On the optimization and the design of ship screw propellers with and without end plates

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Mekelweg 2, 2628 CD Delft Tel:015-786873/Fax:781836

On the Optimization and the Design

of Ship Screw Propellers

with and without End Plates

TECHNISCHE UNIVERSITET Laboratorium vow Scheepshydromechanica

Archlef

Mekelweg 2,2628 CD Delft

-rd.: 015-7U8873- Fax O15-78t33

ON THE OPTIMIZATION AND THE DESIGN OF

ON THE OPTIMIZATION AND THE DESIGN OF

SHIP SCREW PROPELLERS WITH AND WITHOUT END PLATES

PROEFSCHRIFT

ter verkrijging van het doctoraat in de Wiskunde en Natuurwetenschappen

aan de Rijksuniversiteit Groningen

op gezag van de

Rector Magnificus Dr. S.K. Kuipers in het openbaar te verdedigen op

vrijdag 29 november 1991 des namiddags te 1.15 bur precies

door

KAREL DE JONG

geboren op 2 februari 1963 te Geleen

Promotiecommissie: Prof. Dr. Ir. H.W. Hoogstraten Prof. Dr. Jr. G. Kuiper

Prof. Dr. A.E.P. Veldman

Promotor: Prof. Dr. J.A. Sparenberg

providing applied mathematics, beginning with Euler, such a magnificent scope for the highly varied exercise of its power and elegance ? "

Met dank aan de stichtingen TCC en STW voor een bijdrage in de drukkosten van dit proefschrift.

Acknowledgements 1

Preface 3

References Preface 9

Chapter 1. ON THE INFLUENCE OF CHOICE OF GENERATOR LINES ON THE

OPTIMUM EFFICIENCY OF SCREW PROPELLERS 10

Summary 10

1.1 Introduction 'to

1.2 Basic notations 12

1.3 Optimization and efficiency 17

1.4 Symmetry property 21

1.5 Some aspects of the optimum free vorticity 24

1.6 Numerical method 30

1.7 Check on accuracy of numerical method 37

1.8 Numerical results 43

1.9 On the shape of end plates in relation to

viscosity 50

Acknowledgements Chapter 1 56

References Chapter 1 56

Chapter 2. ON THE OPTIMIZATION, INCLUDING VISCOSITY EFFECTS, OF

SHIP SCREW PROPELLERS WITH OPTIONAL END PLATE 57

Summary 57

2.1 Introduction 57

2.2 Preliminaries 61

2.3 Relation between circulation, chord length and

maximum profile thickness 65

2.4 Distribution of vorticity at the reference surfaces 70

2.5 Energy loss and thrust 82

2.6 Formulation and solution of a variational problem 88

2.7 _{Some properties of the potentials Oh 02 and 03 95}

2.8 Iterative determination of some design

2.9 Estimation of the maximum thickness distributions 114

2.10 Some aspects of the optimization method 121

2.11 Some results of optimization 131

Acknowledgements Chapter 2 153

References Chapter 2 153

Chapter 3. ON THE DESIGN OF OPTIMUM SHIP SCREW PROPELLERS,

INCLUDING PROPELLERS WITH END PLATES 154

Summary 154

3.1 Introduction 154

3.2 Basic notation, geometry and cavitation criterion 157

3.3 Outline of optimization and design method 164

3.4 Optimization method 168

3.5 Design method by using lifting-surface theory 176

3.6 Numerical results on small scale and large scale 187

3.7 Numerical checks for propellers without end plates 199

3.8 Final modeling of propellers with end plates 211

Acknowledgements Chapter 3 224

References Chapter 3 224

Appendix A. ASYMPTOTIC EXPANSIONS FOR CALCULATION OF VELOCITY

INDUCED BY TWOSIDED INFINITELY LONG HELICOIDAL

VORTEX 226

Summary 226

References Appendix A 234

Appendix B. SERIES EXPANSION FOR CLASSICAL OPTIMIZATION FOR A

CLASS OF SCREW PROPELLERS WITH END PLATES OR A RING 235

Summary 235

References Appendix B 240

It was a pleasure working on the subject of this thesis. Not in the

least because of the valuable and stimulating contribution of some people. In particular I would like to express my gratitude to Prof. Dr. J.A. Sparenberg who has been an excellent supervisor. It is a privilege to work

closely with someone who has lifetime experience in fields like this. He has

been very willing in transmitting his knowledge and way of thinking to me.

Furthermore I am grateful for carrying out the research as an employee

of the Technology Consultancy Center in Groningen. Especially I am indebted

to Drs. A.J. Postma. He played a key role in getting off to a good start and

saved no time arranging things conveniently for me. His continuous interest

in my work from the viewpoint of commercializing knowledge-based innovations is greatly appreciated.

Also the contribution of' Trig. J. de Vries is gratefully acknowledged.

He has kindly assisted me in preparing his lifting-surface computer code for my applications. This has saved me a lot of time.

The project was carried out at the Department of Mathematics of the

University of Groningen. Starting financial support was provided by the

National Foundation for the Coordination of Maritime Research (CMO). The Dutch Technology Foundation (STW) has financed the project for the greater

part under project number GWI59.0819. For organizational support of the STW I wish to commend Jr. F.C.H.D. van den Beemt.

Through the user committee of the STW-project, a fruitful cooperation was achieved with people from the Maritime Research Institute Netherlands

(MARIN) and LIPS BM. Propeller Works. In particular Dr. Jr. W. van Gent and

Dr. J.A.C. Falco de Campos of MARIN and Jr. T. van Beek and Jr. H.J.A. van de Vorst of LIPS must be mentioned. These persons are heartily thanked for

the many discussions and their efforts, which appeared to be invaluable in

developing ideas and research strategy.

Further thanks go to the reading committee consisting of Prof. Dr. Ir.

H.W. Hoogstraten, Prof. Dr. Jr. G. Kuiper and Prof. Dr. A.E.P. Veldman, for

reading the manuscript and for giving comments; some colleagues of the

Department of Mathematics for their advice; Drs. S.J. Folmer and L. de Lange

LL.M. of the Zernike Science Park for the period of working together; the

Working Group Supercomputers for supplying me with a budget from the fund NFS.

2

-Finally I would like to thank my friends and relatives for their

interest in my work, and my wonderful Janieta for her support and forsharing with me many other pleasures in life.

Groningen, October 1991, Karel de Jong.

PREFACE

A screw propeller with end plates, as considered in this thesis,

is a

screw propeller that has attached to the tips of its blades one or two wings,,

located either at the pressure side of each blade,, or at the suction side or at both sides. In case the wings extend to both sides of the screwblades we speak of "twosided end plates", in case the wings are at only one side we

speak of "onesided end plates".

This thesis is meant to give insight into the question as to whether

optimum screw propellers with end plates can have a higher propulsion

efficiency than optimum propellers without end plates, which we will call

conventional propellers. Therefore optimization theory is an important

subject in this thesis. Furthermore questions are addressed of how to design

conventional propellers and propellers with end plates according

to the

optimization principles.

The thesis consists of three, chapters which were individually prepared

for publication elsewhere. This is

the reason that

in Chapters' 2 and 3'

sometimes repetition of previously discussed material occurs. Especially Section 3.2 is

a recapitulation of

things already discussed' in Chapter 2..

Furthermore, only a few times, notation can be different in the respective

chapters. For instance in Chapter 1 the number of screwblades is 'denoted by IN, in Chapters 2 and 3 by Z.

The theory that is used in first instance is linearized and assumes

'incompressible and inviscid fluid.

Later on in this thesis the influence of

the viscosity of "real water" on the efficiency of a propeller is estimated

by using approximate resistance formulae for the viscous drag of the

screwblades and end plates Also we have included in, later chapters some

nonlinear corrections, in the hope that our results are better applicable for

more heavily loaded screw propellers.

Some further simplifications are as follows. We assume no influence of

the ,ship's hull or the free water surface. Hence the incoming flow to the

propeller will be homogeneous., The hub is modeled in a simple way. In

Chapter 1 by considering it

as a circular shaped ring

with a circulation

distribution in circumferential direction of the ring. This means that in

fact, as follows from the optimization theory, the hub is assumed to be a

4

-circulation distribution of the screwblade has a nonzero value at the root of the screwblade. In Chapters 2 and 3 the hub is omitted and then the optimum

circulation distribution of the screwblade becomes zero at the root of the

screwblade. For a better approach to modeling the hub in the optimization and

the design we refer to for instance [1], [2] and [3]. We expect that our simple modeling of the hub for the purpose of this thesis

is not so bad

because the thrust of the propeller

is mostly delivered by parts

of the

propeller which are not too close to the hub.

We remark that in our theory propellers with end plates are optimized

and designed as one unit. This means that we must calculate the optimum

geometry of the interacting end plates and screwblades. The situation of

mounting end plates

to an

existing conventional screw propeller, is not

considered in this thesis.

The basic principle why a propeller with end plates can possibly be favourable from the viewpoint of efficiency is the following. Compared with a

conventional propeller the total trailing edge of a propeller with end plates

is longer. Then, when the screwblades and end plates are adequately designed,

it becomes possible to spread out the trailing vorticity in the wake of the propeller over a larger area, resulting in lower kinetic energy loss of the propeller.

Furthermore better spreading of trailing vorticity is expected to

reduce propeller noise. Maximizing the propeller efficiency and reducing propeller noise in

a certain sense go hand in hand.

That is, minimizing kinetic energy loss means spreading more evenly the trailing vorticity, which is expected to have a reducing influence on the vorticity induced noise. One can think for instance of the noise generating cavitating tipvortices,

occurring often with conventional propellers. By experiments with a model screw propeller with end plates as discussed in [4], it was hard to succeed in making visible cavitating tipvortices. By reducing strongly the ambient pressure in the cavitation tunnel, a weak kind of tipvortex cavitation could

be established only from the tips of the end plate parts

at the pressure

sides of the screwblades.

The propeller with end plates has some features in common with a

ringpropeller and a ducted propeller. The three types of propellers are each

capable of spreading the trailing vorticity over a larger area than can be

achieved by conventional propellers, provided that they are adequately

is _{carefully optimized and designed. Such a ringpropeller will differ from}

conventional ringpropellers in the sense that the ring will have

circumferentially varying profiles.

The ducted propeller with respect to the spreading of trailing

vorticity has a disadvantage, since the ideal spreading is not attainable.

This is because the duct is not rotating with the screwblades, whereby the circumferentially varying profiles of the optimum ringpropeller can not be

realized. Another drawback of the ducted propeller is that the tip clearance

causes leakage loss.

A ringpropeller

and a

propeller with end plates are capable of

spreading out trailing vorticity in a better way than a ducted propeller, and moreover for these propellers there is no tip clearance. However the fact

that the ring and end plates rotate with the propellers,

also implies a drawback of the ringpropeller and the propeller with end plates. Namely the

ring and end plates

cause, by their large circumferential velocity, extra

viscous energy loss arid hence it requires a larger torque to

rotate a

propeller with end plates or a ring, so that this aspect has an efficiency

decreasing influence.

In this respect propellers with end plates have an important advantage when compared with ringpropellers. Namely the wetted area of the end plates

can be far less than the wetted area of the ring. Hence the viscous energy

loss caused by the presence of end plates can be far less than that caused by

the presence of a ring. To illustrate

this aspect we refer to Chapter 2, where it is found that our "Optimization Including Viscosity" delivers end

plates

with an optimum span of which the

total length is substantially

smaller than the length of a ring.

From the above considerations it seems that propellers with end plates

are worth being investigated. In this thesis we focus on the aspect of the

hydrodynamics of the propeller. Of course

it must be realized that for a

specific application there are a lot more considerations necessary to make a

good balance between the technical and economical factors of different

propulsion units. For instance with respect to aspects of vibrations,

strength, protection and costs of fabrication of the propellers.

Chapter 1 of this thesis discusses the influence of choice of generator lines on the optimum efficiency of screw propellers. The optimization theory

that is used there is the classical linearized optimization theory with

-6

propellers is minimized. The relative simplicity of this theory enables us to

examine without too much complications a wide range of propellersthat can be

generated by different shapes of generator lines of both screwblades and end

plates. The quality number q, which was introduced by Sparenberg and which is

discussed for instance in [5], is used to compare the different classes of

propellers that are considered. It is, as contrasted with the efficiency,

independent of the thrust of a propeller. The concept of quality number is useful in the sense that when both quality number and efficiency

of a

propeller are low, it becomes interesting to search for better propellers.

When either the quality number or the efficiency is high, the search for a

better propeller becomes less interesting.

In Chapter 1 also the numerical method of collocation, used to solve

the optimization problems occurring in this thesis, is outlined. The aim of

the chapter is not to find accurate values of the efficiency of a propeller,

but to find favourable trends. Then small changes in quality number might be important and therefore we try to calculate accurate values of q. Asymptotic

expansions that we used for the calculation of the velocity induced by a

twosided infinitely long helicoidal vortex are derived in appendix A of this thesis. Checks on the accuracy of the numerical method are carried out by comparing our results with analytically obtained series expansions for two special classes of propellers. The first class of propellers is

the one for

which Goldstein in [6] derived series expansions. The second class consists

of propellers with end plates which have such spans that the optimization

theory also applies to ringpropellers. Furthermore propellers of this class

have a twosided infinitely long hub of circular cross section. Appendix B

describes the derivation of the series expansion that we used for checking

our numerical method for this latter class of propellers.

In Chapter 1 a useful symmetry property together with numerical results of calculated quality numbers for different propellers, indicate in which direction we undertake the further search for better propellers.

Therefore, with in mind the results of Chapter 1, we confine ourselves

in Chapter 2 to propellers that

have a

straight generator line of the

screwblade and zero rake angle. The end plate planforms are

part of the

circular cylinder through the blade tips. An extended optimization theory is discussed which, in deriving optimum circulation distributions, minimizes the

sum of the kinetic and viscous energy loss of propellers, instead of the kinetic energy loss only. This optimization theory includes an approximate

the shape of the

trailing vorticity surfaces. Furthermore the problem of

choosing adequate planform shapes of blades and end plates is discussed. This is illustrated by the use of "vorticity stream functions".

The numerical results of Chapter 2 show that in most cases the theory predicts optimum propellers with end plates to have an efficiency gain of the order of a few percent with respect to optimum conventional propellers. The

amount of the gain depends on various design circumstances, such as the types

of end plates that are used.

In the first

two chapters a

detailed discussion of the design of

optimum propellers is not yet given. Design theory is an important subject of Chapter 3. In this chapter first the optimization theory is further extended to include some other nonlinear corrections, related with induced velocities

at the lifting lines. Second the actual design process is treated in this chapter. Because for large scale propellers experiments are often carried out on models of the propellers, in Chapter 3 we also make a comparison between

results of optimization and design for small scale propellers and for large

scale propellers, respectively.

For conventional propellers we checked our optimization and design method by analyzing our designs of optimum propellers by the use of the

existing computer programs ANPRO and PUF3A. Since these latter packages can

only handle propellers without end plates, the checks are not carried out for propellers with end plates. An example of optimizing and designing a screw propeller with end plates is described at greater length in Chapter 3.

Finally in this preface we mention some aspects that are not covered by

the presented theory, but that might be of importance for further research

and applications.

In this thesis the incoming flow to the propeller was considered to be homogeneous. Both our optimization method and our lifting-surface design method does not consider inhomogeneous inflow, which is important when a

propeller is considered behind a ship. _{For propellers with end plates} a

method to deal with inhomogeneous inflow in the optimization method is described in [7]. It is possible to incorporate a similar method in the "Optimization Including Viscosity" as discussed in this thesis. To include the effects of inhomogeneous inflow in the lifting-surface design method it

might be advisable to

extend methods in use already for conventional

-8

propellers, to the case of propellers with end plates.

In this thesis the design problem is addressed. The analysis problem for propellers in off-design conditions is not considered at all. Of course

it is important to have at one's disposal also instruments for the

hydrodynamic analysis of propellers with end plates. Hence for future

research it is desirable to extend existing analysis theories, such as

described for instance in [8], to be applicable also to propellers with end

plates.

Because our strength calculation discussed in Section 2.9 of Chapter 2

was incorporated in the optimization method in an iterative way, it was

necessary to have a simple theory for it. For practical use however, it is

necessary to perform more detailed finite element strength calculations,

dealing among others with the effects of fluctuating loads on the blades and

end plates.

Furthermore it is desirable to have a good procedure for computation of

the rolling-up of the trailing vortex sheets of propellers with end plates. Calculating the rolling-up for conventional propellers at present is not yet in an advanced stage. Often empirical rules or wake models derived from

experimental results are applied for the determination of the geometry of the trailing vortex sheets. For propellers with end plates calculating the rolling-up is possibly more complicated.

Another interesting aspect, possibly worth to be elaborated upon, is

the influence of the ship's rudder, when it is placed not too far downstream of the screw propeller. In Chapter 2 we found that in certain design conditions the most favourable rotational velocity of optimum propellers with end plates is lower than the most favourable rotational velocity of optimum propellers without end plates. Then it might appear that for optimum screw

propellers with end plates more rotational losses can be gained back by

suction forces at the rudder's nose than for optimum screw propellers without

References Preface

Andrews', LEI., Cummings, D.E., "A Design Procedure for Large Hub

Propellers", Journal of Ship Research, Vol. 16, No. 3, Sept. 1972, pp. 167-173.

Braam, H., "Optimum Screw Propellers

with a Large Hub of

Finite

Downstream Length", International Shipbuilding Progress, Vol. 31, 1984, pp. 231-238.

Coney, W.B., "A Method for the Design of a Class of Optimum Marine Propulsors", PhD thesis, Massachusetts Institute of Technology, Department of Ocean Engineering, Sept. 1989.

Sparenberg, J.A., de Vries, J., "An Optimum Screw Propeller with End Plates", International Shipbuilding Progress, Vol. 34, July 1987, No. 395, pp. 124-133.

Sparenberg, J.A., "Elements of Hydrodynamic Propulsion", Martinus Nijhoff Publications, The Netherlands, 1984.

Goldstein, S., "On the Vortex Theory of Screw Propellers", Proceedings

of the Royal Society, London, Series A. 123 (1929), pp. 440-465.

Klaren, L., Sparenberg, J.A., "On Optimum Screw Propellers with End Plates, Inhomogeneous Inflow", Journal of Ship Research, Vol. 25., No. 4, Dec. 1981, pp. 252-263.

Kerwin, J.E., Kinnas, S.A., Lee, J.T., Shih, W.Z., "A Surface Panel

Method for the Hydrodynamic Analysis of Ducted Propellers", Annual Meeting of the SNAME, New York, N.Y., Nov. 1987.

[8]

[11

[4]

[6]

CHAPTER 1

ON THE 'INFLUENCE OF CHOICE OF GENERATOR LINES ONE THE

OPTIMUM EFFICIENCY OF SCREW PROPELLERS I)

Summary

The optimum efficiency of a screw propeller, with or without end plates,,

depends on the geometry of the generator lines of blades and end plates.

Quality numbers of propellers have been calculated for a variety of generator

lines. The shape of the planform of end plates

is discussed in order to

minimize their viscous resistance.

'IL if.. Introduction.

In this paper we generalize in some respect the subject discussed in

[1]. There the influence of an end plate applied to a propeller without rake,, acting in a homogeneous and incompressible parallel flow, is considered and theoretical results are compared with experimental ones. The obtained results

were favourable and the question arose if there would exist still more efficient shapes of propellers, with

rake, different types of end plates or

curved generator lines.

The optimization theory we used for the calculation of the potential theoretical circulation distribution along blade and end plate is, as in [(l], a linear theory. Hence in the first instance the results are valid for

lightly or moderately loaded propellers. However, end plates or rake become more favourable in the case of heavily loaded propellers. The reason is that,,

when T is

the thrust of the propeller, the kinetic energy losses are

proportional to T2, while the viscous losses as we will discuss

(Section 1.9 ), can be taken approximately proportional to' T. This means that

when T is

sufficiently large, an increase in the quality number q of a

propeller which affects the kinetic energy losses, can raise the efficiency

in

spite of an increase of the viscous

resistance caused by rake, by

broadening of the blade tip and by the end plate when the propeller diameter

1)1 Published in the Journal of Ship Research, Vol 34, No. 2, June 1996, pp.. 79-91,, K. de Jong and J.A. Sparenberg.

-is

kept the same. We hope that

linear theory will indicate directions in

which search for a better propeller

in the heavily loaded case, can be carried out. That this is not without prospect follows from the experiments in [1], which showed an increase in efficiency by end plates calculated by a linear theory, of about 1.5 percent in model scale. Because the influence of

the larger Reynolds number in case of a fullscale propeller is favourable,

it can be expected that a higher increase in efficiency will be possible in

that case.

The method of calculation has to be sufficiently flexible to cope with

the various propeller geometries we want to investigate. Moreover the

calculation of the potential theoretical quality number q has to be rather accurate, because a small increase

in q of some percent can be already

relevant for the design of a propeller. This is the reason that we checked

the accuracy of our computer program for some special propellers by comparing

its results with those obtained from Bessel function expansions analogous to

the work of Goldstein [2], or with the results

from a finitedifference

calculation.

It will be shown that the influence of end plates alone is, in a certain

sense, stronger than the influence alone of rake or curvature

of the

generator line of the blade. Besides it turns out that when end plates are

applied, an additional rake or curvature does not contribute substantially to the efficiency anymore. Further, a deviation of the planform of the end plate

from the cylinder through the blade tips seems to worsen the propeller

efficiency.

Finally, the shape of end plates is discussed globally, with respect to three important items. First, in order to reduce their viscous resistance,

their area has to be as small as possible without increasing the danger of

cavitation. Second,

the bound vorticity of a blade has to

be conveyed

smoothly to its end plate. Third, the free vorticity has to be smooth. It

turns out that the best we probably can do, is to place the end plate such

that the blade

tip is in

the middle of the total span of the end plate.

However the planform of an end plate will have relatively shifted halves, each with a chord length at its root which is equal to or somewhat larger than half of the chord length of the tip of the blade, such that the whole

blade tip is covered. The leading edges and the trailing edges of the end plate halves have to be chosen such that the free vorticity does not induce

12

-1.2 Basic notations

For our geometrical considerations we use two types of coordinate

systems, the Cartesian system (x,y,z) and the cylindrical system (x,r,O), see Figure 1.2.1. We consider a screw propeller, rotating with constant angular

velocity Lo in the negative 0-direction around the x-axis, in an

incompressible and inviscid parallel flow U in the positive x-direction. It

has to deliver a prescribed thrust T in

the negative x-direction. At the

trailing edges of the blades free vortex sheets are formed. The linearization assumption

is that the vortex lines of the free vortex sheets are helices,

leaving the considered point of the trailing edge in the positive x-direction. The shape of such a helix is given by

a-- x

const. r=7" (1.2.1)

where const. and rs are appropriate constants. So no roll-up or breakdown of vortex sheets is included in our model. The hub is represented by a two-sided

infinitely long circular cylinder of radius rh, which is the inner radius of

the propeller.

Figure 1.2.1. Hub and generator lines g1 of blade and g2 of end plate.

The surfaces at which the vorticity is located and which float in the linear theory, without change of shape, in the positive x-direction, we call

reference surfaces. In a linear theory the blades and end plates have to move

in a close neighbourhood of these surfaces. First we discuss the reference

surfaces at some moment of time t, say t= 0. For later use they are two-sided infinitely long. Consider in the xy - plane (Figure 1.2.1) the two generator

lines 91 and 92 of a blade and an end plate respectively

x = fi(y) OGX5.X1

rh. y<ri

, fi(rh) = 0 (1.2.2)

y= f2( X ) , X2 S x , T2 < y To (1.2.3)

where f1 and f2 are sufficiently smooth functions and the points (x1,r1),

(x-2,r2) and (x3,r3) in the xy- plane are denoted in Figure 1.2.]. We demand that 92 passes through the point (x,y), (x1,r1) of

The lines gi and 92 are now rotated with constant angular velocity co

around the x-axis in the positive 8-direction and at the same time translated with constant velocity U in the positive x-direction. By doing this gi and 92

will describe the reference surfaces denoted by Hi and H2 respectively, of the screwblade and the end plate under consideration. If the rotation and translation process "started at x= -co", Hi and H2 will stretch from -co towards x = +oo. The surface of the hub is denoted by 113

Hi: 6 LF(±) (f i(r)- x) =0 _{rh< r} (1.2.4)

0---x= -W- X* r =f2(x ) , x2<x <x3 (1.2.5)

T = Th _(1.2.6)

The outer radius rp of the propeller is defined as the greatest r-value which is attained by points (x,r,0) situated on the reference surfaces.

End plates can be "one-sided" or "two-sided". A one-sided end plate has

a generator line 92 (equation (1.2.3)) with x2= xi or x3 = xi, while for the

two-sided end plate x2< xi < x3.

On the surfaces Hz, unit normals are introduced (i =1,2,3) in the

local cylindrical coordinate directions _lo) at the point (x,r70)

under consideration

( , f1(7-.) , 1) 14 711 = (1+ (P.-) +

f

r

)) 2) 1/2 2

(f( X Ualcd

) , 1 ,Uf 2.(XUe/W)

COT 2 rf 2 1/2 (1+ (f;(XUO/W))(1+(c-7.;) )) :A3= (0,1,0) . (1.2.7) (1.2.8) (1.2.9)

The direction of unit normal i is taken positive when it is pointing in

the direction of increasing 0, while the positive direction of -712 and -77.3 is that of increasing r. We require of g2 and the velocities co and U to be such

that H2 Will not cut itself. Then in all points of 112 a unique A2 exists. We

define the "+" side H4t:

of Hi to be oriented in the direction of the unit

normal i (i= 1,2,3). The "" side II; is the opposite side.

If the propeller has N blades, we sometimes add to symbols belonging to blades and end plates a second index j= 1,...,N. Symbols belonging to the hub keep the single index 3. For instance for the reference surfaces with their

unit normals we use: Hid, 112,3, H3 and

respectively. Reference surfaces coincide with each other after rotation over

an angle i2n/N (i= 1,...,N) around

the x axis. If there is no danger of

confusion, the second index will be omitted.

Now we discuss the liftingline system which, in the linear theory, can

represent the screw propeller. We take the lifting lines 11,3 ,12,3

(j=1,...,N) and 13 as the intersection of the reference surfaces ll1,j, 112,j

and If, with the yzplane, see Figure 1.2.2. In this way the hub is

represented by a circular lifting line 13 and can be treated in the same way

as the screwblades and the end plates. So, vorticity that is formed on the

hub, in our model is leaving 13 along helices (equation (1.2.1)) lying at the surface 113. We could have taken the lifting lines in many other ways, for

instance along the generator lines g, and g2 and (y=rh)

in the xyplane.

However the choice we made here, seems to be more palpable.

For the Nbladed screw, along a relevant 1/Nth part (for which we take

j=1) of the liftingline system, length parameters are introduced. Along the

lifting line 13 of the hub the length parameter 473 is increasing from the

point (r,0)= (7-5,0) to (r,0)=(rh, 27/N). Along the lifting line /LI of the

-dr'

Ti(c)= -ffej.1(0,4)

Figure 1.2.2. Lifting lines /Lb 12,11 D3 and length parameters ai

of a 3

bladed screw propeller rotating with angular velocity (4, in the negative

0-direction.

blade we take the length parameter al increasing from (r,0)=-(rh,0), to

(r,O)= (7-1, -coxtIU). If end plates are present the length parameter az is .used,,

increasing along from the point (r,0)= (rz, -wx2IU) to (r,O)= (r3,-cox3111.

Start and end values of ai (i=1,2,3) are denoted by a and

respectively. The cr2 value corresponding to he intersection of and

we denote by 2,bi

Next we define the sign of the bound vorticity along 4, 1/41, 13 and

of the free vorticity along the helices (1.2L1). The bound vorticity Nati)

i(i=1,2,3) which is the circulation around 1 at the place at, is taken

positive with a "right-hand screw" in the positive direction of the parameter

ai. The free

vorticity y(c) situated at Hi

(i =1,2,3) along helices is

reckoned positive when, with a right-hand screw, it has a positive component, in the positive x-direction. For reasons of simplicity we take the free

vorticity -Mai) per unit of length in the (xi-direction (i=1,2,3), which

however is not perpendicular to the helicoidal vortex lines. Then between the

bound vorticity ri(ai) and the free vorticity y(c) we have the relations

!(1.2.105i 12,1

Finally we want to have at our disposal a measure which gives us some

information about the extent to which the fluid "outside" the end plate

reference surfaces H2,i (j=1,...,N) is being separated from the fluid

"inside" these surfaces. Hereto we introduce the ratio of covering k of the

end plates. Suppose the helix through the point (x,r,0)= ( x2,r2, 0 ) cuts the

yz-plane in the point ( x, r, ) = ( 0, r2,02), while the helix through ( x3, r3, 0)

cuts that plane in ( 0, r3, 03 ). For the N-bladed screw propeller with end

plates the ratio of covering k is defined as

k= N(03-02) . 2 r

16

-(1.2.11)

A screw propeller without end plates has k= O. In this paper only screws with 0 <k< 1 are considered.

1.3 Optimization and efficiency

By optimizing, in this paper, we mean minimizing the kinetic energy of

the trailing vortex system, which is formed at the lifting lines,

representing the screw propeller. A slight extension of the classical

optimization procedure of Betz [3] yields the boundaryvalue problem for the potential of the optimum trailing vortex system far behind the propeller. In

words the procedure can be formulated as follows.

In a fluid which is at rest, translate the reference

surfaces H,

(i= 1,2; j=1,...,N) and 113 considered to be rigid and impermeable, with a velocity A in the positive xdirection. The fluid is unbounded and inviscid

while the velocity field is irrotational. For the reference surfaces to

become stream surfaces, they need to have vorticity

(i=1,2; j=1,...,N)

and A73 along helicoidal lines (1.2.1). Then this vorticity is the free

vorticity shed by the optimum propeller. The total circulation around the

free vorticity sheets has

to be zero because

the vorticity of propeller

blades and free vortex sheets is free of divergence. Then far behind the

optimum propeller, a uniquely valued and timeindependent potential A(x,y,z) of the fluid velocities exists outside the vortex sheets. The potential 0

belongs to a unit translational velocity A of the free vortex sheets. Because the N blades with end plates of the screw are alike and equally spaced, this optimum potential will be such that (i= 1,2; p,q = 1,...,N). For -y3

there is an analogous property and therefore only along the relevant 1/Nth

part of the lifting line system (Section 1.2) has the optimum potential to be

found. At that relevant part of the lifting

lines the optimum circulation

distributions Ari(cri) (i =1,2,3) satisfy

i(ai) = [e(o-i) over Hi , (i= 1,2,3) . (1.3.1)

The constant A will be determined directly by the condition that a thrust T

has to be delivered.

So the boundaryvalue problem for the optimum potential 0(x,y,z) becomes

(see also reference [4])

= (71i,j )

ao _{.0 on 113}

i,1

2

= NpoA EJ [or: .i) cirri

i =1 2 = NpA 2

₁

if

_ao

+ 101- 'IS 2 i =I Hi ari. i 1 osxsu

18

-on Ho , (i = 1,2; j=1,...,N) (1.3.3) (1.3.4)

where A is the Laplace operator.

It can be proved [4] that the disturbance velocities following from the

potential function 0, have the property that they are perpendicular to the

helicoidal lines (1.2.1).

By the law of Joukowski we find for the thrust T

2

T =Npu.)A EJ rro

(-7-) dai

i =1

(1.3.5)

where ;..(cri) is the vector pointing from the origin 0 towards the considered

point ai of

The vector t(o) in the yz-plane

is

the unit tangent to

at a=c i and p is

the fluid density. From (1.3.5) follows the unknown velocity A.

When the optimum propeller is moving forward with a velocity U in a

fluid at rest, the unknown velocity A, in the linear theory, is the velocity

by which theoretically the free vorticity sheets infinitely

far behind the

propeller translate backwards by their own induced velocities.

In our linear theory the kinetic energy E of the part of

the trailing

vortex sheets that

is formed per unit of time,

equals the kinetic energy

infinitely far behind the screw inbetween two planes x=const. and x = const.+U. Hence in the optimum case we have

2 +co U

E

=oA=-T-J'Jj- I grad012 dr dydz

co 0

where dS is an elemental surface area of the considered part of H. Using

(1.3.3) the last equation can be reduced to

2

E Np9wA2 E

[0]+_ dai

=1 /ill

Eliminating A from (1.3.5) and (1.3.7) gives

T2 Lac =

2 2

2pU7r( rp-rh)

Ur( r p-rh) 2=12 2 [O]j: (-;'/) da,

Nw Q= i,1 (1.3.7) (1.3.8) (1.3.9)

Then the quality number q, which is independent of the thrust T, is defined by

Ea,

q=

(1.3.10)

The larger q is, the better is the propeller configuration with respect to kinetic energy losses. The quality number has to be smaller or equal to one

because the actuator disk with constant loading is a propeller with the

highest efficiency in the linear theory [4]. For our screw propeller we thus

have

(1.3.11)

By (1.3.11) we find that q is a function of Tp, Th, Cs),

U, N and the

Next we introduce the quality number

of a

propeller. Consider a

rotational symmetric actuator disk with thrust T, constant loading

Tpr(r-r), velocity of advance U and the same inner and outer radius as the

considered propeller. Its

kinetic energy loss per unit of time Ea, has the

value

E.

T2 2 2pcoN E [C+._ (7..?) da, i =1 1 ,1 2 1

TV

It=

=(1+

TU+E 2pU27r(rp2- 7. )q

20

-shape of the generator lines. Then from simple dimension analysis it follows

= q(wrplt. rhlrp, N, 6k) , k= 1,2,... (1.3.12)

where the dimensionless 6k represent the shape of the generator lines.

Although we could have included rhlrp and N in the 6k, we mentioned these explicitly because it evokes a better picture of the propeller under consideration. Using the quality number the efficiency 77 of the propeller can be written as

--1

(1.3.13) g

14 Symmetry property

By the optimization 'procedure of "Section 13 we can obtain a ^useful

symmetry property of optimum propellers. We consider two screw propellers A

and B with the same rotational velocity to around the x-axis, while each is

acting in a parallel flow El in the positive x-direction,, see Figure 1.4.1. A

and B are related to each other in that the lifting lines of B equal those of

A after rotating the latter ones over ir

radians around the y-axis. Length

parameters oh (i = 1,2,3), along the relevant lifting lines of A are

introduced as was done in Section 1.2. The same parameter values occur at

corresponding points of propeller B.

A

Figure 1.4.1. Lifting lines of screw propellers A and B; 3 blades.

Theoptimization procedure is applied to both propellers. So, their

reference surfaces have to be translated with velocity A in the positive x-direction. Suppose the calculated optimum free vorticities for A and B are

ifYi(aiX BYi(ori) = l,2,3 (1.4. r);

respectively. Now we rotate the reference surfaces, the optimum vorticity and

the translational velocity A of screw A over ir radians around the y-axis. After the rotation the reference surfaces of A coincide with those of B. 'Then, changing the sign of the translational velocity and of the vorticity of

22

-the rotated screw A, this vorticity has become identical to -the optimum

vorticity of B, or

Alri(ui)= BYi(ai) (i= 1,2,3) . (1.4.2)

Hence both propellers have the same optimum circulation distributions and as

a consequence the same quality number.

It is not difficult

to show that when we have two generator

line configurations which are symmetric in the xy - plane,

with respect to the

y-axis, the corresponding lifting-line configurations in

the yz-plane are

also symmetric with respect to the y-axis.

As a special case of the foregoing we consider a screw propeller with

generator lines (Figure 1.2.1)

gj: x=0

, (1.4.3)

Y = f2(x) (1.4.4)

where x3= -x2 and f2(-x). f2(x). From (1.4.3) and (1.4.4) it follows that the

lifting line IL, representing a blade coincides with part of the y-axis and

the lifting line 12,1 representing an end plate is symmetric in the 3z-plane,

with respect to the y-axis. Then (1.4.2) implies that the optimum free

vorticity of end plates and hub is symmetrical in the sense

= (i= 2,3) . (1.4.5)

Also we can obtain by our symmetry property some insight in the influence of a rake. Consider a screw with a "rake angle" a, and end plates

which are part of the cylinder r = = rp,

g, : x = y tana (1.4.6)

92: Y = (1.4.7)

where we introduced

4.= (x3-x,)/(xi-x2) , (1.4.8)

which will be called the end plate ratio. Then by our symmetry property such

a propeller has the same efficiency as one with rake angle a. This means that the quality number q or the efficiency ri is a symmetric function of a

with respect to a = 0 deg. Hence a small rake angle will cause only very

small changes of 0(a2) in the efficiency. Of course this result also holds

for a propeller without end plates.

Finally, we remark that from the symmetry property. "ft follows that the quality number is independent of the direction of the rotational velocity t

24

-1.5 Some aspects of the optimum free vorticity

In Section 1.3 we stated that the optimization procedure yielded a

Neumann boundary-value problem. In realizing that the optimization problem

can be considered as a potential-flow problem around the impermeable reference surfaces without thickness we shall derive some a priori knowledge

about the optimum vorticity distributions we are looking for. The resulting

optimum potential flow we will denote by OFF.

In this flow problem we use for the disturbance velocity at each side of a reference surface Hi, (i = 1,2,3), the notation

V+i (ai) va (ci) (1.5.1)

reckoned positive when having a positive component in the positive as-direction. As we remarked below (1.3.4 ) these velocities are perpendicular

to the helicoidal lines (1.2.1) in general, hence also to the helicoidal

lines which form a surface Hi. Then for the optimum free vorticity

which in Section 1.2 we defined per unit of length in the cri direction, we

have

y(o)= (1);(as)V(as)) C°S(Cri,Si) f (1.5.2) where si is the direction on perpendicular to the helices.

Because of their orthogonality with the helices, the velocities v4i. and

v: in the neighbourhood of the intersection of two reference surfaces can be

approximated by the corresponding velocity components of a two-dimensional

potential flow in an angle The plane of this two-dimensional flow is the

flat plane W perpendicular to the common helix of the two considered

reference surfaces. The angle j3 is the dihedral angle, which is the angle enclosed at the angular point by the intersection lines of the plane W with the reference surfaces. The nearer to the common helix, the better the approximation for v.:: and

For instance in Figure 1.5.1 a representation is given of the

intersecting reference surfaces H1

(of blade) and H3 (of hub), having an

angle a (0 <a < ) between the lifting lines 11 and 13 in the yz-plane.

(a)

Figure 1.5.1. (a) Angle a in yz-plane; (b.), dihedral angle fi in plane W.

The dihedral angle $ =(ice) is related to a by

tang= 11 p)11/2 tana ,; _illkice (1.5.3)

'Where we made use of the notation.

Sri= 44-hr- /24 Et(rhY kL5.4)

Later on we will use analogously p,, = p(r). In the plane IW we denote the

velocities of the flow within the angle fi and at its sides by

we))

w301

where the indices are connected to 1, and 13, respectively, and R is the

distance of a point of W to the angular point, see Figure 1.5.1(b).

Approximating v(R) and v+3(R)

by sit(R) and t4(R) respectively

it,

follows from this two-dimensional flow problem that

P-1 11117(R )IIH 1114W/11 R-1 /4(R )111 const.RP R 0, 1.5.

where p=p(P) 'depends on p in the well-known way (for instance 11513.

0

, (1.5.5)

vi(R ) , ,

tr3(a3), h (1-th+1)1/2

According to (1.5.6) we have

v4-3(03) , _{173 t (73,e} or rya 4, /73,0 (1.5.11)

It is easy to derive that cos(a3,s3) in (1.5.2) equals

1 cos(a3,s3)= (11/2t+1) 73(C3) "). 112 h Ph+1 26

p =3-1

to < p coa , (1.5.7)

and const. is unknown. Now it

is clear that at the intersection of the hub

and a reference surface of the blade the velocities

(1.5.1)

tend to zero,

which gives by (1.5.2)

-> 0 , al 4,(71,s (1.5.8)

The same reasoning is valid for the vorticity at HI of a screwblade

ending at a twosided end plate

701)

° .crt t (71,e (1.5.9)

In the OPF the fluid remains at rest in the inside region of Hi.. Therefore we have

(73,85- 173 63,e

'era,. a3 (1.5.10)

(1.5.12)

Substituting (1.5.10), (1.5.11) and (1.5.12) in expression (1.5.2) it follows

that at the intersection of hub and reference surface of the blade y3(a3),

the free vorticity at the hub, will have the behaviour

173 t 173,, or c13 4 173,, (1.5.13)

When the ratio of covering equals one (k= 1) and the reference surfaces H2 of the end plates coincide with the cylinder r=rp, the OFF behaves like an

-P p

72fr2) --

2

Pp+1

'TA)

undisturbed parallel flow in the region outside Hz. Then we find, analogous to (1.5.13)i,

'°:2 C72,,t1

7,(u,)

/lc)

Figure 1.5.2. Behaviour of optimum free vorticities in neighbourhood of

connection of screwblade to hub.

It is seen that y3 has a zero derivative at 4/3

and an infinite derivative . a3,e. This is caused by the relevant dihedral

angle being larger or

smaller than tr/2. The vorticity _Ti has. an infinite derivative at the

intersection of Hi and /13 (Figure 1.5.2(c)). Note that this is a consequence

of the assumption that the hub is a twosided infinitely long cylinder. To find what happens in the theory when the hub downstream of an optimum

propeller has a finite length we refer tor instance to [6]. However, the more

realistic behaviour given there will hardly influence the flow

at the

relevant working parts of the propeller.

The optimum free vorticity at the connection of the reference surfaces of a twosided end plate and a blade exhibits a similar behaviour.

°Is

With the considerations given above we have found limit values of the optimum

free vorticity yi at the helices of intersection of the reference surfaces.

It is not difficult to derive from (1.5.6) the way in which the free vorticity tends to these values. As an example this behaviour of the optimum

free vorticity in the neighbourhood of the common helix of HI and 113 is given in Figure 1.5.2, when 1 cuts 13 under an angle ia tr/2.

Ph 2 C (a) 1 (1.5.14) at

Cons t. 72(az)

(a2,

a)/2

72(az) _{(a2 -0.2,8)v2}Co nst.

Still, a singularity of the free vorticity of another type occurs when a

one-sided end plate is applied. Suppose for instance we have a one-sided end plate ( (xi , ) = ( x2, r2) ) with generator line (1.2.3)

92: y=f2(x) ,

x2x<x3

(xi/ri) (x2/r2) (1.5.18)

In the yz-plane at Q we take a as the smallest angle enclosed by the lifting

lines 11 and 12 (0 <a <ir), (Figure 1.5.3).

28

-At the tip of a reference surface of a blade when the end plate

is

absent, a square-root singularity of the free vorticity will occur

7'1(0.0 cons t,

al t

(1.5.15)

(a1 ,

e1)

Square-root singularities will also arise at the tips of the reference

surfaces of two-sided end plates when these surfaces have no helix in common

(1.5.16)

(1.5.17)

Figure 1.5.3. One-sided end plate with angle a between lifting lines at

corner Q.

2[-ff+P)

i( -1/2< p <04.

x-0I

Using (1.5.2) and (1.5.6) we find

Yidai1P--const.v(oice-ag oi t 01,e )(1.5.19),

y2(0.21, const.(a2- a2,bt)?' , (7.2 .4.C2,s2,bl J(1.5.20)

in which for both expressions the unknown constants' are the same. The order p

now has a negative value depending on fl =9(a),

K1.5.21)"

Hence the free vorticity density becomes infinite at The Junction of the one-sided end plate and the blade. This is caused by the infinite velocities of the OFF which occur at the corner Qi when the fluid flows (Figure 1.5.0

along /1

ff.

From (1.5.8), (8.5.9) (1.5.13)-(1.5.17), (1.5.19) and 1(1.5.20) in combination with (1.2.10)) we obtain information about dri/doi, at the values in question of (for instance see Figure 1.9.1 in Section 1.91.

1

,

(1.5.3),

30

-1.6 Numerical method

The geometry of the reference surfaces that can be constructed with the

generator lines of Section 1.2 can vary rather strongly. To cope with the many kinds of possible boundaries the numerical method for the optimization

problem needs to be flexible. Our numerical method approximates, at the

relevant parts of the reference surfaces Hi, the optimum free vorticity

(i = 1,2,3), by means of piecewise linear functions.

For the vorticity < ai,< (i = 1,2,3; n = 1,...,N),

belonging to the boundary-value problem (1.3.2)-(1.3.4) we can write down

three coupled singular integral equations. As the relevant parts

of the

reference surfaces we choose the part with parameters co, (i=1,2,3), see Figure 1.6.1.

tY

Figure 1.6.1. View in the yz - plane of length parameters cri, and free

vorticity yi, (i = 1,2.3; vi = 1,...,(N = 3)), used in integral equations (1.6.3).

At a point ai,1

C;/ of the line 4, the component, locally normal to

the reference surface Ht,i, of the velocity that is induced by a concentrated

aiti

Using these notations,. we find that the three coupled integral equations become K1, i,i( a i,i,cri,1)1 Y/Iard dui,1 + (a,1Ii,r -a-1,1) 1 3 N Cr i, n C

ai,lis

EE

f

Kic(a"1,1 r C r j,n) Yj( C I j,n)' 1, = (Thi,1 .rx) q(71,) 3 j=/ n=1 Chu)* a /' nt,,S ( 1,1 ) 1,2,3)( *1.6.21 60,e 1,2,4 ty. (1.6,3) The integrals in equations (1.6.3) are the contributions of two-sided

infinitely long vortex sheets induced at the point a11=a-0 of the lifting

line /id

in the locally normal direction to the vortex sheet at the point,

-The kernel functions K1,1,1 (i =1,2,3) and Ki

(i,j =1,2,3; n=

1( j,n) s (,1)), contain in connection with the Biot and Savart law, information

about the helicoidal shape of the vortex sheets, the choice of the generator

lines and the number of blades N. In order to avoid the writing of very

lengthy formulas, we leave the straightforward task of finding expressions for the kernel functions to the reader. The first term of each of the three

equations (1.6.3) is a Cauchy principal-value integral, due to the fact that

this term involves the velocity contribution of a vortex sheet at points of the sheet itself. If the considered screw propeller has no end plates or a helicoidal vortex of unit strength cutting

the yz- plane at the point,

_Zn with 0,70# OM, is denoted by

Xi,j,n(711,1/17j,,n) ,, 0,1= 1,2,3; 1( j,n) (i4)1,)i (1:6.1((

The aforementioned velocity component, but now with X j,n)=0,1)., is denoted

'by K1 _{,1( a ,,107,,LY} (i= . ) 1, e ,

i,1,3 6i,1 (i=

hub with zero radius, 1rh=0), values of i and j equal to 2 and/or 3 do not

Lake part in expressions (1.6.1)-(1.6.3).

If It; is

expected to have no square-root

singularity

of the type

k1.5.15),

(1.5.16) or (1.5.17) and no singularity of the type

(1.5.19) and!

11.5.20), we approximate it by a piecewise linear function of ai along the lifting line i. If however yi does have such a singularity for one or more

values of the index i, we make use of appropriate coordinate transformations.

aim= cri,ii(ri)

iotts LC i,n,e gi,n,s < <CY =°1 .,,N) . 411.6.41

With/ the use of (1.6.4l the left-hand side of (1.6.3) becomes

Ki, 161,17

i,l(r1))

dci,i

Yi.(02,1("1741 (r,) ,dri

kai,dri

ri i s 3 N

EE fK1,5,.(0"1,1

cri,(rila j=1 n=1 nb'

(so)

3' do -Ti(65,451) 1 n )

drj

056.5)

The transformation. chosen such that

defined by

do

7,(r1)=Ye(criAr1)1) (ri) gm=17. -MI a

dri

remains finite for ai<ri<ai. Then instead of approximating Ti(os) by

a piecewise linear function of of, we approximate yi(ri))1 by a piecewise

linear function of ri.

For instance, consider a screw propeller without end plates. The optimum

free vorticity of the blade yi(cri) has at the tip the behaviour (1.5.15). To deal with this singular behaviour an, appropriate coordinate transformation

c=u(t1) is given by

32

-n,e ,

a(r,),

(1.6.4), is (1.6.6)

r (T1 Crl,e)

Cr/ ( T1 ) = 61,.9 +(01,e- crv,$) cos (

2 (a ,

-ai,,))

01,3 5- -501y, az,.5 aro, (1.6.7)

Then the function y-1(71), (1:6.6), will have a finite value for 71 /- 01,e. An analogous transformation can be made to handle the square-root singularities at the tips of the two-sided end plates.

When a one-sided end plate is applied the character of the singularities

at the junction of

blade and end

plate differs from the square-root

singularity. For instance in case we have generator line (11.5.18) we found (1.5.19) and (:1.5.201.. Then we, have applied

a41(T1)=V.1,,e e 610) 2 4 _{(az, e} 02(72) _{0.2,e1- (2p+ 3,) (p2, e} a 2, ), 1 pi 1 71 a 1 a (6/ a)} 0/2((72,4,-Faz,J5 T2 SIT2,e 1(p14. ) (2p+2) ( 2(Tz-az.81 =02,81-(62,e-a2;sj) (2p+ 3) e C72,8 ) 02,6 S V2(62, (12, e)

V.6.4

Next we introduce the parameters oi 0.1,2,3). When no singularity of yi occurs we put tpi = oh when a singularity of yi occurs we put 2Pi = Ti. For the

parameters tpi we introduce the meshes

<i,n0i,e

+(i= 1,2,31 , k1.6.101

lat,s= < IPi,i<

which need not to be equidistant. For the base functions, Figure 1.6.2, of

the piecewise linear approximation of y(c) or of ?1(r1 we choose

Bi,ittPi)=(t-Oi,j-1)/(0i,j-00-1)h 4),(-1<4,1<4id

= (iPi,j+1 - ?Pi )/(thi,j+1 - < < j+1

= 0 elsewhere

010 Crl S a 1,e CA. al, :OA 81

( , { , ,

-r2)

ri = I

for j = raj =.1 and

B NO = (01i, -')/(4'i _4)g)

= 0 elsewhere

'Birni0Pi) IPijni 71 ) tki,M 2Pi,iftra

=0 elsewhere

KEia 1,2,3y (1.6.11),

.°

*Fir) 'Poi Ofteli 9Int,

Figure 1.6.2. Base functions.

The approximation of ydai)

or ?gra, 10= 1 2,,3), gong the' lifting tines

then becomes

i

Cii) or

3(r))=

E (i =1 1,2,3) , a 6.12)

= 0

with appropriate real constants Co ti= 1,2,3; j ,Now we have in total a number of

E(nzi+ (1.6.13r

-1

unknowns Cu. These unknowns have to be determined by the impermeability

conditions (1.3.3) and (1.3.4) and by the zero total circulation condition of

our optimization problem. s.0 , , , < , . 34

As we have seen in Section 1.5 we also have to cope with the local

behaviour of the optimum free vorticity in the neighbourhood of the junctions

of the reference surfaces of the blades and hub, (Figure 1.5.2), or of blades and two-sided end plates. At those places the free vorticity has locally, a large or an infinite derivative. In order to obtain accurate results, we make

use of appropriate 'local refinements of the 0-meshes .(1.6.10). where

necessary. Of course it is possible to use a more accurate method to handle

the last-mentioned behaviour. This is left undone because the singular

behaviour of the derivative appears to have an, extremely local character and, has no influence on the first three digits of the quality number q.

The impermeability condition is satisfied at the lifting lines .0,7

(4..1,2,3) in the middle of the meshpoints L. In Figure 1.6.3 distributions,

of collocation points over the lifting lines are depicted for some screws, when coordinate transformations and refinements are applied as explained..

t

Figure 1.6.3. Distribution of 50 collocation points over lifting lines;

three blades (N =3);

(a) no end plates;

(by one-sided end plates; (c)

two-sided end plates (k = 1).

Each collocation point yields a linear equation for the unknowns Cid. In

each such a point for every base function the velocity induced by the

vorticity on N helicoidal strips stretching from x -co towards x = +co is. determined using the law of Riot and Savart. In determining this influence

the integrations along the parameters tpi over a mesh of the lifting lines are carried out by Gaussian rules For the integrations along the helices a split

36

-has been made. Along the first M windings of a helix (1.2.1) in the positive and negative x-direction, that is for

-Me I <x< MirU , (1.6.14)

diverse numerical integration routines are used adequate for the integrand's behaviour. For the contribution of the remaining windings, stretching towards

x=+oo and asymptotic expansions have been used. These asymptotic

expansions are discussed in Appendix A. The number of windings M that is

numerically integrated we determine in relation with the requested accuracy. It is easy to see that the condition of vanishing total circulation

yields another linear equation for the constants Cid (1.6.12).

The a priori knowledge (1.5.8) at the hub and (1.5.9)

in case of the

presence of two-sided end plates is put into the numerical procedure by

omitting in that case the base functions /30(7p1) and/or /31,,i(*i) (1.6.9) from representation (1.6.12). Then with our choice of the parameters, base

functions and collocation points, the impermeability condition and the zero total circulation condition yield as many linear equations as there are

unknowns C. This is also true if no end plates are applied.

For the case of a one-sided end plate the above paragraph does not hold.

Then the base function Bi ) cannot be omitted because y1(a1) now becomes

infinite at the junction of blade and end plate. This would mean that we have one unknown Cid more than there are linear equations for the Co, (i= 1,2,3; j =0,...,mi). However for this case we can use the extra linear equation for

the unknowns derived by the knowledge that (1.5.19) and (1.5.20) have the

same constant factor.

After solving the system of linear equations, we obtain an approximation for the optimum free vorticity leaving the lifting lines. The optimum

circulation distributions along the lifting lines can be obtained by properly integrating the free vorticities. Using (1.3.11) the quality number q can be

obtained.

1.7 Check on accuracy of numerical method

As we shall see in Section 1.8 the difference in quality number q of

screw propellers with different generator lines will sometimes be small. However it seems not impossible a priori that geometrical shapes which cause small favourable increments in q can be included in the propeller design.

Therefore it is important

to have an

idea how accurately the optimum

circulation distributions and the quality number q can be approximated using the numerical method described in the previous section. We repeat that the

aim of this paper is not to calculate the quality number for a real propeller

with such a high accuracy, but to find trends which we hope will occur also

in reality.

As a

first check we compare =the optimum circulation distributions.

'obtained by our numerical method with the series expansion of Goldstein, see [2]. He considered an NWaded screw, see Figure 1.7.1, without end plates], having, a straight generator line

Wx=11.

with a zero hub radius rh= 0.;

Figure 1.7.1. Wang lines of Goldsteln]s propeller; three blades (N= 3).

In this

special case we can

easily predict with the considerations of

Section 1.5h the behaviour of the optimum free vorticity for cri 4. alp,. That

"is, because aro, here corresponds to the origin 0, the

plane W of

X1.7. l)

38

-Section 1.5, is perpendicular to the x-axis and therefore it coincides with

the yz -plane. The cr1 direction is perpendicular to the helicoidal lines and

the dihedral angle # enclosed, by two screwblades equals (2r/N). Using (1.5.2) we conclude

7'1(1M const.

(c-ic,)("2

cr cr al 4 01,8

Hence at 0 the free vorticity of the blade of an !optimum one-bladed propeller has a square-root singularity. The free vorticity of an optimum two-bladedi screw tends to a finite value at 0 and in case of an optimum screw with more

blades yi(ad tends to

zero at 0. It

is

evident that at the blade tip a

scluare-root singularity of y1(a) occurs.

OUT' numerical method is applied to this propeller +using coordinate

transformations and refinements suited to the expected behaviour, as

explained in Section 1.6. Accurate integrations along Al =12 windings of the helices are used together with asymptotic expansions for the contribution of

the remaining parts of the helices, see Appendix A.

Goldstein's method to

obtain a series

expansion for the optimum circulation distribution along the blade, has been implemented in a computer program in order to yield a more accurate solution than could be given in [21

at that time.

In Figure 1.7.2 the results of the comparison are given for

increasing numbers nb of the collocation' points on the relevant lifting line of the blades. It is seen that good convergence is achieved. Using nb =30

collocation points the optimum circulation distributions of the series

expansions and of our numerical method exhibit a difference of less than 0.1

percent over the entire range 0<p <pp. The quality number 9 is approximated ii?

four digits for nb= 20, as follows from the table in Figure 1.7.2.

lip

Figure 1.7.2. Comparison of numerical method with series expansion;

pp= 5; N=4; series expansion; ...71b=5; for nh= 20 and 74=50 results of both methods coincide.

As a second check we consider another special case, that is a propeller

with a nonzero hub radius rh, straight generator lines, zero rake angle and end plate reference surfaces _112d, (j= 1,...,N), coinciding with the outer cylinder where r = rp.

The ratio of covering equals one, k=1. We adapt

Goldstein's method of solution to this case.

Changing to helicoidal coordinates with y defined by (1.5.4) and

by

numerical method:

ser les expansion:

q=0.73954 (1.7.3) G: , 0<(<71-/N , _(1.7.5) n b q 5 O. 7 3 650 20 0 . 7 3 94 9 50 0 . 7 3 954 we consider (75K42)= (1.7.4)

where ((,p) is the potential of the OFF. We have to determine ri only in the region

because of the symmetry property of Section 1.4, and because is a constant

for /1 lip and 05 /15 .1.th. So, -4. has to be determined from (see for instance [4 ] )

2

a st,

(p, _{)2..q6H-(1-+P.2 =CI}

with boundary conditions (see Figure 1.7.3)

= r/N ( = 0 40

-

2 ao c 2 1+p, ad;

kt=tip: --un

it2

ac

t p2

Figure 1.7.3. Boundary conditions on ac for ,:i6=- ((,p).

Ii

Following Goldstein we take a cosine expansion for in the (direction.

in deviation from his expansion for the open propeller of Figure 1.7.1,

modified Bessel functions of the second kind can not be omitted

for the

propeller considered now, because of the nonzero hub radius. Skipping the analysis for reasons of brevity, we give directly the resulting series solution: PP (1.7.6) (1.7.7) (1.7.8) (1.7.9)

-

--

((/p) = 77N2

t

(Tr,v(VPH-amKv(VPH- bnilv(VP))

co $ vc

,n=o (7n-F1/2)2

where

01.7.11)

The derivation of the series expansion (1.7.10) is given in Appendix B.

In expression (1.7.10) K,, and I,, are the modified Bessel functions as

defined in ,[7], while the Try funtions are the same as those used in [2]..

For every 772, (m=0,1,2,...), the constants am and b, are determined by the

boundary conditions (1.7.9), which give

1.7.12)'

where the primes denote differentiations with respect to the argument up.

On the relevant lifting lines, that is, on those boundaries of G where

Neumann conditions are demanded, the terms of expansion (1.7.10) for values of in which are not too large (for instance m<10) can be calculated directly using (1.7.12) and (1.7.13). It is possible to obtain

as many terms

as desired in series _{(1.7.10). Namely, by using asymptotic expansions (see for} instance I8]3, we find from (11.7.12) and (1.7.13) for larger values of in

v ( v MA? - Ti , v(iv kip) 1 `(11

IL(up)) Ic(v it)

lo,Ku(v 11), (KI:(vith ) I1(111211)) K,;(1/ pp 4 klc(v pt. p)

I(v tip))

-T(utin)1<v(")

1c(44)

Analogously we have

b,I(vp):--I(up)

0.7.0.0)6 0.7.134 (1:7.14)I 7.15)

'The last quotients in 0.7.14) and' (1 7 15)1 "can gain be approximated' thy

Ti,(vp.p) ,

I(vg)

ft

242

asymptotic expansions.

We compared the abovementioned accurate series expansion for problem

(1.7.6)(1.7.9) with the results from the numerical method. It is found that by increasing the number of collocation points, good convergence of the optimum circulation distribution occurs on hub, blade and end plate to the

values found by the series expansion. In Table 1.7.1 the comparison of the quality number q is given. The number of collocation points on the lifting

lines of hub, blade and end plate are denoted by nhnb, and me, respectively.

Table 1.7.1. Comparison of quality number q; N=4, ph= 1, pp 5.

As a double check for the lastmentioned case a finitedifference

approximation, has been made, which also converged to the series expansion in

refining the meshes of the finitedifference grid. The numerical method of

discrete helicoidal vortices was also

tested for this

case. We found that

this method did not give results which we considered to be accurate enough for our purpose..

inh nh ne 2 3 4 4 6 8 8 12 16 16 24 32 expansion

series.

.89243 I 89149 I .89141 I .89145 .89149