Hydrodynamic coefficients of some swaying and rolling cylinders of arbitrary shape

hydrodynamic coefficients of sorne swaying and rolling cylinders of arbitrary shape.

by * Lab. y.

Technische Hogeschoo

,). Dell t R.E,D. Bishop + W,G, Price P. Ternarel *

Kennedy Research Professor in the Univers ity of London and Fellow of University College.

+Lecturer _{Department of Mechanical Engineering, University} College London.

tResearch Assistant, Department of Mechanical Engineering, University Collee LOndCn.

L

.2, fie

C1Fc

2.

c::-,p&.01

17 CKT. 17

ARCHIEF

4

S umiriar

Two dimensional hydrodynamie properties are computed for cylinders swaying and rolling in the free surface of an infinite ideal flujo.. The cylinders may be of arbitrary cross section. A potential flow solution is employed using multipole expansion and conformal mapping techniques. Results are given for triangular, rectangular, chine, fine and bulbous sections and, wherever it is possible, comparisons are made with previous theoretical and experimental findings for these section shapes. To illustrate the versatility of the method, a pair of rudimentary

bilge keel.s are attached to a ship-shaped section and the influence on the hydrodynamic properties is investigated.

Intrc'duct ion.

Since the general formulat ion of a three dimensional

method for the de.t erminat ion of the hydrodynamic prop ert i es of

a ship is difficult and complicated, a simpler two dimensional approach is usually adopted. The ship is divided into a number of slices and the, two-dimensional hydrodynaniic properties

associated with each slice are used in the calculation of the ship's mot ion by means of a form of strip theory.

Ursell (l9Li9a,b) derived a potential flow solution for a circular cylinder oscillating harmonically with arbitrary

frequency in the free surface of an ideal fluid. The solution. is a combination of potential functions, each of which satisfies

Laplace's equation and the linearised free surface condition, while the comb mat jon satisfies the remaining boundary condit ions or.

the circular cylinder. Tasai (1960a) arid Grim (1953, 1959)

generalised the method of Ursell to elliptical and Lewis form

sections. Porter (1960) , Tasai (l960b, 1961) and de Jong (1973) extende

the method to sections of arbitrary shape.

Little experimental verification of the theoretical prcdictiors has been carried out. Porter (1960) and Paulling and Richardson (1962) made force and pressure measuremcrts on

oscillating cylinders while Vugts (1968) perfcrmcd a comprehensive set of experiments in which ship-shaped, triangular and rectangular cylinders of differing sectional area were oscillated. By varying the amplitude and frequency of oscillation and measuring the

with each two-dimensional cylinder were determined for heave, roll and sway motions. The results were compared with theoretical predictions made by de Jong (1967).

S mce the L ewis two-paraet er conformal transformat ion is insufficiently accurate for cylinders of arbitrary shape, as was recently illustrated by Bishop, Price and Tam (1978a,b) more refined conformal transformations have been developed. Landweber and Macagno (1959, 1967, 1975) extended the Lewis transformat ion by including a third parameter; they used a

method propos ed b

form of a Gershgorin integral equation. Smith (1967) used

Fil'chakova's method which is based on the orthogonality condition of a Fourier series, whilst von Kerczeck and Tuck (1969) applied an iterative procedure sirti1ar to the Newton-Raphson method for

solut ion of the paramet ers associated with the inuit iparamet er

conformal transformat ion.

Other methods for the determination of the two

dimensional hydrodynantic properties have been proposed; e.g. see the work of Frank (1967), Maeda (1971, 1975) and S6ding (1973). In the first method (which is a development of a technique

proposed by Kim (1965) for an elliptical cylinder) , the ;ect ion

is represented by a number of points joined together by straight line segments. The velocity potential is obtained by distributing a pulsating source of constant strength over each of these

segments. A numerical application of Frank's close-fit inethod has been performed by Fait insen (1969) . The close-fit method

derived by Maeda (1971, 1975) 'akes Use of stream functions, Inst cad of velocity pot ent jais as in the previous method.

Sding (1973) and Sding and Lee (1975) produced a "quadrpolet1. method which is, in effect, an extension of an earlier approach

developed by Grim (1953) . This method has wide applicability and requireS only a fraction of the computer time needed for the close fit-methods.

In this paper, the hydrodynamic coefficients of a

variety of symmetric sectional shapes - triangular, rectangular, chine, fine, bulbous and also a section with a bilge keel - are comput ed f or sway and roll motions. A mult iparamet er conformal transformation is used to represent the sectional shape and the

transformation parameters are solved in a least square sense by a iethod developed by Peckham (1970).. full description of the sectional shapes adopted has been given by Bishop, Price and Tam (1978a,b)

By mapping the c-plane relationship onto the z-plane, a coordinate transformation is obtained between the Cartesian coordinates (x,y) in the z-plane and the curvilinear coordinates of the -p1ane in such a way that one of the coordinate lines

o) coincides with the contour in the z-plane. That is, a

mapping is found which is defined by

where f() generally is a conplex function and

Z X + íY

ir(01)e1

C

+ in

i(0)ee.

The z- and LC planes are shown in fig. i with Ox, O ' located in the free undisturbed surface and Oy, O' coinciding with the vertical line of symmetry. Angles O and O are measured from the axes Oy and O'rì respectively, as shown in fig. 1.

The usefulness of the conformal mapping lies in the fact that the potential of the fluid mot ion around any sect ion

in the z-plane can be derived from transforming the actual section into a more convenient (and hence easily solved) one

in the -p1ane.

The potential solution for the flow around a circular section oscillating harmonically with arbitrary frequency has been given by Ursell (1949a,b) . The Theodorsen transformation,

(which. i commo±ily used in the flow around aerofoil sections), transforms a unit circle, ¿yl, in the -p1ane to any shape in the z-plane using the following relation

- -T)

z E

CC

nu-1 n

where the constants C are, in general, complex. This ransformaticn reduces to a simpler form for symmetric sections. The parametric equations at the boundary of the symmetric section can be

where a is generally referred to as the scaling factor and the constants a are the transformation parameters. In practice the series has to be truncated to N terms.

For N2 the Lewis donformal transformat ion is obtained, where a, a1 and a3 being uniquely defined in terms of beam, draught and sectional area. Expressions for these parameters, and

rest rictions of the uses of the Lewis transformat ion, are given by von Kerczek and Tuck (1969) . The usefulness of L ewis

transformat ion is, of course, restrict ed by the fact that it cannot produce the actual section shape but only describes a

sect ion with the same b eatn, draught and sectional area.

The parametric relationships in equat icn (1) may be used to obtain a transformed section that is closer to the actual one by choosing a suitable number, N, of transformation parameters. Now the ljmjtat ions of the conformal tran.sformat ion are impos ed by the nature of the parametric equations. It can be _{seen that}

at O ir/2,

O at ORO,

which suggest that the sectior contour must be perpendicular to the free undisturb ed surface and horizontal at the vertical

line of sym try. On the ct.er hand, all higher derivatives

x y a{sjnü + aHose + N-1 N-1

nO

n (-1) a _sin(2n+1)O] 2n+ 1 n+ j (-1) a cos 2n+1 (2n+l)O] } (1)

appear to be continuous, and this suggests that only smooth and Continuous ccntoUrs can be transformed. In practice it has beer shown by Bishop, F rice and Tam (1978a,b) , however, that

discontinuous and sharp contours, _{as occur} in rectangular and chine. sections, may be adequately represented provided that enough trarsfcrrnation parameters are used and sufficient inìpút data are included in the region of the discontinuity. Notice that the discontinuities will be approximated by a continuous curve and that the quality of the representation wiii depend on the input data in that region.

In practice, the transformation parameters (including the scaling factor) are obtained from equations (1) by using section offset points (x,y). Consider the section contour

represented by P offset points. This gives a set of coordinates each having a corresponding angle O in the c-plane for pl,2,...,P. This set of coordinates will provide 2P-2

non linear equations, since trivial solutions exist at positions

x10,010 (at the keel) and yO, OI2 (on the still water beam).

The system of 2F-2 non linear equations

N-1 n x a[sjnO + E (-1) a _sin(2n+l)O p p _{2n+l n}

nU

N-1 n+ i y a[cos8 + (-1) a cos(2n+l)e p _2n+l p n O

for pl,2,.. . P, may be solved to give the (Ni-l) unknown transformation parameters (i.e. the quantities a) and (P-2)

Unknown angles, Op,, in the -'plane provided that there is a sufficient number of equations. That is, the number of offset

points P and transformation parameters N must satisfy the re lat ionship

2P-2N+P-1,

or

PN41.

To avoid discontinuities and to ensure a smooth section contour, a least squares solution of the non linear set of

equations is adopted, as developed by Peckham (1970) . The necessary initial values of the transformation parameters are provided by the L ewis transformat ion and the general set of

equations are solved to a degree of accuracy defined by

expressing the sum of squares of the distance of the P points

from the gerLerat ed contour as a f ract ion of the mean radius

of the section (e.g. see bishop, Price and Tam (1978a)). It is not essential to have equal spacing of the offset points around the contour, but experience shows that uniform spacing of the offset points prcvides a generated contour that is closer to the actual one, especially in sections

itaVifl

sa11 bccm tc draught ratios. TKat is it is desirable fo O be nearly Constant for p1,2,..P.

Potential flow solution

The total velocity potential T(x,y,t) for the swaying and rolling cylinder may be expressed as a sum of a series of linear multipole potentials and a dipole potential situated at

4

thc críï.n. That is

=

jCOst +

E2m2}sinwt]

(2)

T rw[ C

where b is the amplitude of the generated waves at an infinite

distance away from the cylinder

The dipole and multipole

potentials are given by de Jong (1973) as

dipo1e IrU CCOst +

sinwt)

where

-ky

= -Tre

ìnkx,

e_kYcosk e

(kcosy+siny)d

x X

-o) (+k)

k(x2+y2)

for x>O and

N_1(_l)r(2n+1)a _{sin(2n+2m,2)6}

+ 2 n +

i

2m 2zn+l -2m 2m+2n

2m n=l (2m+2n+2)

where the wave number

kw2/g and

1. All the potentials are so

restricted that they satisfy

Laplace's equation,

the linearised free surface boundary conditions, the radiation condition at infinity,

the no-flow condition at infinite depth.

In practice,

the infinite series (2) in the

multipole

expansion is truncated to a finite number M (so that lmM) and,

by setting up sufficient equations and applying the boundary

condition at the contour, the multipole expansion coefficients

velocity potential

T may be determined. By means

of the

linearised Bernoulli equation, the fluid pressure at the boundary of the section may he calculated, from which the total

horizontal hydrodynamic force and rolling moment can be determined. These fluid actions have components in phase with the velocity

and acceleration motions of the cylinder and, from them,

the added mass and damp ing ass oc jat ed with sway mot IOn,

the added inertia and damping associated with roll motion and the cross coupling terms associated with the tWO motions

are determined, all reckoned per unit length.

It is sometimes desirable to express the hydrodynamic coefficients in non-dimensional form as functions of the

non-*

dimensional frequency coefficient

where w is the driving frequency. The factors adopted in this paper for non-dimensionalising are given in Table 1, in which p is the

fluid density, B is the beam at the waterline and S is the sectional

area.

Table 1 Factors used to rend erdrodynamic coefficients non-dimensional.

Coefficient

Sway added mass Sway damping

Roll added inertia Roll damping

Cross coupling mass or inertia Cross coupling damping

Unit s kg Im kg /(m, s) kg .m2Jm 2 kg.m J(m.$) kg .in /m kg .m ¡(in. s)

Dimens ions Fact or

pS pS(2g/B) pSB2 pSB2(2g ¡B) p SB pSB(2g ¡B) M ¡L M /LT ML ML ¡T M M IT

c

The non-dimensional added mass and damping coefficients in sway, the added inertia and damping coefficients in roll,

together with their cross-coupling coefficients have been

calculated for chine-a rectangular-p triangular-, fine- and bi.tlbous sections over a wide range of non-dimensional frequency . The

last two sections are sections 6 and 7 respectively used by

Faltinsen (1969) in a numerical investigation of Frank!s close -fit method. The coefficients have been calculated using a Lewis form fit, the conformal transformation technique and, in some

cases, Frank's close-fit method as well. These are the same sections as those used previously by Bishop, Price and Tam (1978a,b) for

symmetric fluid actions in vertical motion.

Prior to calculating the coefficients for the above sections, the method employed was tested in several ways. The first was to compare calculat ions with previous theoret ical and experimental results derived by Vugts (1968) for three rectangular sections (B/T2,4 and 8) and for a triangular section. The

coefficients so determined agreed very well with the theoretical results? apart from some minor discrepancies due to differing conformal representations of the rect angular section.

The second approach adopted to test the results needed the construct ion of error funct ions defined by the expressions

{2(M p -N q

)2}

/T2

s

00 00

for sway and roll mot iOns respect ively.. The constants M0,

N0,

R' XR, Po and q are defined in the Appendix and the functions are

obtained by equating the energy dissipated by the waves to the work done by the section in one cycle of oscillation.

In general for sway motion coefficients the error is very small at low frequencies and increases with increasing frequency. This is illustrated in fig. 2(a) for a rectangular sect ion using the conformal method. Also the error decreases with

increasing number of multipole expansion coefficients and

2m (i.e. for increasing values of M). 'These findings apply for

both Lewis and multiparameter transformations though some difficulties occurred in the calculations of the appropriate coefficients for

the fine section at high frequencies.

When using a Lewis or multiparameter conformal

transformation method for the rolling motion case a sudden jump in the error function is observed in the low frequency range for sect ions such as rectangular, fine and bulbous. This jump decreases in value with increased number of expansion coefficients M, 'This jump

phenomenon is illustrated in fig. 2(b) for a rectangular section using the conformal method. The jump is much bigger in the Lewis

calculat ion.

The contour curve generat ed by the mu lt iparamet er

conformal transformation does not necessarily pass through all the given offset points but the curve does fit the given data as closely as possible. This is contrary to the Lewis form contour which always passes through the coordinates of the draught and beam

of the section. From the theory, jt can b e shown that the calculat ed value of tR depends on the term

a(e) f[2(e) + y2(e)]- B2}/B2

which is introduced in the surface boundary condition and must, by definit iOn, be zero at Or /2. In the conformal calculat ion this

requirementmay not be satisfied since it is possible for _(7r/2)IO. Although this paramet er is small, it caus es deviations throughout th calculations

/and produces results indicating a large error value. This

difficulty may be overcome by adjusting the scale factor in the multiparameter transformation sud; that at @r/2, x(ir/2)B/2

and c (ir /2) O. V alues of the error funct ion are now greatly reduced

but the function still suffers from thejump phenomenon as previously ment joned.

The cross coupling hydrodynamic coefficients obtained in sway-roll and roll-sway motions must be equal. This is ensured by symmetry and it provides another check on the calculations. This condition was satisfied throughout all calculations.

After the programs had satisfied these tests, the.

hydrodynamic coefficients for the sections chosen were calculated. The results will now be discussed.

i Chine

rectan ular and trigr sect ions

Figs. 3(a)-(f) show the calculated hydrodynamic

coefficients of the chine, rectangular and triangular sections. The curves are shown for calculations using the Lewis forms

and also for the conformal transformation technique. The results for the triangular section show the closest agreement, while the chine

i) Fine and bulbous sections

Figs. 4(a)-(f) show the calculated hydrodynamic

coefficients for the fine and bulbous sections. These have been determined using Lewis forms, conformal mapping and by the

Frank close-fit method. The results in which Frank's method has been used were derived by Fait insen (1969) and it will be

noticed that they suffer from discontinuities at certain irregularly spaced frequencies.

It is seen from the graphs that , for the bulbous sections, the Frank results are closer to the conformal transformation

results than are those of the L ewis form. The same is not true of the f me section in every case. This app ears to b e without

special significance (and values of the error funct ions calculat ed for the conformal method did not reveal unsatisfactory results).

(iii The influences of bil e keels

To illust rat e the versatility of the conformal

transformation method, a simple bilge keel of varying size was attached to a ship-shaped section as illustrated in fig. 5. The section is chosen to satisfy a one-to-one correspondence with its transform. The influences of the size of bilge keel and details of the transformat ion on the values of the hydrodynamic coefficients are illustrated in figs. 6(a)-(f).

For the smaller bilge keel, the results were determined for 17 offsets and 8 paraTfletes and for 24 offsets and 12 parameters. Very little difference was found in the results , indicating a

The larger bilge keel extends to full beam and draught and this section was described by 19 offsets and 10 parameters. Note that the beam and draught of the section are unchanged and that the section area varies only slightly as between the

cases considered; that is to say the b ilge keel has a small

sect jonal area. The Lewis form fit method therefore gave pract ically the same values for the hydrodynamic coefficients in all cases.

The results of the sway coefficients show little dependence on the size of the bilge keel but large differences are observed in the roll coefficients. It will be seen that the bilge keels may increas e the roll added moment of inert ja by a factor gr eat er than 2 and that the roll damping is also changed substantially. Such changes of shape would significantly affect the calculation of the 'roll natural frequency' (which is _{a function of the roll added} moment of inertia),

As one would expect the larger the bilge keel the

greater is the change it makes on the values of the hydrodynamic coefficients. It must be remembered, however, that these

calculations are made under the assumptionof idealised potential flow in which the effects of viscosity,separation, circulation, and so forch are ignored. The results are thus to some extent

art if jCial.

Conc lus ions

A conformal transformat ion mapping method has been

described to calculate the two-dimensional hydrodynamic coefficients for an arbitrary ship-shaped section swaying and rolling in the

free undisturbed surface of an infinite fluid. An idealised

potential theory is used, together with multiparameter and multipole expansions. An error function has been devised for assessing the quality of the results.

In general, agreement between this method and Frank's close-fit method is very good provided that results in the regions of the singular frequencies are discarded. (Further comparisons with other 'extreme' ship-shaped sections, that have not been

discussed in this paper have also indicated very good agreement.) The results presented for the chine-, rectangular-,

triangular-, fine- and bulbous sections and for a section with bilges extended over the non-dimensional frequency range If small values of the error function are to be achieved over this frequency

range, many multipole expansion coefficients are required. Sufficient accuracy may be obtained in the range 2, however, with a smaller number of multipole expansion coefficients. It is clear that the error inherent in the calculations can be reduced by increasing the number of multipole expansion coefficients, but the actual difference in the values of the hydrodynamic coefficients is usually small

except where the error is very large.

y a suitable choice of the parameters P, N and M the hydrodynamic coefficients of a large number of ship-shaped

sections may be determined to any required degree of accuracy by the conformal method. An adequate number of data points is essential to obtain sufficiently accurate values of the hydro-dynamic coefficients. Generally, these offsets are equally

spaced around the contour, but close spacing is required if the section contour is sharp or discontinuous. For simple sections,

jt is not advisable to choose a very large number of transformation parameters because they demand excessive computations. Generally, better accuracy is achieved in the calculations of the hydro-dynamic coefficients in the low frequency range and with larger values of N (i.e. with a greater number of multipole expansions).

If bilge keels are used, a large nunber of offsets and

transforrnation

paraIlet ers are needed b ecasue of the discontinuous nature of the section. The hydrodynamic coefficients associated with roll UOt ions then depend greatly on an accurat e d escript ion

Ref e r e nc e s

Bishop, R.E.D. , Price, W C. and Tam, P.K,Y. 1978(a).

Hydro-dynamic coefficients of some heaving cylinders of arbitrary shape. InC. J. for Nurn,. Methods in Eng. 13, 17-33.

Bishop, R.E.D., Price, W,G, and Tam, P.K.Y. 1978(b). The representation of hull sections and its effects on estimated hydrodynamic actions and wave responses. Trans. RINA. paper W2.

de Jong, B, 1967. Berekening van de hydrodynamische cofficinten van oscillerende cylinders. Neth. Res. Centre TNO. report 174.

de Jong, B, 1973. Computation of the hydrodynamic coefficients of oscillating cylinders. Neth. Res, Centre TNO report 145a.

Fait insen, O. 1969. A study of the two dimensional added mass and damping coefficients by the Frank's close fit method. DeC Norske Ventas report £9-lO-s.

Frank, W. 1967. Oscillations of cylinders in or below the free surface of deep fluids. NSRDC report 2375.

Grim, 0. 1953. Berechung der durch Schwingungen eines Schiffskrpers erzeugten hydrodynamischen Krfte. Jahrbuck der Schíffbautechnischen

Gesellschaft 277-299.

Grim, 0. 1959. Oscillation of buoyant two dimensional bodies and the calculation of the hydrodynamic forces. Hamburgische Schiffbau-Versuchsanstalt report 1171.

Kim, W. D. _{1965. On the harmonic oscillations of a rigid body on} a free surface. J, Fluid Mech. 21, 427-451.

Landweber, L. and Macagno, M.. C. 1959. Added mass of a 3 parameter family of two dimensional forms oscillating in a free surface. J. Ship Res. 2, 36-48.

Landweber, L. and Macagno, M.C. 1967. Padded mass of two dimensional forms by conformal mapping. J. Ship Res. 11, 109-116.

Landweber, L, and Macagno, M.C. 1975. accurate parametric

representation of ship sections by conformal mapping. Proc. ist mt. Con. on Numerical Ship Hydrodynamics, 665-682.

Macagno, M, 1968. A comparison of three methods for computing the added mass of ship sect-ions. J. Ship Res, 12, 279-285.

Maeda,H. 1971. Wave excitation forces on two dimensional _ship of arbitrary sections. Selected papers from the JSNA Japan.

Maeda, H, 1975. }Iydrodynamic forces on a cross section of a

stationary hull. mt. Symposium on the Dynamics of Marine Vehicles and Structures in Waves, ed. R.E.D. Bishop and W.C. Price, 80-90, Lond. Mech. Eng. Pubi.

Paulling, J.R. and Richardson, R,K. _{1962. Measurements of pressure,} forces and radiating waves or cylinders oscillating in a free

surface. Univ. of California, Inst. of Eng. Res., Berkeley, Series 82, no. 23.

Peckham, S,G, 1970. A new method for miniulising a sum of squares without calculating gradients. Computer J. _{13, 418-420.}

OXteX _{W,1960.Pressure distjbutjons, added mass and damping}

coefficients for cylinders oscillating in a free surface. Univ. of California Eng. Pubi. series 82-16.

Smilh, W.l967. Computation of pitch and heave motions for arbitrary ship forms. Neth. Res. Centre TNO report 90s.

Sding, H. 1973. The flow around ship sections in waves. Schiffstechnik 20, 9-15.

Sding, H, and Lee, K.Y. 1975. The two dimensional potential

flow excited by a body oscillating at a free surface. (Description of accompanying program ASYM1). Technische Universitdt Hannover report 12.

Tasai, F, 1960(a) . On the damping force and added mass of ships heaving and pitching. Univ. of California Eng. Pubi. series no. 82.

Tasai, F, 1960(b). Formula for calculating hydrodynamic force of cylinder heaving on a free surface (N-parameter family). Res. Inst. for Appi. Mech., Kyushu Univ., report 31, 71-74.

Tasai, F, 1961. Hydrodynamic force and moment produced by swaying and rolling oscillation of cylinders on the free surface. Res. Inst. for Appl. Mech. , Kyushu J niv. ,

Tir,

No. 35.

Ursell, F, 1949(a), On the heaving motion of a circular cylinder in the surface of a fluid. Quart. J. of Mech. and Appi. Ma-hs. 2, 218-231.

Ursell, F. 1949(b) . On the rolling motion of cylinders in the

surface of a fluid. Quart. J. of Mech. and Appl. Maths. 2, 335-353.

von Kerczek, C. and Tuck, E.O. 1969. The representation of ship

hulls by conformal mapping functions. J. Ship Res. 13, 284-298.

vugts, J.H, 1968. The hydrodynamic coefficients for swaying, heaving and rolling cylinders in a free surface. mt. Shipbldg.

a, a n b g k S Y Not at ion Transformation coefficients.

mplitude of generated wa.ve at infinity. Acceleration due to gravity.

Wave number (w2/g)

Sway added mass per unit length.

Added mass per unit length for the sway motion produced by roll.

Added moment of inertia per unit length for the roll motion produced by sway.

Expansion coefficients.

Coordinat es in the z-p lane.

Physical plane Section beam:

Complex transformation coefficient.

Added moment of inertia per unit length. Number of expansion coefficients.

Number of transformation parameters. Damping per unit length for sway.

Damping per unit length for sway produced by roll. Damping per unit length for roll.

Damping per Unit length for roll produced by sway. Number of offset points.

Number of points along the contour where flow conditions at the boundary are satisfied. Section area. Euler constant. Transform ed plane. 2m x,y z B C n M N N(iS)

N(6)

N(5)

N(S)

P R

Coordinat es in the transformed plane.

0,01

Angles in the

- and z-plane.

Velocity potential.

'j'

Strean function.

w

Circular frequency of driving and of waves.

Durn4y variable.

p

Density of fluid

6

Non-dimensional frequency parameter

w(B/2g)}

e

Error function

App endix

P ot ent

i al flow

as

For a cotnprehensive (and fully coniprehensible) account of the theory; it is necessary to consult the original work of de Jong (1973) and Porter

(1960). They

show that the various

velocity pot ent jais and their corresponding conjugat e stream

funct ions at the contour boundary of the unit radius s etni-circie

(l) are

(e

) '

-

ire

-ky sin

}kx

}

r

+

cos

e

{X}

t ___y

(e

)e{}coskx

{}sinx

k(x2+y2)

Jr

where, Q y +

ln(

k(x2+y2) } +

Jt2._COSnX

OD n ₂

2 n/2

n1

n'.n n 2 2

n/2

u

x--ni

nn

0.577215, the Euler constant and

X tan 1(x/y)

( )4Sifl}(

2inl)O ±ak

r -cos r

Sifl}20

_{N-1 (2n+1)}

_a21

sin

}(2n+2m+2)O

cos

r

cos r +

(-1)"

n1

(2n2rn+2) 2m} 2m

When the series is truncated to M terms and the boundary condition is imposed at R (Mi-1) points on the contour (which

have corresponding angles e1, 02 . . . ,e, in the c-plane)

the following linear equations are obtained

' } '2r

}f

(e ) C

m0

q2 2m r (er) -

Ck/2)

s s) where -

2m°r

2y(e) ¡B (ni0 and sway motion) [4tx2(Or)+y2(0)}_B211B2 (m0 and roll motion)

for 0< er< IT 12 and r'l,2, . . . ,R. These equations may be solved

in a least square sense for the quantities p and q.

In a full description of this theory, de Jong (1973) has shown that, by separating in-phase and quadrature components,

expressions may be found for thehydrodynamic coefficients. They are

sway added mass:

-pB2(Mp+Nq)/2a2

sway da'nping: -pB2w(M0p0-N0cj0)/2c42

roll-sway cross coupling

3 2

added inertia:

m(5)

-pB (Xp+Yq0) /4o

roll-sway cross coupling

3 2

damping:

N()

-pB w(Yp -Xq ) /4cz

roll added inertia:

roll damping:

I()

pB4(Xp

Yq )

/l62

o o

N ()

pB4w(Yp-Xq) J16ß2 sway-roll cross coupling

3 2

sway-roll cross coupling darnping N

₍₎

pB3w(Mp -N q )

/82

0

00

where 2 2 2 sway e

flOti0n

2} - p±q for th roll y (e ) - (e ) + X dO i r

BdO

' 2 r r r r ii. /2 R

f

q

)M(0)do

r N(O )i r o o

r'

lT /2 X .: j1 _{2(0r)M(Or)I0r} B2°

r)

M M(Q ) s(O ) + E I r Ì _,

_ml2mJ

As a check on the computations, the property that the cross coupling calculations are equal ay be used (i.e. tn

y4

qy'

N - N ).

-p lane

F ig.

1.

04

0' 3

U

differing values of M.

Fia. 2(a) Variation of sway error function c (rectangular section) for

4,0

0.2

0cl

0

0.4

0,8

1.2

F ig.

2(b) .

1 ariation of roll error function CR (rectangular section) for differing

values of M.

Mu9IN1.

Ml6

M20

I I -(S t

-3 y 3 2 i rect angular Conformal transformation chine I I t I I I I I I I I I J... I I % Lewis form tr iangu lar

Section chine rectangular triangula:

Form ¡ ¡

/

No. offsets 15 11 11 No. params. 10 8 8 o 2

N' y 1.25 1100

0.75

O 50

0,25

¡ I I I

t r ía ngu lar

I

\

r e c t a ng u i a r \

\'

/

/

/

ch i

ne

I t t I t t I t t t i t

t.

t t I t t I

JJ

0 1 2 3 4

0.3

0.2

0.1

\

chine

N

N

rt.i-iti1r

triangular

-

S

-

-

-_t I I t I I I I I i t I I I i I I I t I o

i

2 3 6

F ig. 3(c)

4

N

t riangular

\

\ chine

\

\

\

\

\

\

\

r ect a ngu lar

\

_\

\ ch i ne

N

r ect angular

3 '5 4

t

I

-t r i angular

F 1g, 3(e)

I t y r

N 4y

triangular

F ig

3(f)

r ec t a ngu lar

t t N N y4 's

y ₅ e

'me

I

4_

s

/

-

/1

V

/

-

/

/

/

2

/

/

,

-..-bulbous

---

Conformal transformation

Lewis form

F rank (fine)

F rank (bulbous)

t--- -

\--Section

fine

bulbous

Form

No, offsets

10 7

No. pararns,

6 5

i

2 3 o

N y

1.5

1.0

0.5

q 'I ¡

¡/1/

¡11f

\

¿1,

bulbous

ri' I

\

_N

r'-\t

I

'3

'

L _'

-

--t' t I t 1 2 '5 4

I

I

0.3

0,2

0.1

-

-

- -

-4- -

- -

_bulbous

,_---..__

fine.

F ig. 4(c)

I I I I I I t t i I I I t I I I t I I t o

_i

2 3 4

N;

0. 010

0.005

/

/

I

/

I

I

/

/fine

/

/

-fine

\

t

I

I/i

I I I ¡

\

\

\

_\

bulbous

//

/

//

_{t I}

bulbous

1/

It.

,/

,

L1

II

t I 1 2 3 o

Fig. 4(d)

t

\

4

0.015

/

/

/

_\

I

\

-O 2

-O 4

¡n qy

-O 6

F 1g, 4(e)

¡n yq r O

i

2 3 4

.4

o

x

offset point or actual hull

generated hull contour

Fig. 5, A "ship-shaped

section of beam 20m and draught 10m

with a simple bilge keel.

in

y

1,5

1.0

Section without bilge keels

Strall bilge keels (17 offsets, 8 param.)

Small bilge keels (24 offsets, 12 param.)

Large bilge keels

I I t I I I I I I I I I I I I I I I I

F 1g. 6(a).

6 4 1 2 3

0.5

o

y 1.00

0,75

0, 50 O 25 O I t I I t J I t I I I I I I I J I t

i

F 1g, 6(b)

2 3 4

0. 06

0. 04

\

I I t t t I I t I I I I t I I t I t t

0. 02

o

i

2 ₃ 6 4

i

2 3 4

V--0. 05

-o io

-o 15

-O 20

3 4 I I I I I I o 2

-0,075

-0.025

-0 .0 50

-o. ioo

-0.125

o r-I ---r i i i i i r i i

F ig, 6(f)

i t N N yq 4y 3 S 4 i I i r 1 2