Hydrodynamic coefficients of some heaving cylinders of arbitrary shape

HYDRODYNAMIC COEFFICIENTS OF SOME HEAVING

CYLINDERS OF ARBITRARY SHAPE

R. E. D. BISHOP. W. G. PRICE AND P. K. Y. TAM

Department of Mechanical Engineering, University College London, England

SUMMARY

The fluid forces acting on a uniform cylinder with an infinitely long axis, heaving in the free surface of an infinite ideal fluid, are described in terms of its 'added mass' and 'damping coefficient'. The techniques of multipole expansion and multiparameter conformal transformation are adopted for such calculations and applied to shapes which cannot be adequately represented by the conventional, and more rudimentary, 'Lewis form fit'. The shapes referred to are both relevant to ship bows, one being a 'fine section' and the other a 'bulbous section'. The parameters which influence the accuracy of the solution are examined. Results for these two sections are computed and compared with results based on (a) the 'Lewis form approximation' and (b) the 'Frank's close fit method' which employs a singularity representation.

INTRODU'TION

Conformal transformation and mapping techniques are extensively used in problems of fluid mechanics and are fully discussed by Mime-Thomson' and Batchelor.2 Their use in cal-culations of added mass and damping for ship shaped sections is due to the pioneering work of Lewis.3 Ignoring the presence of the free surface, he developed a method whereby the added mass of an infinitely long cylinder of ship shaped cross section oscillating in a fluid of infinite depth could be obtained from that of a semi-circle of unit radius by means of the conformal transformation

z =x +iy = a(-l-a,(1 +a3(3).

In this expression '= i e'° describes the semi-circle which is mapped into the ship section described by the co-ordinates (x, y) in the z-plane and a is a scale factor having the dimension of length. The coefficients a1 and a3 are constants which must be determined for a given section. The family of forms generated by such transformations is commonly referred to as being of 'Lewis form'.

Lewis's results were extended to the three parameter form by Prohaska,4 Landweber and Macagno,5 and to more general N-parameter forms by Landweber and Macagno.6 The effect of the free surface, which gives rise to the frequency dependence of the hydrodynamic coefficients was introduced by Ursell,7'8 who treated the case of heaving and rolling circular sections by the multipole expansion technique. Applying a conformal transformation method to Ursell's analysis, Porter,9 Tasai'° and Grim" obtained the hydrodynamic coefficients for a family of 'Lewis forms'. In addition, Porter9 and Tasai12 extended the theory to multi-parameter families obtaining expressions for the added mass and damping coefficients. For general sectional shapes therefore, the determination of the appropriate coefficients in the multiparameter transformation is essential.

0029-5981/78/01 13-0017$O1.00 Received 16 November 1977

© 1978 by John Wiley & Sons, Ltd.

Various methods are used to calculate the required transformation coefficients for a given section. Landweber,'3 Landweber and Macagno'4 applied Bieberbach's method of inverse transformation and later15 presented another solution in the form of a Gershgorin integral equation. Smith16 and de Jong'7 used FiI'chakova's method which is based on the ortho-gonality condition of a Fourier series. Solution by an iterative procedure similar to the NewtonRaphson method was employed by Smith16 and von Kerczeck and Tuck.'8

A completely different method for the evaluation of hydrodynamic properties of two dimensional cylinders was provided by Frank.19 The method is essentially numerical. The section outline is approximated by a series of straight line segments in which a uniformly distributed pulsating source of unknown strength is situated. By satisfying the flow condition at the section boundary, the unknown source strengths may be obtained as the solution of a set of integral equations. A numerical investigation of a series of sections using Frank's method was presented by Faltinsen.2°

In this paper, the hydrodynamic coefficients of heaving 'fine' and 'bulbous' symmetric sectional shapes are computed based on the multipole expansion technique with multi-parameter conformal mapping. The conformal transformation is solved in the form of a set of non-linear simultaneous equations in the least square sense by Peckham's method.2' The theory involved is simple and easy to apply, requiring only the minimum amount of informa-tion about the secinforma-tion shape. The results computed in this manner are compared with the

'Lewis form' solution and the 'Frank's close

fit' method solution of Faltinsen.2° The parameters which influence the accuracy of the solution are examined.

CONFORMAL MAPPING

Consider a mapping of the -pIane onto the z-plane given by the complex function

z=f()

(1)

where

z = x + iy = ir(4)e

=e+iii

=j(9)_IO

such that the exterior of the shaded region in the e-plane maps onto the exterior of the shaded region in the z-plane as shown in Figure 1. The axes Ox and O' are located in the undisturbed free surface and Oy and O'r coincide with the vertical centre-line plane. Angles and O are measured from the axes Oy and O'i respectively.

Let

w()='t+i'I'

(2)

be the complex potential of the fluid motion in the e-plane with respect to some boundary conditions. Provided that w() is analytic, then at corresponding points ¿ and z satisfying equation (1) and subject to equivalent boundary conditions the function w takes the same

value in both planes (e.g. see Mime-Thomson'). The analytic function w(z) may be obtained by either eliminating from equations (1) and (2), or treating as a parameter in the complex potential function. It follows that the potential solution of the fluid motion in the z-plane may be found by conformally transforming the actual shape onto a convenient one such as a unit circle in the e-plane where the flow is easier to solve. Given the Ursell7 solution for a circular section it remains to find the appropriate transformation relationship (1).

(4)

- plane z - plane

Figure 1. Systems of axes in the ¿ and z-planes

It is common in the study of flow around aerofoil sections to use the general Theodorsen transformation (e.g. see Abott and Doenhofl22), which is expressed in the form ofan infinite

Laurent series expansion of the unit circle, ¿= 1, ¿= i e' and Z

= (3)

where the constants C are, in general, complex. For symmetric sections, equation (3) reduces to a simpler form

z =

a[+

a2n_i((2_1)]

nl

where the constants a are real numbers.

By restricting the number of parameters N to 2, the 'Lewis form approximation' is obtained. Given the section's area, S, beam B and draught T, the constants a, a a3 are uniquely defined by von Kerczek and Tuck18 as

a =B/2(1+aj+a3),

a1 = (1 +a3)(H-1)/(H+1)

C1+3+V'(9-2C1)

a3= where

c1=(3++(1_

_ì

_\

_iH+1)'

'H-1\2

s

o=

H=f

The fact that Lewis forms only conform to the actual area, beam and draught of the section imposes restrictions on its usefulness. Bulbous sections, fine sections and sections with dis-continuities cannot be treated adequately. Furthermore, not all forms generated are useful and the restrictions which have to be imposed on the parameters o and H are as suggested in Figure 2. The limitations may be relaxed, however, by introducing extra terms in the Laurent series. For example, N = 3 in the three parameter family suggested by Landweber and Macagno.6 In principle, with a suitable choice of N, one could approximate most shapes to a required accuracy. Sectional area coefficient s

är

20 I0 00 No Lewis form max Conventional hulls

Lewis forms avouable

No Lewis form min (o)

Figure 2. Restrictions on the use of Lewis forms and the suggested ranges of their applicability are as shown

Alternatively, equation (4) at the boundary of the contour may be re-written in a parametric

form:

=a[sin

+

(-1)"'a2_1 sin (2n - 1)8]

(Sa)

y

=a[cos o+

(-1)a2_1 cos

(2n_1)8J (5b)

which can be interpreted as an attempt to solve a general hydrodynamic problem using a set of orthogonal curvilinear co-ordinate axes (, O), where (5a) and (5b) represent their relationship with the Cartesian co-ordinates (x, y). The Cartesian co-ordinates are specified as 'offsets'.

In order to maintain consistency with subsequent ship response calculations, the section profiles that we shall examine are expressed in terms of the axis system adopted for the ship as a whole. That is to say the origin is located at the point of trisection of the undisturbed water plane, the vertical plane of port and starboard symmetry and the stern, OZ being positive upwards and OX pointing in the direction of forward motion.

20

10

and

TREATMENT OF DISCONTINUITIES

It is obvious from inspection of the set of parametric equations (5a) and (5b), that all higher derivatives are continuous for all values of O. Therefore, in principle, only smooth and continuous contours can be represented by such a transformation. In particular the tangents to the contour at the keel and the waterline must necessarily be horizontal and vertical

respec-tively since

ax ir

=O atO=,

/

/

Discont,nuities

Figure 3. Discontinuities occurring at maximum beam and draught in a double ship section

mapping techniques are suitable for sections where discontinuities occur (e.g. when there is a hard chine). In practice, however, there is no serious problem since the potential flow solution is only applicable at some distance from a discontinuity. In fact, equations (5a) and (5b) are the

Fourier series of the piecewise continuous functions x = x(4'), y = y(), and so long

as

sufficient terms are included a discontinuous contour may be represented adequately. A consequence of truncating the series to a finite number of terms is that corners are necessarily rounded.

It is of interest to note that the given offsets need not be specified at equal spacing along the hull contour. For a better description of the contour in the region of rapid change of curvature and discontinuity, closely spaced points are required.

Alternative ways of treating discontinuities by successive transformation or modified

Gershgorin integral equations are discussed by Landweber'3 and by Landweber and

Macagno.5 From an engineering point of view, however, the method proposed in this paper appears more than satisfactory when employed with the sections of actual ships, and is much less expensive of time and effort.

ao at O =0

A sloping side and keel are a form of discontinuity when one considers the double ship section as sketched in Figure 3 and it is therefore a matter of speculation whether or not conformal

POTENTIAL FLOW SOLUTION

The total velocity potential 4)T(X, y, t) for the heaving cylinder may be expressed as the sum of

a series of multipole potentials, symmetrical about O = O, and a pulsating source located at the origin. That is

gb r - r

4TIPc+

Pzm4)2mIosct)t+Is+

q2m4)2m]sinú.t} (6)

irctL

J L m1

where b is the amplitude of the generated waves at an infinite distance away from the oscillation and for shapes defined by (5a) and (5b), 4)2n, sour are given by Porter9 and

Tasai12 as

cos2mO cos(2m-1)O N

_{(-1)"'(2n-1)a2n_icos(2m+2n-1)O}}

4)2m - m

+Kaj(2

- 1)m1

+ (2m +2n - 1)m+2h_1

source= 1c cos wt + Sin wt,

= ir e' cos Kx

''

eX

ir

sinKx_J

_{K2+z,2' cos ¡'yK sin vy)dv,}

K =2/g is the wave number and = 1.

The quantities 4) are constructed so that they satisfy

the linearized free surface condition, the no flow condition at the bottom and the radiation condition at infinity.

By satisfying the boundary condition at a number of points along the section contour, Laplace's equation can be solved. The infinite series (6) is truncated to a finite number of terms

M (i.e. i

m M) and by setting up sufficient equations through the boundary condition at the contour the unknown Fourier coefficients q2,,. can be solved in a least square sense. The velocity potential may now be determined. By means of Bernoulli's equation, the fluid pressure at the boundary of the section may be calculated from which the added mass and damping coefficients can be determined.

At high frequencies, the added mass approaches an asymptotic value given by Landweber and Macagno5 a

1 2

m=pira

{(1+ai)2+

(a2n+i)2(2n+1)}

,t1

The added mass rn(S) of a section may be non-dimensionalized by the factor irpB2/8 such that

Cv

m(S)/

where

23

The damping coefficient is expressed as a function of the ratio of the amplitude of the generated wave to the amplitude of the heaving oscillation in the form

N(ö)=ÇA2

(L)

A brief summary of the analysis is included in Appendix I.

NUMERICAL CONFORMAL TRANSFORMATION

Let the section outline be represented by P points spacednot necessarily equally spaced-around the contour starting from the keel, in such a way as to best describe the section shape (see Figure 4). Due to symmetry, only half of the submerged section need be considered.

o (x _Y)

Figure 4. Offset data description

Let 01, 0, . . .

, Ori,

0, be the corresponding angles in the i-plane for each of the P points.

When the co-ordinates (x9, y,,) are substituted into equations (Sa) and (5b), 2F-2 equations

are found, since x1, Y,, generate trivial equations. These equations are in the N+P i

If P=N+1, the set of 2F-2 simultaneous non-linear equations

_{may be solved by the}

NewtonRaphson method for the sets of quantities a and 9. Since the series is truncated at the Nth term, however, solutions obtained in this way may lead to a poor result, in the sense that the contour so generated can only be expected to pass through the P data points, whereas

substantial irregularities may occur between them as suggested in Figure 5. This defect is easily

overcome by increasing P so that p > N + 1; then the set of 2F-2 equations can be solved in a least square sense. A smooth profile is thus ensured though it may not necessarily pass through all the data points. Equations (7a) and (7b) are solved by Peckham's method2' which is unknowns a, a1...a2_1, 02, 93,. . . _, satisfying the equations

xp=a[sin

0+

r N

(_1Y''a28_isin(2n-1)0]

_p=2,.

.

. ,P

(7a)

y,, = a[cos 0, +

Figure 5. Possibk genera ed section passing through offset points but not describing profi'e

essentially a variation of the GaussNewton method but does not require the calculation of the Jacobian matrix. The accuracy of the method may be judged by expressing the sum of squares of the distance of the P points from the generated contour as a fraction of the mean radius.

NUMERICAL POTENTIAL FLOW SOLUTION

If the infinite series function (6) is truncated at the Mth term, convergence of -r is dependent

on a suitable choice of M. In general, convergence may be tested by examining the way in which P2m and q2,,, decrease with increasing m. Alternatively an error function s, defined by the expression

s

=(Moa N0ß

may be examined, & M0, N0, a and ß being defined in Appendix II. The relationship M0cs -N0ß = .2/2 is obtained by equating the energy dissipated by the waves to the work done by the cylinder in one cycle of oscillation. Figure 6 illustrates the variation of the error function e over a range of frequencies for a circular section, taking 3 values of M For uniform accuracy, more terms are required at higher frequencies.

HYDRODYNAMIC COEFFICIENTS

The fine and bulbous sections to be referred to here are sections 7 and 6 respectively used by Faltinsen2° in a numerical investigation of Frank's close fit method. The added mass and damping ratio for the two sections are computed for a range of non-dimensional frequency

8(=w2B/2g), using a conventional Lewis form approximation, the multiparameter fit and Frank's method. A comparison is made of the various results obtained.

Fine section

Figure 7(a) shows that the multiparameter conformal mapping method produces a far better generated contour than the Lewis fit which even fails to distinguish the basic characteristics of the original section. The conformal mapping fails to fit the original shape in the region

E x103 30 2-0 IO 00 0ff sels Y(,n) Z(m) 00 -10-00 08 -970 I-2 -840 12 -610 14 -430 22 -210 3-5 -I 20 50 -055 65 -015 7-4 000 0235 M6 MI0 MI5 20 3-0 40 50 6-0 7-0 80 90 lO-O w 812g

Figure 6. Variation of error function with frequency for circular section

y (a)

Figure 7. (a) A fine section determined by the offsets quoted and represented by the marks .. The full line curve is obtained with a 6-parameter fit and the dotted curve is the appropriate Lewis form. (b) The variation of dimensionless added mass with the quantityw2B/2g.The full line curve corresponds to a 6-parameter fit and the dotted curve the Lewis form. The chain dotted curve is found using Frank's method. (c) The variation of A2 withw2B/2g.The chain

dotted curve is found using Frank's method

N Lewis Conformal 0 685 6957 o I-898xI0 -288Ixl0 0 2701x101 2-945d0 I-53lxI0 07 l228xltY2 09 -5-553x102 -3357xIO2 C 0750 0-729

A I 8 6 4 2 I0 08 06 cv 20 6 paran, fit (b)

/

'FronêsmethOd I I I 8 IO 12 4 w 2g Lewis form (c)

Figure 7. (continued from the previous page)

For w2B/2g < 1, the values of non-dimensional added mass calculated by the Frank and the multiparameter conformal methods are in close agreement and are higher than those of the Lewis form fit, as shown in Figure 7(b). In the higher frequency range, too, the Lewis form fit produces lower values than does the conformal fit, but it will be seen that the results derived by Frank's method are then plagued with discontinuities at discrete irregu'ar frequencies. Faltin-sen indicated that this latter defect may be overcome by either increasing the number of line segments with frequency or by interpolation over the irregularities. Significant discrepancies are found in the damping values, as illustrated in Figure 7(c).

lO 2 14

2

2g

04

Bulbous section

The five parameter conformal mapping again produces a sectional shape that is in close agreement with the original, but the Lewis fit shows little resemblance to the original as shown in Figure 8(a). A three parameter conformal transformation gives a better approximation than the Lewis form but fails miserably in the region Z = 80 m to Z = 100 m. The offsets used in the calculation for this and the previous shape were those given by Faltinsen. The number of offsets determines the upper limit of the number of disposable parameters in the conformal mapping. For û2B/2g < 1, the differences in the added mass and damping values shown in Figures 8(b) and (c) respectively are small. But substantial disparities appear at higher frequencies and, while the Frank method provides support for the five parameter fitting it again suffers from discontinuities in the higher frequency range.

CONCLUSIONS

By a suitable choice of the parameters P, N and M the hydrodynamic coefficients for a variety of shapes may be determined to any required accuracy, using multipole expansion and conformal mapping techniques. Bulbous- and fine-sections can be treated provided sufficient data points are available. Other sections, too, can be dealt with in a like manner; a hard chine, a triangular or a rectangular section all submit to this approach whereas they effectively defeat the Lewis form technique. In all cases it is preferable to have ample data points near regions of rapidly changing slope.

As for the hydrodynamic coefficients, better accuracy is generally obtained in the lower frequency region and

with larger values of M

0ff sels Y(m> Z(m) o o -io o 5 -97 23 -8 4 30 -61 3.3 4.3 43 -21

74 00

as -2 -4 -6 -8 Io f

4.

o i y /V Lewis Cociformoi 00 7266 01 -1 789xI0 -3 08xi0 03 i973xI0 I 200xI0 05 I093x101 a7 48635102 09 2407x102 C 0763 0 611 (a)

Figure 8. (a) A bulbous section determined by the offsets quoted and represented by the marks .. The full line curve is obtained with a 5-parameter fit and the chain dotted curve with a 3-parameter fit. The dotted curve is the appropriate Lewis form. (b) The variation of dimensionless added mass with the quantity w2B/2g. The full line corresponds to the

5-parameter fit and the dotted curve to the Lewis form, while the chain iotted curve is found using Frank's method. (c) The variation of A2 with co2B/2g. The chain dotted curve is found using Frank's method

A 2 26 24 22 20 I8 16 '4 2 Io 8 6 4 2 cv IB 2 4 6 5 parom. fit j"". Franks method I

---:'---'I

Lewis form

\

8 0 12 (4

ja

2g

(e)

Figure 8. (continued from the previous page)

The former may be attributed to the slower convergence of the source potential at higher frequencies. The Lewis form approximation and the multiparameter fit give significantly different hydrodynamic coefficient values over the entire frequency range.

Frank's close fit method and the multiparameter fit method may both be used for general, ill-conditioned sections. The former imposes little restriction on the geometry of the section and poses little difficulty in the calculation of the hydrodynamic coefficients for submerged or asymmetric sections. But the method breaks down at certain discrete freq'uencies as a consequence of breaking the section down into linear segments. The behaviour of the hydro-dynamic coefficients at these irregular frequencies is illustrated in Figures 7 and 8. The

16 '.4 12 I.ò 08 Frank's mettod LewiS form 06 /

- -

* Asymptotic value 5 porom. fit 04

\,

--:

0 2 ./

/

ySj

8 IO 12 14_{we B} ap (b)

position may be improved by increasing the number of line segments or by suitable inter-polation. Computation based on this method can be very time consuming, however.

The multipole expansion and conformal mapping techniques, as presented here, do not cater for submerged or asymmetric sections. But in principle, similar approaches may be suitably modified; e.g. see Ogilvie23 on submerged cylinders and MimeThomson' on

asym-metric aerofoils. The method produces continuous hydrodynamic data over the entire

frequency range since the section is approximated by a smooth and continuous profile. Computation is generally faster in that the transformation coefficients are, frequency indepen-dent and therefore need only be computed once.

APPENDIX I

Potential flow analysis

Following de Jong,'7 the various velocity potential components and their corresponding conjugate stream functions at the contour boundary ( = 1) are given by

cos (O,) . 2mOr+Ka Ç1'2m 51fl -Ky COS Kx sin 29 cos cos (2m-1)Or _N

(2n-1)a2_,

(2m+2n-1)6.

sin sin +

(-1)"'

2m-1

_,,,

(2m+2n-1)

I cos sin

\

(v

,

vv+K

uy)

(6)

=±lre

-Ky Sfl Kx e sin cos

COS Jo K2±v2 dv

Numerical computations of 1s and 'I's as shown above converge very slowly, (see Porter9) and an alternative series form may be used:

-Ky COS Kx ± U Kxl

(Or)e

L sin cos _J where (Kr)"

Qv+logKr+ z

cosn

n=L n!n (Kr)"

U+z

_{ni n!n}

sinnx

In this expression y = O'577215664901 is the Euler constant, and

-1

x=tan

-,

y r =

'J(x2+y2)

When the series is truncated to M terms and the boundary condition is satisfied at R points on the contour (which have corresponding angles Or,. .. , On.. ., in the -p1ane), the

following linear equations are obtained:

m1 P2nf2m (Or) =

'l'c(0)

4J!1 i'()

q2,J2,,,(0) = S(9r)

m B0 21

where Bo=x(O=rr/2)=05B and

x(O,)

-

¿I'2m(Or)

f2rn (Or) = B0

for 0< Or<r/2, r- 1,2,..

These equations may be

where F {Fr,n} is a matrix

and

,R.

conveniently expressed in a matrix form F(p, q)= ('I'c 'l's) of order R X M, p={p2,p4,. . q ={qz, q4, .. . ,q2M},

v(O1)-

i')]

c(OR)_x(0c()t

B0

2)

x (Or) B0 2 "C = x(OR) Iir\

)

B0 \21 Frm f2rn(Or).

If R is chosen such that R > M, equation (9) may be solved in a least square sense for the quantities p and q. Hence the total velocity potential and the applied fluid forces may be

calculated. By separating in-phase and quadrature components, expressions may be found for the added mass and damping coefficients; they are

M0ß + N0a 2

2 2 2pBo,

a +ß

N(S)

Moa

2pBo.

Further, the ratio of the generated wave amplitude to the amplitude of the motion is given by

A-

KBair

In these results, M0= f aM(Or)W(OT)dO i0 B0 ,r/2 N0=

I

aN(Or)W(Or)dO B0 N

W(Or)CO5 Or+

(-1)'1(2n - 1)a2_i cos (2n-1)8,.,

n1

M

M(9r)I(O,)+

q2,t2,fl(9), m-1 M

N(O,)I'c(6,)+

_ml

a i'c()

=

- +

M _P2rnII2rn(H m=1 2ì /3 = +

The non-dimensional added mass Cv and frequency parameter 8 are defined by: m(8)

C=

2

rrpB /8

5w2BKB

2g 2

By equating the work done by the cylinder in one cycle of oscillation to the energy radiated by the generated waves, it is found that

M0a - N0ß = (10)

Hence the damping coefficient may be expressed as a function of the damping ratio A; that is,

pBûrn-2 pg2 2

N-2 2-3A.

_a+/3

_w

Equation (10) therefore provides a check on the accuracy of the solution. APPENDIX II

Notation

a, = transformation coefficient

b = amplitude of generated wave at infinite distance g = acceleration due to gravity

i = .J(- 1)

m(&)= added mass per unit length

Pzm, q2,,, = expansion coefficients

r = distance from origin in physical plane w = complex potential

x, y = co-ordinates in the physical plane

z physical plane

A = ratio of amplitude of generated waves to amplitude of heaving motion B = section beam

C,. = complex transformation coefficients

H = half beam to draught ratio

K wave number (=w2/g)

M = number of expansion coefficients N(S) = damping coefficient

N = number of transformation coefficients P = number of data points

R = number of points along the contour where flow condition at boundary is satisfied S = section area

T = section draught = Euler constant

C= transformed plane

= co-ordinates in the transformed plane = angles in the z -plane

= angles in the e-plane

CI = velocity potential

= conjugate stream function

= area coefficient

= circular wave frequency i' = dummy parameter

p = density of fluid

S = non-dimensional frequency parameter = ø2B/2g e = accuracy parameter

Axis system

OXYZ = equilibrium axis of the ship

Oxy local section axis system

corresponding axis for the transformed section

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