The flow around ship sections in waves

TECHNISCHE UHWERSITET Laboratorium voor

The Flow Around Ship Sections in

Waves sheepsiydromechanIca

frrch!ef

Publication of the onderforschungsbereich Schifstechnik und Schiffbau

1. The Problem

Ship motions and loads on the ship structure innatural sea-ways may be calculated by integrating over the ship length

the hydrostatic and hydrodynamic forces and moments exerted

on the different ship sections by regular waves. Usually, the flow around ship sections is approximated as a superposition

of a constant longitudinal flow due to the ship's speed. and a periodical two-dimensional flow in the section planes due to the wave action and the periodical ship motions. in 1949, Ursell [1] published a method for calculating this

two-dimen-sional periodical flow around circular sections oscillating at

the free surface with arbitrary frequencies. On the other hand,

in 1950, Wendel [2] developed a method forarbitrary section shapes, but only for high frequencies (vibrations). A method suitable for arbitrary frequencies and for a variety of section

forms (for Lewis sections) was given in 1953 by Grim [3].

Re-cently, essential improvements of this method are being

deve-loped [4. 5, 6]. Their goal is:

Suitability for complicated section shapes and for sec-tions with extremely low or high breadth-to-draft ratio. This would allow to tackle for instance the following problems:

in-fluence of bulbous bow on seakeeping, additional resistance and loads in the seaway; design of floating bodies with mini-mum motions in waves: design of section forms for "floating harbours' for optimum leeward wave damping; computation

of rolling motions of heeled ships for determining the necessary transverse stability.

Suitability for determining the exact pressure distri-bution over the section. This is necessary for the computation of the stress distribution in the bottom and the transverse frames of ships. It will also allow to calculate and optimize the response amplitude operator of shipborn wave recorders

which depend on pressure measurements.

For these problems, more sophisticated methods have to be applied than for the determination of hydrodynamic forces

only on well-proportioned sections. The method developed by

Maeda [4] seems to be suitable on principle for all the above listed problems. The method described in the following aims at equal accuracy and wide applicability with only a fraction

of computer time being required. This diminution of compu-ter time seems to be indispensable for complex optimization procedures of the seakeeping behaviour in natural, short-crested seaways. The method is in effect an extension of Grim's method.

2. Potential Flow Types Used for the Solution

The goal is the determination of the two-dimensional poten-tial flow around a body (ship section) which heaves, sways or

rolls sinusoidally at the fluid surface, and the flow around a fixed body in regular surface waves. The amplitude of the body motions and of the waves are assumed to he small corn-pared with the body dimensions and the wave length; the fluid is assumed to be incompressible and without viscosity and rotation. For a wide range of applications, the validity of these assumptions has been established: in other cases, for instance for calculating extreme values of motions and loads

in severe sea conditions or for swaying motions of ships, some adjustments appear advisable.

In the following, tite x-axis is assumed to coincide with the mean fluid surface; the y-axis points downward. The flow will

Mekeiweg 2, 2628 CD De!ft

By H. Söding 'T'eLO157ß6873-FaXO15-.781838

be characterized by its potential (x, y, t); in the case of a fixed body in waves, denotes the difference between the

actual potential and the potential of the wave without anybody

present. has to meet the continuity equation =O

as well as the following boundary conditions: Downward fading of the flow potential:

That implies the assumption of an unlimited fluid depth. Waves running away from the body in great distance at the fluid surface:

ILm 4(x,O,t) Acos(ut- kx +c

ftm +(xO,t)3tos(utkxß)

with positive w and k and arbitrary A, B, a and 3. (The wave amplitudes A and B may be zero.)

Constant pressure at the fluid surface. From titis condition

follows:

1x0. t) - c _{lx, 0, t) O}

with g = acceleration of gravity.

No fluid penetrating the body surface. This condition will

be dealt with later on.

Just as the exciting body or wave motion, the flow potential

and the associated stream function P will be sinusoidal

functions of time:

The two symbols one above the other are used to indicate that two equations may be generated by taking either the upper or the lower symbol of each pair. U is a real constant;

wis the circular frequency of the body or wave motion: and

W are complex functions of x and y. Instead of these coordi-nates, in the following the dimensionless values X = kx and

Y =ky will be used frequently. k

=

w0/g is the wave number.

There exist several types of functions and W such that

the corresponding function satisfies the equation of conti-nuity and the above listed boundary conditions i to 3. Such

types of functions are:

dc 7-icr (KX

- urn e i,..lcx)

r=

This potential function has been used frequently for related problems. It corresponds to a flow symmetrical to the y-axis and may he interpreted as the effect of a periodical vertical force acting on the water at the point (0,0). For X - ± oc, regular waves leaving the point of excitation are generated. The function may be computed according to Grini [10] from the polar coordinates R and (Fig. 1) in the following way:

,where sn,(n) fo R6 [CR)

i.iL

costp)) _sgn(x)1re [cix]

'6

SR suntopi 1c os X) (1) - 9 - Schiffstechnik Bd. 20 - 1H3 - Heft 99

Fig. I

The antimetric counterpart of the foregoing potential.

used already by Tamura [8] for similar calculations:

th (x 1 e-er sw-cKX)_dK

-1

VbJ_{I jis +}

+ r

J -C) t-cj

A generalization of a potential used by Grim:

f((x-x-ï),

Y-L -21x-X)(Y-Y) ) 1- X) where ((x-x,.,)- (y-,. (r -H -2(X-x,)(Y,j

ix-L)

- (x-X,,,)2+ C Y- r, (o- x,J+ (y+'r.,)

This potential has two singularities: one at an arbitrary point Xm, Y15, the other at the point X

, -

Y5. The singu-larities are quadrupoles: In the horizontal direction fluid is sucked in, while in the vertical direction fluid is ejected. The function has real values only and generates no waves far off

from the singularity.

The proof that these functions satisfy the condition of con-tinuity and the boundary conditions 1-3 is presented for the potential types i and 2 in [7] and [9]; for type 3, the validity

of these conditions may be confirmed by simple arithmetic.

3. General Formula for the Flow Potential

For the description of the periodical flow around ship sec-tions, a linear combination of the three potential types listed

above is chosen:

A1.J Bt -i1)B).

Obviously, this expression satisfies also the condition

of

continuity and the boundary conditions 1-3 for

ar-bitrary complex constants m' A and B. The boundary

condi-tion on the body surface will be met by an appropriate choice

for these constants.

The reasonig behind formula (2) is as follows: In most cases,

a body oscillating at the free surface will generate surface

waves far off from the body the amplitudes and phase relations of which on the left and right side depend on its section form. As with a linear complex combination of the potential types 1

and 2 arbitrary amplitudes and phase relations of the waves

far oíl from the singularity may be obtained, the two potentials

of type i and 2 (or similar potentials) seem to be necessary

and sufficient to describe the wave field far-off from the body. However, to describe the flow field in the vicinity of the body more exactly, the wave-free potentials of type 3 seem more

ad-visable than, for instance, extensions of the types i and 2, as type 3 potentials require only a small fraction of computer

time for their numerical evaluation. Contrary to the wave-free

potentials with singularities in the mean fluid surface, which were used by Grim and which may be computed even faster, the singularities of type 3 may be arranged near the body sur-face. As in many other potential flow problems, this is a

pre-(2)

Schiff stechnik Bd. 20 - 1973 - Heft 99 ₁₀

-requisition for accurately satisfying the boundary condition

on complicated contours.

Fig. 2

Therefore, the quadrupoles generating the flow potential of

type 3 are arranged on a sequence of straight lines roughly parallel to the body surface (Fig. 2). The distance between the singularities and the body contour may be about 1.5 to 3 times the distance between the singularities themselves. A

great number of such singularities is necessary to prevent oscil-lations of the flow along the section contour, and corresponding

oscillations of the pressure distribution. For such an arrangement, the intensities of the singularities the factors 0m -will varr from one quadrupole to the next one in a somewhat fair manner. Therefore, the accuracy with which the

boun-dary condition on the section contour can be satisfied, will not

essentially be impaired, if only some of the values Um - per-haps every fifth one - is computed from the boundary condi.

tion assuming that the intensities of the other singularities may be interpolated linearly between the computed ones. The poten.

tials of such a "fifth" singularity and the 2 times 4 singulari-ties adjacent on each side of the straight line sequence are combined to one single potential function in the following

manner:

That means: The intensities of the nine singularities com-bined to a single potential function decrease linearly from the "middle" singularity k until zero for the (k 5)th and

(k + S)th singularity. (Naturally, for k i or k = M, the sums contain 5 terms only with m = k until k ± 4 or k 4 until k, respectively.) Then, a linear combination of the

poten-tial functions I with arbitrary complex constants a5 is iden-tical with a linear combination of the functions m such that

the intensities of the singularities vary linearly between thc

first, 6th, lS singularity. The general form of the polen.

tial then becomes N

'PI-

_{QS1JtAíJ+B[).}

w

By proper definition of the terms with indices N ± i and N + 2, one obtains:

1J

2°:J

(3)

Instead of using a set of 9 discrete singularities for one po-tential IJ), a continuous line singularity of quadrupoles could

also be used. However, the numerical effort would increase

substantially without essential improvement of the solution.

4. Boundary Conditions on the Body Surface

For a two-dimensional potential flow, the volume per time

interval and per fluid depth (measured perpendicular to the x.y.plane) passing between two fixed points P and k equals

the difference of the stream function at these two points. For the stream functions given above, 'Pk - WØ (the index here designates the respective point) is positive, if - looking from P0 to P - the flow is directed to the left.

X X. y wove prapQtiOfl y Contour Ct Tirr t

Body Contour at Time t.t

Re(Uev)t)At Fig. 3

Fig. 3 shows a part of the surface of a body heaving

periodi-cally with the velocity Re (Uewt) (positive downward). Between

the two fixed points P0 (x0, y0) and Pk (xk, yk) on the body surface at time t, during the small time interval At a volume

per depth of Re (Uelu)t) -At (XoXk) is pushed through. There-fore, the flow caused by the heaving motion satisfies the condi-tion

With (1), there follows the boundary condition on the body sirf ace for the heaving motion with velocity amplitude U:

k5

For a body swaying horizontally with the velocity Re (U e»t) (positive in the x-direction), by similar reasoning, there follows the boundary condition on the body surface,

k-4

For rolling motions of the section around the origin of the coordinate system with the angular velocity Re (Uekut)

(posi-tive, if directed according to the arrow in Fig. 3), one obtains

the boundary condition

---

-Ì1?

with rk and r0 designating the distances of Pk and P) from

the centre of

rotation. - As the motion amplitudes are

assumed to be small, the points P0 and k may be arranged on

the mean body surface instead of on the momentary surface.

Further, the boundary condition for a fixed body in a

regu-lar wave has to be determined. The wave with amplitude h and

circular frequency w is assumed to propagate according to

Fig. 4 in the direction u. Without any disturbance by the body, the potential of the wave would be

tf. !Re[exp(-r-- i{Xsnj -cosj.

FIg. 4

Between the two points P0 and Pk. this potential delivers

per time intervall and per depth in z-direction thefluid volume

_4

dx +

y

Here, the indices x and y designate partial derivatives; the integrations have to be performed along the body contour in

the x-y-plane. (As the wave constitutes a threedimensional flow,

in this case no stream function exists.) The presence of the body produces a secondary flow , which - together with prevents the penetration of fluid through the body

sur-face. The (two-dimensional, as assumed) potential Ï' alone,

therefore, pushes the same volume per time and depth through

the body surface between P0 and k as but in the oppo-site direction:

jdx-dy

By inserting the potential . as given above and defining

U

=

h w Re (e_iZ COO ) (U is the vertical velocity of orbital

motion in the fluid surface at the point (0,0, z) or t

=

0). with equation (1) one obtains the following boundary condition on

the body surface:

nj'exp(X0m

dy.

In the following, the right-hand sides of all these boundary conditions on the body surface are abbreviated as R.k: j de-fines the mode of motion (j

=

1: heaving, j =2: swaying,

j = 3: rolling, i 4 until J: fixed body in waves of

direc-tion i); k designates the respective contour point k As

R

=

0, the boundary conditions may be written as

5. Equations Defining the Coefficients of the Flow

Potential

A simple method for the determination of the coefficients a, of equation (3) results from satisfying the boundary equations (4) on the body surface at N ± 2 points on the body contour:

k 1.., (5)

Here, the first index of the coefficients a , designates the

respective mode of motion. The presentation of this system

of N ± 2 linear equations and the calculation of its coefficients may be simplified by expanding it to N ± 3 equations:

k'.O,.., (6)

The stream function W, - appearing here for the first time - is defined as i in the whole x-y-plane, while the

cor-responding potential I is set to 0. This type of potential satis-fies the conditions given in chapter 2; it lacks physical meaning, as it describes a motionless flunid. That the systems (5) and (6) are equivalent becomes obvious if one subtracts in the system

(6) the first equation (k

=

0) from all the other ones.

Practical application of the system (6) for the determination of the a shows that, albeit the boundary conditions hold

exactly at the points k, between these points there are often

very large deviations. That means, the flow defined by these coefficients a belongs to a body contour passing with wide

oscillations through the points k While this may have only minor influence on the overall forces and on the moment exerted on the body, the pressure distribution is greatly in error. These oscillations, however, can be eliminated, if the boundary conditions are satisfied approximately only on a

greater number K of contour points. K should be not less than

about twice the number of coefficients a,). The best fit is de-- 11 de-- Schiffstechnik Bd. 20 1273 - Heft 99

fined here as the set of an which minimises the sum S of the squares of deviations from the boundary conditions at the K

points on the body surface:

K '2

The minimum of S is found in the usual manner by requiring that the partial derivatives of S to the coefficients a are zero.

If real and imaginary parts of complex values are characterized by indices r and i, respectively, this condition results in

--- ='!QL ,A,,

B- C- 0,

n- O...w.z ,ry .o 2 j,L,L AL,- =0, n - 0,..., Nr2, (7) where =&

-C 4 (j,4fl + R,j',3) D, +Rh,j'n,)

This is a system of 2 (N + 3) linear equations with J dif-ferent "right-hand sides" C and D , from which the real

quantities a and ai li may be determined. In many cases,

systems developed like this one from the condition of minïmu.m

sum of deviations squared are strongly ill-conditioned. In this case, however, for N up to at least 30 (that is, more than 60 equations), no numerical difficulties arise, because the

stream functions of the quadrupoles assume substantial values only at a few contour points and different ones for different quadrupoles. So, the elements in the neighbourhood of the main

diagonal of the coefficient matrix are much greater than the

other elements.

6. Inversion of the System of Linear Equations*)

The stream functions are real-valued for n = O until N;

only and 1N0 are complex. Therefore, the coefficients

Bin are zero if 1 as well as n are< N. To obtain continuous blocks of zero coefficients, the equations (7) and the unknowns

in each equation are rearranged as shown in the following

matrix notation:

A0 . .

. A0

O . . - o A,..10 B4O z,e

A,

. O

AN AN,O

B,,,.

.0-

0

IA. .

. . A_{N,0 N,r, n} -

A.

ANr 1,0 O . . . Ô I Ao . .

.A

A,

A0

N3l,4ri BNI,N+2 E.1, n+ AN,l,Ñ+I A0,4+, AN.,r..2 (8) 12

-The matrices will now be partitioned along the indicated

lines; the submatrices are designated as follows (0 = zero

matrix)

t_t, C? A2 , R,

Cr C.!,C!3 k.

a4 (A a

By performing the matrix multiplication on the left-hand

side explicitely, one obtains:

ot =

Vt, B3 r a3 30,

014 30, rCl,B+

From the first and second of these equations follows:

i (9)

(10)

Substituting these expressions into the third equation results

in

f3=(-ot*a,cz- (..eO(0t * Oî'-(,-o'R,-ut,or'R3) (11)

To demonstrate the matrix operations necessary for the

nu-merical evaluation more clearly, new matrices until are

defined by

(12)

Substituting these into equations (11), (9) and (10) gives

'f (l - Ot,l,

Ï

0t1. tY,.51j (13)

Instead of inverting a system of 2 (N + 3) equations, the

system (12) of N + 1 equations is solved first. With the results 2),the coefficients of the 4 times 4 system (13) are computed. From the solution of this system, the matrices and

con-taining the rest of the coefficients

and a0

are found

by simple multiplications and additions according to (14) and

(15).

With the coefficients anr and the potential (I in

equation (3) is calculated at the given contour points for the

different modes of motion. From this potential. the pressure at the contour points,

and by integration along the section contour, the vertical and horizontal forces V and H and the moment M per zylinder

length due to the potential I. are easily obtained.

') The method gtven here was deveLoped independently for a

similar problem by Dipl-Ing. Seiffert, HSVA.

A - -3. Bo,W2.. -BON.-1 I AN,.z B,,...2 ... BN,Nr2 N,N* i

A,rri ... A, r i

BN, r1+l I An,Ni-2-- Ae,Nr2 I A,, ,.+., ANN., A,.1 NI-3 AN,lN.2

i - 3N+lnl 3N,l,N,2

Br.,rri

"a1 O r . . a O fc1,0 C3,0 04 O,,. . . a7, N r a QL C.l,N D10 C3, D7,0 a1, N2_ - --N-lIT . . Q,NIt,T .. i, . a3 . a3 NrZ,T DIN

C,,ri

NrZ D C3,Nr, C , \ a1W+2. . . . . D, N+1 D,,N.Z D3, ,_, D30+z, Schiflstechnik Bd. 20 - 1973 Heft 99

In the following, the indices r and i used in connection with

the quantities p, V, H and M designate the values at wt = (xtI2)

or wt = 0, respectively; that means, r indicatesvalues in phase

with the body or wave amplitude h, i indicates values in phase

with the velocity U = wh.

7. Examples of Results

The variations of the results obtained for the same section

with different numbers and arrangements of contour points k

and different singularity arrangements may be regarded as an indicator for the accuracy of the results. Fig. 5 shows such

results for a Lewis section, calculated with N 7 to 17

(with-Out making use of the symmetry of the section) with various contour point arrangements between 10 and 19 points on either

side of the section. In one such arrangement, the origin ofthe

coordinate system and with it the position of the singularity of the potentials 4A and (I was shifted from the center line to point 0. The results obtained for all these arrangements fell within the hatched areas. The broken lines indicate results ob-tained by Maeda [4] for the same section. The curves show the following dimensionless quantities:

,'ydbo ,e. lo,,q#h 8 V,

K0- _{"/2 - (3,zJ°} ,9P° 8.

for heaving motion;

o ol j_kIl.

A _{QUto0 'I body ".01...'} 590

for heaving motion;

ei. _h(B/2) and ei.'9h''s,o

for waves running perpendicular to the cylinder axis. V.K

de-signates the Froude-Kriloff force on the body, that is, the force

caused by the undisturbed pressure fIeld of the wave.

0'. 0.6

Fig. 5

Fig. 6 shows for the same section the pressure distribution

for heaving motion, including hydrostatic head.

FIg. 6

01

Ob

02

Fig. S

Fig. 9 shows sorne results for the horizontal forces and

mo-ments on a swaying and rolling Lewis section. The broken

lines are results obtained for the same section by Tamura [8].

Tho symbols used in the figure are defined as follows: 000.4 048 p0, l..'qlh H,

for swaying motion:

addtd of i,,e-t'e Fr Lenqtfl 8M,.

K4 - 5.,. rb F90.o' 4 - 13 - Schiftstechrsik Bd. 20 1973 Heft 99 'A 9//20.020 p.0644 ' _e' e'5

y6;/

J

V.'

V-(3.0907

\i1

, 0'. as Fig. 7

Fig. 7 is a comparison of the above defined quantities

cal-culated by Maeda [4] (broken lines) and tise author, for a much more "difficult" Lewis section. Fig. 8 shows the dimen-sionless vertical force

excited by regular waves on a so-called wave excitationless section form; that is a section form on

whichtheoretically---waves of a certain frequency do not exert anyvertical force. From the differences between the results calculated by Maeda (broken lines) and the author, and the measured values, it may be concluded that the numerical stability of the two calculation

methods is quite good; however, the common assumption of

potential flow seems to impair the results for such sections, in which wake formation has to be expected.

ai al 03 C. .0 e as l-a ai 0.6 0.2 .0 0.8 0.6 54 0.

I'

02

o'

for rolling motion;

A6

for swaying motion;

006 004 002 000 002 -GOL ygOB 00 -0l -02 0.3 O,opL.vde Of ,vd,oId -'000

OmpI.IvO,of body ,ofltoO

a,eS&t000 of .-odst.d osoo J -k t1

A6 O,ept4:ve.00v-rpOvu

j h(8/2)

for rolling motion;

ever 01 ogLed "on loo" Ooot.,Ii3O N, IrT

for swaying motion.

Figs. 10-13 show the hydrodynamic force F2 parallel to the bottom line and the hydrodvnamic moment M due to heaving motions (Figs. 10 and 11) or longitudinally running waves (Figs. 12 and 13), respectively, for a symmetrical

s-tion heeled to 200. _{The section is shaped like a ship's midship}

section with bilge radius/B = 0.088; the breadth-to-draft ratio varies between 1.5 and 3.5. If similar calculations are

performed for various sections of a ship and if a suitable strip method for unsymmetrical ship forms is applied, the influence

026 kB/O 06

B/0

of the hydrodynamic interaction between ship and waves on

the moment of the transverse stability may be estimated. From

the values given in Figs. 10-13, the influence of this inter-action seems to be at least in the range of one decimeter of stability lever and may be much greater. Calculations on this influence are among the applications envisioned for this

pro-gram. 0g 2çhB alo 0,00 025 oso Fig. 11 Fig. 12

u--u

ff/

Loee voovee 076 IX k BIO l-50 b B/O 8/2 kB/O 1.60 i4

'-u

44U

B/loll. 06 0e

-p

A6 B/I OBI OX oto 020 OIS i 2.5 os 025 050 0.75 .03 525 1.5 00 os

205--

F,

Bj l*Ovng 'fOSan_vl.O

I 026 050 C'S tOO 25 .5 R9i 000 002 -000 002 -05k 025 02.0 075 1.00 .25 1.02

w

M LOOMICI 00006 'e/O IS - SII 025 0,60 675 00 525 Il

Schiff stechnik B. 20 - 1973 - Heft 99 _{- 14}

-Fig. 30 Fig. 13 020 000 0.76 l-00 .25 52 03 Fig. 9 too 025 000 075 25 0,75 loo 025 150 025 12 lo ce 0.6 04

8. Acknowledgment

The program developed for these calculations is an exten-sion of a program made by Dipl.-Ing. K.-P. Beier, Berlin. whose accurate work on this subject is highly appreciated.

9. List of Symbols

A, B amplitudes of symmetrical or antimetrical, resp..

radiation potential (complex)

A n,B. n1_{see after equation (7)} i,n, i.e J

AS, AR amplitude ratio between radiated wave and body motion for heaving, swaying or rolling, resp. a5 , a constants for the superposition of. flow potentials

(complex)

CR, C1 real and imaginary parts of

e', e'c, e'5 coefficients of vertical force in waves F1, F5 forces perpendicular or parallel, resp., to ship

bottom

Hr, H1 horizontal force on the body not including static and Froude-Kriloff force

h wave or body motion amplitude

= V i

i index for imaginary part of complex values or

for component in phase with velocity

j number or index, resp., of body or wave motions

K integration variable

KH, K5, KR added mass coefficients for heaving, swaying or rolling, resp.

k number or index, resp., of contour pointa

k wave number w/g

LS lever of added mass for swaying I, n indices of the potentials 'D

Mr, M1 moment around z-axis not including Static and Froude-Knloff moment

m number or index, resp., of the potentials 'D n number or index, resp., of potentials 'D

k points on body contour

R =kr

R k right-hand side of boundary conditions

r distance from the origin of the coordinate system

r index for real part of complex values or for

com-ponent in phase with motion amplitude S sum of deviations squared from the boundary

condition

SR, S1 real and imaginary parts of WA

if velocity amplitude

Vr V vertical force on the body not including static

and Froude-Kriloff force X, Y, Z kx, ky, kz resp.

x, y, z horizontal and vertical coordinates in flow plane,

and horizontal coordinate perpendicular to flow

plane, resp.

lia.. ,3l matrices of coefficients (equation (8)) IL . .IL matrices of right-hand sides (equation (8))

L.. .3ll matrices of a _n(equation (8)) pt.. . D.. see equation (12)

- 15 - Schiffstechnlk Bd. 20 - 1973 - Heft 99

constants for the superposition of flow potentials (complex)

a. phase angles

fit time interval

i fictive viscosity parameter (real) wave direction

'D, 'V flow potential and stream function, resp. (real;

not including wave potential)

't', 'V amplitudes of 'D or '8', resp. (complex)

4'A' ''A 'DB, 'B amplitudes of flow potential or Stream function,

resp., of symmetrical (index A) or antimetrical

(index B) radiation potential (complex)

'D wave potential

'De, W5 amplitudes of a superposition of several double-quadrupole flow potentials and stream functIons. resp. (real); 'Dy * =

'Dy .., = 'DB (complex);

'D = o, W, = i

= arc tan xiy

amplitudes of flow potential and stream function, resp., of double-quadrupoie flow (real)

10. References

if r a e li, F.: On the Heaving Motion of a Circular Cylinder

on the Surface of a Fluid. The Quarterly Journal of Mechanics and Applied Mathematics 1949.

W e n d e 1, K.: Hydrodynamische Massen und hydrodyna-mische Massenträgheitsmomente. Jahrbuch der Schiffbautech-nischen Gesellschaft 1950.

G r i ni , O.: Berechnung der durch Schwingungen eines

SchifSskörpers erzeugten hydrodynamischen Kräfte. Jahrbuch der Schiff bautechnisc'nen Gesellschaft 1953.

M a e d a , H.: Wave Excitation Forces on Two Dimensional

Ship of Arbitrary Sections. Selected Papers from the Journal

of the Society of Naval Architects of Japan 1971.

F r s n k , W.: Oscillations of Cylinders In or Below the Free

Surface of Deep Fluids. NSRDC Report 2375, October 1967.

R u b io , A. R.: Calculo numerico de los movimentos del

buque de las cargas hidrodinamicas. Ingenieria naval 1972. K i r s c h , M.: Die Berechnung der Bewegungsgrölien der

ge-koppelten Tauch- und Stampfachwingungen nach der

erwei-terten Streifentheorie von Grim und die Berechnung der

Wahrscheinlichkeit für das tYberschreiten bestimmter Schran-ken durch diese Größen. Report 241 of the Institut für

Schiff-bau, Hamburg.

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