Three dimensional potential flow and potential wake

TRANSACTIONS ÖFTHE

DANISH

ACADEMY OF TECHNICAL SCIENCES

A.

195.4

s.

s

No.2

CONTENTS

SVENDAAGE

HARVALD, D.Sc

Three!.Dimensional

Potential Flow and

Potential Wake

AKADEMIET FOR DE TEKNISKE VIDENSKABER

I KOMMÌSSION HOS G. . C. GAD VIMMELSKAFTE.T 32 KØBENHAVN IC.

Tranà. Dan. Acad..

Techa. Sci. No.

2,

1954

AIKADEMI1T

FORDE

TEKNISKE VIDENSKABER

ACA,DM1E

DES SCiENCES TECHNIQUES

AKADEMIE

DER TECHNISCHEN WISSENSCHAFTEN

ACADEMY

OF ÏECHNICAL SCIENCES

KØBENHAVN

DAN MARK

PREFACE

The investigatiöns described in this paper were carried out during stùdies

in the Ship-building Department of The Technical University of Denmark, Copenhagen, and the Ship Model Basin of The Technical University of

Norway, Trndheirn, where the author was a guest for three months in

1950.. The author is indebted to the heads of the two departments,

Profes-sor C,. W. PROHASKA, D. Sc., and ProfesProfes-sor J. K. LUNDE, for advice and assistance.

Besides, the author wishes. to thank the Academy of the Tehnical

Sciences, Copenhagen, for having enabled him to have the paper translated

and published.

The translation into English has been done by messrs. HANS H.

HIN-RICHSEN and HELGE HANSEN, Official Translators, to whom the author's

than1 are due.

Kgs. Lyngby, January 1954.

SVEND AAGE HARVALD.

CONTENTS

Page

A Summary 5

Introduction 6

Chapter I: Three-dimensional Flows

Basis of calculation 7

One source and one sink S

Two sources and two sinks 11

Three sources and three sinks 16

Five sources and fivè sinks 16

25 sources and 25 sinks 19

Lineiform sources and sinks 20

Planiform sources arid sinks 21

Point of stagnation ...24

Chapter II: Potential Wake

Basis of calculation / 25

Potential wake in the case of U-form and in the case of V-form 27

Potential wake in the case of an ellipsoid 30

Comparison of calculated and measured wake coefficients 31

Symbols and units 34

A. SUMMARY

The potential flow about a number of bodies of different forms has been determined with the application of sources and sinks.

In the first instance, the sources and sinks _{were punctiform, but latcr} on it proved expedient to adopt lineiform and planiform soUrces and

sinks.

Furthermore, the velocity distribution in the vicinity of the propeller

was calculated for two forms of ships, viz, a pronounced U-form and a

pronounced V-form.

In order to get an idea of the magnitude of the potential wake with_these

forms of ships, a comparison with the flow about an ellipsôid was made.

o

B. INTRODUCTION

In hi treatise "Wake of Merchant Ships" [2] the auth2r carried out a

number of calculations of the potential flow about cylindrical bodies,

bodies of revolution, and ellipsoids.

The calculation of the two-dimensional flows about the cylindrical

bodies was made with the application of sources and sinks according to

the method developed by RANKINE [3] and later simplified by D. W.

TAYLOR [4]. For the majority of the cylinders used, the sections had the

form of water lines, but also cylinders of effiptic cross-sections wereused.

In the case of the latter forms the velocity distribution was determined

directly, since the development of formulae for the stream function and

the velocity potential function is comparatively simple These forms, afford

good possibilitiès for determining the dependence of the flow on the main dimensions of the form: in this case the axes of the effipses.

Sources and sinks have also been applied in the calculation of the three-dimensional flows about bodies of revolution, the method suggested by

D. W. TAYLOR in 1895 [5] having been applied. Also the three-dimensional potential flows afford possibifities of direct determination of the flow

velo-cities in the vicinity' of certain bodies. This applies to ellipsoids, and it is

therefore possible to compare the two-dimensional flow about effiptic

cylinders and the three-dimensional flow about effipsoids. As such a

corn-parison' shows that the two-dimensional and the three-dimensional flows

are widely different, it would appear to be reasonable to make the three-dimensional flows the subject of a closer study.

Such an investigation can be made by using sources and sinks placed in different ways in the space. The flow between the sources and the sinks is linked up with a rectilinear flow, a parallel stream with constant

direc-tion and velocity, whereby are obtained partly a range of flow where only

flow from sources to sinks takes place and partly a range for the

by-pass-ing flow. Without disturbby-pass-ing the flows, an impermeable shell can be placed between the two ranges of flow. Variations in the source-sink distribution,

its intensity, Or the direction or velocity of the rectilinearflow, will cause

the form of the shell to change. If the flow about some body is to be

de-termined, the source-sink distribution and the rectilinear flow having a

boundary shell which is identical with th surface of the bodywill have to

be determined first.

In this paper the flow js determined first about simplesource-sink

distri-butions and later about more complicated distridistri-butions, the main task

1954

CHAPTER I: THREE-DIMENSIONAL FlOWS

C. BASIS OF CALCULATION

The calculations of the potential flows are based upon the fo1lçwiS

assumptions:

1. The movement takes place in an ideal flow which is incompressible and

homogeneous. Besides, the flow takes place without friction and irro-tationally.

2 The ideal flow extends infinitely in all directions.

As, in thé case of three-dimensional flows, it is :geuierally impossible to

define a stream fimction, the calculations have to be carried out with the use of the velooity potential function.

The velocity potential for a punctiform source (sink) is

Ç01=m/r (C-1)

where m is the strength of the source and r the distance from the source to the points considered. If the source has unit strength, and if it is found

in the pOint (x1, y, z1), the velocity potential will be

i

(C-2)

V(x- x1)2

+

(y -- y1)2

±

(z

--if there are seeî'al sourCes and sinks, the velocity potential will be

(C-3)

V(xx8j2

₊

(y y8)°- ± (zz8)2

the sinks being reckoned to be negative souÈáes.

If the question is one of lineiform sources, i.e. an iñfinite number of

inInitely small soùrces placed on a line section, the potential will be

9 =

- dl (C-4)

If the question is one of planifoim surces, i.e. some surface with an

infimte number of infimtely small sources placed on it, the potential will be

"A

- da

r

(C'5)

ni8

The source-sink fields are combined with the field from a uniform stream

parallel to the X axis with the velocity potential

(C-6)

sc that the final velocity potential function becomes

=1'+92.

'

(C-7)

Based on the course of the surfaces for constant 9 the lines of flow are determined, the latter having a coûrse at right angles to the equipotential

surfaces At the same time the boundary shell between the flow from sources

to sink and the transformed uniform stream are obtained.

The velocity at an arbitrary point is determined by the equation

ax (C-8)

In the subsequent sections, descriptions are given of the practical per formanôe of the various mathematical operations.

D. ONE SOURCE AND ONE SINX

Although, on account of the symmetry, the field about one sourbe and

one sink, both placed on the X axis, can most easily be calculated with

the use Of the stream function, the alculations were made 'with. the use of the elocity potential function This method of calculation was chosen with the object of developing a method which would also be applicable in the case of the complicated distributions.

The source and the siak'were placed at the pOints (60,0,0) and ( 60,0,0), and the potential

il

i.

i

(D-l)

r1 r2

was determinéd at a large number of points in' the X Y piane, sothat the

diagram represented by Fig. i might be constructed, As there is symmetry about the Y axis, 'ç is calculated only for positive x values. The abscissa

in the diagram is the x value of the points, the ordinate is the velocity potential, and the curves correspond to constant y values of the points

To these potentials must be added the potential from the uniform stream

çv2=Ux.

Equipotential surfaces are then obtained by drawing across the curves

to the lineçt2 = Ux The intersections of one of the lines and the ço curves

give a number of points in the X Y plane with equal potential, so that equipotential lines can. be drawn. The field of flow in the X Y plane is

then easily determmed, smce it is merely necessary to place curves winch everywhere intersect the equipotential lines at right angles. For reasons

Fig.. i. Vélocity potential for, a source (60,0,0) and a sink (-60,0,0).

of symmetry, a line of flow in the X Y plane will never leave the latter, and the complete field of flow is obtained by rotatmg the X Y plane about

the X axis, whereby the lines of flow are turned into stirfaôes of flow. Also

the boundary shell is determined by means of the equipotential surfaces,. the line of flow winch has been running in the X axis thvidmg at the pomt

of stagnation so as to enclose the source and the sink, whereby it will

form the boundary shell..

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-By varying the velocity of flow, i.e. by varying U, it is possible to form different fields of flow, but all the boundary shells will be effipsoidal (see Fig. 2, where the ratio of the velocities is 1: 2 :4).

or. 10_ 20 fiò r 2 r 20 X r I o io ,v0 I2X

Fig. 2. Equipotential lines for the flow around a source and a sink at three velocitiès.

Out-hhe of boùndary shell drawn in.

On the basis of the formula ([1] page 208) for the stream function, vhich can be defined for this flow:

çlr = (cos 02 cos O) - r Uy2 (D-2)

'where m is the strength of the source, O the angle between the X axis :and,

the line connecting the source (sink) and an arbitrary point P, U is the velocity of the parallel flow, and y the ördinate of the point P, various

conclusions may be drawn. If the distance between the source and the sink

is put at 2 a and the largest diameter of the boundary shell at 2 b, (D-2)

can be rewritten as under, the stream function assummg the value of nil in the boundary shell:.

mf 2a

\

O =

-

j U b2 . (D-3)

2 \Va2 + b2J

r

m a

V+

b2

U b2. (P-4)

With the application of this expression the following can be inferred: If the strength of the source and the velocity of flow are made n times

as great as originally, a field of flow identical to the original one is

obtained.

If the distance between the source and the sink is made n times longer and the strength of the source n2 times greater, similar fields of flow are obtained.

If a is great in proportion to b, inverse proportionality exists between the width of the shell and the square root of the velocity of flow, and there is direct proportionality between the width of the shell and the

square root of the strength of the source As our principal object is

the treatment of flows about ship-shaped bodies, the length of which

is large in proportion to the width, the condition will geheially be

satisfied.

3 involves that with constant strength cif source and constant velocity

of flow the width of the shell becomes independent of the distance

between sOurce and sink.

From Fig. 2 will be seen that the width of the shell opposite the source is about two-thirds the full width.

E. TWO SOURCES AND TWO SINKS

The next thing to be investigated is the flow about two sources and two

sinks.

First, the sources are placed at the poiíits (60,5,0) and (6Q-5,0) and the sinks at the points (-60,5,0) and (-60,-5,0). As, in the first place, it

is the flow in the X Y plane which is of interest, the course of the flow is

determined only in this plane. As mentioned before, lines of flow in a plane

of symmetry will never leave that plane. Consequently, we need

only-determine the potential at various points of the X Y plane in order to have the flow determined. A diagram in analogy with that in Fig. i is construct-ed In the new diagram, the 9 curve for y = 5 will tend to mfimty when x approaches 60, whilè in Fig. i it is the curve for y = 0.

Fig 3 shows the course of some equipotential lines in the vicinity of the

sources. It will be noted that with the vèloóity çhoséíi foÏ te parallel flow, a flow about two boundary shells is obtained. The. latter are also ellipsoidal for this source-sink distribution, though they are slightly

z-57

Fig. 3. quipotetitial lines in the tY

plane for the flow about 2 sources and

2 sinks. Symmetry about the X axis.

Outline of boundary shells drawn in.

Fig. 5. Velocity potential for 2 sources ((60,0,0) and (64,0,0)) and 2 sinks ((--60,0,0) and (-64,0,0)).

-3

5? A,

Fig. . Equipotential lines in the X Y plan for the flow about 2 sources

and 2 sink'. 3 2 I o -I -2 5- á1

asyinínetrical. The smaller the velocity of the parallel flow is made, the more asymmetrical will the two boundary shells become, and for very

small velocities the two boundary shells merge into one If the twosources

are moved nearer the X axis, somethIng similar will happen, and in Fig. 4

the course of theequipotential lines in the vicinity of the sources is shown

4

r

L

L

iuïiIvahilhicLii

Fig. 6. Equipotential lines for the flOw about 2 sources and 2 sinks at three veloëities of. the

parallel flow.

-for a flow _{about two sources and two sinks placed at (60,1,0), (60,-1,0),}

(-60,1,0) and

(-60,-1,0),

respectively. The two bodies almost merge

into one, but it will still be so that the lines offlowhaving their course

in the XZ plane will remain in this plane, and the X axis will be a line

offlow separating the two bodies.

Next, both sources and sinks are placed on the X axis. The sources are

placed at the points (60,0,0) and (64,0,0) and the sinks at (-60,0,0) and (-64,0,0). The velocity potential is determined at a number of points in the X Y plane, and 9 curves for constant y are constructed (see Fig. 5).

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It will bè notéd that at two places the curves have vertices. With the aid

of these curves, the equipotential lines shown in Fig. 6 are constructed.

The three velocities of the parallel flows have the ratio of 1: 2: 4. Since

all sources and sinks are placed on the X axis, the boundary shells become

bodies of revolution, and for each flow two boundary shells will hèobtained,

Àii,iiiii'

dJIJ&,,a

IdÍf!1!Y'

Fig. 7. Equipotential lines for the flow about 2 sources and 2 sinks at three velocities of the

parallel flow. The strength of one source is twice that of the other.

ohe lying inside the other. The flow främ the source (60,0,0) to the sink (-60,0,0) takes place solely inside the inner shell while the flow from the

source (64,0,0) to the sink (-64,0,0) takes place only in the region between

the two shells.

Fig 7 shows fields where the strength of one source sink pair (±60,0,0)

is doublé that of the other (±64,0,0). Also in this case the ratio of the

three velocities is 1: 2 : 4.

To the diménsions of the shells similar conditions apply as in the case

of the fields with one source and one. sink (points

1-5).

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iLIj

57

iPJIRti

5 ¡

ITI

III'I»iI

i6MrdJII!JPP1

ìiifluipii

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Fig. 8. Equipotential lines in the XY plane for the flow about 3sources ((i20,0,0), (60,2,0)

and (60,-2,0)) and 3 sinks ((-120,0,0), (-60,2,0) and (-60,-2;0)).

Fig. 9. Equipotential lines in the XY

plane for the flow about 3 sources and 3

sinks at three velocities.

Fig. 10. Eqüipotential lines in the XY

plane for the flow about 3 sources and 3 sinks at three velocities.

1954 ₁₅ 3

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F. THREE SOURCES AND THREE SINKS

The distribution investigated first was: sources at the points (120,0,0),

(60,2,0), (60,-2,0) and sinks at the points (-120,0,0), (-60,2,0), (-60,

2,0).

A section of the equipotential lines in the X Y plane is shown in Fig. 8.

In this case the flow will take place about two effipsoidal bodies and a very

long and slender body with a reduced cross=section near the middJe

Next, the source situated farthest away and the sink situated farthest

away were moved towards zero, resulting in the following distribution:

sources at the points (64,0,0),(60,2,0), (60,-2,0) and sinks at the points

(-64,0,0), (-60,2,0), (-60,-2,0). All sources and sinks were of the same strength, and the equipotential lines were constructed in the X Y plane for three different velocities of the ratio 1: 2: 4 (Fig. 9). At the lowest

veloCity, two of the boundary shells lay inside the thiid, while the flow at the two high velocities took place about three shells which did not touch each other.

Once more the source-sink pair situated farthest away was changed, the new distribution being: sourCes at (60,-2,0), (60,0,0), (60,2,0) and

sinks at (-60,-2,0), (-60,0,0), (-60,2,0). Also this time the potential

lines were plotted in the X Y plane at three different velocities. At all

velocities, the boundary shells were clear of each other (Fig 10).

G. FIVE SOuRCES AND FIVE SINKS

The preceding investigations show that with the use of 1, 2 or. 3

source-sink pairs, rather symmetrical bodies with ellipsoid ends are generally

obtained. In order to get bodies with less symmetry and of a less

pronoun-ced effipsoid form, also the flow about five sources and five sinks was

investigated.

-The first. distribution is asfóllows :- -The sources were placed at (64,0,0),

(60,-2,0), (60,0,-2), (60,2,0), (60,0,2), and the sinks were placed at.

(-64,0,0), (-60,-2,0), (-60,0,-2), (-60,2,0), (-60,0,2). With this

distribution, the fields of flow are alike in the X.Y planes and the XZ

planes. In Fig. 11 the equipotential lines are constructed in these planes for three different velocities of the parallel flow. By comparing the

dia-grams with the corresponkling ones inFig. 9, it will be seen that the only

resuÏt brought about by the two additional source-sink pairs is that the

shells i the YZ plane have been forced farther away from (0,0,0) by the

In order to obtain a great asymmetry, the five sourdes and the five

sinks were placed on two straight lines passing th.iòugh (±60,0,0) and at

right angles to the X axis. The distribution was: sources at (60,0,0),

(60,0,+2), (60,0,±4) and sinks.at (--60,0,O), (-60,0,±2), (-60,0,±4).

For this distribution, the equipotential lines were determined partly in

1 L

Ii Villi PA Vil II li II

WIV!1IiMII!

!I1UhiV1iiWJ/iA

IVb

1

uII

___._IIlI&t -4 r

Fig. 11. Equipotential lines for the flow about 5 sources and 5 sinks at three velocities.

the X Y plane and partly in the XZ plane, diagrams for the variation of

ç with (z, y) and (x, z) bemg constructed first in the same way as m the

case of the diagram, Fig. 1. In the XZ planè, thé 4' curves for z = 0, z = +2 and z = ±4 will tend to infinity when x approaches 60, while. in

the X Y plane only the curves for y = O will tend to infimty for z ap

proaching 60.

-In Fig. .12, equipotential lines have been drawn in the twO planes in the vicinily of the sources for three different velocities of the parallel flow.

1954- .

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ìttijjì

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Fig. 12. Equipotential lines in the XZplane and the XY plane for the flow about 5 sources

and 5 sinks at th±èe velocities.

Atall three velocities a boundary sh1i is found around each source-sink pair, but at the lowest velocities the distancè between tiie boundary shells

is very small. . ,

62 .,(

H. 25 SOURCES AND 25 SINIS

In. örder to ascertain whether it s possible to produce almost. plane boundary shells with a finite number of sources and sinks, the flow about

25 souròes and 25 sinks, was calculated. All the sources were placed in the

plane x = 60 and all the sinks in the plane x = 60. In these planes the

sources and sinks were placed at the following points (y, z):'

(-4,-4), (-2,---4), (0,-4), (2,-4), (4,-4).

The eqmpotential lines m the X Y plane were constructed for four velo

cities of the parallel flow. The ratio of the velodities was 1: 2 :4: 16 (Fig. 13).

X

Fig. 13. Equipotential lines at four velocities of the flow about 25 sources and 25 sinks.

It will be observed that only at the lowest veloèity. have the shells

merged. 'They are lying inside eaçh other in a rather complicated manner. Also in this case will the outermost shell be ellipsoidal, however. At all

the other velocities the parallel flow wili pass between the different source-sink pairs, and at all high velocities the boundary shells will be effipsóidaL.

1954 19

(-4,4), (-2,4), (0,4), (2,4), (4,4), (-4,2.), (-2,2), (0,2), (2,2), (.4,2), (--.4,0),

Thüs, tho investigation shows that t is impossible to bùild up a plane

surface of a finit number of sources and sinks. At high velocities the

par-allel flow will pass between the sources and the sinks, and at low velOcities

only effipsoidal forms can be produced. In order to be able to proceed, it

is therefore necessary to change to the adoption of imeiform and plamform

sources and sinks.

I. LINEIFORM SOURCES AND SINKS

For the purpose of these investigations a lineiform soiircè shall mean a line section with an infinite number of infinitely small sources placed on

it The strength mtensity of a source may vary or be constant along the line. In the example calculated herein the strength has been constant.

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ut

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so 65 A

Fig. 14. Equipotential lines in the XZplane at two velbeities of the flow about a lineiform

source and sink.

Fig. 15. Equipotential lines in the XY plane for the flow about a lineiform source and sink

at two velocities.

y

J

IillIhÁ

4

The source and the sink have the length 8 and have been placed in the XZ'

plane, syrnmetriòally about and at right angles to the X axis at (60,0,0) and (-60,0,0).

The method of calculation was as follOws : First, curves fôr the

contri-bution of the indlividual line elements to the potential have been

construct-ed for a number of points in the XZ plane and the X Y plaie, the lineiform source (sink) being imagined to be divided up into an infinite number of

infinitely small sources (sinks). These curves have been integrated by

means of an mtegrimeter, whereby the potential at the various pomts has

been determined Thereupon a diagram in analogy with that u1 Fig. 1

could be constructed both for the XZ plane and the X Y plane, and with the aid of these diagrams the equipotential lines were determined. In the

diagram for the XZ plane all the p curves for y values situated between o and 4 would tend to infinity when z approached 60.

Fig. 14 and 15 give the equipotential lines in the immediate vicinity

of the source at two velocities (1: 2) in thé XZ plane and the XY plane,

respectively.

It will be observed that this time the boundary shefl has become very

asymmetrical. In the X Y plane the boundary curve is elliptical, while the boundary curve in the XZ plane can be made to "deflect heavily at

the terminal points of the source by the application of high velocities of the

parallel fiw.

After working with lineiform sources and sinks it was natural tO

inves-tigate the conditions when the sources and the sink were made plani

form.

J. PLANIFORM SOURCES AND SINKS

A planiform source shall mean a small surface with an infinite number of infinitely small sources placed on it. In this Section the source intensity is reckoned to be the same every-where on the surface.

A source having the form of a plane square surface with a side of 8 units is placed at the point (60,0,0) at right angles to the X axis, so that the sides are parallel to the Y axis and the Z axis, and so that thre is symmetiy about the X axis. A sink of the same form is similarly placed at the point (-60,0,0).

The velocitypotential was determined at various points in the XZ plane,

with the application of the calculations from the preceding Séction. Having

there summated m the direction of the Z axis, it was then only necessary

to summate in the direction of the Y axis. Also this summation was made

by first constructing a curve of the contributions to the potential from lines

at right angles to the X Y plane, after which the curves were integrated

Fig. 16. Velocity potential in the IZ plane for a planiform source and sink.

by means of an integrirneter. In Fig. 16, velocity potential ciïrves in the XZ plane were plotted for the source and the sink, and in Fig. 17 equipo-tential lines were constructed in the XZ plane in the vicinity of the source

II-!11IAWIVAU

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NIhiiiU

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Fig. 17. Equipotential lines in the XZ plane for a pinriiforrn source and sink at two

velo-cities.

at two velocities of the parallel flow (1: 2). It will be observed that the

boundary shell was very close to the planiform source and that the flow

about the source-sink pair deflected vry heavily at the ontour of the

source.

g'

000

10

In order to try and obtal

ship-shaped óiidy shell, calculations

were carried out for a form of source as shown in Fig. 18, where

the:sur-Fig. 18. Planiform source..

face is broken so that it has a sharp edge in the XZ plane The sink had

a corresponding fOrm

In the calculatión of the ç' curves,' the summations in the direction of the Z axis from the preceding Section were applied agam, after which a summation along the broken ime was made graphically and with the aid of the integrimeter.

56 57 ' 50 60 of 0 ' 6.? 66 ¿

Fig. 19. Velocity potential in the XY plane for a planiform source and sink (see Fig. 18).

The 9 curves fòr the XY plane are shown in Fig. 19, and it will be ob served that all the (ç curves corresponding to y values between O and 3 tend 'to iiifirúty at diffèrent x valuês.

Finally, Fig. 20 shows equipotential lines in the X Y plane at two velo-cities (1: 2) of the parallel flow. With this source-sink form, the boundary

4 1l111I11I1II1IIi1III

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Fig. 20. Equipotential lines iì the XY plane fora planiform source and sink at two velocities. Some lines of flow drawn in.

shell gets â shipshaped form. It is therefore possible, by using planiform

sources and sinks suitably distributed, to determine the potential flow

about ships.

K. POINT OF STAGNATION

In the majority of the flow thagrams constructed the X axis is the axis of symmetry, and sources and sinks are placed on it. The effect of this is that. the line of flow running in the X axis will divide when striking the

shôll, spreading over it. This division takes place at the point of stagnation.

Behind the body the flow will colleçt again into a line of flow continuing its course in the X axis. Where the flow takes place about several divjded parts of bodies there will be one set of stagnation points at each parti of

the body.

If the lines of flow across the surfaces of a ship are determined by

model tests, the lines of flow will appear to start at a point near the stém. This point is generally near the surface of the water (compare the fields of flow from Taylor's series of tests, [6] page 25). In case a calculation of the potential flow in the vicinity of these ship models is to be made, 'one

would let the X Y plane be the water-line plane and the XZ plane the

centreline plane of the ship. Since the potential flow might be imagined to. take place without wave formation, the XZ plane might be regarded

as an impenetrable surface or a plane of symmetry. It would then b

found that with this flow the point of stagnation would be the point where

the X axis meets the stem, which is in good agreement with the model

CHAPTER II: POTENTIAL WAKE

L. BASIS OF. CALCULATION'

In order to get an idea of the dependence of the velocity distribution

in the vicinity of the propeller on the shape of the ship, a calculation was made for two forms of ships, one with pronounced U-formed frames, the other with pronoúnóed V-formed frames.

The calculations just referred to have shown that it is necessary to work

with planiform sources and sinks when the field of flow about a ship-shaped

body is to be determined, Further, it was possible to conclude on the basis

of the calculations that the desired surface, the desired shell, was best

obtained when the source and the sink had the same form as the desired surface, and when, at the same time, a high velocity of the parallel flow

was applied. As, furthermOre, it is not necessary to place sources and sinks

on surfaces which are parallel to the uniform stream, and as it isnot neces-sary either for surfaces forming some angle with the flow to be covered

with sources of the same strength as those on sUrfaces forniing right angles with the direction of flow, it is possible to let the strength of the

source be proportional to sin a1 sin a2, where a1 and a2 are the angles which the lines of intersection of the surface element with the XZ plane and the X Y plane form with the X axis Thereby is òbtained. that the shell follows the source-sink surface in a higher degree than if the latter had the same strength everywhere. Accordingly, the strength of the source is unity

for surfaces at right angles to the direction of flow, while it is nil for surfaces

parallel to the direction of flow.

As previously, the following assumption are made

The movement takes place in an ideal flow which is homogeneous and incompressible. Besides, the flow takes place without friction and

irro-tationally.

The ideal flow is infinite in all directions.

3 The surface of.the water remains plane during the movements of the

body.

For the water's surface, the X Y plane is chosen, which, since the cal-culations are made as if the ship were symmetrical about this plahe, will remain plane. Accordingly, ve calculate the flow about a ship whose upper

part is similar to its lower part, and we reckon on this body being complete

ly submerged in a flow which is infinite in all directions. Thereupon we

imagine the surface of this ship to be coated with an infinite number of

sources and sinks. The sources are placed in the forward halfand the sink

in the after part, and their strength is varied proportionally with sin a1

sin a2. As the form of the forebody of the ship has no influence on the

con-ditions of flow in the vicinity f the propeller [2], the forebody and tbe

aft-body are reckoned tò be identical. The velocity potential

(L-l) is then determiried t a numberof points in two planes situated near each other, parallel to the YZ plane, and situated at the place of the propeller.

The velocity potential is determined in the way that summation is done

first along the water-lines and, next, from keel to keeL

where Hence,

r,

'r2

-dx

b

Fig. 21. The vlooity potential as a function of x.

cp2 in Fig. 21 being the potential from the uniform stream, and ç the

potential from the source-sink field, the coefficient of wake is determined by the formulá

w =

(Ç

a + a1 - b - b1

dx b1 - a1

w=

_ab

and

where thó difference b1 -r-- a1 is the differenöe in the sourcesink potential

m two neighbourmg planes, while a - b is the difference m the umform

stream potential in the same two planes.

Now, we form the differences b1 a1, using the values just determine4

in the two planes. in the vicinity of the propeller. The differehces a b,

however, need not be formed, because they are similar for all the points,

and b1 a1 will thus give an idea of the variation of the potential *ake over the própeller disc.

M. POTENTIAL WAKE IN THE CASE OF U-FORM AND IN

THE CASE OF V-FORM

In order to be able to compare the calculated values with values de

termined by model tests, two of Yamagata's model forms [7] have been

WI WI WI. WA WI5 WI. WI.

Fig. 22. Body plan fo U-form.

used in the calculation. The models used ae Nos. 198 and 197 with pro-nounced U-formed and V-formed aft-body frames, respectively. The tw body plans are shown in Fig 22 and 23

Fit. 23. Body plan for Vform.

1954 27 .

\ \" " ".

./ / t / i

ii \

iiui

iiiairiiiirn

iiiiuiiiiiii

-III"-"

\.

/ / /

I .

/ / i

-

//

/ / / / I,

\ N N

\. \

N

w_

Ï

A

.1W7 j

₄

This model was incorporated in the co-ordinate system as follows: The

A.P of the ship was placed at x = 100, the F.P. at x = 100, and q'

was determined by x = 97 and x = 98 at the points indicated in

Fig. 24.

WI WI

Fig. 4. Range of the determination.

As mentioned.in the.preceding Sect., the velocity potential was

determin-ed by summating, first, along the water-lines and, next from keel to keel. In order to himt the work, the summations were done with the application

of the trapezoidal rule, the interval length at the propeller being 1 unit,

some distance from there 2, and farther away 5 and 10 umts Some auxiliary cuïves were likèwise constructed; thus, curves giving the strength of source

sin a1 jii a2 along the water-lines were determined.; furthermore, curves

for Vz2 + y2 for different values of z suitable for the water-line distances

(

P

7 ô S

ôpjil IO O ¿6(5

The main data of the models are the following:

Length betweên perpendiculars 6.000 m

Breadth 0.800m

Draught 0.355 m

Displacement 1,265.6 kg

Block coefficient 0.743

Prismatic coefficient - 0.754

Midship section coefficient 0.986.

Distance of centre of buoyancy forward of 0.072 m

Vertical prismatiö coefficient: U-form 0.948

- : Vform 0837

and curves for 1 ¡r as a function of x The results of the calculations of the differences between the velocity potentials in the two planes are indicated

in the following tab1es

U-/orrn, Point 1 2 3 4 .1123 .0859 .0625 .0479 Point 5 6 7 8 .1102 .0821 .0595 .0447 Pöint 9 10. .1013 .0738 Point 13 14 .0784 .0588 Point 27

4çv

.0477 V-/orni Point 1 2 3 4 .0994 .0648 .0457 .0321 Point 5 6 7 8 4L'q .0829 .0575 .0412 .0301 Point 9 10 11 12

JE

.0605 .0451 .0323 .0249 Point 13 14 15 16 z12 .0380 .0313 .0248 .0196 Póint 17

JEç

.0222 WL 5 k' 4 WLe'

Fig. 25. Distributión of the potential wake for U-form and V'form

1954 . . 29 9f 9Q(n9 1 ¿I FORM

'ii!áiiii

'fl-i

6(1W 6 S(4d

11111111

AÓIJVA

O/JO) V FORM 6/lW f?Of9' 44 11 12 .0530 .0405 15 16 .0448 .0342 h'L f WL 1

Based on these values, curves for constant wake were constructed (see Fig; 25). Ït will be noted that the curVes, have a quite smooth course, and

that, for the U-form, thé curves aré more vertical that for the V-form.

The figures written on the curves indicate relative values, those in

paren-thesis are possible absolute values.

The absolute values were determined by comparisonwith the flow about

an ellipsoid of the same mam dimensions as the U form and the V-form,

the flow being determined partly in thé same way as described herein, partly by. direct caiçulation as described in [2].

N. POTENTIAL WAKE IN THE CASE OF AN ELLIPSOID

The main dimensioiis of the ellipsoid being:

Length between perpendiculars ...200 units

Breadth ...26.66 Units

Draught...11.83 units

its serniaxes were a = 100, b = 13.33 and o = 11.83.

Jjjthe same way as for thé two forms of ships a source was placed on the

forebody and a sink on the aft-body., The strength of source was varied

by sm a1 sm a2, and the same auxthary curves were apphed as m the

preceding Section.

Fig. 26. Distribution of the potential wake for an ellipsoid.

Thevelócity potential was at the points 1, 5, 9 and 13, and

was foufld equal to .0517, .0276, .0168 and .0111, respectively. On the

.//.

,SOS)

- h'L.I

basis of thesé values, the wake distribution curves in Fig. 26 were

construct-ed, and at the same timé the wake values from the investigation in [2] are indicated in parentheses.

On the assumption of proportionality existing between the potential

differences and the wake coefficiénts for the U-form, the V-form and the ellipsoid, the potential wake oefficients for the two firstmentioned forms

can be determined. The valuès thus found are indicated in parentheses

in Fig 25 If the velocity of flow is very high, the parallel field will be sure to come very close to the source and the sink, and the desired flow wifi be obtained. If, furthermore, the velocity of the flow is the same in the three

cases, a comparison would be possible.

Fig. 27. Variation of the potential wake with the propeller diameter.

Thereupon an integration of the wake (volume integration) over the

propellerdisc was made for the distributions in Fig. 25 and 26 with different diameter-length ratiOs, and the result is shown in Fig. 27. The height of the

propeller axis over the keel is E 0.4 d

O. COMPARISON OF CALCULATED AD MEASURED WAKE COEFFICIENTS

In Fig. 28 the wake distributions determined experimentally by Yama-gata are outlined. A comparison of the figure with Fig. 25 will show 'that

the potential wake curves have a smooth course, while the _{cOurse of the}

experimental curves is very irregular. This must be due to the fact that

the friction belt has different thicknesses at differeñt levels over the keel,

which is very likely since the lines of flow, startmg at the forward point

of stagnation, will have passed through different distances before reaching

the propeller disc. Consequently, the small friction belts built up by the

1954 ₃₁ 20 .10 --01 - (LL/PSOß .02 .03 04

indivjçlual. lines of flow vill have unequal widths. As the flow in the case

of the U form is almost two thmensional, unlike the flow in the case of

the Vform, it is, farther, natural that in the case of the U-form the

friction belt gets a relatively large width under the propeller axis

- WL5 U FORM WL ' WI 3 WL ê WL f WL 5

Fig. 28. The wake distribtion for Yamagata's models Nos. 198 and 197.

If a quantitative investigation is to be made, it is necessary first to

imagine the propeller axis in the calculated examples lowered to the same

level as in the model tests, after which correction will have to be madefor

the frictional wake For the two sizes of propellers, the calculated potential wakó will be.

D/L = 0.03 D/L = 0.04

U-form .365 .345

V-form .205 .195

With the two diameter-length ratios, the frictional wake may be put

at (cfFig.4iÍi [2])

.175 .140

Hence, the total wake will be

U-fOrm .540 .485

'V-form .380 , - .335

In Yamagatá's tests the following wake coefficients were measured:

U-form .47 : .41

In the case of the V-form there is thus fair agreement between the

meas-ured and the calculated values, while the deviations m the case of the U-form are comparatively great. The causes of the deviations may bè partly that the method of calculation is not quite crrect, prtly that the assessment of the frictional wake is uncertain, and partly that also the Pitot tübe readings are attended with some experimental mstabthty

Finally, the wave wake may have some effect From Fig 28 it will be seen that for the V-form,. wake coefficients of 0.60 are measured, while in the

case of the U-fórrn only 0.50 is reached, which must either be due to

diffeience in the buildingup of the frctlon belts or instabifity òf the measurc

-ments.

In the calculation of the potential wake the strength of source was

allowed to vary by sin a sin a where ai and a2 were the angles formed

by the surface element with the direction of flow. Perhaps it would be sùfficient to allow the strength to vary by sin ai, or a constant strength

òf the source over the whole surface might be reckoned with, because in

the comparisons a very high - almost infinitely high - velocity df flow

is used. At the high velocity the shell will lie close up to the source-sink surface, and the distance between the two surfaces will be almost

inde-pendent of the variations in the source-sihk strength The application of

a constant source-sink strength will make the calculations slightly simpler.

The investigation shows that it is possible to calculate the potential field about an arbitrary body - particularly about a ship if oi1y a

source-sink surface of the same form as that of the body is used. When the potential field has been determined, the velocity of flow can easily be

determined.

The methOd of calculation indicated will be applicable in systematic

investigations of the flows about ships and may perhaps contribute towards

the solution of the problems occurring m connection with the friction belt, especially the question of the dependence of the frictiön belt on the form of the ship.

Symbol

Ship model Propeller

D metres or feet Propellér diámeter

r metres or feet Distance from axis to section in propeller bladè

E metres or féet Height of propeller a±is over kèel

Velocities

y and V metres sec or knots Speed of propulsion

ve metres sec' Speed of advance

w = (y - ve)/v Wake coefficient U metres sec' Velodity of flow Potential field

x, y, and z metres Co-ordinates

ç rn2 sec' Velocity potential function

m2 sec' Stream function

m2 sec Summated potential

4 Z q, m' sec' Potential difference

r metres Distance between points

m m' sec Strength of source

O and a Angles'

Ellipsoid

a, b and ,c metres Semiaxes

Q. BIBLIOGRAPHY

Duii.&n, W. F., '.'Aerodynarnicl'heory", Berlin 1934, Vol. I. Hvw, Sv.. AA., "Wake of Merchant Ships", Copenhagen 1950.

R&z.xrex M On plane water lines in two dimensions Philosophical Transaetion 1864, p. 369.

TAyLO1, D. W, "On ship,shaped stream forms", TINA 1894, p. 385.

TAYLOR D W On solid stream forms and the depth of water necessary to avoid

abnormal resistance of ships", TINA 1895, p. 234.

TAYLoR D W The Speed and Power of Ships 1933 (1910)

YAMAGATA, M., "Model experiments of the combined effect of aft-body forms and pro-peller revolutions upon the propulsive economy of single screw ships TINA 1934

p.386.-

-Abbrëviétions:

TINA: Transactions of the Institute of Naval Architects, London. L metrès or feet Length

P. SYMBOLS AND UNITS

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10,50).

-Berge Dunn: Sorne Experiments with Sleeve Beäring Metals. A Report and some Notes. (Pris kr. 18,00).

M. Andrea8en og Lottrup Knudsen: The Theory and plane reflectors in micro-wave antenna systems. (Prie kr. 7,50).

Ernst Hellmers: Bacterial Leaf Spot of Polargonium. (Pris kr. 4,50).

L. T. Muus and R. W. Asmussen: Moisture Regain Determinations of Textiles from Dissipation Factor Measurements at 100 kc/e. (Pris hr. 3,00).

1953

R. E. H. Rasmussen: On the Examination of Approximately Straight Edges (Rules) and

of Approximately Plane Surfaces. (Pris kr. 3,00).

H. Højgaard Jensen og Asger Nielsen: Ferromagnetism of thin Nickel 'ihns. (Pris kr. 3.00).

Frank Engelund: On the L minar and turbulent flows f groundwater through

L. T. Muu: Dielectrical Investigations of Cellulose with a special view to the Cellophane.

Water System. (I tryk).

Chr. Winther: A Chromatèd Gelatin Interference Plate. Preparation, Properties and Theory. (Pris kr. 6.00).

Uhr. Wínther: A Chromated Gelatin Interference Plate. Treatment with Water and Water Vapour, Measurement of the Depression of Vapour Pressure. (Pris kr. 3.00).

1954.

0. Heie : Third Report of the Virus Committee. (I tryk).

S. Aa. Uarvald: Three-Dimensional Potential Flow and Potential Wake. (Piis hr. 4.50).

J. Arn1wosen and 0. KoJoed-Hansen: Ionization Chambers with high RO-Values. (I tryk).