The scattered velocity potential for cylinder i within each set of zero normal flow boundaries (shown as dotted lines) is equal to the corresponding potential for cylinders i + 2, i + 4, i - 2, i - 4 etc. The scattered potentials for cylinder i + 1, i -+ 3 etc are the mirror images of the potential for cylinder i, as reflected in the boundary equidistant between the corresponding cylinders. Under these circumstances, the following symmetries are found: S
equation (8) as
3
2
i
m=-~
X [C~m cos(m + n) Ooi + (--1)iCfin sin(m + n) Ooi] i=1
× (--1)mpm +n(n/loi) Qn(gia) + CTmP'n(Kia) = Bin, Figure 1. Geometry of reference cylinder and its images in walls of wave tank
68
Applied Ocean Research, 1985, Vol. 7, No. 2
--~
(17)
Mean drift forces on an articulated column oscillating in a wave tank: R. Eatock Taylor and S. M. Hung
(-i)n+lJ'n(ka),
/=0
B/n =
(18)
~0,
Fx Fy
(¢z + ~bo)\sin 0 /
f>~l 21r
For the radiation problem the right hand side is
( My ] = icop Xmx!
0
f :(z) Z/%z) dz --el B/n =
, n=+l 0
;! z}(K/z) dz (19)
0,
n=0, lnl>l
From the antisymmetric components (terms in sinn0o in equation (14)) we obtain, for both radiation and diffraction problems:
~_
~ [CTmsin(m+n)Ooi-(-1)iC~m cos(m+n)Ooil
m=--** i=1 I
$
t
(-- 1)mpm +n(t¢]loi) Qn(gja) + C}mPn(gja) = 0 (20) If we truncate the various series, we may write these equations in matrix form. Thus we assume that we may retain only J terms in the series over ], and M terms in the series over the Fourier harmonics m and n. The number of image cylinders is taken to be I. Equations (17)-(20) are therefore expressed in the form [Arc A]c
ArS l [ C]C] : A]s [ C / sj [B/]
(21)
where 1
CC
t
A/nra = 8mnP'n(K/a) + • (--1)mPm+n(r]loi) i=1
x Q'n(r/a) cos(m + n) Ooi 1
ff o
t
ic°s°o
(q~z + q~D)(z + d) k-- sin 0 ]
dO dz (23)
It is clear that only the first Fourier harmonics (n = -+ 1) in the potentials ¢I and q~o contribute to the forces and moments. The added inertia and damping terms are obtained in a similar manner from the radiation potential q~R. These terms also involve only the first Fourier harmonics (n =-+ 1), although it should be noted that solution is required for all of the coefficients C/m: this in turn re~luires formulation of equations (21) and (22) over the full range of values of n. Finally the wave induced response of the articulated column (i.e. the tilt angle in the direction of propagation of waves down the tank) may be evaluated by direct solution of the equation of motion. MEAN DRIFT FORCES A method has been given by Pinkster 13 for evaluating mean drift forces and moments on floating bodies. This involves direct integration of the hydrodynamic pressure on the instantaneous submerged surface of the body, in conjunction with a perturbation expansion of the governing variables in terms of wave steepness. Retention of terms up to second order in the expansions, and evaluation of the time averages, leads to the mean drift forces and moments. The approach has been extended by Drake, Eatock Taylor and Matsui4 to include the second order contribution due to motion of an articulated column at the seabed. A well known alternative method for obtaining horizontal mean drift forces, and moment about a vertical axis, uses the far field momentum approach.12 Although simpler to formulate this is not suitable for the present work. We thereforefore adopt the former approach to write the mean drift forces and overturning moments on a pair of cylinders and on an articulated cylinder in a wave tank. The expressions involve the total first order potential ~, where
.
~b= ~z + q~B
i=1
(24)
and equations (16) and (14) are used to obtain ~I and ~B respectively. The latter includes contributions from both diffraction and radiation potentials. For the general case where a column is free to undergo small rotations about two orthogonal horizontal axis at the base of the tank, we may define a complex rotation vector ~ = (f2x, £2y, 0), corresponding to harmonic response at frequency co. We thus obtain
1
Ajn m = ~" (-1)m pm+n(giloi) Q'n(g/a) sin(m + n) Ooi i=1 I
A/n m = 8mnPn(gja)-- ~ (--1)i+mpra+n(rjloi) i=1
X Qn(gja) cos(m + n)Ooi
0
-a
CS A/n m = ~ (--1)i+mp m +n(t~jloi)Qn(gja) sln(m + n) Ooi
$c
dO dz
0 -el
(22)
The matrices A/, B/, and C/are of order M x M, M x 1 and M x 1 respectively, and the subscript ] runs from 0 to J. The subscripts m and n also take both positive and negative values, as seen from equation (17) and (20). For each value o f / , equation (21) may be solved for the diffraction and radiation problems, giving in each case the coefficients C/C m and Cfm. These may then be substituted into equation (14) to give the potentials q~o and q5R. Hence for a cylinder hinged at z = - d the first order complex hydrodynamic forces and moments may be obtained from
if(2)/T(" x2, = ~ Re [ ( - -Pg 2
cBJ +off so
• CW
It/+
Xxf2,--X,~2x,2(nXldl \ nyl \ ny
{½1V~12+iw(Rx~2)'.V¢*}(nx) ds ] ny
(25)
Applied Ocean Research, 1985, Vol. 7, No. 2 69
Mean drift forces on an articulated column oscillating in a wave tank." R. t:'atock Tavh)r and S. M. Hzmg NUMERICAL
jl,~x(2), = 2 Re [ - - ~ -
flrT+Xxg2y--Xy~2xIRzl \ - - n y /} dl cw
--ioop fo*(Xx~2y--Xyg2x)R z ( n ~ , ) d , cB ~r +;ll
{½lv$12+i¢°(Rx ~)'V¢*} so
x (Rznx--Rxnz) ds] \Rzny Rynz/
(26)
where r/is the first order free surface elevation, given by -
at
(27)
Oz
X = (Xx, Xy, Xz) is the position vector (relative to the origin in the mean free surface) of a point on the submerged surface S of the body; R = (Rx, Ry, Rz) is the vector from the point about which the moment is evaluated to the point of integration; n x, ny and n z are direction cosines; C w and Cs are the intersections between So and horizontal planes a't the mean free surface and the seabed respectively (it is assumed in the above that n z = 0 around C w and 6'3). In the case of the circular cylindrical column of radius a, pitching dnly in a plane parallel to the direction of wave propagation, we obtain (in scalar form) 2rr
2[Re-7Pg of [l~+mkcosOl2]acosOdO 277
-- i¢op j
[q~*ff]z = _ a a 2 cos20 dO
0 2rr
0
0
--d
+i°3(z--zA)~O¢*sinO} ] a O c°sOadOdz O
(28)
2n
m~
= 2 Re
-- 7
.
[Ir/+aqJ cos0 121==o aza cos 0 dO
o 2Tf
--icop f
[O*~]z=_da 2 cos2,0(d+zA)dO
0 2rr
0
0
In order to implement the preceding analysis it is necessary to truncate the various series at a finite number of" terms. We consider here some of the implications of" this truncation. From equation (10), it can be seen that if the Fourier series in 0 is included up to the nth harmonic, then both the Hankel function of the second kind 11(2) and the modified Bessel function K need to be calculated up to order 2n. These have as argument the term g/loi. We recall that the •i are the generalised wave numbers ('the roots of equation (6)), and loi is the distance from the reference cylinder to the ith image. From tables 9.4 and 9 . l l of Abramowitz and-Stegun, 1 it is clear that the high order ( > 5 0 ) Bessel functions ./ and K with small arguments can easily have values outside the normal numeric range of computers. Thus Jso(1) = 2.9E--80 and Ks0(1 ) = 3.4E77. This computational limitation has been found to preclude the use of more than 25 harmonics in 0 under general conditions. Fortunately, numerical experiments described below confirm that in most cases, only a small number (~<6) of harmonics is needed for first and second order forces to converge. Exceptions to this rule have only been found for cases where the cylinders are almost touching each other. Under these latter circumstances, the highly confined flow near a cylinder may require higher harmonics for convergence. Similarly, a larger number of harmonics is required for accurate calculation of the free surface elevation at large distances from the cylinder. This is associated with using the Bessel function Jn(kr) to describe a plane progressive wave, for which clearly the radial coordinate is not well suited. Let us now consider the series in ]', for the radiation problem (since / = 0 is the only term required for the diffraction problem). Due to the orthogonality of the eigenfunctions Z/ in equation (2), the coefficients of this series (for each value of m) are uncoupled. This permits the contribution of each successive term in the series to be efficiently monitored (e.g. in evaluation of the hydrodynamic coefficients). In this way it has been found that convergence of results is generally rapid. The remaining series that needs to be truncated relates to the number of image cylinders employed on each side of the reference cylinder: this is the series over the index i in equation (7) for the wave tank problem. If we consider the transformed expressions obtained after use of the Bessel addition theorem, equation (8)-(10), we see that the function Pm+n(Kjloi) is a measure of the influence of cylinder i on the reference cylinder. Except for the term corresponding to ] = 0, Prn+n is a modified Bessel function K. The rapid decay of this function with large argument implies that contributions from cylinders which are far away may safely be neglected. The term ] = 0 corresponds to a progressive wave component, and Pm+n is a Hankel function of the second kind. Ensuring convergence of these terms, for increasing i, is more troublesome. To obtain converged results efficiently, it is necessary to use the asymptotic form of the Hankel function, namely
--d
•O~* sinO }cosOa(z--,A)dO dz](29) + ico(z+za)~a30 where ~ is the complex pitch amplitude and (0, 0, ZA) denotes the centre of rotation on the seabed pivot.
70
CONSIDERATIONS
Applied Ocean Research, 1985, Vol. 7, No. 2
H(Z)(r)
•v rrr
exp i •
2
+-2
r
(30)
The convergence is then inversely proportional to the square root of the distance to the image cylinder. This
Mean drift forces on an articulated column oscillating in a wave tank: R. Eatock Taylor and S. M. Hung asymptotic approximation may therefore be used in the evaluation of the terms Ajmn in (22). We have, for example (replacing i by j as the summation index):
assuming that the series converges. This is transformed using the formula in Appendix B, with
B*
8 =-kBz
v=-+z; i Hg{,(kloi) j=1
+ ‘2’
(36)
B2
H,$,(kl,,j)
j=l
(31) The infinite sum is then evaluated by transforming it to integral, using the formula of Lipschitz given in Appendix Reference to Fig. 1 shows that there are three cases be considered, so that all image cylinders are included the series. These are given by
an B. to in
loj = B1 + jB*
(32)
Bz = 2(b* + b,)
(33)
where
and
The resulting integral may be transformed into a finite range and evaluated numerically. An exception to the general case of a cylinder placed asymmetrically in the tank arises when kBz is equal to an integer multiple of 2n. The infinite series then becomes
j;o+P2 (v
which does not converge. This arises when the tank width (b, + b2) is an integer multiple of the half wavelength h, and corresponds physically to the excitation of standing waves across the width of the tank. In reality the amplitude of these waves must be limited by non-linear and viscous effects. As far as our irrotational flow theory is concerned, the solution is singular at these values of wavelength. For the case of a cylinder placed on the centreline of the tank (b, = b2), the general solution given above is slightly modified. We take
(34) loi = jB for the three cases respectively. In each case we may write
=$$
exp[ -ikB2(;+j)]
(35) -+r+i B2
Table 1. First order wave forces on one cylinder of cz two cylinder pair (ka = 0.4), compared with reference 14
lla
2.500 3.250 4.000 5.000 6.250 7.500
7.0122 (6.7246) 5.1843 (5.1817) 4.6415 (4.6394) 4.4069 (4.4049) 4.2456 (4.244 1) 4.1497 (4.1477) 4.1141 (4.1121)
0.5542 (0.5542) 0.4753 (0.4753) 0.4181 (0.4181) 0.3794 (0.3794) 0.3371 (0.3371) 0.2887 (0.2887) 0.2459 (0.2459)
where B = 2bI is the tank width. The corresponding series is then singular when kB is an integer multiple of 2n; i.e. the tank width is a multiple of the wavelength. Physically the difference from the preceding case is that antisymmetric standing wave modes cannot be excited when the cylinder is placed symmetrically in the tank.
($+j)-“’
=~exp[-ik~2(~+~~]~~~~(-ik~2~!,*
2.000
(37)
156.8 (157.9) 163.6 (163.7) 166.0 (166.1) 167.4 (167.5) 168.9 (168.9) 170.4 (170.4) 171.6 (171.7)
78.1 (78.2) 67.2 (67.3) 50.5 (50.6) 32.6 (32.6) (77::) -23.7 (- 23.8) -53.8 (-53.8)
1.6550 (1.5880) 1.2237 (1.2236) 1.0955 (1.0955) 1.0402 (1.0402) 1.0021 (1.0021) 0.9794 (0.9794) 0.9710 (0.9711)
DISCUSSION OF RESULTS
The theory presented above has been employed to investigate the drift forces on an articulated column on the centreline, and offset from the centre, in a wave tank. First, however, we examine two closely spaced cylinders in the open sea. Our results have been obtained by using the same theoretical formulation as for the wave tank problem, and enable some comparisons to be made with predictions obtained in an alternative way?4 In Table 1 values are given for first order forces on one of the two cylinder pair, corresponding to equation (23). The tabulated results are parameters R and 6, defined according to ref. 14 by
It may be recalled that F, is the first order force in line with the direction of wave propagation, and F,, is the force along the line of the c linders. Values are given for R,, R,, 8) between F, and the corresponding S,, a,,, and the ratio ~1,
Applied Ocean Research, 1985, Vol. 7, No. 2
71
Mean drift jbrces on an articulated column oscillating in a wave tank: R. Eatock Taylor and S. M. llung Table 2. Mean drift Jbrees on one cylinder of a two cylinder pair (ka = 0.4)
2.000 2.500 3.250 4.000 5.000 6.250 7.500 10.000 100.000 1000.000
0.78440 0.50644 0.42732 0.38232 0.33196 0.27538 0.23012 0.18118 0.21407 0.20875
9.41225 0.62951 0.22961 0.12042 0.05450 0.01179 -0.01038 -0.02319 -0.00052 -0.00145
3.72875 2.40704 2.03132 1.81741 1.57803 1.30905 1.09389 0.86127 1.01762 0.99232
force on a single cylinder in the open sea. Thus /~(1) is a direct measure of the interaction effect. Results in Table 1 are limited to the single frequency considered in ref. 14, corresponding to ka = 0.4, over a range of spacings l between the two cylinders. The values in parentheses are from Spring and Monkmeyer 14 (corrected by multiplying the values in Table 1 of the cited reference by the factor 2cosh 1.6). Our results are obtained from equation (14), using Fourier harmonics up to +100 in equation (7). Except when the cylinders are touching (l/a = 2.0), the results from the two methods are seen to be virtually identical. Differences may be assumed to arise from truncation of the series at different numbers of Fourier harmonics - i t is not clear from ref. 14 how many terms were employed. In Table 2 we give the total mean drift forces fix(z) and /7(2) on one of the two cylinder pair, and the ratio #(2) between/7(2) and the corresponding drift force on a single cylinder in the open sea. Again these results are for/ca = 0.4. It is found that the influence of hydrodynamic interactions on the drift forces persists over much wider separations than is the case for first order wave forces. Indeed substantial magnification of drift forces can arise from interactions between widely separated cylinders. Additional results, to be presented in the future, suggest that for separations greater than about five radii the magnification/a(2) tends to a value close to two, as ka tends to zero: while under the same conditions/a(J ) tends to one. For smaller separations, the drift force magnification is substantially greater than 2 at low frequencies. This behaviour associated with hydrodynamic interaction effects has important implications for the mean drift of structures such as tension leg platforms involving several closely spaced columns. We turn now to results for the cylinder in a wave tank. Our calculations of first order force on a fixed cylinder at the centre line of a tank agree to within one part in the fifth significant digit with those given in Table 2 of ref. 15. The results cover the non-dimensional frequency range 0.1 < k a < 1.0 and tank width B in the range given by 0.628 < kB < 30.8: they fall within the extremes B/a = 2.5 and B / a = 3 0 0 . We employed 100 image cylinders and Fourier harmonics up to + 100 for these calculations. The same procedure was then adopted in obtaining the drift pivot moment on a model articulated column in a wave tank. The geometry is that used for the experiments by Drake, Eatock Taylor and Matsui. 4 The cylindrical column having a radius of 100 mm was tested in a tank of width 2.2 m and in water of depth 1 m. The column was initially free to articulate in a plane parallel to the length of the tank, about a simple hinge surrounded by a flexible
72
Applied Ocean Research, 1985, Vol. 7, No. 2
collar. The centre of rotation was 825 mm tinder tile mean free surface, and below this point was attached a rigid section of cylinder as shown in Fig. 2. As described in ref. 4, a vibration absorber was then incorporated into the test arrangement so that the column was prevented from responding to the dynamic force in regular waves. It could, however, adopt a steady tilt, from which the mean drift force could be obtained. Further details have been given by Drake. s Let us first consider the theoretical predictions for this column. Figure 3 provides an indication of the convergence of the numerical results for second order drift moment on the column when placed on the centreline of the tank. The mean overturning moment about the hinge has been non-dimensionalised by the quantity ogA2ad, and plotted
20
~i
i
I o
I
__t
/
/
/
/
/
/
I / I/
I
Figure 2.
o6't
/
m
I
1
I
I
/1111
1
I
Model o f articulated column
---~---N
varying)
I=15
-
vorylng,
N=15
~
- I
° 62
o
.60.
% C71 CD
If-/
^\ IS--
~ 58
ogB
J
°54
M, I Figure3. Convergence o f mean drift m o m e n t about hinge f o r f i x e d column at eentreline o f wave tank, ka = 1 (M = number o f Fourier harmonics, I = number o f image cylinders)
Mean drift forces on an articulated column oscillating in a wave tank: R. Eatock Taylor and S. M Hung .8_ +
,6_
%: UT~
/
,4_
+
\ ..]Y',if, /
.2_
+
+1
/
X10-1 k~
Figure 4. Mean drift overturning m o m e n t about hinge for fixed column: - theory for column on tank centreline; - - - theory for column in open sea; + experimental averages for column on tank centreline
1.0.
,8.
-t5
~<
.6_
+
ffL, IZ2
.4.
.2_
/
"/
/
XIO-I
ka
Figure 5. Mean drift overturning m o m e n t about hinge for fixed column: - - theory for column offset from tank centreline; - - - theory .for column in open sea; + experimental averages for offset column.
against the number of terms in the relevant series. M corresponds to the number of Fourier harmonics, and I corresponds to the number of image cylinders. Results are given for the frequency at which ka = 1. Convergence is seen to be rapid. Figure 4 shows the mean drift overturning moment about the hinge plotted against non-dimensional wave number ka. Theoretical results are given for the column fixed on the tank centreline, and for an identical column in open sea conditions. Large jumps are apparent in the theoretical predictions for the wave tank arrangement, in the region of the values ka = 0.29, 0.57, 0.86. These correspond to wavelengths of 2.2m, 1.1m, 0.73m and are associated with the frequencies of symmetric standing waves across the 2.2 m width of the tank. It will be recalled from the previous section that the theoretical solution is in fact singular at these precise frequencies.
For a cylinder placed asymmetrically in a tank, the theoretical solution is singular when the tank width is an integer multiple of half the wavelength. As an example of the theoretical behaviour, Fig. 5 compares results for the previously considered cylinder in the open sea with those for the same cylinder located 238 mm off the centreline of the tank. Additional jumps in the mean drift moment are indeed observed in the region of values ka = 0.43, 0.71, 1.0, corresponding to the frequencies of asymmetric standing waves across the tank width. Figures 4 and 5 also display the experimental values of mean drift overturning moment on the column, placed on the centreline of the tank and 238 mm offset respectively. The results plotted correspond to averages from different runs with different wave heights. In both figures the experimental points, with the exception of the two lowest frequencies, are all closer (or at least as close) to the tank theoretical results than to the open sea predictions. The wave tank confinement effects appear to provide at least a partial explanation of the apparent scatter of the experimental averages about the smooth predictions obtained for the cylinder in the open sea. It must also be noted, however, that there is considerable difficulty in obtaining experimental data of this type, particularly at low frequencies: both the small magnitude of the second order f o r c e s compared with first order wave frequency excitation - and the strong influence of any reflections from the ends of the tank, impose serious constraints on the experimental technique. Further insight into the confinement effects associated with the wave tank walls may be sought from the theoretical solutions. It is instructive to examine the amplitude of the wave surface elevation in the vicinity of a rLxed cylinder, first under open sea conditions and then in the wave tank. Figure 6 shows the open sea surface elevation within a rectangle 2.2 m wide and 2 m long, centred on the axis of the cylinder of radius 100 mm. The co-ordinate at each point is the amplitude of the total wave elevation at that point: thus the plot should not be interpreted as an instantaneous snap shot, but rather as an indication of the disturbance of the incident wave which has a constant ordinate of unity. The particular condition illustrated in Fig. 6 is for a wavelength of 1.101 m. Figure 7 shows at the same scale the corresponding surface when the same regular wave is diffracted by the same cylinder placed at the centreline of the 2.2 m wide wave tank. The striking difference between these two figures is associated with the presence of near standing waves across the tank in the latter case. The free surface is seriously disturbed, both in the region between cylinder and tank walls as well as upstream and downstream of the cylinder. The incident wave amplitude is increased by some 50% due to the near resonant condition (exact resonance occurring for an incident wave 1.1 m long: but the theory breaks down in this condition). Instantaneous cross sections of the scattered wave elevation, for a wavelength 1.101m, are shown in Figs. 8 and 9. These cross sections are taken parallel to the incident wave crests, at the centreline of the cylinder, for successive time steps of one eighth of the wave period. The conditions of Figs. 8 and 9 correspond to the open sea and wave tank conditions of Figs. 6 and 7 respectively. The sharp contrast between the standing wave behaviour of Fig. 9, and the radiating wave of Fig. 8, is readily apparent. The standing waves are characterised by antinodes at the tank walls and the cylinder surface, with nodes in between. With the cylinder placed at the centreline of the tank, this pheno-
Applied Ocean Research, 1985, Vol. 7, No. 2
73
Mean drift forces on an articulated column oscillating in a wave tank: R. Eatock Taylor and S. M. Hutl~
Figure 6.
Free surface amplitude around column m open sea ()~ = 1. l O1 m)
Figure7.
Free
74
surface amplitude
around column at c e n t r e l i n e o f t a n k ( B = 2 . 2 m , )~= l . l O l m )
Applied Ocean Research, 1985, Vol. 7, No. 2
Mean drift forces on an articulated column oscillating in a wave tank: R. Eatock Taylor and S. M. Hung X10-1 15.
--t=
0
---t-
]/8
....... t=
2/8
T T
.... t =
3/8
T
---t-
~/8
T
.... t-
5/8
T
10.
g ~
5
d
0
~
~_
.....
t=
8/8
T
....
t=
7/8
T
-5
-10
-15 -15
10
-5
0
5
10 xi0-I
15
Y
Figure 8. 1.101 m).
Scattered wave cross sections in open sea (X =
X10-I 15
t=0 t-
1/8
T
t=
2/8
T
10 .........
g ~
/.
5
~
o
-:-g,-:-:-'-.,.t-,
\
'"
. '
--
t=
3/8
T
----
L=
4/8
T
-
--t=
5/8 T
r
-1~ -15
-10
-5
0
5
10 x10 I
15
T
Figure 9. Scattered wave cross sections in tank (k = 1.1 O1 m, column central)
Figure 10.
menon will occur whenever the incident wavelength is close to an integer submultiple of the tank width (in this case a factor of 1/2). Another condition is represented by Figs. 10 and 11, showing total wave amplitude and instantaneous scattered wave cross sections at the same scales as Figs. 6 and 8 respectively, and for the same spatial arrangement of the cylinder in the tank but for a wavelength of 1.465m. Figure 11 shows that the scattered waves are reflected from the tank walls as they propagate down the length of the tank; but there is no evidence of standing waves. The behaviour illustrated by Figs. 6-11 relates to open sea and wave tank conditions giving rise to the drift moments in Fig. 4. Specifically they correspond to results at ka = 0.571 and ka = 0.429. We now consider the cylinder offset by 238 mm from the tank centreline, namely the geometry leading to the results of Fig. 5. For the wavelength 1.465 m (ka = 0.429) the resulting total wave amplitude and instantaneous scattered wave cross sections are shown in Figs. 12 and 13 respectively (again at the same scales). Once again we see the characteristic standing wave behaviour, although the only difference between Figs. 11 and 13 is that the cylinder has been offset from the tank centreline. The introduction of a small asymmetry into the geometrical arrangement of the cylinder in the tank is seen to give rise to resonant type behaviour when the half wavelength is close to an odd integer submultiple of tank width (in this case 1/3). This is in addition to the even submulfiples which alone cause standing waves for symmetric geometry. The occurrence of the submultiples is not related to the magnitude of the cylinder offset, merely to its existence.
CONCLUSIONS An analytical solution has been developed for calculating mean drift forces and moments on a circular cylindrical
Free surface amplitude around column at centreline o f tank (B = 2.2 m, X = 1.465 m).
Applied Ocean Research, 1985, Vol. 7, No. 2
75
Mean drift forces on an articulated column oscillating in a wave tank." R. l/atock Taylor and S. 34. Hung X18 1 15.
7<
t::
8
t
1~9
T
t
2,8
T
t
3/8
]
18.
.,
5i
,±j
g~
LE :> <
i
I
c]d LL'
[5~
•
t- 5-8
-
t:
-
--
- t :
T
6/8
7/8
Io
ally modified even for large separations between the t w o cylinders. In the case of the cylinder centrally placed in a wave tank, major differences from open sea behaviour can arise if the tank width is an integer multiple of incident wavelength. Equally important discrepancies also arise when the tank width is an odd multiple of half wavelength, if the cylinder is offset from the centreline of the tank. This behaviour exists irrespective of the dimensions of the cylinder relative to the tank and is not a conventional problem of 'blockage'. These findings have serious implications for the investigation of drift response by testing models in a wave tank.
-15 -15
Llg
25
g
5
lg X10-1
15
¥
X~8
1 -
Figure 11. Scattered wave cross sections in tank (X = 1.465 m, column central)
L=
/ ¢'
articulated column, confined within a parallel sided wave tank. The solution procedure has also been applied to the problem of two closely spaced circular cylinders in line with the crests of an incident wave system. In each case results have also been obtained for first order (linear) wave forces, which are found to be virtually identical to theoretical results published elsewhere} 4' 15 The emphasis of this paper is on obtaining drift force results. These illustrate a striking influence of hydrodynamic interaction between the two cylinders or the single cylinder and tank walls. Even though first order forces and moments may not be significantly affected, drift forces are substanti-
Figure 12.
76
>~:
"4 //7"
:u
// i
-" /
,x /
f
/
b
/
/ / '• ~
,
"~\
--(J ,~'.._..G'
.5
cm k±J ca/
" :,.:.,
_
'
//
,/ ,4~,
,,k " ',,\
1/8
T
t=
2/8
7
t=
3/8
7
- t~
-5/8
%=
6,'8
1
'/8
,-
/
tO
<< 15
_._ 15 ]
Figure 13. Scattered wave cross sections in tank (X = 1.465 m, offset column)
Free surface amplitude around offset column (bl = 1.338 m, b2 = 0.862 m, X = 1.465 rn)
Applied Ocean Research, 1985, Vol. 7, No. 2
-
;3
t=
Mean drift forces on an articulated column oscillating in a wave tank." R. Eatock Taylor and S. M. Hung Experimental results obtained in the 2.2 m wide tank at University College London have highlighted this phenomenon. It is vital to avoid investigations of drift force at frequencies corresponding to the occurrence of standing waves across the tank. In view of current interest in slowly varying drift responses in irregular seas, it is desirable to be able to measure quadratic transfer functions by testing in regular bichromatic waves. The avoidance of standing waves in these conditions is more demanding than in monochromatic incident waves, and adds to the importance of understanding the interaction phenomenon.
APPENDIX A Transformation o f co-ordinates by Bessel addition theorem Looking at Fig. A I , it is known that (Watson 18) exp(v/~) Cu(,o) =
~. Cv+m(Z) ~3re(z) exp(miq~) m=-**
such that/3 = J for C=Jor
Y
and/3 = I for
C=K
ACKNOWLEDGEMENTS The authors are indebted to Dr K. R. Drake for the provision of experimental data for the articulated column. The work was supported by the Science and Engineering Research Council and the Cohesive Programme on the Dynamics of Compliant Structures, through Grant GR/C/ 7137.8 to the London Centre for Marine Technology.
furthermore
p = x / Z ~ + z 2 - 2Zz cos From Fig. A2, we have __
2
2
rg - x/lp~ + rp - 2lpgrp cos(0p
-
Opg)
so that putting
lpg=Z, rp=Z,
~):Op--Opg, ~):~--Ogq-Opg
we obtain
REFERENCES 1 Abramowitz, M. and Stegun, I. A. Handbook of Mathematical Functions, Dover, 1972 2 Bateman, H. Higher TranscendentalFunctions, 1, McGraw-Hill, 1953 3 Chakrabarti, S. K. Wave forces on multiple vertical cylinders, J. Waterway, Port, Coastal and Ocean Div., ASCE May 1978, 147 4 Drake, K. R., Eatock Taylor, R. and Matsui, T. The drift of an articulated column in regular waves, Proc. The Royal Society 1984, A394, 363 5 Drake. K. R. Dvnandcs of large diameter articulated columns in waves, Thesis submitted in partial fullfillment of the requirement for the Ph.D. degree, London University, 1984 6 Havelock, T. H. Forced surface-waves on water, Philosoph. Mag. 1929, 8, 569 7 Kyllingstad, A. A low-scattering approximation for the hydrodynamic interactions of small wave-power devices, Appl. Ocean Res. 1984,6, 132 8 Massel, S. R. Interaction of water waves with cylinder barrier, J. Waterway, Port, Coastal and Ocean Div., ASCE May 1976, 165 9 Masumoto, A., Yamagami, Y. and Sakata, R. Wave forces on multiple floating bodies, Appl. Ocean Res. 1982, 4, 2 10 Mclver,P. and Evans, D. V. Approximation of wave forces on cylinder arrays, Appl. Ocean Res. 1984, 6, 101 11 Miles,J. W. Surface wave diffraction by a periodic row of submerged ducts, J. FluMMech. 1983, 128, 155 12 Newman, J. N. The drift force and moment on ships in waves, J. Ship Res. March 1967, 51 13 Pinkster, J. A. Low frequency second order wave exciting forces on floating structures, PubI. No. 650, Netherland Ship Model Basin, Wageningen, 1980 14 Spring, B. H. and Monkmeyer, P. L. Interaction of plane waves with vertical cylinders, Proc. 14th Int. Con]: on Coastal Engineering, Chapter 107, pp. 1828-1849, 1975 15 Spring, B. H. and Monkmeyer, P. L. Interaction of plane waves with a row of cylinders, Proc. 3rd Speciality Conf. on Ovil Engineering in the Oceans, ASCE, Chapter 2, pp. 979-998, 1975 16 Srokosz, M. A. Some relations for bodies in a canal, with an application to wave-power absorption, J. Fluid Mech. 1980, 99, 145 17 Standing, R. G., Dacunha, N. M. C. and Matten, R. B. Mean wave drift forces: theory and experiment, National Maritime Institute Report NMI R124, 1981 18 Watson, G. N. A Treatise on the Theory of Bessel Functions, Cambridge University Press, 1944 19 Wehausen,J. V. and Laitone, E. V. Surface Waves, Handbuch der Physik, IX, Springer-Verlag, 1960
exp [vi(rr --Og + Opg)] Cu(rg)
Cv+m(lpg) /3rn(rp) exp [mi(Op - Opg)] m=-~
Z
Figure A1.
Geometry for Bessel addition theorem
rg
rp O~
Opg
lpg
Figure A2. ~r~, o~)
Relationship between coordinates (rp, Op} and
Applied Ocean Research, 1985, VoL 7, No. 2
77
Mean drift ]brces on an articulated column oscillating in a wave tank. R. tfatock Taylor aJ~dS. M. 1tung Multiplying throughout by exp
exp(viOpg) yields
If we put z = exp(iO ), we may obtain Lipschitz's to, mula:
[ui(n - Og)] Cv(rg)
exp(ijO)
oo
=
y"
Cv+m(lpg)~m(rp)exp[imOp--i(m+v)Opg]
1 ~
m~--~
t s-1 e x p l - v t )
= r(s~- J i ~ oxpiiO) expi ~ t; d,
since
0
exp(vin) = (--1) ~, exp(-- viOg)= cos vOg--i sin vOg Hence finally
1 i
COS
Cu(rg)
sin
VOg= (--1) v
_+
~ m=--o
Cu+m(lpg)~m(rp) +
cos(m
t s-1 e x p ( - u t ) [ ( e x p ( t ) - e x p ( - i O ) ] [ ~ e x p ) = t + io)l [ e x p ( t ) ~ e x p ( ~ t ~
P(s)
~o
0
sin
1;
]
mOp + sin (m + p) Opgeos mOp ] + v) Opg cos sin
t s-1
e x p ( - v t ) [exp(t) - exp(-iO)]
p(s)
2(cosh t - cos O) 0
=
APPENDIX B
Summation of the asymptotic form of Hankel function
1 {f
ts-lexp[-(v-1)t]
2F(
cosh t -- cosO 0
From ref. 2 we use the result
~o--°° cb(z, s, u) =
78
i=
zj _ 1 f t s-lexp(-vt) (v + j) s P(s) 1 z exp(- t) dt 0
Applied Ocean Research, 1985, Vol. 7, No. 2
f l 0
-- cososh 0
dt
dt
at