On optimum sculling propulsion of ships

Scheepshydramechanica

Archief

Mekelweg 2, 2628 CD Delft

Te1:015-786873/Fax:781836

I

ON OPTIMUM SCULLING PROPULSION OF SHIPS

i

TECHNISCHE UNIVERSITEIT

Laboratorium voor

ScheepshydromechanIca

Archlef

Mekelweg 2, 2628 CD Delft

Tel.: 015 - 786873 - Fax: 015 - 781835

ON OPTIMUM SCULLING PROPULSION OF SHIPS

PROEFSCHRIFT

ter verlcrijging van het doctoraat in de

Wiskunde en Natuurwetenschappen

aan de Rijksuniversiteit te Groningen

op gezag van de

Rector Magnificus Dr. E. Bleurnink

in het openbaar te verdedigen op

maandag 18 mei 1987

des namiddags te 4.00 uur

door

WILLEM POTZE

geboren te Emmen

CONTENTS

Acknowledgements ibrd

Preface xi

CHAPTER I W. !Potts and J.A. Sparenberg: ON THE EFFICIENCY OF

FINITE AMPLITUDE SCULLING PROPULSION'', International

Shipbuilding Progress, vol. 3O, 1983, pp. 238-244

Introduction 1

2. Formulation of the problem 1

3. The lost kinetic energy 2

4. Small values of h

a

5c Finite values of h and h =co 4

6. Numerical results 5

Acknowledgement

References V'

CHAPTER II..

0.

Potzev mON OPTIMUM SCULLING PROPULSION",

Journal of Ship Research, vol. 30, 1986, pp. 221-241 a

I. Introduction

Some preliminary considerations! 8

3. Base motion 9

k, Optimum shed vorticity

al

5. Lost kinetic energy

Added motion

ra

7,. Force and pitching moment acting on. a profile 15

8. Numerical results 18

9). Concluding remarks 25

Acknowledgement % 28

References 28

Metric conversion factors 28,

1

7

8

CHAPTER III. W. Potze: "ON LARGE AMPLITUDE SCULLING PROPULSION

BY WINGS OF FINITE SPAN"

1. Introduction

2, Description of the problem, 3. The vortex and source-sink layers

29

30,

32

ar

4. The efficiency 51

5. The numerical method 51

6. The accuracy of the mathematical model 61

7. "Optimal' angles of incidence 64

Acknowledgements 89

Appendix 90

References 92

CHAPTER IV. W. Potze: "ON THE INFLUENCE OF VISCOSITY ON THE

'OPTIMUM' EFFICIENCY OF LARGE AMPLITUDE SCULLING PROPULSION"

1. Introduction 55

2. The inviscid fluid flow around the Wings 97'

S. The forces acting on the wings 102

4. The efficiency of the sculling propeller 108

5. The numerical method 110

k. 'Optimum" motions of the, wings 112

Acknowledgements 131

References 132

SAMENVATTING 133

ACKNOWLEDGEMENTS

I want to express my gratitude to:

Prof.dr. J.A. Sparenberg for giving me the opportunity to write this thesis, for his continuous interest and valuable suggestions. It was a

pleasure to work under his guidance.

The "promotie commissie", consisting of prof.dr.ir. P.J. Zandbergen,

prof.dr.ir. A.I. van de Vooren and prof.dr.ir. H.W. Hoogstraten, for

reading the manuscript and their comments.

The "gebruikers commissie" of the Technology Foundation, consisting of

prof.dr.ir. W.M.J. SchlOsser (TUE), prof.ir. M.C. Meijer (TUD), ir. H.A.

van der Hoeven (Kon.Mij."de Schelde"), prof. ir. N. Dijkshoorn (TUD), ir.

R.M.R. Muijtjens (TUE) and ir. F.C.H.D. van den Beemt (STW), for their

interest and suggestions.

The secretaries of the Mathematical Institute for doing the typework. In particular Ineke Kruizinga and Pietje Risselada, who have typed the major parts of the manuscript in an excellent way. Ineke Kruizinga also helped to prepare it for reproduction.

Friends and relatives for their interest, while I was working on this thesis.

The drawing on the cover has been made at the Eindhoven University of

Technology.

This research was sponsored by the Technology Foundation, project no.

Preface.

A ship has to be propelled by some propulsive device in order to move, with a desired velocity, through the water. The propulsive device or propeller has to deliver a thrust which equals the drag caused, among others, by the viscosity of the water and the wave making. Energy has to be put into the propulsive device to generate the needed thrust. It is

obvious that this energy should be as small as possible. Then an

optimization problem occurs for the hull of the ship and the propeller.

The hull and the propeller affect each other. However, in order to avoid

large complications, the influence of the ship's hull is neglected in

this thesis, in which an attempt is made to solve the optimization

problem for the propeller alone, which in our case is a sculling propeller.

A sculling propeller consists of one or two wings mounted vertically

behind the ship (figure The wings move sideways back and forth, while their angles of incidence are adjusted such that water is pushed in the

backward direction. By this the momentum and kinetic energy in the water increase. This type of propulsion is called unsteady in contrast with the

uniformly rotating screw propeller.

The optimization problem for the sculling propeller is to determine angles of incidence of the wings such that the energy loss is minimum,

subject to the constraint that a prescribed mean value of thrust is

delivered. The optimization problem is divided into four subproblems, which are discussed in the four chapters of this thesis. Each chapter is

self-contained. Chapter I and II are reprints of published papers.

In chapter I the optimum efficiency of the sculling propeller is

determined within the framework of a linear three-dimensional potential-flow theory. The water is assumed to be inviscid. The finite span of the wings and their interaction are taken into account. The optimum vorticity shedded by the wings is calculated, and this determines the efficiency of the propeller. The exact motion of the wings such that the prescribed mean value of thrust is delivered needs not to be known at this level of

xii

of the propeller wings is assumed to be infinite. Furthermore the wings are assumed to have zero thickness. A linearization procedure is carried

out with respect to the delivered thrust. The wings have to shed

vorticity in order to produce a thrust. In the neighbourhood of the wings this shed vorticity is assumed to be in motion. The shed vorticity is

optimized and is coupled to the prescribed value of thrust by the

increase of momentum in the water. Besides the optimum angles of

incidence, the strongly unsteady force and pitching moment acting on each wing and the power needed to move the wings are determined.

In chapter III the influence of the finite span of the propeller wings on the efficiency is investigated by means of a linear three dimensional

potential-flow theory. The thickness of the wings is taken into account.

The wings' motion is almost the same as that derived from the

two-dimensional optimization theory (chapter II), however, with a slight modification in order to allow for the effect of finite span.

In chapter IV the influence of the viscosity of the water on the

efficiency is investigated. The wings have finite span and their

thickness is taken into account. For given amplitude of the motion of the wings the frequency is determined such that the efficiency is maximum. As in chapter III the wings move almost according to motions derived from

the two-dimensional optimization theory of chapter II, again with a

CHAPTER I

ON THE EFFICIENCY OF OPTIMUM FINITE AMPLITUDE SCULLING PROPULSION

by

W.tome andJ.A. Sparenberr

Summary

Using a linear theory the highest possible efficiency is considered for sculling propulsion by means of one wing

or by two wings. The span of the wings is allowed to be finite and their interaction in case of two wings is taken

into account. For a number of motions of the wings the quality number is calculated.

ii .Introduction

The type of propulsion we consider is by one wing

or by two wings, mounted vertically behind a ship. The

wings are moving sideways back and forth while their angles of incidence are adjusted such that a thrust is

created. For practical purposes two wings are ap-propriate because theninconvenient moments acting

at the ship can be avoided. The fluid in which the propeller is working is assumed to be inviscid and

in-compressible.

We will try to obtain insight in the maximum values of the efficiency which can be obtained by this kind of propulsion. It is expected that its efficiency will be larger than the efficiency obtainable by a

con-ventional screw propeller because the amount of water' that can be influenced by the wings is larger than the amount of water' influenced in general by a screw propeller. Another advantage of the wings is that their velocity with respect to the water is constant over their span, while the velocity of the profiles of a screw propeller blade increases towards the tip. Of course a disadvantage is the complexity of the

mech-anism needed to move the wings.

This hydrodynamical efficiency investigation is part of a more comprehensive one proposed by

By

Kon.

MU. 'De Schelde', about the feasibility of unsteady

propulsion. That investigation is also directed towards the construction of the required machinery.

We calculate the highest efficiency that can be ob-tained for three types of motions of the wings. There are however some more or less serious approximations and assumptions we will make. First, our theory is linearized, this means that it does not hold for propel-lers which are heavily loaded. Second, we assume the wings to work in an undisturbed fluid which fills the whole three dimensional space. Hence we neglect the influence of the free water surface and of the ship's hull both as a boundary of the region in which the propeller is working. Also we neglect the influence of the ship's hull as a causeofan inhomogeneous inflow.

It is not difficult to adapt the theory to an inflow with a small inhomogeneity. when however a large amount State University Groningen, Departmentof Mathematics. Groningen. The Netherland,

of incoming vorticity is admitted more serious

dif-acuitiesoccur (for instance Ill). Third. we calculate in the following the free vorticity which has to be shed by the propeller when it is an optimum one. The question is, to which extent can this free vorticity be created by rigid wings. This has to be a subject of

further research.

2. Formulationof the problem

A Cartesian coordinate system is embedded

in an inviscid and incompressible fluid. The undis-turbed fluid is at rest with respect to this coordinate system. First we consider one wing IV moving with a

mean velocity U in the positive x direction. The wing is assumed to remain in a close neighbourhood of a

reference surfaceHwhich is decribed by

H :y = g(x)= g(x + 61= g(xl

lz1<4

, (2.1)

where g(x) is a sufficiently smooth function with

period b.

Figure 2.1. Reference surfaceHand wingW.

Hence in the case of rigid wings a necessary condition for the theory to be valid is that the chord length of the wings is sufficiently small with respect to the smallest radius of curvature of g(x). Of course the theory is valid for wings with arbitrary chordlength of which the profiles are flexible and can follow by this closely the surface H. The reach A of the motion is defined by

A = max I(1)

g( 2)1

,

2<h

. (2.2)

The wing W has to deliver a thrust with a mean value

T, with; respect To time, in the positive x direction. This thrust has to be generated at the cost of as little

energy as possible, then the wing is called optimum.

Carrying out its task the optimum wing leaves behind free vorticity which, because our theory is

linearized, lies at H and which can be characterized as

follows (2). The reference surface H is placed as an impermeable and rigid surface in a homogeneous flow with velocity X in the positive x direction. Then we need vorticity ; on H in order to make it a stream surface. This vorticity. when X is chosen correctly, is the free vorticity left behind by the optimum wing. The way in which we calculate X will be described later on. Hence in order to find we have to solve an

obvious boundary value problem for the potential function

$(x.y.2)= XCe +#(..c.y.z)),, (2.3)

where 14(.x.y/z) is the disturbance potential caused by the surface H. In terms of 4,(x.y.z) the boundary

value problem reads.

32 ,3.2 o2

(

a.r2 dy2+ ,p(x,y.z) e 01 ,t .(2t4)

a:2/=

_cos(n.x)

e

an x

Urn _{a(n.y.z)= 0}

where ;; is the unit vector in the x direction and-4 is the unit normal at H. pointing from the

side (H)

towards the + side(H) of H(Figure 2.1).

Note that in order that a potential function so exists in the whole space with the exception of II, in is necessary and sufficient that the total circulation around H is zero. This makes .p a 'single valued

func-tion' outside H and causes the kinetic energy belonging to to to be finite.

We also consider the probably more impOrtant case

of two wingsWI and 142 moving with a mean velocity

U in the positive x direction along two different sur-faces Hi and H2 (Figure 2.2). The mean value, with

H2

(ci Is/

Figure 22. Two wings; (a).intersectingreference surfaces, (b) non intersecting reference surfaces.

respect to time, of the thrust of both wings together Is also prescribed to be . We assume Hi andH2 to: be placed symmetrically with respect to the (x.z)

plane. They are described by

H1'ye (-1 )1,g(x)le

i( I )1g(x +b)' ( I )1g(x)

(17) 112.1C . 1=11,2 .

2

We distinguish between two different cases, the

refer-ence surfaces Hi and H2 intersect each other or they

do not intersect, as drawn in Figures 2.2(a) and 2.2(b)!

respectively. In case (a) thewings and 1V2 cannot move side by side but have to pass one after the other

the lines of Intersection of If, and H2 . In case tb) there is no objection that W, and W2 move side by

side, although then a gap has to exist in order to avoid

that the profiles come too close to each other. It will turn out that such a gap is from the point of view of efficiency unfavourable. The disturbance potential to'

has to satisfy in this case (2.4), (2.6) and

Tra = .71 cos(n.x)

(x.y,z)e , (j = 2) . 1(-213)1

Here an analogous remark can be made with respect

to the potential being a single valued function and to the kinetic energy per period b being finite, as has been done in the case of one wing. In this case the total circulation around H, and around H2 separately

has to be zero.,

3. The lost kinetic energy

Suppose the disturbance potential to is known by solving one of the boundary value problems mentioned

in the preceding section. then we need to determine the unknown coefficient X. This scalar follows from the condition that the mean value of the throat is T.

We first consider the propeller with one wing W.

The period of time r of the motion of W is given by

.1:12

A

ts A H1(7? 1.'1; (2:5) (2.6) = 1,

r = b/U , (3.))

The potential Xe, is the potential of the flow far behind

the optimum propeller induced by the shed free vor-ticity at H. Hence the condition from which X follows

reads

T = T(b) , (3.21

where p is the density of the fluid and p1(b) is the momentum of the fluid far behind It/over a period of length b in the x direction induced by 0. So we

have to calculate 1(b)

1(b)=If' 117 dy dz (3.3)

o - ax

Because the disturbance velocity field is a periodic function of x with period b and because ; is single valued and for (r2 + :2) we have ; 0. it is not difficult to argue thatalso No is periodic. Then we can rewrite (3.3) as

1(b)= -

If

[viii

--A) de . (3.4)

11(b) -

-where 11(b) is one period of H

and [et

is the jump of the potential over H

1,011 = (3.5)

and do is an element of area

of HI.

Using (3.4) the coefficientx follows from (3.2).

Next we calculate the mean value of the kinetic energy lost per unit of time by the propeller

E.P42

2r a

.PX2 1 =

-If (grad 012d Vol =

div GPgrad 47) d Vol =

2.7 o

b.f. 2 1(b)

2p U1(b) (3.6)

By his we find for the efficiency

TU 11 b I

(3.7)

T U E

I 2p U21(b)

We now consider instead of the propeller an ac-tuator disk with the same rectangular working area Ail as the working area of the propeller. The disk has constant load / in the negative x direction such that its total thrust is T, hence

(3.8)

1'7'1A h

and its velocity in the x direction is U. The loss of kinetic energy per unit of time of this disk equals

E

-

(3.9)

" 2pAhU

The quality number q of the propeller is then defined

by

Eac

q il . (3.10)

E A hb

The inequality in (3.10) follows from the fact that the propeller and the disk are assumed to work in an undisturbed fluid (no wake of a ship) and the kinetic energy loss of the actuator disk is the smallest pos-sible loss compared with any propeller with the same mean velocity of advance U. the same mean thrust andthe same working area Ah. Using q the efficiency

can be written as

+

fill I)

2p U2 qA h

Next consider two wings W and W2. In this case we

define the reach A of the wing motions by

A = 2 max ig(x)I , 0 x < b . (3.12) Themomentum HObecomes

1(b)= -E [kPl+ x 7.1) do .

2

(b)

(3.13)

It is seen that we have to replace A and 1 in the fore-going formulas by (3.12) and (3.13) respectively in order to obtain the corresponding quantities for two

wings.

Hence also in the case of Figure 2.2(b) we take as the working area Ah although there is a gapbetween' andH,. However this gap is an unavoidable draw-back of a propeller ot' which the wings move side by side, so it seems more realistic to reckon this gap also to the working region because it could have been used by a slightly different propeller. In this way we es-timate the penalty on letting the wings move side by

side.

4. Small valuesof

In ease that the span Is of the wing or wings is small

with respect to the period b, the stroke .4 and the radii

of curvatureof px) we can apply slender body theory in order to calculated the potential 0.

Again we consider lust the case of one wing moving along the reference surface H. The potential 0 is rhen determined by (2.4). (2.5) and (2.6). In agreement with slender body theory we consider each vertical cross section of H to be placed separately in a parallel flow normal to it, with velocity (--j, ). Then it is well known that, when the total circulation around

the cross section is zero, the vorticity becomes

73.(z)= - 2(lx if) z 1(i)2 - z2 (4.)) y, is reckoned positive when it has, with a right hand

screw, a positive component in the positive x direction.

Here we assumed in order to avoid complications, that g(x) is a one valued function of x. The vorticity y, is an approximation of the component of the vorticity at H parallel to the plane: = 0.

-I

An approximation of the vorticity component On H in the z direction can be obtained from (4.1) by the demand that a vorticity field is free of divergence. 'Because in (3.4) we only need [vile we do not

cal-culate ;. From (4.1) we find

[01: = 7012

ek

z 1(4) =2(e. x I 2 !Hence 1(b) becomes

to)= -

2 122

I (;,,

:it )2 I 2)2

r

(x)2

rdx

dz = =

±2 (e.702 ft +1(421)%dr..

_{4 x}

Then q and n follow from (3.10) and (3.11).

In case we have two wings Wi and W2 of smaltspani we can simply replace 1(b) (4.3) by

21(b) , (4.4)

because by the smallness of there is upto the ac-curacy of the theory, no interaction between the

vor-ticity at I/5 and at Ho .

Possibly the part of the theory of this section as given for one wing, can be used for estimating the

highest possible efficiency of the large amplitude

swimming of eel like fishes. 5. Finite values of h, and +

When the span of the wing or wings is not small we will use a vortec lattice method [3], [4) in order to obtain an approximation of the basic quantity

ho] , which occurs in the formula for 1(b) which in

its turn determines the efficiency p and the quality number q. We first indicate the method for one wing,

hence for one reference surface H.

From the symmetry properties of gtx) (2.1) and from the symmetry of 11 with respect to the (x,y) plane it follows that only the vorticity at the part

0 x C lib. 0 z C Kit of /I need to be considered as

unknown. Along the line of intersection of H with the (x,y) plane we introduce a length parameter s, with,

s0 at x = 0 and s= at x = b. On half of this in-terval we choose Nt 1 points sn (n = 0.1 N). The x coordinate of so equals zero and the x coor-dinate of sN equals Sib. On the z axis for DC z <1/2h. we choose Mt 1 points zrn (in = 0,1 Al). with

zo = 0 and zm =1/2h. The points in the s direction as well as the points In the z direction need not to be equally spaced. Their position for a number of cases

will be mentioned later on.

Next we draw on H the straight lines parallel to the z axis, through the points so and the curved lines y = g(x), z = zn. These lines intersect at (N+ 1)(14 I)

points aim. Each point tit,m is connected by straight line segments with its neighbouring points in the r. direction and in the s direction. In this way a lattice is constructed which is in the neighbourhood of H. The rectangles of this lattice, which have one side

along the upper edge of H (z = 4) are slightly changed. The uppersides of these rectangles are brought

down-wards over a distance of one quarter of their height in the: direction, hence over a distance 1/4(zw 11m-1).

Around each of the N.M rectangles of this lattice

we put a concentrated vortex of still unknown strength

rn., (n = 0,1....

; in n0.1 M I). These are the N.M unknowns of the problems This lattice is now extended in an obvious way over the whole of H. Note that by using the closed rectangular vortices of

strength ['man we automatically satisfy two

condi-tions. First, the divergenge of the vortex lattice is zero.

Second, the total circulation around H Is zero, as was

demanded previously.

met

artm fn,n,

Figure 5.1. Closed rectangular vortex

When the whole of H is covered by the lattice Of concentrated vortices, we calculate by the law of Riot and Savart the induced velocities at the midpoints cio.o, of the vortex rectangles (Figure 5.1). glen the component of these velocites in the direction of it is

equated to the prescribed value at H. In this

way we obtain N.M linear equations for the Half unknown vortex strengths m41,,. Although the vortex lattice at H stretches from x = towards x we take into account only a finite number? of periods,

to the right and to the left of the considered part of H.

We now discuss briefly the case of two wings. Then

we cover both reference surfaces 1 and H2 with a grid of the type we discussed before, while the grid on H2 is the reflected one of Hi with respect to the (x,z) plane. Note that the number of unknown cm in the case of two wings is the same as in the case of one wing when the fineness of the vortex lattices in

both cases ,is the same. This is caused by the symmetry of 111 and H2 with respect to the (x.z) plane. by

1 (4.2) (4.3) I + g' h, It

,N

an.1,m4 =

which only the vorticity on forinstance Hi has to be

considered as unknown.

In both cases, one wing or two wings, the problem becomes two dunentional for h=a. Then we need(

only a grid which consists of lines parallel to the z axis..

In this case the problem becomes easier to handle,

from the numerical point of view.

For information we mention the values of At M and P and the distribution of the points an and rk which. by carrying out some tests, turned out to give

satis-factory results for a number of specific cases. We take

g(x)= 0.25 cos 2 kx , (5.1) for the case of one wing and also for the case of two wings with intersecting reference surfaces Hi and H2

(Figure 2.2(a)). For two wings with non intersecting

Hi andH2 (Figure 2.2(b)) we take

g(x)= 0.1375 + 0.125 cos 4 . (5.2)

In the latter case the maximum slope of 'rice reference surfaces is the same as the maximum slope of g(x) given by (5.1), however the length period of the motion is halved. To denote these three cases we in-troduce a parameter a, the value a = I denotes one wing: a = 2 denotes two wings with intersecting

refer-encesurfaces Hi and H2 and a = 3 two wings with non

intersecting'', and H2.

Further we take in some rises the points sn or, zn to be equidistant and in other cases as given in Figure 5.2. We introduce a parameter # with P = I to denote 'the 'points sk to be spaced equidistantly and 0 = 2 for the points to be spaced as in Figure 5.2. A parameter is introduced analogously for the points r, when' = 1 the zn are equidistant, when 7 = 2 the zn are

spaced as in Figure 5.2.

Then it turned out that the schemes of Table 5.1 for calculating the quality number q were satisfactory. A criterion was that q became stable rather soon with respect to a refinement of the vortex lattice, hence for

an increase of the numbers N and Al.

We also compared two ways for the calculation of the momentum 1(b) (34) from the found discrete vortex system. First, we determined rr directly 'by

AS AS 12' 5N-3i az az' AS as 2 2 SN..2 Sti-1 Sit 20 Zi _{2ei-3 44-2} Zm

Figure 5.2. Distributions of points in s direction and in z

three-ton.

Table 5.1

Schemes for calculating the qualitynumber q

means of the concentrated vortices hence the integrat-ion was carried out over a step functintegrat-ion. Second, the discrete vortex system was replaced by acontinuously

distributed vortex system of the shape

1/2, f

(x)I

c,.(x)(

z)lk

2 1

, (5.3)

a

k= " 2

where K is a suitable chosen number, anyhow smaller than Al. The coefficients ck (k a 0,1 K) were

cal-culated by means of least squares. The differences of these two methods however were neglicible, hence the first method, which is the more simple one, has been

used in the following. 16, 'Numerical results

We now give a number of values of q which is the 'only unknown quantity needed to calculate the ef-ficiency ep (3.11) for a certain propeller. In the case of one wing. (a = 1) or in the case of two wings with in-tersecting reference surfaces (a = 2) we take

g(x)= a cos 2ex , (b = 1) . (6.1)

In the case of two wings with 'non intersecting refer-ence surfaces (a = 3) we take

1 1

g(x)= a

+ (a 5) cos 4sx

(t. (625,

2 2

In all three cases the reach A of the wing motions equals 2a. We consider three values of a. a = 0.12, a = 025 and a = 038. For the half width 5 of the gap between the two reference surfaces (6.2) (case a = 3), we take five values, 6= 0.8 = 0.01, 6 = 0.02,6 = 0.04 and 6 = 0.06. The span of the wings ranges from It = 0 towards h = 3, while at the right hand sides of 'the

figures the values of q for It = are given.

It follows from the results that the qualitynumber increases in all cases with It and with a. The quality number decreases strongly in the ease a = 3 of two wings with the increase of S hence with the increase of the width of the gap in between the two reference surfaces. This means that in the case of Figure 2.2(b) the optimum efficiency (3.11) will be sensitive for the space which is needed us between the two profiles when they are at their closest position.

When we keep 7', U and A constant it follows from the graphs in connection with (3.11)1 that in leases

h arc ha 11 h = 0.1 a 1 2 3 1 2 3 I 2 3' 18t N 1 30, I 30" 1, ;30 1 1,5 2 15 2 la 2 1121 J 1 12 11 2 12' I 128 128, 18 & 18 8 12 8 1.3 2 1,2 2 12, 2. AS AZ 7 P

r

0.80 -cc 0.60 0. 0 00.20 a0

P---°0.00 0.50 1.00 1.50 2.00 2.50 1.00 co SPAN 0.80 0.60 0.40-0 0.40-0.20.40-0 1.00 0.80 UJ in 0.60 4: 0.40 .r 3 0.20 a.0.38 0.25

nit

Figure 6.1. Quality numberq, ISwing, case a=If.

s.0.38 0.00 0.00 0.50 ii .00 012 0.01 0.02 0.04 0.05

Figure 6.2. Quality number q, 2 wings, case a = 2.

4.0.00

-0.648 0.545 0.334 0.861 0.823 0.755 0.670 - 0.409 - 0.314 0.188 0.103

050r/

0.00 0.50 1.00 1.50 2.00 2.50 3.00/ co SPAN

Figure 6.3. Quality number q, 2 wings, case a= 3,a = 0.12.

= I and a = 2 for increasing values of h with it< 0.5 b the efficiency17 will strongly increase. For values

of is> 0.5 b the efficiency will increase more smoothly with it. In case a = 3, because there the length period is halved, we have to replace the inequalitiesby ft< b

and h a b. When the wings behind the ship pierce the free surface it is not clear what we have to take for the value of the span. To a certain extent the free surface acts as a boundary of the fluid and possibly the 'ef-fective' span is longer than the underwater part of the wing, by which a larger value of q will be more

ap-propriate.

The slender body theory of section 4 yields the

1.00 0.80 UI in I 0.60 03.0 3 0.20 0.60

I

111 z 0.60 1.00 0.80

t

0.60 0.40 a 0.20 0.00 0.00 0.50 1.00 -4.0.00 0.01 grip 0.04 0.06 1.00 1.50 2.00 2.50 3.00 SPAN

Figure 6.4. Quality numberq, 2wings, casea 3.a= 0.25.

4.0.00

1.00 1.50 2.00 2.50

SPAN

CO

Figure 6.5. Quality numberq,2 wings, case a = 3, a= 0.38.

00 0.774 0.601 0.537 0.444 0.368 eon 0.699 0.653 0.585 0.528 0.0000000 0.50 00 1.50 2.00 2.50 00 co SPAN

Figure 6.6. Comparison of quality numberq, a = 0.38; case

a= 2 C3Sea 3,5 = 0,

slopes of the graphs at the origin of the Figures 6.1

-6.6, hence for sufficiently small values of h. The results

of that theory are given in Tables 6.1 and 6.2. In Table 6.1 the second row is a factor 2 times the first row. This is because slender body theory does not perceive the interaction between the two reference surfaces H1 and H., in the casea= 2, hence the results

for one wing are simply doubled. In Table 6.2. which also refers to slender body theory, the quality number q changes for one value of a. This is because when 5 changes and the total reach A of the two wings together is kept the same (6.2), the geometry of Hi

and H2 changes. 3.00 50 2.00 2.50 SPAN

-a 0.40 0.20 0.00' 0.00 0.50 3.

Table 6.1

,Quality number q slender body theory I < 0,04

Ja\,a 0.12 0.25 028

II 0.782h El 75h I.339h

5.564h 23501: 2.6781s

Table 6.2

Quality number q, slender body theory, a = 3S h C 0.04

a\s 0

0.01 0.02 0.04 0.06 0.12 1.564h 1346h 1.137h 0.759h 0.442h 0.25 2351h 2.220h 2.090h 1 .832k I .578h 0.38 2.677h 2.590h 2.5031, 2328A 2.154h Acknowledgement

This research is partly sponsored by the 'Elichting voor de Technische Wetenschappen% project or. GWI 22.0277,

References

I. Weissinger, Lineansierte Profiltheone be hog° ichformIger Anstromung, I, Unendlich dunes Profile (Wirbel und,Wirbel1 belegungen), Acta Mechanics 10,1970.

Sparenberg, IA., 'Elements of hydrodynamic propulsion Martinus NijhoffPubl., 1983.

Beloiserkovskii. S.M., 'The theory of thin wings is subsonic flow', Plenum Press. 1967.

4, Kerwin, 11., tie solution of propeller lifting surface.

problems by vortex lattice methods', MAT.. Naval, Architec-ture Department Report, 1961.

Using a semillnear and two-dImensional theory. the highest possible efficiency is considered for sculling propellers consisting of twowings,the interaction of which is taken into account. The force and pitching moment acting on each wing and the power needed to move the wings are determined. The influence of the variation of a number of parameters is calculated.

I. Introduction

IN THIS PAPER we consider optimum large-amplitude sculling propulsion. The type of propeller discussed consists of one or two wings, mounted vertically behind a ship, and which move

sideways back and forth. The angles of incidence of the wings are adjusted so that a prescribed thrust is delivered. For

practi-cal purposes two wings are appropriate because inconvenient

moments acting at the ship can be reduced to a minimum. Also.

the efficiency obtainable by a propeller consisting of two wings can be greater than that obtainable by a propeller with one wing

[ t -31.2

This type of propulsion has certain advantages as compared

with a conventional screw propeller. Its efficiency is expected to

be greater than the efficiency of a screw propeller because the volume of water influenced by the wings can be made larger than the volume of water influenced in general by a screw

propeller. Another advantage is that the velocity of the wings

with respect to the water is constant over their span, while the

velocity of a screw propeller blade increases toward the tip. Of course. disadvantageis the complexity of the mechanism need-ed to move the wings.

In the case of one wing the ship is propelled in the manner of a

fish. The tailfin of a fish oscillates in order to deliver a mean

thrust. There have been several studies on the efficiency of

fishtail propulsion (or propulsion by one oscillating wing) [4-81. In those papers the motion of the wing is prescribed, and then the thrust coefficient and efficiency are calculated. In the present work, however, we will prescribe the mean value of thrust and to

a certain extent calculate the optimum motion of the wing(s).

We will calculate the highest possible efficiency and the angles of

incidence that will achieve this maximum efficiency. In the case of two wings we take the interaction on each other into

account.

In order to formulate a mathematical model we will make

some approximations and assumptions. First, the fluid in which

the propeller is working isbesides being

incompressibleas-sumed to be inviscid. Second, we assume the spans of thewings

Mathematical Institute, University of Groningen, Groningen, the

Netherlands.

2Numbers in brackets designate References at end ofpaper.

Manuscriptreceived at SNAME headquarters December 2.0, 1984: revised manuscript received February 2.8, 1986.

DECEMBER 1988 0022-4502J86/3004-0221$00.6710

to be large in comparison with the length period of the motion

and with the chordlengths of the wings. This allows us to use a two-dimensional theory (see (39 as will be discussed in Section S.

Third, we neglect the influence of the ship's hull both as a

boundary of the region in which the propeller is working and as a

cause of an inhomogeneous inflow. And fourth. we assume the

propeller to be not heavily loaded.

2. Some preliminary considerations

-A Cartesian coordinate system (s,y) is embedded in an inviscid and incompressible fluid. The undisturbed fluid is at rest with respect to this coordinate system. We consider two rigid and infinitely thin flat profiles Pi and Pa. Both profiles have the

same chordlength 21, because, as we will see later on, the load of the wings is about the same.

On each profile P, we have a pivotal point Q. around which

the profile can oscillate. The points 0, have a constant velocity El in the positive x-direction and move along periodic paths H, with period b

y f1(x) A(r b), i 1,2 (1)

where we assume the fAx) to be sufficiently smooth. On each profile P, we introduce a length parameter s with s e at the

leadingedge ands= '-Pat the trailing edge. The pivotalpoints Qthave length parameter s = a. hence they are chosen the same

on each profile.

In the following we consider three possible cases of shapes of

the H as drawn in Fig. 1. First, we consider only one path, Hi; hence our sculling propeller consists of one wing. In this model one profile moves through the fluid [Fig. 1(a)] and causes two-dimensional disturbances,

Second, we study two paths Hi and Hpwhich do not intersect each other. In this case the profiles P,and Pscan move side by

side. Then, if thepaths if are symmetric with respect to the z-axis, a gap between Hiand Hphas to exist in order to prevent the profiles from colliding [Fig. 1(b)]. In I2j and [3] it is shown that, from the point of view of efficiency, such a gap is unfavorable.

Third, which is the ease of our main interest, we consider two

paths Hi and Hpwhich do intersect each other [Fig. 1(c)]. Now the profiles cannot move side by side but have to pass one after

the other the points of intersection. Then the z-component of

Journal Of Ship Research, Vol. 30, No. 4, Dec. 1986, pp. 221-241

Journal of

CHAPTER II

Ship Research

On Optimum Sculling Propulsion

W. Potzel

bAlk,,az.P2

collo

Fig. 2 Profiles Pt moving airing their intersecting paths II, tat

lb)

X

Fie. ii Three possible cases of paths Hi Cal one path H,, lb) two

nonintersecting paths H, and H2, and (C) two intersecting paths H, and 1-4

the distance between the pivotal points Q, is given a fixed value The profiles have to deliver a prescribed thrust per unit of span of 0(e) in the positive x-direction. with a mean T. with respect to time. Here e is is parameter with respect to which the theory is linearized. The angles between the profiles and the tangent to the paths have to be adjusted so that this thrust is delivered at the cost of as little energy loss as possible. These angles are called optimum.

Because of the semi-linear theory we will derive, we split the optimum angles into two parts, dat) and Wt. The first part ad) is of ow) and is the angle belonging to the so called -base motion." This is a motion whereby the profiles do not shed vorticity into the fluid and hence no mean thrust with respect to time is delivered; see, for instance, The second part i(t) is of 0(e) and is the angle which defines the so called "added motion... which is responsible for the delivered thrust and by which vortic-ity is shed into the fluid. In Fig. 2 a possible configuration is drawn for the case of intersecting Hi.

Considering only periodic motions of the profiles, the follow-ing relations are valid for (s((t) and 131(t)

a,(t) a,(t + r), 1-1.21,2 (2)

(t + r), 1 1,2 (3)

where r is the time period of the motion

T5

(4)

In the following we will need concepts such as momentum of the fluid, thrust, lost kinetic energyall per unit of span of the propeller wings. The addition per-unit-of-span will be omitted in general and is understood whenever necessary.

3. Base motion

The angles csi(t) follow from the fact that the impermeable and

rigid profiles do not shed free vorticity into the fluid while

carrying our their base motion.

In this ease we find for the unit normal on each profile

/stn(a)t)+ a1(t)J\

ti(t) (5)

cos(a(t I + a1(t)l

where =dr) is the angle between the positive x-axis and the tangent to the path ft, at the pivotal point 0, at time t. The profiles have locally velocity components in their

normal-direc-tions. These normal velocities are

Is ana,,(t)+ cy(t)l V1(t) sinkt1(01,1s1 e = 1.2 i6) where V,(f) is the velocity of (, along the path H,. The first part of n(3,t) is the normal velocity caused by the rotation around Qo The second part is caused by the translation of Q, along H,. The angle cr,,,(t) and the velocity V(t) follow from

df,

au(t) = aretan{dx)x,,,,(t)) (7)

V,(t)

_{.o/ +}

_{ctfd [xc),(t)1}2' U} (8)

where ki,(r) is the x-coordinate of Q, at time t.

In order to let the fluid flow along the profiles we need

vorticities tht(s,t) =s 1 or 2) on them. The index b denotes that this vorticity belongs to the base motion. Using the law of Biot and Sayan these vorticities have to satisfy

e r,,4,t)d

tle

t),,(34).

r(7.

t)

2x. $ 2ir

IT v(i,j,s,o.e) sinkriit) + a1it)1+ crx4a1(t) + exp(t)ilda

71(i,j,s,a, t I

-I-.Isi e (9) where

j'=2ifii l,p 1 i1i=2

"T(t,f,s,cr,t) ro.(t) xg,(t)+ (s a) cosia,(t) + oat)]

a) costal(i) +ni(t)) (10)

= f,[xo(t)1

f

,frys(t)1 + (s a) sinke,(t) + 01401 a) sin{ c At) + cx,,(t)] (11) The Kutta condition at the trailing edges of the profiles is

e)

has to be satisfied, so

rb,( = 0,

i1,2

(12)

The circulation around each profile is constant, otherwise ac-cording to the theorem of Kelvin PI the profiles would shed free vorticity. For these constants we take the value 0. because then we may hope that the neglected tip vortices, which occur in reality when the wings have finite span. are small

JOURNAL OF SHIP RESEARCH

-(I) -e 1 Ts(i,j,s,a, j,s.a.t)

(a

-Ler b,(s.t)ds = 0. i = 1,2 (13)

We first consider the case that the distance between the pro-files is infinite: then the second term in (9) vanishes. This is the same situation as if each profile moves alone in the fluid. Then

the solution for rb,(3,t) of (9) which satisfies (12) is

rb,(s,t)= l[ixo + (t))(a+

e -

Si

+ V(t) sin a(t)( 2 e i = 1,2 (14)

e -

s

Substituting (14) in (13) we find an ordinary differential equa-tion for ode)

(a,(t) + ix(i)ge + 24) + 2V(t) sin( ai(t)I 0 (15) It is seen that if the pivotal point Q, is the three-quarter chord point (a = -e/2d, then

a,(t) 0 (16)

Hence at each moment the three-quarter chord point has only a velocity component in the chord direction of the profile. In this case of infinite distance between the profiles, we have the same base motion as discussed in [101.

If the distance between the profiles is finite we have to take the second term of (9) into account. To solve (9) in this case, we write 1),.(s,t) in the form

r

Re, + O)(a 4. e - s)

e+ s A

+ V,sinad + rb,(s,t) (17)

e - s

Substituting (17) into (9), (12) and (13) we obtain, together with (2), four equations for the four unknowns T(s.t) and ai(t) It

1,2):

fe rbfk,i.or, coda, + did T sin(a, + (kunda

s J-1

If (18)

0 (19)

16,W+ (t)Ie(t + 2a) + 2ireV1(t)siniat1(t) +ra,i)da 0 -e

(20)

a,(t) cqt + r) (21)

We use an iterative procedure to solve equations (18)-124 Starting with some chosen ("mar) we calculate from (20) by means of a shooting method an cii(t), which satisfies (21). Next we substitute r(a,t), using (17), and the calculated chit) into the right-hand side of (181 and solve for a new T b,(a.t), which satis-fies (19). With this rh,(03) we restart the calculations This procedure is carried out until the maximum change in ot(t) per iteration becomes smaller than a prescribed value. We begin with r'Nia.t) T. O.

This iterative procedure can be interpreted as follows. We start moving the profiles as if they were infinitely far away from each other, while their circulation has the constant value zero, As the real distance of the profiles is finite, they influence each other. Then we have to correct the motions in order that the circulations remain constant. Now the influence on each other has changed. So again we have to correct the motions. This has to be continued until the corrections become insignificant.

We now describe how equations (18) and (20) have been

DECE5481ER 1986

solved numerically. First we discuss (18) when its right-hand side is supposed to be known. .Although it is possible to write down the exact solution of this equation [111, we preferred, in order to avoid two-dimensional integrations, the following collo-cation method.

We introduce the transformation

a = e - sin2e cos2e,

0'

(22)

-

2 Instead of solving (18) for f(a,t), which is likely to posess a square-root singularity at the leading edge a = e, we solve for

r;',,(0,t) 11b,(a,t) sine coup, a cos2e (23) The function fL(e.t) is approximated by a cubic spline 112, 13 on an equidistant mesh 0 = 00 < el < < / 2 of Kt w/2]. On this mesh a basis for the cubic splines is formed by the 8-splines j13] (Fig. 3) ea 4 [Al 1

/0 -

,)3

1

II

(40 ,.,)3 lai

)

3(4' v4)2 (c °'k+2)+ 1.sek_, < sog.wk

Bow) w 4 [Al lAl

4 [Al [Al 3_{(c° - "1- ('' - 'Pk 2)+ 1,.ci < ,g,} ,Pi+1 sok+1 < sok+a (24) where , - 00 - 214,1. 0_, - 00- [Al, 0,+142il 2p and 00.2"Op,+ 21AI

In the following we write 11(03), making no distinction be-tween its exact and approximated values, as a linear combination of these cubic 8-splines

p+1

11(4p,t) cd,(t)Bfk,) (25)

For some number of moments t of the time period we deter-mine the unknowns Aden) with a collocation method. In the collocation points the integral equation (18) is satisfied. For these points we take

01,2 <

= k 1 k I, , p + 2 (26)

2(p + 2)

(

2),

.pk-2 55-1 Ork 04.1 h1.2

Fig. 1 Cubic B.spline

SO44.2

8 s01.t

-woke of first profile

strongly oforrneo woke

\

\

wake of test prof,.

wokei of seccnd orof.ke

Fig. Deformationof a wakeby

"Intersection-Together with the Kutta condition (19) we have p + 3 equations for the p + :3unknowns calm).

Next we discuss the solution of (20) when Niter / is given. As

mentioned before, the angles a5(t) can be determined by a shoot-ing method. If e +ga> 0, we start with some choseninitial

crate) and solve cg(t) on one period Ito, to + r) by means of a

Runge-Kutta method. Then we repeat this procedure with auto + r) as the new initial value until the difference (ago + 7)

-a,kt5)1 becomes insignificant. However, if + 2a <0. we start with an initial value arts1- r) and integrate, by means of a

Runge-Kutta method, in the reverse direction towards to. Then

we take in the iterative procedure ctAto) as the new initial value.

If awi -en,when the pivotal point (3/, is the three-quarter chord point, no shooting method is needed and hence equation (20) is solved directly.

4. Optimum shed vorticity

While carrying out their base motion the profiles induce finite disturbance velocities in the fluid. These velocities follow from the bound yorticities onthe profilesas calculated in the preced-ing section. For the x-component ukx,y,t and y-component

c(x.y.t) we find, using the law of Biot and Savart:

We define the wake behind a profile to be situated at the fluid particles which have passed the trailing edge of the profile carry-ing out the base motion. When the profiles together deliver a prescribed mean value of thrust they shed free vorticity. This vorticity is shed at the trailing edges of the profiles and carried away with the fluid particles which pass these trailing edges. As

the free vorticity and the added motion are of0(f)we assume the free vorticity to be situated at the previously defined wakes of the

profiles. This is the linearization assumption we make with

respect to the shed vorticity.

If the two paths (1) intersect, one of the profiles, say P2,will

-cut- the wake of the other one. However, because of the

stagnation pointon, Ps it cannot really cut the wake of Pi. A

Fig. 5 wakes 04 intiniteiy far behind profiles in the case of intersecting

paths

small part of the wake is strongly deformed; it moves on with the

-cutting-profile Pc(see Fig. 4). The free vorticity on this small part of the wake is spread over a great distance due to the strong deformation of the wake. Using Kelvin's theorem the vorticity at this strongly deformed part becomes very small; in the follow-ing, therefore, we neglect these strongly deformed parts. By this we assume also the pierced wake to be a smooth and one-valued function of x and t.

In the neighborhood of the profiles the wakes are time-depen-dent. As the profiles move on, the wakes assume a fixed periodic

shape. Thisis seen from (1). (2), (27) and (28) by an asymptotic consideration. In order to calculate points of the wake, we determine for a number of moments t the coordinates [xT(t),

y-e(t)) of the trailing edge of a profile. By solving the coupled

differential equations (27) and (28) with kost), yr(t)I as initial

values, we know for t > t the position of the fluid particle which

has passed the trailing edge at time t. Doing this for a number

of moments we approximate the shape of the wake W, of the

profile P, by interpolating the calculated points with a cubic

spline. Making no distinction between the exact and the approx-matedwake, we write in the neighborhood of the profiles:

417 y Cx.t) (31)

where an asterisk denotes that !his part of the wake is still in

motion. Infinitely far behind the profiles the wake has become time independent and periodic, with period b:

W,, ,g,(x) g,(x + (32)

To avoid numerical difficulties in the integrations in and (28) we carry out the same transformation (22) as in the previous

section.

Infinitely far behind the profiles, the optimum free vorticity is characterized asfollows (II. The two-dimensional wakes W, are

placed as impermeable and rigid surfaces in an homogeneous

flow with velocity A in the positive x-direction. Then we need

vorticity Ay, on W, in order to make them streamlines. This

vorticity, when A is chosen correctly, is the free vorticity left

behind by the optimum profiles.

In order to find X-r, we have to solve an obvious boundary'

value problem for the potential function

Aix + ip(x,y)) (33) which is defined outside of the wakes and where Ap(x.y) is the

disturbance potential caused by the W. The boundary-value

problem for co(x,y) reads

a2 Coo(x.Y)

(at-az , +

) (xY)

0 (34) x ay' dot

a.

_{(r,y) - cos(m,x), y g5(x) (35)} an,

JOURNAL Of SHIP RESEARCH

i(x,y,t)=oix.y,r)=

-

, n,(3.01ds 2/r

t

x 515.1112 + (y -E,cs.t)ids

I

(27) ysx.y.e) v(r.y.t) = -t (x - + [y (28) where

&(s,t) = x9(t) + (s - a) costa,(t) + et(t)] (29)

and

Nom f,Exc,,(t)( + (s - sin( a5(t) + au(t)I (30) n,(3.012. 1 _rb(s.t)ix

-a) H, b)

do Ow

(x.y). (4.0 .-111: 1v1 (38)

Ox ay

whereñ, is the unit normal at W, pointing from the"-' side Wr

toward the -4- side W; and .er Isthe unit vector in the x-direction (see Fig. 5).

Because of the periodicity of W,, it follows thatalso is

periodic)

78(X ± b) = 7,(x) (37)

and because the total strength per period of 7, is zero, it follows that also kofx,y) is periodic with period b. The velocity induced by an infinite equally spaced row of concentrated vortices, with distance b, of equal strength 11 Is given by [141:

r

[sum - is): sin (x -El]

b b

,_, 2r , 2r

2h _coshZr_{(y - 0- cos '"(4 - Er}2r

b b

Using (38), the equation that yr has to satisfy becomes (35):

'where

(COa

b n,.4)) (-A'krs

cos(nry) =

i")

+ 44/2

Because the mean value of the free shed vorticity is zero, (39) has to be solved with the condition

.4(0,11 + g(02tle w 0, I w 1,2

Then indeed the disturbance velocities (ato/a4)(4.0_{and (No/} Ziy)(x.y) vanish as lyi ot (36).

In order to calculate thevorticity 71(x) (Iw 1,2) we introduce a cubic spline in the following way. In the foregoing we have calculated thecoordinates rAt,,),8,((..), of points of the wake Wi infinitely far behind the profiles. One period of W, consists of N

intervals (x4(4,),i(t..1)1. These intervals are not of equal length: however, this is not a hindrance On this mesh a basis for the cubic splines is formed by the cubic B.spline

ft Jo

DECEMBER lose

1

(F) {sin T- - 0 costnky) - stash .tbr jg,(x)- gi(k)] cognor)141 g,(;)gde

;

21, ja 2r 27r cogn,.4)) a° 1,2.

tvi

cosh

Val - cosy(' 0

(38)

(40

1

o ,r -4 i,(tk_k), z > Spk.,2)

aukir - ;44-2)13' ,i,(4_5)< x Itir(tk._,)

81.(k.x)= aids - ;Ma + aukfx - -A,(tk)12 + tri,k[a - iMkn+ I, ;1(k-i) < z _iiled aNkix - it(tx)13 + aulz - i(k)j2 + aikkix - i,(4)1+ 1, i,(tkl< x

awls - ir(tk+2)11 _{..71141.1) < 4 3 iXtt+2/}

satisfied strictly. With this constraint we can express for each. s one unknown ba into the remaining ha, (n rA k). This decreases the total number of equations and unknowns by two. The remaining 2N linear equations for the 2(N - 1) unknowns are solved by means of a minimum least-squares method. The still unknown factor A (33) will be determined in the next section.

5. Lost kinetic energy

The scalar A follows from the condition that the mean value, with respect to time, of the thrust Is?. The equation from which A follows is the balance of momentum

rt = -Ap 1(6) (44)

where p is the density of the fluid and p1(b) is the momentum of the fluid far behind the profiles over a period oflength binthe r. direction induced by so. So we have to calculate

ru-

:C431 (42)

to 02

ihradwrdyi dz Pf:

tE div

gradr4dYli

-

ob),- - 674 Zr 2pUI(b)

Because for yI = we have (avt/Dx), (acioo) 0, and as we' mentioned already to is periodicwith periodb, wecan rewrite

(45) as

=

r7 kodilk. .)1A,Pdx

o 46)

where isthe jump ofthe potentialover

(Veil= " tot - on W, (47)

In terms of y, and with the use of (32), we can rewrite (46) as

2 b

ri(b)

{It

+ g(E)2(it -142,11(0)181(x)4s

.2)

-r4ogi(r)Vi i-g,(x)2dr (48)

where xi( i)+

-

N.,. a and the have to be determined such Using (48)thecoefficient follows from (44).

'that Bi(k.z) and its first andsecond derivativesare continuous. Next we determine the men value of the kinetic energy1losty, Inthe followingwewrite 'Mr),making no distinction between by the propeller per unit of tim

the exact nd the approximated vorticity, as a linear combination of these cubic 8-splines

71(X)c.. b*L3,(k.4c K43).

We determine the unknowns ba again by means of a collocation' method. In the midpoints of the intervals [x,(4). xt(brai)] the integral equation (39) is satisfied, taking j (j 1,2), With (41) we have 2(N + 1) linear equations for the 2N unknowns ba,.

As the total amount of the shedfree vorticityis zero, (41) has to be By this wefind for the efficiency

;(49)1 (um) [g1(x) -= (41) 1(1') (39) A

DU 7' 1-1

i = - _ 1 b1. (50) TV + E 2pU'l(b).1

We now consider. instead of the propulsive device, consisting of two profiles, an actuator disk [If For the working area of the actuator disk, we take the strip perpendicular to the x-axis with sides perpendicular to the lay) plane and of width A, where A is defined by

A max Iggi) - g(E2)1 (51)

which is also the working area per unit of span. The disk has constant load f in the negative x-direction such that its total thrust per unit of length along the sides of the working area (which are perpendicular to the z,y plane in Fig. 1) is T; hence

=

Its velocity is the same as the mean velocity of advance of the profiles: hence equals U in the a-direction. This actuator disk leaves behind, per unit of span and per unit of time, an amount of kinetic energy E0. which equals

D2

2pAU

The quality number q of the propeller is defined by

E 1(b)

q = = (54)

Ab

The inequality in (54) is caused by the following conditions. With respect to the shed vorticity the theory is linear and the propeller and the disk are assumed to work in an undisturbed

1+,,(f) ),;(E,t)Ity,(s.t) - 0,01 sm(a, + au + 0,)+ (x,(s,t)- fl cas(oti + + J3d1

2Xir

-

(XAS,t) El2 tY4(s.t) 4. g./"E't )2dt

fluid. Then an actuator disk with a constant and time-indepen-dent load has the smallest possible energy loss compared with any propeller with the same mean velocity of advance II, the same mean thrust T, and the same working area A. Hence q

Using q the efficiency can be written as

{1+

t

1-,

2pU2qA

Introduction of the thrust coefficient Cr, defined as

Jr

I',(cr,t)lly,(s,t) yl(a,t)Isin(a, + + r.?,) + [x,(s,t) zi(o.1)]coga, + all + ddida 00(5,1) =

_e tx,(s,t) x1(cr,t)I2+(y,(s.t)Y1(c.t)12

=

1 p U2A 2

allows us to write the efficiency as r, =

6. Added motion

The angles $1(t) which determine the added motion, and which have to be superimposed on the angles a,(t), have to be

calculated such that a prescribed thrust is delivered at the cost of as little energy as possible. In other words the .it(e) have to be determined such that vorticity is shed at the trailing edges of the profiles, which Infinitely far behind the profiles equals the vor-betty calculated in Section 4. These angles follow from the normal component of the velocities at the profiles and the Kutta condition at the trailing edges.

The shapes of the wakes are known (Section 4) for each x and for cache. In the neighborhood of a profile the shape of its wake W7 and the optimum free vorticity are time dependent. In the following we denote the time-dependent optimum free vorticity by X-y:(x,t) The vorticityey;(x,t) follows from 'ear) by the theorem of Kelvin

Yids; "nds, (58)

where ds; is an element of length along W7 and ds, is that element of length along W, into which ds; is transformed by the base motion.

The unit normals at the profiles are -sin(ce, + all + gls) =

cos(a, + +13,)) (59)

The normal component of the velocity of a moving profile is given by

(a - a)(a,+ + () - V1(t)sia(a, + (60)

We need vorticities nu) on the profiles in order to let,

together with the optimum free vorticity in the fluid, the fluid flow along them. The normal component of the velocity on the ith profile induced by the optimum free vorticity on the jth wake W; is, using the law of Blot and Savart

where

(61)

s1(3,t) z0(t) + (s - a) cos(a, + a. + 3,), (62)

y,(s.t) f iixo,(01 +

-

a) + + et,) (6()

and xr,(t) is the x-coordinate of the trailing edge of the jth

(55) profile. The contribution to the normal component of the veloc-ity on the ith profile caused by the vorticveloc-ity I on the other profile is

The normal component of the velocity induced by the vortIcity on the same ith profile is given by

r

,Icr.t)da

u.(s,t) = (65)

21r s

-The normal component of the velocity on a profile induced by all vorticities has to equal the normal velocity caused by the motion of the profile, so

- a)(a, + a + - V1(t) sin(a, + 3() on + °us + utzt + °so (66) The vorticity X-r,' and the angle 0, are small of 0(e). When only terms of 0(1) and 0(e) are considered, we can replace (66) by

JOURNAL OF SHIP RESEARCH

(52)

E

+

1;it 41') v t(e)(sinati + Arcosad iriP) = >

L

ij(E.1)1C)(1.1.31F)04 + Ficr.ollc(rk.s.o.f )+ 0,1C3(t.k.s.cht)+ )5kIcli:k.s.ce.tnda 2r -e. do, 0.14 (67), 2r s - a 'inert* K10,14.0)

e[y;(1.t)-gi(f.t)) sin a, +au)+ l4(s,t)- El coga, + a) [4(3.1)- + Iy7(s,r)- g;(k,t)12

+ gi(Of 1(68)

IC2(1,k.s,e.t)

- y(er.1)1 =la? + au) +140.0 - x(cr.t)1,cos(att+a,i)

lxr(s,t)- x,(7.01 + 00.012

(69)

FC.5(ck.s,a.t)

[K(s.t)- y(o.0)coa(a,+ cy,) - kr(s.0- 4(e.t)) Mita, + ord'

)44.0 - 4(6412+ (Y7(3,0 - liger.012

1[;,*(a,t) - xi(o,t)I' - 14(a - YZ(a.t)121(a - a) sin2lai + aid

asr(s.i) -4(tr,t)12 +

-

14(001212

2)x;(s.t) - 4(o,t)IIY;(3,t) _{0E01(a - a) aaa2Cm'}

_aj

(70) Ilx;(s.t) - 4(a,k)r + (47(s.k)- 4leht)12/2

K4(1.k,s,a.t)

1E4(0 -Yi(a4 -14(a.t)-410.019(a- a)na(a,+ a <irk+ aki)

lis7(3.t) -4(0.012+ v(ait) - 4(c.1)11t

+214M- 4(c.t)11Y74.0 - ygg.01(ff - a) coda, + au + oh) Ilx;(3,t)- 4(a.t)]2 +(y7(s.t)- aga.f)1212

P1) with

C(s,t) xo:(t)L+ (a - a) con, +aid 72)

y:(a,t) f,(x9.(0),+ (a - a) suaa,.+ a) (73)

We split the bound vorticity F,(s,t) into two parts

Tp,t) Tb,(s,t)+ "4,(s.t (74)

where I' bi(s.t) is of 0(1) and is the vorticity belonging to the base motion as discussed in Section 3 and xr,,3, ) is of 0(e) and is the extra vorticity caused by the added motion. With (9) we can eliminate all terms of 0(1), becauseTE.,(5,1) satisfiesthe equation' for the base motion.

The remaining equation has only terms of 0(e):

DECEMBER, IEEE

M- 4-

,(t) cos.; iriU) ,;(E.tiKt(t,b,Leal4

r

fl rbk(c,f );(1.k,r.ctfriti Fbk(o:t) rt K4((,k,s,a.t)do + ;k(a,t);(ilca.a.t)da 21r -r x _{1%,(a.t) do.} 2r s

The Kutta condition at the trailing edges yields. using (12) and ,(74)

t

1,2 (76) Furthermore, we have another condition which follows from the 'theorem of Kelvin

d

r,fr,od.

--ri(xt(t),t)lit Troy, dt

(77) where Vr,(t) is the velocity of the trailing edge of profile I), and tir,(t) is the velocity at its trailing edge, induced bythe vorticity

rb,(s.t) andthe vorticity Ihw(s.t) of the other profile Pk. In order to solve the linear equation (73) withthe condition

(76) we write r1(3,0 as follows

131 i + s 01(t)

1;js.t) w (t)2(e + a - s) +

t -

A

{

tI 2Vt(s)casoir)

IF"

+ C(s.t) +

tiq

G2,(s.t)

e - s X

+ Gu(s,t)+ C48,t)+ GN(s,t), iki* ,I '(78)

gu(3;t1=71tee,(0./11,-e mix

Gu( - 2,0 . C2A co - Ca- t.t) w Cu( - e so = 0 (80)

where

and

The vortmities C,(s.t)(n 1,2,4,5)0 = 1.2) and angles di(t)(ft''II 1,2) have to satisfy

1 g Cu(r.e)d 4

it

2r

s-

e ar f-e

1

2rfee.;As -II) de' 2WW

I-

t

ri I' ur(a,t)K4(1,1c.s.a.Ocia, 1

it cipm)

1

I

.40 do -6(ka)KI(i,J.s.E,t)cli, 2r Ile s - a I-I _e a - a fe G,44(e,t) de

il

Gula.tIK,(04,a,t)da

-

--2r 2r -e 1 Jr 0 i (81c), ,k (75) sk(cht)1C3(l,k,s,o,t)do,. l'k1r6 (8.1a)l Jr (81b) (s - + +

-r

[y(8.1) [y;(s,t)

--

+ + +

-r,(-e,t)

-i 1,2

-

i

-

i s

I it Cstor.t) 1

it

liftk2( i + a

a)/i +

do. = 2rr r_e s _{27r ..e L X} \t

_i

+ '- -[2V,eosct,.,,s& a+ Ot

4- G,itt,t) + C. ib(tt,t) + Gu(a,t) K2(i,k.s.a.t)da, X

k (31d)

It is seen that the vorticity Go,(a,t) (79) is chosen such that the total vorticity is continuous in the neighborhood of the trailing edge on the wake and the profile. Then the normal velocity in the right-hand side of (81c) is finite at the trailing edge of the profile.

Integration of (77) yields two coupled differential equations for 3,(t):

-I

,itrete + 2a1 + itl, {27reV,cola, +jGi1(s.r)d,s1

t t + t3k f G2,ks,tkis + N le Gai(s.this + X 1 G 4,(s,t)ds

-t

+

XIt

G5,(s.t)ds = X {- j xi-,(a))V7-.(a)da + C,ito)},

-t

( 1 1,2, k * i (82)

where XC,(to) is the circulation around the ith profile at time to. The right-hand side of (82) is the circulation around the ith profile at time i. The constants CAto) are determined such that the mean value with respect to time of the circulation around each profile vanishes:

C,(to) 14.." -f:Exa)iv 7.,ta)do.dt, t 1,2 (83)

r t,

Again we hope that by this choice the tip vortices which occur ill reality, where the wings have finite span, are small.

The vorticities C,(3.t) In are expected to have a square-root singularity at the leading edge s = e. To avoid numerical difficulties we carry out the same transformation (22) as we used in Sections 3 and 4. Analogous to (23) see solve in (31), instead of Cm(s.t), for

G,;(4,,t) Cns(s,t) sirup cow, s cos2rp i 1,2. n 1,2,4,5 (841

Equations (81) and (82) with condition (80) are solved by means of an iterative procedure analogous to the one used in Section 3. We solve from (81a,b,c), G71(0,t), C.1(so.t) and GI,4,1), where we approximate the unknown functions Gn.,(.,a.t) (i 1.2) (a = 1,2,4,5) by cubic splines (see Section 3). We begin the iterative procedure with CL(ca,t) a 0 and 431(t) eØ, Substitut-ing CGp,t)(t= 1,2), in = 1.2.45) and /3(f) in the right-hand side of (81d) we solve for new G(e',t), With the new G;(,,a,t) we solve from )82) new periodic functions 110). by means of the shooting method described in Section 3. Then we restart the calculations with these G;(so,t) and Oat). The procedure is carried out until the maximum change in di(t) (I = 1,2) becomes smaller than a prescribed value. In the case a = -12, that is, when the pivotal points Q, are the three-quarter chord points, no shooting method is needed and hence we can solve dd(t) directly from t82).

lithe propeller consists of one wing, (78) simplifies to

e+ s

r.,(s.t) 2(e 4- a - s) + 2V(t) cosai(t) 12212

e X e

+ c,,(s,t)+ G ace), I I (85)

where Gu(s,t) is defined in (79) and Geas,t) has to satisfy the condition (80). The vorticity Ge,(s.t) follows from the equation (8 Lc). which in this case is simplified to

1 41 Giiker.t(da I 1,1")_{7, (Z,t)Kiii,t,s, ,t1dE}

27 s - a 27 _..,

1 G31(a.t) da

I (86)

2.7r r_e s - a

The differential equation (82), from which we solve the angle 8.(t), changes into

71,e( + 24) 4- /3,271V, costs, + is C34(s,t)ds + X C,,,(s,t)ds

-e

AI-

1.:[xr,(0)1V1.(olda +Co,)} ;87)

Now no iterative procedure is necessary to solve Ct (86) and 0(i). We solve G(9,t) as discussed before and determine 131(t) by means of the shooting method.

7. Force and pitching moment acting on a profile

The motion of a profile is time dependent, therefore the force and the pitching moment acting on it are also time dependent. The profiles together deliver a mean value, with respect to time, of thrust T. This thrust is the mean value with respect to time of the component in the x-direction of the forces acting on both profiles. In order to determine the force we split it into its

component F5(t) in the normal direction and Fa(t) in the chord-wise direction, which is the suction force. The normal force as well as the pitching moment are caused by the pressure differ-ence between the two sides of the profile. The suction force is caused by the flow around the leading edge.

We first discuss the suction force F5(t) of the tat profile; it is known to be [II:

F5(t)= lim [r ,(s e 038)

4 ,-e

In the previous section the vorticity on each profile is calculated. Using (74) and (88) we find

Fa(t)lim Itr (5,1) + xr.,(s,t)ist2

4

pie

Inn (rh,(s,t)ve 312+ len bi(s,t)

4 2

+ X2 111Llim (r,(s.r)42

4

r 40) + XF ub(t) 4- W.F. (89) We use a sernilinear theory with respect to the prescribed thrust. so linearizing the suction force means neglecting the last term on the right-hand side of (89). If we do, however, the suction force can become negative at some moments. As this conflicts reality, we take all terms into account. The second-order term also becomes significant when the suction force F(t) caused by the base motion becomes small and thus all terms in (89) become of the same order.

JOURNAL OF SHIP RESEARCH

-

Integration over the ith profile of the pressure :difference between the two sides of it yields the normal force Fin(lP

F,(t)t IP17.(g.Ods (P-(8.1)- pli.048 (90) where p is the pressure in the fluid. The and -" stand for

taking the value at the and "-- side of the profile and the 'index I means the ith profile. The pitching moment WO

around the pivotal point Q, acting on the lilt profile is ,A,I,(0=J (s - a)(037.(s,t)d, a sip17.(s,t)ds - aF,(t)r 1(91);

The pressure jump over a profile follows from, Bernoullia equation for unsteady flow. When w*(x,y,t) is the time-dependent velocity potential in the fluid, we find

alw;11-(s,t)

00.7.(3.0 = +

2

livietnati

(92)

We want to express this pressure difference in the known 'hound and free Tamales. Therefore we introduce a time-dependent system;, i,,I of which the ;-axis is parallel to profile P. the yeaxis is normal to P,, and the origin coincides with pivotal point Qt. Hence

.ii(x.r,t) fr-xo(t)icos(a + a + 0,)

+ -./A4(1)1sitga + a, + 0,) 47644,0"

-

ro,(0isnarls + a1 + Oil

t) ±

- at(0.01rsin(ra + + 0i) - trt(3.0 - 91,(a,t)( coach, + + OtHda'

at(s,

V&A

2. 1-t ifx,(a,t)- xh(a.t)12 + iY1(3,t)- yge.011

± x

firti° ,4(2s)'llA(2.0 - jI Maui, + a5 + /9,)-

- Kum] co(., + aV

l+ I

Lit + EY5(3.0- gi(V)12

+ C(0)14

k (96)

+ - ilfz4(t)1 cea + a, +

i,wt

(93) where lAp,(t).11(x9,(01 describes the path of the pivotal point Further, we introduce the function co; (x.,,y,,t,), which gives the values of the potential function 0. (1. y.t) in this time-dependent system; hence

;(i.

sefr.y.f) (99)

DECEMBER 11188

The derivatives ekosini, and Otis/OE/Late the velocity compo-nents of the fluid parallel to the; and y, axis. as measured In the x.y,t system.

Using this time-dependent coordinate system, we find for the pressure difference at the a h profile

10,1.7(8,1) p{[(efit- V, cosaA + I3,) + [0, sink, +

=1(e + a5 +

+

[(ef;?)2

(W)211

(84

f2.lcrI°1:t- %floosie; +0/)[41-i'l

31x,

t[(t:)2 (Z]+-Pa't)

+ 1;(3,t)V,(t)cosiab) + 8,(t)1 at

,(s,t)u5(s,01 (95) where tiara) is the tangential velocity at the Ith profile induced by the shed free vorticity in the fluid and the bound vorticity at the other profile. The tangential velocity uu(s.1), expressed In the known vorticities. reads

where

x,(s.t) = mo(f) Is - a)mac', + - (3,)1

y,(s,t) hti0a)1+ - a/ Mach, + at + 33,) (98) and xr,(t) is the x-coordinate of the trailing edge of the ith profile at time I. Linearization of (96) with respect to quantities which are of OW yields

.55(3

t) - sties))/ smax + - (03,0- yi(o.)1cos(ccu + ;Ada

.0

2. J_t _{144.0 - 4(a.012 + [Y7(s.i)- Yila.012}

- 4(1.1)1 malau + aJ - b(s.1)- Orr/10,0a + adlaIr ).± r r skce.t) '

2. -t

(zi(s.t) zgr,011 + (117(1.1) rgsCf)12

r r

/co 14(3,1)-4(e.01 car at, + + (y7(s,t) yge.01 sin(ao + ar)4

-Le

_[0.0

_{4(at)? + [OA - YVaM12}

+ 2(s a)

do- +

fl (s.t) - 4(ent)1 =la/ + - 14(ct) - Oct.t)) costa + [41s,t)- 4ga.0 + iY(st) Y(a.t)18

iler,wation (99) continues content], -e (y -(x Q,. = + + =

-a, a, [y,(s,t) [x,(s,t) -(5 +

-a,) a) X

-a,) ad}2]

-[v (97)