A contribution to the solution of some specific ship propulsion problems: A reappraisal of momentum theory

\

A CONTRIBUTION TO THE SOLUTION

OF SOME SPECIFIC

SHIP PROPULSION PROBLEMS

A REAPPRAISAL OF MOMENTUM THEORY

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5153

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A CONTRIBUTION TO THE SOLUTION

OF SOME SPECIFIC

SHIP PROPULSION PROBLEMS

- A REAPPRAISAL OF MOMENTUM

THEORY-PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGE-SCHOOL DELFT, OP GEZAG VAN DE RECTOR MAGNIFICUS IR H.B. BOEREMA, HOOGLERAAR IN DE AFDELING DER ELEKTROTECHNIEK VOOR EEN COMMISSIE AANGEWEZEN DOOR HET COLLEGE VAN DEKANEN TE VERDEDIGEN OP

WOENSDAG 12 DECEMBER 1973 TE 14.00 UUR

DOOR

LEONARD ANTHONIE VAN GUNSTEREN

SCHEEPSBOUWKUNDIG INGENIEUR GEBOREN OP 9 OKTOBER 10.^8 TE "S-t.RAVt-NHAGE

DIT PROEFSCHRIFT IS GOEDGEKEURD DOOR DE PROMOTOR:

TABLE OF COWTENTS

Page

Introduction 7

1. Free actuator disk 10

1. Slipstream o f a heavily loaded free actuator disk 10

2. Lightly loaded actuator disk 14

3. Numerical procedure 15

^. Numerical results 17

5. Conclusions 22

2. Counter-rotating propeller design 2*»

1. Introduction Zk

2. Theory 27 The velocity field of an infinitely bladed

propeller with arbitrary radial load distribution 27

Design procedure 35

3. Numerical results kk

k. Conclusions 51

3. Cavitation inception of shaft brackets 53

k. Shrouded actuator disk 57

1. Slipstream of a heavily loaded shrouded actuator disk 57

2. Numerical results 61 3. Conclusions 8O

5. Performance calculation of heavily loaded ducted propellers 81

1. Introduction 81 2. Theory 82 3. Numerical results 93

Nozzles without propeller 93 Pressure distribution at finite thrust coefficient 95

Open-water characteristics 96

li. Conclusions 98

6. Effect of a nozzle on steering characteristics 99

1. Introduction 99 2. Mathematical model of turning 100

Turning capacity 103 Response time lO't Rudder-nozzle force coefficients (C., C.) lO^t

3. Prediction of lift forces on rudder and nozzle 106

't. Numerical results n't 5. Ful1-scale tests 118

Turning circle tests 120

6. Discussion 122 7. Conclusions 123 7. Slotted nozzles 125 1. Introduction 125 2. Open-water characteristics 129 Geometrical characteristics 129 Test results 132 3. Comparison with conventional propulsion devices l'l3

Performance at non-zero advance ratios 145 Performance in bollard condition 152

4. Conclusions 156

3. Ring propeller with ring stator 157

1. Introduction 157 2. Open-water test results 158

3. Conclusions 161 Final remarks 162 References 164 Nomenclature 169 Overzicht l8l Dankwoord 183 Curriculum vltae 184

II^TROÜUCTIÜN

The present approach in dealing with propeller problems is using the concept of singularities, the hydrodynamic load being represented by vortices and the thickness of the lifting surfaces by a source-sink distribution. The vortex theory of propellers, as introduced by Prandtl, Betz and Goldstein at the beginning of this century, has been developed during the last decades, when high-speed computers became available, to a high degree of perfection.

The classical momentum theory of propellers, on the contrary, as formulated for the first tine by Rankine and Froude, Ref. [ l ] , is generally used only to gain insight in the main action of the propul-sion system. The reason for this is that momentum theory provides no detailed information on the flow field. As a result, it gives little indication of how a particular propulsor should be shaped, which is the purpose of any design procedure. Vortex theory does provide such information, and in sufficient detail for use In any blade shaping process.

In general, the application of rigorous lifting surface theories is limited to linearized solutions, i.e. to lightly loaded propellers, because the numerical work involved in non-linear iterative solutions would increase computing costs to an unacceptable level. To extend the

range of application to moderately loaded propellers, lifting line theory can be introduced as a non-linear refinement- As in pure linearized theory, the hydrodynamic pitch of the free vortices is assumed to remain constant downstream, however, the velocities induced at the lifting lines are taken

into account. This concept, combined with the allowance for thickness effects, Ref. [ 2 ] , has proved to be very effective in practice, Refs. L3j, L**]. [5]- With respect to mathematical rigour, however much can be argued against it. The justification for adopting it nevertheless, lies in the practical results it provides. Similarly, the way in

which wake effects are usually taking into account in propeller design is justified only by its practical results: obviously, the ship speed corrected for the wake at the propeller disk does not prevail both upstream and downstream, as Is assumed in all current propeller design methods.

Similarly, it will be shown that various forms of momentum theory and the related actuator disk concept can be used to extend the application of linearized (vortex theory) solutions to the range of practical interest. It is realized that this necessitates some sacrifice of mathematical rigour, but here again the justification lies in the practical results that are obta i ned.

The solutions to ship propulsion problems treated here, have In common that momentum or actuator disk theory is an essential step towards the achievement of useful results. Chapter 1 presents a free actuator disk theory which allows

for slipstream deformation.*) The numerical results obtained with this theory show that the application of linearized actuator disk theory should be limited to thrust coefficients below 2.0 (the typical range of C.|.-values for large tankers being 2.0 - 3 - 0 ) .

In Chapter 2, the variable load linearized actuator disk concept is applied to the design of counter-rotating propellers, Ref. [ 6 ] . Chapter 3 discusses its use in calculating the inception of cavitation of shaft brackets, Ref. [ 7 ] .

Chapter 4 presents a shrouded actuator disk theory which takes into account slipstream deformation. In Chapter 5. the results of this theory are applied to calculation of the performance of heavily loaded ducted propellers.

*) Slipstream deformation or deterioration should not be confused with slipstream contraction. Slipstream deformation refers to the downstream variation of both the radius and the pitch of the free vortices shed from the propulsor, whereas slipstream contraction refers only to their rad i us.

Chapter 6 deals with the effect of a nozzle on steering characteristics, Ref. [ 8 ] . The results of momentum theory, for both open and ducted propellers, are used to allow for the effect of the propeller upon the flow onto the

lifting surfaces which determine the steering characteristics.

As mentioned earlier, momentum theory provides insight into the main action of a propulsor, in particular in regard to the various losses which determine the efficiency. These are made up of axial, rotational and frictional losses and in the case of non-optimum design, additional losses

in these three categories occur. The axial and rotational losses of any propulsion device can conveniently be estimated with momentum theory, which can be useful in evaluating the prospects of new types of propulsion. Momentum theory can also provide other guidance in the development of new propulsion concepts. The innovations discussed in Chapters 7 and 8, relating to the application of slotted airfoil sections to propeller shrouds, were developed on this basis, Refs. [ 9 ] , [ 1 0 ] .

The practical applications discussed in this work show that momentum theory and the related actuator disk concept contribute significantly to the solution of various ship propulsion problems. In the light of these applications, there appears to be good reason for a thorough reappraisal of momentum theory in the

1. FREE ACTUATOR DISK

1.1. SLIPSTREAM OF A HEAVILY LOADED FREE ACTUATOR DISK

The slipstream boundary of a heavily loaded actuator disk consists of a tubular vortex sheet which is of the nature of a close succession of vortex rings, Ref. [ n ] . This concept corresponds to the assumption of an infinite number of blades and uniform loading over the actuator disk, rotating with an infinite angular velocity. The latter coincides with the case that the actuator disk or the propeller rotates with

infinite angular velocity while a stator is used to eliminate the rotational losses.

> - ^

Fig. 1. Slipstream of a heavily loaded actuator disk

Fig. 1 represents a section of the slipstream. At axial station x the velocity just inside the slipstream is v. and just outside the slipstream v^. The strength per unit length of the vortex sheet representing the slipstream boundary is:

Y = V. - v+ . (1.1)

The fluid element which constitutes this element of the vortex sheet moves along the slipstream boundary with a velocity:

V = i (v- + v+) (1.2)

The vorticity of this fluid element remains constant as it passes along the slipstream boundary, although its length is increasing, and hence the vortex strength and velocity along the vortex sheet are governed by the equation:

Y V = constant . (1.3)

The axial and radial velocities, u and v, induced at field points (x, r) by the tubular vortex sheet are:

u (x, r) = 2TI 5=0

rJi^Jll. ü ^^^-^ , -1 dC

P Y \ P P / (1.4) and V (x, r) = 2TI i=0 Y (f.. P) . \7 /x - S I Y \ P P dC . (1.5)

The Influence functions, U and V , for an isolated vortex ring, at

Y Y s>

the origin (C = 0) and of unit radius (p H 1 ) , can be written,

Ref. [12], as: _ 1 U (x, r) = V x ^ + (r + 1)2 K (k) - [1 + 2 (r - 1) -| ^ (|,j x2 + (r - 1)2 (1.6) and V (x, r) Y r V x 2 + (r + 1)^ K (k) - [1 + 2 r x2 + (r - 1)-] E (k) (1.7) where: x2 + (r + 1)=

and K (k) and E (k) are complete elliptic integrals of the first

and second kinds.

For field points (x, r) on the vortex sheet the integrals of Eqs. (1.4) and (1.5) are to be taken in the sense of the Cauchy principal value.

The velocity V on the vortex sheet is:

V (x, R) = \/ [y + u (x, R)]2 + V (x, R)2 (1.8)

Furthermore, as the slipstream boundary must be a streamline:

d R ^ V (x, R) (1,3) ^ '^ V^ + u (x, R)

Application of Bernoulli's law from far upstream to just forward of the actuator disk and from just behind it to far downstream provides the relation between the thrust coefficient and the jet velocity far downstream: ''•'^' 1 , (1.10) where: ^T and jp "A V. = jet velocity, T = thrust,

R„ = actuator disk radius.

The corresponding strength y per unit length of the vortex sheet far

downstream is:

Eqs. (1.3) to (1.11) completely describe the slipstream of a heavily loaded actuator disk. One Is primarily interested in determining the

slipstream vortex distribution y (x) and the shape R (x) for any

thrust coefficient C-. For this purpose an iterative solution was programmed for a high-speed computer. Since better convergence was

expected when using the law of continuity, Eq. (1.9) was replaced by: rR (x)

[v. + u (x)] r dr = constant . (1.12)

The iterative process starts with the assumption of no slipstream deformation:

^ ''^^ = ^" • (1.13) R (x) = R Q .

In each cycle, the velocities induced on the slipstream are calculated for a number of axial stations, using Eqs. (1.4) to (1.8). The resulting velocity distribution along the slipstream boundary, together with Eq- ( 1 ' 3 ) . provides the new estimate for the vortex distribution y ( x ) . The axial velocities induced in the slipstream (at the same axial

stations) are calculated from Eqs. (1.4) and (1.6). The law of continuity, Eq. (1.12), then provides the new estimate for the slipstream shape R ( x ) , and the process is repeated until sufficient convergence is attained.

In classical one-dimensional momentum theory is assumed that the unknown pressure Integrals over the control surfaces vanish. The law of

momentum then yields:

where:

The relation between the contraction * and thrust coefficient C_ Is obtained by elimination of V. from Eqs. (1.10) and (1.l4):

. 1 "^T

2 ,, ,, • (1.15)

(- 1 + Vl + Cp) Vl + Cj.

The contraction ratio approaches unity when the thrust coefficient tends to zero. On the other hand, when approaching the static

condition, C , tends to infinity, and i, according to Eq. ( 1 . 1 5 ) , to 1/2.

1.2. LIGHTLY LOADED ACTUATOR DISK

The first cycle of the Iterative procedure corresponds to the classical momentum theory in which the ring vortices are assumed to be of

constant strength and diameter (Eq. (1.13)).

This assumption is applicable only to the special case of light propeller loading. The semi-infinite integration of Eqs. (1.4) and (1.5) along the C'axis (Fig. 1) can then be performed analytically. The solutions can again be expressed in complete elliptic integrals, Ref. [ l 3 ] , and are for a vortex cylinder of unit strength and unit radius:

u (x, r) = u* + 2TI V x 2 + (r + 1)2 where: [ K (k) (r - 1) (r + 1) V. (a2. k) ] (1.16) u* and V (x, r) 1/2 = 0 1/4

^

for r < 1 r > 1 r = 1 2 Ti k2 V x ^ + (r + 1) = [ E (k) - (1 - ^ ) K (k) ] (1.17)

Here K (k), E (k) and fl (a2, k) are complete elliptic integrals of the first, second and third kinds, respectively; k is defined as in Eqs. (1.6) and (1.7). and a by:

4 _{for ^2 I ,} (r + 1)^

For field points (x, 0) on the axis of the seml-infInite vortex cylinder having strength y and radius R one obtains:

u (x/R„, 0) = 1 f W , , ^^'° = ) . (1-18)

^ V V 1 + (x/Ro)V

The limiting values of Eq. (I.l8) are exactly as predicted by momentum theory. Combining Eqs. (1.11) and (1.18) yields the velocity induced

on the axis at x by a lightly loaded actuator disk at thrust coefficient Cy. It will prove useful to define this velocity as a reference velocity with respect to which average axial induced velocities in the slipstream

are normalized:

V ' + ( ^ / R Q ) ' where:

( - 1 + V 1 + Cy )

In this way, the effects of loading (Cj.) and of distance from the actuator disk (x/R ) are for a great deal incorporated in the

reference velocity according to Eq. (1.19), so that the normalized average velocities in the slipstream remain fairly constant up- and downstream.

1.3. NUMERICAL PROCEDURE

A computer was programmed to perform the iterative process described in Section 1.1. The purpose of the program was to determine the vortex distribution, Y (x), and the slipstream boundary, R (x), of the actuator disk, and also to calculate the induced velocities at a number of locations of practical Interest.

The slipstream was assumed to have reached Its final shape at ten times the disk radius downstream. The following l8 axial stations were selected in which Eqs. (1.3) and (1.12) were to be satisfied:

x/R = 0.0, 0.1, 0.15, 0.2, 0.3, 0.4, 0.5, 0.6, 0.8, 1.0, o

1.5, 2.0, 3-0, 4.0, 5-0, 6.0, 8.0, 10.0. (1.20)

The vortex strength, y (x), and slipstream shape, R ( x ) , at other axial

locations were obtained by a 4-point Lagrangian logarithmic interpolation: 4 4 ( I n x - l n x . )

f (In x) = I Vl ^— • f (In x,) . (1.21)

i=l j=1 (In X| - 1 n x J)

The selection of Eqs. (1.20) and (1.21) was based on the consideration

that the variables involved decay more or less logarithmically with the distance from the actuator disk.

For the case of velocities induced on the slipstream boundary, Eqs. (1.6) and (1.7) become singular for (x - C)/p = 0, r/p = 1. Therefore, an

X - £

interval - 0.002 < ^ < 0.002 was excluded from the integrations; the contribution of the excluded vortex sheet to the radial Induced velocity is zero, that to the axial velocity was computed taking k = V'*/(0-001^ + 4) in the evaluation of the elliptic Integrals.

The integrations were performed by using Simpson's rule in a way similar

to that described in Ref. [l4]. The integrations are approximated by a

summation of algebraic terms: 1 '^

I = ^ I \ [ f iKu.) + 4 f (Ck + h,^) + f (f,k + 2 h,,) ] . (1.22)

^ k=l

The sub-intervals were increased in width in geometric progression. The first term was taken as:

Here (C., p.) is the closest point of the integration Interval to the field point (x, r ) . Subsequent terms were taken as:

\ = q h^ ,

where:

^

Sufficient numerical accuracy could be obtained by taking:

q = 1.5 for the axial Induced velocities q = 1.2 for the radial induced velocities

Semi-infinite integrations were terminated whenever a term was encountered which was smaller than 10 .

The number of cycles was limited to 5, since the results of Ref. [l4] indicate that little improvement should be expected from running more.

The accuracy of the numerical procedures was checked by comparing the first cycle results with independent results for the semi-infinite vortex cylinder obtained with the formulas described in Section 1.2.; the results were identical, to four figures.

The convergence becomes more critical the larger the thrust

coefficient Cj.. At C = 100 the convergence was such that the radius of the ultimate slipstream according to the 5th cycle remained within 0.5% as compared with the value according to the 4th cycle. The largest variation in the product y V was found to remain within 61 in the 5th cycle. To improve this figure a finer spacing of the control points on the slipstream than that given by Eq. (1.20) would be

necessary, which would, of course, require correspondingly more computing time.

1.4. NUMERICAL RESULTS

Fig. 2 shows how the shape, R ( x ) , of the slipstream boundary varies with the thrust coefficient, Cj.; Figs. 3 and 4 show the corresponding variation of the vortex distribution, y ( x ) .

17

1.0 o ^—o C ' o <l -<) () 1—o ,. ,.

t—

^<^y~~~p-—o-I;

^ — <

^~r^

o—,;

• — 1 ~ ;,

> -0 o —: > c

_l

'—f—'

1 ' — o 0 0 0 1 — o

1

-(1 <) () () 0.5 2. 5. 10. 20. 50. 100. 0.2 O."! 0.6 0.8 1.2 1.'4 l.fc 1.3 2.0 2.2

Fig. 2. V a r i a t i o n w i t h thrust c o e f f i c i e n t o f a c t u a t o r d i s k s l i p s t r e a m shape

0.2 a.k 0.6 0.8 1.0 1.2 1.1. 1.6 1. 2.0 2.2

Fig. 3. V a r i a t i o n w i t h thrust c o e f f i c i e n t of v o r t e x d i s t r i b u t i o n on a c t u a t o r disk s l i p s t r e a m b o u n d a r y (range of low C )

1

\

\

V

k

^

^

' c CT = t o o . 50. < 2 , 0 2 . 2 F i g . k. V a r i a t i o n w i t h t h r u s t c o e f f i c i e n t o f v o r t e x d i s t r i b u t i o n o n a c t u a t o r d i s k s l i p s t r e a m b o u n d a r y ( r a n g e o f h i g h C_)

A comparison between tlie c o n t r a c t i o n according to the (two-dimensional) i t e r a t i v e procedure and the value according to Eq. (1.15) of

one-dimensional momentum theory is presented in F i g . 5- i t is apparent t h a t , as a r e s u l t of the assumption t h a t the pressure i n t e g r a l s over the c o n t r o l surface v a n i s h , Eq. (1.15) underestimates the c o n t r a c t i o n at very high t h r u s t c o e f f i c i e n t s .

—

-—

. - ^

ONE s a -DIMEM ICEDURE . 7 1

\ ^

TERATIVE

K

_^

N

_^

7

^

10

\

)

•

^ \

0 . s

—

^

7

—

. IDS F i g . 5 . V a r i a t i o n w i t h t h r u s t c o e f f i c i e n t o f f r e e a c t u a t o r d i s k s l i p s t r e a m c o n t r a c t i o n 19

0.2 0.4 0.6 0.8 I.O 1.2 1.4 1.6 1.8 2.0 2.2

" -'"„

Fig. 6. Variation with thrust coefficient of mean axial induced velocity in actuator disk slipstream

The variation with thrust coefficient C of the average velocity induced in the slipstream is given in Fig. 6. These data are useful for to determining the approximate effect of the propeller on the flow onto rudders placed in the slipstream. The results of classical actuator disk theory have been used in ship steering problems by several authors. The present results extend the application to the range of high thrust coefficients (low manoeuvring speeds).

Classical linearized actuator disk theory, either in its original uniform load concept or as the variable load disk, Ref. [ 1 5 ] , is used In most current theories for ducted propellers. To determine the range of propeller thrust coefficients where the assumption of light loading is justified, the velocities induced at the cylinder r/Rjj = 1.05 surrounding the actuator disk were calculated for various propeller thrust coefficients. The results are given in Figs. 7 to 9.

It appears that linearized actuator disk theory considerably

over-estimates the induced velocities at higher propeller thrust coefficients. Therefore, whenever the propeller thrust coefficient exceeds approxi-mately 2.0, slipstream deformation should be taken into account.

Fig. 7. V a r i a t i o n w i t h thrust c o e f f i c i e n t of a c t u a t o r - d i s k - i n d u c e d axial v e l o c i t i e s on c y l i n d e r f^„/R = 1.05 (forward of a c t u a t o r d i s k ) Ü o — ^ ^ ^ i : : : : ^ *--^ X / R Q = 0 . 1 2 5 ^^^i^J^f-^ • • - • ^ . 5 -~.^.7i

,

. . J L

- ^ — ^ 3 ^ > :5 > VA

\ . , ^ ^ - FREE ACTUATOR DISK \ N ^ ^ - FIELD POINTS AT r/R = 1.05

\

\

\

\

^

\ ^ ^ v V : : : ; ^ : : : : : ^ ^ ^ ^ - ^ = ^

Fig. 8. V a r i a t i o n with thrust c o e f f i c i e n t of a c t u a t o r - d i s k - i n d u c e d axial v e l o c i t i e s o n c y l i n d e r R„/l^ = 1.05 (aft of a c t u a t o r d i s k )

' D o

0.1 Q.2 0.5 1. ï. 5. 10. 20. 50. 100.

Fig. 9. V a r i a t i o n w i t h thrust c o e f f i c i e n t of a c t u a t o r - d i s k - i n d u c e d radial v e l o c i t i e s on c y l i n d e r R„/R = 1.05

D o

The results of Figs. 7 to 9 can be used in ducted propeller design and analysis to allow for slipstream deformation, provided the effect of the duct induced velocities upon the slipstream shape and vortex distribution can be neglected. This implies that, although the propeller thrust coefficient, C , may be high, the duct thrust coefficient, C , should be low, or T = C_p/C_ = 1. This situation is applicable to ducted or ring propellers with an extremely low chord-diameter ratio. For usual ducted propeller configurations, h o w e v e r , the effect of the duct induced velocities upon the slipstream deformation at high propeller thrust coefficients is significant. This will be elaborated in the shrouded actuator disk theory presented in Chapter 4.

1.5. CONCLUSIONS

1. Slipstream deformation becomes significant at thrust coefficients exceeding approximately 2.0. Since the range of interest of shrouded propellers extends to far higher propeller thrust coefficients, linearized actuator disk theory does not suffice

2. Linearized actuator disk theory should be used only at propeller thrust coefficients below approximately 2.0. Since, in such lightly loaded (high-speed) applications, the radial load distri

butlon is generally of interest, the variable load actuator disk model, as discussed in the next chapter, is preferable.

3. The numerical results presented in Section 1.4. (Fig. 6) are useful in assessing the approximate effect of a heavily loaded propeller upon rudders placed in its slipstream.

4. The results presented in Figs. 7 to 9 can be used in short-chord heavily loaded shrouded propeller theory.

COUNTER-ROTATING PROPELLER DESIGN

INTRODUCTION

The trend towards higher powers has given considerable impetus to research in counter-rotating propellers. This type of propulsion is especially attractive for high-speed vessels because of its advantage in efficiency and cavitation properties. Nevertheless, widespread adoption of counter-rotating propellers has, until now, been prevented by mechanical difficulties. Although the major problems seem to be on the mechanical side, the hydrodynamic aspects, to which our discussion is limited, are also extremely important because the expected saving in fuel costs and in particular the better cavitation properties may play a decisive role in future applications.

Since a comprehensive review of the state of the art has been given by Hadler, Ref. [ 1 6 ] , we shall limit the following discussion to some other pertinent references.

The basic problem in the design of counter-rotating propellers consists of the calculation of the mutually induced velocities. The most widely used design method is the induction factor method of Lerbs, Ref. [17]> which has been elaborated by Morgan, Ref. [ l 8 ] . The ratio between the velocity induced at a finite axial distance and the self Induced velocity at any radius of an infinitely bladed propeller is expressed

in a distance factor. It is assumed in Ref. [l7] that the distance factors can be calculated with the actuator disk model; i.e. the distance effect of the propeller with arbitrary radial load distribution is assumed to be the same as that of a uniformly loaded disk. The distance effect depends in fact on the radial circulation distribution of the propeller. This has been clearly shown by Tachmindji, Ref. [ 1 9 ] , who calculated the distance factors for the optimum infinitely bladed propeller; these factors were used in Ref. [ l 8 ] .

Since counter-rotating propellers find their application at high speed and high power, cavitation is an extremely important matter.

The local loading of the blade sections should therefore be predicted as precisely as possible, which can be done only when the actual radial

load distribution is taken into account in the calculation of the distance effect.

The idea of using design methods for single propellers combined with appropriate corrections on intake velocities is elaborated in the method of Van Manen and Sentic, Ref. [ 2 0 ] . In this method the mutually

induced velocities are determined on the basis of measurements. Its validity, therefore, is limited to circumstances similar to those of the measurements. Whenever conditions, such as distance between the two propellers and radial load distribution, are different from those of the tests, substantial deviations in local propeller loading can be expected. The tangential velocities induced at the aft propeller are taken into account by a correction on the rotational speed of the shaft. This implies that the tangential velocities are assumed to increase

linearly towards the tip. In reality, however, there is a decrease towards the tip which makes a reduction of pitch necessary at the root sections of the aft propeller.

An attempt to modify the method of Lerbs so that specific design method for wake-adapted single propellers can be used for the design of each propeller has been made by Glover, Ref. [ 2 1 ] . The solution for dealing with the radially varying rotational speed of the aft propeller applies only to the particular design method for single propellers that was used. As in the method of Lerbs, use is made of the standard distance factors discussed previously.

A lifting surface theory for counter-rotating propellers has been formulated by Murray, Ref. [ 2 2 ] , but numerical results are not yet available. In Ref. [23] a linearized lifting surface theory is given which is based on the so-called quarter point method.

The following considerations are relevant to the method presented in this chapter:

1. The slipstream contraction and in particular the change in static pressure due to the interaction between both propellers should be taken into account (see the results of the cavitation tests of Ref. [2^1]). Momentum theory provides a simple means to allow for these non-linear effects. We therefore apply momentum theory for the calculations of the mean mutually induced velocities and pressures.

2. The radial distributions of the mutually induced velocities may then be calculated with a linearized lifting line concept, such as the generalized actuator disk of Ref. [ 1 5 ] .

The present discussion is limited to these aspects, lifting surface correc-tions on mutually induced velocities being left out of consideration.

In principle, chordwise effects should be taken into account in the calculation of the mutually induced velocities because of the blade areas envisaged for counter-rotating propellers. This can be done by extending the present method with lifting line - lifting surface corrections, based on linearized theory as in the design of single propellers. These corrections on the mutually induced velocities consist of circumferential averages of the velocities at various chordwise stations

induced by:

(i) the lifting lines of the other propeller, minus the values at the lifting line itself,

(ii) the vortex distribution representing the difference between the lifting surface and the lifting line vortex distribution of the other propeller (this vortex distribution is limited

to the blade area of the other p r o p e l l e r ) , (iii) the thickness effect of the other propeller.

2.2. THEORY

I!3§.Xël9£iJï_fifl^-2f.SC-iDfi!3iïiiï-^I§^?^-BÜ2Bêllf E_tfi£!3-§t^|ïn§nï

When designing a set of counter-rotating propellers use is made of the concept of an infinitely number of blades. Owing to the finite number of blades In a practical propeller the mutually Induced velocities are unsteady. When determining the shape of the propeller only the steady components, which are the same with a finite or an infinite number of blades, are of interest. It should be noted that the circumferential

irregularity of flow due to both the wake of the ship and the presence of a second propeller should be considered when determining margins against cavitation.

A description of the vortex system of an infinitely bladed propeller with arbitrary radial load distribution and its velocities in an

ideal fluid was given by Hough and Ordway, Ref. [ 1 5 ] . Results of their variable load actuator disk theory which pertain to our discussion can be summarized as follows.

The vortex system of an infinitely bladed propeller is made up of:

An infinite number of horse-shoe vortices consisting of radial bound vortices in the propeller plane and free rectilinear vortices trailing aft from the ends of the bound vortices. These vortices Induce tangential velocities only.

A semi-infinite tube of ring vortices. These vortices induce axial and radial velocities.

For the present discussion the deterioration of the vortex system due to the induced velocities themselves is neglected. As before, axial and

27

radial coordinates are denoted x, C and r, p for field points and

singularities respectively (Fig. 1 ) . The strength of the free rectilinear vortices per unit disk area is:

, d r (p) d P 2 ÏÏ p (2.1) where: r = blade circulation, Z = number of blades, $ = angular coordinate.

The bound vortices do not contribute to the induced velocities. By Integration of the Blot and Savart law or by applying Stokes's law the tangential velocity field is found to be:

U-p (r) Z r (r)

U T ( O

2 IT r V „

= 0

in the siipstream

outside the slipstream

(2.2)

The strength of the ring vortices per unit area (in the x-r-plane) is:

^ d r (p) d p

TTTïïr

(2.3) where: P. = hydrodynamic pitch.

The radial velocities induced by the ring vortices determine the slipstream contraction. Since the slipstream contraction can more easily be calculated with the law of continuity, only the axial velocities are of interest. According to the Blot and Savart law an elementary ring vortex at C with radius p and of strength y dj dp

d U„ (x, r) Y^ dC dp

—Trr~

2 TT (p2 - r p coS(t) d (fi ( 2 . M {(x - 5)2 + (r - p)2 + 2 r p (1 - cos*)}3/2

Integration along the x-axis yields the axial velocity induced by a semi-infinite vortex cylinder with radius p:

d U^ (x, r) = u (x/p, r/p) • y^ "^P (2.5) where: u (x, r) = h TT C=0 d5 . 2 IT (1 - r cosiji) {(x - 5)2 + (r - 1)2 + 2 r (1 - cos*)} 3/2 (2.6)

The function u (x, r) gives the ratio between the axial velocity induced at the field point (x, r) and the velocity induced at infinity within the vortex cylinder. As mentioned in Section 1.2, Eq. (2.6) can be reduced to complete elliptic integrals (Eq. (1.16)). A high-speed computer was programmed for numerical evaluation of the function u (x, r ) ; Fig. 10 shows some results, which agree with calculation of others, Refs. [12], [13].

! . 2 5 2 • —

_{- ^}

^ 5

>v^^ \ s

^

5

" ^

^

f

V

%

c

r/o) r/0

-4

1

-

^

^

> - — y'^'

.'.

.,

^ 0 . 0 ^ 0 . 6 ^ 0 . 8 ^ 1 . 2 ^ 2 . 0 *~~--~.^ ^ 5 - ; - 2 5 - 3 .

F i g . 1 0 . A x i a l v e l o c i t y f i e l d o f a hal f-i nf i Tii te v o r t e x c y l i n d e r

The axial velocity field of an infinitely bladed propeller is obtained by integration of the contributions of semi-infinite vortex cylinders at various radii which, In conjunction with Eq. ( 2 . 3 ) , yields:

U , (x, r) \^A r^ u (x/p, r/R) u (x/p, r/p) d r (p) P - h ^ P; (p) dp dp ^h • " ^^/^h' ^/^h^ (2.7) w h e r e :

R = propeller tip radius, and

the subscripts h and t refer to hub and tip respectively.

Eqs. (2.2) and (2.7) completely describe the tangential and axial velocity field of an infinitely bladed propeller and could be directly used in counter-rotating propeller problems. However, for the reasons mentioned earlier, we shall first establish a link with momentum theory, using the following definitions of mean values:

Mean c i r c u l a t i o n

r ( r ) r d

r d r

'J

( 2 . 8 )

Mean axi a 1 veloc i ty :

UA, ^R U^ (r) r d r r d r

\

(2.9)

Mean tangential velocity :

U.^ (r) r2 d

r^ d r

(2.10)

In the case of an actuator disk in which the circulation is

constant from radius zero to the tip, only the first term in Eq. (2.7) Is left. Furthermore, the integration In Eq. (2.6) can be performed analytically for field points on the propeller axis (x, 0 ) , which

yields Eq. (1.18). Therefore, the velocity induced on the axis according to Eq. (1.19) will also be used here as a reference velocity U» , (x)

"ret to which w e shall relate the axial velocities at arbitrary field points

(x, r) induced by an arbitrary circulation distribution. The reference velocity Is here defined as the velocity at (x, 0) on the propeller axis Induced by an actuator disk having the same mean circulation, i.e. producing the same thrust per propeller disk area. With the assumption of constant hydrodynamic pitch, Eq. (2.7) then can be written a s :

U^ (x, r) "ref u (x/R, r/R) R d ^ <P) mean dp u (x/p, r/p) dp u (x/r^, r/r^) /u (x/R, 0) , (2.11) where: IT (x/R, 0) = (1 + x/R

V l + (x/R)'

It should be noted that Eq. (2.11) Is a simplification of Eq. (2.7) and

strictly valid for lightly loaded propellers only. An iterative design procedure using Eq. (2.7) instead of (2.11) could be programmed without much difficulty. Nevertheless we consider It appropriate to use

Eq. (2.11) because counter-rotating propellers are especially attractive at high speeds and consequently light loading. For such wide-bladed propellers the inaccuracies due to neglecting chordwise effects In the calculation of the mutually induced velocities can be expected to be more important than those resulting from the simplifications in Eq. (2.11).

Eq. (2.11) indicates that the axial velocity induced at an arbitrary field point depends on the reference velocity U» • (x) and the radial

"ret

circulation distribution. If the hub diameter is zero, the reference velocity becomes: Ufl f (x/R) "ref \/A 1 + x/R

V ' + (x/R)'

(2.12) where: and (-1 + V l + Cy)

ip V I °'

If the hub diameter is not zero, then the available propeller disk area and the actuator disk area defining the reference velocity are, for usual hub sizes, slightly different. With equal mean pressure jumps for both the propeller and the actuator disk, the thrust of the actuator disk is larger. We allow for this effect by defining the thrust constant C.^ as fo 11ows: ^T where: (2.13) ip V A 2 i Ü 2 1 - (d/D)' d = hub diameter D " propeller diameter.

We shall now consider the mean axial velocity induced in a disk at x. The ratio U. /U^ , can be computed with Eqs. (2.9) and (2.11), from which It can be seen that this ratio depends only on the radial

circulation distribution. We have computed U A /l^Ar f °" ^ high-speed computer for several circulation distributions (constant, elliptical,

Increasing towards the tip, decreasing towards the tip, etc.) at various positions x. The type of circulation distribution proved to have little Influence on the mean axial induced velocities, the largest differences obtained having been within '4^. Consequently, the

ratio U A / U A , as a function of the axial coordinate x may be "mean "ref

calculated once and for all starting from a standard circulation distribution. The result of such a calculation, carried out for an elliptical circulation distribution and a hub diameter ratio 0.2, is given in Fig. 11.

2. 1.5 1. .5 O -.5 - 1 . -1.5 - 2 . . ^ x/R

Fig. 11. Ratio between mean induced axial velocity and reference velocity

The mean axial velocity is larger than the reference velocity behind the propeller and smaller in front of it, which is In agreement with Fig. 10. Keeping in mind the order of magnitude of the induced velocities as compared with the main stream, we may state that the error in the intake velocity introduced by using the results of Fig. 11

for arbitrary circulation distributions is within 0.51. T h u s , the mean Induced velocities may be calculated with axial momentum theory and the function F (x/R) as given in Fig. 11. The radial distributions of the induced velocities are of course strongly dependent on the

radial circulation distribution and are to be calculated with Eq. (2.11).

Similar reasoning holds for the tangential velocity field. The mean rotation is determined with momentum theory, Ref. [ 2 5 ] , in which the assumption is made that the slipstream behind the propeller rotates

like a solid cylinder. The maximum value of the tangential velocity,

Uj , wi11 then occur at the slipstream boundary. The propeller torque, Q, equals the moment of momentum impressed on the fluid passing through

'max where: n Cn = -^ , (2.11.) iP V,2 1 D3

The relation to the mean induced tangential velocity is:

U T = 3/^* U T . (2.15) 'mean 'max

The radial distribution of the tangential velocities can be calculated with Eq. (2.2).

The induced velocities have been obtained neglecting the effect of slipstream contraction. In order to allow for this, we assume the radial coordinate to be expressed as a fraction of the radius of the slipstream boundary at the axial position under consideration. In this way vortices that are at the outside or, conversely, the Inside of a field point in the linearized concept so after allowing for contraction. As mentioned before, the slipstream contraction is determined by using the law of continuity.

9ê5l9D-B!I9£ê^yr§

Having established the basis of counter-rotating propeller design, namely the velocity field of the infinitely bladed propeller, we can now develop our design procedure. A review of the procedure is given in flow-chart form in Fig. 12.

ï

ï

1«

7

S

CST-'JiLlSM bASlC DLSir.!, P.-.r/.-LTiir'.

LSTir/,Tt; TunuSTs

rj-At. A / I A L MUTUALLr liUUCLÜ V t U I X l T I L S VilTK t U S . C2.1C) AIJJ ( 2 . 1 7 J

SLIPSTRC//; COf-TWCTIOt. W[TN t C . ( 2 . 1 3 3

MUTüTiLLY IlilXJCEU PItESSURCS I.ITII t y s . C i J . T j ) Ailü C ï . 2 0 )

f

KA^'l-L . i i l l L . _ i l , - . i -'• K.XI AL MUTUALLY li.UUi.L_' VLL;.i.l : i t ' . . CLi,. C M C ) OP

1

lUTAKE VL-LOCirrcS CORP-LXTED FOP hAW: AliJ lUTEkFEREIiCE EFFrCTS U I T l l Et>. C ï . 2 1 )

1

ÜtSIGI. FÜRW/WJ PROPELLER

THRUSTS AID CIKCULATIOTJ DISTRIBUTiÖ.5 SLT ACCORUIIJT, TC L LOCKS a MV 10 riOT SATISFACTOPY 10 n 12 13 T . T

COP.RcCTic;. t l , ^.r.r. OF /i^T p^i-'PrLLrr FOR SLIPSTREAM P0TATI0t:5 WITH EQS. C 2 . 2 2 ) A;.D ( 2 . 2 3 )

DESJOr. AFT PPCPLLLEW

1

RADIAL D I S T R I E U T i a , CF TAIT-ErjIAL VELOCITIES AT AFT PPOPELLEP PJDUCfD CY F0R1ARD PKüPELLER /'ITH EC. C2.2't3

1

CORRECirCf. or: PITCH DISTr.lF.UTIOi: OF AFT PROPELLER FCP. RADI/^L DISTRIPUTICii OF TAJGDITIAL VELOCITIES IViTH EC. C2.25)

„„--''''CGMPARE THKUSTS^""---^^^ W Ü PADIAL ClftCULATIOt: ^ " - ^ DISTRIBUT10>;S KITH ^ . . . ' ' ^ ''~~~'-~~.J^\J-\ïXi V A L U E S , - ' - ' ' ' ^ J SATISFACTORY PfiUlT RESULTS

Fig. 12. Flow chart of counter-rotating propeller design procedure

At first the basic design parameters have to be established (block 1 of Fig. 1 2 ) :

advance velocity and wake distribution,

diameter of forward propeller,

power and r.p.m. of forward and aft propeller,

axial distance between the propellers,

single-propeller design parameters, such as number of blades, static pressure at shaft height, margins against cavitation, admissible static stresses, etc..

Advance velocity and wake distribution are determined as in the case of a single propeller. If the rotational speed Is not prescribed, the diameter of the forward propeller should be chosen as large as can conveniently be fitted In the aperture. According to momentum theory the rotational losses are zero if the torque of both propellers Is the same. Then the propellers will also have approximately the same optimum rotational speed. The usual practice of selecting equal power and r.p.m. for the two propellers is therefore justified with regard to propeller efficiency. The present design method has nevertheless been developed for arbitrary distribution of power and rotational speed. Numerical calculations with the present method for systematically varied distributions of power and r.p.m. Indeed showed the highest efficiency at equal power and rotational speed. However, this conclusion seems to hold true for the open water condition only; it is reported In Ref. [l6] that more power should be assigned to the forward propeller for optimum efficiency in the behind condition. Apparently, hull-propeller interaction

Is Important In this respect.

.75 .70 .65 .60 .55 .50 .•(5 .3 .'t • .5 .6 .7 .8 .9 1- 1.1 » ~ J = n Di

Fig. 13- D i a g r a m for s e l e c t i n g o p t i m u m r.p.m. and e s t i m a t i n g e f f i c i e n c y of c o u n t e r - r o t a t i n g p r o p e l l e r s

Optimum diameter or optimum r.p.m. can most conveniently be determined with a systematic propeller series, because the optimum design points are to only a slight extent dependent on blade area ratio, number of blades, etc.. Diagrams for this are given In Refs. [2'(] and [26]. Usually the r.p.m. are selected slightly above the optimum value for which the diagram given in Fig. 13 can be helpful. The diagram Is based on the series given in Ref. [21] supplemented by some unpublished data. Any optimum choice can of course be checked by making several complete designs and comparing the efficiencies. The choice of the number of

blades should be based entirely on the vibratory output of the two propellers and Its interaction with shafting and gearing. It should be

noted that there Is no need to assign the higher number of blades to the aft propeller in the interest of efficiency (see Table 3 of the next section).

In order to be able to apply the momentum theory relations an estimate

has to be made of the thrust of forward and aft propeller (block 2 of Fig. 12); this can be done with the aid of Fig. I3. As an error of

even 15% appears to have no noticeable impact on the design, this estimate need only be rough. The factors F (S/R ) and F (- S/R ) relating the mean axial mutually Induced velocities to the reference velocities are obtained

from Fig. 11, assuming at first R_ = R (block 3 of Fig. 12). The mean axial mutually induced velocities are calculated using momentum theory relations: C •^1,2

'''' ' ^^ViV2^ . - ( ^

a,_2 = i ('1 ^ V l + Cj^ 2' ' UA ^^„ 1 / S/R, "mean 1 /, 2 V A \ V l + (S/R2)2 F, (- S/R^) , (2.16)

u

Amean 2 \/A S/R, 1 + V l + (S/R,) 2 F, (S/R,) (2.17) where:

S = distance between forward and aft propeller, and the subscripts 1 and 2 refer to forward and aft propeller respectively.

The diameter of the aft propeller is calculated with the law of

continuity (block 't of Fig. 1 2 ) . Taking Into account first order effects only, we obtain: '1 • 1 + ai + U A , / V . _{"mean 1 A} (2.1i 1 + ao + Ua mean 2 A

The mutually induced pressures are of Interest for the cavitation calculation (block 5 of Fig. 12). The forward propeller operates in a region of negative differential pressure due to the aft propeller and conversely the aft propeller operates in a region of positive

differential pressure due to the forward propeller. The static pressure used for the cavitation calculation has therefore to be corrected with the mutually induced pressures. The correction can be derived from the velocity field. In general, the static pressure is of Interest only at 0.8R. At this radius the mutually induced velocities are approximately equal to the mean mutually induced velocities. We therefore derive the corrections on the static pressure from the mean induced velocities. From Bernoulli's law and taking Into account first order effects only, we obtain: S/R. 1 + V l + (S/R2) F, (- S/R2) (2.19) P VA S/R, 1 + V l + (S/R,)2 (S/R,) (2.20) and: 39

( P Q - e ) , _ 2 = P Q - ^ + P i , 2 •

where P - e denotes the static pressure without allowance for mutually

Induced effects.

From numerical calculations it appears that the allowance for mutually Induced pressures has an Increasing effect on the blade area of the forward propeller and, conversely, a decreasing effect on that of the

aft propeller, both of approximately S%- The radial distributions of

the axial mutually Induced velocities U. (r), ,/U, , . are obtained A 1,2 Amean 1,2

with Eqs. (2.9) and (2.11) from the given or assumed circulation

distributions r (r)^ ^/T ^^^^ ^ ^ (block 6 of Fig. 12). Distributions

associated with an elliptical circulation distribution which can be

used as a first approximation are given in Figs. ]k and 15.

1. .9 .8 .7 .6 .5 .'t .5 .2 .2 .Ii .6 .8 1. 1.2 m ^ "A f'-^'UA mean ' ^ '^O/T mean

Fig. II». Distribution of axial velocity induced at forward propeller by aft propeller with elliptic circulation distribution

F i g . 15. D i s t r i b u t i o n of a x i a l v e l o c i t y induced at a f t p r o p e l l e r by forward p r o p e l l e r w i t h 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

The mean m u t u a l l y Induced v e l o c i t i e s , t h e r a d i a l d i s t r i b u t i o n s as c a l c u l a t e d In b l o c k 6 o f F i g . 12 and t h e g i v e n wake d i s t r i b u t i o n , w ( r ) , d e t e r m i n e t h e a x i a l i n f l o w v e l o c i t y ( b l o c k 7 o f F i g . 1 2 ) : ^^A (--) _{1 , 2} = 1 - w ( r ) L'A " r t 1,2 ^A ( ^ ) l , 2 ^mean 1,2 ( 2 . 2 1 )

Now the forward propeller can be designed using any available design procedure for single propellers (block 8 of Fig. 1 2 ) . The aft propeller operates in the rotating screw race of the forward propeller, which

Implies that the rotational speed with respect to the incoming flow is higher than the rated r.p.m.. The correction on rotational speed, n ,

corr is obtained using momentum theory (block 9 of Fig. 1 2 ) :

-Q = Qi iP V A 2 ^ D , 3 1 + a, TT D l (2.22)

The moment of momentum should be preserved throughout the contracting slipstream, which gives:

(2.23)

where n denotes the nominal rotational speed of the aft propeller.

The aft propeller can then be designed according to a procedure for single propel

of Fig. 1 2 ) .

single propellers, using the increased rotational speed n (block 10

The radial distribution of tangential velocities induced by the forward propeller does not increase linearly towards the tip, as implied by a correction on rotational speed, but is related to the circulation distribution of the forward propeller. The distribution of the tangential velocities follows from Eqs. (2.2) and (2.10) (block 11 of Fig. 1 2 ) : r (r), Uj (r) rhJ R d r /R (2.24) r ( r ) , r d r

If the circulation distribution of the forward propeller is not known a priori, an elliptical distribution can be assumed as a first approximation (Fig. 1 6 ) .

~ ~ ~ ^ ^ ^ —-**'^ •^ >

\

\

r CrVr].„ CELLIPTIO y

^.A

^

1 /^

k

A

V

1

/

/

r d/D =0.2 "T <:r)/U„„„ ^

— —

\ ,

\ \ \

- ^

1. 1.2 l.it 1.6 Uj Cr)/i>r „ . „ . r C r V r „ , „

Fig. 16. Distribution of tangential velocity induced at aft propeller by forward propeller with elliptic circulation distribution

The actual distribution of the tangential velocities is taken into account by correcting the radial pitch distribution of the aft propeller (block 12 of Fig. 1 2 ) :

P ( r ) , n^/n, 1 + (n2/n2 - 1) ^ U T (r) 4 U. -/(r/R) P (r). (2.25) where:

P (r) denotes the pitch distribution of the aft propeller as resulting from the single propeller design procedure.

The thrusts and circulation distributions are then compared with the assumed values. If the agreement is not satisfactory we repeat the procedure starting with block 3 of Fig. 12. Usually no iteration is necessary when the design procedure for single propellers is based on

a prescribed circulation distribution; only one iteration is usually needed when the design procedure for single propellers is based on a prescribed pitch distribution.

2.3. NUMERICAL RESULTS

The method was applied in a computer program for the design of counter-rotating propellers In which the distribution of power and r.p.m. may be arbitrary. Lips's single propeller design procedure was used. This is a lifting line method with corrections on induced velocities for the following effects:

number of blades (and thickness e f f e c t ) ,

stream curvature (lifting line - lifting surface c o r r e c t i o n ) ,

propeller loading (slipstream c o n t r a c t i o n ) .

The corrections have been derived from a regression analysis of the Wageningen B-series, with the principal advantage that the mean pitch according to this method is in agreement with the value obtained from

the propeller series. This feature enabled us to check whether the present method yields correct power absorption. For this purpose we redesigned the systematic series of counter-rotating propellers that are given in Ref. [ 2 6 ] . The results of the calculation are

compared with the data from the experiments in Table 1. It appears that the agreement as to power absorption and efficiency is satisfactory for the higher advance ratios, which is the range of interest for counter-rotating propellers.

Set no. of counter-rotat ing propel Iers ace. to Ref. [26] 1 2 3 It All propellers; <T„^^^ = "-SS J 0.373 0.523 0.708 1.027 Q2/Q1 (n|=n2) 0.960 0.9'l1 O.9I19 0.958 '^Tcalc-0.376 0.379 0.366 0.368 contract ion Dj/D, meas. 0.891 0.912 0.935 0.958 calc. 0.852 0.895 0.92'! 0.953 dev. in t -li.lt -1.9 -1 .2 -0.5 pitch forward P, /D, '.7R ' meas. 0.893 0.972 1.104 1 .360 calc. 0.9111 1 .052 1 .122 1 .400 dev. in % 5.4 8.2 1.7 2.9 pi tch aft ^2.7R^°' meas. 1 .192 1.199 1.275 1.465 calc. 1 .204 1.189 1 .227 1 .425 dev. in % 1 .0 0.8 -3.7 -2.7 efficiency meas. 0.392 0.500 0.615 0.713 calc. 0.389 0.502 0.598 0.691 dev. in % -0.8 0.3 -2.8 -3.1

Table 1. Comparison of measured and calculated power absorption and efficiency

Since counter-rotating propellers are most attractive at large power and high speed, a 33 knot twin-screw container vessel with 60 000 h.p. per shaft was selected as a design example. These values of speed and power can be regarded to be not far from the limit in view of cavitation. Counter-rotating propeller designs for equal power and r.p.m. at aft and front propeller have been made for varying r.p.m.. The efficiency as a function of r.p.m. is plotted in Fig. 17, from which a rotational speed of 100 r.p.m. has been selected. The number of blades has been chosen as five and four for forward and aft propeller respectively. All other design parameters were the same as for the design of the 60 000 h.p. single propeller.

CALCULATED WITH D = 7.00 M STAMJARD RADIAL DISTRIBUTIONS V = 33 KNOTS OF HrrmLLY ItCUCED VELOCITIES w = 0.112

- ^

70 80 90 100 111

» . R.P.M.

F i g . 17. E f f i c i e n c y - r . p . m . r e l a t i o n s h i p f o r 60 000 h . p . c o u n t e r - r o t a t i n g p r o p e l l e r arrangement

Al 1 propel 1ers: Vg = 33 knots w = 0.112 Power [h.p.] R.p.m. Number of blades Diameter [m] Mean pitch [m] Pitch at 0.7R [m] Blade area ratio Weight of blades [kgf]

(Ni-Al-Bronze) Distance between propellers [m] Open water efficiency

Counter-rotatir Forward 30 000 100 5 7.000 10.916 11.077 0.539 16 067 1 . 0. g P ?5 761 ropellers Aft 30 000 100 4 6.778 11.155 11.492 0.485 14 140 Single pro-pel ler 60 000 135 5 7.000 8.436 8.547 0.909 28 314 0.687

Table 2. Comparison of 60 000 h.p. counter-rotating propeller design with characteristics of single propeller

A c o m p a r i s o n o f the p a r t i c u l a r s o f the c o u n t e r - r o t a t i n g p r o p e l l e r s w i t h those of the s i n g l e p r o p e l l e r is m a d e in T a b l e 2 . A f t e r c o r r e c t i o n for m u t u a l l y Induced p r e s s u r e s , all p r o p e l l e r s have the same c a v i t a t i o n m a r g i n . It can be seen that a s i g n i f i c a n t reduction in b l a d e area p e r p r o p e l l e r is o b t a i n e d by d i s t r i b u t i n g the power o v e r two p r o p e l l e r s . This shows that the c a v i t a t i o n p r o b l e m , w h i c h is the limiting factor as to power and speed for fast c o n t a i n e r ships and naval v e s s e l s , can be relieved to a great extent by a d o p t i o n o f c o u n t e r - r o t a t i n g p r o p e l l e r s

A - SIIJGLE PROP. B = FORliARD PROP. C = AFT PROP. B - ^

1

/

/

/

A

Lc

f PITCH

4.

\

I

MAKE

1

.2 .4 .6 .8 1. 1.2 .9 1. 1.1 ^ P Cr/Ryp„,j„ — 1 - w Cr)

Fig. 18. Radial distributions of pitch and wake of 60 000 h.p. configuration

The w a k e distribution and the pitch distributions are given in Fig. 18. The pattern of the velocities induced by the forward propeller leads to a pitch d i s t r i b u t i o n of the aft propeller which is increased at 0.3 radius and reduced at the very tip and the root o f the blades. The effect of the velocities induced by the aft propeller on the pitch distribution of the forward propeller is a slight decrease at the tip and a slight increase at the root. The circulation d i s t r i b u t i o n s ,

together with the associated distributions of mutually induced v e l o c i t i e s , are given in Figs. 19 and 2 0 .

- —

^

. ^

1^

y

^~~"—.

\

r c r ) ,

/

r

\

x\>^

\

è

/

/

/

j /

v^

\

\

/

UA C O J " A meanj UT ( r ) r (r), .3 I. U A C r ) , 1.2 U T Cr)

"••"I ^ mean2 ' iiean

Fig. 19. c i r c u l a t i o n distribution of forward propeller of 60 000 h . p . counter-rotating propeller arrangement and associated distributions of induced velocities at aft propeller

Fig. 20. circulation

distribution of aft propeller of 60 000 h . p . counter-rotating propeller arrangement and associated distributions of induced velocities at forward propeller

^

_^

y

^

'

r c r ) , r mean2

/

^

\

\

>

/

/

y

\

\

\

\

A

₁

V

/

\

/

"A CD, UA _{" mean,} ^A " I

49

Number of blades Forward 4 5 3 6 Aft 5 4 6 3 Eff ic iency

(calculated with standard radial distributions of mutually induced veloc i t i es)

0.7575 0.7530 0.7479 0.7483

Table 3- Impact of assignment of blade numbers on efficiency of 60 000 h.p. counter-rotating propeller arrangement

Design data: V = 20.5 knots w = 0.25 For wake r.p.m. aft/forward 1. 1.1 0.9 1. 1.

°1

r.p.m.^ = 5.22 m ^1 = "• 105 Z^ = 5

distribution see 'cargo liner' of Ref. [24]

torque aft/forward 1. 1. 1. 1.1 0.9 Efficiency

(calculated with standard radial distributions of mutually induced velocities)

0.6398 0.6387 0.6347 0.6381 0.6366

Table h. Impact of distribution of torque and r.p.m. on efficiency of 16 000 h.p. counter-rotating propeller arrangement

The effect on efficiency of alternative combinations of blade numbers, keeping the total number of blades constant, is presented in Table 3-There does not appear to be any reason to assign the higher number of blades to the aft propeller for reasons of efficiency as stated in Ref. [ 2 4 ] . The effect of open water efficiency of the division of power and r.p.m. between forward and aft propeller is shown in Table 4. Equality of power and r.p.m. appears to be the optimum choice, even though the differences found were small in the range of +_ ]0% deviation from equal power and r.p.m. that was considered. The indication of Ref. [16] that more power should be assigned to the forward propeller appears to hold true only in the behind condition.

Finally, suppression of the correction of static pressure due to the mutually induced pressures led to the conclusion that the mutually

induced pressures have an increasing effect on the blade area of the forward propeller and a decreasing effect on that of the aft propeller, both in the order of 5%.

CONCLUSIONS

By comparison with previous methods for the design of counter-rotating propellers, the method outlined in this chapter has the following features:

1. Momentum theory is used for the calculation of mean mutually induced velocities and pressures. Consequently, the design of the propellers is straightforward and an iterative process is not necessary.

2. Vortex theory (according to the variable load actuator disk model) is used for the calculation of the radial distributions of the mutually induced velocities, taking into account the actual load distribution.

3- The mutually induced pressures are allowed for in the cavitation calculation.

4. Mutually induced effects are separated from self-induced effects in such a way that each propeller can be designed according to an established procedure for single propellers.

The theory has been applied in a computer program which permits the division of power and r.p.m. between the two propellers to be arbitrary. From the numerical results obtained the following conclusions can be drawn:

1. Comparison with experiments shows that the method yields the correct power absorption, provided a reliable method for the design of single propellers is used.

2. The pattern of the velocities induced by the forward propeller leads to a pitch distribution of the aft propeller which is increased at 0.8 radius and reduced at the tip and the root of the blades. The velocities induced by the aft propeller affect

the pitch distribution of the forward propeller so as to call for a slight decrease at the tip and a slight increase at the root.

3- The mutually induced pressures call for an increase in the blade area of the forward propeller by about S% and a decrease, by the same amount, in that of the aft propeller.

4. The highest open water efficiency is obtained when the torque and r.p.m. of both propellers are equal.

5. There is no reason to assign the higher number of blades to the aft propeller in the interest of efficiency.

6. The cavitation problem, which is the limiting factor to the powering of high speed propellers, is greatly reduced by distributing the power over two counter-rotating propellers.

CAVITATION INCEPTION OF SHAFT BRACKETS

The straightforward presentation of the flow field of an infinitely bladed propeller as discussed in the preceding chapter, can be used to determine the effect of the propeller upon the flow around any object located up- and downstream. In this chapter we will use it for calculating the inception of cavitation on an upstream shaft bracket, Ref. [ 7 ] .

For high-speed applications not only cavitation on the propeller but also on the shaft bracket forward of it, is of interest. Cases have been encountered in practice, where the propeller was free from

cavitation, but the shaft bracket cavitated heavily, causing cavitation erosion on both the bracket and the propeller. The cavitation inception characteristics of the shaft bracket are also important when avoidance of noise radiation is a primary concern.

The problem consists of calculating the pressure distribution and cavitation number of the bracket. The propeller induced velocities and pressures, which have an unfavourable effect on the cavitation phenomena at the bracket, have to be taken into account in calculating the inception of cavitation.

The following assumptions are made:

The number of blades is infinitely large, which implies that only average effects are taken into account.

The effect of the propeller upon the flow at the bracket is considered only at mid-chord.

The effect of the bracket upon the flow at the propeller is neglected (if necessary this effect can be allowed for by calculating with an increased propeller thrust coefficient).