Hydrodynamic coefficients of vertical cylinders of arbitrary section

r r-M. Isaacson Professor. T. Mathai Graduate Student. Department of Civil Engineering, University of British Columbsi, Vancouver, British Columbia, Canada

I Introduction

A significant requirement in the design of an offshore structure is the estimation of anticipated hydrodynamic loads acting on the structure. Such loads on a lärge offshore struc-ture are generated whenever the strucstruc-ture is set in motiOn, for

eÂample, by ice impact or by earthquake-induced base mo-tion. lfthe dimensions of the structure are large compared to the relative distance traveled by fluid particles, appreciable flow separation should not occur and viscous effects may be neglected, so that the problem can be treated by potential flow theory. The corresponding nonlinear problem may be simplified further by a linearizing approximation due to the assumption of small motions. The resulting hydrodynamic

forces may then be expressed conveniently in terms of added

masses and damping coefficients corresponding to the fôrce components in phase with the acceleration and velocity of

the structure, respectively.

For certain structural configurations, these coefficients may be evaluated analytically by methods based on eigenfünction

expansions. Examples include a horizontal circular cylinder

(Ursell, 1949), averticalcircularcylinder(Yeung, 1981;

Isaac-son et al., 1990) and a floating hemisphere (Hulme, 1982). More generally, numerical methods applicable to structures

of arbitrary geometry are based on the finite element method or the method of integral equations (e.g., Sarpkaya and

Isaac-son. 1981). However, full three-dimensional solutions are

cumbersome and simplified treatments, which are much more

economical, may be considered for more restricted

geome-tries. These include two-dimensional formulations in the

ver-tical plane for floating horizontal cylinders; and methods applicable to vertical axisymmetric structures (BaL 1977;

Isaacson. 1982).

Contributed by the OMAE Division for publication in the JOURNAL OF

OFFSHORE MECHANICS AND ARCTIC ENGINEERING. Manuscript received by the OMAE Division. December 21. 1989: revised manuscript received August 7,

990. TECHNISCHE UN!VERSITEtT Laboratorium voor ScheepshydromechafllCa Archlof Mekelweg 2, 2628 CD Deift TeL 015- 786873- Fax: 015- 781836

Hydrodynamic Coefficients of

Vertical Cylinders of Arbitrary

Section

The calculatiOn ofadded masses and dampIng coefficients ofa large surface-piercing

vertical cylinder of arbitrary section extending to the seabed and undergoing

harmonic oscillations is described. The linearradiation problem in three dimensions is reduced lo a series of two-dimensional problems in the horizontal plane by the

use of appropriate eigenfunctions that represent the variation ofthe velocity pot enliai

in the vertical direction. Each of these is solved by a numerical approach based on the method of integral equations. Comparisons are made with an analytic solution available for the case of a circular cylinder. Results are also provided for square cylinders, and the application to typical offshore structures subject to base motions

isdiscussed.

In the present paper. the linear radiation problem of a vertical cylinder of arbitrary section extending to the seabed

is considered. The method is an extension of the approach

adopted by Isaacson (1978) to solve the analogous scattering

problem of a vertical cylinder of arbitrary section in waves. The boundary value problem for the radiated potentiàl in three dimensions is reduced to a series of two-dimensional problems in the horizontal plane by the use of appropriate eigenfunctions that represent the variation of the velocity

potential in the vertical direction. These are then solved by a

numerical approach based on the method of integral equa-tions. Results for the special case of a circular cylinder are compared with the analytic solution of Yeung (1981) and

Isaacson et al. (1990). In addition, added masses are calculated for a square cylinder and the application to a typical offshore structure subject to base motions is discussed.

2 Theoretical Formulation

2,1 Governing Equations. A vertical surface-piercing

cyl-inder of arbitrary section extends to the seabed in water of constant depthdas indicated in. Fig. I. Fixed Cartesian and cylindrical coordinate systems, (x, y, z) and (r,O, z), respec-tively, as indicated in the figure are used. The structure may

oscillate with six degrees of freedom, with the translational motions in the y and z directions (surge sway and heave respectively) and the rotational motions about the x, y and

z axes (roll, pitch and yaw, respectively) denoted by the subscripts 1, ..., 6 in the foregoing order. Each mode of motion is taken to be harmonic and expressed as

ak =

ke''

fork =

1, 2...6

(1)

where akis a displacement fork = 1, 2, 3 and a rotation for k = 4, 5, 6, i = is angular frequency, t is time and

is the complex amplitude of each component motion.

b=

d=

fk

-J'" Gm

HO' =

K,, = k,, = + + = o ôx2 ôy2

a2

Fig. i Definition sketch

It is assumed that the body motions are of small amplitude and that the fluid is incompressible and inviscid and that the

fluid motion is irrotational Consequently the flow may be descnbed by a velocity potential which satisfies the Laplace equation within the flùid region, linearized kinematic and

dynamic boundary conditions at the free surface, a radiation conditioñ in the far field and suitable kinematic conditions at

the seabed as well as on the equilibrium body surface. The

potential 4) k due to body motion in the kth mode is harmonic and proportional to the corresponding motion amplitude, and thus may be expressed in the form

= (2)

The boundary value problem forV' is then given as follows:

_Nomenclature

radius of circular cylinder side of square cylinder still water depth

hydrodynamic force or mo-ment injth mode due to body motion in kth mode

source strength distribution

function Green's fOnction gravitational acceleration

Hankel function of first kind

and zeroth order

modified Bessel function of

second kind and zeroth order

wave number of propagating

mode

within the fluid region (3)

where

km = wave number of mth

evanes-cent mode

M = no. ofterms considered in

ei-genfunction expansion

N = no. of segments

n = distance in direction of n n = unit vector normal to

equilib-rium body surface and di-rected into fluid

= component of n in x direction

n = component of n in y direction p6 = hydrodynamic pressure due to

body motiOn in kth mode r, O, z = cyliiidñcá.1 coordinates

S = horizontal length along body surface

S,, = equilibrium body surface

-

,k _{= O}

9z g

Lt

h,,(k0z) = - cosh(k,,z)

Here g is acceleration due to gravity, S,, is the equilibrium body surface and k is real positive and satisfies the linear

dispersion relation

w2d

k,,dtanh(k,,d) -

(8)

In (6), n denotes distance in the direction of the unit vector n normal to the equilibrium body surface and directed into the fluid, andn6 for the kth mode may be obtained from

_fflk'

k=l,2,3,6

n6

- lzn6'

k = 4, 5 (9) where nl' = n),; ns'

_{= fl;}

n3' = O; n4' = n5' n,,' = xnr - yn (10)

Here n, and n, are the components of n in the x and y directions respectively It is noted that the six boundary value

problems corresponding to k = 1. 2...6 differ only by the

right-hand side of the boundary condition (6).

2.2 Eigenfunction Expansion. Since the structure has vertical sides 4,& may be expressed in the form of an eigen function expansion which satisfies the seabed and free surface boundary conditions, (4) and (5)

y, z) = 4,k( v)/z,,(k,,z)

(12)

110 ¡Vol. 113, MAY 1991 Transactions of the ASME

at z = d (4) at z = (5) on S,, (6) ik,,ØL) ₌ (7) time X, y, z = Cartesian coOrdinates = _{translation or rotation of the}

body in kth mode

angle of orientation of square = _{amplitude of motion in kth}

mode

Xi6 = damping coefficient

.LJk = added mass p= fluid density

4)/C = velocity potential due to body

motion in kth mode = see equation (2) = see equation (11) = angular frequency Ç2

f)

where the potential derivatives are evaluated on S, the

hori-zontal contour of the body surface. Thus.

m = 0. 1. 2. ...

on S (20) where

fh,,,(krn:) dz

k = 1, 2 3. 6

For k = 4 and 5

The integrals in (21) may be evaluated by substituting for h,,,

from (12) and (13). Thus, for k = 1, 2. 3 and 6

The three-dimensional problem. for has now been

re-duced to a series of two-dimensional problems in the

horizon-tal plane for the potentials

,,,,", m = 0,

1, 2...

wth governing equations for each mode of motion k given by (17), (18) and (20), together with radiation conditions

similar to (7).

2,3 Green's Function Representation. Adopting the method of separation of variables in cylindrical coordinates,

(17) and (18) yield Bessel's equation and Bessel's modified equation, respectively. The general solution of the former involves the Hankel functions of the first and second kinds. Of these, only the Hankel function of the first kind satisfies the far field conditiOn. The general solution of the latter

involves the modified Bessel functions of the first and second

kinds. Of these, the modified Bessel function of the second kind vanishes for large values of the argument, whereas the

modified Bessel function of the first kind increases

exponen-tially with distance and so is discarded. Accordingly,

appro-pnate Green functions may be chosen and the potentials

may then be represented by means of a continuous

diatribu-tion of sources as follows:

frnk( flrepreseíits a source strength distribution function, x is a general point (X, y) within the fluid region, dS denotes a

differential length talçen around the body section and th integral is taken over the points ¿ on the horizontal contour

of thebody surface. G,,, is given as

m=0

Grn(X, E) (25)

K,,(krnR) m 1

in which R = I

x - E

I. It remains for the source strength

function to be chosen so as to ensure that the body surface

boundary condition is satisfied. This condition given by (20),

taken together with the representation for t'rn' given by (24), then gives rise to the following integral equation for the

unknown functionf,/:

+ ffrnk(E) -" (x, E) dS = flk'(X)brn (26)

(21) Here x is the pOint at which the boundary condition is applied

and lies on the horizontal contour of the structure and n is

measured from x.

2.4 Numerical Approximation. In the numerical solu-tion of (26), the contour S is discretized into a finite number of segments, with the functionfm& _{taken as uniform over}

each segment. Equation (26) is then applied at the center of each segment in turn and the integral equation is reduced

to a set of linear algebraic equations

B7ff,,k()

_{= flk(Xi)brnk}

_{for i = I...N}

₍₂₇₎

x1 and ¿ are the values of x and E at the centers of the ith

and

j

th segments, respectively. The coefficientsB7 are given as

B'=

'

(XE)dS

(28)

'.s än

where is the length of the

j

th segment. When i

_j,

the

integrand in (28) is täken as constant over and thus B7]

b,,,L = sinh(k,d)

m=0

m?:1

(22) k,,N,, sin(k,,,d) krnN,,, h,,,(k,,:) = cos(k,,,z) (13) N,, =

+ i,d

sinh(2kd))] (14) N,,, =

[(i

+ Ed

sin(2k.d))] (15)

The variables h,, and h,,, are eigenfunctions in the vertical direction and k,, and k,,, are eigenvalues which are real and positive. The former satisfies the linear dispersion relation

given in (8) and the latter satisfies instead

krndtan(krnd)= --

m = 1,2,...

(16) g

The governing equations for rnk_{can be developed by} substi-tuting (11) into (3). This leads to

8x2

+ -- +

_ôy- = O (17)

+ k,,,ço,,,k =O

8x2

ôy-The orthogonality properties of the functions h,,, can now be utilized to determine appropriate boundary conditions to

be satisfied on the body surface. In order to do so, (11)

is substituted into (6), both sides are multiplied by h,(k,z), ¡ 0, 1, 2, ... in turn and integrated with respect to z

over (O, d)

fd

h,,(k,,z) + hrn(kn)}hi(kiZ) dz = _{nkhl(kIz) dz (19)} k,,2 N,, [k,,d sinh(k,d) - cosh(k,,d)+ 1] b,,,A = (23) [k,,,d sin(k,,,d) +cos(k,,,d) - I]

m1

m = 1. 2. (18) Ç5k(x)

T/G,(x,

_{E) dS} (24) b,,, = fZh,,,(k,,,z) dz k =4. 5

E o 2.0

-5.0 k0a k0o

Fig. 2 Hydrodynamic coefficients for sway motion of a circular cylinder as a function of k0a for various values of a/d(a) added mass,

(b) damping coefficient. Present results: O, aid = 0.5; A, aid 1; +, aid = 2; X, aid = 5. Isaacson et al. (1990): -, aid = 0.5, 1, 2, 5.

2 3 6 3 In 4 o , 2 o o/d = 0.5 2 (b)

112/Vol. 113, MAY 1991 Transactions of the ASME

1 2 ₂ 3

k0a k0o

Rg. 3 Hydrodynamic coefficients for roll motion of a circular cylinder as a function of ka for various values of aid(a) added mass, (b) damping coefficient. Present resUlts: O, aid = 0.5; A, aid = 1. Veung (1981): , aid = 0.5, 1.

is approximated as of the jth segment on its own center, so that B7,' is taken as

B7=1

(30)

B7

Sm(xi, E)

(29)

Once the matrix B'j has been evaluated, the source strength

distribution functiOns fm' may be obtained by a standard When ¡ = j the first term of (26) accounts for the influence matnx inversion procedure The potentials themselves may

then be obtained from a discretized form of(24)

= _{A7Jj,1'(,) i= I}

_....N

₍₃₁₎

J_I in which

=

f

E) dS (32)

As a consistent approximation, .4' may be evaluatedas A') = S1G,,,(x1, ₎

for i j

₍₃₃₎

s,

1[k1..s]}

(34) The numerical solution will have to be repeated for n=0, i.

until sufficient number of terms in the series expansion

ford1' in (11) have been obtained for each mode of motiòn indicated by k.

2.5 Hydrödynamic Coefficients. Once the potentials due to body motion in the kth mode are known, the resulting

hydrodynamic pressure and subsequenti the loads on the body may be determined. The hydrodvnamic pressure at any point in the fluid is given by a linearized form of the unsteady Bernoulli equation

ph = (35)

in which p is the fluid density. Applying (2) to (35), the components of fluid force or moment acting on the body are

Ff1' =

forj

= 1.

. .., 6

(36) Here FA, F21', F31' denote the force components in thex, y, z directions, respectively, and F41', F51, F61' denote themoment

components about the x, y, z axes, respectively, these applying

to each mode of motion indicated by k. These may in turn be described in terms of added masses . and damping

coefficients Aa., which are real. In order to do

so, F

is

expressed as

F1'

= (2

+ iwA1k)ke' (37)

Equating the right-hand sides of (36) and (37) and

evalu-ating the integral with respect to z over (0, d), one obtains

1'Jk

p Re1

(38)

AJ1' = _{PQ (39)}

where Re and Im{ _{represent real and imaginary parts.} respectively, and J, is given by

15A =

_{b.,/ f}

1' n' dS (40)

lt is noted that the potentials _m= 1, 2... are real, and

hence do not contribute to the damping coefficient. In a

numerical scheme, the series expansion in (40) is truncated

to a finite number of terms and the integral in (40) is replaced by an appropr ate summation as follows:

Jfk= b,,,J

k(X)flI(X)S}

IIIII L

3 Results and Discussion

A computer program based on the method described in the

preceding section has been developed and applied to struc-tures of various fundamental configurations and the

corre-sponding results are presented here and discussed.

3.1 Circular Cylinder. Initially, results have been

ob-tained for a circular cylinder for which a closed-form solution is available. The added masses and damping coefficients of_a

circular cylinder, expressed in dimensionless form, are

pre-sented in Figs. 2-4 as functions of k0a for various values of (41)

2 3 2 3

k0o _k0o

Fig. 4 _{Coupling hydrodynamic coefficients between surge and pitch ola circûlar cylinderas afunctionof ka for various values of a/d(a) added} mass, (b) damping coefficient. Present results: O, a/d = 0.5; , a/d = 1; +, aid = 2. Yeung (1981): -, id = 0.5, 1, 2.

aid,wherea isthe radius of the cylinder. In the computations,

64 segments were used to describe the circulàr section and M = 9 was used in the approximation to the infinite series

expansion given in (41) for the added mass.

Figure 2 shows the sway added mass and damping

coeffi-cient fork,,aranging from O to 3 and for aidratios of 0.5, 1, 2 and 5. Results based on the closed-formsolution of Isäacson

et al. (1990) are also given for comparison. Figure 3 shows

the roll added moment of inertia and damping coefficient for k,aranging from Oto 3 and for_aidvalues of 0.5 and 1. along

with the corresponding analytical solution based on Yeung

(1981). Similar results for the coupling added mass and

damp-ing coefficient between surge and pitch are given in Fig. 4. The agreement with the closed-form solution is seen to be very good, even though relatively few terms of the infinite

series expansions have been used.

The effect of number of segments on computed values of hydrodynamic coefficients has also been investigated.

Figure 5 shows the ratios of the sway added mass and damping

coefficient calculated using various numbers of segments N

to the corresponding results of the complete solution fOr the

particular case ofk,,a = 0.5 andaid = 0.5. Also shown are

the damping coefficients derived from the corresponding

ex-citing forces using the Haskind relation, where the exex-citing

forces themselves have been calculated numerically following lsaacson (1978). Using the previous method, for the particular choice of

ka

and aid, the added mass is estimated to within 6 percent of the complete sòlution using as little as 16 seg-ments, whereas the evaluatiOn of the damping coefficient

requires at least 128 segments for the same accuracy. Thus, it would seem to be advantageous to solve the scattering

prob-lem and then derive the dämping coefficient from the

corre-sponding exciting force. However1 in order to apply the

Has-kind relation for añ arbitrary section, the scattering problem

would have to be solved for a series of incident wave directions

and the results then integrated numerically with respect to

wave direction, so that such an approach would not be particularly efficient. b/d = 0.5 1;0 2.0 5.0 (o)

3.2 Square Cylinder. The added masses of a square

cyl-inder are presented in dimensionless form in Fig. 6. In the calculations, 64 segments were used and once more M 9

was used in theapproximation t the infinite series expansion

given in (41) Results for two orientations corresponding in turn to motions parallel to a pair of sides, ß = O deg. and motions parallel to a diagonal, = 45 deg, where f is the angle between a pair of sides and the direction of motion,

2 o ,L. o,( A ,Q o o A A A

114 ¡Vol. 113, MAY 1991 Transactions of the ASME

2 4 6 4 6

k0 b k0b

Fig. 6 Added masses of a square cylinder as a function of kb for various values of bld; ß = O deg or 45 deg(a) sway, (b) roll

S 16 32 64 128

N

Fig. 5 Ratio of the hydrodynarnic coefficients of a circular cylinder

com-puted using various numbers of segments N to the corresponding analytical

results for k0a arid a/d = 0.5. 0, ; , À; x, A from óiciting forcé.

(b) b/o= 0.5

1.0

have been found to be identical. Figure 6(a) shows the sway added mass as a function of k,b, where b is the side length, ford = O deg or 6 = 45 deg and for bld ratiòs ofO.5, 1,2 and 5. As k,,b approaches zero, the dimensionless sway added mass 22/pb3 approaches a value of 1.19 d/b, which corre

sponds to the value for a two-dimensional flow about a square

section. Corresponding results for the roll added moment of

inertia are given in Fig. 6(b) for bld ratios of 0.5. 1 and 2. 3.3 ApplicatiOn. In a number of engineering situations,

the representation of an offshore structure as a vertical struc

turc of uniform section may be useful iñ order to examine the influence of sectional form on the hydrodynamic coeffi, cients. As an example of this, the Molikpaq structure which

has been deployed in the Beaufort Sea has an octagonal

section as indicated in Fig. 7. Although the Molikpaq does not have vertical sides, a comparison of results for a vertical structure of this octagonal section with those for a vertical circular cylinder of the same cross-sectional area is useful as a means of indicating the influence of sectional shape on

the hydrodynamic coefficients. Figures 8(a) and (b) show the

o

N N 3 2 o E E k2

Fig. 7 Typical crOss section of the Molikpaq structure

sway added mass and roll added moment of inertia, respec-tively, of a vertical cylinder of the same cross section as a

function of frequency for a typical water depth of 20 m

calculated using the first 9 terms of the series expansion of (41) and 64 segments to descnbe the octagonal section Ana

lytical results for a vertical circular cylinder of the same cross-sectional area are included for comparison. This indicates that

at low frequencies the octagonal shape increases the added

mass by as much as 5 percent compared to that of the equivalent circular cylinder.,

4 Conclusions

A theoretical and numerical approach is described for cal

culating the hydrodynamic coefficients of a large

surface-piercing vertical cylinder of arbitrary section extending to the

seabed The linear radiation problem in three dimensions is

reduced to a series of two-dimensional problems in the hon-zontal plane by a proper choice of eigenfunctions in the

vertical direction. Thus, a series of line integral equations is obtained in place of the usual surface integral equation. The solution of only one of these is required in order to obtain the damping coefficient, whereas the solutions of a seres of

these are required in order to obtain the added mass. However.

for the range of parameters considered, the infinite series

representation of the added mass is found to converge rapidly. and hence only a few integral equations need to be solved in

order to obtain sufficient accuracy. In a numerical solution,

there is considerable economy of effort, both in the

complex-ity of programming as well as in computer time and storage requirements when the method outlined here is applicable

and is adopted in placed of a more general one used for bodies of arbitrary geometry.

Comparisons of the present results have been made with a

previous analytical solution for a circular cylinder and good agreement obtained However, in general a lärger number of segments have been found necessary in order to predict the damping coefficients to a given level of accuracy, than are

3 2 N E -

o-o

o

Journal of Offshore Mechanics and Arctic Engineering MAY 1991, Vol. 113/ 115

0.05 o lo 0.05 o io

Frequency (Hz) Frequency (Hz)

Fig. 8 Added masses of vertical structures of different cross sections as a function of frequency; d = 20 m--(a) sway, (b) roll. -, octagoñal

needed for calculating the added masses to the same accuracy. Results are also presented for a square cylinder and the

application to a typical offshore structure subject to base

motion is discussed.

References

Bai. K. J.. 1977. "Zero-Frequency Hydrodvnamic Coefficients of Vertical

Axisymmetric Bodies at a Free Surface." Journal lf Hidronaulics. VoI. il.

No. 2. PP. 53-57.

Hulme. A.. L 982. "The Wave Forces Acting on a Floating Hemisphere

Undergoing Forced Periodic Oscillations." Journal of Fluid Mechanics.

Vol. 121. pp. 443-463.

_Readers of

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lsaacson. M.. 1978. "Vertical Cv lindera of Arbitrary Section in Waves." Journal of Hateritai. Pori. Coastal, and Ocean Dir ision. ASCE. VoI. 104.

No. WW4. pp. 309-324.

lsaacson M 1982 "Fixed and Floating Axis mmetnc Structures in V aves

Journal of Hateri*'av. Pori. Coastal, and Ocean Division. ASCE. Vol. 108.

No. WW2. Pp. 180-199.

lsaacson. M.. Mathai. T.. and Mihelcic. C.. 1990. "Hydrodvnamic Coeffi-cients of a Vertical Circular Cs linder - Canadian Journal of Cit il Enzneering Vol. 17. No. 3. PP. 302-3 LO.

Sarpkasa. T.. and lsaacson. M.. 1981. .4lechanics of' JFai'e Forces on Offshore Structures. Van Nostrand Reinhold. New York.

Ursell. F.. 1949. 0n the Heaving Motion of a Circular Cylinder on the

Surface of a flúid." Quarierl.i' Journal qf Meckanics and .4pplied Mathematics.

Vol.2. pp. 218-231.

Yeung. R. W.. 1981. Added Mass and Damping of a Vertical Cylinder in Finite-Depth Waters. Applied Ocean Research. Vol. 3. No. 3. Pp. 119-133.