O N T H E C A L C U L A T I O N O F D U C T E D P R O P E L L E R

P E R F O R M A N C E I N A X I S Y M M E T R I C F L O W S

J. A. C. F A L C A O D E C A M P O S

C A L C U L A T I O N OF DUCTED PROPELLER P E R F O R M A N C E IN A X I S Y M M E T R I C FLOWS

BIBLIOTHEEK T U Delft

P 1734 3404

821780

O N T H E C A L C U L A T I O N O F D U C T E D P R O P E L L E R

P E R F O R M A N C E I N A X I S Y M M E T R I C F L O W S

T E R V E R K R I J G I N G V A N D E G R A A D V A N D O C T O R IN D E T E C H N I S C H E W E T E N S C H A P P E N A A N D E T E C H N I S C H E

H O G E S C H O O L D E L F T OP G E Z A G V A N D E R E C T O R M A G N I F I C U S , P R O F . IR. B. P. T H . V E L T M A N

V O O R E E N C O M M I S S I E A A N G E W E Z E N D O O R H E T C O L L E G E V A N D E K A N E N T E V E R D E D I G E N OP D I N S D A G

14JUNI 1983 T E 14.00UUR

JOSÉ A L B E R T O C A I A D O F A L C Â O D E C A M P O S

P R O E F S C H R I F T

D O O R

E N G E N H E I R O M E C Á N I C O G E B O R E N T E L I S S A B O N

H. V E E N M A N E N Z O N E N B.V. - W A G E N I N G E N

Dit proefschrift is goedgekeurd door de promotoren

Prof. Dr. Ir. J. D. van Manen Prof. Dr. Ir. P. J. Zandbergen

CONTENTS

1. INTRODUCTION 1

2. ANALYSIS OF THE FLOW PAST A PROPELLER DUCT 6 2.1. Introductory remarks 6 2.2. P o t e n t i a l flow a n a l y s i s 9

2.2.1. Formulation of the problem and boundary conditions 9 2.2.2. Numerical s o l u t i o n 15 2.2.3. C a l c u l a t i o n of the duct c i r c u l a t i o n . F i r s t i n v i s c i d

approximation and the Kutta c o n d i t i o n 21 2.3. C a l c u l a t i o n of the duct viscous layers 25 2.4. V i s c o u s - i n v i s c i d coupling 28 2.5. Results i n uniform flow and comparison with experiment 30

2.6. C a l c u l a t i o n of the duct steady load f o r a duct with p r o p e l l e r 38 2.6.1. P r o p e l l e r model and p r o p e l l e r induced v e l o c i t i e s on

the duct 38 2.6.2. Remarks on viscous e f f e c t s on the duct f o r a ducted

p r o p e l l e r 49 2.7. Numerical r e s u l t s and comparison with experiment 51

3. DUCTED PROPELLER IN AXISYMMETRIC SHEAR FLOW 70

3.1. Introduction 70

3.2. Governing equations 73 3.3. I t e r a t i v e s o l u t i o n by a d i s c r e t e vortex sheet method 82

3.3.1. Vortex sheet approximation to the v o r t i c i t y i n the flow 82

3.3.2. F i r s t approximation to the actuator disk vortex sheets and the flow stream surfaces 85

3.3.3. C a l c u l a t i o n of the flow stream surfaces and d i s c r e t i z a t i o n of the vortex sheets 87

3.3.4. C a l c u l a t i o n of the strength of the vortex sheets 95

3.3.5. I t e r a t i v e procedure 97 3.4. Numerical r e s u l t s and comparison with experiment 98

4. INTERACTION STUDIES BETWEEN A DUCTED PROPELLER AND THE STERN FOR AX I SYMMETRIC FLOWS 4.1. Introduction 4.2. A p p l i c a t i o n to the c a l c u l a t i o n of the stern flow f o r an

axisymmetric body

4.3. I n t e r a c t i o n between a ducted p r o p e l l e r and the stern

4.4. Discussion of the r e s u l t s

5. DUCTED PROPELLER DESIGN 5.1. Introduction

5.2. Design procedure

5.3. P r o p e l l e r induced v e l o c i t i e s 5.4. Duct thrust and duct induced v e l o c i t i e s 5.5. The design with the induction f a c t o r method 5.6. Results and di s c u s s i o n

6. CONCLUSIONS

Appendix 1 Appendix 2 References Nomenclature Summary Samenvatting Acknowledgement Curriculum v i t a e

114 114

119

126

130

139 139

140 142 145 147 152

161

163 167 168 175 188 190 192

193

1. I n t r o d u c t i o n

In ship propulsion, for a t t a i n i n g high propulsive e f f i c i e n c y or reducing the r i s k of c a v i t a t i o n on p r o p e l l e r s , the ducted p r o p e l l e r became i n the l a s t decades, a widely used propulsion device.

With regard to i t s purpose or i t s basic working p r i n c i p l e two main types of ducts may be discerned. The a c c e l e r a t i n g type, which finds i t s f i e l d of a p p l i c a t i o n i n improving the e f f i c i e n c y of h e a v i l y loaded p r o p e l l e r s and the d e c e l e r a t i n g type which i s employed to reduce the extent of c a v i t a t i o n on p r o p e l l e r s , as i n the case of pump j e t s .

In recent years, with the permanent increase of power i n s t a l l e d on

ships, high levels of v i b r a t i o n induced on the h u l l may occur p r i m a r i l y due

to unsteady c a v i t a t i o n phenomena on the p r o p e l l e r blades operating i n the

highly non-uniform ship's wake. In c e r t a i n cases, as suggested f o r example by Oosterveld (1971),

a p p l i c a t i o n of a non-axisymmetric duct may lead to a decrease of the extent of c a v i t a t i o n on the p r o p e l l e r and reduce i t s induced v i b r a t i o n and radiated noise l e v e l s .

In view of an i n c r e a s i n g number of a p p l i c a t i o n s of the ducted propeller, the development of t h e o r e t i c a l models f o r d e s c r i b i n g i t s hydrodynamical performance has received a t t e n t i o n of many authors.

Ear l y t h e o r e t i c a l work on ducted p r o p e l l e r s aiming at the evaluation of i t s performance and the set up of adequate design methods were concentra­ted on the uniform flow case. In such i d e a l i z e d conditions the hydrodynamic problem f o r the ducted p r o p e l l e r c o n f i g u r a t i o n presents i t s e l f as an i n t e r f e r e n c e problem between p r o p e l l e r and duct. Under the assumptions of i d e a l f l u i d and flow i r r o t a t i o n a l i t y the flow past p r o p e l l e r and duct may be represented by s i n g u l a r i t y d i s t r i b u t i o n s on the duct and p r o p e l l e r surfaces and the corresponding t r a i l i n g vortex sheets representing t h e i r wakes.

Under a d d i t i o n a l l i n e a r i s i n g assumptions regarding the magnitude of

the disturbances introduced by p r o p e l l e r and duct on the undisturbed flow i n a reference frame r o t a t i n g with the p r o p e l l e r , the s i n g u l a r i t i e s and the p o t e n t i a l flow boundary conditions can be t r a n s f e r r e d to s p e c i f i c reference surfaces: a c y l i n d r i c a l surface f o r the duct and a set of h e l i c o i d a l vortex sheets f o r the p r o p e l l e r .

The basic ideas f o r the a n a l y s i s of the steady performance of the duc­ted p r o p e l l e r in a x i a l and uniform flow, were already contained i n the work of Dickmann and Weissinger (1955). They represented the duct with an e l l i p t i c a l d i s t r i b u t i o n of r i n g v o r t i c e s on a c y l i n d e r of constant radius and the p r o p e l l e r was modeled by an actuator disk of constant load with i t s corres­pondent slipstream r i n g v o r t i c i t y .

Theories introduced subsequently by Ordway et a l (1960),and Morgan (1961) made use of l i n e a r i z e d r i n g a i r f o i l theory, o r i g i n a l l y developed by Weissinger (1955,1957), and included a l i f t i n g l i n e model f o r the p r o p e l l e r . Ordway et al considered a l i g h t l y loaded p r o p e l l e r , while Morgan used the induction f a c t o r method as introduced by Lerbs (1952) for moderately loaded p r o p e l l e r s . These theories could be applied to the evaluation of both steady and non-steady loading on a r o t a t i o n a l symmetric duct under the influence of the p r o p e l l e r . Discussion of non-steady duct performance i s outside the scope of the present work. For an account on the l i n e a r i z e d theories mentioned above we r e f e r to the review work of Weissinger and Maas (1968).

A fundamental r e s u l t which followed from those i n v e s t i g a t i o n s and which i s of importance f o r the evaluation of the duct's steady performance, i s the equivalence between the time averaged flow f i e l d induced by a r o t a t i n g set of bound r a d i a l l i f t i n g l i n e s together with t h e i r t r a i l i n g vortex sheets and the axisymmetric flow induced by an i n f i n i t e l y blade number p r o p e l l e r model, the actuator disk.

This equivalence was demonstrated by Hough and Ordway (1965) i n the s t r i c t l y l i n e a r i z e d case of a l i g h t l y loaded p r o p e l l e r for which the h e l i c o i d a l vortex sheets are assumed to have a constant p i t c h determined by the undisturbed advance and r o t a t i o n a l v e l o c i t i e s .

In the case of the moderately loaded p r o p e l l e r l i f t i n g l i n e model the equivalence does not hold. However, the actuator disk model has been used

2

as an approximation to study non-linear e f f e c t s of co n t r a c t i o n and p i t c h v a r i a t i o n of the h e l i c o i d a l vortex l i n e s taking place i n the slipstream of moderately and heavily loaded p r o p e l l e r s as considered by Van Gent (1976).

Studies of the e f f e c t s of slipstream c o n t r a c t i o n on duct performance i n ducted p r o p e l l e r a p p l i c a t i o n s using s i m p l i f i e d actuator disk models have been c a r r i e d out by Chaplin (1964) and Van Gunsteren (1973).

Apart from such refinements of the p r o p e l l e r models, improvement of the representation of the duct has been achieved by a p p l i c a t i o n of surface v o r t i c i t y models which take i n t o account i n a rather accurate way,the duct's geometry. This has been done by Lewis and Ryan (1971).

In general the ducted p r o p e l l e r operates i n the ship's wake i n the

proximity of the h u l l and the water surfaces. The fa c t that the propulsor works i n a flow region with high concentra­

t i o n of v o r t i c i t y b a s i c a l l y i n v a l i d a t e s the assumption of p o t e n t i a l flow which underlies uniform flow theories.

Nevertheless, the success enjoyed by the design of wake adapted p r o p e l l e r s based on the induction f a c t o r method of Lerbs (1952) has ensured the a p p l i c a t i o n of p o t e n t i a l flow theories i n a s l i g h t l y modified form to the general non-uniform flow case. The modifications introduced i n the theory of wake adaption consist i n considering the flow perturbations induced by the bound and t r a i l i n g v o r t i c i t y to be added to the undisturbed l o c a l inflow v e l o c i t i e s to the p r o p e l l e r disk assumed to vary with the r a d i a l coordinate. In addi t i o n the p i t c h of the h e l i c o i d a l vortex l i n e s assumed constant i n a x i a l d i r e c t i o n i s determined at the p r o p e l l e r plane by the l o c a l t o t a l v e l o c i t i e s .

The l o c a l inflow v e l o c i t i e s i n t o the p r o p e l l e r i . e . the t o t a l v e l o c i t i e s minus the p r o p e l l e r perturbations, are known as e f f e c t i v e v e l o c i t i e s and i t s knowledge i s considered indispensable i n wake adapted p r o p e l l e r design. They d i f f e r from the nominal v e l o c i t i e s which occur behind the ship's h u l l i n the absence of the operating p r o p e l l e r .

Such d i f f e r e n c e i s regarded as a consequence of the p r o p e l l e r - h u l l i n t e r a c t i o n phenomena.

The previous considerations i l l u s t r a t e some of the problems involved i n the a p p l i c a t i o n of the a v a i l a b l e theories and point out the need of considering the problem of ducted p r o p e l l e r h u l l i n t e r a c t i o n .

S p e c i f i c a l l y i t i s thought necessary to a s c e r t a i n to what extent some

3

of the most relevant e f f e c t s of the i n t e r a c t i o n phenomena may influence the d e t a i l e d performance of the propulsor.

Recent studies by Huang et a l (1976, 1977), Schetz and Favin (1979), on the i n t e r a c t i o n between a conventional p r o p e l l e r and the stern have been con­centrated on axisymmetric bodies. In view of the considerable s i m p l i c i t y of the stern flow when compared with the s i t u a t i o n behind the ship, o f f e r e d by the flow axisymmetry, these studies employ models which attempt complete pre­d i c t i o n s of the flow f i e l d around the stern with the p r o p e l l e r i n operation.

On the other hand, with the advent of Laser-Doppler anemometry, the measurement of the v e l o c i t y f i e l d i n the close v i c i n i t y of the operating p r o p e l l e r , (Huang et a l , 1976, 1977), has enabled the d e t a i l e d comparison with the t h e o r e t i c a l p r e d i c t i o n s .

Attempts to adress the problem from a d i r e c t numerical s o l u t i o n of the f u l l Navier-Stokes equations have been undertaken by Schetz and Favin (1979). From the comparisons with experimental data the authors recognized some of the short comings of the turbulence model employed and the need of i t s refinement i n order to improve the t h e o r e t i c a l r e s u l t s . The approach of Huang et a l (1976, 1977), makes use of a c a l c u l a t i o n method of the viscous flow on the stern region based on boundary layer theory brought i n t o i n t e r ­action with the outer p o t e n t i a l flow i n an i t e r a t i v e scheme. The influence of the p r o p e l l e r i n the boundary layer i s exerted through a m o d i f i c a t i o n of the external p o t e n t i a l flow.

A novel feature of the approach i s , however, the c a l c u l a t i o n of the flow f i e l d i n the close v i c i n i t y of the p r o p e l l e r by an i n v i s c i d r o t a t i o n a l flow model based on the Euler's equations of motion. Rather accurate p r e d i c t i o n s of the t o t a l v e l o c i t i e s ahead of the p r o p e l l e r are obtained with such model.

The need f o r an e l u c i d a t i o n of some of the fundamental aspects of ducted p r o p e l l e r h u l l i n t e r a c t i o n and the development of c a l c u l a t i o n tech­niques appropriate to the non-uniform flow s i t u a t i o n when the v o r t i c i t y of the incoming flow i s taken i n t o consideration has motivated the present i n v e s t i g a t i o n .

The basic approach pursued i n t h i s study assumes that the i n t e r a c t i o n flow between ducted p r o p e l l e r and h u l l which u l t i m a t e l y determines the

4

performance of duct and p r o p e l l e r i s i n v i s c l d i n nature and therefore may adequately be treated by the consideration of the Euler's equations of motion.

Although the i n v i s c i d a n a l y s i s might be t h e o r e t i c a l l y j u s t i f i c a b l e or

experimentally v a l i d a t e d when dealing with the gross e f f e c t s of the i n t e r ­

a c t i o n problems to the ducted p r o p e l l e r , viscous e f f e c t s i n the boundary

layers on the various components of the ducted p r o p e l l e r system may be of

primary importance i n determining the o v e r a l l forces acting on the system.

Viscous e f f e c t s on p r o p e l l e r blades and t h e i r i n f l u e n c e on p r o p e l l e r

c h a r a c t e r i s t i c s have been i n v e s t i g a t e d f o r years. On the other hand, the

influence of v i s c o s i t y on the duct performance has received much less

a t t e n t i o n i n s p i t e of being a source of serious scale e f f e c t s on

f u l l - s c a l e p r e d i c t i o n s when separation flow phenomena occur on model s c a l e .

Therefore, the second chapter i s concerned with the ana l y s i s of the

viscous flow past an axisymmetric duct e i t h e r i n uniform a x i a l flow or when

regarded as being a part of the ducted p r o p e l l e r . In the t h i r d chapter, the flow past an annular a e r o f o i l and a ducted

p r o p e l l e r i n axisymmetric shear flow i s considered and approximate numerical s o l u t i o n s of the Euler's equation by a d i s c r e t e vortex method are given.

Chapter four deals with the a p p l i c a t i o n s of the methods developed i n chapter two to the i n t e r a c t i o n problem of a ducted p r o p e l l e r behind a r e v o l u t i o n body.

In chapter f i v e some considerations on the design of ducted p r o p e l l e r s are given.

The r e s u l t s of the basic flow models i n the f i r s t and second chapters are v e r i f i e d by c o r r e l a t i o n with experiment.

5

2. A n a l y s i s o f t h e f l o w p a s t a p r o p e l l e r d u c t

2.1. INTRODUCTORY REMARKS

For the c a l c u l a t i o n of duct performance i t i s necessary to have a method for the evaluation of the pressure d i s t r i b u t i o n and f o r estimating i t s f r i c t i o n a l drag. This i s normally accomplished by combined p o t e n t i a l flow and boundary layer c a l c u l a t i o n methods.

In general, i n ducted p r o p e l l e r a p p l i c a t i o n s , f o r ducts with sharp t r a i l i n g edges or with small radius of curvature at the t r a i l i n g edge and operating near design conditions, a p o t e n t i a l flow c a l c u l a t i o n ignoring the presence of the boundary layer already gives r e l i a b l e values of the o v e r a l l forces a c t i n g on the duct provided that flow separation occurs from the duct's surface only i n the v i c i n i t y of the t r a i l i n g edge. In t h i s case, the viscous drag i s rather small when compared with the a x i a l force a c t i n g on the duct.

The p o t e n t i a l flow s o l u t i o n obtained as a f i r s t approximation by disregarding the presence of the boundary layer and wake, or f i r s t i n v i s c i d approximation, i s assumed to s a t i s f y the Kutta-Joukowsky con d i t i o n f o r the flow at the t r a i l i n g edge.

The a p p l i c a t i o n of the Kutta-Joukowsky c o n d i t i o n f o r the c a l c u l a t i o n of the p o t e n t i a l flow on a p r o f i l e i s not a t r i v i a l matter and i t s various i n t e r p r e t a t i o n s and corresponding implementations may bear considerable influence on the s o l u t i o n .

Gostelow (1974) , gives a review of the a p p l i c a t i o n of t r a i l i n g edge conditions on two-dimensional and turbomachinery blade se c t i o n s . For p r o f i l e s with blunt t r a i l i n g edges, a t r a i l i n g edge con d i t i o n equivalent to the c l a s s i ­c a l Kutta-Joukowsky con d i t i o n cannot be formulated, and any hypothetical f i r s t i n v i s c i d approximation based on an a r b i t r a r y value of c i r c u l a t i o n i s deprived from p h y s i c a l meaning. C l e a r l y , f o r the s o l u t i o n of t h i s problem, viscous e f f e c t s have to be considered. In r e l a t i o n to p r o p e l l e r ducts t h i s subject w i l l be considered more extensively l a t e r i n t h i s Chapter.

6

A c l a s s i c a l approach c o n s i s t s i n s o l v i n g the p o t e n t i a l flow and the boundary layer problems i t e r a t i v e l y . On each i t e r a t i o n step the p o t e n t i a l flow s o l u t i o n provides the required boundary conditions f o r the boundary layer c a l c u l a t i o n s and these, i n turn, f u r n i s h the required boundary conditions f o r the p o t e n t i a l flow problem.

One of the methods of coupling the two s o l u t i o n s through t h e i r correspondent boundary conditions c o n s i s t s i n d i v i d i n g the flow f i e l d i n t o two well-defined regions: an outer i n v i s c i d p o t e n t i a l flow region and a v o r t i c a l region dominated by the e f f e c t of v i s c o s i t y . These two flow regions are separated by some boundary surface where the so l u t i o n s of the two flow problems should be matched by proper s p e c i f i c a t i o n of the respective boundary conditions. Such an approach requires an adequate d e f i n i t i o n of the boundary surface which n a t u r a l l y could be taken as the surface s p e c i f i e d by the boundary layer and the wake thicknesses and should be determined as a part of the s o l u t i o n , (Rom, 1977).

Such a procedure i s , however, disadvantageous from the point of view of computational e f f i c i e n c y , since the i n v i s c i d part of the computation has to be performed with boundary conditions prescribed on a surface changing i t s p o s i t i o n during the i t e r a t i o n process.

An a l t e r n a t i v e approach, which circumvents t h i s problem, c o n s i s t s i n t r a n s f e r r i n g the matching conditions to the body's surface by assuming that the outer i n v i s c i d flow may be continued down to the body's surface.

For t h i n shear l a y e r s the truncation e r r o r of the Taylor expansion about the points on the o r i g i n a l matching surface i s i n general small except near s e p a r a t i o n . L i g h t h i l l , (1958), showed that such a coupling procedure between the two flows may be formulated i n terms of an equivalent source d i s t r i b u ­t i o n on the body's surface.

The s o l u t i o n of the viscous flow problem at high Reynolds numbers by so l v i n g i t e r a t i v e l y the outer i n v i s c i d flow and the boundary layer forms the c l a s s i c a l "weak i n t e r a c t i o n " theory.

One of the main d i f f i c u l t i e s encountered i n the a p p l i c a t i o n of the theory l i e s on the f a c t that the procedure may break down i n regions of "strong i n t e r a c t i o n " of the boundary layer and the i n v i s c i d flow such as near a separation point or at the t r a i l i n g edge region.

In f a c t , i t might be impossible to continue the boundary layer c a l c u l a ­t i o n s beyond separation, using d i r e c t methods i . e . with prescribed pressure

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d i s t r i b u t i o n at the edge of the l a y e r .

Rigorous a n a l y s i s of the l o c a l flow i n a region of strong viscous-i n v i s c i d i n t e r a c t i o n f o r laminar flows at the t r a i l i n g edge of cusped and wedged a i r f o i l shapes based on asymptotic theory,have been given by various authors: R i l e y and Stewartson, (1969), Brown and Stewartson, (1970), Chow and Melnik, (1976). Estimates for the c o r r e c t i o n due to the e f f e c t of v i s c o s i t y on the c i r c u l a t i o n as determined by the Kutta condition have been given i n those analyses. A new method of general a p p l i c a t i o n i n regions of strong i n t e r a c t i o n , has been rece n t l y proposed by Veldman, (1979), (1981).

The a n a l y s i s based on the "weak i n t e r a c t i o n " theory i s c l a s s i c a l (see

Thwaites, 1960), and i n many, cases i s capable of providing p r e d i c t i o n s of

sec t i o n l i f t and drag forces with engineering accuracy. In the a p p l i c a t i o n to p r o p e l l e r ducts one such a method has been

considered. In t h i s respect the following remarks should be made:

- Laminar separation phenomena occurs frequently on p r o p e l l e r ducts at model s c a l e , Dyne (1977). Therefore the treatment of laminar separation bubbles i n the c a l c u l a t i o n method was f e l t neces­sary .

- Due to operation requirements p r o p e l l e r ducts have, rather often,

considerably t h i c k round t r a i l i n g edges. Accordingly, the e f f e c t s of

t r a i l i n g edge bluntness had to be considered.

This chapter i s divided i n t o three parts. In the f i r s t part the p o t e n t i a l flow analysis i s given and numerical r e s u l t s i n uniform flow are presented and compared with experimental data. In the second part extension of the method to include the e f f e c t of viscous l a y e r s i s given and experimental v e r i f i c a t i o n i s supplied. F i n a l l y , i n the t h i r d part the case of the duct with p r o p e l l e r i s examined i n uniform flow by means of p o t e n t i a l theory. For that purpose an actuator disk representation of the p r o p e l l e r i s used. The r e s u l t s are compared with experimental data.

An attempt i s made to include viscous e f f e c t s i n the duct a n a l y s i s i n the presence of the p r o p e l l e r .

8

2.2. POTENTIAL FLOW ANALYSIS

2.2.1. Formulation of the problem and boundary conditions

We consider the flow of an i n v i s c i d and incompressible f l u i d past a duct

i n an onset flow.

The disturbance p o t e n t i a l s a t i s f i e s Laplace equation

V2<f> = 0. (2-1)

The r e g u l a r i t y c o n d i t i o n at i n f i n i t y

it) -*• Q at i n f i n i t y , (2-2)

and the boundary co n d i t i o n at the impermeable duct surface D

m where «— denotes the d i f f e r e n t i a t i o n with respect to the normal to the on duct's surface taken p o s i t i v e outwards and <|> i s the p o t e n t i a l of the onset

flow. For a duct i n uniform flow with v e l o c i t y U . the undisturbed p o t e n t i a l o

<|>o i s simply the p o t e n t i a l of the uniform stream

<f> = U X . (2-4) r O O

For a duct placed i n the flow induced by a p r o p e l l e r advancing with constant speed U the p o t e n t i a l <J> i s o o

<t> = U x+d> , (2-5) y o o p

i> being the p r o p e l l e r disturbance p o t e n t i a l which has to be determined P

subject to the correspondent boundary conditions and i s assumed to be known for the present purposes.

In the l i f t i n g case we are considering, there i s c i r c u l a t i o n around a contour e n c i r c l i n g a s e c t i o n of the duct and the p o t e n t i a l i s discontinuous on a surface W extending from a l i n e on the duct's surface to i n f i n i t y ( F i g . 2-1). 9

The disturbance p o t e n t i a l at a point P outside the surfaces D and W i s given i n terms of i t s boundary values (Lamb, 1952), by

(P) = • A ƒ ƒ (Z^L ^ D+W 8 N 3n ' i d s + i ƒ ƒ (c

R 4 7 1 D+W (2-6)

where (J) , T T 1 — and d> , T T — denote the values of the p o t e n t i a l and i t s normal dn on derivatives,respectively on the outer and inner sides of the surfaces D and W,

andR=|Rj i s the distance between the f i e l d point P and the point on the

surface where the i n t e g r a l i n (2-6) i s being evaluated.

Equation (2-6) gives the representation of the p o t e n t i a l i n terms of a source d i s t r i b u t i o n on the surface with strength a equal to the d i s c o n t i ­nuity i n the normal d e r i v a t i v e

3<j) 3n

3<j> 3n (2-7)

and a dipole d i s t r i b u t i o n with axes d i r e c t e d along the normal to the surface

and with i t s strength y equal to the d i s c o n t i n u i t y of the p o t e n t i a l

(2-8)

With eq. (2-7) and (2-8), eq. (2-6) writes

D+W D+W (2-9)

Uo

Fig. 2.1. Schematic representation of the flow past a propeller

duct section.

10

By assuming c o n t i n u i t y of the p o t e n t i a l on the surface,we obtain a representation f o r the p o t e n t i a l i n terms s o l e l y of the source d i s t r i b u t i o n and by assuming c o n t i n u i t y of the normal d e r i v a t i v e s on the surface we obtain a representation i n terms of a dipole d i s t r i b u t i o n only. In the l i f t i n g case, for which the p o t e n t i a l i s discontinuous the l a t t e r represen­t a t i o n or a combination of the two as i n (2-6) has to be adopted.

As stated i n the previous s e c t i o n , i n the i n t e r a c t i o n between viscous and i n v i s c i d flow regions, the displacement e f f e c t s due to the boundary layer and wake can be represented by a source d i s t r i b u t i o n on the surfaces D and W i n the manner suggested by L i g h t h i l l (1958). The Neumann boundary cond i t i o n (2-3) should then be applied on the surface B displaced from the o r i g i n a l surfaces D and W by the boundary layer and wake displacement t h i c k ­ness.

However, as remarked before, the boundary con d i t i o n on B can be replaced by the Neumann boundary con d i t i o n on the o r i g i n a l surface D+W s p e c i f y i n g the value of the normal d e r i v a t i v e f o r the outer p o t e n t i a l at the surface. The normal d i s c o n t i n u i t y i s equal to the source strength on the surface

From (2-7) we obtain

| i _ + O = 0 on D , (2-11)

since d> and i t s d e r i v a t i v e s are continuous on D. o Using Green's theorem, we conclude from (2-11) that the p o t e n t i a l i s

constant i n s i d e D:

$• + $ = C , (2-12)

where C i s an a r b i t r a r y constant.

Equation (2-12) c o n s t i t u t e s a D i r i c h l e t boundary con d i t i o n f o r the

inner p o t e n t i a l which may be employed instead of the Neumann cond i t i o n (2-10).

11

As i n the present a n a l y s i s the source d i s t r i b u t i o n i s only used to represent boundary layer and wake displacement thickness e f f e c t s , i t s strength follows from the shear l a y e r flow s o l u t i o n s . In such case, a p p l i c a ­t i o n of the boundary conditions (2-10) or (2-12) leads to i n t e g r a l equations for the dipole d i s t r i b u t i o n .

Applying the Neumann co n d i t i o n (2-10), a Fredholm i n t e g r a l equation of the f i r s t kind i s derived

9 (J)

^ D + W P ( q ) ^ ) d S = ~ 3 ^ H a ( P ) + 4 V n 7 w

a ( q ) ^ ( I ) d S

p D+W q p D+W p

(2-13)

where p denotes the point where the boundary co n d i t i o n i s stated and the terms i n the source d i s t r i b u t i o n have been placed on the right-hand side to emphasize the fa c t that they are considered to be known.

A l t e r n a t i v e l y , we may d i f f e r e n t i a t e equation (2-12) along any d i r e c t i o n tangent to the surface r e q u i r i n g the v e l o c i t y component i n that d i r e c t i o n on i t s inner side to vanish. Such co n d i t i o n i s expressed by the following vector equation

n x V (<J> +<J>_) = 0 . (2-14) —p p Y o

By using a well-known equivalence between the p o t e n t i a l due to a dipole sheet and a vortex sheet, i t i s u s e f u l to show that equation (2-14) leads to the surface v o r t i c i t y formulation of the p o t e n t i a l flow problem.

To do t h i s we f i r s t evaluate the v e l o c i t y induced by the dipole d i s t r i b u t i o n which, i n any case, i s required for a p p l i c a t i o n of e i t h e r boundary co n d i t i o n (2-13) or (2-14).

We have

W V P " * * > ? 5 - < i > d s = i ¥ / / w ^ ) v p [ n ^ . v ( | ) ] d s (2-15) D+W q D+W M ^

where we use Vp to denote the gradient with respect to the f i e l d coordinates and the gradient with respect to the coordinates of the point q on the surface.

12

Since

1 ^ V . ' • 7

we obtain

R R V p [ n q . V q ( i ) ] = ( n q . V p ) ^ + n q x ( V p x - 3 ) .

For an a r b i t r a r y vector A the following r e l a t i o n holds

n x ( V x A ) = ( n x V ) x A - ( n . V ) A - n(V.A)

and (2-16) becomes

R

To integrate (2-15) by parts we note that

(2-16)

V p [ n q . V q ( | ) ] = - ( ^ x V q ) x ^ n g V q ( i ) . (2-17)

R 5 5: y ( n x V J x - T = (n x V ) x ( y - ^ - ) - ( n x V y ) x - ~ - (2-18)

~ q q R 3 - q q R 3 - q R 3

and the i n t e g r a l i n (2-15) writes

1 5 + -r- // (n x V u x-7 dS +

4 ^ D+W "«3 Q R 3

+ T= ti y ( q ) n V 2 ( I ) d S . (2-19) 4 7 T D+W Q Q R

By a variant of Stokes' theorem the f i r s t i n t e g r a l on two right-hand side of (2-19) can be transformed i n t o

R R ƒ ƒ (n x V ) x ( y - T ) d S = tfr d i x p ^ (2-20) D+W Q Q R R

where C i s a contour enclosing the surface D+W. This i n t e g r a l gives the

13

c o n t r i b u t i o n to the induced v e l o c i t y by the boundary edges of the surface D+W. In the present case such edges are i n e x i s t e n t and the i n t e g r a l vanishes.

When i d e n t i f y i n g the strength of the vortex sheet y as

1 = -n x Vy (2-21)

the second i n t e g r a l expresses the f a m i l i a r r e s u l t obtainable from B i o t -Savart law.

F i n a l l y , the l a s t i n t e g r a l i n (2-19) vanishes i n view of the fact that 2 1 V (—) i s zero everywhere except when the point P coincides with the point q. R

With the previous r e s u l t (2-19), (2-20) and (2-21), equation (2-14) y i e l d s the f ollowing i n t e g r a l equation f o r the surface v o r t i c i t y d i s t r i b u t i o n

R

(2-22)

D+W " P R" E D+W

For the determination of the v o r t i c i t y d i s t r i b u t i o n , equation (2-22) must be resolved i n t o i t s components i n a c u r v i l i n e a r coordinate system. We note, however, that the two components of the surface v o r t i c i t y d i s t r i b u t i o n are r e l a t e d through equation (2-21).

In axisymmetric flow, the d i p o l e strength i s independent of the circum­f e r e n t i a l coordinate and the surface v o r t i c i t y vector has only circumferen­t i a l component. Moreover, the dipole d i s t r i b u t i o n i s constant i n the surface W which implies that no v o r t i c i t y i s shed i n t o the wake.

Let x=x(s) and r=r(s) be the parametric equations of the duct contour , and s the arc length measured anti-clockwise from the t r a i l i n g edge, F i g . 2-1.

Writing y=y(s)ia

a n d taking the cross product of (2-22) by n we obtain the -0 -p following i n t e g r a l equation

- | Y ( S ) + $ s y ( s 1 ) k ( s , s ' ) d s ' = f ( s ) (2-23)

The kernel k(s,s') represents the induced v e l o c i t y tangent to the contour at the point s due to a r i n g vortex located at s=s' and i s given by

14

k ( s , s ' ) = ( x - x 1 ; r , r ' ) ~ - (x-x ' ; r , r ' ) | | (2-24)

where u and v are the a x i a l and r a d i a l v e l o c i t i e s induced at the point Y Y

x=x(s), r=r(s) by a vortex r i n g at x'=x(s'), r'=r(s'). The right-hand side i n equation (2-23) i s

f Cs] = | | $ a ( s I ) T ( S / s ' ) d s ' (2-25) dx ds dr ds s + w

where T(s,s') i s a coupling function f o r the source d i s t r i b u t i o n

T ( s , s ' ) = u a ( x - x ' ; r , r ' ) | | - v Q ( x - x ' ; r , r ' ) | | (2-26)

with u (x-x';r,r') and v q ( x - x ' ; r , r ' ) r e s p e c t i v e l y the a x i a l and r a d i a l

v e l o c i t i e s induced by a source r i n g .

Expressions f o r the functions u , v , u , v are given i n terms of y y 0 a

complete e l l i p t i c i n t e g r a l s by Kiichemann and Weber, (1953). We note that, a surface v o r t i c i t y formulation could be used i n combi­

nation with the Neumann boundary c o n d i t i o n leading to a Fredholm i n t e g r a l equation of the f i r s t kind. Equations of second kind are, however, advanta­geous from the point of view of t h e i r numerical s o l u t i o n .

2.2.2. Numerical s o l u t i o n

Numerical s o l u t i o n procedures for i n t e g r a l equations of the type of equation (2-23) of the l a s t s e c t i o n e i t h e r , f o r two-dimensional or axisymme-t r i c p o t e n t i a l flow problems,have been given by many authors.

The great majority of the s o l u t i o n techniques employs a c o l l o c a t i o n method. In the c o l l o c a t i o n method the equation i s only exactly s a t i s f i e d at a d i s c r e t e set of points and the number of points i s chosen equal to the number of knots employed i n the numerical quadrature used to approximate the i n t e g r a l . For l i n e a r i n t e g r a l equations t h i s procedure leads to a l i n e a r system of equations having as unknowns the values of the function at the knot l o c a t i o n s .

15

Two basic approaches have been followed:

A p p l i c a t i o n of a transformation to the i n t e g r a t i o n v a r i a b l e , p r i o r to the a p p l i c a t i o n of the c o l l o c a t i o n method,or d i r e c t s o l u t i o n of the o r i g i n a l equation by c o l l o c a t i o n having the arc length as independent v a r i a b l e .

The o r i g i n a l version of the surface v o r t i c i t y method, developed by Martensen (1959), for two-dimensional a i r f o i l s i s o l a t e d or i n cascade, belongs to the f i r s t category. The method was subsequently developed by Jacob and Riegels (1963), Wilkinson (1967), and others and applied to p r o p e l ­l e r ducts by Lewis and Ryan (1971).

A l l these methods make use of a transformation of the arc length i n t o the polar angle with respect to a point i n s i d e the contour and employed tra p e z o i d a l i n t e g r a t i o n to approximate the i n t e g r a l .

The basic d i f f e r e n c e s between the various methods regard the d i s t r i ­bution of c o l l o c a t i o n points and the numerical procedures used to evaluate the geometrical parameters of the contour.

As remarked by Wilkinson (1967), o r i g i n a l l y recommended p i v o t a l point d i s t r i b u t i o n s , (Jacob and R i e g e l s , 1963), included the t r a i l i n g edge as a c o l l o c a t i o n point and gave u n r e l i a b l e r e s u l t s f o r p r o f i l e s with sharp t r a i ­l i n g edges. The p o t e n t i a l flow problem i s not uniquely determined by the s a t i s f a c t i o n of the kinematical boundary con d i t i o n on the contour and the c i r c u l a t i o n must be given to s p e c i f y the s o l u t i o n . The main d i f f i c u l t y arose i n the a p p l i c a t i o n of the Kutta co n d i t i o n i n the transformed v a r i a b l e . Due to the properties of the transformation at the t r a i l i n g edge the implemen­t a t i o n of the Kutta co n d i t i o n i n the transformed v a r i a b l e d i d not imply zero loading at a sharp t r a i l i n g edge.

To minimize the er r o r s introduced by the t r a p e z o i d a l i n t e g r a t i o n s a "back diagonal c o r r e c t i o n " was normally applied to the o r i g i n a l matrix which rendered i t s i n g u l a r .

Wilkinson showed that the system of equations a f t e r the a p p l i c a t i o n of the Kutta co n d i t i o n became i l l - c o n d i t i o n e d at smaller t r a i l i n g edge r a d i i . The problem has been circumvented by a l t e r n a t i v e implementations of the Kutta condition,(Wilkinson, 1967), and by d i f f e r e n t choices of p i v o t a l point locations,(Lewis and Ryan, 1971).

16

Although these type of methods may require a rather small number of knots f o r an "optimum" transformation, the main drawback l i e s i n the s e l e c ­t i o n of an adequate transformation.

Methods of the second category employ the arc length as independent v a r i a b l e . They may d i f f e r on the order of approximation used to d i s c r e t i z e the p r o f i l e contour and the surface s i n g u l a r i t y d i s t r i b u t i o n . F i r s t order approximations d i s c r e t i z e the contour by s t r a i g h t elements and assume a con­stant s i n g u l a r i t y d i s t r i b u t i o n on each element.

Higher order approximations, d i s c r e t i z i n g the contour and the s i n g u l a ­r i t y d i s t r i b u t i o n s by polynomials of higher degree, can reduce s u b s t a n t i a l l y the computation time by reducing the number of elements required f o r a prescribed accuracy at costs of a d d i t i o n a l a n a l y t i c work.

The e f f e c t i v e n e s s of several second order methods r e l y i n g on various formulations of the p o t e n t i a l flow problem i n combination with d i f f e r e n t numerical approximations of the contour and s i n g u l a r i t y d i s t r i b u t i o n s , h a s been r e c e n t l y assessed by Labrujere (1979), f o r two-dimensional flows on p r o f i l e s . Although surface v o r t i c i t y formulations were contemplated,they have been only examined i n combination with the Neumann boundary c o n d i t i o n and a boundary co n d i t i o n i n terms of the stream function. In t h i s respect,we w i l l be l i m i t e d to a comparison with methods of the f i r s t category or experimental r e s u l t s .

The second order method proposed by Hess (1973, 1974), i n connection with h i s surface source panel method has been adopted i n the present numeri­c a l s o l u t i o n .

The duct i s panelled by elements of p a r a b o l i c shape where v o r t i c i t y i s d i s t r i b u t e d according to a polynomial function. Constant, l i n e a r and parabo­l i c functions have been used.

The v o r t i c i t y d i s t r i b u t i o n i s expanded about a c o n t r o l point chosen as the mid point of the element i n the form

Y j U ) = Y< 0 ) + Y ^ C + yfh2 (2-27)

where Y j ^ > Yj a n < i Yj are, r e p e c t i v e l y the strength of the vortex sheet, i t s f i r s t and h a l f the second d e r i v a t i v e s evaluated at the c o n t r o l point j

17

X

Fig. 2.2. D e f i n i t i o n of a parabolic panel on the duct's section contour.

and E, i s the arc length on the element measured from the c o n t r o l point, F i g . 2.2.

The values of the f i r s t and second d e r i v a t i v e s at the c o n t r o l points

are obtained by a divided d i f f e r e n c e scheme, as given by Hess and Martin (1974)

We note that when using such scheme, d i s c o n t i n u i t i e s i n the v o r t i c i t y d i s t r i b u t i o n are introduced at the junction points between elements.

The parametric equations of the arc element admit a s i m i l a r expansion

2 x _.(£;) = x_. +cosct_.5 - C j sina_.5 (2-30)

- 2 (^) = r_. +sina_.£ + c_. cosou? (2-31)

where (x^, r^) are the coordinates of the c o n t r o l point, a. i s the slope of the chord on the element and c. i s the element curvature.

J

18

The boundary c o n d i t i o n of zero v e l o c i t y tangent to the contour on the inner side applied at N c o n t r o l points leads to the l i n e a r system of equations

N (0) I k . • Y j = f , i = 1, (1) ,N (2-32)

j = l 3 3

The matrix of influence c o e f f i c i e n t s k.. i s only a function of the duct's

geometry and i s given by

k. . = h&. • + c o s a . X . . + s i n a . Y . . , i = l , ( l ) , N , j = 1 , ( 1 ) , N

(2-33)

6 i s the Kronecker d e l t a 6. . = 1 for i=j and 6. .=0 f o r The matrix X. .

and Y . are a x i a l and r a d i a l induced v e l o c i t y matrices , and f o r t h e i r

evaluation we r e f e r to Hess and Martin (1974).

The right-hand side i n equation (2-32) i s obtained by evaluating equation (2-25) at the c o n t r o l points and includes the undisturbed flow and the disturbance v e l o c i t y caused by the boundary layer and wake displacement thicknesses. The determination of these v e l o c i t y f i e l d s i s the subject of the subsequent sections.

The computational advantages of accounting f o r the infl u e n c e of the boundary l a y e r on the p o t e n t i a l flow by an a d d i t i o n a l disturbance v e l o c i t y to the basic onset flow, become evident from the form of equation (2-32). The matrix of inf l u e n c e c o e f f i c i e n t s k _ which depends only on the geometry of the duct, does not need to be changed i n the v i s c o u s - i n v i s c i d i t e r a t i o n process.

As a consequence of the non-uniqueness of the p o t e n t i a l flow problem

eigen s o l u t i o n s of equation (2-25) representing c i r c u l a t o r y flows may be

added to a p a r t i c u l a r s o l u t i o n without a f f e c t i n g the boundary c o n d i t i o n on

the surface.

19

S t r i c t l y speaking at i n c r e a s i n g number of elements the matrix k ^ becomes i l l - c o n d i t i o n e d . C l e a r l y any s o l u t i o n of the system (2-32), i f made poss i b l e through the d i s c r e t i z a t i o n , does not, i n general, s a t i s f y the Kutta-Joukowsky co n d i t i o n of smooth flow at the t r a i l i n g edge.

Such c o n d i t i o n has to be s p e c i f i e d i n a d d i t i o n to the system obtained from the c o l l o c a t i o n method. In the l i t e r a t u r e various implementations of the Kutta c o n d i t i o n have been considered. Also there i s considerable freedom i n the way the a d d i t i o n a l conditions are coupled to the e x i s t i n g system of equations.

Regarding the form of the Kutta c o n d i t i o n , Mangier and Smith (1969), showed that, for sharp t r a i l i n g edges, i n two-dimensional i n v i s c i d flow,the stagnation streamline should leave the t r a i l i n g edge along the b i s e c t o r to the t r a i l i n g edge angle. In f i r s t order methods t h i s c o n d i t i o n can be approx­imated, f o r example, by equating the vortex strength of the f i r s t and l a s t c o n t r o l points, (Lewis and Ryan, 1971), or by computing the v e l o c i t y at a point outside the surface close to the t r a i l i n g edge and g i v i n g i t the d i r e c ­t i o n of the b i s e c t o r . This approach may obviously be of low accuracy i n the case of loaded t r a i l i n g edges. In second order methods extrapolated forms of the Kutta c o n d i t i o n to the t r a i l i n g edge i t s e l f become p o s s i b l e , which would improve the p r e d i c t i o n s f o r loaded t r a i l i n g edges.

With the increase of the t r a i l i n g edge radius these forms of Kutta co n d i t i o n becomes of questionable a p p l i c a t i o n i n view of the d i f f i c u l t y of d e f i n i n g a s u i t a b l e " b i s e c t o r " to the t r a i l i n g edge angle. In such case, d i s c r e t i z a t i o n of the t r a i l i n g edge i t s e l f and s p e c i f i c a t i o n of stagnation point as done by Martensen (1959) can be used.

With respect to the coupling to the system of equations various choices are p o s s i b l e . S t r i c t l y speaking, the system of equation should be rendered s i n ­gular p r i o r to adding the Kutta c o n d i t i o n . Martensen (1959), makes the sys­tem s i n g u l a r i n a least-squares sense. In the method of Lewis and Ryan (1971), the s o - c a l l e d "back diagonal" c o r r e c t i o n i s the most obvious choice of ma­king the e x i s t i n g system s i n g u l a r , since i t replaces the coupling c o e f f i c i e n t i n each column most a f f e c t e d by t r a p e z o i d a l i n t e g r a t i o n e r r o r s by the one needed to'make the sum of a l l column elements vanish.

20

In methods which accurately compute the coupling c o e f f i c i e n t s such choice i s not unique. In the present method the overdetermined system of equations which r e s u l t s from adding an a d d i t i o n a l equation expressing the Kutta condi­t i o n , to the o r i g i n a l system, i s solved by a least-squares method. In t h i s case the Kutta condition i s s a t i s f i e d only approximately with the same degree of accuracy as the equations expressing the boundary co n d i t i o n on the c o n t r o l points. Stronger implementations which r e s u l t from s a t i s f y i n g exactly the Kutta c o n d i t i o n and only approximately the other equations have been consider­ed but d i d not introduce discernable changes i n t o the r e s u l t s .

2 . 2 . 3 . C a l c u l a t i o n of the duct c i r c u l a t i o n . F i r s t i n v i s c i d approximation

and the Kutta c o n d i t i o n .

In the v i s c o u s - i n v i s c i d i t e r a t i o n process, the s o l u t i o n of the poten­

t i a l flow problem i s obtained by s p e c i f y i n g the c i r c u l a t i o n around the duct's

s e c t i o n .

The c i r c u l a t i o n i s defined i n a contour c o i n c i d i n g with the outer side of the duct's contour.

T = (5 Y ( s ) d s ( 2 - 3 4 )

and i s approximated by

N r = l g . y . 0 ) ( 2 - 3 5 )

j = l : 3

where the c o e f f i c i e n t s g^ are taken as

g . = 25'. ( 2 - 3 6 ) j :

The c i r c u l a t i o n has to be determined from l o c a l flow conditions at the t r a i l i n g edge.

A rigorous a n a l y s i s of the flow on the t r a i l i n g edge i s rather elabo­rate and requires the abandon of the conventional weak i n t e r a c t i o n scheme i n the neighboorhood of the t r a i l i n g edge.

A simpler approach i s based on the assumption that f o r non-separated

21

flow the pressure gradient normal to the streamlines at the t r a i l i n g edge l o c a t i o n i s rather small and can be neglected. This leads to the e q u a l i t y of the pressure on the inner and outer sides and can be expressed by

C = C (2-37) r o u t r i n n P-P Q

where C= — i s the pressure c o e f f i c i e n t . Equation (2-37) provides a good * 4 p U o

approximation for a cusped t r a i l i n g edge where the streamlines leave the t r a i l i n g edge p a r a l l e l to i t . At i n c r e a s i n g t r a i l i n g edge angle streamline curvature e f f e c t s become important and condition (2-37) i s i n p r i n c i p l e l e s s accurate (see Thwaites , 1960).

For separated flow at the t r a i l i n g edge,Thwaites (1960) shows that, under the assumptions of f i r s t order boundary l a y e r theory, a s i m i l a r c o n d i t i o n to (2-37) holds, provided that the pressure i s taken at the boun­dary layer separation points on the outer and inner surfaces:

( c p ) o u t = ( C p ' inn <2"38>

P -p *sep o with C = and p being the pressure at separation.

P s *PU 2

o

Equation (2-38) may be used to determine the p o t e n t i a l flow s o l u t i o n when the l o c a t i o n s of the separation points are known from experiment. In

the v i s c o u s - i n v i s c i d i n t e r a c t i o n , equation (2-38) i s not d i r e c t l y implemented in t o the p o t e n t i a l flow s o l u t i o n . It i s , however, s a t i s f i e d i n the converged s o l u t i o n by p r e s c r i b i n g the sequence of c i r c u l a t i o n Y according to the

n r e l a x a t i o n formula, (Dvorak et a l , 1979).

r ,. = r + p[ (c ) -(c ). 1 , (2-39) n + l n L p o u t p i n n J ' v ' r s e p r s e p

where the r e l a x a t i o n factor f> has a value comprised between 0.1 and 0.3.

To s t a r t the i t e r a t i o n and i f the l o c a t i o n of the separation points are not known the f i r s t i n v i s c i d approximation i s considered to conform with the c l a s s i c a l Kutta condition. For a sharp t r a i l i n g edge the occurrence of

22

i n f i n i t e v e l o c i t i e s at the t r a i l i n g edge has to be precluded. This implies that the stagnation streamline leaves the t r a i l i n g edge with a d i r e c t i o n between the tangents to the outer and inner surfaces of the duct, (Mangier et a l , 1969 ). By applying B e r n o u l l i equation to the flows on the outer and inner surfaces, i t i s seen that equation (2-37) implies the stagnation streamline to leave the t r a i l i n g edge i n the d i r e c t i o n of the b i s e c t o r to the t r a i l i n g edge angle. This form of t r a i l i n g edge c o n d i t i o n , which i n the i n v i s c i d a n a l y s i s of Mangier et a l (1969), implies the inexistence of v o r t i c i t y i n the wake, i s consistent with the more general form of t r a i l i n g edge co n d i t i o n (2-38), which i n the viscous flow a n a l y s i s implies a zero net discharge of v o r t i c i t y i n t o the wake.

For a round t r a i l i n g edge the previous form of the Kutta c o n d i t i o n cannot be applied. We simply s p e c i f y the p o s i t i o n of the t r a i l i n g edge stagnation point.

To i l l u s t r a t e the a p p l i c a t i o n of t h i s l a t t e r form of the t r a i l i n g edge condition and to examine the s e n s i t i v i t y of the r e s u l t s to the p a r t i c u l a r choice of the stagnation point, p o t e n t i a l flow c a l c u l a t i o n s i n uniform flow have been c a r r i e d out f o r two ducts.

The f i r s t duct i s the duct NSMB 19A with length-diameter r a t i o of 0.5. The duct se c t i o n geometry i s given by Van Manen and Oosterveld (1966). The duct p r o f i l e and the computed pressure d i s t r i b u t i o n are shown i n F i g . 2-3. Gibson's (1972) t h e o r e t i c a l r e s u l t s with another surface v o r t i c i t y method and h is experimental r e s u l t s are also shown i n F i g . 2.3. His t h e o r e t i c a l c a l c u l a t i o n r e s u l t s corrected f o r tunnel wall i n t e r f e r e n c e are included as w e l l .

The present c a l c u l a t i o n s have been performed with 44 elements on the

contour and the stagnation point at the t r a i l i n g edge i s located at the

i n t e r s e c t i o n point between the b i s e c t o r to the angle formed by the inner and

outer surfaces and the t r a i l i n g edge c i r c u l a r arc f a i r i n g .

It can be seen that the d i f f e r e n c e s between the two c a l c u l a t i o n s are, i n t h i s case,marginal except near the t r a i l i n g edge presumably due to d i f f e r e n t f a i r i n g procedures. It should be noted that both c a l c u l a t i o n s s a t i s f y approximately equation (2-37) at the pressure minima at the t r a i l i n g edge implying a zero load i n the t r a i l i n g edge region.

23

Fig. 2-3. Pressure distribution on duct NSMB 19A in uniform flow.

The comparison with experiment i s poor on the outer surface due to the occurrence of leading edge laminar separation followed by turbulent reattach­ment. The c i r c u l a t i o n i s also not well predicted by the t r a i l i n g edge condi­t i o n and t h i s i s the cause f o r the discrepancy of the pressure l e v e l on the inner s i d e .

The s e n s i t i v i t y of the c a l c u l a t i o n to the p o s i t i o n of the stagnation point on the t r a i l i n g edge has been i n v e s t i g a t e d on the duct NSMB 37. This duct has a length diameter r a t i o of 0.50 and a blunt t r a i l i n g edge to improve astern operation, Oosterveld (1971).

24

The r e s u l t s are shown i n F i g . 2-4 and depict an enormous change of the duct c i r c u l a t i o n with small v a r i a t i o n s of the stagnation point l o c a t i o n .

Fig. 2-4. Effect of the location of t r a i l i n g edge stagnation point

on the pressure distribution on duct NSMB 37.

2.3. CALCULATION OF THE DUCT VISCOUS LAYERS

In t h i s s e c t i o n we w i l l r e f e r to the c a l c u l a t i o n of the boundary layers on the duct surface i n axisymmetric steady flow.

S t a r t i n g from the stagnation point on the duct's nose the laminar

boundary layer i s c a l c u l a t e d by the axisymmetric version of Thwaites' method,

due to Rott andCrabtree, (see Rosenhead, 1963).

25

The method gives the momentum thickness i n terms of the v e l o c i t y V at the edge of the boundary l a y e r by

0 2 0.45 2 5 ƒ r V ds v

(2-40) o

where V i s the kinematic v i s c o s i t y and s i s the arc length measured along the contour from the stagnation point.

The Curie and Skan modifications to the o r i g i n a l Thwaites u n i v e r s a l functions for the shape f a c t o r and ski n f r i c t i o n suggested f o r two-dimensio­nal flows have been adopted. The laminar boundary l a y e r c a l c u l a t i o n i s persued u n t i l the occurrence of laminar separation or the onset of t r a n s i t i o n i s p r e d i c t e d .

The c r i t e r i o n for laminar separation has been kept to a value of the d v e 2

pressure gradient parameter m=—.— equal to -0.09. d s y

At present no experimental data on natural t r a n s i t i o n of the boundary layer flow on p r o p e l l e r ducts i s a v a i l a b l e . On bodies of r e v o l u t i o n i n axisym-metric flow various methods have been assessed by Kaups (1974). The methods of Michel, (1951)and G r a n v i l l e (1953) have been used i n the present work. The method of Michel has been used i n i t s transformed version as recommended by Kaups. The constants appearing i n Mangier's transformation have been set for each surface on the duct equal to the correspondent mean radius. C a l c u l a ­t i o n s performed on the duct NSMB 37 have shown that the i n f l u e n c e of choice of the constant on the Mangier's transformation on the t r a n s i t i o n p r e d i c t i o n i s small. Experimental confirmation regarding the adequacy of the previous methods for the p r e d i c t i o n of t r a n s i t i o n on ducts has not been gathered i n t h i s study.

If laminar separation i s predicted before t r a n s i t i o n the c a l c u l a t i o n of the laminar separation bubble and the p r e d i c t i o n of the eventual turbu­lent reattachment conditions are performed.

The c a l c u l a t i o n of the laminar part of the separation bubble i s done according to the method proposed by Van Ingen,(1975) . The c a l c u l a t i o n makes use of an inverse version of Thwaites i n t e g r a l method. The pressure d i s t r i ­bution i s not prescribed after separation but the shape of the separation streamline i s given instead. In the present c a l c u l a t i o n s the separation

26

streamline i s assumed to be a s t r a i g h t l i n e making an angle y with the surface given by the empirical r e l a t i o n used by Oskam, (1979),

t a n y = 1 5 R

(2-41) 9 s e p

where R, '6 V

i s the Reynolds number based on the momentum thickness sep

at separation. The s o l u t i o n y i e l d s the reversed boundary layer flow i n t e g r a l

parameters i n ad d i t i o n to the pressure d i s t r i b u t i o n , and i s termina­

ted i f t r a n s i t i o n i s predicted. A f i r s t approximation to the p o s i t i o n of

t r a n s i t i o n on the bubble may be found by using the t r a n s i t i o n c r i t e r i o n .

The c r i t e r i o n used f o r the p r e d i c t i o n of reattachment was given by the i n t e r s e c t i o n between the S t r a t f o r d ' s zero skin f r i c t i o n l i m i t i n g pressure d i s t r i b u t i o n and the i n v i s c i d pressure d i s t r i b u t i o n .

This simple c r i t e r i o n has given reasonable p r e d i c t i o n s of the length

of the separation bubble i n some of the cases i n v e s t i g a t e d by Van Ingen,

(1975) and Oskam (1979).

The turbulent boundary layer c a l c u l a t i o n i s performed using the i n t e ­

g r a l method of Head and P a t e l , (1968).

This method i s based on the s o l u t i o n of the momentum i n t e g r a l equation together with an equation f o r the entrainment rate of the boundary layer with allowance f o r departure from e q u i l i b r i u m conditions and an a u x i l i a r l y expression f o r the skin f r i c t i o n . The Cumpsty-Head skin f r i c t i o n formula as given by Head and Patel (1968) has been used. The method has proved to make rather accurate p r e d i c t i o n s of the boundary layer i n t e g r a l parameters i n two-dimensional flows f o r a wide v a r i e t y of pressure d i s t r i b u t i o n s .

The turbulent boundary layer c a l c u l a t i o n i s st a r t e d using the momentum thickness at t r a n s i t i o n obtained from the laminar boundary l a y e r c a l c u l a t i o n and an empirical r e l a t i o n f o r the shape f a c t o r given by Dvorak et a l (1979), i f n a t u r a l t r a n s i t i o n has been predicted.

s. - s t r s e p 8 s e p

3 . 6 x l 0 4

0 s e p (2-42)

27

2.4. VISCOUS-INVISCID COUPLING

As mentioned before the coupling of the viscous and i n v i s c i d c a l c u l a t i o n s i s e f f e c t e d , i n what concerns the p o t e n t i a l flow c a l c u l a t i o n , by a surface source d i s t r i b u t i o n representing the boundary layer displacement e f f e c t . With respect to the boundary layer c a l c u l a t i o n the coupling with the outer p o t e n t i a l flow i s r e a l i z e d by using on each i t e r a t i o n as boundary co n d i t i o n at the edge of the boundary layer an updated i n v i s c i d v e l o c i t y d i s t r i b u t i o n evaluated from the previous p o t e n t i a l flow s o l u t i o n at the duct's surface.

From each s o l u t i o n of the boundary layer flow the l o c a t i o n of the separation points on the outer and inner surfaces of the duct are obtained and the pressure c o e f f i c i e n t at the separation points follows from the correspondent i n v i s c i d pressure d i s t r i b u t i o n .

In a d d i t i o n , from the boundary l a y e r displacement thickness, the strength of the equivalent source d i s t r i b u t i o n can be calculated, as we w i l l ex­p l a i n i n short, a f t e r . The subsequent p o t e n t i a l flow c a l c u l a t i o n i s performed with an onset flow according to equation (2-25) and the c i r c u l a t i o n given by equation (2-39).

In the present a n a l y s i s of the e f f e c t of displacement thickness on the p o t e n t i a l flow the following assumptions have been made:

- Wake displacement e f f e c t s have been neglected. In p r i n c i p l e , t h e viscous layer c a l c u l a t i o n s should be prolonged into the wake. The wake displacement e f f e c t s could then be represented f o r a t h i n wake by a source d i s t r i b u t i o n located on the wake's centre l i n e .

- The displacement e f f e c t s of eventual laminar separation bubbles have been disregarded.

This l a s t assumption may seem rather crude i n view of l o c a l values of the displacement thickness on a separation bubble which can be one order of mag­nitude l a r g e r than the displacement thickness of the attached boundary l a y e r s . However, f o r not too long bubbles, the p o t e n t i a l disturbance, which has a dipole character, i s p r i m a r i l y l o c a l i z e d and has, i n general, a small e f f e c t on the pressure d i s t r i b u t i o n on the flow regions f a r from the separated region.

Yet, i t i s conceivable that, besides bearing a large influence on the p r e d i c t i o n of the bubble extent and the development of the reattached turbu­lent boundary l a y e r , the disturbance on the p o t e n t i a l flow caused by the

28

bubble may influence the p r e d i c t i o n of the l o c a t i o n of laminar separation i t s e l f . In many s i t u a t i o n s laminar separation on the leading edge occurs i n v a r i a b l y on a r e l a t i v e l y sharp pressure peak at the leading edge and i s i n the f i r s t place determined by the nose geometry.

We note that the c o r r e c t p r e d i c t i o n of the length of the bubble may be c r u c i a l f o r the p r e d i c t i o n of the l i f t on the s e c t i o n but i t s e f f e c t on the turbulent separation at the t r a i l i n g edge may be l e s s c r i t i c a l .

According to L i g h t h i l l , (1958),the source strength per uni t area f o r

axisymmetric flow i s

0 ( S ) = J | g <V r 6 * ) (2-43)

where 6* i s the displacement thickness defined by

oo <5* = ƒ ( l - ^ ) d z (2-44)

o

with u the streamwise v e l o c i t y i n the boundary l a y e r , and z the coordinate

normal to the surface.

Within the present d i s c r e t i z a t i o n of the contour,the i n t e g r a l repre­senting the induced v e l o c i t y due to the source d i s t r i b u t i o n i n equation (2-26) , evaluated at the c o n t r o l point i , i s

N 6„o{s' ) T ( s , s ' ) d s ' = E (cosct. U . . + s i n c t . V..) (2-45) S ' j = l ^3 i i - l

where the coupling c o e f f i c i e n t s U j . and give the a x i a l and r a d i a l v e l o c i ­

t i e s induced at the c o n t r o l point i by the source d i s t r i b u t i o n on panel j .

The evaluation of the coupling c o e f f i c i e n t s U.. and V.. has been given by

Hess and Martin, (1974), for constant, l i n e a r and parabolic source d i s t r i b u ­

t ions on a panel.

In the evaluation of the source strength and i t s d e r i v a t i v e s at the p i v o t a l point l o c a t i o n s , s p l i n e functions are used. The fun c t i o n (Vr6*) i n equation (2-43) i s approximated by weighted l e a s t squares cubic s p l i n e s to achieve various degrees of smoothing i n d i f f e r e n t flow regions, i n the manner suggested by Oskam, (1979).

29

This procedure i s used to eliminate the sudden v a r i a t i o n s of the source strength associated with the sharp v a r i a t i o n s of displacement t h i c k ­ness caused by modelling d e f i c i e n c y of the processes of natural t r a n s i t i o n or the reattachment of separated turbulent layers.

In the same way, the displacement e f f e c t of separation bubbles have been treated, by providing a smooth connection between the displacement thickness at separation and reattachment.

2.5. RESULTS IN UNIFORM FLOW AND COMPARISON WITH EXPERIMENT

To assess the c a p a b i l i t i e s of the method described i n the previous sections a set of c a l c u l a t i o n s were c a r r i e d out f o r the duct NSMB 37 i n uniform a x i a l flow and compared with experimental data. As noted before i n Section 2.2, t h i s duct has a rather blunt t r a i l i n g edge and the accuracy of i n v i s c i d c a l c u l a t i o n s s u f f e r s from a large degree of uncertainty.

A duct model with 20 cm diameter was tested i n the Large C a v i t a t i o n

Tunnel of the NSMB, at three d i f f e r e n t Reynolds numbers. The experiments comprised the measurement of the forces a c t i n g on a

duct se c t i o n and the v e l o c i t y f i e l d around the duct.

For the experimental determination of the forces a c t i n g on the section

the t o t a l drag force and the force i n c i r c u m f e r e n t i a l d i r e c t i o n a c t i n g on a

meridional cut of the duct have been measured. The a x i a l and r a d i a l v e l o c i t y components at various a x i a l l o c a t i o n s

upstream i n s i d e and downstream of the duct have been measured with the NSMB Laser-Doppler velocimeter wake f i e l d scanner.

A short d e s c r i p t i o n of the t e s t arrangement and the experimental techniques employed, i s given i n the Appendix 1.

The character of the boundary layer on the duct model has also been

in v e s t i g a t e d at the same Reynolds numbers i n the Deep Water Basin by paint

t e s t s . The tests apply a s i m i l a r technique to the one used to i n v e s t i g a t e the

boundary l a y e r on p r o p e l l e r models, Kuiper (1981).

The paint t e s t s were run i n the deep water basin at the advance speeds

30

1.25, 2.50 and 3.75 m/s corresponding to Reynolds numbers based on the duct's 5 5 5 length r e s p e c t i v e l y of Re c=l.10x10 , Rec=2.20x10 and Rec=3.29x10 .

The r e s u l t s of the v i s u a l i z a t i o n of the l i m i t i n g streamlines by the paint are shown f o r the highest speed of 3.75 m/s on Figures 2-5 to 2.8. In a l l these Figures the undisturbed stream i s from l e f t to r i g h t .

A schematic representation of the boundary layer character suggested by the observed paint patterns i s given i n F i g . 2-9.

Fig. 2-5. Paint pattern on the Fig. 2-6. Faint pattern on the

outer surface of duct 37. outer surface of duct 37.

Leading edge region. Trailing edge region.

Re =3.29xl05. Re =3.29xlOS. c a

In the same f i g u r e the l o c a t i o n s of the reference s t r i p s , which may be discerned i n F i g s . 2-5 to 2-8, are given.

The occurrence of a separation bubble on the outer surface i s c l e a r l y

31

Fig. 2-7. Faint pattern on the inner surface of duct 37. Leading edge

region. Re =3.29x10^.

32

L E A P W 6 E D G E 'LAMINAR S E P A R A T I O N

TURBULENT SEPARATION OUTER S U R F A C E

L LAMINAR OR TURBULENT SEPARATION INNER S U R F A C E

evidenced i n F i g . 2-5 where a region of reversed flow i s to be seen. Turbu­lent separation occurs near the t r a i l i n g edge as shown i n F i g . 2-6. We note that both patterns have been strongly influenced by g r a v i t y force e f f e c t s leading to the transverse component on the paint streaks i n F i g . 2-5 and the accumulation of paint downwards at the t r a i l i n g edge.

Figs . 2-7 and 2-8 show a laminar boundary layer approaching a region of low skin f r i c t i o n at about x/c=0.40 and a c l e a r separation l i n e at the t r a i l i n g edge at about x/c=0.82. Whether the laminar boundary l a y e r separates at x/c=0.40 forming a separation bubble i s not c l e a r from the paint patterns.

From the paint t e s t s c a r r i e d out at the two other speeds,no s i g n i f i c a n t Reynolds e f f e c t has been found on the pressure d i s t r i b u t i o n s and therefore on the l o c a t i o n s of leading edge laminar separation or t r a i l i n g edge separation.

Yet, the p r e c i s e streamwise extent of the leading edge laminar separa­t i o n bubbles on the outer surface could not be c l e a r l y e s t a b l i s h e d by the paint t e s t s i n any of the t e s t conditions.

T h e o r e t i c a l c a l c u l a t i o n s and other experimental studies (Van Ingen,

1975), reveal that the Reynolds number has a marked e f f e c t on the extent of

33

the bubble by i n f l u e n c i n g the onset of t r a n s i t i o n on the separated shear layer and therefore determining the reattachment l o c a t i o n .

The t h e o r e t i c a l c a l c u l a t i o n s have been performed for the three d i f f e ­rent Reynolds numbers corresponding to the test c o n d i t i o n s .

The computed pressure d i s t r i b u t i o n s are given i n F i g . 2-10 together with the i n d i c a t i o n of the p r e d i c t e d l o c a t i o n of separation and reattachment points of the correspondent boundary l a y e r s .

The d i f f e r e n c e s between the pressure d i s t r i b u t i o n s on the inner surface are very small. On the outer surface the extent of the laminar separation bubble considerably decreases with the increase of Reynolds number.

Fig. 2-10. Computed pressure distributions on duct 37 in uniform flow.

Trailing edge laminar separation on the inner surface.

34

In order to i n v e s t i g a t e the e f f e c t of t r a i l i n g edge turbulent separa­t i o n on the duct loading the c a l c u l a t i o n s have been c a r r i e d out with "stimulated" t r a n s i t i o n assumed to take place at x/c=0.39 on the inner sur­face of the duct. The r e s u l t s are shown i n F i g . 2-11.

Fig. 2-11. Computed pressure distributions on duet 37 in uniform-

flow. Trailing edge turbulent separation on the inner

surface.

In t h i s case separation from the t r a i l i n g edge on the inner surface i s

s h i f t e d to x/c=0.96 staying, however, independent of Reynolds number f o r the

cases considered.

Comparison of the r e s u l t s of F i g s . 2-10 and 2-11 reveals a considerable e f f e c t on the duct loading due to the "turbulence s t i m u l a t i o n " on the inner surface. Although the change of character of the inner boundary l a y e r i n f l u ­ences the pressure d i s t r i b u t i o n on the outer surface, the extent of the separation bubbles i s not s i g n i f i c a n t l y changed. We note that the determina­t i o n of reattachment with reasonable accuracy i n the present c a l c u l a t i o n s i s

impaired by the quasi p a r a l l e l course near reattachment of the curves

representing the S t r a t f o r d recovery f o r the separated turbulent shear layer

and the " i n v i s c i d " pressure d i s t r i b u t i o n .

The duct s e c t i o n l i f t c o e f f i c i e n t s obtained from the measurement of the

c i r c u m f e r e n t i a l force on the duct's meridional cut are given i n F i g . 2-12.

The measurements of c i r c u m f e r e n t i a l force included not only the integrated e f f e c t of the r a d i a l pressure forces a c t i n g on the duct's inner and outer surfaces, but also the force component due to the pressure forces acting on the duct sections on the cut, (see Appendix 1).

A r a d i a l force c o e f f i c i e n t C i s defined as R

F C = 2 — S

R i 2„2 ^ P U Q R

where F i s the r a d i a l force a c t i n g on the duct per unit radian, n

Fig. 2-12. Effect of Reynolds number on measured and calculated duct

section lift coefficients. Duct 37 in uniform flow.

36

0.3

0.2

0.1 -

C A L C U L A T E D

_ L A M I N A R S E P A R A T I O N " I N N E R S U R F A C E

T U R B U L E N T S E P A R A T I O N " I N N E R S U R F A C E

M E A S U R E D

C A V I T A T I O N T U N N E L

J I I L _ 10. 20. 30. 50. 60

R e c »10"4

Fig. 2-13. Effect of Reynolds number on the duet's section drag

coefficient, duct 37 in uniform flow.

An equivalent two-dimensional l i f t c o e f f i c i e n t i s defined by

c - i f a / c L 2 c ^R

where c i s the duct length. To enable the comparison with the experiment r e s u l t s the duct s e c t i o n

l i f t c o e f f i c i e n t was corrected f o r the e f f e c t of the duct's f i n i t e t h i c k ­ness by assuming a l i n e a r v a r i a t i o n of pressure on the cut between the outer and inner surfaces.

We note the remarkable e f f e c t of the Reynolds number on the l i f t which changes sign i n the range of speeds considered. This e f f e c t has been found both f o r the measurements performed i n the deep water basin and i n the c a v i t a t i o n tunnel, though a c l e a r s h i f t between the two l i f t curves i s found.

The c a l c u l a t e d r e s u l t s with assumed turbulent separation on the t r a i l i n g edge seem to c o r r e l a t e reasonably with the measurements i n the c a v i t a t i o n tunnel. However, the character of the boundary l a y e r on the inner surface f o r the t e s t s i n the c a v i t a t i o n tunnel has not been e s t a b l i s h e d and no d e f i n i t e conclusions i n t h i s respect can be drawn.

37

The paint t e s t s in the Deep Water Basin i n d i c a t e d the occurrence of separation at a t r a i l i n g edge l o c a t i o n which c o r r e l a t e d rather well with the t h e o r e t i c a l p r e d i c t i o n s of t r a i l i n g edge laminar separation. Yet, considerable discrepancies are found i n the l i f t c o e f f i c i e n t . A po s s i b l e reason f o r such discrepancies may l i e on the d i f f e r e n c e s on the extent of the laminar separation bubbles which are responsible f o r the high slope of the t h e o r e t i ­c a l l i f t curves at the lowest Reynolds number.

The r e s u l t s f o r the v e l o c i t y f i e l d around the duct are shown i n F i g s . 2-14 to 2-17.

For the purposes of comparison, the v e l o c i t i e s c a l c u l a t e d with the f i r s t i n v i s c i d approximation are included. These c a l c u l a t i o n s have been c a r r i e d out using a t r a i l i n g edge co n d i t i o n of equal pressure, (see eq. 2-37), applied at the pressure minima on the t r a i l i n g edge. It has been found from the paint t e s t s that the l o c a t i o n of turbulent separation on the outer surface and laminar separation on the inner surface c o r r e l a t e d rather well with l o c a t i o n of the pressure minima of the f i r s t i n v i s c i d pressure d i s t r i b u t i o n at the t r a i l i n g edge.

The r e s u l t s of the viscous flow a n a l y s i s show a s a t i s f a c t o r y agreement with the experiments except i n s i d e the boundary l a y e r and wake. The agreement i n s i d e the duct i s p a r t i c u l a r l y good. No attempt has been made to c a l c u l a t e v e l o c i t y p r o f i l e s i n s i d e the boundary l a y e r s and wake and therefore the pre­sent comparison i s only meaningful i n the outer p o t e n t i a l flow regions.

Because of the neglection of the displacement e f f e c t s of the separation bubbles, the c a l c u l a t e d p o t e n t i a l flow v e l o c i t i e s do not agree with the measurements on the outer surface i n the leading edge region. However, the agreement obtained downstream of reattachment suggests the l o c a l character of such e f f e c t .

2.6. CALCULATION OF THE DUCT STEADY LOAD FOR A DUCT WITH PROPELLER

2.6.1. P r o p e l l e r model and p r o p e l l e r induced v e l o c i t i e s on the duct.

When a p r o p e l l e r rotates i n s i d e a duct, the flow past the duct i s

unsteady. Within i n v i s c i d flow theory, i f the duct i s axisymmetric and the

ducted p r o p e l l e r i s placed i n an axisymmetric onset flow, the flow through

the ducted p r o p e l l e r becomes steady i n a coordinate system r o t a t i n g with the

38

X

p r o p e l l e r . On the other hand, i f viscous e f f e c t s are considered, the flow i n s i d e the duct's boundary layer i s time dependent i n a coordinate system f i x e d to the duct and w i l l have, i n general, a p e r i o d i c character stemming from the p e r i o d i c nature of the p r o p e l l e r induced flow.

At s u f f i c i e n t l y small clearances between the p r o p e l l e r blade t i p s and the duct, the duct boundary layer flow w i l l f u r t h e r i n t e r a c t with the p r o p e l l e r blades and the associated t i p clearance flows, a r i s i n g from the pressure d i f f e r e n c e between the pressure and suction sides of the p r o p e l l e r blades.

Due to the great complexity of such i n t e r a c t i o n flow, various approxi­

mations have to be introduced i n the t h e o r e t i c a l treatment.

The time-mean forces and pressure d i s t r i b u t i o n on the duct can be

approximated by considering the time-mean v e l o c i t y f i e l d induced by the

p r o p e l l e r on the duct. Obviously, the determination of such v e l o c i t y f i e l d

i s equivalent to the c a l c u l a t i o n of the c i r c u m f e r e n t i a l mean flow induced

by the p r o p e l l e r blades i n a coordinate system r o t a t i n g with the p r o p e l l e r .

As i t has been observed i n the i n t r o d u c t i o n , i f the l i f t i n g l i n e representation of the p r o p e l l e r i s adopted, the equivalence between the c i r c u m f e r e n t i a l mean of the induced v e l o c i t y f i e l d and the v e l o c i t y f i e l d of an actuator disk model with the same c i r c u l a t i o n d i s t r i b u t i o n , has been est a b l i s h e d by Hough and Ordway, (1965), for l i g h t l y loaded p r o p e l l e r s , i . e . under the assumption that the h e l i c a l v o r t i c e s shed from the l i f t i n g l i n e s representing the p r o p e l l e r blades, are convected by the undisturbed r e l a t i v e flow.

For moderately or h e a v i l y loaded p r o p e l l e r s the equivalence does not hold. The F o u r i e r a n a l y s i s of the v e l o c i t y f i e l d applied by Hough and Ordway may e a s i l y be extended to moderately loaded p r o p e l l e r s , f o r which the p i t c h of the shed h e l i c a l v o r t i c e s i s modified to include the a x i a l and tan g e n t i a l perturbation v e l o c i t i e s induced by the p r o p e l l e r . The r e s u l t f o r the zeroth harmonic leads to the v e l o c i t i e s induced by a p r o p e l l e r with an i n f i n i t e number of blades, with the correspondent slipstream vortex system being b u i l t up of an i n f i n i t e set of h e l i c a l v o r t i c e s with a p i t c h d i s t r i b u -

43

t i o n i d e n t i c a l to the one of the f i n i t e blade number model, (see Morgan 1961, 1965).

In the i n f i n i t e blade number l i m i t the vortex sheets degenerate i n t o a volume d i s t r i b u t i o n of v o r t i c i t y with a t a n g e n t i a l component or r i n g v o r t i c i t y inducing a x i a l and r a d i a l v e l o c i t i e s and an a x i a l v o r t i c i t y component, which together with the r a d i a l bound v o r t i c i t y at the disk and the hub a x i a l v o r t i c i t y , induces t a n g e n t i a l v e l o c i t i e s i n s i d e the slipstream.

For the c a l c u l a t i o n of the c i r c u m f e r e n t i a l l y averaged a x i a l and r a d i a l v e l o c i t i e s induced at the duct by the p r o p e l l e r , i t i s simpler to make use of the expressions for the v e l o c i t i e s induced by a s e m i - i n f i n i t e r i n g vortex c y l i n d e r of constant radius.

This v e l o c i t y f i e l d may be r e a d i l y obtained from Hough and Ordway expressions i n terms of Legrendre functions. Equivalent expressions i n terms of e l l i p t i c i n t e g r a l s have also been obtained by Gibson, (1974), d i r e c t l y from the a p p l i c a t i o n of Biot-Savart law to the s e m i - i n f i n i t e r i n g vortex c y l i n d e r .

If V A ( x , r ; r ' ) and V R ( x , r ; r ' ) denote the a x i a l and r a d i a l v e l o c i t i e s induced at a point (x,r) by a s e m i - i n f i n i t e r i n g vortex c y l i n d e r of unit strength, of radius r' and i t s basis located at the plane x=0 of a c y l i n d r i c a l coordinate system, we have

V ( x , r ; r ' ) = V* + X _ [ K ( k ) - ^ 11 ( a 2 \ k ) ]

(2-46)

2 V ( X , r ; r ' ) = —--~ \ - r [E (k) - ( K (k)] (2-47)

k [ ( p - r + ( p - r p 1

2 where K(k), E(k), H(a \k) are, r e s p e c t i v e l y the e l l i p t i c i n t e g r a l s of f i r s t second and t h i r d kind with modulus

. 2 4 r r 1

k = - s j , (2-48) x + ( r + r 1 )

and parameter

44

2 4 r r ' a = ö . (2-49) 2 '

+r' ) ( r + r ' )

The fun c t i o n V * ( r , r ' ) i s

V ^ ( r , r ' ) = 1 r < r ' = h r = r ' = 0 r > r ' (2-50)

Let F (r)=2lTr w+ be the c i r c u l a t i o n of the p r o p e l l e r with i n f i n i t e °j o

number of blades, w being the c i r c u m f e r e n t i a l mean ta n g e n t i a l v e l o c i t y o immediately downstream of the p r o p e l l e r disk, x=+0.

The a x i a l and t a n g e n t i a l v o r t i c i t y components are r e s p e c t i v e l y

U = - - J - (2-51) x 2irr d r

and

1 dX^ 2-nr t a n g . d r l

(2-52)

where 3. i s the p i t c h angle of the h e l i c a l v o r t i c e s . According to the assump­tions of moderately loaded p r o p e l l e r theory, the p i t c h of the h e l i c a l v o r t i c e s remains constant i n the p r o p e l l e r s l i p s t r e a m and therefore, the v o r t i c i t y components O J ^ and given by (2-51) and (2-52) are constant i n the a x i a l d i r e c t i o n . For an a r b i t r a r y c i r c u l a t i o n d i s t r i b u t i o n the p i t c h

P. (r)=2irr tang, i s a function of the radius. l i -p

Introducing the non-dimensional c i r c u l a t i o n d i s t r i b u t i o n Gaa

=^^}~ we o

obtain f o r the ta n g e n t i a l v e l o c i t y

G + CO _ _ = r < 1 (2-53) o r

= 0 r > 1 In the sequel, as i n (2-53), the v e l o c i t i e s are made non-dimensional

by the undisturbed a x i a l v e l o c i t y U q and the coordinate lengths by the pro­

p e l l e r radius.

45

The a x i a l and r a d i a l v e l o c i t i e s are then obtained by i n t e g r a t i n g the

contrib u t i o n s of the vortex c y l i n d e r s from the hub to the t i p i n the form

1 dG G = - 1 P t i H ê T d ï ^ V ( x . , r ; r ' ) d r ' (2-54)

r h

1 ! d G c o V = - 1 r ' t a n g . d F ^ V R (x , r ; r ' ) d r ' (2-55)

r h As mentioned before, the a x i a l and r a d i a l v e l o c i t y components given by

(2-54) and (2-55) represent the c i r c u m f e r e n t i a l average of the induced v e l o ­

c i t i e s of the p r o p e l l e r with f i n i t e number of blades provided that 3. i s the

hydrodynamic p i t c h of the h e l i c a l v o r t i c e s shed from the l i f t i n g l i n e s

and the c i r c u l a t i o n d i s t r i b u t i o n G i s taken as

G^ = ZG , (2-56)

where G i s the c i r c u l a t i o n of the l i f t i n g l i n e s and Z the number of blades.

If Lerbs' (1952) moderately loaded theory i s considered,the p i t c h of the h e l i c o i d a l vortices is taken as the hydrodynamic p i t c h angle at the l i f t i n g l i n e . This approach has been followed by Morgan, (1961), f o r ducted p r o p e l l e r s . When considering the v e l o c i t i e s induced by the duct and hub on the pro p e l ­l e r , the hydrodynamic p i t c h angle at the l i f t i n g l i n e i s ( F i g . 2-18).

1+u +u,+u, t a n g . = P d h (2-57) i Trr

—r- - W - W , J P d where J=U /nD i s the advance r a t i o , u , u, and u the a x i a l v e l o c i t i e s at the o p d h l i f t i n g l i n e induced by the p r o p e l l e r duct and hub r e s p e c t i v e l y , and w and

P w the tange n t i a l v e l o c i t i e s át the l i f t i n g l i n e induced by the p r o p e l l e r and duct. The tangential v e l o c i t y induced by the hub, being very small, has been discarded i n eq. (2-57).

In equation (2-57) the a x i a l v e l o c i t y induced by the duct includes, not only the c o n t r i b u t i o n of the steady load on the duct, but also the co n t r i b u ­t i o n of i t s f l u c t u a t i n g part. Moreover, the ta n g e n t i a l v e l o c i t y w, induced by

d the duct i s e n t i r e l y due to the f l u c t u a t i n g part.

Therefore, i t s determination requires the s o l u t i o n f o r the higher blade harmonics of the duct's c i r c u m f e r e n t i a l load d i s t r i b u t i o n as i n d i c a t e d by

46

Morgan, (1961). Such procedure e n t a i l s a lengthy i t e r a t i o n process between

duct and p r o p e l l e r i n v o l v i n g on each i t e r a t i o n step the c a l c u l a t i o n of the

blade harmonics of the p r o p e l l e r and duct induced v e l o c i t y f i e l d .

Considerable s i m p l i f i c a t i o n of the c a l c u l a t i o n i s achieved i f the

higher blade harmonics are neglected and, i n a d d i t i o n , the hydrodynamic p i t c h

i s determined by the v e l o c i t i e s induced by an i n f i n i t e l y bladed p r o p e l l e r

model.

We notice that, when assuming an i n f i n i t e l y bladed p r o p e l l e r which

d e l i v e r s the same thrust as the f i n i t e bladed p r o p e l l e r , the correspondent

c i r c u l a t i o n d i s t r i b u t i o n does not comply with equation (2-56), which only

holds i n the l i m i t Z-*o°. This can r e a d i l y be seen, from the a p p l i c a t i o n of

the law of Kutta-Joukowsky which gives f o r the p r o p e l l e r with i n f i n i t e number

of blades

C T = 4 ƒ - | ) G „ r d r (2-58)

P r h

and for the f i n i t e bladed p r o p e l l e r

1

C m = 4Z ƒ (—• - w) Gr d r (2-59) P r h

where w i s the ta n g e n t i a l v e l o c i t y induced at the l i f t i n g l i n e

W = W + W-, P d

With regard to the hydrodynamic p i t c h f o r the vortex system of an

i n f i n i t e l y bladed p r o p e l l e r , Dyne, (1967),showed that the a x i a l induced

v e l o c i t i e s f a r downstream as c a l c u l a t e d from general momentum theory of the

actuator disk, (see Glauert,1935) , and from equation (2-54), are i n agreement,

provided that the hydrodynamic p i t c h angle i s taken i n the ultimate wake

(Fi g . 2-19):

1+U CO

t a n g . = . (2-60) 1 » irr Goo

J r

Equations (2-58), (2-60), (2-54), (2-55) together with the r e l a t i o n s

(2-46) through (2-50), completely determine the a x i a l and r a d i a l induced 47

2 7 t n r

Fig. 2-18. Velocity diagram at a l i f t i n g line in the moderately

loaded theory of a ducted propeller.

v e l o c i t i e s on the duct and hub i f the advance r a t i o , p r o p e l l e r loading c o e f f i c i e n t and c i r c u l a t i o n d i s t r i b u t i o n are known, without i n v o l v i n g the duct induced v e l o c i t y f i e l d e x p l i c i t y .

The duct flow problem i n the presence of a general axisymmetric poten­t i a l flow f i e l d such as the p r o p e l l e r induced flow described above, has been treated i n se c t i o n 2.2 and w i l l not be considered here any f u r t h e r .

2 T t n r

Fig. 2-19. Velocity diagram in the ultimate wake in the moderately

loaded actuator disk theory. 48

2.6.2. Remarks on viscous e f f e c t s on the duct f o r a ducted p r o p e l l e r .

The i n v i s c i d flow theory presented i n the previous sections has been a powerful t o o l to p r e d i c t duct performance and to provide design guidance. Nevertheless, i t might be of l i m i t e d use i n the cases where viscous e f f e c t s , which c o n t r o l the c i r c u l a t i o n on the duct,cannot be adequately represented by the c l a s s i c a l Kutta-Joukowsky c o n d i t i o n . This f a c t has been i l l u s t r a t e d i n t h i s Chapter f o r axisymmetric steady flow past p r o p e l l e r ducts with blunt t r a i l i n g edges.

When the p r o p e l l e r i s operating i n s i d e the duct, the boundary layer flow on the duct i s a f f e c t e d by three-dimensional and unsteady e f f e c t s which a r i s e from the i n t e r a c t i o n with the r o t a t i n g p r o p e l l e r blades. For most of the cases of p r a c t i c a l i n t e r e s t i n which the duct e x h i b i t s a form of t r a i l i n g edge separation, the question regarding the p r e d i c t i o n of the steady compo­nent of c i r c u l a t i o n and forces on the duct, i s r e l a t e d to the extent to which the p r o p e l l e r induced three-dimensional and unsteady e f f e c t s i n f l u e n c e the o v e r a l l separation pattern at the t r a i l i n g edge.

When considering the boundary l a y e r flow from the point of view of an

observer f i x e d to the duct, the outer p o t e n t i a l flow may be regarded as

composed of a mean meridian flow with a superimposed three-dimensional

o s c i l l a t i n g disturbance flow. The amplitude of the o s c i l l a t o r y disturbance w i l l strongly depend on

the proximity to the p r o p e l l e r .

On the other hand, on the d i f f u s e r part of the duct, the mean outer

flow includes a c i r c u m f e r e n t i a l component r e s u l t i n g from the s w i r l imparted

by the p r o p e l l e r to the f l u i d i n the slipstream.

Also, the flow around the p r o p e l l e r t i p s which i n t e r a c t s with the boundary layer flow on the duct's inner w a l l , may be of importance i n deter­mining the downstream development of the boundary layer i n the d i f f u s e r . Such flows have been studied by several authors (see f o r instance Hutton, 1958, Gearhart, 1966 , Lakshminarayna, 1970), i n view of t h e i r importance i n asses­sing the correspondent t i p clearance losses or i n r e l a t i o n to the determina­t i o n of the blade t i p c a v i t a t i o n performance.

49

Most of the models proposed are i n v i s c i d flow models which neglect the existence of the wall boundary l a y e r and are p r i m a r i l y concerned with the flow conditions at the gap and t h e i r i n f l u e n c e on blade performance.

The response of a two-dimensional boundary l a y e r to unsteadiness i n the free-stream has been studied experimentally and t h e o r e t i c a l l y f o r both laminar and turbulent boundary l a y e r s . For a review we r e f e r to the paper of T e l i o n i s , (1979).

In p a r t i c u l a r , the unsteady separation of an o s c i l l a t o r y two-dimensio­

nal boundary layer from the t r a i l i n g edge of an a i r f o i l - l i k e body has been

examined experimentally by Despard and M i l l e r , (1971). Their r e s u l t s supplied further evidence f o r a now commonly known f a c t

that steady separation c r i t e r i a such as the vanishing of the wall shear at separation,cannot be applied to unsteady boundary l a y e r flow.

In f a c t , occurrence of reversed flow at a c e r t a i n downstream l o c a t i o n on the boundary layer could be observed near the wall during the greatest part of the o s c i l l a t i n g c y c l e without being associated with flow breakdown from the wall at that l o c a t i o n .

In a d d i t i o n , t h e i r r e s u l t s i n d i c a t e that the d e v i a t i o n of the l o c a t i o n flow separation from the wall, implying the formation of a wake i n the case of unsteady flow, from the l o c a t i o n of separation i n steady flow with the same mean pressure gradient, decreases with i n c r e a s i n g Reynolds number and f r e ­quency of the o s c i l l a t i o n , b e i n g influenced to a much l e s s degree by the am­p l i t u d e of the o s c i l l a t i o n .

We notice, that the mean pressure gradient used by Despard and M i l l e r i n t h e i r experiments are much weaker than the ones which can be expected on the inner surface of a duct i n the presence of a p r o p e l l e r , except at very l i g h t loadings, and the amplitude d i s t r i b u t i o n being uniform along the chord, d i f f e r s from the one occurring on the duct surfaces.

The range of Reynolds numbers and frequency parameter i n v e s t i g a t e d can yet be considered as representative of the flow on the duct at model s c a l e .

The previous considerations suggest that by neg l e c t i n g unsteady and

three-dimensional e f f e c t s as a f i r s t approximation, one may obtain an e s t i ­

mate of the steady component of c i r c u l a t i o n considering the steady boundary

50

l a y e r flow subject to the time-mean pressure d i s t r i b u t i o n .

The method f o r steady flows described i n Section 2.3 has been applied to the ducted p r o p e l l e r case making use of a pressure d i s t r i b u t i o n obtained by the method of the previous s e c t i o n . The r e s u l t s of these c a l c u l a t i o n s are presented i n the next s e c t i o n .

2.7. NUMERICAL RESULTS AND COMPARISON WITH EXPERIMENT

In order to v e r i f y experimentally the various t h e o r e t i c a l models which have been employed i n the c a l c u l a t i o n of the duct steady performance and, i n pa r t i c u l a r , to assess the accuracy of the n o n - i t e r a t i v e method based on the actuator disk theory presented i n t h i s chapter, a set of experiments with ducted p r o p e l l e r s has been c a r r i e d out i n uniform flow.

The experiments have been se l e c t e d to provide, to the degree required by the t h e o r e t i c a l models used i n the c a l c u l a t i o n s , d e t a i l e d information on the flow f i e l d around the ducted p r o p e l l e r .

The measurements included the o v e r a l l forces a c t i n g on duct and propeller, pressure d i s t r i b u t i o n on the inner side of the duct and the a x i a l and r a d i a l v e l o c i t y components of the v e l o c i t y f i e l d upstream and downstream of the ducted p r o p e l l e r , i n c l u d i n g the p r o p e l l e r Slipstream. In this i n v e s t i g a t i o n the measurements were focussed on the time-mean values and no attempt has been made to gather information on the turbulent s t r u c t u r e of the flows i n the duct and p r o p e l l e r wakes. Also the d e t a i l s of the boundary layer flow on the duct have not been considered.

The f o l l o w i n g ducted p r o p e l l e r configurations have been considered:

- Duct NSMB 19A with a p r o p e l l e r of the KA-4-70 s e r i e s . Open-water characte­r i s t i c s and the r a d i a l force component ac t i n g on the duct s e c t i o n , d e r i v a ­ble from the measurement of the ta n g e n t i a l force a c t i n g on a duct meridio­nal cut, have been measured i n the Deep Water Basin.

- Duct NSMB 37 with a four-bladed c o n t r o l l a b l e p i t c h p r o p e l l e r of Kaplan type denoted here by p r o p e l l e r A. Open-water c h a r a c t e r i s t i c s and pressure d i s t r i b u t i o n on the inner side of the duct at three d i f f e r e n t p r o p e l l e r loadings have been measured i n the Depressurized Towing Tank ( i n atmospheric c o n d i t i o n ) .

51

- Duct NSMB 37 with a five-bladed p r o p e l l e r denoted by p r o p e l l e r B: open-

water c h a r a c t e r i s t i c s , r a d i a l force measurements on the duct se c t i o n and

v e l o c j t y f i e l d measurements with Laser-Doppler velocimeter at various

l o c a t i o n s upstream and downstream of the duct with operating p r o p e l l e r

were c a r r i e d out i n the Large C a v i t a t i o n Tunnel at three d i f f e r e n t p r o p e l l e r

loadings.

The p a r t i c u l a r s of the p r o p e l l e r of the KA-series can be found i n

Oosterveld,(1971). The geometry of the p r o p e l l e r s A and B are given i n the

Appendix 2.

In a d d i t i o n , the character of the boundary layer on the duct NSMB 37 with the p r o p e l l e r B has been i n v e s t i g a t e d i n the Deep Water Basin by means of a paint t e s t technique.

The r e s u l t s of the c a l c u l a t i o n s with the actuator disk a n a l y s i s of the present chapter f o r the forces a c t i n g on the duct NSMB 19A with the KA-4-70 se r i e s p r o p e l l e r model are compared with the measurements i n F i g . 2-20. The r e s u l t s are presented i n terms of a K - J diagram. In order to enable an

T immediate comparison of the magnitude of the r a d i a l and a x i a l force compo­

nent ac t i n g on a duct's s e c t i o n , a r a d i a l force c o e f f i c i e n t K , s i m i l a r to R

the thrust c o e f f i c i e n t K„ has been defined as T d

2TT F R K R 274 pn D

where F i s the r a d i a l force per unit radian, acting on the duct. R

The c a l c u l a t i o n s were performed with an assumed load d i s t r i b u t i o n on

the actuator disk given by Gco=K(r-r ) / l - r and a clearance between the edge

of the disk and the duct's surface of 0.5%. The p r o p e l l e r loading c o e f f i c i e n t

was taken i d e n t i c a l to the experimental one.

Viscous e f f e c t s were neglected and the Kutta co n d i t i o n was implemented

i n the conventional way by p l a c i n g the stagnation point on the b i s e c t o r to

the t r a i l i n g edge angle.

52

Fig. 2-20. Comparison of duct force coefficients for duct NSMB 19A

with propeller KA 4-70, P/D-1.0.

From F i g . 2-20 i t can be seen that the agreement i n duct thrust

between c a l c u l a t i o n and experiment i s good except at high advance r a t i o s .

Also at high p r o p e l l e r loadings a trend f o r the theory to p r e d i c t duct

thrusts higher than the experimental ones can be discerned.

At low p r o p e l l e r loadings the i n v i s c i d theory obviously p r e d i c t s a p o s i t i v e duct thrust as long as the p r o p e l l e r also d e l i v e r s t h r u s t . Therefore, the discrepancies i n t h i s range of loading may be ascribed to the e f f e c t s of v i s c o s i t y . In p a r t i c u l a r the large measured values of duct drag can be a t t r i ­buted to the occurence of laminar separation on the outer surface of the duct.

At high p r o p e l l e r loadings non-linear e f f e c t s on the p r o p e l l e r s l i p -

53

Fig. 2-21. Calculated pressure distributions on duct NSMB 19A with

propeller KA 4-70. P/D=1.0 at various propeller loadings.

stream become important and may be responsible f o r the overestimation of the p r o p e l l e r induced v e l o c i t i e s .

With regard to the r a d i a l force the c a l c u l a t i o n s consistently underesti­mate the measured values. We note that the t h e o r e t i c a l values have been corrected for the pressure force a c t i n g on the duct meridional cut assuming, as i n the case of uniform flow treated before, a l i n e a r pressure v a r i a t i o n between the outer and inner surfaces. The corrected c o e f f i c i e n t i s to be compared with the experimental c o e f f i c i e n t K^.

Considering the good c o r r e l a t i o n on the duct t h r u s t , the discrepancy i s expected to be caused by the p r e d i c t i o n of the pressure d i s t r i b u t i o n on the inner side of the duct, downstream of the p r o p e l l e r .

54

Fig. 2-22. Effect of tip clearance on the pressure distribution on

duct NSMB 19A. J=0.60, =1.245. V

The c a l c u l a t e d pressure d i s t r i b u t i o n s are given i n F i g . 2-21. Apart from the sudden pressure r i s e due to the vanishing chord actuator disk modelling, the pressure d i s t r i b u t i o n s on the d i f f u s e r are independent of the pr o p e l l e r loading which i s rather unexpected.

The e f f e c t s of t i p clearance and l o c a t i o n of p r o p e l l e r plane on the

ca l c u l a t e d pressure d i s t r i b u t i o n s are shown i n F i g s . 2-22 and 2-23, r e s p e c t i v e ­

l y .

The r e s u l t s f o r the duct, p r o p e l l e r and t o t a l thrust on the duct NSMB 37 with the p r o p e l l e r A are given i n F i g . 2-24.

The c o r r e l a t i o n on duct thrust of the i n v i s c i d c a l c u l a t i o n i s consider­ably worse than f o r the duct NSMB 19A. The discrepancies are r e l a t e d to the d i f f i c u l t i e s encountered i n p r e d i c t i n g the duct c i r c u l a t i o n from the a p p l i ­c a t i o n of a steady form of the Kutta c o n d i t i o n to t h i s duct. The i n v i s c i d c a l c u l a t i o n s have been c a r r i e d out using the cond i t i o n of equal pressure at the points on the t r a i l i n g edge where the pressure minima occur. Such c r i t e r i o n has proved to give a reasonable estimate of the c i r c u l a t i o n on the duct 37 i n uniform flow without p r o p e l l e r .

55

- 3 0

Fig. 2-23. Effect of location of propeller plane on the pressure

distribution on duct NSMB 19A. J=0.50, Cm =1.245. J-p

Contrary to the case of duct 19A the I n v i s c i d c a l c u l a t i o n tends to

give smaller t h r u s t s on the duct than the experimental ones at higher pro­

p e l l e r loadings and the correspondent underestimation of the c i r c u l a t i o n

around the duct can be r e l a t e d to the inaccuracies i n the pressure d i s t r i b u ­

t i o n on the inner side of the duct downstream of the p r o p e l l e r . This f a c t i s i l l u s t r a t e d i n F i g s . 2-25 to 2-27 where the c a l c u l a t e d

and measured pressure d i s t r i b u t i o n s on the inner side of the duct are shown. Since, at i n c r e a s i n g p r o p e l l e r loading, the l e v e l of the pressure minimum i s underestimated by the l i n e a r i z e d theory, the c i r c u l a t i o n i s also underestimated.

For the highest p r o p e l l e r loading the e f f e c t of the a p p l i c a t i o n of a t r a i l i n g edge co n d i t i o n on the pressure d i s t r i b u t i o n are shown f o r c a l c u l a ­tions performed with the l i n e a r i z e d and non-linear t h e o r i e s .

The non-linear theory used i n the c a l c u l a t i o n s solves the exact equa­tions of motion f o r axisymmetric i n v i s c i d flow subject to the boundary conditions on the duct and hub by a d i s c r e t e vortex method and i s considered

56

Fig. 2-24. Correlation for the invisoid and viscous calculations of

the thrust acting on the duct 37 with propeller A.

i n Chapter 3. In comparison with the l i n e a r i z e d model presented i n t h i s Chapter, i t can be s a i d that the e f f e c t s of c o n t r a c t i o n of the stream surfaces and vortex p i t c h deformation i n the p r o p e l l e r slipstream under the i n f l u e n c e of the duct-hub induced v e l o c i t i e s are taken i n t o account.

Note worthy i n the r e s u l t s of F i g . 2-27 are the d i f f e r e n c e s between the pressure d i s t r i b u t i o n c a l c u l a t e d by the l i n e a r i z e d and non-linear theories downstream of the p r o p e l l e r plane. Except near the leading edge and i n the very v i c i n i t y of the p r o p e l l e r plane, the pressure d i s t r i b u t i o n c a l c u l a t e d with the non-linear theory agrees well with the measured pressure d i s t r i b u t i o n i f a proper t r a i l i n g edge con d i t i o n i s chosen. The smaller values of the measured pressure at the leading edge might be r e l a t e d to the occurrence of a separation bubble on the inner surface and are responsible

57

Fig. 2-26. Pressure distribution on duct 37 with propeller A. J=0.40

C =3.50. V

LINEARIZED THEORY LINEARIZED THEORY STAGNATION POINT x / C . 90 .9 V . OUTER SIDE

0.2 0 . 4 0.6 0.8 1.0 x / c

Fig. 2-27. Pressure distribution on duct 37 with propeller A.

J=0.20. CT =16.36. P

for the d i f f e r e n c e s between the measured and the c a l c u l a t e d values of duct thrust c o e f f i c i e n t .

The steady v i s c o u s - i n v i s c i d i n t e r a c t i o n model has been applied to c a l c u l a t e the flow past the duct under the i n f l u e n c e of the propeller, using the l i n e a r i z e d actuator disk model to compute the p r o p e l l e r steady induced v e l o c i t y f i e l d . The c a l c u l a t i o n s were only performed at the lowest loadings, due to the i n v a r i a b l e p r e d i c t i o n of separation near the p r o p e l l e r plane at increasing p r o p e l l e r loading. This f a c t might be r e l a t e d to the use of an actuator disk model and, presumably,a model which accounts f o r f i n i t e chord loading e f f e c t s would lead to d i f f e r e n t r e s u l t s .

The r e s u l t s are shown i n F i g . 2-24 and the pressure d i s t r i b u t i o n f o r

the c o n d i t i o n J=0.60, C =1.10 i s compared with the i n v i s c i d c a l c u l a t i o n i n p

F i g . 2-25. Although the viscous c a l c u l a t i o n leads to a smaller value f o r the c i r c u l a t i o n f o r J=0.60, C =1.10 the c a l c u l a t e d duct thrust i s c o r r e c t . As i t

P can be seen from F i g . 2-25 a considerable l o s s of thrust i s caused by the leading edge separation bubble on the outer surface.

59

The r e s u l t s of the paint t e s t s c a r r i e d out on duct 37 with p r o p e l l e r B i n the Deep Water Basin are shown from F i g . 2-28 to F i g . 2-39. In these Figures the d i r e c t i o n of the undisturbed stream i s from l e f t to r i g h t . Concerning the i n t e r ­p r e t a t i o n of the paint patterns obtained,the following remarks can be made:

- At the smallest p r o p e l l e r loading J=0.625 there i s a laminar separation bubble on the outer surface of the duct shown i n F i g . 2-28 and 2-29. The separation point l i e s at about x/c=0.02 and the turbulent boundary layer reattachment l o c a t i o n i s at about x/c=0.10 ( i d e n t i f i e d by the angle between the paint streaks at reattachment caused by the e f f e c t s of g r a v i t y ) . Separation of the turbulent boundary l a y e r from the t r a i l i n g edge i s seen to occur between x/c=0.92 and x/c=0.96.

The paint patterns on the inner surface, ( F i g . 2-30), reveal an attached boundary l a y e r from the leading edge, a region at mid-chord where the paint apparently d i d not stream, presumably due to a low value of the time average skin f r i c t i o n near the l o c a t i o n of the p r o p e l l e r plane, and a rather i r r e g u l a r "separation region" near the t r a i l i n g edge. Comparison with the correspondent t r a i l i n g edge separation pattern i n uniform flow without p r o p e l l e r at the same Reynolds number based on the undisturbed stream i n F i g . 2-11, which shows a c l e a r separation l i n e , i n d i c a t e s the p o s s i b l e presence of a region of o s c i l l a t i n g reversed flow.

- At the intermediate p r o p e l l e r loading J=0.417, the boundary layer on the outer surface remains attached up to the t r a i l i n g edge as shown i n F i g . 2-31 and 2-32. The l i n e of attachment of the flow to duct surface can be discerned i n F i g . 2-31 at about x/c=0.03 and the l i n e of separation at the t r a i l i n g edge l i e s at about x/c=0.98. The character of the boundary layer on the inner sur­face i s shown i n F i g . 2-33 and 2-34. The leading edge pattern shows an attached boundary layer and the mid-chord t h i n pattern seems to i n d i c a t e a removal of the paint by the action of p r o p e l l e r blades. The t r a i l i n g edge pattern shows again an i r r e g u l a r "separation region" with an apparent t h i n i n g e f f e c t

of the paint which has streamed under the action of g r a v i t y .

- The t e s t at the highest p r o p e l l e r loading has been performed at J=0.22.

The paint patterns on the outer surface are shown i n F i g . 2-35 and 2-36.

It may be concluded that the v e l o c i t y i s too low to obtain a reasonable

paint pattern with the present technique. The pattern on the leading edge

i n d i c a t e s attachment of the flow well on the outer surface. The paint

60

Fig. 2-28. Outer surface. Fig. 2-29. Outer surface,

leading edge region.

Fig. 2-30. Inner surface.

Figs. 2-28 - 2-30. Paint patterns on duct 37 with propeller B.

J=0.625, U-°! = 3. 29x1 05 .

61

Fig. 2-31. Outer surface,

leading edge region.

Fig. 2-32. Outer surface,

mid-chord and trailing

edge regions.

Fig. 2-33. Inner surface,

leading edge and

midchord regions.

Fig. 2-34. Inner surface,

trailing edge region.

Figs. 2-31 - 2-34. Paint patterns on duct 37 with propeller B.

J=0.417, 2.19x10"

62

leading edge and trailing edge region,

mid-chord regions.

Fig. 2-39. Leading edge (front view).

Figs. 2-35 - 2-39. Paint patterns on duct 37 with propeller B.

J=0.22 , = l.lOxlO5. v

63

patterns obtained on the inner surface are f a r more r e v e a l i n g and they are shown i n F i g . 2-37 to F i g . 2-39. No i n d i c a t i o n of separation at the leading edge i s to be seen i n F i g . 2-37 and 2-38. A region of reversed paint flow has been observed extending from about x/c=0.60 to x/c=0.33. A s i m i l a r paint pattern on the t r a i l i n g edge as i n the case J=0.417 has been obtained.

The r e s u l t s of the i n v i s c i d c a l c u l a t i o n s with the l i n e a r i z e d actuator disk model and using a t r a i l i n g edge con d i t i o n based on the e q u a l i t y of pressure at the l o c a t i o n of separation points which could be i n f e r r e d from the paint tests, are shown i n F i g . 2-40.

The assumed l o c a t i o n s are: x/c=0.973 on the outer surface and x/c=0.901 on the inner surface f o r J=0.625; x/c=0.981 on the outer surface and x/c=0.871 on the inner surface f o r J=0.417; x/c=0.973 on the outer surface and x/c=0.822 on the inner surface f o r J=0.208.

The c o r r e l a t i o n on duct thrust i s reasonable except at lowest p r o p e l l e r loading. The i n c l u s i o n of viscous e f f e c t s i n the c a l c u l a t i o n s considerably improves the c o r r e l a t i o n both on thrust and on s e c t i o n r a d i a l force c o e f f i ­cient .

The a x i a l and r a d i a l v e l o c i t y d i s t r i b u t i o n s measured upstream and downstream of the ducted p r o p e l l e r x/R=-0.80, -0.53, 0.53 and 0.90 with r e s ­pect to the p r o p e l l e r plane are shown i n F i g . 2-41 to F i g . 2-48 and compared with c a l c u l a t i o n s by l i n e a r i z e d and non-linear theories f o r J=0.417 and J=0.202.

The c a l c u l a t i o n s have been performed with the same t r a i l i n g edge condi­t i o n s f o r the i n v i s c i d l i n e a r i z e d and non-linear models i n correspondence with the r e s u l t s of F i g . 2-40.

Fig. 2-41. Axial velocity profiles upstream of the propeller at

x/R=-0.80. Duct Z7 with propeller B.

65

Fig. 2-42. Axial velocity profiles upstream of the propeller

x/R=-0.53. Duct 37 with propeller B.

At the lowest loading, J=0.635 the c a l c u l a t i o n s have been c a r r i e d out with l i n e a r i z e d theory with and without considering viscous e f f e c t s on the duct.

66

i o EXPERIMENT J 1 0 . 6 2 5 w J =0.417 a J : 0 . 2 0 8

LMEARIZED THEORY

WITH DUCT VISCOUS EFFECTS

M

NONLINEAR THEORY

—-• 0

Aj />/

i i 0 I 1 1 1 1 1

O 1.0 20 3 . 0 *0 S O Ui

Fig. 2-43. Axial velocity profiles downstream of the propeller at

x/R=0.S3. Duct 37 with propeller B.

r /R

2 0

o E X P E R I M E N T J s 0 . 6 2 5 » J s 0 . 4 1 7 D J- 0 . 2 0 8

L I N E A R I Z E D T H E O R Y

WITH DUCT V I S C O U S E F F E C T S

WUo

Fig. 2-45. Radial velocity profiles upstream of the propeller at

x/R=-0.80. Duct 37 with propeller B.

Fig. 2-46. Radial velocity profiles upstream of the propeller at

x/R=-0.53. Duct 37 with propeller B.

68

o E X P E R I M E N T J : 0 . 6 2 5 J : 0 . 4 1 7 J : 0 . 2 0 8

L I N E A R I Z E D T H E O R Y

WITH DUCT V ISCOUS E F F E C T S

Fig. 2-47. Radial velocity p r o f i l e s downstream of the propeller at

x/R=0.53. Duct 37 with propeller B.

1.5

E X P E R I M E N T J : 0 . 6 2 5 J : 0 . 4 1 7 J s O . 2 0 0

• L I N E A R I Z E D T H E O R Y

WITH DUCT V ISCOUS E F F E C T S

Fig. 2-48. Radial velocity profiles downstream of the propeller at

x/R=0.90. Duct 37 with propeller B.

89

3 . D u c t e d p r o p e l l e r in a x i s y m m e t r i c s h e a r f l o w

•3.1. INTRODUCTION

In general, ducted p r o p e l l e r s work i n a highly non-uniform flow f i e l d i n the ship's wake. When placed i n a non-uniform flow, away from a d d i t i o n a l disturbances such as the one caused by the presence of the h u l l , the ducted p r o p e l l e r performance may considerably d i f f e r from the one obtained i n a uniform flow. The presence of v o r t i c i t y i n the incoming flow and i t s i n t e r ­a c t i o n with the p r o p e l l e r and duct vortex systems, are the d i s t i n c t i v e fea­tures of the behaviour of the ducted p r o p e l l e r i n a non-uniform flow.

The study of the r a d i a l non-uniformity of the inflow,assumed to have only a x i a l v e l o c i t y component, i s of importance when designing wake-adapted p r o p e l l e r s . For ducted p r o p e l l e r s , i n a d d i t i o n to the d i f f e r e n t conditions i n which duct and p r o p e l l e r operate, i t s i n t e r a c t i o n may be f u r t h e r modified by the e f f e c t of incoming flow v o r t i c i t y .

It i s , i n general, assumed that the main e f f e c t s of v o r t i c i t y may be properly described by i n v i s c i d flow theory. The study of such e f f e c t s under t h i s general assumption, i s the p r i n c i p a l aim of t h i s chapter.

The study of the flow disturbances to p a r a l l e l shear flows has been considered by many i n v e s t i g a t o r s . E a r l y work on a e r o f o i l and wing theory has been done by Tsien, (1943), who considered the flow past a symmetrical Joukowsky a e r o f o i l placed i n a uniformly sheared stream. Von Karman and Tsien (1945), developed a l i n e a r i z e d theory f o r small disturbances caused by a l i g h t l y loaded l i f t i n g l i n e i n a non-uniform flow.

To s i m p l i f y the generally d i f f i c u l t task of s o l v i n g the i n v i s c i d non­l i n e a r Euler equations of motion, two basic approaches can be used, as described by Hawthorne (1966). In many cases, the v o r t i c i t y transported by the undisturbed stream i s large while the disturbances produced to the stream are r e l a t i v e l y small. This assumption allows a treatment based on a l i n e a r i z e d form of the Euler equations of motion, obtained by n e g l e c t i n g second order

70

terms i n the perturbation v e l o c i t i e s . On the other hand, i n many cases, the v o r t i c i t y i n the undisturbed stream

may be considered small, although the stream may be subject to large d i s t u r ­bances. The theory based on t h i s assumption, or secondary flow theory, c o n s i ­ders that the secondary v o r t i c i t y i s obtained by allowing the v o r t i c i t y of the undisturbed stream to be transported, according to the v o r t i c i t y t rans­port equation, by the primary flow which i s determined by assuming a poten­t i a l flow disturbance to a uniform flow. The t o t a l flow i s the superposition of the primary and secondary flows, the l a t t e r being the one associated with the secondary v o r t i c i t y .

The f i r s t approach, based on the large shear, small disturbance theory i s s u i t e d f o r studying disturbances introduced to shear flows by bodies with forms of aerodynamic i n t e r e s t .

L i g h t h i l l (1957), studied the three-dimensional flow produced by a point source set i n a two-dimensional p a r a l l e l shear flow. The correspondent funda­mental s o l u t i o n i s found by reducing, with F o u r i e r transform techniques, the two-dimensional boundary value problem to a problem i n v o l v i n g an ordinary d i f f e r e n t i a l equation.

Using s i m i l a r a n a l y t i c a l methods, Weissinger (1970), (1972) developed a l i n e a r i z e d two-dimensional theory f o r p r o f i l e s i n shear flow which includes e f f e c t s of camber and angle of attack (1970) and the a d d i t i o n a l combined e f f e c t s of thickness (1972).

Overlach (1974), extended t h i s work to annular a e r o f o i l s i n axisymmetric

shear flows, and obtained numerical r e s u l t s from the theory f o r i n f i n i t e l y

t h i n r i n g a i r f o i l s i n various types of axisymmetric shear flows.

The problem of a two-dimensional a i r f o i l i n non-uniform flow has also r e c e n t l y been attacked with numerical methods, using f i n i t e d i f f e r e n c e s by Chow et a l (1970), f i n i t e elements by Van der Vooren and Labrujère (1973) and d i s c r e t e vortex methods as the vortex sheet method of Gliick (1979).

These l a s t methods attempt the s o l u t i o n of the f u l l y non-linear Euler equations taking into account the exact shape of the a i r f o i l and are i t e r a t i v e by nature.

In contrast with the a i r f o i l problem,the disturbance flow due to a

p r o p e l l e r i n a shear flow has not been studied so extensively. In a r a d i a l l y non-uniform flow, the model of an i n f i n i t e l y bladed

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p r o p e l l e r , o r actuator disk,has been i n v e s t i g a t e d by Goodman (1979), assuming large shear, small disturbance.

For the he a v i l y loaded ducted p r o p e l l e r t h i s l a s t type of assumption i s not l i k e l y to hold with a reasonable approximation, when dealing with the three­f o l d i n t e r a c t i o n between the p r o p e l l e r , the duct and the non-uniform stream. In a d d i t i o n , even at l i g h t to moderate loads, the duct thickness i s l i k e l y to have strong non-linear i n t e r a c t i o n e f f e c t s with the incoming v o r t i c i t y i f the duct i s placed in a region of strong shear. Non-linear actuator disk theory f o r p r o p e l l e r s has been dealt with by Wu (1962), Greenberg (1972) , Van Gent (1976) and Coesel (1979).

The present non-linear approach i s concerned with an extension of the previous methods to consider the combined e f f e c t s of the duct and an actua­to r disk representing the p r o p e l l e r placed i n a non-uniform stream.

A v o r t i c i t y stream f u n c t i o n formulation i s employed and the v o r t i c i t y associated with the non-uniform inflow and shed by the actuator disk i s d i s -c r e t i z e d i n t o a f i n i t e set of vortex sheets.

The boundary conditions on the duct and hub surfaces are s a t i s f i e d by introducing surface vortex sheets and s o l v i n g the correspondent inner poten­t i a l flow problems.

In order to determine the l o c a t i o n and strength of the vortex sheets representing the two v o r t i c i t y f i e l d s associated with the non-uniform stream and the actuator disk, a r e l a x a t i o n procedure i s used, i n which from an estimate of the l o c a t i o n of the vortex sheets,the corresponding strength i s obtained from the v o r t i c i t y transport equation. The l o c a t i o n s of the stream surfaces are obtained with a streamline t r a c i n g procedure,based on the stream function induced by the v o r t i c i t y f i e l d of the previous i t e r a t i o n . P r o v i s i o n i s made that, p r i o r to t r a c i n g the stream-surfaces, the boundary conditions on duct and hub are properly s a t i s f i e d .

The basic formulation and the d e s c r i p t i o n of the numerical s o l u t i o n procedure are given i n sections 3.2 and 3.3. Numerical r e s u l t s which show the convergence and accuracy of the numerical method and i l l u s t r a t e the ef­fect of incoming v o r t i c i t y on the ducted p r o p e l l e r performance are reported i n s e c t i o n 3.4. F i n a l l y the experimental i n v e s t i g a t i o n s with a ducted p r o p e l l e r i n a r a d i a l l y non-uniform flow together with a comparison with the t h e o r e t i c a l c a l c u l a t i o n s are presented.

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3.2. GOVERNING EQUATIONS

Consider the flow of an i n v i s c i d and incompressible f l u i d through a ducted p r o p e l l e r system modelled as follows:

- An axisymmetric duct with an a r b i t r a r y meridional s e c t i o n of contour D and an axisymmetric c e n t r a l body of meridional contour H representing the p r o p e l l e r hub.

- An actuator disk of n e g l i g i b l e thickness exerting a x i a l and ta n g e n t i a l forces on the f l u i d .

The ducted actuator disk system i s placed i n a r a d i a l l y v a r i a b l e a x i a l stream with a x i a l v e l o c i t y U ( r ) .

The Euler equations of motion are

(u.Vu) = -V(£)+F (3-1)

where u i s the v e l o c i t y vector, p the pressure and IF the external body force per unit mass.

The equation of c o n t i n u i t y i n the absence of sources or sinks

V.u = 0 (3-2)

can be i d e n t i c a l l y s a t i s f i e d i n axisymmetric flow, by introducing the Stokes' stream function V ^ r ) which per d e f i n i t i o n i s r e l a t e d to the v e l o c i t y by

1 3 f 1 3 ? ,„ „, U = r dr" ' V = " F 3 x ' < 3" 3 )

where (u,v,w) are the v e l o c i t y components i n a c y l i n d r i c a l coordinate system (x, r, 6 ) .

The v o r t i c i t y vector tü=V x u has components

1 3 ( r w ) 3w l . S 2 * 1 9H' 3 2 , r \ , „ „ N

D X D 17

We introduce a c u r v i l i n e a r coordinate system (s, n, 9), s being measured along the stream-surfaces on a meridional plane and n being measured along the normal to the stream-surfaces.

73

In t h i s coordinate system,the v o r t i c i t y vector has components

,1 3(rw) 3w .

In the absence of external forces the equation of motion (3-1) writes

u x u = VH (3-6)

where

H = | + \ u 2 (3-7)

i s the t o t a l head. The dot product of (3-6) by u y i e l d s

i f = °' ( 3 " 8 )

which means that i n the absence of external forces the t o t a l head i s constant along the stream-surfaces. Therefore, we may write

H = H(¥) (3-9)

Also the dot product of (3-6) with (o shows that the streamtubes coincide with the vortex tubes and from (3-5) we conclude that

^ H^=0, (3-10)

which means that the angular momentum i s conserved along the stream-surfaces

rw = f(¥) . (3-11)

Using equations (3-8) and (3-10), the equation (3-6) i n the c u r v i l i n e a r

coordinate system i s w r i t t e n

£ X ( r w ) - u W f l = p (3-12) r 3 n s 8 3n

74

2 2 — where u =(u +v ) z i s the meridional v e l o c i t y .

S d 1 8 Using the r e l a t i o n —— = TT— the equation (3-12) becomes r u s 3n

W 9 dH ^ (rw) d , . ,„ — = " d T + :T~ d T ( r w ) ( 3 " 1 3 )

r

and with (3-4), (3-13) becomes

1 ,9 2 ,r 1 3? 3 2 ? , dH (rw) d , . , „ , „ , — - ? s í + — 2 3 = d f — r d ¥ ( r w ) • ( 3 - 1 4 > r 3x 3 r r

The previous equation governing i n v i s c i d axisymmetric flow i n the absence of external forces, i s an e l l i p t i c p a r t i a l d i f f e r e n t i a l equation f o r the stream function. The operator i n the left-hand side i s l i n e a r while, i n general, the right-hand side i s a non-linear f u n c t i o n of ¥.

In the absence of s w i r l , w=0, equation (3-14) takes the form

showing that the quantity — i s constant along the stream-surfaces.

Equation (3-14) i s v a l i d everywhere i n the flow f i e l d except at the actuator disk where external forces act upon the f l u i d .

If the undisturbed stream i s free of s w i r l , the p a r t i c u l a r form (3-15) of the equation (3-14) holds everywhere i n the flow f i e l d , except at the disk i t s e l f and the slipstream c o n s i s t i n g of the flow region downstream of the disk comprising a l l the streamtubes which have crossed the disk.

In order to determine the right-hand side of equation (3-14) i n the disk slipstream, one has to consider the form which the force f i e l d takes at the disk.

From the equation of motion, r e t a i n i n g the body force term, we have

u x u = VH-F. (3-16)

By taking the dot product of (3-16) by u we get

U = A (U.F) (3-17) S

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and the 8 component of (3-16) i s

E ( r w ! = Fe • <3"18> s

The body force f i e l d F being concentrated at the disk, has a Dirac d e l t a

function behaviour. In the streamline coordinate system i t takes the form

F = ( f x m 6 (s) , 0 , f e m<$ (s) ) , ¥<<P(R) (3-19)

where R i s the disk radius, s=0 denoting the point where the streamtube i n ­t e r s e c t s the disk.

Using (3-19) and i n t e g r a t i n g (3-17) between -°° and s>0, we obtain

rw = 0 , s<0

rw = — f Q ( ¥ ) , s>0, (3-20) s

fe which implies that the c i r c u m f e r e n t i a l v e l o c i t y has a jump — at the disk.

u s The jump i n t o t a l head at the disk can be found from a matching condi­

t i o n with the flow i n s i d e the blade row at the disk. Assuming an i n f i n i t e number of blades r o t a t i n g with angular v e l o c i t y

fi =Qi , the co n d i t i o n expressing the normality between the force acting on — —x

the f l u i d and the r e l a t i v e v e l o c i t y reads

(u - i g f i r ) .F = 0 (3-21)

and from (3-17) and (3-18), we obtain

IS = !r Fe = ^ ( r w > ( 3"2 2> s

Integrating (3-22) from s=-°° to s=s we have

H e n = H 0 e n , f o r s<o o r y ^ t o ^ )

H ( ¥ ) = H Qei')+f2rw, f o r s>0 and ( 0 , R ) , (3-23)

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where HgCf), being the t o t a l head before the disk,has to be determined from

the conditions at i n f i n i t y upstream.

Using (3-23) i n (3-14) we f i n a l l y , o b t a i n , f o r the flow i n the slipstream,

the equation

1 ,3 2¥ 1 3Y . 3 2¥, d H 0 ^ , n rw. d , , — { T 2 - r ^ + -2] = -dT + ( f i—2>d? ( r w )' ( 3' 2 4 )

r 3x 3r r

which i s a g e n e r a l i z a t i o n to the non-uniform inflow case of the equation

obtained by Wu (1962) f o r the uniform inflow.

Outside the slipstream, equation (3-15) writes

1 ,32V 1 3T , _ d H o ...

r 3x 3 r

Equations (3-24) and (3-25) have to be solved subject to the p a r t i c u l a r

boundary conditions f o r the present problem, namely:

- At i n f i n i t y upstream we have P = P Q and u=U(r)i^. The t o t a l head i s

P() 1 2

H Q m = -j- + |U (3-26)

and the stream function i s V = ƒ U ( r ' ) r ' d r ' . (3-27)

U 0

The v o r t i c i t y vector has only c i r c u m f e r e n t i a l component

w e 0 - - I F ( 3 " 2 8 )

and the boundary c o n d i t i o n at i n f i n i t y upstream writes

¥ = \ J J Q as x •> -oo (3-29)

At i n f i n i t y as x-M-°° and r-*», we have

P- -> 0 x + +°° , r - + o o . (3-30) 3x

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- On the duct's surface the boundary c o n d i t i o n i s simply

4f = C on D, (3-31)

where C i s an unknown constant to be determined from the Kutta c o n d i t i o n at the duct's t r a i l i n g edge.

- At the hub the boundary c o n d i t i o n i s

¥ = 0 on H. (3-32)

If the r a d i a l d i s t r i b u t i o n of c i r c u l a t i o n

r m ( r ) = 2-rrrwQ = 2nrw(s=0 + )

i s known at the disk, the r o t a t i o n a l momentum becomes known and the forces

ac t i n g on the f l u i d are determined from equation (3-20). In the present problem, the r a d i a l d i s t r i b u t i o n of c i r c u l a t i o n and the

shape of the duct and hub are assumed to be known.

The main d i f f i c u l t y i n s o l v i n g equations (3-24) to (3-32) l i e s i n the fact that, the right-hand sides of (3-24) and (3-25) being non-linear func­tions of ¥ are not known at the outset i n the whole flow f i e l d , but only at p a r t i c u l a r surf aces, namely, at i n f i n i t y upstream s=-°° and at the disk s=0.

By a p p l i c a t i o n of Green's theorem, the previous boundary-value problem, c o n s i s t i n g of equations (3-24) to (3-32) , can be transformed i n t o a set of coupled non-linear i n t e g r a l equations, r e s p e c t i v e l y , on the flow domain e x t e r i o r to the duct and hub surfaces, and on i t s boundary, c o n s i s t i n g of the duct and hub surfaces themselves.

Although the theory may be applied to the ducted p r o p e l l e r placed i n a general non-uniform stream, we w i l l l i m i t ourselves to ducted p r o p e l l e r s i n wake flows.

In such case we are dealing with, of a ducted p r o p e l l e r i n an axisymme-t r i c wake f i e l d , the non-uniformity of the incoming flow i s of l i m i t e d r a d i a l extent and we may therefore assume that, at r a d i i greater than the radius R ,

78

the inflow i s uniform with v e l o c i t y U^.

We make the flow q u a n t i t i e s non-dimensional using the disk radius R and the v e l o c i t y U Q, as follows: r*=r/R, x*=x/R, u*=u/UQ, U*=U/UQ> r=V/iVQH ),

^ 0 / ( U 0 R 2 ) , ^=R<,e/U0, 03* H S = V U 0 - J = ™ 0/ f l R -

With the a s t e r i s c omitted, equations (3-25) and (3-24) become:

1 ,3 2¥ 1 3Y . 8 2¥, d H 0 , ,. -,. , —=•( 7T — H X-) = outside the slipstream 2 . 2 r 9 r 2 d f

and

r 3x 3 r (3-33)

1 ,Z2V 1 3¥ d2V> d H 0 . /TT ( r w ) . d , . . . .. - j l J gT7 + = !W ( - 2 ~ ) d f ( ' l n s l d e t h e slipstream. r 3x 3 r J r ( 3 _ 3 4 )

We may continue the flow f i e l d i n t o the region e x t e r n a l l y bounded by the duct and hub surfaces i n an a r b i t r a r y way, provided that these surfaces remain stream-surfaces of the flow. If we assume that the v e l o c i t y vanishes i d e n t i c a l l y i n that region, the v e l o c i t y t a n g e n t i a l to the surface w i l l become discontinuous.

Denoting by V+ and ¥ the stream function of the external and i n t e r n a l flows r e s p e c t i v e l y , we have

- ? [ ^ - ^ ] = - = v ( s ) o n D a n d H ' ( 3 - 3 5 )

where V(s) i s the outer meridional v e l o c i t y on D and H. In t h i s way, the common representation of the duct and hub surfaces by

vortex sheets with strength y(s)=V(s) , as r e f e r r e d to i n the previous Chapter i s obtained.

The r e l a t i o n between the stream f u n c t i o n V(x,r) induced by a given v o r t i c i t y f i e l d U)g(x,r) i s

V ( x , r ) = If G ( x - x \ r , r ' ) u e (X ' , r ' ) d x ' d r ' , (3-36) K

i n which the i n t e g r a t i o n i s performed over the region K f o r which 0)g(x,r)^0. The function of Green G ( x - x 1 , r , r ' ) , associated with the operator

79

1 3 2 1 3 3 2

— ( — TT- + — ; r ) appearing i n (3-33) and (3-34), represents the stream r ~ A r or ~ £ ox 9r function induced by a r i n g vortex with unit c i r c u l a t i o n . It has been given i n various ways by several authors (Lamb (1952), Greenberg (1932)). The re­presentation i n terms of e l l i p t i c i n t e g r a l s , as derived by Kiicheman and Weber (1953), i s used here:

G ( x - x ' , r , r ' ) = / ( x - x ' ) 2+ ( r + r ' ) 2 '[( 1-^-) K (k) -E (k)] . (3-37)

The modulus of the e l l i p t i c i n t e g r a l s K(k) and E(k) i s

,2 _ 4 r r ' ~~ 2 2 * ( x - x ' ) + ( r + r ' )

(3-38)

The t o t a l stream function can be wr i t t e n as the sum of the three c o n t r i b u ­tions :

Y ( x , r ) = * w ( x , r ) ( x , r ) +4>d ( x , r ) . (3-39)

<p (x,r) i s the stream function induced by the v o r t i c i t y f i e l d w

dH we = " r " d T ' ( 3 - 4 0 )

w l|Jp(x,r) i s the stream function induced by the v o r t i c i t y f i e l d

,rTr .d(rw) u Q = - ( — - w) d¥ (3-41)

and <i> i s the stream function induced by the duct and hub surface v o r t i c i t y d Y(s) .

We note that co„ i s d i f f e r e n t from zero i n the flow region external to w

the duct and hub surf aces, s a t i s f y i n g li'(x,r)<lr'(x,R„) , while w0 i s d i f f e r e n t 0 o

from zero i n s i d e the disk slipstream ¥(x,r)<li'(x, 1) with x>0. P It should also be noted that, though one could i n t e r p r e t e 10. as the v o r t i c i t y r e s u l t i n g

°w from the transport of the v o r t i c i t y of the incoming stream and 0) Q the v o r t i -

9 p c i t y shed from the actuator disk, t h i s d i s t i n c t i o n i s not an e s s e n t i a l one when s o l v i n g the non-linear problem, because the two v o r t i c i t y f i e l d s are coupled through the dependence on the t o t a l stream function V.

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When taking into account the various components to the stream function i n equation (3-35) expressing the boundary c o n d i t i o n on the duct and hub surfaces, one obtains the i n t e g r a l equation on the surface v o r t i c i t y strength:

- 4 Y ( S ) + ƒ y ( s 1 ) k ( s , s ' ) d s ' = + (3-42) 2 D+H r 3 n ó n

since ill and Ui are continuous on D and H. w p The constant C, appearing i n equation (3-31), i s not involved i n equa­

t i o n (3-42). However, as discussed i n Chapter 2, equation (3-42) i s s a t i s ­f i e d by an i n f i n i t e number of s o l u t i o n s , corresponding to each p o s s i b l e value of the constant C. The Kutta c o n d i t i o n has to be added to s p e c i f y the s o l u t i o n .

S u b s t i t u t i o n of (3-39), (3-40) and (3-41) i n t o (3-36) y i e l d s an i n t e ­

g r a l equation on the stream function ¥ which has to be solved together with

the i n t e g r a l equation (3-42).

The non-linear character of the problem asks f o r an i t e r a t i v e approach,

and the f o l l o w i n g has been adopted:

1. We depart, as a f i r s t approximation, from a s o l u t i o n based on small shear, large disturbance approximation to the flow f i e l d , i . e . we assume that the v o r t i c i t y of the undisturbed stream i s convected along the p o t e n t i a l flow stream-surfaces of a primary flow.

The stream-surfaces of such primary flow are obtained from a l i n e a r approximation to the v o r t i c i t y shed from the actuator disk i n the presence of the undisturbed stream, together with the vortex sheet strength required to make from the duct and hub stream-surfaces of the flow. The l i n e a r approximation to the v o r t i c i t y shed from the actuator disk i s

d r 1 ÜT7T T co

we ( r ) " - a f ^ T - V u t r i + u (r) -&F ( 3 " 4 3 )

P J Poo where u i s the a x i a l v e l o c i t y induced f a r downstream by the shed v o r t i c i t y

Poo U)„ (r) . This approximation corresponds to the vortex system of a moderate-

9 P l y loaded actuator disk with continuous d i s t r i b u t i o n of c i r c u l a t i o n and

the p i t c h of the vortex l i n e s determined i n the ultimate wake and

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has been i n v e s t i g a t e d i n Chapter 2 .

2 . The stream function i s computed from equations ( 3 - 3 6 ) and ( 3 - 3 7 ) .

3 . With the t o t a l stream function known, new estimates f o r the v o r t i c i t y d i s t r i b u t i o n s OJg (x,r) and U)„ ( x , r ) , follow from equations ( 3 - 4 0 ) and ( 3 - 4 1 ) . W P

4 . The s o l u t i o n of the i n t e g r a l equation ( 3 - 4 2 ) y i e l d s Y ( s ) o n D a n d H -

Steps 2 . to 4 . are repeated u n t i l convergence i s achieved. The various steps of the i t e r a t i v e s o l u t i o n using a d i s c r e t e vortex sheet method w i l l be examined i n the next s e c t i o n .

3 . 3 . ITERATIVE SOLUTION BY A DISCRETE VORTEX SHEET METHOD

3 . 3 . 1 . Vortex sheet approximation to the v o r t i c i t y i n the flow

In the present approximate numerical s o l u t i o n to the problem described i n the previous s e c t i o n , we assume the undisturbed stream to be approximated by a piecewise constant v e l o c i t y d i s t r i b u t i o n . As a con­sequence, the v o r t i c i t y of the incoming flow i s d i s t r i b u t e d i n a f i n i t e num­ber of vortex sheets ( F i g . 3 . 1 ) .

Let the inflow v e l o c i t y be piecewise constant, given by

U ( r ) = U m f o r r Q < r < r Q , m = l ( l ) , N , ( 3 - 4 4 ) m m+1

with rQ =0,r„ =R and U ..=Un being the constant v e l o c i t y outside the wake. 1 N+l 0 N+l u

The stream function at i n f i n i t y upstream i s

V r > T* r 0 - r 0 , 3 V W^l ) U m f o r r 0 ^ r 0 • 1=1 i+l I m m m+1 ( 3 - 4 5 )

At the d i s c r e t e r a d i i r n , we have um m - 1

¥ = i> ( r ) = z j(r - r ) U , m = l , ( 1 ) , N . ( 3 - 4 6 )

At i n f i n i t y upstream, the v o r t i c i t y vanishes everywhere, except at the d i s ­crete r a d i i r ^ , m=l,(l),N, where i t has a Dirac d e l t a function type behav-

m iour.

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The f u n c t i o n H^CV), which coincides with the t o t a l head of the flow outside the slipstream, (see eq. 3-23), i s constant between the vortex sheets corresponding to the stream surfaces , m=l,(l),N. At the vortex

m sheets themselves, the function has a jump associated with the d i s c o n t i ­

nuity of v e l o c i t y across the sheet. th

The jump at the m vortex sheet i s

A H = H Q - H 0 - | < U r a + 1 - U m ) = U m Y 0 , m = l , ( l ) , N , (3-47) m m+1 m m

where U =i(U _+U ) i s the mean v e l o c i t y at the sheet, at i n f i n i t y upstream, m m+1 m and

Y„ = U ,.-0 (3-48) '0 m+1 m m

i s the strength of the vortex sheet at i n f i n i t y upstream. The f u n c t i o n dH^/d^, becomes

d H N - g f = E A H Q fi(¥-«F ) , (3-49)

m = l m

where 6 i s the d e l t a function of Dirac. The strength of the correspondent vortex sheet can be deduced by i n t e ­

g r a t i n g the tang e n t i a l component of v o r t l c i t y on a small area element with

thickness 2e normal to the sheet, enclosing a vortex sheet element with

length ds, and l e t t i n g £ approach zero. In such case, we get

+e + c d H Q

y ( s ) = l i m ƒ d i . d n = l i m ƒ (-r -,„, ) d n (3-50) e^O - e 0 £+0 -£

Using (3-49), and changing the i n t e g r a t i o n v a r i a b l e to f, the strength of th

the m vortex sheet becomes

A ^ O f - A f m m m

(3-51)

Using (3-47) and noting that the meridional mean v e l o c i t y at the vortex sheet .1(31)

m m 1 S U s = - 7 ( 8 n " ) ^ ' W e h a V C

83

Y ( s ) = —5i- Y n • ( 3 - 5 2 ) s m m

The same equation could be obtained by applying B e r n o u l l i equation at both sides of the vortex sheet and using the conditions of c o n t i n u i t y of the pressure across the sheet.

As remarked before, the N vortex sheets with strength given by equation (3-52), do not represent the t o t a l v o r t i c i t y i n s i d e the slipstream. The addi­t i o n a l c o n t r i b u t i o n of the v o r t i c i t y from the actuator disk and given by equation (3-41) has to be accounted f o r .

Let the c i r c u l a t i o n at the disk be approximated by a piecewise constant function, so that the correspondent shed v o r t i c i t y i s d i s t r i b u t e d on a number of vortex sheets. The inner vortex sheet i s assumed to be bound to the hub and the outer vortex sheet, forming the outer boundary of the slipstream, i s shed from the disk' s edge ( F i g . 3-1).

Fig. 3-1. Schematic representation of the discrete vortex model for

a ducted propeller in non-uniform wake flow.

84

We have

r j r ) = r k f o r r k < r < r k + 1 , k = l , ( l ) , N r . (3-53)

with r = r , being the hub radius at the disk and r =1. 1 h ° Np+1 If Ar =r, -T , with r =0, we have k k+1 k' '

N r d - g ^ - = 27 Z A r k 6 ( ¥ - ¥ k ) , k = l , (1) ,N , (3-54)

k = l

and i n t e g r a t i n g (3-41) with (3-54) we obtain f o r the vortex strength of the kth v o r t e x sheet shed from the actuator disk, the following r e s u l t :

V A ¥ Ar y (s) = l i m ƒ [- r ] S ( M ) d¥, (3-55)

k

i r,+r, A T ,

V s ) = - - 2 2 } — • < 3 " 5 6 )

2 J 8TT r s, k 1 3f 1

where U g = ~ " r ~ ( ' g ^ a n c * Crw>u;—tj/ =~4TI^ a r e t h e l i m i t i n 6 values f o r

the meridional v e l o c i t y and the r o t a t i o n a l momentum at the vortex sheet. Equation (3-56) agrees with the equation given by Greenberg (1972) f o r the vortex strength of the slips t r e a m of a he a v i l y loaded actuator disk i n u n i ­form f l o w . Comparison of equation (3-56) with (3-52) evidences the r o l e of s w i r l i n the slipstream represented by the second term between brackets i n (3-56).

3.3.2. F i r s t approximation to the actuator disk vortex sheets and the flow stream surfaces.

Using the previous r e s u l t s the l i n e a r approximation of equation (3-43)

to the v o r t i c i t y shed from the actuator disk can be derived. The r e s u l t , f o r the strength Y of the s e m i - i n f i n i t e r i n g vortex

k ( l ) c y l i n d e r s shed from the disk r a d i i r, , a f t e r some c a l c u l a t i o n , i s

k

85

(1) k + l (l) £ = k + l (1) J 4TT

k = 1, (1) ,N p . (3-57)

The v e l o c i t y f i e l d induced by a s e m i - i n f i n i t e r i n g vortex c y l i n d e r has been given i n Chapter 2. The r e s u l t f o r the corresponding induced stream function i s deduced by i n t e g r a t i o n of the Green's function, equation (3-37), from x'=0 to to x'=°°, (Coesel, 1979):

2 l o BC5*n,,,r») = r - ( A ( n ) + — § 7{ ( n - i ) 2 [ K ( k n ) - n ( a ^ k )] +

4 1 T S

z + ( n + D

[ c 2 + ( n + D 2 ] [ K ( k 1 ) - E ( k 1 ) ] } ) (3-58)

where 2

A(n) = wn f o r n<i , (3-59)

= 7T for T1>1 , (3-60)

k 2 = , 4 " =• , (3-61) r + ( n + D

2 _ 4n

(Tl + 1)' a = ' (3-62)

r X 1 r 5 = — - and n=—r • r ' 1 r'

Using the a x i a l and r a d i a l v e l o c i t i e s induced by the vortex system (3-57), the i n t e g r a l equation (3-42) f o r the f i r s t i t e r a t e on the surface v o r t i c i t y y ^ (s) on the duct and hub i s solved and a f i r s t approximation to the stream surfaces of the flow follows. -The corresponding t o t a l stream fun c t i o n i s

86

f , . ( x , r ) = i> n ( r ) + i K ( x , r ) + Z y v * ( 5 , n , r ) (3-63) U ) 0 d ( l ) k = l KCL) k

where 0 i s the stream function induced by the duct and hub v o r t i c i t y y .j,, (1)

3.3.3. C a l c u l a t i o n of the flow stream surfaces and d i s c r e t i z a t i o n of the

vortex sheets.

In the i t e r a t i o n process, the p o s i t i o n s of the vortex sheets, represen­t i n g e i t h e r the wake v o r t i c i t y or the slipstream v o r t i c i t y , are obtained by t r a c i n g on an E u l e r i a n g r i d the stream-surfaces V=¥ , m=l,(l),N and ,

m k k=l, (1) ,N

The g r i d has constant stepsize i n the r a d i a l d i r e c t i o n and v a r i a b l e stepsize i n the a x i a l d i r e c t i o n , (see F i g . 3-2). The constant r a d i a l step-s i z e i s introduced f o r obtaining higher accuracy i n the computation of a x i a l v e l o c i t y p r o f i l e s from the stream function knot values by numerical d i f f e r e n ­t i a t i o n , and the v a r i a b l e s t e p s i z e i n the a x i a l d i r e c t i o n to permit adaption to the decay rate of the v o r t i c i t y disturbances with the distance from the duct and actuator disk.

On each r a d i a l s t a t i o n x=x^, i = l , ( l ) , M , three point Lagrange i n t e r p o l a ­

t i o n i s used to obtain the radius r. of the m t h vortex sheet f ^ f from the lm m

stream f u n c t i o n knot values f, .. The same set of points (x ,r ) i s used to d i s c r e t i z e the corresponding

th vortex sheet i n t o p a r a b o l i c elements. The parametric equations of the I

th element on the m vortex sheet are

2 x. (£) = x. + c o s a . 5 - c . s i n a . 5 l m l m i n lm l m

r . (5) = r . + s i n a . 5+c. c o s a , £ 2 , i = l , ( l ) , M - l , (3-64) l m l m l m l m l m

with (x ,r ) as the ordinates of the parabola vertex, a. the angle made im lm lm by the element chord with the x-axis, c the element curvature and 5 the

i m arc-length measured from the point (x. ,r. ). " im im

87

UPSTREAM BOUNDARY CONDITION DOWNSTREAM BOUNDARY CONDITION

L

L - 1

VORTEX SHEET,

3 2

Fig. 3-2. Computational grid, equispaced in v with variable step in

x, for streamline tracing and discretization of vortex sheets

We further assume that, at s u f f i c i e n t l y large distances from the

o r i g i n , say f o r x<x, and x>x , the vortex sheet has constant radius. Accord = 1 — m i n g l y , i n the i n t e r v a l s (-°°,x^) and (x ,+<») , outside of the computational domain, i t may be replaced by s e m i - i n f i n i t e r i n g vortex c y l i n d e r s with con­stant strength.

In the computational domain and on each element the vortex strength i s

assumed to vary l i n e a r l y .

The basic procedure for the c a l c u l a t i o n of the stream function on the

knots of the computational g r i d i s the evaluation of the stream function

induced by a vortex sheet element, (3-64), with a l i n e a r l y varying vortex

strength. If Y. ' and Y 5 ^ are, r e s p e c t i v e l y , the values of the vortex km km

88

strength and i t s f i r s t d e r i v a t i v e at the parabola vertex (x, ,r, ), we ob-km km

t a i n , f o r the stream f u n c t i o n induced by the element on the g r i d point

i 5i ( 0 ) ' k m

= T u f , G ( x . - x , ( 5 ) , r . , r . (£))d?+ 1 j km km _ ' I km ' j km ^' •km

+ C ( ^ ^ k m ^ ^ . , ^ ) ) ^ < 3" 6 5> ^km

where £, denotes h a l f the arc length of the element, km The i n t e g r a l s i n equation (3-65) are regular since r ?T , and can be evalu-

km j ated numerically to any degree of accuracy without d i f f i c u l t i e s . In the pres­ent a p p l i c a t i o n , a 10 point Gaussian and a 21 point Kronrod formula has been applied f o r that purpose.

However, i f the distance of the f i e l d point to the element vertex i s large compared with the c h a r a c t e r i s t i c arc length of the element, m u l t i -pole expansions of the Green's fun c t i o n about the vertex (x, , r, ), can be

km km used.

A Taylor expansion about the point (x, , r ), y i e l d s km km

G ( x . - x . ( E ) , r . , r . (£)) = G ( x . - x , , r . , r . ) + l km s ' j' km s I km' j' km

+ G ,(x.-x, , r . , r , ) Tx. (£)-x, 1 + x' I km' 3 ' km L km s km-1

+ G , ( x . - x . , r . , r . ) f r , ( p - r , l + r ' 1 km j ' km L km * km-1

1 - - - 2

+ T T { G . , ( x . - x , , r . , r , ) fx. (£)-x. 1 + 2 x'x' 1 km' j' km L km ^ kmJ

+ G , ,(x.-x. , r . , r . ) fx, ( ? ) - x , 1 f r , (£)-r. 1 + x ' r 1 1 km' j ' km 1 km ^ km J L km s km J

2 + G . , (x.-x, , r . , r . ) f r . ( 5 ) - r , (3-66) r ' r ' 1 km' 3 ' km L km ^ kmJ

where

8 9

G x , ( x - x ' , r , r ' ) - - ¿ ^ ' { K ( k ) +

/ ( x - x ' ) z + ( r + r ' ) ¿

[1+ - ] E ( k ) } , (3-67) ( x - x * ) 2 + ( r - r ' ) ¿

G , ( x - x ' , r , r ' ) = ¿ , f ,{K(k) + / ( x - x ' ) 2 + ( r + r ' ) ¿

[1 2 r ' r r ' } =-]E(k)}, (3-68) ( x - x ' ) + ( r - r ' )

x ' x ' 2 T r [ ( x - x ' ) 2 + ( r + r ' ) 2 ] 3 / 2 [ ( x - x ' ) 2 + ( r - r ' ) 2 ]

( [ ( x - x " ) ( r + r ' / ) + ( r - r

2 2 1 +k

K ( k ) + { 2 r r ' ( x - x ' ) + 1-k

- ( r + r ' ) 2 [ ( x - x ' ) 2 + ( r 2 + r ' 2 ) ] } E ( k ) ) , (3-69)

G ( x - x ' r r ' ) = — ( x - x ' ) x , r , X X , r , r 2 ï ï ^ x _ x l ) 2 + ( r + r l ) 2 j 3 / 2 ^ ( x _ x I ) 2 + ( r _ r I ) 2 j

{ [ ( x - x ' ) 2 - ( r 2 - r ' 2 ) ] K ( k ) +

+ [ 2 r ( r - r ' ) ( ï ± E l + ^ _ ) + 1-k

- ( x - x ' ) 2 + ( r 2 - r ' 2 ) ] E ( k ) } , (3-70)

G r ' r ' ( X ~ X ' ' r ' r ' )

1 2 T r [ ( x - x ' ) 2+ ( r + r ' ) 2 ] 3 / 2 [ ( x - x ' ) 2+ ( r - r ' ) 2 ]

{ ( x - x ' ) 2 [ ( x - x ' ) 2 + ( r 2 + r ' 2 ) ] K ( k )

-{ ( x - x ' ) 2 [ ( x - x ' ) 2 - ( r 2 - r ' 2 ) ] - 2 r ( r + r ' )

9 O O r r 1 9 9 ( r - r , z ) + i T [ ( x - x ' ) + ( r - r ' ) ] E ( k ) }.(3-7l)

1-k 90

The modulus of the e l l i p t i c i n t e g r a l s i s

2 4 r r ' k Z = 5- (3-72)

(x-x') + ( r + r ' )

The functions G , and G , represent the stream function induced by r i n g x r vortex dipol e s of uni t strength, with t h e i r axes d i r e c t e d i n the a x i a l and r a d i a l d i r e c t i o n s , r e s p e c t i v e l y . The functions G , ,, G , , and G , , are

x x ' r ' r ' x r ' the stream functions induced by unita r y r i n g vortex quadrupoles with t h e i r p a i r of axes, r e s p e c t i v e l y , a x i a l l y d i r e c t e d , r a d i a l l y d i r e c t e d and orthogo­nal i n both a x i a l and r a d i a l d i r e c t i o n s . These functions are given i n F i g s . 3-3 to 3-7.

When the multipole expansions are used, the stream function il. ., i n l j k m

equation (3-65), becomes

* = T < 0 ) G ( ° > n ' 1 ^ 1 1 ' (3-73) y l 3 k m 'km l j k m 'km ï 3 km

wit

G f = 2GE,' + [~c, ( - s i n a , G ,+cosa, G ,) + l j k m km L km km x' km r '

4 ( c o s 2 a k m G x ' x • + 2 s i n a k m C O S a k m G x <r<) +

+ s i n 2 a k m G r , r , ] | 4 3 , ( 3 _ 7 4 )

and

G..? = ( c o s a , G , + s i n a , G ,)^,3 , (3-75) 13km km X 1 km r 1 3 km

where the short-hand notation G stands f o r G(x.-x , r . , r , ) and s i m i l a r l y to 1 km j km

the other functions. The c r i t e r i o n used to e s t a b l i s h i n which flow region f a r from the

element multipole expansions should be used, depends on the accuracy desired. From numerical c a l c u l a t i o n s i t turned out that, f o r a wide range of element geometric parameters, l i n e s of equal e r r o r i n the approximation (3-66), could be reasonably approximated by s e m i - e l l i p s e s . Therefore, multipole expansions

9 1

'Fig. 3-3. Stream function induced

by an axially directed

ring vortex dipole.

Fig 3-4. Stream function induced

by a radially directed

ring vortex dipole.

Stream function induced by

a ring vortex quadrupole.

Two axial axes.

Fig. 3-6. Stream function induced by

a ring Vortex quadrupole.

One axial axis and one

radial axis.

Fig. 3-7. Stream function induced by a ring vortex

quadrupole. Two radial axes.

93

were used when

'km and

1 / - 2 - 2 -p-.— / (x. -x. ) +c . ( r . - r . ) > c for r . / r . < 1 Ç' l km l j km y km

1 / - 2 - 2 •p-.— / ( x . - x , ) +c ( r . - r , ) < c f o r r . / r . >1 l km e ' j km j ' km

The values of the numerical constants have been chosen as c =7.6, c =1.0 l e

and c=6.0.

By summing the c o n t r i b u t i o n s of a l l vortex sheets, the t o t a l stream functions at the mesh knots writes

N+N^+l , , r p (m) f . . = £ £ ill . .. +R. . (3-76) i n i l , i n k m i n J m=l k = l J J

th where p(m) i s the number of elements of the m vortex sheet. The summation over the vortex sheets i n equation (3-76) includes the vortex sheets of the incoming wake flow, the vortex sheets of the disk s l i p s t r e a m and the duct and hub boundary vortex sheet. R represents the c o n t r i b u t i o n s from the re­mainder of the flow f i e l d outside the computational domain and i s given by

N R , . = I [yn ¥ (-Ç. , n n , r )+y ¥ (ç ,n , r )] + jM L 0 0 0 0 0 O 0 0 CO CO CO CO -1

J m=l m m m m m m m m

+ 1 Y m * (5 ,n , r ) (3-77) 'CO CO CO ' CO CO

k = l k k k k

x.-Xj^ r . x.-x„ with r = — , n =—^— , C =—-——, n =—^—. r and r are the asymptotic O r O r 0 0 r 0 3 r 0 0 0 0

m O m O m °°_ m » m k m m r a d i i of the wake vortex sheets and s l i p s t r e a m vortex sheets, r e s p e c t i v e l y ,

and y œ , y œ the correspondent vortex strengths, m k

94

3.3,4. C a l c u l a t i o n of the strength of the vortex sheets

In each i t e r a t i o n step, when the l o c a t i o n of the vortex sheet becomes

known from the streamline t r a c i n g procedure and the d i s c r e t i z a t i o n of the

vortex sheet i n t o i t s various elements has been e f f e c t e d , the vortex strength

on each element s t i l l needs to be evaluated.

Concerning the wake vortex sheets and the slipstream vortex sheets, the c a l c u l a t i o n of the vortex strength can be, e a s i l y , c a r r i e d out from equations (3-52) and (3-56), provided that the mean meridional v e l o c i t i e s u s and us^

For the computation of the v e l o c i t y f i e l d , two methods are, i n p r i n c i p l e a v a i l a b l e : Biot-Savart i n t e g r a t i o n over the vortex sheets or numerical d i f ­f e r e n t i a t i o n of the stream function.

With respect to computational e f f i c i e n c y , the second method i s to be

preferred to the f i r s t , though i t i s , c l e a r l y , of l e s s accuracy.

Since the slope of the stream-surfaces i s already known from the stream­l i n e t r a c i n g procedure and the d i s c r e t i z a t i o n of the stream-surfaces, one only needs to compute the a x i a l v e l o c i t i e s .

From the values of stream function at the g r i d knots,axial v e l o c i t y p r o f i l e s , at a l l the a x i a l s t a t i o n s , are derived using a three-point Lagrange d i f f e r e n t i a t i o n formula (see, f o r example, Isacson and K e l l e r , 1966).

The values of the a x i a l v e l o c i t y are, i n the f i r s t place, evaluated at the g r i d knots and, secondly, i n t e r p o l a t e d to the correspondent vortex sheet r a d i i . We note that, f o r p a r t i c u l a r choices of the g r i d spacing i n the r a d i a l d i r e c t i o n , t h i s procedure may lead to continuous a x i a l v e l o c i t y d i s t r i b u t i o n s by smoothing the v e l o c i t y d i s c o n t i n u i t i e s across the vortex sheets.

If u„. and u c . . denote the meridional v e l o c i t i e s on the extremi-°i,m Bi+l,m ^ t i e s of the i t h element of the m wake vortex sheet, obtained from the a x i a l

v e l o c i t i e s and vortex sheet slopes at the same points, we get

are known.

(0) im u

1 + u 1

) (3-78) m s . i + 1 ,m s .

I ,m

95

= i V o ^ < 3" 7 9> m s . , . s. 1+1,m i,m f o r the wake v o r t i c i t y on the element.

S i m i l a r l y , i f u s ^ + 1 k a n t l u s i k

a r e t n e meridional v e l o c i t i e s on the extremities of the i * * 1 element of the k**1 slipstream vortex sheet, we have

W W 0) l.„ , i + l , k i , k . .„ 0 „ %

Y i k = 2 A r k ( u ^ — + ^r—> ' ( 3 - 8 0 )

s i + l , k s i , k

w W = i A r k ( ^ ± ^ - ^ - )

s i + l , k s i , k

f o r the slipstream v o r t i c i t y on the element, where

r +r W - J - - k k + 1 ( 3 - 8 2 i i , k - „ - „ 2 2 " < 3 8 2 )

< J B I T r i k

The duct and hub v o r t i c i t y i s found from the s o l u t i o n of the i n t e g r a l ' equation (3-42), when the a x i a l and r a d i a l v e l o c i t i e s induced on the duct and hub are known. Since the numerical d i f f e r e n t i a t i o n procedure tends to be less accurate near the flow boundaries i . e . the duct and hub surfaces, B i o t -Savart i n t e g r a t i o n i s used to compute the v e l o c i t i e s on the p i v o t a l points on these surfaces.

It must be noted that, the strength of the boundary v o r t i c i t y on the duct and hub surfaces has much steeper gradients than the wake and s l i p s t r e a m vortex sheet strengths,due to the presence of stagnation points on the boundary. Therefore, the d i s c r e t i z a t i o n of the boundary surfaces i s not coupled to the d i s c r e t i z a t i o n of the other vortex sheets and,generally, admits a l a r g e r number of elements concentrated near the fore and a f t e r stagnation po i n t s .

The a x i a l and r a d i a l v e l o c i t i e s induced by constant, l i n e a r and para­

b o l i c vortex d i s t r i b u t i o n s on p a r a b o l i c elements and s e m i - i n f i n i t e r i n g

vortex c y l i n d e r s , have been already discussed i n Chapter 2 when dea l i n g with

the surface v o r t i c i t y method f o r the duct as well as the s o l u t i o n of the corres­

pondent i n t e g r a l equation, and therefore w i l l not be f u r t h e r considered here.

96

3.3.5. I t e r a t i v e procedure

The i t e r a t i v e procedure used to c a l c u l a t e the flow i n s i d e the computa­t i o n a l domain can now be summarized as follows:

1. From the p a r t i c u l a r vortex sheet d i s c r e t i z a t i o n adopted for the wake at the upstream computational boundary, the values of the stream f u n c t i o n for the vortex sheet stream-surfaces are determined. On the hub the stream function i s zero.

2. Neglecting vortex sheet deformation i n the computational domain, and from the f i r s t approximation to the actuator disk vortex sheets, (Section 3.3.2), and corresponding induced v e l o c i t i e s , the strength of the surface vortex sheet on the duct and hub i s determined from the con d i t i o n that the inner v e l o c i t y tangent to the surface must vanish, together with the Kutta c o n d i t i o n on the duct t r a i l i n g edge.

3. From the values of the surface vortex sheet on the duct and hub, the stream function induced at the knots of the computational g r i d i s computed.

4. The streamlines corresponding to the wake vortex sheets and the stream­l i n e s corresponding to the slips t r e a m vortex sheets are traced i n s i d e the computational domain.

5. The strength of the wake vortex sheets i s determined from the form taken by the v o r t i c i t y transport equation f o r axisymmetric flow without s w i r l (eq. 3-78 and 3-79), and the strength of the sl i p s t r e a m vortex sheets i s determined from the p a r t i c u l a r form of the v o r t i c i t y transport equation f o r axisymmetric flow with s w i r l (eq. 3-80, 3-81 and 3-82).

6. The stream function induced at the knots of the computational g r i d by a l l the vortex sheets i s computed.

7. The a x i a l and r a d i a l v e l o c i t i e s induced by a l l vortex sheets at the p i v o t a l points on the duct and hub contour are computed.

8. The boundary co n d i t i o n on the duct and hub surfaces together with the Kutta c o n d i t i o n on the duct t r a i l i n g edge i s s a t i s f i e d s o l v i n g the correspondent inner p o t e n t i a l flow problems f o r the surface vortex sheets.

Steps 3 to 8 are repeated u n t i l convergence i s attained.

97

3.4. NUMERICAL RESULTS AND COMPARISON WITH EXPERIMENT

In order to a s c e r t a i n the e f f e c t of the wake v o r t i c i t y on ducted pro­

p e l l e r performance, a s e r i e s of c a l c u l a t i o n s were c a r r i e d out for the

ducted p r o p e l l e r c o n f i g u r a t i o n duct 37 with p r o p e l l e r B i n uniform and non­

uniform flow. This c o n f i g u r a t i o n was tested i n the NSMB Large C a v i t a t i o n Tunnel both

in uniform and non-uniform flow conditions and experimental data became a v a i l a b l e f o r d e t a i l e d comparison with the t h e o r e t i c a l c a l c u l a t i o n s .

Some of the r e s u l t s concerning the uniform flow case f o r t h i s configura­t i o n have already been discussed i n Chapter 2, f o r the purpose of compari­son with the p r e d i c t i o n s of a l i n e a r i z e d model. Further r e s u l t s regarding the duct pressure d i s t r i b u t i o n s and the flow stream-surfaces w i l l be introduced here as they are needed f o r a f u l l a p p reciation of the e f f e c t of the wake v o r t i c i t y .

In f a c t , due to the non-linear i n t e r a c t i o n between the incoming wake v o r t i c i t y and the slipstream and duct vortex systems, i t i s not p o s s i b l e to separate the associated induced v e l o c i t y f i e l d s and we must resort to the uniform flow case at the same chosen o v e r a l l loading parameter, f o r the pur­poses of comparison.

For the uniform-flow case, only the slipstream v o r t i c i t y needs to be d i s c r e t i z e d . The c a l c u l a t i o n s were performed at a l l p r o p e l l e r loadings with 48 elements on the duct and 21 elements on the hub. The computational g r i d was chosen with 10 a x i a l s t a t i o n s at x/R=0.0, 0.231, 0.442, 0.533, 0.643, 0.754, 0.905, 1.307, 1.751, 3.00, implying 10 elements on each slipstream vortex sheet and 15 r a d i a l s t a t i o n s with 0.175 as constant r a d i a l s t e p s l z e .

In order to assess the e f f e c t of the number of vortex sheets on the nu­merical r e s u l t s , the slipstream v o r t i c i t y has been d i s c r e t i z e d f i r s t with 5 and then with 9 vortex sheets. From preliminary c a l c u l a t i o n s i t turned out that underrelaxation was required to ensure convergence. A constant r e l a x a ­t i o n f a c t o r of 0.5 was used when computing new i t e r a t e s for the t o t a l stream function on the knots of the computational g r i d . For such case, the conver­gence properties of the r e l a x a t i o n method are i l l u s t r a t e d i n F i g . 3-8, i n

98

4 0 . 0 A C T d - 5 V O R T E X S H E E T S

' C R - . .

A C T ( J -9 V O R T E X S H E E T S

' C R -

3 0 . 0

2 0 . 0

1 0 . 0 -

0 2 3 4 5 6 7 8 9 I T E R A T I O N N U M B E R

Fig. 3-3. Convergence of i t e r a t i o n for the duct thrust and radial

forces, ducted propeller in uniform flow J=0.208,

Cy =12.48. Relaxation factor 0.5. P

which the r e s u l t s f o r the a x i a l and r a d i a l forces a c t i n g on the duct i n each i t e r a t i o n step are shown for the p r o p e l l e r loading J=0.208, CXp=12.48. It i s seen that 6 i t e r a t i o n s were s u f f i c i e n t to get converged r e s u l t s with an accu­racy of 0.01. The e f f e c t of number of vortex sheets used to d i s c r e t i z e the slipstream v o r t i c i t y appears to be rather small i n what concerns the forces on the duct. In f a c t , the change from 5 to 9 vortex sheets i s responsible for v a r i a t i o n s of the r a d i a l force of l e s s then 3% and of the a x i a l force l e s s than 1%.

In the non-uniform flow case the c a l c u l a t i o n s were performed f o r the experimental conditions corresponding to the p r o p e l l e r loading c o e f f i c i e n t s Kj^=0.204, 0.171 and 0.101 r e s p e c t i v e l y . We note that the co n f i g u r a t i o n has been investigated i n uniform flow at the same p r o p e l l e r loadings (see Chapter 2).

The axisymmetric wake v e l o c i t y p r o f i l e i n front of the p r o p e l l e r was generated by means of wire meshes. In t h i s way a reasonable strongly sheared wake v e l o c i t y p r o f i l e with a r a d i a l extent of 2.4 times the p r o p e l l e r radius could be obtained. It was found that the v e l o c i t y p r o f i l e was f a i r l y constant behind the mesh screen f o r distances of about two p r o p e l l e r diameters and was

99

r l R

2 4

2 0

1 6

1.2

0 8

0 4

° 0 0 .2 0 .4 0 .6 0 .8 1.0 U / U o

Fig. 3-9. Approximation of wake velocity profile.

reasonably axisymmetric. The v e l o c i t y p r o f i l e derived from the Laser-Doppler measurements behind the screen i s shown i n F i g . 3-9. In t h i s Figure the wake v e l o c i t i e s are made dimensionless by the measured v e l o c i t y outside the wake f i e l d . This v e l o c i t y appeared to be considerably l a r g e r than the tunnel speed,respectively 4.13 m/s and 3.75 m/s. This increase of free-stream v e l o ­c i t y due to displacement of the wake screen and the presence of the tunnel

walls, was taken into account i n the c a l c u l a t i o n s by r e f e r r i n g the disk ad-_ - 2 2 vance r a t i o J=Up/nD and the disk thrust c o e f f i c i e n t CXp=T^/(JpU^irR ), to the

measured v e l o c i t y outside the wake instead of the tunnel speed. No other c o r r e c t i o n s due to the tunnel "wall e f f e c t " were considered.

The v e l o c i t y p r o f i l e was d i s c r e t i z e d i n t o 10 or 11 vortex sheets. D i f f e r ­ent d i s c r e t i z a t i o n s were employed f o r each p r o p e l l e r loading i n order to obtain a high number of vortex sheets i n the v i c i n i t y of the duct's stream-surface. As an i l l u s t r a t i o n , F i g . 3-9, shows the d i s c r e t i z a t i o n used at the highest p r o p e l l e r loading, J=0.78, 0^=0.87, f o r which the radius at i n f i n i ­ty upstream of the duct's stream surface was about 1.70. The need f o r a f i n e d i s c r e t i z a t i o n of the inflow v e l o c i t y p r o f i l e i n the v i c i n i t y of the duct's

100

stream-surface radius at i n f i n i t y upstream, a r i s e s from the f a c t that, as expected, the p r e d i c t i o n of the force on the duct appeared to be rather s e n s i t i v e to the prec i s e l o c a t i o n of the vortex sheets nearby. Moreover, s i t u a t i o n s arose, i n which the method f a i l e d to converge because the nearest vortex sheet to the duct was a l t e r n a t i n g l y traced outside and i n s i d e the duct from one i t e r a t i o n step to the next.

To i l l u s t r a t e the e f f e c t of p r o p e l l e r r a d i a l load d i s t r i b u t i o n s , two t y p i c a l disk c i r c u l a t i o n d i s t r i b u t i o n s were used i n the c a l c u l a t i o n s and are shown i n F i g . 3-10 together with the corresponding piecewise constant approximations leading to d i s c r e t i z a t i o n s of the slipstream v o r t i c i t y i n t o f i v e vortex sheets.

Because of the v a r i a b l e curvature of the stream-surfaces i n the v i c i n i ­ty of the duct's nose depending on the p r o p e l l e r loading two g r i d designs were used. The f i r s t employed 24 a x i a l s t a t i o n s unequally spaced and 15 r a d i a l s t a t i o n s with a t y p i c a l s t e p s i z e of 0.175 and i t was applied at the highest p r o p e l l e r loadings. The second employed 15 a x i a l s t a t i o n s unequally spaced and 15 r a d i a l s t a t i o n s with the same ste p s i z e and i t was used at the

r/r.

Fig. 3-10. Discretization of disk circulation distributions.

101

smallest loading. With t h i s p a r t i c u l a r r a d i a l stepsize,which i s of the order of the mag­

nitude of the duct's thickness, the rather steep v e l o c i t y gradients near the nose and the t r a i l i n g edge cannot be accurately described. Therefore, l o c a l e r r o r s i n the strength of the vortex sheet are introduced i f the vortex sheet i s traced i n the close v i c i n i t y of the duct's surface. The problem could be probably obviated by a refinement of the r a d i a l s t e p s i z e which should be accompanied by an increase of the number of vortex sheets i n order to keep the e f f e c t i v e n e s s of the numerical d i f f e r e n t i a t i o n scheme f o r the a x i a l v e l o c i t y p r o f i l e s . Another way of minimizing such errors i s through refinement of the number of vortex sheets passing c l o s e to the duct's sur­face because of the r e l a t i v e decrease of the strength of the nearest vortex sheet. The numerical r e s u l t s f o r the force on the duct i n the various i t e r a ­t i o n steps revealed that f o r a given d i s c r e t i z a t i o n of the inflow v e l o c i t y p r o f i l e the forces a c t i n g on the duct were not p a r t i c u l a r l y s e n s i t i v e to small v a r i a t i o n s of the l o c a t i o n of the vortex sheets i n the v i c i n i t y of the nose, provided that the dynamic pressure of the duct's stream-surface d i d not change. On the basis of these r e s u l t s the e f f e c t of the dynamic pressure on the duct force was sought predominant to the e f f e c t of v o r t i c i t y . Therefore no attempt has been made to f u r t h e r r e f i n e the r a d i a l stepsize i n the pre­sent i n v e s t i g a t i o n . I t must be, however, recognized that small v a r i a t i o n s of c i r c u l a t i o n around the duct due to small d i f f e r e n c e s i n the strength of the vortex sheets may cause large v a r i a t i o n s of the force i f a s h i f t of the duct to a d i f f e r e n t dynamic pressure i s implied.

In a l l cases the c a l c u l a t i o n s were performed with a r e l a x a t i o n f a c t o r of 0.5. The r e s u l t s for the duct thrust and r a d i a l force obtained on each i t e r a t i o n step are shown i n F i g . 3-11. It i s seen that i n general, s i x i t e r a ­t i o n s were s u f f i c i e n t to obtain converged r e s u l t s within 2% f o r the duct r a d i a l force and 1% f o r the duct t h r u s t .

The l o c a t i o n s of the wake vortex sheets obtained from the l a s t i t e r a t e s

at the various p r o p e l l e r loadings are shown i n Figs . 3-12 to 3.14. The c i r c u l a t i o n

d i s t r i b u t i o n 1 has been assumed i n these c a l c u l a t i o n s . Small inaccuracies can

be found on the l o c a t i o n of the vortex sheets close to hub' s nose which can be

ascribed to the small number of elements used to approximate i t .

102

Fig. 3-11. Convergence on duct thrust and radial force. Duct 37 with

propeller B in shear flow .

In order to make evident the e f f e c t of the wake v o r t i c i t y on the flow stream-surfaces, i n F i g . 3-15, the stream-surfaces corresponding to the vortex sheets shed from the p r o p e l l e r disk, with the exception of the vortex sheet bound to the hub, are shown f o r both uniform and shear flows at the same p r o p e l l e r loading. Both c a l c u l a t i o n s have been done with the same Kutta co n d i t i o n on the duct's t r a i l i n g edge and the same disk r a d i a l d i s t r i b u t i o n

103

Fig. 2-12. Location of wake vortex sheets. Duct 37 with propeller B

in shear flow. 1=1.92, CT =0.07.

r -

X

i — i - 2 0 - 1 0 0 1 0 2.0 3.0 a / R

Fig. 3-13. Location of wake vortex sheets. Duct 37 with propeller B

in shear flow. 1=1.29, CT =0.26.

o i > - 2 0 - 1 0 0 1.0 2 0 3.0 „ , „

Fig. 3-14. Location of wake vortex sheets. Duct 37 with propeller B

in shear flow. 1=0.78, =0.87. P

104

Fig. 3-15. Location of slipstream vortex sheets. Duct 37 with propeller

B. Comparison between uniform flow and shear flow.

of c i r c u l a t i o n . It i s i n t e r e s t i n g to note the much lower rate of decay of the r a d i a l v e l o c i t i e s with distance from the p r o p e l l e r plane obtained i n the shear flow case i n comparison with the uniform flow.

The pressure d i s t r i b u t i o n s on the duct i n shear flow are given i n F i g . 3-16 to 3-18 and compared with pressure d i s t r i b u t i o n s i n uniform flow at the same p r o p e l l e r loadings i n F i g s . 3-17 and 3-18. From these f i g u r e s i t may be concluded that the e f f e c t of v o r t i c i t y tends to decrease the thrust r a t i o between p r o p e l l e r and duct at the same p r o p e l l e r r a d i a l load d i s t r i b u ­t i o n .

In order to estimate the e f f e c t of the disk r a d i a l c i r c u l a t i o n d i s t r i ­bution on the duct pressure d i s t r i b u t i o n , computations were made with the load d i s t r i b u t i o n 2 of F i g . 3-10 and the corresponding pressure d i s t r i b u t i o n s are shown i n F i g . 3-17 and 3-18. The e f f e c t of the r a d i a l c i r c u l a t i o n d i s t r i ­bution on the computed duct thrust amounted to a change from CT^=0.59 to Cf,=0.51 and on the computed r a d i a l force c o e f f i c i e n t C from 1.70 to 1.43, d R r e s p e c t i v e l y f o r the c i r c u l a t i o n d i s t r i b u t i o n s 1 and 2 with J=0.78, C T p=0.87. For J=1.29, CTp=0.26, the duct thrust c o e f f i c i e n t changed from 0.16 with c i r c u l a t i o n d i s t r i b u t i o n l , t o 0.13 with c i r c u l a t i o n d i s t r i b u t i o n 2 and the corresponding change of the r a d i a l force c o e f f i c i e n t was from 5^=0.52 to

C =0.45, (see F i g . 3-11). R

105

- 0 6

- 0 . 5

- 0 . 4

- 0 . 3

0 2

- 0 1

0 1

0 2

ft

K T p = 0 . 1 0 1 J =1.92 C T p = C . 0 7

-

1 1 1 T " f I t I _ ^ - - T T 0.1 0 . 2 0 . 3 0 . 4 0 5 0 . 6 0 . 7 0 . 8 0 . 9 \

j 1 0 x , C

Fig. 3-16. Pressure distribution on duct 37 with propeller B in

shear flow.

As mentioned before, to check the numerical c a l c u l a t i o n s a set of mea­

surements on the c o n f i g u r a t i o n duct 37 with p r o p e l l e r B were performed i n

the Large C a v i t a t i o n Tunnel, which included Laser-Doppler measurements of the

a x i a l and r a d i a l v e l o c i t y components and measurements of the thrust and i n t e ­

grated r a d i a l force a c t i n g on the duct.

These measurements have been reported by Luttmer and Janssen (1982) and

a b r i e f d e s c r i p t i o n of t e s t equipment and procedure can be found i n Appen­

dix 1.

An a n a l y s i s of the v e l o c i t y measurements i n the p r o p e l l e r s l i p s t r e a m

showed a c l e a r s h i f t of the l o c i of maximum a x i a l v e l o c i t y from r/R=0.7 i n

uniform flow to r/R=0.6, 0.5 and 0.4 i n shear flow at decreasing p r o p e l l e r

loadings. This f a c t i s i l l u s t r a t e d i n F i g . 3-19 where the perturbation a x i a l

106

Fig. 3-17. Pressure distributions on duct 37 with propeller B in

uniform and shear flow.

v e l o c i t i e s derived from the Laser-Doppler measurements at 3 mm downstream of

the duct's t r a i l i n g edge and analysed at the p r o p e l l e r thrust i d e n t i t y are

shown for the uniform and non-uniform flow cases. The s h i f t towards the hub of

the point of maximum v e l o c i t y i s associated with the change of p r o p e l l e r load

d i s t r i b u t i o n i n the wake f i e l d f o r the various p r o p e l l e r loadings.

The d i f f e r e n c e i n the magnitude of the perturbation v e l o c i t i e s at the

smallest p r o p e l l e r loading could p a r t i a l l y be due to the way of analysing the r e s u l t s . Since the measured duct thrusts considerably d i f f e r from each other at t h i s p r o p e l l e r loading, being r e s p e c t i v e l y K =0.043 and K„ =0.001

i<3 1 d i n uniform and non-uniform flow, t o t a l thrust i d e n t i t y would lead to a

higher e f f e c t i v e v e l o c i t y .

107

Fig. 3-18. Pressure distributions on duat 37 with propeller B in

uniform flow and shear flow.

For the intermediate loading, the values of the duct thrust c o e f f i c i e n t are K =0.060 and K =0.058 for the uniform and non-uniform flow, respec-

d d t i v e l y , and the t o t a l thrust i d e n t i t y v i r t u a l l y y i e l d s the same e f f e c t i v e v e l o c i t y , which suggests a remarkable e f f e c t of the i n t e r a c t i o n with the wake v o r t i c i t y i n the induced v e l o c i t y f i e l d . We note that the depicted perturbation v e l o c i t i e s i n the non-uniform flow case include the disturbance v e l o c i t y induced by the change i n the wake v o r t i c i t y . For the highest loading the d i f f e r e n c e i n the duct thrust c o e f f i c i e n t i s s i g n i f i c a n t , (Kxd=0.134 i n uniform flow and KTd=0.111 i n non-uniform flow) and t o t a l thrust i d e n t i t y would lead again to a higher e f f e c t i v e v e l o c i t y .

108

Fig. 3-19. Measured axial perturbation velocities at 3 mm downstream of

the t r a i l i n g edge (x/B= 0.53). Duct 37 with propeller 3 in

uniform and non-uniform flow at propeller thrust identity.

The comparison between the c a l c u l a t e d and measured v e l o c i t y f i e l d i s given i n F i g s . 3-20 to 3-22, where the a x i a l v e l o c i t y p r o f i l e s at three d i f f e r e n t l o c a t i o n s upstream and downstream of the ducted p r o p e l l e r are shown. It may be concluded that the agreement between the measured and c a l c u l a t e d p r o f i l e s i s reasonable f o r the highest p r o p e l l e r loading, but worsens as the p r o p e l l e r load decreases. As remarked before, the c i r c u l a t i o n d i s t r i b u t i o n on the p r o p e l l e r at decreasing l o a d i n g , i n c r e a s i n g l y deviates from the c i r c u l a t i o n d i s t r i b u t i o n 1 assumed i n the c a l c u l a t i o n s . For the sake of comparison, the r e s u l t s of the c a l c u l a t i o n s with c i r c u l a t i o n d i s t r i b u t i o n 2 for the two highest loadings, are shown also i n the Figures.

109

o E X P E R I M E N T

J = 1 . 9 2 C T p = 0 . 0 7

Fig. 3-20. Measured and calculated axial velocity -profiles at

x/R=-0.53. Duct 37 with propeller B in axisymmetric wake

flow.

It can be seen that the e f f e c t of the p r o p e l l e r load d i s t r i b u t i o n on the

induced v e l o c i t i e s i s remarkable i n s i d e the p r o p e l l e r slipstream. The

measured v e l o c i t y p r o f i l e s depict steep gradients i n the v i c i n i t y of the

duct surface and deviate considerable from the t h e o r e t i c a l ones. With the

r a d i a l s t e p s i z e employed i n the c a l c u l a t i o n s such strong gradients,

s p e c i a l l y at the nose and the wake peak downstream of the duct t r a i l i n g

edge, could not be resolved and therefore, the c a l c u l a t i o n s are not

expected to be accurate i n those regions.

The comparison between the c a l c u l a t e d and measured force a c t i n g on the duct i s shown i n Table 3-1. It may be concluded that the c o r r e l a t i o n

110

Fig. 3-21. Measured and calculated axial velocity profiles at x/R=0.53.

Duct 37 with propeller B in axisymmetric wake flow.

i s poor, even at the highest p r o p e l l e r loading f o r which the p r e d i c t i o n of the v e l o c i t y f i e l d i s reasonable. Although the e f f e c t of v i s c o s i t y on the forces i s not known there i s a tendency towards consistent overestimation of the p r e d i c t e d values i n comparison with the experimental r e s u l t s . The change of the p r o p e l l e r load d i s t r i b u t i o n , although considerably a f f e c t s the v e l o c i t y f i e l d , appears to have a small e f f e c t on the duct forces.

The most l i k e l y cause of the discrepancies i s the e f f e c t on the duct

forces of the dynamic pressure of the duct stream-surface. In f a c t , f o r the

p a r t i c u l a r d i s c r e t i z a t i o n s used i n the c a l c u l a t i o n s , the d i f f e r e n c e i n the

duct forces i f the duct stream-surface i s traced i n a consecutive i n t e r v a l

111

of dynamic pressure i s of the order of magnitude of the discrepancies between c a l c u l a t e d and measured r e s u l t s .

We r e c a l l that, f o r the sake of comparison, the Kutta c o n d i t i o n was implemented i n the same way i n the c a l c u l a t i o n s i n uniform and non-uniform flow. As remarked i n the second Chapter, t h i s form of Kutta c o n d i t i o n gave reasonable p r e d i c t i o n s of the forces on the duct at the two higher p r o p e l l e r loadings i n uniform flow.

Fig. 3-22. Measured and calculated axial velocity profile at x/R=0.90.

Duct 37 with propeller B in axisymmetric wake flow.

112

J P

EXPERIMENT THEORY

J P

EXPERIMENT C i r c u l a t i o n d i s t r i b u t i o n 1

C i r c u l a t i o n d i s t r i b u t i o n 2

J P \ 5R \ eR % 5R

1.92 0.07 0.00 0.035 0.04 0.16 - _

1.29 0.26 0.09 0.34 0.16 0.52 0.13 0.45 0.78 0.87 0.47 1.19 0.59 1.70 0.51 1.43

Table 3-1. Comparison of measured and calculated forces on duct 37

with propeller B in axisymmetric wake flow.

113

4. I n t e r a c t i o n s t u d i e s b e t w e e n a d u c t e d p r o p e l l e r a n d t h e s t e r n f o r a x i s y m m e t r i c f l o w s

4.1. INTRODUCTION

This Chapter deals with the a p p l i c a t i o n of the methods presented i n Chapters 2 and 3 to the c a l c u l a t i o n of stern flows and the i n t e r a c t i o n between the ducted p r o p e l l e r and the stern i n axisymmetric flow.

When placed close behind a body, an operating p r o p e l l e r introduces a disturbance to the e x i s t i n g flow around the stern. This disturbance flow leads to an increase of v e l o c i t y i n the stern region which r e s u l t s i n a decrease of pressure and an increase of skin f r i c t i o n on the body's surface. The correspondent increase of body r e s i s t a n c e at a c e r t a i n speed,when ex­pressed as a f r a c t i o n of the thrust to be d e l i v e r e d by the propulsor to p r o p e l l the body at the same speed, i s known as the thrust deduction f r a c t i o n

The v e l o c i t i e s occurring at the p r o p e l l e r plane i n the absence of the p r o p e l l e r are known as the nominal v e l o c i t i e s . The d i f f e r e n c e between the nominal v e l o c i t i e s and the body's speed expressed as a f r a c t i o n of that speed i s known as the nominal wake.

It i s generally considered that, due to the i n t e r a c t i o n between the p r o p e l l e r induced flow and the stern flow, the inflow to the p r o p e l l e r i s thereby modified and the correspondent v e l o c i t i e s at the p r o p e l l e r plane d i f f e r from the nominal v e l o c i t i e s . These inflow v e l o c i t i e s are known as e f f e c t i v e v e l o c i t i e s and t h e i r d i f f e r e n c e to the body's speed as a f r a c ­t i o n of that speed i s known as the e f f e c t i v e wake.

In model t e s t i n g the mean e f f e c t i v e v e l o c i t y to the p r o p e l l e r i s ob­tained by the thrust or torque i d e n t i t y method from the open-water character i s t i c s of the propulsor.

In the c a l c u l a t i o n of wake adapted p r o p e l l e r s , the e f f e c t i v e v e l o c i t y d i s t r i b u t i o n at the p r o p e l l e r plane i s an important input parameter. A com­monly accepted s u i t a b l e d e f i n i t i o n f o r the e f f e c t i v e v e l o c i t i e s i d e n t i f i e s them with d i f f e r e n c e between the t o t a l v e l o c i t i e s p r e v a i l i n g behind the body

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with the operating p r o p e l l e r and the p r o p e l l e r induced v e l o c i t i e s . In such d e f i n i t i o n , under the concept of p r o p e l l e r induction i s understood the v e l o c i t y f i e l d induced by the assumed p r o p e l l e r vortex model.

In view of the r e l a t i v e s i m p l i c i t y of t h e i r stern flow, axisymmetric bodies i n the deeply submerged s t a t e , have been i n v e s t i g a t e d f o r the purpose of g e t t i n g i n s i g h t into the i n t e r a c t i o n phenomena and providing checks on new a n a l y t i c a l and computational techniques. The c l a s s i c a l approach to the i n t e r a c t i o n problem, f o r a deeply submerged body, separates the t o t a l flow i n t o i t s p o t e n t i a l and viscous components, r e s p e c t i v e l y associated with the displacement e f f e c t s of the body and the body's boundary l a y e r and wake system.

It i s not s u r p r i s i n g that e a r l y a n a l y t i c a l studies on axisymmetric bodies,considering both p o t e n t i a l and viscous effects,were based on c l a s s i ­c a l p o t e n t i a l flow and boundary l a y e r theories, (Korvin-KroukovsKy, 1956, Tsakonas and Jacobs, 1960).

In these studies the body was represented by a l i n e source-sink d i s t r i ­bution and the p r o p e l l e r by a sink disk with constant strength. Korvin-Kroukovsky (1956),used experimental boundary layer thickness data, together with an assumed power v e l o c i t y law d i s t r i b u t i o n , to estimate the e f f e c t of the boundary l a y e r on the thrust deduction by accounting f o r the e f f e c t s of the boundary layer displacement thickness.

Tsakonas and Jacobs (1960), evaluated various methods a v a i l a b l e at that time to compute the boundary l a y e r on the body. In t h e i r work a mathematical model f o r the i n t e r a c t i o n between the p r o p e l l e r and the boundary layer was i n v e s t i g a t e d . In t h i s model only the displacement effects of the boundary layer and wake were assumed to i n t e r a c t with the p r o p e l l e r . As a conse­quence the p r o p e l l e r was considered to work i n a p o t e n t i a l flow region where Dickmann's, (1938), c l a s s i c a l r e l a t i o n s h i p s between p o t e n t i a l thrust deduction and wake f r a c t i o n hold. Since, as i t i s well known, the f r i c t i o n a l c o n t r i b u t i o n to the e f f e c t i v e wake i s an important one,they were led to the conclusion that the f r i c t i o n a l c o n t r i b u t i o n to the thrust deduc­t i o n , f o r the p a r t i c u l a r bodies of r e v o l u t i o n investigated,was s i g n i f i c a n t . Such conclusion i s d i f f i c u l t to accept and has not been confirmed by further studies.

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More recent work on the p r e d i c t i o n of thrust deduction has been based on

p o t e n t i a l flow theory. Beveridge (1969), applied a surface source panel

method to represent the body and a sink disk with v a r i a b l e r a d i a l strength

to represent the p r o p e l l e r . This p r o p e l l e r model i s formally equivalent i n the region outside the

p r o p e l l e r slipstream, to the actuator disk model c o n s t i t u t e d by a d i s t r i b u ­t i o n of r i n g vortex cylinders (see Chapter 2), and,as stated before, cor­responds to the c i r c u m f e r e n t i a l mean flow of a l i f t i n g l i n e representation of the p r o p e l l e r . Further developments include the representation of the p r o p e l l e r by l i f t i n g surface theory, as reviewed by Cox (1977).

Recently, Huang et a l (1976), Huang and Cox (1977) and Huang and Groves (1980), presented a d e t a i l e d i n v e s t i g a t i o n on the i n t e r a c t i o n between an open p r o p e l l e r and various axisymmetric bodies. This work included measurements of afterbody pressure and skin f r i c t i o n d i s t r i b u t i o n s , a n d a x i a l v e l o c i t y p r o f i l e s with Laser-Doppler velocimeter,with and without p r o p e l l e r i n operation. To tr e a t the d i f f e r e n t aspects of the i n t e r a c t i o n problem various a n a l y t i c a l procedures were proposed and checked by comparison with the measurements.

To c a l c u l a t e the flow on the afterbody,Huang et a l (1976), (1977) employed an i t e r a t i o n scheme based on p o t e n t i a l flow and t h i n boundary l a y e r c a l c u l a t i o n methods. The scheme appeared only to give s a t i s f a c t o r y p r e d i c t i o n s of pressure d i s t r i b u t i o n and a x i a l v e l o c i t y p r o f i l e s f o r the f i n e s t afterbody (fineness r a t i o L^/D=4.31). For the two other afterbodies (L^/D=2.25 and 1.48),the p r e d i c t i o n s were considerably poorer. Huang et a l (1976), also showed that when using, i n the i t e r a t i o n scheme, the pressure d i s ­t r i b u t i o n on the body, modified to account f o r the p r o p e l l e r induced flow, un­r e l i a b l e p r e d i c t i o n s of the v e l o c i t y p r o f i l e s i n front of the p r o p e l l e r were obtained. With an a l t e r n a t i v e approach, based on i n v i s c i d flow theory to describe the i n t e r a c t i o n between the p r o p e l l e r and the thick stern boundary la y e r , p r e d i c t i o n s of the a x i a l v e l o c i t y p r o f i l e s ahead of the propulsor were i n excellent agreement with the Laser-Doppler v e l o c i t y measurements. The input c o n s i s t s of the nominal v e l o c i t i e s at the p r o p e l l e r plane and the p r o p e l l e r induced v e l o c i t i e s as c a l c u l a t e d by a p r o p e l l e r induced v e l o c i t y f i e l d program. The nominal v e l o c i t i e s were taken from the measured

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v e l o c i t y p r o f i l e s without the operating p r o p e l l e r . The greatest disadvantage of Huang's method l i e s i n the f a c t that the

accuracy of the t o t a l and e f f e c t i v e v e l o c i t i e s u l t i m a t e l y depends on the accuracy of the nominal v e l o c i t i e s . The c a l c u l a t i o n of the flow i n the t a i l region of an axisymmetric body seems,therefore,to be of v i t a l importance f o r the complete a n a l y t i c a l p r e d i c t i o n of the i n t e r a c t i o n between p r o p e l l e r and stern.

Before c l o s i n g these remarks on the various approaches to the i n t e r a c ­t i o n between a conventional p r o p e l l e r and the afterbody, the c a l c u l a t i o n method used by Schetz and Favin (1979), should be mentioned. They solved numerically the f u l l Navier-Stokes equations f o r axisymmetric flow and obtained reasonable agreement with experiments.

In contrast with the large amount of work on p r o p e l l e r - h u l l i n t e r a c t i o n fo r conventional p r o p e l l e r s , s i m i l a r studies f o r ducted p r o p e l l e r are scarce i n the l i t e r a t u r e . Perhaps, the most complete a n a l y s i s of ducted p r o p e l l e r -h u l l i n t e r a c t i o n i s the one due to K r i e b e l (1964). Assuming p o t e n t i a l flow, K r i e b e l represented the h u l l by a source-sink l i n e d i s t r i b u t i o n , the duct by r i n g vortex s i n g u l a r i t i e s on the reference c y l i n d e r and the p r o p e l l e r by an actuator disk with constant strength. The wake of the body was represented, i n a s i m p l i f i e d way, by a number of s e m i - i n f i n i t e vortex c y l i n d e r s . The strength of the vortex c y l i n d e r s , assumed constant downstream,was determined by matching the correspondent a x i a l v e l o c i t y d i s t r i b u t i o n to the nominal v e l o c i t i e s at the p r o p e l l e r plane.

P o s s i b l y , the most d r a s t i c s i m p l i f i c a t i o n introduced i n h i s mathematical model, i s r e l a t e d to the way the wake was allowed to i n t e r a c t with the ducted p r o p e l l e r and h u l l . S p e c i f i c a l l y , the v e l o c i t i e s induced by the wake vortex c y l i n d e r s on the duct were taken into account but the i n f l u e n c e of the h u l l , duct or p r o p e l l e r s i n g u l a r i t i e s on the wake was disregarded. By evaluating the i n t e r a c t i o n forces between the various s i n g u l a r i t i e s , t h e r e l a t i v e merits of various duct camber shapes f o r short and long ducts were demonstrated.

The above-mentioned studies,and i n p a r t i c u l a r the work of Huang et a l ,

suggest that any attempt to compute the v e l o c i t y f i e l d i n the v i c i n i t y of a

p r o p e l l e r operating i n s i d e the stern boundary layer, should contemplate the

117

i n t e r a c t i o n between the p r o p e l l e r induced flow and the flow i n the boundary lay e r . Apparently, the basic i n t e r a c t i o n mechanism of the two flows i s i n v i s c i d i n nature and may be properly described by the Euler equations of motion.

The f a i l u r e of the conventional boundary-layer p o t e n t i a l flow i n t e r a c ­t i o n scheme used by Huang et a l i n c a l c u l a t i n g the flow with p r o p e l l e r i n ope­ra t i o n , and the r e l a t i v e l y poor p r e d i c t i o n s of the st e r n flow without propel­l e r f o r the two f u l l e s t forms,indicate that the pressure gradients and vel o ­c i t i e s normal to the surface are not n e g l i g i b l e and should be considered.

F i n a l l y , Huang et a l found the f r i c t i o n a l component of the thrust deduction f o r the three models i n v e s t i g a t e d to be n e g l i g i b l e . The pressure component of the thrust deduction was well p r e d i c t e d by the p o t e n t i a l flow c a l c u l a t i o n s although questions were r a i s e d regarding the adequacy of the approach,when a region of separated flow was present as i n the case of the bluntest afterbody.

This Chapter i s centered on the i n t e r a c t i o n problem between a ducted p r o p e l l e r and the stern i n axisymmetric flow. An i n v i s c i d approach to c a l ­culate the flow on the stern with and without propulsor based on the Euler equations of motion i s presented. Apart from the approximations involved i n the numerical solution, no other approximations are introduced i n the governing equations. The basic numerical s o l u t i o n procedures have already been dealt with i n Chapters 2 and 3 . Therefore the emphasis i s placed on the under­l y i n g assumptions and the c a l c u l a t i o n r e s u l t s .

In Section 4.2 an a p p l i c a t i o n to the computation of the flow i n the s t e r n region of a body of r e v o l u t i o n i s presented.

In s e c t i o n 4.3 the r e s u l t s of computations f o r two ducted p r o p e l l e r c o nfigurations behind the same body of r e v o l u t i o n are given.

In s e c t i o n 4.4 the r e s u l t s are discussed and the e f f e c t i v e wake and thrust deduction are estimated using two d i f f e r e n t methods of a n a l y s i s : the duct as a part of the h u l l and the duct as a part of the propulsor.

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4.2. APPLICATION TO THE CALCULATION OF THE STERN FLOW FOR AN AXISYMMETRIC BODY.

In t h i s s e c t i o n the method presented i n Chapter 3 i s applied to the

c a l c u l a t i o n of the flow i n the stern region of an axisymmetric body.

The basic assumption underlying the c a l c u l a t i o n procedure i s that the flow may be considered i n v i s c i d everywhere i n s i d e a flow region, c a l l e d the computational domain, which comprises the stern and the near wake regions. For non-separating flows, t h i s assumption i s l i k e l y to hold as a good approx­imation for the r o t a t i o n a l flow region i n s i d e the boundary l a y e r and wake with exception of a r e l a t i v e l y t h i n layer close to the wall where the e f f e c t s of v i s c o s i t y become important.

A considerable amount of experimental and t h e o r e t i c a l work performed on axisymmetric bodies gives support to t h i s supposition. S p e c i f i c a l l y , the measurements done by Pa t e l et a l (1974), (1978) on two axisymmetric bodies revealed that the Reynolds stresses were rather small i n the thick stern boundary layer and near wake regions. Also,the t h e o r e t i c a l studies of Dyne (1978) and G e l l e r (1979), r e s p e c t i v e l y on the f i r s t and se­cond , axisymmetric bodies studied by Patel et a l , provided f u r t h e r evidence regarding the adequacy of the i n v i s c i d approach by r e v e a l i n g a rather good agreement between predicted and measured v e l o c i t y p r o f i l e s down to the t a i l . It should be noted that Dyne's (1978), streamline curvature method includes the losses i n t o t a l head along the streamlines due to the e f f e c t of viscous and Reynolds st r e s s e s , but such losses appeared to be small over about ninety per cent of the boundary layer thickness i n the stern region. More rece n t l y , the work of Larsson et a l (1982), with an extension of the stream­l i n e curvature method to ship stern flows, suggests that the i n v i s c i d assumption might be a p p l i c a b l e to more general three-dimensional stern flows.

The computational domain i s shown i n F i g . 4-1, (fo r the purpose of the ducted p r o p e l l e r - h u l l i n t e r a c t i o n studies to be considered i n the next sec­t i o n the p o s i t i o n s of duct and p r o p e l l e r are included). It i s selected such that i t includes i n the streamwise d i r e c t i o n the

119

Fig. 4-1, Schematic representation of computational domain for

the stern flow.

afterbody's region extending from a s t a t i o n on the p a r a l l e l middle body (or an upstream s t a t i o n where the flow i n the boundary l a y e r i s approximately a x i a l l y d i r e c t e d and the boundary l a y e r i s thin) downstream to the t a i l and part of the wake. In the r a d i a l d i r e c t i o n , i t i s d e l i m i t e d by a c y l i n d r i c a l surface of constant radius.

Since the method of Chapter 3 i s based on a Green's function formula­t i o n to the stream fu n c t i o n boundary-value problem, the boundary con d i t i o n at i n f i n i t y i n the r a d i a l coordinate i s automatically s a t i s f i e d . Therefore, the radius of the boundary c y l i n d r i c a l surface can be kept small, provided that no f l u x of v o r t i c i t y takes place through that p a r t i c u l a r surface.

The v o r t i c i t y contained i n the computational domain i n s i d e the boundary layer and wake i s d i s c r e t i z e d i n t o a number of say H, vortex sheets: N-l vortex sheets representing the outer region of the boundary l a y e r and one vortex sheet representing the wall l a y e r . The N-l outer vortex sheets are assumed to be governed by the i n v i s c i d v o r t i c i t y transport equation for axisymmetric flow without s w i r l (see Chapter 3). The vortex sheet represen­t i n g the wall layer i s assumed to be bound to the body's surface with i t s strength determined from the c o n d i t i o n that the v e l o c i t y at the inner side of the surface must vanish. This procedure can be sought to be equivalent i n viscous flow to the s a t i s f a c t i o n of the non-slip c o n d i t i o n on the body's surface, by a wall l a y e r of vanishing thickness.

Because a l l the v o r t i c i t y i n the flow f i e l d gives non-zero c o n t r i b u -

120

tions to the stream function i n s i d e the computational domain, o r , i n other words, the i n t e g r a l representation f o r the stream function extends over the t o t a l v o r t i c a l region, the v o r t i c i t y outside the computational boundaries and contained i n the boundary layer on the forebody and the wake, has to be approximated.

The l a t t e r v o r t i c i t y f i e l d i s simply represented by N r i n g vortex c y l i n d e r s of constant strength textending from the downstream computational boundary to i n f i n i t y and having the correspondent r a d i i i d e n t i c a l to the computed vortex sheet r a d i i on the l a s t a x i a l s t a t i o n . In a s i m i l a r way, the former v o r t i c i t y f i e l d , i s s u b s t i t u t e d by a set of semi-i n f i n i t e r i n g vortex c y l i n d e r s with t h e i r strength matched to the v e l o c i t y p r o f i l e i n s i d e the boundary layer at the f i r s t s t a t i o n of the computa­t i o n a l domain. Such an approximation deviates considerably from the actual s i t u a t i o n on the forebody,in which a l l the v o r t i c i t y i s contained i n s i d e the boundary l a y e r growing from the body's nose. However, the flow around the forebody appears to have a small e f f e c t on the flow around the a f t e r ­body and the approximation can be accepted. Moreover, by extending the p a r a l l e l middle body, the nose can be put s u f f i c i e n t l y f a r upstream i n order to have a n e g l i g i b l e i n f l u e n c e on the afterbody flow. The h i s t o r y of the flow on the forebody i s brought into the computational domain through the upstream v e l o c i t y matching c o n d i t i o n .

The i t e r a t i v e procedure used to c a l c u l a t e the flow i n s i d e the compu­t a t i o n a l domain, adapted from Chapter 3 f o r t h i s p a r t i c u l a r case, can be summarized as follows:

1. From the p a r t i c u l a r vortex sheet d i s c r e t i z a t i o n adopted f o r the shear layer i n the upstream computational boundary, the values of the stream function for the vortex sheet stream-surfaces corrected for the presence of the body, are determined. "At the body's surface the stream function i s zero.

2. Neglecting any vortex sheet deformation i n the computational domain, the strength of the surface vortex sheet representing the wall layer i s determined from the co n d i t i o n that the inner v e l o c i t y tangent to the surface must vanish.

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3. From the values of the surface vortex sheet on the body's boundary,the stream function induced at the knots of the computational g r i d i s computed.

4. The streamlines corresponding to the outer vortex sheets are traced i n s i d e the computational domain.

5. The strength of the vortex sheets i s determined from the v o r t i c i t y t r a n s ­port equation.

6. The stream function induced at the knots of the computational grid,induced by the outer vortex sheets i s computed.

7. The a x i a l and r a d i a l v e l o c i t i e s induced at the body p i v o t a l points by the outer vortex sheets are computed.

8. The boundary co n d i t i o n at the body's surface i s s a t i s f i e d s o l v i n g the inner p o t e n t i a l flow problem f o r the strength of the surface vortex sheet.

Steps 3 to 8 are repeated u n t i l convergence i s attained.

The procedure described above has been applied to the c a l c u l a t i o n of

the flow i n the stern of a body of r e v o l u t i o n . The choice of the body was

d i c t a t e d p r i m a r i l y by considerations r e l a t i n g to the problem of i n t e r a c t i o n

with a ducted propeller, which i s treated i n the next section, and more s p e c i ­

f i c a l l y , by the need of having an afterbody with a considerably f u l l form.

Therefore, the afterbody i n v e s t i g a t e d by Huang et al (1976) and designated

by afterbody 3 i n the o r i g i n a l p u b l i c a t i o n , was chosen. The form of the

afterbody i s given by a cosine curve f o r the dimensionless body radius r /L:

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] 1 1 1 1 1 1 1 1 r r ; R L/Db=10.975

0.7 0.8 0.9 1.0 X /L

Fig. 4-2, Geometry of axisymmetrio body. {From Huang et al, 1976).

U D , E b 1 8182

LM / Db 7 6727

L A / D b 1 4836 0 646 0 416

C 0 862 p

TTD2/4L2 0 006521

1TDg/4S 0 02540 V2/3/ g 0 123 L/D b " 10 9745

S(m 2) 2 408

L(m) 3 066

Table 4-1. Particulars of axisymmetrio body (From Huang et al, 1976).

123

r / r m a x • E X P E R I M E N T S H U A N G (1976)

A P P R O X I M A T E VELOCITY 1 2 0 - P R O F I L E

i 1.16 -

1.12 -

1 0 8 -

1 .04 -

1 . 0 0 -

0 0 . 2 0 . 4 0 . 6 0 . 8 1.0 U / U 0

Fig. 4-3. D i s c r e t i z a t i o n of boundary layer velocity profile at the

p a r a l l e l middle body.

^ = a - b c o s ( C ~ * / L 7 r ) (4-1)

for 0.864811=x/L=0.977137 with a=0.025886, b=0.0196736, c=0.977137 and d=0.112326. The p a r t i c u l a r s of the afterbody and the h u l l model taken from Huang et a l (1976), are given i n Table 4-1 and shown i n F i g . 4-2.

A rather coarse g r i d has been applied i n the f i n a l c a l c u l a t i o n s using 10 a x i a l s t a t i o n s unequally spaced and 10 r a d i a l s t a t i o n s . The f i r s t a x i a l s t a t i o n was located, at the p a r a l l e l middle body at x/L=0.802 and the l a s t a x i a l s t a t i o n at x/L=1.10. The a x i a l s t a t i o n s on the stern were sel e c t e d at lo c a t i o n s coincident with p i v o t a l points on the body. The r a d i a l s t e p s i z e which was constant on each s t a t i o n , was adapted to the body's geometry by an equal p a r t i t i o n of the distance between the p i v o t a l point on the body and the outer computational boundary.

F i g . 4-3 shows the a x i a l v e l o c i t y p r o f i l e i n s i d e the boundary l a y e r measured by Huang et a l (1976), at a Reynolds number based on the body length

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of Re=5.9xl0 , and the d i s c r e t i z a t i o n into 5 vortex sheets applied at the f i r s t a x i a l s t a t i o n , x/L=0.802. It can be seen that the present d i s c r e t i z a ­t i o n assumed a v e l o c i t y at the edge of the wall layer equal to 0.667 the v e l o ­c i t y at the edge of the boundary l a y e r . We note that the v e l o c i t y at the edge of the boundary layer at the f i r s t a x i a l station,has been taken as reference v e l o c i t y . In the present case,the p o t e n t i a l flow c a l c u l a t i o n r e ­vealed that i t s d i f f e r e n c e with the free stream i s l e s s than 2 per cent.

As reported by Huang et a l (1976) , at the Reynolds number

Re=5.9xl0^ separation occurred on the afterbody at about x/L=0.92 followed

by reattachment at about x/L=0.98.

The computed pressure d i s t r i b u t i o n on the afterbody i s compared with

the p o t e n t i a l flow s o l u t i o n and the measured values on F i g . 4-4. It i s seen

that the present c a l c u l a t i o n agrees rather well with the experiments up to

the separation region. Also i t considerably d i f f e r s from the p o t e n t i a l flow

c a l c u l a t i o n .

C P • E X P E R I M E N T H U A N G E T A L ( 1 9 7 6 )

P O T E N T I A L F L O W

0 6 " P R E S E N T I N V I S C I D M E T H O D

R E A T T A C H M E N T ( H U A N G E T A L 1 9 7 6 )

- 0 . 6 0 8 0 0 8 2 0 8 4 0.86 0 8 8 0,90 0 9 2 0.94 0 .96 0 9 8 »00 X / L

Fig. 4-4. Pressure distribution on the afterbody. Re=5.9xl0 .

125

Fig. 4-5. Axial velocity profiles on the afterbody. Re=5.9xl0 .

The a x i a l v e l o c i t y p r o f i l e s are shown i n F i g . 4-5. Up to the separation region the agreement with the experimental p r o f i l e s i s good except near the wall, where, as expected,the c a l c u l a t e d v e l o c i t y p r o f i l e " f a i l s " to go to zero. For the two s t a t i o n s within the separation region the agreement i s poor also i n the outer part of the boundary la y e r . In the l a s t s t a t i o n , downstream of separation, the shape of the measured v e l o c i t y p r o f i l e on the outer part of the boundary l a y e r i s well described.

4.3. INTERACTION BETWEEN A DUCTED PROPELLER AND THE STERN

For the c a l c u l a t i o n of the flow f i e l d i n the stern region i n the presence of a ducted p r o p e l l e r , the disturbances i n the flow introduced by the p r o p e l l e r and duct need to be considered. As i n the previous Chapter, the duct i s represented by a surface v o r t i c i t y d i s t r i b u t i o n with the strength determined from the boundary co n d i t i o n on the surface and the p r o p e l l e r i s

126

represented by an actuator disk with v a r i a b l e r a d i a l load d i s t r i b u t i o n , which i n the present a p p l i c a t i o n , i s considered to be known.

The i t e r a t i v e scheme,employed i n the c a l c u l a t i o n of the stern flow

without propulsor i n s e c t i o n 4-3,is modified to account f o r the i n t e r a c t i o n

with p r o p e l l e r and duct,as follows:

- The duct i s treated together with the afterbody when s a t i s f y i n g the boun­dary conditions on both surfaces.

- The vortex sheets shed by the actuator disk i n the slips t r e a m are

treated together with the vortex sheets representing the boundary layer

and wake, but t h e i r strength i s determined from the i n v i s c i d v o r t i c i t y

transport equation f o r axisymmetric flow with s w i r l .

To s t a r t the i t e r a t i o n , the strength of the duct and afterbody vortex sheets i s computed i n the presence of the p r o p e l l e r disturbance flow,as given by the l i n e a r i z e d actuator disk model of Chapter 3.

A hypothetical ducted p r o p e l l e r c o n f i g u r a t i o n i s considered behind the afterbody of s e c t i o n 4-2. The p r o p e l l e r i s located at x/L=0.98 and a diameter of 54 per cent of the maximum diameter of the afterbody i s assumed.

DUCT B C/D=0.35, d = 20° MODIFIED NACA 6417

1.0

0.5

0 - 0 . 0 4

- 1 . 0 - 2 . 0 x / R - 0 . 7 4

Fig. 4-6. Afterbody with ducts A and B.

127

Fig.4-7. Computed pressure distributions on the afterbody.

In order to i n v e s t i g a t e the e f f e c t of duct angle of attack two d i f f e r e n t ducts have been considered: duct A with a p r o f i l e of the type NACA 6417 with a modified t r a i l i n g edge and set at an angle of attack of 15 degrees with

respect to the axi s ; duct B with an i d e n t i c a l p r o f i l e but at an angle of

attack of 20 degrees. Both ducts have a length-diameter r a t i o of 0.35 and

are placed ahead of the p r o p e l l e r plane with an a x i a l clearance of x/R=0.04. The two configurations behind the afterbody are shown i n F i g . 4-6.

The same boundary layer p r o f i l e at the upstream computational boundary g

and corresponding to the length Reynolds number Re=9.5xl0 has been assumed throughout the c a l c u l a t i o n s . A t y p i c a l parabolic c i r c u l a t i o n d i s t r i b u t i o n at the p r o p e l l e r disk has been used and 5 vortex sheets were taken to repre­sent the p r o p e l l e r slipstream.

128

C p

- 6 . 0

- 5 . 0 -

- 4 . 0 -

1.0 -

0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 x / C

Fig.4-8. Computed pressure distributions on duot A.

The r e s u l t s for the computed pressure d i s t r i b u t i o n s on the afterbody with duct without p r o p e l l e r and with the ducted p r o p e l l e r at two d i f f e r e n t p r o p e l l e r loadings are shown i n F i g . 4-7 f o r both con f i g u r a t i o n s . The p r o p e l l e r loading c o e f f i c i e n t s C T g and advance r a t i o s J g are based on the ship speed which i s approximated by the v e l o c i t y at the edge of the boundary layer at the f i r s t a x i a l s t a t i o n on the p a r a l l e l middle body. The correspondent pressure d i s t r i b u t i o n s on duct A and B are shown i n F i g s . 4-8 and 4-9, r e s p e c t i v e l y . The a x i a l v e l o c i t y p r o f i l e s along the afterbody down to the p r o p e l l e r plane f o r the two configurations are shown i n Figs . 4-10 and 4-11. For the sake of comparison the v e l o c i t y p r o f i l e s obtained f o r the stern alone, are included i n the Figures.

129

Fig. 4.9. Computed pressure distributions on duct B.

4.4. DISCUSSION OF THE RESULTS

Upon inspection of the r e s u l t s presented i n the previous section, some conclusions regarding the performance of the two configurations can already be drawn:

- Both ducts, when placed behind the afterbody without p r o p e l l e r , have a d e c e l e r a t i n g e f f e c t on the flow on the afterbody. This d e c e l e r a t i n g e f f e c t i s l a r g e r f o r the duct B which has a higher angle of attack than f o r the duct A. As a consequence, the pressure on the afterbody i s increased i n comparison with the case of the h u l l alone. This would lead to a decrease of the pressure r e s i s t a n c e component on the h u l l . On the other hand, a pressure drag force acting on the duct r e s u l t s from the i n t e r f e r e n c e between the duct and the afterbody. It must be borne i n mind that the dynamics of the wall l a y e r on the afterbody.which i s dominated

130

131

. 4-11. Axial velocity p r o f i l e s on the afterbody with duct B.

132

by the e f f e c t s of v i s c o s i t y and determinant f o r the occurrence of separa­ti o n , has been l e f t aside. Also, viscous e f f e c t s on the duct boundary layer are neglected. Accordingly, the conclusion reached above regarding the pressure on the afterbody could be d i f f e r e n t i f a c l e a r e f f e c t of the duct on separation at the stern i s implied.

Examination of the i n v i s c i d pressure d i s t r i b u t i o n on the ducts A and B without p r o p e l l e r reveals the p o s s i b l e occurrence of flow separation at the pressure minima on the outer side of the ducts.

- With i n c r e a s i n g p r o p e l l e r loading at C-r =0.95 and 2.50,both ducts d e l i v e r s

a thrust force. Duct A i s considerably more loaded than duct B and at &J S=2.50 i s operating near i t s optimum. Duct B appears to have an exces­s i v e l y large angle of attack i n t h i s p a r t i c u l a r range of loading conditions.

- At i n c r e a s i n g p r o p e l l e r loading,the pressure on the afterbody decreases

considerably,but the r e l a t i v e trends of the pressure d i s t r i b u t i o n f o r the

two ducted p r o p e l l e r c o nfigurations remain the same: the pressure de­

crease i s l a r g e r for the c o n f i g u r a t i o n with duct A than f o r the configura­

t i o n with duct B. A tendency for the attenuation of the d i f f e r e n c e , a t the

highest p r o p e l l e r loading,can be discerned.

From the previous r e s u l t s the conventional parameters characterizing the

i n t e r a c t i o n between the ducted p r o p e l l e r and the h u l l i . e . the thrust deduc­

t i o n , wake f r a c t i o n together with t h e i r d i s t r i b u t i o n s and the thrust r a t i o

parameter can be estimated.

For bodies f i t t e d with ducted p r o p e l l e r s there are, i n p r i n c i p l e , two

d i f f e r e n t methods of analyzing the propulsion f a c t o r s thus, of

d e f i n i n g the thrust deduction and the wake f r a c t i o n : the duct can be

considered as a part of the h u l l or as a part of the propulsor. Both methods

w i l l be considered.

- The thrust deduction.

Since the f r i c t i o n a l forces a c t i n g on the body and the duct are neglected,

we r e s t r i c t our d i s c u s s i o n to the pressure component of the thrust deduc­

t i o n . Huang et a l (1976) , found that the f r i c t i o n a l part of the

thrust deduction amounts to l e s s than 5% of the pressure component f o r

the p a r t i c u l a r axisymmetric bodies considered.

133

If the duct i s considered as part of the h u l l , the thrust deduction

f r a c t i o n i s defined as

p hd (4-2)

where T i s the p r o p e l l e r t h r u s t , R, , i s r e s i s t a n c e of body plus duct, p hd Huang et a l (1976), obtained rather good r e s u l t s f o r the thrust deduction on the axisymmetric bodies with a conventional p r o p e l l e r by i n t e g r a t i n g the pressure d i f f e r e n c e on the afterbody f o r the two cases with and without p r o p e l l e r . Using the same procedure,the d i f f e r e n c e T -R, . can be

p hd estimated as follows:

r p hd

max 2TT ƒ Ap, r d r - T ^ + D ^ ,

0 r l d dh' (4-3)

where T i s the duct thrust i n the presence of the p r o p e l l e r , D i s the d dh drag on the duct i n the presence of the h u l l and i n the absence of the p r o p e l l e r , and Ap^ the d i f f e r e n c e of pressure on the afterbody between the cases of h u l l with duct and h u l l with ducted p r o p e l l e r . Introducing non-dimensional variables,we have

T _ 2 W R ) 2 > * / R

P l C T 0 AC . r / r . d ( r / r )• p^ max max

c -c d s dh (4-4)

where

2 2 kpV TTR

s

(4-5)

"T 2 2 dS JjpVgTTR (4-6)

d h d h

ï j p V g T r R 2

(4-7)

134

AC = C -C (4-8) p l p h u l l + d u c t p h u l l + d u c t e d p r o p .

and C i s the pressure c o e f f i c i e n t . P

If the duct i s considered as a part of the propulsor, the thrust deduc­t i o n f r a c t i o n i s defined as

T +T,-R, = p d h

Po T +T, ' 1 J )

^2 p d

where R i s the re s i s t a n c e of the h u l l alone. The d i f f e r e n c e T +T ,-R, can be h p d h

estimated as r max

T +T,-R, = 2TT ƒ A p „ r d r , (4-10) p d h Q F 2

where Ap^ i s the d i f f e r e n c e between the pressure on the afterbody of the h u l l alone and the pressure on the afterbody with ducted pro­p e l l e r . In terms of dimensionless quantities,we obtain f o r the thrust deduc­t i o n r a t i o

2 ( r / R ) 2 r m a x / R

t = - — 5 ^ ƒ AC ( r / r ) d ( r / r ) (4-11) p 2 c T + C T Q P2 m a x m a x

s ds

The e f f e c t i v e wake

If the duct i s considered as a part of the h u l l , the e f f e c t i v e v e l o c i t y d i s t r i b u t i o n i s defined as

u ( r ) = u n ( r ) - u ( r ) (4-12)

where uQ ( r ) and " ^ ( r ) are, r e s p e c t i v e l y , the t o t a l and p r o p e l l e r induced

v e l o c i t i e s at the p r o p e l l e r plane.

If the duct i s regarded as a part of the propulsor, the e f f e c t i v e v e l o ­c i t y i s defined as

u g ( r ) = u Q ( r ) - u p ( r ) - u d ( r ) (4-13)

135

whereu (r) i s the v e l o c i t y induced by the duct at the p r o p e l l e r plane, d

The effective wake i s evaluated by i n t e g r a t i n g over the p r o p e l l e r disk

2 1

w = 1- - ƒ u ( r ) r d r (4-14) e , 2 e h h

where r. i s the non-dimensional hub radius, h

By considering the vortex systems representing the h u l l , i t s boundary layer and wake, the duct and the p r o p e l l e r , the correspondent induced ve l o ­c i t i e s at the p r o p e l l e r plane can be evaluated and the e f f e c t i v e wake d i s -

1 D U C T P A R T O F T H E H U L L

2 D U C T P A R T O F T H E P R O P U L S O R

E F F E C T I V E W A K E _ _ - J s = 0 . 6 6 . C T s = 0 . 9 5 1* . , = 0 . 2 0 . W . 2 = 0 2 2

J s = 0 . 5 0 . C T , = 2 . 5 0 . W „ = - 0 . 0 4 . W„ 2 =0 .17

N O M I N A L W A K E _

Fig. 4-12. Nominal and effective wake distributions. Configuration

with duct A.

136

Fig. 4-15. Nominal and effective wake distributions. Configuration

with duct B.

t r i b u t i o n s can be c a l c u l a t e d . For the two configurations with duct A and B at the d i f f e r e n t p r o p e l l e r loadings,they are shown i n F i g s . 4-12 and 4-13. In the f i g u r e s the nominal wake d i s t r i b u t i o n s f o r the h u l l with and without duct and the duct induced v e l o c i t i e s are included as w e l l .

It can be seen that the e f f e c t i v e wake d i s t r i b u t i o n s when the duct i s regarded as a part of the propulsor have the same type of behaviour as the e f f e c t i v e wake d i s t r i b u t i o n s found i n s i m i l a r studies f o r conventional screws (Huang et a l , 1980, Dyne, 1981), with i t s greatest d i f f e r e n c e i n r e l a t i o n to the nominal d i s t r i b u t i o n near the hub. The e f f e c t i v e wake d i s t r i b u t i o n s , when the duct i s considered as a part of the h u l l , are considerably d i f f e -

137

rent, as expected, because of the e f f e c t of the duct induced v e l o c i t i e s . The

greater s e n s i t i v i t y of the e f f e c t i v e wake d e f i n i t i o n (4-12) with the propel­

l e r loading i n comparison with the d e f i n i t i o n (4-13) i s remarkable.

The r e s u l t s of the thrust deduction computations from equations (4-4)

and (4-11), the wake f r a c t i o n s from equations (4-12, (4-13), (4-14) and

the thrust r a t i o parameter are given i n Table 4-2. It can be seen that, when considering the duct as a part of the h u l l ,

the h u l l e f f i c i e n c y appears to be rather i n s e n s i t i v e to v a r i a t i o n s of pro­p e l l e r and duct loading. Yet, the correspondent thrust deduction and e f f e c ­t i v e wake f r a c t i o n s are strongly a f f e c t e d by the v a r i a t i o n of p r o p e l l e r loading. When considering the duct as a part of the propulsor, the h u l l e f f i c i e n c y decreases with p r o p e l l e r loading which i s p r i m a r i l y associated with the decrease of wake f r a c t i o n .

1 DUCT AS A PART

OF THE HULL

2 DUCT AS A PART

OF THE PROPULSOR

CASE t P

w e

1-t n = E t

P W e

1-t n P

T T- P CASE t

P w e

t P

W e h 1-w e

T +T , P d

DUCT A

J =0.66 s C =0.95 T

s 0.22 0.20 0.98 0.15 0.22 1.09 0.86

DUCT A J =0.50 s C =2.50 T

s 0.00 -0.04 0.96 0.15 0.17 1.02 0.79

DUCT B

J =0.66 s C =0.95 T

s 0.25 0.23 0.98 0.09 0.25 1.22 1.00

DUCT B J =0.50 s

C =2.50 T s 0.02 0.01 0.98 0.13 0.17 1.04 0.83

Table 4-2. Computed thrust deduction, wake fraction and thrust

ratio for the two afterbody-ducted propeller configurations.

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5. D u c t e d p r o p e l l e r d e s i g n

5.1. INTRODUCTION

In the previous Chapters various flow models f o r the evaluation of ducted p r o p e l l e r performance both i n uniform flow and r a d i a l l y v a r i a b l e inflow have been i n v e s t i g a t e d . E f f e c t s of v i s c o s i t y on the duct flow and e f f e c t s of heavy loading on the i n t e r a c t i o n between p r o p e l l e r and duct have been considered. Most of the computational procedures developed to account for such e f f e c t s are i t e r a t i v e and rather time consuming.

In many cases, for ducts with a d i f f u s e r downstream of the p r o p e l l e r ,

e f f e c t s of slip s t r e a m c o n t r a c t i o n can be small even at heavy loadings.

In the axisymmetric model f o r the i n t e r a c t i o n between p r o p e l l e r and duct,other non-linear e f f e c t s such as the a x i a l v a r i a t i o n of vortex p i t c h i n the p r o p e l l e r slipstream can be accounted f o r , i n an approximate manner, by a s u i t a b l e choice of the vortex p i t c h (assumed constant i n the a x i a l d i r e c ­t i o n at each radius) . The simple l i n e a r i z e d actuator disk model considered i n Chapter 2 incorporates these features and i s well su i t e d f o r design purposes .

Although the i n t e r a c t i o n between p r o p e l l e r and duct can be treated assu­

ming axisymmetric flow, when designing the p r o p e l l e r f i n i t e blade number

e f f e c t s need to be considered.

In t h i s Chapter,Lerbs'(1952) induction f a c t o r method f o r wake adapted conventional p r o p e l l e r s i s used to design the p r o p e l l e r i n s i d e a given duct together with the simple axisymmetric propeller-duct i n t e r a c t i o n model men­tioned above. The e f f e c t s on the design of various assumptions regarding the choice of the vortex p i t c h i n the i n t e r a c t i o n between p r o p e l l e r and duct are inv e s t i g a t e d .

139

5.2. DESIGN PROCEDURE

In the design method the f o l l o w i n g assumptions have been made:

- An axisymmetric duct and hub c o n f i g u r a t i o n i s represented by a surface v o r t i c i t y method on the assumption of i n v i s c i d and incompressible i r r o t a -t i o n a l flow (see Chapter 2).

- When accounting f o r the i n t e r a c t i o n between p r o p e l l e r and duct the propel­l e r i s represented by an actuator disk with a r a d i a l d i s t r i b u t i o n of c i r c u l a t i o n i d e n t i c a l to the f i n i t e bladed p r o p e l l e r .

- The duct i n v i s c i d thrust i s c a l c u l a t e d by i n t e g r a t i o n of the pressure d i s ­t r i b u t i o n and the disk i n v i s c i d thrust from Kutta-Joukowsky law.

- The p r o p e l l e r i s designed by Lerbs' induction f a c t o r method,(1952). In the p r o p e l l e r l i f t i n g l i n e c a l c u l a t i o n s the p i t c h of the vortex l i n e s i s deter mined at the p r o p e l l e r plane under the i n f l u e n c e of the average duct induced v e l o c i t i e s . A t i p clearance i s assumed and the p r o p e l l e r blade c i r c u l a t i o n i s considered to vanish at the hub and the t i p .

- F i n i t e blade propeller-duct i n t e r f e r e n c e e f f e c t s are neglected. The inf l u e n c e of the f i n i t e blade number i n t e r f e r e n c e e f f e c t s on the hydro-dynamic p i t c h has been accounted f o r , i n an approximate way, by Minsaas (1978),by m i r r o r i n g the p r o p e l l e r h e l i c a l vortex sheets on the c y l i n d e r prolonging the inner surface of the duct. He concludes that t h i s e f f e c t i s l o c a l and tends to reduce the p i t c h towards the t i p .

- Viscous e f f e c t s are accounted f o r by c a l c u l a t i n g the viscous drag on the duct and s p e c i f y i n g the blade s e c t i o n drag c o e f f i c i e n t .

- Corrections to camber from l i f t i n g surface theory as deduced by Morgan et a l (1968),for conventional p r o p e l l e r s may be applied.

Given the t o t a l thrust, p r o p e l l e r r e v o l u t i o n rate,propeller diameter and mean speed of advance, wake d i s t r i b u t i o n , duct and hub shapes, number of blades, blade c i r c u l a t i o n d i s t r i b u t i o n and c a v i t a t i o n safety margins, the method determines the thrust on p r o p e l l e r and duct, the duct and hub pres­sure d i s t r i b u t i o n s , the p r o p e l l e r blade area and the camber, thickness and p i t c h of the p r o p e l l e r blade s e c t i o n s .

The design procedure can be considered to be performed i n three steps:

1. From an i n i t i a l estimate of the thrust r a t i o between p r o p e l l e r and

140

duct and an allowance f o r viscous e f f e c t s on the p r o p e l l e r thrust, an i t e r a ­t i o n between p r o p e l l e r and duct i s performed using the actuator disk model to determine the time-mean p r o p e l l e r induction on the duct. Each i t e r a ­t i o n step departs from the p r o p e l l e r i n v i s c i d t h r u s t , c a l c u l a t e s the p r o p e l l e r r i n g vortex system, the p r o p e l l e r induced v e l o c i t i e s on duct and hub, duct and hub pressure d i s t r i b u t i o n , duct i n v i s c i d thrust and duct viscous drag. The i t e r a t i o n s are continued u n t i l convergence on the thrust r a t i o i s achieved. From the strength of the duct-hub vortex d i s t r i ­bution the a x i a l v e l o c i t i e s at the p r o p e l l e r plane can be c a l c u l a t e d .

2. With the a x i a l v e l o c i t i e s induced by the duct and hub incorporated i n the inflow v e l o c i t i e s to the p r o p e l l e r , the p r o p e l l e r i n v i s c i d thrust and c i r c u l a t i o n d i s t r i b u t i o n , the l i f t i n g l i n e c a l c u l a t i o n s are performed g i v i n g the p r o p e l l e r s e l f - i n d u c e d v e l o c i t i e s and the hydrodynamic p i t c h .

3. The d i s t r i b u t i o n s of chord length, camber and thickness of the p r o p e l l e r blade sections are c a l c u l a t e d on the basis of c a v i t a t i o n and strength requirements and c o r r e c t i o n f a c t o r s from l i f t i n g surface theory.

The p r o p e l l e r t h r u s t , torque and e f f i c i e n c y are f i n a l l y c a l c u l a t e d taking into account the viscous drag of the blade se c t i o n s .

A f i n a l check on the assumed i n v i s c i d thrust f o r the pr o p e l l e r - d u c t

i n t e r a c t i o n c a l c u l a t i o n s i s performed and, i f necessary, the whole design

computation i s repeated.

M o d i f i c a t i o n s of t h i s basic design procedure are required i f the design i s to be based on given hydrodynamic p i t c h or given power. In such case, the s t a r t i n g point i s an estimate of the duct induced v e l o c i t i e s at the p r o p e l l e r plane which are d i r e c t l y r e l a t e d to the thrust r a t i o . Next, the p r o p e l l e r i s designed to match the thrust or torque requirements and the c i r c u l a t i o n d i s t r i b u t i o n on the p r o p e l l e r blades i s determined as a r e s u l t .

The p r o p e l l e r induced v e l o c i t i e s on the duct are computed from the actuator disk approximation and the duct thrust and the duct induced v e l o ­c i t i e s on the p r o p e l l e r plane are determined. The i t e r a t i o n between propel­l e r and duct proceeds u n t i l convergence on the duct induced v e l o c i t i e s i s obtained.

Both design procedures are summarized i n the flow charts of F i g s . 5-1

and 5-2.

141

INPUT DESIGN DATA SM IP SPEED. PROPELLER REVOLUTIONS,

PROPELLER DIAMETER . WAKE DISTRIBUTION. TOTAL THRUST, CIRCULATION DISTRIBUTION, NUMBER OF BLADES DUCT AND HUB

GEOMETRY

DESIGN INPUT DATA : SHIP SPEED I PROPELLER REVOLUTIONS

PROPELLER DIAMETER , WAKE DISTRIBUTION TOTAL THRUST (POWER) ,HYDROOYNAMIC PITCH DISTRIBUTION NUMER OF B L A D E S .

DUCT AND HUB GEOMETRY.

FIRST ESTIMATE PROPELLER IDEAL THRUST

ACTUATOR DISK MODEL PROPELLER STEADY INDUCED VELOCITIES

ON THE DUCT

CALCULATION OF DESIGN PREDICTIONS BLADE SECTION DRAG.

FINAL CHECK ON PROPELLER THRUST.

ESTIMATE OF DUCT INDUCED VELOCITIES AT THE PROPELLER

P L A N E

PROPELLER LIFTING LINE MODEL OESIGN FOR ASSUMED P R O P E L L E R THRUST

(OR GIVEN TORQUEi CALCULATION OF CIRCULATION DISTRIBUTION

AND PROPELLER IDEAL THRUST

ACTUATOR DISK MODEL PROPELLER STEADY INDUCED VELOCITIES

ON THE DUCT

DUCT SURFACE VORTICITY MODEL DUCT INDUCED VELOCITIES ON THE PROPELLER

DUCT THRUST AND DUCT DRAG

B L A D E SECTION DESIGN CAVITATION AND STRENGTH CALCULATION

CALCULATION OF DESIGN PREDICTIONS

Fig. 5-1. Flow chart of design for Fig. 5-2. Flow chart of design for

given thrust and circula- given thrust (or power) and

Hon distribution. hydrodynamic pitch distribu­

tion.

5.3. PROPELLER INDUCED VELOCITIES

In Lerbs* induction f a c t o r method the p r o p e l l e r i s represented by l i f t i n g l i n e theory, under the assumption that the p r o p e l l e r i s mode­r a t e l y loaded.

Accordingly, the t r a i l i n g v o r t i c e s of h e l i c a l shape, shed from the

l i f t i n g l i n e s , are considered to l i e on c y l i n d e r s of constant radius and

have a constant p i t c h i n the a x i a l d i r e c t i o n . If G(r)=r(r)/7TDV A denotes the

142

non-dimensional c i r c u l a t i o n d i s t r i b u t i o n of the l i f t i n g l i n e s , with denoting the mean advance v e l o c i t y and D the p r o p e l l e r diameter, the a x i a l and t a n g e n t i a l p r o p e l l e r induced v e l o c i t i e s at the l i f t i n g l i n e s are given by

U p ( r ) 1 \ . dG d r '

r h

and

! P U ) 1 \ • ^ d r '

r h

where r i s the non-dimensional radius coordinate, r, i s the hub radius and h

i and i ^ are, r e s p e c t i v e l y , the a x i a l and t a n g e n t i a l induction f a c t o r s . The induction f a c t o r s are dependent on the r a t i o r / r ' , the number of

blades and the hydrodynamic p i t c h angle $ . Expressions f o r i t s evaluation have been r e f i n e d by Wrench (1957), and are given by Van Oossanen (1973).

The hydrodynamic p i t c h i s P. ~ = Tfr t a n B ± (5-3)

EL i s given by ( F i g . 5-3),

V g ( r ) ud ( r ) u p ( r )

v 7 ~ + ~\T + v t a n e ± = —±- ^ (5-4)

IT

at the l i f t i n g l i n e s .

The hydrodynamic p i t c h f a r downstream i s

V ( r ) u ( r ) e + 2-2-

V V A t a n g i • w~TrT ( 5 ' 5 )

°° 77 - i P J V A

The a x i a l wake v e l o c i t y V (r) i s e

143

Fig. 5-3. Definition of hydrodynamie pitch angles and velocities at a

propeller blade section.

V ( r ) = (1-w ( r ) ) V (5-6) e e s

where V i s the ship speed and w (r) the e f f e c t i v e wake f r a c t i o n at radius r. s e The mean wake v e l o c i t y i s V ^ = ( 1 - W

T ) VS w n e r e wT i s the Taylor wake f r a c t i o n ,

u.(r) i s the duct induced a x i a l v e l o c i t y and J=V./nD i s the advance r a t i o , d A

The v e l o c i t i e s induced by the p r o p e l l e r on the duct and hub are obtained from the v e l o c i t y f i e l d induced by an actuator disk with the same load d i s ­t r i b u t i o n as the f i n i t e bladed p r o p e l l e r . They are given i n Chapter 2 (eq. 2-54 and 2-55) and r e c a l l e d here f o r the sake of completeness

u 1 1 d G

= - 1 r ' t a n g . d T ^ V A ( x ' r ' r ' ) d r ' ' <5"7> h

1 1 d G c o k = - 1 r ' t a n g . d r ^ V x ' r ' r ' ) d r ' , (5-8) A r h

144

where G (r) i s the dimensionless c i r c u l a t i o n on the actuator disk. For the o o x '

meaning of the function V and V we r e f e r to Chapter 2. A R

When accounting f o r the i n t e r a c t i o n between p r o p e l l e r and duct the pr o p e l l e r i d e a l thrust i s determined from Kutta Joukowsky law:

TT T* CO

C T = 4 1 " I - ) G o o d r f ( 5 " 9 )

P i r h 1 2 2

where C T =T /(TTPV.TIR ) . P i P i 2 A If the c i r c u l a t i o n d i s t r i b u t i o n f o r the f i n i t e bladed p r o p e l l e r and

the p r o p e l l e r i d e a l thrust are known e i t h e r from the design input data or from the l i f t i n g l i n e c a l c u l a t i o n s , eq. (5-9) gives the strength of the c i r c u l a t i o n d i s t r i b u t i o n on the actuator disk.

To complete the evaluation of p r o p e l l e r induced v e l o c i t i e s on the duct, the p i t c h angle i n eq.(5-7) and (5-8) s t i l l has to be known. A simple assumption which seems to work rather w e l l , i s to consider that the p i t c h i s determined i n the ultimate wake (see Chapter 2, eq. 2-65):

oo

t a n 6, = ~ - (5-10) CO Tf JJ" oo

Other p o s s i b i l i t i e s for basing the p i t c h and t h e i r i n f l u e n c e on the design r e s u l t s w i l l be i n v e s t i g a t e d l a t e r i n t h i s Chapter.

5.4. DUCT THRUST AND DUCT INDUCED VELOCITIES

With the knowledge of the v e l o c i t i e s induced by the p r o p e l l e r on the

duct and hub, the s o l u t i o n of the i n t e g r a l equation expressing the boundary

co n d i t i o n on the surface y i e l d s the duct and hub surface v o r t i c i t y d i s t r i b u ­

t i o n Y(s) • The pressure d i s t r i b u t i o n on the duct i s

V

C (s) = ( ^ d ) 2 - ( ï l ^ ) 2 (5-11) p d A A

where V i s the v e l o c i t y at i n f i n i t y upstream on the same stream-surface. e d

145

The duct i n v i s c i d thrust i s

d. A l If flow separation does not occur on the duct, the duct viscous drag i s rather small compared with the t h r u s t . As f o r two-dimensional a i r f o i l s the duct's viscous drag c o e f f i c i e n t can be estimated as

C D = [ l + 2 ( c ^ ) 2 + 6 0 ( - ^ ) 4 ] (2C ) (5-13) d

where

d 2 p - V m - 2 ï ï R d C

and D, i s the duct viscous drag, t , i s the maximum thickness of the duct, c d d i s the duct length, R. i s the duct radius, V i s a mean v e l o c i t y between d m the outside and i n s i d e of the duct and i s the f l a t plate s k i n - f r i c t i o n

c o e f f i c i e n t . The s k i n - f r i c t i o n c o e f f i c i e n t i s a f u n c t i o n of a duct Reynolds

number defined as

c V R e c = (5-15)

where V i s the kinematic v i s c o s i t y .

The t o t a l thrust i s

V = c t „ " c d „ < v f ) 2 - ( ! r > - ( 2 R > ( 5 " 1 6 )

d d. d A l The v e l o c i t i e s induced at the p r o p e l l e r plane by the duct and hub configura­

t i o n can be c a l c u l a t e d from the surface v o r t i c i t y d i s t r i b u t i o n . Only the

a x i a l v e l o c i t y i s of i n t e r e s t f o r the design of the p r o p e l l e r with the

induction f a c t o r method

u , = ƒ u ( - x ' ; r , r ' ) I ( s ' ) d s ' (5-17)

A S,+S, ' A d h

146

5.5. THE DESIGN WITH THE INDUCTION FACTOR METHOD

When the duct induced v e l o c i t i e s at the p r o p e l l e r plane become known, the p r o p e l l e r can be designed i n a s i m i l a r way as f o r conventional wake-adapted p r o p e l l e r s to match the required thrust or power.

From Kutta-Joukowsky law, the p r o p e l l e r i d e a l thrust c o e f f i c i e n t f o r a

f i n i t e bladed p r o p e l l e r i s

1 w C T = 4Z ƒ G ( r ) (J£ - = E)dr (5-18)

p. r, A r i h and the p r o p e l l e r i d e a l power c o e f f i c i e n t i s

A r, 1 1-w ( r ) u u , C = i ^ Z ƒ G ( r ) ( § + =E + ^ ) r d r (5-19)

p i l - w T A A

with

C p = P / ( ^ p V ^ r r R 2 ) . i

In the design with given c i r c u l a t i o n d i s t r i b u t i o n G(r)=kF(r), eq.(5-18) or eq. (5-19) are used to determine the c i r c u l a t i o n strength k i n the way i n d i c a t e d by Lerbs (1952). With the p r o p e l l e r s e l f - i n d u c e d v e l o c i t i e s given by eq. (5-1) and (5-2) and s u b s t i t u t e d i n (5-18) or (5-19), the c i r c u l a t i o n d i s t r i b u t i o n and the induction f a c t o r s are expanded i n a F o u r i e r s e r i e s and a quadratic equation for k r e s u l t s . Since, at the outset, the induction f a c t o r s are not known, fo r the hydrodynamic p i t c h has not yet been deter­mined, a method of successive approximation i s u s e d , s t a r t i n g , as a f i r s t approximation, with tanB =tan|3.

For the design with given hydrodynamic p i t c h , from the v e l o c i t y diagram

at a p r o p e l l e r blade s e c t i o n ( F i g . 5-1), the f o l l o w i n g r e l a t i o n i s derived

147

u u , w t a n B . V v f + v f + t a n B . ( ^ ) = (5-20) A A A A

With 3 s p e c i f i e d i n advance, when s u b s t i t u t i n g (5-1) and (5-2) i n (5-20),

the f o llowing i n t e g r a l equation f o r the c i r c u l a t i o n d i s t r i b u t i o n i s obtained

1 , , , t a n g . V u , ƒ ^ ( i + t a n @ . i J _ ) ^ - r =2(- - j i - 1 ) ^ - 2.-y3. (5-21) d r ' a I t r - r 1 t a n g V, V, r , A A h Also, the s o l u t i o n of t h i s equation i s performed i n the way suggested

by Lerbs. A f t e r expansion i n Four i e r s e r i e s of the induction f a c t o r s and the c i r c u l a t i o n d i s t r i b u t i o n , a set of l i n e a r equations i n the Four i e r c o e f f i ­c i e n t s of the c i r c u l a t i o n d i s t r i b u t i o n i s obtained and the c i r c u l a t i o n values are e a s i l y determined at the various p r o p e l l e r r a d i i . In order to obtain agreement on the required i d e a l thrust or power given by (5-18) and (5-19) with the c a l c u l a t e d c i r c u l a t i o n d i s t r i b u t i o n , the previous computa­t i o n has to be performed at d i f f e r e n t values of the constant f a c t o r k

P defined by

t a n g . k f ( r ) = -—=ji (5-22) p p t a n g

where f (r) i s the assumed d i s t r i b u t i o n of hydrodynamic p i t c h . P

The f i n a l value of k can be obtained by i n t e r p o l a t i o n on the assumed P

k values. In both design cases, when the c i r c u l a t i o n d i s t r i b u t i o n becomes P

known, the l i f t forces a c t i n g on the blade sections given by the product of the blade sec t i o n l i f t c o e f f i c i e n t C and the sect i o n chord c are c a l c u l a t e d from

C c 2TTG c o s g . — = 1 Wp (5-23)

t a n g V A

With the values of C^c/D, -the hydrodynamic p i t c h g and the a x i a l and

tan g e n t i a l induced v e l o c i t i e s , the determination of the chord length, camber,

thickness and p i t c h of the p r o p e l l e r blade sections can be performed on the

basis of c a v i t a t i o n c r i t e r i a and a strength c a l c u l a t i o n .

148

When c a l c u l a t i n g the c a v i t a t i o n number at the various p r o p e l l e r r a d i i one has to consider the decrease of s t a t i c pressure induced by the duct and hub. Assuming the p r o p e l l e r to have an i n f i n i t e number of blades and apply­ing B e r n o u l l i equation between i n f i n i t y upstream and the p r o p e l l e r plane, we obtain, neglecting the t a n g e n t i a l v e l o c i t i e s , ( F i g . 5-4):

P 0 - P l rV r ) . l U ~ ( r ) +

U d ( r )n 2 rVr,-,2

7 — 2 - - [-yT + 2 ^ T — + - v — J * L - v — ] ( 5 " 2 4 )

^ p V A A A A A

Applying B e r n o u l l i equation between the p r o p e l l e r plane and i n f i n i t y

downstream we get

PQ-P 2 _ R V R ) l U » ( r ) u d ( r )l 2 r

V e ( r ) u « ( r ^ l 2

:—a— l~v— + 2~v— + — J ~ L ~ V — + ~ v — - 1 ( 5 _ 2 5 )

hpV^ A A A A A

The mean s t a t i c pressure p , at the p r o p e l l e r plane i s me an

P 2 - P l

Pmean = P 2 " ( 5 " 2 6 )

Using (5-24), (5-25) and (5-26), the mean s t a t i c pressure becomes

v 2 u d ( r )r

V e ( r ) 1 u d ( r ) 1 5 - ( r ,n „2 l R V R )

L 2

A A A A A

Fig. 5-4. Definition of the flow through an i n f i n i t e l y bladed ducted

propeller.

149

The l a s t term represents the mean pressure decrease f o r an open p r o p e l l e r . Therefore, the induced pressure decrease due to the duct i s

u (r) V (r) u (r) u ^ r ) A pd = p VA - V ^ + I "V+1 (5_27)

which coincides with the expression given by Dyne (1967).

The c a v i t a t i o n number i s

p - e - A p , - r R p g o ( r ) = — ~ (5-28)

where p i s the s t a t i c pressure at the centre l i n e of the p r o p e l l e r shaft, o e i s the vapour pressure at the p r e v a i l i n g temperature, p i s the water density, g the g r a v i t y a c c e l e r a t i o n and V i s the r e s u l t a n t v e l o c i t y to the blade se c t i o n at radius r :

w i r r p J V A

V = - (5-29) A COSg. 1

If m i s the margin against c a v i t a t i o n , then the minimum pressure c o e f f i c i e n t C of the blade sections i s given by p . min

C = m ( r ) a ( r ) (5-30) p m i n

The r e l a t i o n between the minimum pressure c o e f f i c i e n t f o r shock-free entrance at the blade s e c t i o n and the q u a n t i t i e s C^c/t and i s dependent on which type of camber and thickness d i s t r i b u t i o n i s adopted f o r the p r o p e l l e r sections. In the design method used by Van Gent and Van Oossanen (1973), f o r open p r o p e l l e r s , a Walchner "set B" thickness d i s t r i b u t i o n with p a r a b o l i c camber l i n e was applied, from about r=0.5 to the p r o p e l l e r t i p , while a Gutsche type of p r o f i l e i s adopted from r=0.5 down to the hub. For the modi­f i e d Walchner p r o f i l e the f o l l o w i n g r e l a t i o n holds at shock-free entrance;

C C C c C =(0.622-£- + 2 . 7 4 ) - - [~0.2 ( - i f - ) 2 + l . 04] (-) 2 (5-31) p . t c L t - c

where t i s the maximum thickness of the blade s e c t i o n . Eq.(5-31) y i e l d s f o r

150

the chord c

c = t=—i { 0 . 3 3 1 C T c + 1.37t +

+ / ( 0 . 3 1 1 C T c + 1 3 7 t ) 2 + C [0.2 ( C T c ) 2 + 1 0 4 t 2 l } (5-32) • ^ r t i i n

A strength c a l c u l a t i o n i s required to give an a d d i t i o n a l r e l a t i o n f o r the chord and maximum thickness of the blade sections. In the present method a s i m p l i f i e d strength a n a l y s i s i s used and such r e l a t i o n i s given by

t 2 . c = W(r) ( 5 _ 3 3 )

0.087 c o s e

where W(r) i s the se c t i o n modulus of the blade s e c t i o n at radius r and e i s the rake angle. The se c t i o n modulus i s c a l c u l a t e d by the fol l o w i n g formula

2 2^5 1 C T c c o s 2 ( B ! - B ) W ( r ) = p " n ƒ (-£-) D ' 2n ! a T r h c o s 6

c o s ^ - g ^ ) r ' ( r ' - r ) d r ' (5-34)

where a i s the maximum allowable t e n s i l e s t r e s s l e s s the t e n s i l e s t r e s s due T to c e n t r i f u g a l forces and 3^ i s the value of 3^ at r=r'.

The l i f t i n g surface c o r r e c t i o n s derived from open-propeller s e r i e s are not v a l i d f o r ducted p r o p e l l e r s due to the a d d i t i o n a l e f f e c t s induced by the duct loading and thickness. However, lacking more appropriate c o r r e c t i o n f a c t o r s , one may re s o r t to i t s a p p l i c a t i o n when designing a ducted p r o p e l l e r . In such cases, the geometric camber of the blade sections i s c a l c u l a t e d by

C L C f n = K — — (5-35)

0 C 4 T T C 0

where K i s the l i f t i n g surface c o r r e c t i o n f a c t o r to camber due to loading c derived by Morgan et a l (1968). Polynomials f o r i t s evaluation as function

of the hydrodynamic p i t c h , expanded blade area r a t i o and radius of the pro-

151

p e l l e r blade s e c t i o n (r=0.3 up to r=0.9) are given by Van Oossanen (1973),

fo r p r o p e l l e r s with 4, 5 and 6 blades and zero skew. i s a c o r r e c t i o n

f a c t o r f o r the e f f e c t of p r o f i l e thickness on the l i f t and i s given by

C = l + 0,4TTT- + (-)21 o L c c J (5-36)

The f i n a l p i t c h angle i s obtained as

(5-37)

where a i s a p i t c h c o r r e c t i o n given by

a = 0.10 C T

Li (5-38)

The d e l i v e r e d thrust of the p r o p e l l e r i s

T (5-39)

and the required torque i s

(5-40)

5.6. RESULTS AND DISCUSSION

For the purpose of checking the basic assumptions used i n the design

method with respect to the i n t e r a c t i o n between p r o p e l l e r and duct, an

e x i s t i n g p r o p e l l e r o r i g i n a l l y designed by the method of Van Manen and

Superina (1959), to operate i n s i d e the duct 19A i n a wake f i e l d , has been

redesigned by the present method.

In the design method proposed by Van Manen and Superina, the p r o p e l l e r

i s designed according to a x i a l flow pump design theory. The v e l o c i t i e s indu­

ced by the duct on the p r o p e l l e r are c a l c u l a t e d assuming an e l l i p t i c load

d i s t r i b u t i o n on a c y l i n d e r with the same length and inner diameter as the

152

duct and the p r o p e l l e r induced v e l o c i t i e s are c a l c u l a t e d assuming an i n f i n i t e number of blades.

Table 5-1 gives the input design data and the wake d i s t r i b u t i o n .

r 1-•w (r) e m a T

(Nnf 2 )

0.20 0 362 1 0 5 2 x l O 7

0.30 0 404 1 0 5 5 x l O 7

0.40 0 431 1 0 5 7 x l O 7

0.50 0 444 1 0 5 75xl0 7

0.60 0 450 0 696 5 8 5 x l 0 7

0.70 0 466 0 735 5 90xl0 7

0.80 0 511 0 764 6 0 x l O 7

0.90 0 611 0 760 6 1 x l O 7

1.0 0 742 0 730 6 2 x l O 7

- T o t a l thrust T = 1286407 N - P r o p e l l e r rev.N = 87 rpm.

- Ship speed = 16.23 knots

- P r o p e l l e r diameter D = 7.20 m

- E f f e c t i v e s t a t i c pressure at p r o p e l l e r shaft (p Q-e) = 169713 Nm~

o - Rake angle = 0 - Taylor wake f r a c t i o n w =0.49.

T fable 5-1. Input design data.

In the o r i g i n a l design, named here design D, the thrust r a t i o between p r o p e l l e r and duct was obtained from the open-water r e s u l t s of the Ka-4-55 s e r i e s . In the present method the following a l t e r n a t i v e s regarding the hydrodynamic pitch,when determining the p r o p e l l e r induced v e l o c i t i e s on the duct, have been considered:

1. A non-uniform flow to the p r o p e l l e r i s assumed and the strength of the

p r o p e l l e r r i n g vortex system i s determined i n the ultimate wake.

2. Uniform flow to the p r o p e l l e r i s assumed and the strength of the p r o p e l ­l e r r i n g vortex system i s determined i n the ultimate wake.

153

3. Uniform flow to the p r o p e l l e r i s assumed and the strength of the propel­l e r r i n g vortex system i s c a l c u l a t e d at the p r o p e l l e r plane,.

The designs corresponding to the three a l t e r n a t i v e s are named designs 1, 2 and 3, r e s p e c t i v e l y . A l l the designs are based on given t o t a l thrust and c i r c u l a t i o n d i s t r i b u t i o n . An e l l i p t i c c i r c u l a t i o n d i s t r i b u t i o n G=K/(r-0.2)(l-r) has been assumed for the three cases.

Table 5-2 reviews the c a l c u l a t i o n r e s u l t s and compares them with the

open-water r e s u l t s of the Ka-4-55 s e r i e s .

Design J C T C T P \ T

D 0.418 4.737 3.411 1.326 - 0 720 1 0.418 4.737 3.681 1.056 0.0066 0 777 2 0.418 4.737 3.435 1.302 0.0069 0 725 3 0.418 4.737 3.631 1.106 0.0067 0 767

D - Experiment with Ka-4-55 s e r i e s - uniform flow

1 - Non-uniform flow and p i t c h i n the ultimate wake 2 - Uniform flow, p i t c h i n the ultimate s l i p s t r e a m 3 - Uniform flow, p i t c h at the p r o p e l l e r plane

Table 5-2. Review of calculation results.

It i s seen that the c a l c u l a t i o n 2 based on uniform flow and p i t c h i n the ultimate wake agrees rather c l o s e l y with the experimental values f o r the thrust on the p r o p e l l e r and duct. As expected, the e f f e c t of s l i g h t v a r i a ­tions of the p r o p e l l e r blade load d i s t r i b u t i o n on the thrust r a t i o i s small.

If the non-uniform inflow i s accounted for i n the determination of the hydrodynamic p i t c h , the r i n g v o r t i c i t y shed from the p r o p e l l e r disk i s strengthened at the inner r a d i i of the disk and weakened at the outer r a d i i , leading to a higher thrust r a t i o than -the one obtained i n uniform flow. Concerning design 3 we see that the duct induces at the p r o p e l l e r plane, r e l a t i v e l y higher v e l o c i t i e s i n comparison with the p r o p e l l e r s e lf-induced v e l o c i t i e s and therefore, the choice of the hydrodynamic p i t c h at the propel­l e r plane gives a higher thrust r a t i o .

The pressure d i s t r i b u t i o n s on the duct are shown i n F i g . 5-5.

154

- 8 . 0 1. NON-UNIFORM FLOW. PITCH IN THE ULTIMATE WAKE 2. UNIFORM FLOW. PITCH IN THE ULTIMATE WAKE 3. UNIFORM FLOW. PITCH AT THE PROPELLER PLANE

C p - 7 . 0

- 6 . 0 X, - 5 . 0

\ \

- 4 . 0

- 3 . 0

- 2 . 0

- 1 .0

O .5 x / R

1.0

Fig. 5-5. Calculated pressure distributions on duct 19A.

Concerning the duct induced v e l o c i t i e s at the p r o p e l l e r plane, shown

i n F i g . 5-6, considerable d i f f e r e n c e s occur when comparing with the v e l o c i t y

d i s t r i b u t i o n assumed i n design D, e s p e c i a l l y towards the p r o p e l l e r t i p .

The p r o p e l l e r a x i a l and ta n g e n t i a l s e l f - i n d u c e d v e l o c i t i e s are compared

with the i n f i n i t e l y bladed p r o p e l l e r induced v e l o c i t i e s (design D) i n

F i g . 5-7. The d i f f e r e n c e s concerning the a x i a l induced v e l o c i t y are small.

However, i t should be noted that design D i s based on a d i f f e r e n t p r o p e l l e r

load d i s t r i b u t i o n with f i n i t e loading at the hub and t i p given by

The p a r t i c u l a r s of the designed p r o p e l l e r s are given i n Table 5-3. The f i n a l thrust and torque d e l i v e r e d by the p r o p e l l e r have been determined by assuming a constant drag c o e f f i c i e n t C^=0.0075 f o r a l l the p r o p e l l e r blade sections. It can be seen that the designs 1, 2 and 3 are f a i r l y s i m i l a r . Although they are based on d i f f e r e n t thrust r a t i o s ( f o r instance comparing

155

P r o | •peller 1

r V e

(r)/V A

C tang tanß , l

C L c / d t (m) c (m) 1/K í e c o

f (m) o

P s ( « ) dT/dx dQ/dx

0 .2 0 .704 0 0 0 468 1 568 0 0 0 294 1 402 0 489 0 7 .095 0 0

0 3 0 . 784 0 0612 0 348 0 958 0 158 0 244 1 683 0 753 0 119 7 .444 643877 681167

0 4 0 .837 0 0801 0 278 0 719 0 160 0 194 1 908 0 895 0 108 7 .370 1230060 1307040

0 5 0 .863 0 0895 0 230 0 577 0 145 0 151 2 077 0 878 0 103 7 300 1809100 1942560

0 6 0 .875 0 0925 0 194 0 486 0 126 0 121 2 190 0 814 0 099 7 304 2308560 2536490

0 7 0 .906 0 0895 0 172 0 423 0 105 0 097 2 248 0 750 0 091 7 324 2661630 3012270

0 8 0 994 0 0801 0 165 0 378 0 082 0 073 2 232 0 664 0 081 7 385 2762790 3266560

0 9 1 188 0 0612 0 176 0 352 0 055 0 050 2 024 0 521 0 071 7 626 2399250 3077090

1 0 1 443 0 0 0 192 0 342 0 0 0 026 0 493 0 352 0 0 7 744 0 0

C T = 3.794 J T P i t = o °

P . , = 7 . 4 4 0 m P N / D

87 r .p.m.

p -e = 169713 Nm T = 1421510 N ° P

N Welght= 16839 Kg

Propel ler 2

V e

<r)/VA G t a n ß t a n ß t C L c / d t <m> c (m) 1/K xc c O

1 0

(m) P N On) dT/dx dQ/dx

0 2 0 . 704 0 0 0 468 1. 504 0 0 0 288 1 374 0 499 0. 0 6 802 0 0

0 3 0 .784 0 0566 0 348 0 954 0 143 0 239 1 637 0 763 0. 107 7 . 346 608533 642314

0 4 0 .837 0 0741 0 278 0 715 0 146 0 190 1 848 0 903 0 097 7. 281 1152350 1219750

0 5 0 .863 0 0828 0 230 0 574 0 133 0 148 2 008 0 885 0 094 7 230 1687160 1807180

0 6 0 .875 0 0855 0 194 0 484 0 116 0 119 2 116 0 821 0 090 7 240 2148190 2355500

0 7 0 906 0 0828 0 172 0 421 0 096 0 095 2 172 0 759 0 083 7 266 2472850 2795720

0 8 0 .994 0 0741 0 165 0 380 0 076 0 072 2 162 0 673 0 074 7 399 2572440 3068620

0 9 1 . 188 0 0566 0 176 0 355 0 051 0 049 1 967 0 532 0 064 7 653 2222170 2885190

1 0 1 443 0 0 0 192 0 350 0 0 0 026 0 505 0 361 0 0 7 920 0 0

c T

Pi = 3.534 J = 0.418 V

s = 8.277 -1

ms N = 8 7 r .p.m. D 7.20 >

E 0 ° p o " e 169713 2 T P

1323170 N Q = 19890.3 Nm V A o = 0.534

P N = 7.417 P N /D = 1.030 Weight= 16029 Kg

Table 5-3. Particulars of designed propellers.

156

Propel ler 3

r V e

<r)/VA G tanß t a n ß j C c/ä t Im) c <m> 1/K c cx a f (m) o P dT/dx dQ/dx

0 2 0 .704 0 0 0 468 1 562 0 0 0 292 1 396 0 490 0 0 7 066 0 0

0 3 0 .784 0 0602 0 348 0 958 0 155 0 243 1 675 0 755 0 117 7 428 636809 673969

0 4 0 .837 0 0788 0 278 0 719 0 157 0 193 1 899 0 896 0 105 7 359 1214330 1291060

0 5 0 .863 0 0881 0 230 0 577 0 142 0 150 2 067 0 879 0 101 7 290 1783890 1916510

0 6 0 875 0 0910 0 194 0 486 0 124 0 120 2 180 0 815 0 097 7 296 2275510 2501580

0 7 0 906 0 0881 0 172 0 423 0 103 0 097 2 237 0 751 0 089 7 318 2623060 2971550

0 8 0 994 0 0788 0 165 0 378 0 080 0 073 2 223 0 665 0 080 7 387 2721700 3224760

0 9 1 188 0 0602 0 176 0 353 0 054 0 050 2 019 0 523 0 069 7 637 2362620 3041500

1 0 1 443 0 0 0 192 0 344 0 0 0 026 0 496 0 353 0 0 7 777 0 0

C T P ] 3.740 J = 0.418 V

s = 8.277 ms L N = 87 * P m D 7.20 m

o -2 E = 0 P Q -e = 169713 Nm T p = 1401180 N Q = 21032.8 Nm A^/A^ = 0.548

P N = 7.440 P N /D = 1.033 Weight= 16687 Kg

Design D

r V e

( D / V A Ap

(*)

tanß t a n ß 1 C L c / d t <m> c (m) 1/K xc c

f (m) o

Ph (m) P N ( f ina l )

0 2 0 .704 14459 0 468 1 383 0 105 0 320 1 455 - 0 045 6 996 6 .968

0 3 0 .784 23299 0 348 0 955 0 115 0 285 1 644 - 0 053 7 165 7 . 165

0 .4 0 .837 30152 0 278 0 739 0 112 0 245 1 829 - 0 054 7 328 7 . 317

0 5 0 .863 34743 0 230 0 604 0 103 0 203 2 000 - 0 052 7 424 7 .427

0 6 0 875 37244 0 194 0 513 0 092 0 166 2 150 - 0 048 7 503 7 . 507

0 7 0 906 37792 0 172 0 451 0 080 0 120 2 275 - 0 043 7 631 7 .567

0 8 0 994 36764 0 165 0 414 0 067 0 081 2 310 - 0 037 7 932 7 .619

0 9 1 188 34880 0 176 0 400 0 056 0 045 2 182 - 0 031 8 591 7 .671

1 0 1 443 32345 0 192 0 399 0 046 0 026 1 300 - 0 026 9 701 7 .722

C T = 3.536 J = 0.418 V = 8.277 ms N = 8 7 r . p . m . D = 7.20 • T P i »

t = 0 ° p -e = 169713 Nm"2 T = 1267719 N ° P

( » ) Thrust d i s t r ibut ion

Table 5-3. (Continued). Particulars of designed propellers.

157

1. N O N - U N I F O R M F L O W . P I T C H IN T H E U L T I M A T E W A K E 2. U N I F O R M F L O W . P I T C H IN T H E U L T I M A T E W A K E 3. U N I F O R M F L O W . P I T C H AT T H E P R O P E L L E R P L A N E

DESIGN D

0.4 0 . 5 0.6 0.7 0 . 8 0 .9 1.0

Fig. 5-6. Duct induced velocities at the propeller plane.

1 and 2), they lead to a rather s i m i l a r p i t c h i n g of the p r o p e l l e r . C l e a r l y the increase of p r o p e l l e r induced v e l o c i t i e s due to the higher p r o p e l l e r loading i n cases 1 and 3 i s "c a n c e l l e d " by the decrease of duct induced v e l o c i t i e s .

Concerning the comparison with design D two main d i f f e r e n c e s can be noted. In the f i r s t place the p i t c h d i s t r i b u t i o n obtained i n the design D shows a pronounced increase towards the t i p which i s not obtained i n the present designs. F i g . 5-8 shows the non-faired p i t c h d i s t r i b u t i o n obtained for d i f f e r e n t designs and compares i t with the f i n a l adopted p i t c h d i s t r i ­bution of design D. Secondly, the camber of blade sections i n design 1, 2 and 3 i s considerably higher than the f i n a l camber adopted i n the design D. (see Table 5-3). I t should be noted that i n the design D d i f f e r e n t blade sections are used, namely the NACA a=0.8 mean l i n e with a modified NACA 66 thickness d i s t r i b u t i o n , and no c o r r e c t i o n f a c t o r s from l i f t i n g surface theory are a p p l i e d .

158

1. N O N - U N I F O R M F L O W . P ITCH IN THE ULTIMATE WAKE 2. UNIFORM FLOW PITCH IN THE ULTIMATE WAKE 3. UNIFORM FLOW. PITCH AT THE P R O P E L L E R P L A N E

DESIGN D

0.7 -

0.1 -

" I l I I I I I I I 0 2 0 3 0 4 0 5 0 .6 0.7 0 8 0.9 1.0

r

Fig. 5-7. Axial and tangential propeller induced velocities.

It can be e a s i l y seen that the d i f f e r e n c e s i n camber f o r the present designs i n comparison with design D a r i s e mainly from the a p p l i c a t i o n of l i f t i n g surface c o r r e c t i o n f a c t o r s . Considering the good agreement with experiments f o r the f i n a l design D i t can be concluded that the a p p l i c a t i o n to ducted p r o p e l l e r s of camber c o r r e c t i o n f a c t o r s as derived by Morgan f o r open p r o p e l l e r s , should be faced with care.

The blade contours of p r o p e l l e r s 1, 2 and 3 are f a i r l y s i m i l a r . They are compared with the blade contour of p r o p e l l e r D (Kaplantype) i n F i g . 5-9.

159

Fig. 5-9. Propeller blade outlines.

160

6. C o n c l u s i o n s

In t h i s study, some of the most important e f f e c t s determining ducted p r o p e l l e r performance both i n uniform and non-uniform axisymmetric flows have been demonstrated.

In general, i t can be concluded that the i n v i s c i d flow models c o n s i ­dered i n Chapters 1 and 2 give s a t i s f a c t o r y p r e d i c t i o n s of the flow f i e l d and duct performance i n a wide range of p r o p e l l e r loadings provided that the c i r c u l a t i o n around the duct p r o f i l e can be accurately determined. As shown by the cases i n v e s t i g a t e d , i n the establishment of c r i t e r i a f o r the determination of the duct c i r c u l a t i o n , d e t a i l e d account of viscous e f f e c t s on the duct i s required. The methods developed have been s u c c e s s f u l l y applied to describe the i n t e r a c t i o n between afterbody and ducted p r o p e l l e r and to the design of ducted p r o p e l l e r s .

From the various parts of t h i s work i n p a r t i c u l a r , the following main

conclusions can be drawn:

- A p o t e n t i a l flow c a l c u l a t i o n method, applied to ducts with a thic k round t r a i l i n g edge, revealed an extreme s e n s i t i v i t y of the computed pressure d i s t r i b u t i o n to the l o c a t i o n of the stagnation point on the t r a i l i n g edge.

- An analysis method f o r the viscous flow past a p r o p e l l e r duct i n an uniform flow, based on a v i s c o u s - i n v i s c i d i t e r a t i o n scheme, gives good p r e d i c t i o n s of the flow f i e l d around the duct. The trends of the e f f e c t of the Reynolds number on the l i f t force a c t i n g on the duct s e c t i o n are well p r e d i c t e d by the c a l c u l a t i o n s . The chordwise extent of laminar separation bubbles on the outer surface of the duct and the character of the flow separation from the t r a i l i n g edge, appeared to be determinant to t h i s e f f e c t .

- In the c a l c u l a t i o n of o f f - d e s i g n performance of a duct f o r a ducted p r o p e l l e r at l i g h t p r o p e l l e r loadings, a p p l i c a t i o n of an approximate viscous a n a l y s i s to the flow past the duct considerably improves the c o r r e l a t i o n on the duct forces. More work i n t h i s area i s needed i n order

161

to f u l l y c l a r i f y the r o l e of v i s c o s i t y when the p r o p e l l e r i s placed i n s i d e the duct.

- Experiments with a ducted p r o p e l l e r i n a r a d i a l l y non-uniform flow showed a small e f f e c t of the incoming v o r t i c i t y on the thrust r a t i o between p r o p e l l e r and duct. Although the i n t e r a c t i o n with the v o r t i c i t y induces a d d i t i o n a l negative r a d i a l v e l o c i t i e s on the duct, i t tends to diminish the i n t e r f e r e n c e between p r o p e l l e r and duct.

- The curvature of the streamlines ahead of the ducted p r o p e l l e r i s smaller i n shear flow than i n uniform flow.

- Reasonable p r e d i c t i o n s of the flow f i e l d f o r a ducted p r o p e l l e r i n axisymmetric shear flow can be obtained with the method of Chapter 3. For the correct p r e d i c t i o n s of the forces on the duct, the a p p l i c a t i o n of the Kutta c o n d i t i o n appears to be e s s e n t i a l .

- The r e a l i s t i c estimates f o r the propulsion f a c t o r s obtained i n the Chapter 4,point out that the most important e f f e c t s of the i n t e r a c t i o n between a ducted p r o p e l l e r and the stern f o r an axisymmetric body can be described within an i n v i s c i d approach. By r e v e a l i n g the d e t a i l e d pressure d i s t r i b u ­t i o n on the duct, the method proposed i n Chapter 4 may be use f u l i n the design of afterbody adapted ducted p r o p e l l e r s .

- The design of a p r o p e l l e r i n s i d e a given duct appears to be rather i n s e n s i t i v e to the assumptions regarding the p i t c h of the vortex l i n e s shed from a moderately loaded actuator disk, when accounting f o r the i n t e r a c t i o n between p r o p e l l e r and duct.

- For the determination of the duct induced v e l o c i t i e s on the p r o p e l l e r , i t i s important to know with good accuracy the duct chordwise load d i s t r i b u ­t i o n . Neglection of t h i s f a c t may lead to an erroneous p r o p e l l e r p i t c h d i s t r i b u t i o n .

- I t i s po s s i b l e to design p r o p e l l e r s operating i n s i d e the duct with an assumed vanishing c i r c u l a t i o n at the t i p .

162

APPENDIX 1: EXPERIMENTAL SET-UP

A short d e s c r i p t i o n of the experimental set-up used i n t h i s study for the measurements on duct 37 with p r o p e l l e r B i n the Large C a v i t a t i o n Tunnel of the NSMB i s given. For a d e t a i l e d account on the r e s u l t s of the measurements we r e f e r to Luttmer and Janssen (1982).

1. Force measurements on the duct

To obtain the force a c t i n g on a duct s e c t i o n the a x i a l and r a d i a l force components are determined. The a x i a l component i s obtained from the t o t a l a x i a l force a c t i n g on the duct measured with a s t r a i n gauge mounted on the fastening between the duct and the tunnel w a l l . The r a d i a l force component i s deduced from the measurement of the tang e n t i a l force which i s

Fig. Al-1. Position of windows and force transducers on the duct 37.

(dimensions are given in mm).

163

measured by s t r a i n gauge transducers placed on a cut of the duct surface by a meridional plane.

The l o c a t i o n of the force transducers i s shown i n F i g . Al-1, and the geometry of the duct model i s given i n F i g . Al-2.

2. V e l o c i t y measurements

The a x i a l and r a d i a l components of the v e l o c i t y f i e l d around the duct and ducted p r o p e l l e r were measured by a Laser-Doppler velocimeter placed around

Dimensions of duct in nta

X y 2.0 13.80 4.0 11.20 6.0 9.20 8.0 7.70

10.0 6.11 14.0 4.20 18.0 2.70 22.0 1.60 26.0 0.81 30.0 0.44 34.0 0.20 38 .0 0

straight 66.0 0 70.0 0.20 74.0 0.36 78.0 0.76 82.0 1.40 86.0 2.36 90.0 3.80 92 .0 4.70 94 .0 5.80 96.0 7.20 98.0 8.80

r 2 3.34 r 3 101.00 hj 12.42 h 2 18.33 c 100.00

Fig. Al-2. Geometry of duct model.

164

Fig. Al-3. Laser-Doppler velocimeter placed around the cavitation

tunneI.

the test s e c t i o n of the large c a v i t a t i o n tunnel ( F i g . A l - 3 ) . The Laser-Doppler velocimeter operates i n the forward s c a t t e r and reference beam modes. Due to the presence of large windows on both sides of the tunnel, i t was pos s i b l e to scann the flow f i e l d upstream and downstream of the duct. The loc a t i o n s of the various a x i a l s t a t i o n s where v e l o c i t y traverses were performed are given i n F i g . Al-4. In the same f i g u r e i t i s also shown the l o c a t i o n of the screen used to generate the axisymmetric non-uniform flow used i n the measurements on the ducted p r o p e l l e r i n the wake f i e l d .

165

Fig. Al-4. Location of measuring station and coordinate system.

The measuring volume of the Laser-Doppler velocimeter has dimensions of 0.214 mm i n the x and z d i r e c t i o n s and 16.34 mm i n the y d i r e c t i o n ( F i g . A l - 4 ) . This f a c t a f f e c t e d the accuracy of the measurements of the v e l o c i t y p r o f i l e s i n s i d e the duct where, without p r o p e l l e r , the traverses were e f f e c t e d on the x-y plane. For these p a r t i c u l a r measurements "windows" were opened on the perspex duct model. The l o c a t i o n of the "windows" are shown i n F i g . Al-1.

166

APPENDIX 2: GEOMETRY OF PROPELLERS A AND B.

Fig. A2-1. Geometry of propeller A. (Dimensions are given in mm).

E X P A N D E D B L A D E A R E A R A T I O A E / A Q = 0 . 6 0 8

C H O R D L E N G T H Q 7 / D I A M E T E R cQ?/D = 0 . 2 6 1

T H I C K N E S S / C H O R D L E N G T H o.7 , / C 0 7 " 0 . 0 6 3

Fig. A2-2. Geometry of propeller B. (Dimensions are given in mm).

167

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174

NOMENCLATURE

A(r|) function

AE / A

0 expanded blade area r a t i o

C duct drag c o e f f i c i e n t d

C duct force c o e f f i c i e n t behind the afterbody without p r o p e l l e r dh

C^ f l a t p l a t e s k i n f r i c t i o n c o e f f i c i e n t

C l i f t c o e f f i c i e n t L C pressure c o e f f i c i e n t P

C duct pressure c o e f f i c i e n t P d

C pressure c o e f f i c i e n t at the t r a i l i n g edge on the inner surface P i n n

C minimum pressure c o e f f i c i e n t P • min

C pressure c o e f f i c i e n t at the t r a i l i n g edge on the outer surface P o u t

C pressure c o e f f i c i e n t at separation p sep

AC pressure d i f f e r e n c e c o e f f i c i e n t on the afterbody between the P l

cases: h u l l with duct and h u l l with ducted p r o p e l l e r

AC pressure d i f f e r e n c e c o e f f i c i e n t on the afterbody between the P2

cases: h u l l alone and h u l l with ducted p r o p e l l e r

C prismatic c o e f f i c i e n t P

C forebody prismatic c o e f f i c i e n t

C afterbody prismatic c o e f f i c i e n t PA

C duct r a d i a l force c o e f f i c i e n t R

C duct thrust c o e f f i c i e n t T , C i d e a l duct thrust c o e f f i c i e n t

C T * duct thrust c o e f f i c i e n t based on the ship speed

g

C p r o p e l l e r thrust c o e f f i c i e n t P

C i d e a l p r o p e l l e r thrust c o e f f i c i e n t T P i

175

p r o p e l l e r thrust c o e f f i c i e n t based on the ship speed

c o r r e c t i o n f a c t o r for the e f f e c t of p r o f i l e thickness on

the l i f t

duct r a d i a l force c o e f f i c i e n t i n the non-uniform flow case

based on U Q

duct thrust c o e f f i c i e n t i n the non-uniform flow case based

on U Q

p r o p e l l e r thrust c o e f f i c i e n t i n the non-uniform flow case

based on

-duct length

-blade s e c t i o n chord

curvature of element on the duct and hub th th curvature of the i element on the m vortex sheet

- p r o p e l l e r diameter

-actuator disk diameter

body diameter

duct viscous drag

force on the duct behind the afterbody without p r o p e l l e r

numerical d i f f e r e n t i a t i o n c o e f f i c i e n t

complete e l l i p t i c i n t e g r a l of the second kind

vapour pressure

numerical d i f f e r e n t i a t i o n c o e f f i c i e n t

normalized c i r c u l a t i o n d i s t r i b u t i o n

external body force per u n i t mass

duct r a d i a l force per u n i t radian

components of i n the coordinate set (x,r,8)

right-hand side of i n t e g r a l equation

176

f. right-hand s i d e of system of equations

f (r) normalized hydrodynamic p i t c h d i s t r i b u t i o n P

f x > f g a x i a l and t a n g e n t i a l components of the concentrated force

on the actuator disk

fg geometric camber of a blade s e c t i o n f numerical d i f f e r e n t i a t i o n c o e f f i c i e n t J

G non-dimensional c i r c u l a t i o n of a l i f t i n g l i n e

G(x-x',r,r') Green's fu n c t i o n G „G .,G . ,, p a r t i a l d e r i v a t i v e s of the Green's fu n c t i o n x" r " x'x'

x r r ' r ' G non-dimensional c i r c u l a t i o n d i s t r i b u t i o n on the actuator disk CO

g g r a v i t y a c c e l e r a t i o n

g\ numerical d i f f e r e n t i a t i o n c o e f f i c i e n t

H t o t a l head

Hp t o t a l head of the undisturbed stream

H Q t o t a l head of the piecewlse constant d i s c r e t i z e d non-uniform m

stream

h. numerical d i f f e r e n t i a t i o n c o e f f i c i e n t

i i a x i a l and t a n g e n t i a l induction f a c t o r s a t i , i ,i„ u n i t vectors of the coordinate set (x,r,9) —x r W

i numerical d i f f e r e n t i a t i o n c o e f f i c i e n t

J advance r a t i o

J advance r a t i o based on the ship speed s J advance r a t i o i n the non-uniform flow case based on U .

K(k) complete e l l i p t i c i n t e g r a l of the f i r s t kind

K c o r r e c t i o n to camber from l i f t i n g surface theory

177

K duct r a d i a l force c o e f f i c i e n t R

K T t o t a l thrust c o e f f i c i e n t

K duct thrust c o e f f i c i e n t T d

p r o p e l l e r thrust c o e f f i c i e n t P

k,k^ modulus of e l l i p t i c i n t e g r a l s

k(s,s') kernel function

k. . matrix of i n f l u e n c e c o e f f i c i e n t s kp constant i n the hydrodynamic p i t c h d i s t r i b u t i o n

L body length

length of afterbody

L length of forebody E L length of middle body M

M number of a x i a l s t a t i o n s

m -boundary layer pressure gradient parameter

-margin against c a v i t a t i o n

N - p r o p e l l e r r.p.m.

-number of elements on the boundaries

-number of vortex sheets i n the d i s c r e t i z a t i o n of the

non-uniform stream

number of slips t r e a m vortex sheets

n -coordinate of streamline based coordinate set (s,n,6)

- p r o p e l l e r r a t e of revolutions

P p r o p e l l e r p i t c h

P -hydrodynamic p i t c h

- p r o p e l l e r i d e a l power

P^(r) nominal p i t c h at radius r

T

178

mean nominal p i t c h

p pressure

p mean s t a t i c pressure at the p r o p e l l e r plane mean

Pg pressure at i n f i n i t y upstream

p^ pressure before the p r o p e l l e r plane

p pressure a f t e r the p r o p e l l e r plane

Ap pressure decrease induced by the duct at the p r o p e l l e r d

Ap^ pressure d i f f e r e n c e on the afterbody between the cases:

h u l l with duct and h u l l with ducted p r o p e l l e r

Ap^ pressure d i f f e r e n c e on the afterbody between the cases:

h u l l alone and h u l l with ducted p r o p e l l e r

Q p r o p e l l e r torque

R - p r o p e l l e r radius

-distance between two points

-actuator disk radius

R vector radius between two points

R, duct radius d Re Reynolds number

Re Reynolds number based on the duct's length c R r e s i s t a n c e of h u l l alone h R r e s i s t a n c e of h u l l with duct without p r o p e l l e r hd

R c o n t r i b u t i o n to the stream f u n c t i o n on a mesh knot due to the i j

v o r t i c i t y outside the computational domain

R Reynolds number based on the momentum thickness of the y

boundary layer

R Reynolds number based on the momentum thickness at U sep

separation

179

R Q r a d i a l extent of the inflow non-uniformity

r r a d i a l coordinate

r body radius b r non-dimensional hub radius h

th r. radius of the m vortex sheet at x=x. îm l

r a d i a l coordinate on element j th

r ^ radius of the k slips t r e a m vortex sheet at the actuator

disk r maximum body radius max

th radius of the m vortex sheet at i n f i n i t y upstream

m o k

th

r m radius of the k slipstream vortex sheet at i n f i n i t y down­

stream th

r radius of the m vortex sheet at i n f i n i t y downstream m "im r a d i a l coordinate of the vertex of the i * * 1 element on the

t h * V. * m vortex sheet

th

r^ r a d i a l coordinate of c o n t r o l point on the j element

S -area

-length of the duct contour

s coordinate of the streamline based coordinate set (s,n,6)

distance on the surface at separation measured from the nose

stagnation point

sep

s distance on the surface at t r a n s i t i o n measured from the nose t r

stagnation point

T t o t a l thrust

T(s,s') coupling function for the axisymmetric source d i s t r i b u t i o n

T_, duct thrust d T p r o p e l l e r thrust

180

T p r o p e l l e r i d e a l thrust P i

t maximum thickness of blade s e c t i o n

t_ vector tangent to the duct's s e c t i o n contour

t , duct thickness d t pressure component of the thrust deduction: duct as a part

P l of the h u l l

t pressure component of the thrust deduction: duct as a part P2

of the propulsor

U(r) a x i a l v e l o c i t y of the undisturbed non-uniform stream

u f f experimental mean e f f e c t i v e v e l o c i t y ( p r o p e l l e r thrust

i d e n t i t y )

U. . a x i a l v e l o c i t y induced at the i * * 1 c o n t r o l point by a source th

d i s t r i b u t i o n on the j element

U v e l o c i t y of the piecewise constant d i s c r e t i z a t i o n of the m

non-uniform stream

Up -uniform flow v e l o c i t y

-uniform flow v e l o c i t y outside the wake

th U mean a x i a l v e l o c i t y at the m vortex sheet at i n f i n i t y m

upstream

u a x i a l v e l o c i t y component i n the coordinate set (x,r,8)

_u v e l o c i t y vector

u (r) a x i a l v e l o c i t y induced by the duct d

u (r) e f f e c t i v e v e l o c i t y : duct as a part of the h u l l e l

u (r) e f f e c t i v e v e l o c i t y : duct as a part of the propulsor 62

u.(r) a x i a l v e l o c i t y induced by the hub h

u (r) a x i a l v e l o c i t y induced by the p r o p e l l e r P

u (r) a x i a l v e l o c i t y induced f a r downstream by the f i r s t approxima­ted

t i o n to the slipstream v o r t i c i t y i n the non-uniform flow cast

181

u ,u ,u„ components of u i n the streamline based coordinate set (s,n,6) s n B — th

u meridional v e l o c i t y at the m vortex sheet evaluated at s. i ,m

S i , k

(x.,r. ) i lm th

meridional v e l o c i t y at the k slips t r e a m vortex sheet

evaluated at (x ,r.,) i l k

u meridional v e l o c i t y at the k**1 slipstream vortex sheet s k th u meridional v e l o c i t y at the m vortex sheet s m

Ug(r) t o t a l a x i a l v e l o c i t y at the p r o p e l l e r plane

Uy a x i a l v e l o c i t y induced by a vortex r i n g with u n i t c i r c u l a t i o n

u^ a x i a l v e l o c i t y induced by a source r i n g with u n i t c i r c u l a t i o n

u a x i a l v e l o c i t y induced by the p r o p e l l e r on the duct

u^Cr) a x i a l v e l o c i t y induced f a r downstream by the actuator disk

vortex system i n the uniform flow case

V r e s u l t a n t v e l o c i t y to the blade s e c t i o n

V(s) v e l o c i t y outside the boundary layer

V. mean advance v e l o c i t y A

V^(x,r;r') a x i a l v e l o c i t y induced by a s e m i - i n f i n i t e r i n g vortex c y l i n d e r

with u n i t strength

V * ( r , r ' ) a x i a l v e l o c i t y induced f a r downstream by a s e m i - i n f i n i t e r i n g

vortex c y l i n d e r with u n i t strength

e f f e c t i v e v e l o c i t y derived from Taylor wake f r a c t i o n

V v e l o c i t y at i n f i n i t y upstream on the duct stream-surface

V. . r a d i a l v e l o c i t y induced at the i * * 1 c o n t r o l point by a source th

d i s t r i b u t i o n on the j element

V mean v e l o c i t y between the inner and outer sides of the duct m V ( x , r ; r ' ) r a d i a l v e l o c i t y induced by a s e m i - i n f i n i t e r i n g vortex

c y l i n d e r with u n i t strength

182

V ship speed s V v e l o c i t y outside of the boundary layer at separation sep v r a d i a l v e l o c i t y component i n the coordinate set (x,r,0)

v^ r a d i a l v e l o c i t y induced by a r i n g vortex with u n i t c i r c u l a t i o n

v^ r a d i a l v e l o c i t y induced by a r i n g source with unit strength

v r a d i a l v e l o c i t y induced by the p r o p e l l e r on the duct

W blade s e c t i o n modulus

W. , function i,k w t a n g e n t i a l v e l o c i t y component i n the coordinate set (x,r,9)

w ta n g e n t i a l v e l o c i t y induced by the duct d w (r) e f f e c t i v e wake f r a c t i o n d i s t r i b u t i o n w (r)=l-u (r) e e e w (r) e f f e c t i v e wake f r a c t i o n d i s t r i b u t i o n : duct as a part of the e..

'1

"2

h u l l w (r)=l-u (r) e l e l

w (r) e f f e c t i v e wake f r a c t i o n d i s t r i b u t i o n : duct as a part of the e„

propulsor w (r)=l-u (r) e2 e2

w (r) nominal wake f r a c t i o n d i s t r i b u t i o n N w (r) nominal wake f r a c t i o n d i s t r i b u t i o n : duct as a part of the h u l l

w (r) nominal wake f r a c t i o n d i s t r i b u t i o n : duct as a part of the N2

propulsor

w tang e n t i a l v e l o c i t y induced by the p r o p e l l e r P

w Taylor wake f r a c t i o n T 1

w mean e f f e c t i v e wake f r a c t i o n e w mean e f f e c t i v e wake f r a c t i o n : duct as a part of the h u l l _ e l

w mean e f f e c t i v e wake f r a c t i o n : duct as a part of the propulsor

W Q t a n g e n t i a l v e l o c i t y immediately downstream of the actuator

disk X a x i a l induced v e l o c i t y matrix

i j

183

x a x i a l coordinate

x^ a x i a l coordinate of mesh knots

x^(5) a x i a l coordinate on the element j

x a x i a l l o c a t i o n of p r o p e l l e r plane measured from the duct's P

th th x. a x i a l coordinate of the vertex of the i element on the m ïm

vortex sheet

Xj a x i a l coordinate of c o n t r o l point on the j * * 1 element

Y. . r a d i a l induced v e l o c i t y matrix

Z p r o p e l l e r number of blades

z coordinate normal to the surface i n the boundary layer

a -parameter of the complete e l l i p t i c i n t e g r a l of t h i r d kind

- p i t c h c o r r e c t i o n

th th a. slope of the i element on the m vortex sheet ïm th

a. slope of the j element J B - r e l a x a t i o n f a c t o r

-advance angle

p\ hydrodynamic p i t c h angle

B. hydrodynamic p i t c h angle f a r downstream oo

r - c i r c u l a t i o n around a duct s e c t i o n

- c i r c u l a t i o n of a l i f t i n g l i n e strength of "the piecewise constant d i s c r e t i z e d c i r c u l a t i o n

d i s t r i b u t i o n on the actuator disk

F n t h i t e r a t e to the c i r c u l a t i o n around a duct s e c t i o n n

T c i r c u l a t i o n on the actuator d i s k oo Y strength of a vortex sheet i n axisymmetric flow

Y vector strength of a vortex sheet

184

y j I Y J , Y ^ ^ s t r e n g t h of vortex sheet, i t s f i r s t and h a l f the second J J J

(1)

d e r i v a t i v e s evaluated at the c o n t r o l point j

Y^^.Y.^1^ strength of vortex sheet and i t s f i r s t d e r i v a t i v e evaluated km km th th at the vertex of the k element on the m vortex sheet

Y^ strength of the s e m i - i n f i n i t e r i n g vortex c y l i n d e r s i n the

f i r s t approximation to the slipstream vortex sheets

th Y strength of the m vortex sheet m th Y strength of the m vortex sheet at i n f i n i t y upstream o m th Y strength of the k slipstream vortex sheet f a r downstream

°°k th Y^ strength of the m vortex sheet f a r downstream

k

Y#ij f i r s t approximation to the duct and hub surface v o r t i c i t y

6 Kronecker d e l t a

6(x) Dirac d e l t a function

6* boundary layer displacement thickness

e rake angle D non-dimensional coordinate

n, h u l l e f f i c i e n c y h 6 -angular coordinate i n the coordinate set (x,r,6)

-momentum thickness of the boundary layer

9 momentum thickness at separation sep

5 element arc length measured from c o n t r o l point

5'. h a l f the arc length of the j t h element

£' h a l f the arc length of the k**1 element on the m*"*1 vortex sheet km

U. dipole strength

Y kinematic v i s c o s i t y

II complete e l l i p t i c i n t e g r a l of the t h i r d kind

p f l u i d s p e c i f i c mass

185

p

0 -source strength

- c a v i t a t i o n number

0 maximum allowable t e n s i l e s t r e s s l e s s the t e n s i l e stress T

due to c e n t r i f u g a l forces

T thrust r a t i o T=T /(T + T J p p d

t> perturbation p o t e n t i a l

p r o p e l l e r perturbation p o t e n t i a l

outer and inner p o t e n t i a l s

¥ t o t a l stream fu n c t i o n

., stream function i n f l u e n c e c o e f f i c i e n t ljkm ¥ stream function of the m t h vortex sheet m

^ooC? >~ >r' ) stream function induced by a s e m i - i n f i n i t e r i n g vortex

c y l i n d e r with u n i t strength

f i r s t approximation to the t o t a l stream function

ip -perturbation stream function

- p i t c h angle

ij^ stream function induced by the duct and hub surface v o r t i c i t y

i j j ^ f i r s t approximation to the stream function induced by the (1)

duct and hub surface v o r t i c i t y

tjjp stream function induced by the actuator disk v o r t i c i t y lOg

4) stream function induced by the wake v o r t i c i t y 0)„ w 6 w

4j stream function of the undisturbed non-uniform stream

P. p r o p e l l e r angular v e l o c i t y

U) v o r t i c i t y vector

u) , u , u Q components of 0) i n the coordinate set (x.r.8) x r 0 —

U)„ v o r t i c i t y of the undisturbed non-uniform stream 90

186

actuator disk v o r t i e i t y

wake v o r t i e i t y

non-dimensional coordinate

SUMMARY

In t h i s study a t h e o r e t i c a l model f o r the c a l c u l a t i o n of duct p e r f o r ­mance f o r ducted p r o p e l l e r s working i n both uniform and r a d i a l l y v a r i a b l e inflow, i s in v e s t i g a t e d . A p p l i c a t i o n of t h i s model to the problem of i n t e r a c t i o n of a ducted p r o p e l l e r and the stern i n axisymmetric flow i s considered. A s i m p l i f i e d model to account f o r the i n t e r a c t i o n between p r o p e l l e r and duct i s incorporated i n t o a design method f o r ducted p r o p e l ­l e r s .

In Chapter 2 the l i m i t a t i o n s of a p o t e n t i a l flow analysis based on an axisymmetric surface v o r t i c i t y method when applied to ducts with round t h i c k t r a i l i n g edges are demonstrated. Incorporation of viscous e f f e c t s through an i t e r a t i o n scheme using boundary layer and p o t e n t i a l flow c a l c u l a t i o n methods, e s s e n t i a l l y removes such l i m i t a t i o n s . The v a l i d i t y of the approach i s confirmed by comparison with experiments c a r r i e d out f o r a duct i n uniform flow. A s i m p l i f i e d model f o r the i n t e r ­action between p r o p e l l e r and duct, based on moderately loaded actuator disk theory, i s used to evaluate duct o f f - d e s i g n performance for ducted p r o p e l l e r s i n uniform flow. Comparison with r e s u l t s of the non-linear ducted actuator disk model developed i n Chapter 3 and d e t a i l e d measurements of the flow f i e l d i s given. E f f e c t s of v i s c o s i t y at l i g h t p r o p e l l e r loadings are accounted f o r , i n an approximate way, by an extension of the viscous analysis developed for the duct without p r o p e l l e r i n uniform flow.

In Chapter 3 a d i s c r e t e vortex sheet method i s used to c a l c u l a t e the axisymmetric flow through a ducted p r o p e l l e r placed i n a shear flow. The convergence of the method i s shown. The method i s used to analyse the e f f e c t of v o r t i c i t y of the incoming stream on the i n t e r a c t i o n between p r o p e l l e r and duct. Detailed comparison of flow f i e l d c a l c u l a t i o n s with experiments i l l u s t r a t e s the c a p a b i l i t i e s and l i m i t a t i o n s of the method.

In Chapter 4 the numerical method of Chapter 3 i s applied to compute

the flow i n the stern region of a body of r e v o l u t i o n and i s used to t r e a t

the i n t e r a c t i o n problem between a ducted p r o p e l l e r and the stern i n a x i -

188.

symmetric flow. P r e d i c t i o n s of pressure d i s t r i b u t i o n on the afterbody and duct and v e l o c i t y p r o f i l e s up to the p r o p e l l e r plane are given. The s e n s i t i v i t y of the thrust deduction and e f f e c t i v e wake to p r o p e l l e r loading and duct geometry are i l l u s t r a t e d f o r two d i f f e r e n t methods of looking at the i n t e r a c t i o n : duct as a part of the h u l l and duct as a part of the propulsor.

In Chapter 5, the c a l c u l a t i o n procedure based on moderately loaded actuator disk theory and given i n Chapter 2, i s incorporated i n a design method f o r ducted p r o p e l l e r s based on Lerbs'induction f a c t o r method. The sen­s i t i v i t y of the design to various assumptions regarding the p i t c h of vortex l i n e s used i n the procedure, i s i n v e s t i g a t e d . An i l l u s t r a t i v e example i s included.

Conclusions are presented i n Chapter 6.

189

SAMENVATTING

In deze studie wordt een t h e o r e t i s c h model voor de berekening van de werking van schroef-straalbuissystemen i n homogene en r a d i a a l ongelijkma­t i g e aanstroming onderzocht. Het t h e o r e t i s c h model i s toegepast op het probleem van de i n t e r a c t i e tussen een schroef-straalbuissysteem en de romp i n axisymmetrische stroming. Een vereenvoudigde berekeningsmethode voor de i n t e r a c t i e tussen schroef en s t r a a l b u i s wordt gebruikt i n een ontwerpmetho­de voor de schroef i n de s t r a a l b u i s .

In Hoofdstuk 2 worden de beperkingen van een analyse gebaseerd op p o t e n t i a a l theorie en wervelverdelingen op de s t r a a l b u i s aangetoond, wan­neer die worden toegepast b i j s t r a a l b u i z e n met een grote afronding aan de uittredende kant. Verbeterde r e s u l t a t e n kunnen worden bereikt met behulp van een i t e r a t i e schema gebaseerd op grenslaag en potentiaal berekenings­methoden. Goede c o r r e l a t i e met experimentele r e s u l t a t e n wordt b e r e i k t i n uniforme stroming. Een eenvoudige berekeningsmethode op basis van matig belaste actuator s c h i j f t h e o r i e wordt toegepast om de hydrodynamische eigen schappen van een s t r a a l b u i s i n " o f f - d e s i g n " c o n d i t i e te berekenen. De r e s u l t a t e n worden vergeleken met r e s u l t a t e n van een n i e t - l i n e a i r actuator s c h i j f model, gegeven i n Hoofdstuk 3 en verder met g e d e t a i l l e e r d e metingen van het stromingsveld. Een benadering voor de viskeuze e f f e c t e n voor l i c h t e schroefbelastingen wordt gegeven met de methode die ontwikkeld i s voor s t r a a l b u i z e n zonder schroef.

In Hoofdstuk 3 i s een d i s c r e t e wervel vlak methode toegepast b i j de de berekening van de axisymmetrische stroming rond een s c h r o e f - s t r a a l b u i s -systeem geplaatst i n een r a d i a a l ongelijkmatige aanstroming. De convergenti van de methode wordt aangetoond. De methode wordt gebruikt om het e f f e c t van de v o r t i c i t e i t i n de aanstroming op de i n t e r a c t i e tussen schroef en s t r a a l b u i s te analyseren. G e d e t a i l l e e r d e v e r g e l i j k i n g e n met experimentele r e s u l t a t e n tonen de mogelijkheden en de beperkingen van de methode aan.

In Hoofdstuk 4 wordt de numerieke methode toegepast b i j de berekening

van het,stromingsveld achter een omwentelingslichaam en b i j de i n t e r a c t i e

190

tussen het schroef-straalbuissysteem en de romp. Drukverdelingen op het lichaam en de s t r a a l b u i s en s n e l h e i d p r o f i e l e n t o t aan het schroefvlak worden gegeven. De gevoeligheid van de zoggetal en de e f f e c t i e v e volgstroom voor de s c h r o e f b e l a s t i n g en s t r a a l b u i s geometrie wordt gegeven voor twee ve r s c h i l l e n d e beschouwingswijzen van de i n t e r a c t i e . De s t r a a l b u i s wordt beschouwd a l s een deel van de romp of de s t r a a l b u i s wordt gezien a l s een deel van de voortstuwer.

In Hoofdstuk 5 wordt het berekeningsschema gebaseerd op de matig be­l a s t e actuator s c h i j f t h e o r i e van Hoofdstuk 2 gebruikt i n combinatie met een ontwerp procedure volgens Lerbs' i n d u c t i e f a c t o r methode. De invloed van de v e r s c h i l l e n d e keuzen voor de spoed van de afgaande wervels op de ontwerpresultaten worden onderzocht. Een i l l u s t r a t i e f voorbeeld wordt gegeven.

Tenslotte worden i n Hoofdstuk 6 conclusies getrokken.

191

ACKNOWLEDGEMENT

The author wishes to express his gratitude to the Management of the Netherlands Ship Model Basin f o r the opportunity given to perform t h i s study.

The author i s deeply indebted to Dr.Ir. W. van Gent f o r h i s guidance and advice during the various stages of t h i s work. S p e c i a l thanks are extended to the SR-group of the NSMB, and p a r t i c u l a r l y to I r . B.R.I. Luttmer, for t h e i r c o n t r i b u t i o n s to the i n v e s t i g a t i o n reported here.

Thanks are given to Mrs. G.P.M. Swint-Jongsma f o r her constant assistance i n the preparation of t h i s d i s s e r t a t i o n , to Mr. F.A.J. Janssen f o r h i s c o n t r i b u t i o n to the experimental work, to Mr. G. van de Weerd and Mr. G.J. Seves f o r the execution of the fig u r e s and to Mr. B. Millecam f o r the photographic work.

This study was performed while on leave from " I n s t i t u t o Superior Técnico" and p a r t i a l l y supported by grants of "Junta Nacional de Investigaçao C i e n t i f i c a e Tecnológica" which i s g r a t e f u l l y acknowledged.

A f i n a l word of gratitude i s given to Professor A.F. de 0. Falcao of Lisbon Technical U n i v e r s i t y f o r h i s encouragement and support before and during the stay i n the Netherlands.

192

S T E L L I N G E N

1. Wu's formulation for the flow through a heavily loaded actuator disk may be general­ized to treat the problem of a propeller with finite hub, ducted propellers and a pro­peller in a radially non-uniform free stream.

Wu, T. Y . 'Flow through a heavily loaded actuator disk' Schiffstech-nik, Vol , 9-1962.

2. Paint test techniques provide a most useful means of assessing the boundary layer character on a duct with and without propeller.

3. For most of the ducts of the acceleration type having a diffuser downstream of the propeller, the effects of slipstream contraction are of less importance than the effects of vortex pitch deformation in the slipstream. When considering the interaction be­tween propeller and duct the latter effects can be rather well approximated by a reason­able choice of the pitch.

4. The arguments to justify the design of a propeller with zero circulation at the hub can be employed to make a design of a ducted propeller with an assumed zero circu­lation at the tip acceptable.

5. The experimentally found trends for the propulsion factors for ships fitted with ducted propellers indicate that the thrust deduction is considerably more influenced by changes of duct loading than by changes of propeller loading. For the wake fraction the same trends cannot be discerned.

M I N S A A S , K . J . , G . M . J A C O B S E N and H. O K A M O T O . 'The design of large ducted propellers for optimum efficiency and manoeuvrability' R I N A Symposium on ducted propellers. Paper no. 11, London 1973.

6. Laser-Doppler velocimetry gives new possibilities of theoretically studying propeller-hull interaction.

7. Assuming an ideal fluid it is possible to show on theoretical grounds that diffusion of the propeller's slipstream improves the efficiency of a ducted propeller. One possi­ble proof may be given by applying surface vorticity techniques.

8. The availability of a high speed computer and appropriate computational techniques associated with the need for a relatively quick answer to a technical problem offers to the researcher an alternative which may exclude the set-up of an analytical investiga­tion of considerable interest.

9. The increasing influence of technology in the forming process of political decisions makes it necessary that an increasing attention should be paid to the divulgation of the technical sciences. Significant contributions to such a task could be made by the technical universities.

10. The particular demands from the industry for specific profiles of engineers should be contemplated when defining or reviewing the curricula of the correspondent univer­sity courses.

11. The acceptance of a hierarchical principle relating to collective and private transport in urban and sub-urban areas as it is already done in many cases, constitutes a first step toward a more rational solution for traffic problems.

12. It is not justified to think that a great diversification of educational programmes is inefficient and promotes social inequalities. Instead it contributes to the insertion of the schools in the society at the professional and cultural levels and helps the students to find their jobs.

J . A. C . F A L C A O DE C A M P O S

Delft, 14 June 1983

CURRICULUM VITAE

The author of t h i s work was born i n Lisbon on A p r i l 11, 1952. Frequented Grammar School from 1962 to 1969. From 1969 to 1975 studied Mechanical Engineering at the Technical U n i v e r s i t y of Lisbon. In 1974 joined the Mechanical Engineering Department, se c t i o n of Applied Thermodynamics and i n 1975 became a member of the research group of NEEM-CTAMFUL (Centro de Termodinámica Aplicada e Mecánica dos Fluidos das Universidades de Lisboa). Since October 1977 worked at the Netherlands Ship Model Basin on the subject of ducted p r o p e l l e r s . The r e s u l t s of t h i s work are presented i n t h i s t h e s i s .