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

ARCHIEF

A CONTRIBUTION TO THE SOLUTION

OF SOME $PECIFIC

SHIP PROPULS O PROBLEMS

- A REAPPRAISAL

df MOMENTUM THEORY

-LA. VAN GUNSTERIN

1a1

y0

Sdieepsboáwkundé

Technische

Hog.sGhed

t-STELL ING EN

De door de schroef geinduceerde snelheden ter plaatse-van de

straal-buis zijn slechts bij lichte belasting (CTp <2) bij benaderiñg lineair

met de stuwkràchtcoefficiënt C1, zoals wordt verondCrsteld in vrijwel

alle huldlge methoden voor ontwerp en. analyse van

mantèldeschroeven.

In het belángrijke gebied van matige en zwarè belasting dient de invloed

van dé vérvorming van de schroèfstraal ¡n rékeñiÌg te worden gebracht.

-

Hoofdstukken 1,

1

en 5 van dit proefschrift.

"Slots!' en "slats" als beschréven ¡n de hoofdstukken 7 en 8 van dit

proefschrIft kunneñ het toepassingsgebied vañ ommantéldé schrbeven ¡n

belangrljke mate verruimeñ.

BIndel en Garguet hebben geen verklaring kunnen vinden voor het door hun

gemeten verloop van het effectieve volgstroomgetal en het zoggetal tijdens

een stoprnanoeuvre. Een plausibele verklaring ¡s echter te geven door

Dickznann's axiale lmpulstheorie uit te breiden tot negatieve en zware

schroefbelasting en Lagally's theoreme toe te passen.

-

H.E. Dlckmánn

:

(I)

Ingenieur Archiv 1938

(ii)

Jahrbuch der Schiffbautechnischen

Gesellschaft 1939.

-

S. Bindel én M. Garguet

:

"Qiaelques aspects du fonctionnement des

hélices pendani les manoeuvres d'arrêt

des navires", Association Technique

Maritime et Aéronautique 1962.

L.A. van Guns tereñ

4. De toenemende wenselijkheid achterwaártse "skew11 toe te passen teneinde de trlllingsexcitatie door de scheepsschroef te verminderen kan vrstrek kende gevölgen hebben voor de ontwikkel Ing van verstelmechanismeh van verstelbare schroeven.

- Discussi

W.B. Morgan enR.J. Boswell, Transactions of the Society of Naval Architects and Marine Engineers 1972.

5. Het uitblazen ar, lucht door gaten nabij de intredézijde van bladen van verstelbare schroeven kan, naast ceo vermindering van het geruisniveau, leiden tot een aanzienfljke afname van het hydrodynamisch verstelkoppel hetgeen de mogelijkheden verruimt voor de toepassing van "skew" alsmede vah lichteen compacte verstelmechanismen.

Deze kenmerken

ten volle rechtvaardigen.

Reduction of blade spindle torque by ventilation",

International Shipbuilding Progress, No. 171, Vol.

15, 1968.

6. De rechtvaardlglng van de toepassing vãfl contra-roterende schroèven moet niet worden gezocht In de rendementsverbetering, maar In de omstandigheid dat de door schroefcavIatIe opgelegde begrenzingen aan de scheepssnelheid eh het vermogen per asleiding aanzienlijk gunstiger liggen dan bij

conventlônéle schroeven.

- "Marine propellers for large power and high: speed", Marine Engineers Review, January 1972.

bijdrage tot "Highly Skewed Propellers", döor R.A. Curivning,

zöuden verdere ontwikkeling door de Koninklijke Marine

De stuwkracht-schroef dienen

of vermogenscoefficiënt en het cavitatiegetal van de te worden betrokken blj het scheepsvöorontwerp teneinde de waarschijnl jkheid van cavitatie- en trlllingsproblemen en de mogé-lijke noodzaak het vermogen over meerdere schroeven te verdelen tijdig vast te stellen.

- "Propeller design concepts", Second Lips Propeller Symposium, Internat onal Shipbuilding Progress, No.

227, Vol. 20, 1973.

De I.S.O. Class I toleranties voor scheepsschràeven zijn niet toereikend

voor schroeven van snelle containerschepen.

"Propeller production coflcept.ioì,s", First Lips Propeller Symposium,

International Shipbuilding Progress, No. 199, Vol.

18, 1971.

Voor de numerieke integratie van statische momenten en traagheldsmomenten met behulp van een computer dient men andére intégratieregels té ge-bruiken dan bij de berekening van oppervlakken.

"Numerical calculations of areas and moments", Internätional Shipbuilding Progress, No. 157,

Vol. 14, 1.967.

Het is gebruikelljk de kroriine van instromend lekwäter ten behoéve van de schottenkrornme te berekenen met behulp van de Bonjean-kromen. Het is echter lógisch däärtoe de waterlijnen van het bovenwaterschip te gebruiken.

Zur Schóttkurve; Berechnung der Kurve der éiñströménden Leckwassermengén durch Integration von Schwimmlàgenänderungen", Schiffstechnik, Heft 67, Bd. 13, 1966.

Een regelsysteem voor de stuurmachine waarbij het ingangssignaal niet, zoáls gebruikelijk, rechtstreeks is afgeleid van de. stuurstand, maar ¡s samengesteld uit het stuurstandsignaal en een signaal dat de hoeksnelheid van het schip weergeeft, heeft belangrijke voordelen wanneer manoeuvreren bij grote roerhoeken een vereiste ¡s..

- "Werkwijze'voor het besturen van schépeñ", Nederlandse Octrooiaanvragè No.

66.03583, 18

maart 1966.

Door administratief en technisch rekenwerk te verwérken met dezelfde còmputer hardware kunnen belangiijke besparingen eh andere vòordeIen worden bereikt ten opzichte vah de in véle bedrijven doorgevoerde strikte scheiding van deze àctiviteiten, zu het dat additionele organisatorische maatregelen noodzakel ijk zijn.

9

13. Het ontwérp van een zeilénd schoolschlp ten behoeve van de gezameli'jke Neder)àndse Opleidiñgen voor de zeevaart zou niet moeten w rden afgeleid van de conceptie van de zeilschepen uit de vorige eeuw, maar zou gericht moeten zijn op een zo gunstig mogelijke expIoitatie d.w.z. het zou

in overeensteniming moeten zijn met de modernste technische inzichten.

-. "Het moderne schoolschlp", Mrineblad, oktober 1968.

1i. In de provincie Noord-Brabànt zou de overheid de subsidies ten behoeve van het A.M.V.- enblokflultonderwijs beter kunnen aanwenden ter onder-steuning van fanfare- en harmonieorkesten

A CONTRIBUTION TO THE SOLUTION

OF SOME SPECIFIC

SHIP PRQPULSION PROBLEMS

A REAPPRAISAL OF MOMENTUM THEORY

-PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE

TECHNISCHEWETENSCHAPPENAAN DE TECHNISCHE HOGE-SCHOOL DELFT, OP GEZAG VAN DE RECTOR MAGNIFICUS

IR H.B. BOEREMA. HOOGLERAAR IN DE AFDELING DER

ELEKTROTECHNIEK VOOR EEN COMM ISSIE AANGEWEZEN

DOOR HET COLLEGE VAN DEKANEN TE VERDEDIGEN OP

WOENSDAG 12 DECEMBER 1973 TE 14.00 UUR

DOOR

LEONARD ANTHONIE VAN GUNSTEREN

SCHEEPSBOUWKUNDIG INGENIEUR

DIT PROEFScHRI FT I S GOEDGEKEURD DOOR DE PRctOTOR:

TABLE OF CONTENTS

Page

IntrOduction 7

i. Free actuator disk

Slipstream of a heavily loàded frée actuator disk 10

Lightly loaded actuator disk 14

Numerical procedure 15

Numerical results 17

Conclusions 22

2. Counter-rotating propeller design 24

Introduction 24

Theory 27

The velocity field òf an infinitely bladed

propeller with arbitrary radial load distribution 27

Design procedure ₃₅

Numericäl results 1414

1 _{Conclusions 51}

3. Cavitation inception of shaft brackets 53

4. Shrouded actuator disk ₅₇

Slipstream ofa heavily loaded shrouded actuator disk ₅₇

Numerical results 61

3 Conclusions 80

LI

6. Effèct of a nozzle on steering characteristics 99

1. Introduction 99

2. Mathematical model of turning 100

Turning capacity 103

Response time 104

Rudder-nozzle force coefficients (C1, C2) 104

3. Prediction of lift forces on rudder and nozzle 106

4. Numerical results 114

5. Full-scale tests 118

Turning circle tests 120

6. Discussion 122 7. Conclusions 123 7. Slotted nozzles 125 1. Introduçtion 125 2. Open-water characteristics 129 Geometrical characteristics 1,-29 Test results 132

3. Comparison with conventional propulsion devices 14-3

Performance at non-zero advance ratios 145

Performance in bollard condition 152

4. Conclusions 156

Performance calculatioñ 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-wäter characteristics 96

8. Ring propeller with ring stator 157

Introduction 157

Open-water test results 158

Conclusions 161 Final remarks 162 References 16x4 Nomenclature 169 Overzicht 181 Dankwoord 183 Curriculum vitae 18h 5

INTRODUCTION

The present approach in dealing with propeller problems is using the concept of singularities, the hydrodynàmic load being represented by vortices and the thickness of the lifting surfaces by a source-sink distribution. The vortex thèory of propellers, as introduced by Prandtl, Betz and Goldstein at the beginning of this century, has been developed during the last decades, whèn 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 time by Rankine and Froude, Ref. [i], is generally used only to gain insight in the main action of the propul-sion system. The reason for this is that momentum theory providesno 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 ¡n any bladé 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 propel 1ers, lifting line theory can be introduced as a non-linear refinement. As in pure linearized theory, the hydrodynamic pitch of the free vortices ¡s 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. [s], [14], [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

dòwnstream, 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. lt is realized that this necessitates sorne sacrifice of mathematical rigour, but here again the justification lies in the practical results that are obtained.

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 deformationi*) The numerical results obtäined with this theory show that the application of linearized actuator disk theory should be limited to thr1ust coefficients below 2.0 (the typicäl range of C1avalues for large tanker 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 shàft brackets. Ref. [7].

Chapter 4 presénts a shrouded actuator disk theory which takes into account slipstream deformation. In Chapter 5, the results of this theory are applied to calculationof 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 vørt ices shed from the propulsor, whereas slipstream contraction réfers only to their radius.

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 för 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 frictioñal losses ànd in the casè 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 cari also provide other guidance in the development of new propulsion côncepts. 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), [iO].

The practical applications discussed in this work show that momentum theory and the related actuator disk concept contribiite 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 field of marine propellers.

1. FREE ACTUATOR DISK

.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. [ii].. This concept corresponds to the assumption of

lo

vi

an infinite number of blades and uniform loading over the actuator disk, rotat ng 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.

The fluid el moves along

V =

(v.

+

v)

y

Fig. 1. Slipstream öf a heavily loaded actuator disk

Fig. i represents a section of the slipstream. At axial statioñ 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 =

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

ement which constitutes this element of the vortex sheet

the slipstream boundary with a velocity:

(1.2)

R r R0

and

y (x, r). =

The influence functions, U and V , for an isolated vortex ring, at 'r 'r

the origin ( a O) and of unit radius (p a i), can be written, Ref. [12j, as: U (x, r) = and (x, r) =

Y (. P)

_u

(x , d ( )

-fr

d . (1.5) P

Y\

P PJ Vx2 + (r + 1)2 { K (k)

-V x2

+(r + 1)2 { K (k) - [i + where: k2 + (r + 1)2 + 2 (r - 1) _J E (k) } + (r - 1)2

and K (k) and E (k) are completé elliptic integrals of the first and second kinds.

(1 .Li)

(1.6)

= constant. (1.3)

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

2r

J E (k)

(1.7)

12

For field points Cx, 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 (x, R) =

Furthermore,

downstream i

V on the vortex sheet is:

as the slipstream boundary must be ä stréamiine:

-p VA2 R02

= jét velocity, e thrust,

e actuator disk radius.

The correspOnding strength y per unit length of the vortex sheet far s:

= _1+VI+CT

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 shapé R (x) for any thrust coefficient CT. FOr this purpose an iterative sòlution was programmed for a high-speed computer. Since better convergence wâs

V

[VA + u (x, R)]2 + y (x, R)2 (1.8)

d R. y (x, R) (1.9)

dx

_{VA+u(x, R)}

Application f Bernoúlli's law from far upstream to just forward of the äctuator disk and from just behind it to far downstream provides the relation between thè thrust coefficient and the jet velocity far

downs t ream:

()2

-I (1.10) where: T and V. J T R 1= VA

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

-R (x)

[VA + u

(xfl

r dr = corìtant (1.12)

o

the Iterative process starts with the assumption of no slipstream deformät ion:

y (x)

₌

'ç

,

(1 13)

R(x) = R0

In each cycle! the velocities induced on the slipstream are calculated for a number of axial stat:ions, using Eqs. (1.4) to (1.8). The resulting velocity distribution along the slipstream bôundary, together with

Eq. (.1.3), prôvidès the new estimate for the vortex distribution y (x). The axial velocities induced iñ 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 ¡s attained.

In classical one-dimensional momentum theory ¡s assumed that the unknown pressure integrals over the control surfâces vanish. The law of

momentum then yields:

CT =

2(!1-VA) \VA

where:

(2

\, R)

The relation between the contraction and thrust coefficient C1 is obtained by elimination of from Eqs. (1.10) and (1.14):

CT (1.15) 13 2. - i

+ Vi +

_CT)

Vï+

_CT 1) (1.14)

14

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

condition, C1 tends to infinity, and , according to Eq. (1.15), 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. (1.1+) and (1.5) alOng the axis (Fig. i) can then be perforthed analytically. The solutiOns ca n again be expressed in complete elliptic integÑls, Ref. [13],

a vortex cy inder of unit strength and unit radius:

(x, r,)

u"+

where: and

(x,

r) -Here K (k), a2 = L4 (r +11)2

[ K (k)

(r - 1)_{(r' + i)}

(a2, k)

]

for

2\/x2 +

(r + 1)2

r<1

r>1

r= I ii k2

Vx2 +

(r

f)2

'L E (k)

(i

X

k2'))]

first, second and third kinds, respectively; k is defined as and (1.7), and a by:

for 2

(1.17)

in Eqs. (1.6) E (k) and JI ('a2, k) are complete elliptic integrals 6f thè

and are for

For field points (x, O) on the axis of the semi-infinite vortex cylinder having strength y and radius R one obtains:

u (xIR0, O)

= 1

₂

(1

xIR0

Vi +

(x/R0)2

The limiting values of Eq. (1.18) are exactly as predicted by momentum theory. Combining Eqs. (1.11) añd (1.18) yields the velocity induced on the axis at x by a lightly loaded actuator disk at thrust cóefficient

C1. _{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: Uref(X) _def. a i

+ Vi

+ (/R0)2 x/R0 } where: a =

_(-

Vi +

In this way, the effects of loading (CT) and of distance from the actuator disk (x/R0) 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. NIJIERICAL PROCEDURE

A computer was programed 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 downstreäm. The following 18 axial stations _were selected in which Eqs. (1.3) and (1.12) were: to be satisfièd:

(i .18)

(1.19)

16 x/R0 = 0 k-hk = q where: q constant. 0, 0.1, 0.15, 0.2, 0.3, 0.4, 0.5, 0.6, 0.8, 1.0, 5, 2.0, 3.0, 4.0, 5.0, 6.0, 8.0,. 10.0.

The vortex strength, y (x), and si lpstream shape, R (x), at other axial locations were obtained by a 4-point Lagrangian logarithmic interpolation:

4 (in x - in x.)

f (in x) = n f (in x1) (1.21)

i=1 j=i (in x1 - in x)

ii

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) becöme singular for Cx - = 0, r/p = 1. Therefore, an interval - 0.002 x < 0.002 was excluded from the integrations; the contribution äf the excluded vortex sheet to the radiai induced velocity is1 zero, that to the axial velocity was computed taking

k=

V4/(10.0012 + 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. [14]. The integrations are approximated by a summätion of algebraic terms:

= _{hk [}

() +

(c + hk) + f (k + 2 bk)

I

. (1.22)

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

hi =

(i -

+ (r

- Pi)2

Here l'

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

h1

Sufficient numerical accuracy could be obtained by taking: q = 1.5 for the axial induced velocities

q - 1.2 fr the radiai 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. [ik] 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 formuläs described in Section 1.2.; the results were identical, to four figures.

The convergence becomes more critical the larger the thrust

coefficient C1. At CT = 100 the convergence was such that the radiûs of the ultimate slipstream according to the 5th cycle remained within O.5 as compared with the value according to the Iith cycle. The largest variation ¡n the product y V was found to remain within 6 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.

l.k. NLJ'IERICAL RESULTS

Fig. 2 shows how the shape, R (x), of the slipstrèam boundary varies with the thrust cOefficient, C1; Figs. 3 and 4 show the corresponding variation of the vortex distribution,

_{y Cx).}

18 3. 2. 2. 0 2 04 0 8 1.0 12 - 1:0 1 6 1.8 2.0 2 2 /80

Fig. 2. Variation with thrust

coefficient of

actuator disk slipstream shape

0.2

disk slipst

o:'. 0.6 08 1.0

/Ro

Fig. 3. Variation with thrust coefficient of vortex distribution on actuator

.2

earn boundary (range of low C1)

.6 1 8 2.0 2.2 20 50 100 05 5. 10. Ct io. -J. _o .0 0.8 0.6 0.'.

0.9

Fig. 5. Variation with thrust coefficient of free actuator disk slipstream contraction

19

0.2 0.6 0.8 1.0 .2 .6 98 2.0 2.2

IRo

Fig. 4. Variation with thrust coefficient of vortex distribution on actuator

disk slipstream boundar9 (range of high CT)

A comparison between the contraction according to the (two-dimensional) iterative procedure and the value according to Eq. (1.15) of one-dimensional momentum theory is presented in Fig. 5. It is apparent that. as a result of the assumption that the pressure integrals over the control

surface vanish, Eq. (1.15) underestImates the còntractlon at very high thrust coefficients.

i. . 1.3 :,< 09 ¡ 20 disk theory The present

coefficients (low manoeuvring speeds).

Classical linearized actuator disk theory, either in its original unifòrm load concept or as the variable load disk, Ref. [15], is used in most current theories for ducted propellers. To deterine

the range of propeller thrust coefficients where the assumption of light loading is justified, the velocities induced at the cylinder r/R0 1.05 surrounding the actuator disk were calculated for various propeller thrust coefficients. The results are given in Figs 7 to 9.

It appears thät linearized actuator disk theòry 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. 6. Variation with thrust coefficient of mean axial induced

velocity in actuator disk slipstream

The variation with thrust coefficient C1 of the average velocity induced in the slipstream Is given in Fig. 6. These data are useful fòr to determining the approximate effect of the propeller on the flow onto rudders placed in the slipstream. The results of classical actuator

have been used in ship steering problems by several, authors. results extènd the application to the range of high thrust

21 ACrIARrOR 7011115 Ar - FREE - FIELD DISK /0 1.05 - 0.75 125 AEVII0006 POINTS AT VA FOCE - FIELD OLDO IR .05 02 0.5 20. 50. IDO. CT

Fig. 7.

Variation with thrust coefficient of actuator-disk-induced axial

velocities on cylinder R0/R = 1.05 (forward of actuator disk)

02 05 IO. 20. 50. loo.

Cj

Fig. 8.

Variation with thrust coefficient of actuator-disk-induced axial velocities on cylinder RD/R = 1.05 (aft of actuator disk)

0.0 O 07 0.00 O 05 0 03 O 02 O 01 0.00 0.07 0 06 0.05 0.04 0.03 0 02 0.0l

22

- CT

Fig. 9. Variation with thrust coefficient of actuator-disk-induced

radial velocities on cylinder RD/Ro 1.05

Thé results of Figs. 7 to9 can be úsed in ducted propéllèr 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, ál:though the propeller thrust coefficient, CTp. may be high, the duct thrust coefficient, CTD, should be low, or

t

= C1/C1 1. This situation is applicable to ducted or ring propellers with an extremely löw chord-diameter ratio. For usual ducted propeller configurations, howéver, the effect of the duct induced velocities upon the slipstream deformation at high propeller thrust coefficients is significant. This will be elaborated in the shràuded actuàtor disk theory presented in Chapter 1

1.5. CONCLUSIONS

I. Slipstream deformation becomes significant àt thrust coefficients exceed ng 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 in practical dùcted propeller design and analysis.

CTDAT P011ITO DISK AT nR - FREE FIELD .05 125 2D - 0.5

-

TO 20. 50. IDO. D. 28 0.24 0.22 0.16 0.12 ¡ 0.08 0.04 0.

Linearized actuator disk theory should be used only at propeller thrust coefficients belbw approximately 2.0. Since, in such lightly loaded (high-speed) applications, the radial load distri-bution is generally of interest, the variable load actuator disk model, as discussed in the next chapter, is preferable.

The numerical results presented in Section 1.. (Fig. 6) are useful in assessing the approximate effect of a heavily loaded propeller upon rudders placed in its slipstream.

. The results presented in Figs. 7 to 9 can be usèd in short-chord

heavily loaded shrouded propeller theory.

2. COUNTER-ROTATING PROPELLER DESIGN

2.1.

INTRODUCTION

The trend towards higher powers has given considerable impetus to research in counter-rotating propellers. This typé 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 appl icat.ions.

Since a comprehensive review of the state of the art has been given by Hadler., Ref. [16], we shall limit the following discussiOn to some other pertinent references.

The basic problem in the design of counter-rotating propellers consists of the calculatión of the mutually induced velocities. The most widely used design méthod is the Induction factor method of Lerbs, Ref. [17], which has been elaborated by Morgan, Ref. [18]. 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. lt is assumed in Ref. [17] that the distance factors can be calculated with the actuator disk model; i.e. the distance effect of the propeller with arbitráry 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 thè propeller. This has been clearly shown by Tachmindji, Ref. [19], who calculated the distance factors fòr the optimum lñfiñitely bladed propeller; these factors were used in Ref. [18].

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 distañce 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. [20]. In thi.s 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. [21]. The solution for dealing with the radially varying rotational speed of the aft propeller applies only to the particular design method for single propel 1ers that 'was used.. As in the method of' Lerbs, use is made of the standard distance factors discussed previously.

'A lifting surface theory for counterrotating propellers has been formulated by Murray, Ref. [22], but numerical results are not yet available. In Ref. [23] a linearized lifting 'surface theory is given which ¡s based on the so-called quarter point method.

26

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

The slipstream c7ntraction 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. [2k]). Momèñtum theory provides a simple means to allôw for

thesé non-linear effects. We therefore apply momentum theory for the ca cülations of the mean mutually induced velocities and préssures.

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. [15].

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

In princip e, 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:

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

the vôrtex distribution representing the difference between the lifting surface and thelifting line vortex distribution of the other propeller (this vorteA distribution is limited

to the blade area of the other propeller), (iii) the thickness effect of the other própellér.

2.2. THEORY

The velöcit field of an infinitelX bladed 2ro2el 1er with arbitrary radial load distribution

When designing a set of counter-rotating propellers Lise 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 velocitiesare unsteady. When determinln the shape of the propeller only the steady components, which are the same with a fiflite or an infinite number of blades, are of Interest. lt 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 bladéd propeller with arbitrary radial load distribution and its velocities ¡n an

Ideal fluid was given by Hough and Ordway, Ref. [15]. Results of their variable load actuator disk theory which pertain toour discussion can be sumarized 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 eñds 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

28

radial coordinates are denoted x, and r, p for field pòints and

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

d r (p) = Id p (2.1) where: r = blade circulation, = 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 appiyiñg Stokès's law the tangential velocity field is found to be:

U-i- (r) Z r (r)

(2.2)

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

where:

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 àre of interest. According to the Biot and Savart law an elementary ring vortex at with radius p and of strength

y d dp

induces an axial velocity d UA (x, r) at (x, r):

VA

2rVA

- ¡n the slipstream

= O outside the slipstream

VA

d UA (x, r) = '2 ir d dp ir

(p2 -

r p cose) d ((x - + (r -

p)2 +

2 r p (1 - cos)}3/2

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

d UA (x, r) = (xip, rip:) y dp where: (x, r) = j d

2rr

(i - r cos) d {(x, -

)2 +

(r -

1)2 +

2 r (1 - cos))3"2

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

(2.)

29 (2.5)

,,Ip

.-Fig. 10. Axial velocity field of a half-infinite vortex, cylinder

The axial vélocity 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:

30 P.

_'t

rh

(x/rh, n/rh)

P. -

'h

where:

R = propeller t:ip 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 öf an infinitely biaded 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,

üsing the fo lowing definitions of mean values:

p=rh R

J (x/p, rip)

d r

(p) Pj _(p) dp dp (2.7)

VA (x,

r)

VA VA

Mean circulation

r =

mean

Mean axial velocity

r

Mean tangential velocity

UT mea n R Ï' (r) r d r r

rdr

R UA (r) r d r

rdr

rh 'R UT (r) r2 d r R r2 d r rhJ (2.8)

In the case of an actuator disk ¡n which the circulation ¡s

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 UArf ( to which we shall relate the axial velocities át arbitrary field points (x, r) induced by an arbitrary circulation distribution. The reference velocity ¡s 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) thêncän be written as:

31

rh

(2.10)

32 UA (x, r) (x) f where:. (x/R, o)

lt 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) cquldbe programed without much difficulty. Nevertheless we consider it appropriate to use

Eq. (2.11) because counter-rotating propellers are especially attractive at high speeds and coñsèquently Fight loading. Por such wide-bladed propel:lers the inaccuracies due to neglecting chordwise effects in the calculation of the mutually Indùced velocities can be expected to be mOre Important than those resulting from the simplifications in Eq. (2.11).

Eq. (2.11)

field point dépends on the eference velocity LiAr

f (x) and the radial

circulätion di.stribìjtiòn. If the hub diaÑeter is zero, the reference velocity becomes:

UAf

VA where: a = : (-1 and r mea n rh u (x/p, rip) dp - r u (xirh, r/rh) mean x/R

Vi +

(x/R)2

Indicates that the axial velocity Induced at an arbitrary

= a (i +

+ VI + CT)

(x/R, nR) xiR i + (xiR)2 p=rh R d r mean dp /t (x/R, o) (2.11) (212)

T

CT

-p VA2

If the hub diameter is not zero, then the available propeller disk area and the actuator disk area defining the reference velocity are, fèr usual hub sizes,, slightly diffèrent. With equal mean pressure jumps for both the propeller and the actuator disk the. thrùst of the actuator disk Is larger: We allow for this effect by defining the thrust coñstánt C1 as fol 1 c (2.13) P VA2 D2 i - (dID)2 where: d = hub diameter D = propèller diameter.

We shall now cñsider the mean axial velocity induced in a disk at x

The ratio Uea/UArf can be computed with Eqs. (2.9) and (2.11),. from which It cân be seen that this ratio depends only on the radial

circulation distribution. We have computed Uean/UAref on a high-speed computer for several circulation distributions (constant, ellitical, Increasing towa.rds the tip, decreasing towards thetip 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%. Conseqiently, the

ratio UA /U1 - 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 húb diameter ratio 0.2, is given in Fig. il.

3

x/R

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

The mean axial velocity is. larger than the reference velocity behind the propeller and smaller ¡n front of ¡t, 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. li for arbitrary circulation distributions ls within O.5?. Thus, the mean induced velocities may be calculatèd with a5dal moméntum theory and the function F1 (x/R) as given Ín Fig. 11. The rädlal 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 thfory, Ref. [25], in which the assumptIon is made that the slipstream behind the. propeller rotates like a solid cylinder. The maximum value of the tângential velocity, UTmax will then occur at the slipstream boundary. The propeller torque, Q, equals the moment of momentum impressed on the fluid passing through

it per unit time, which yields pon integration:

ELLIPTICAL C1RCU1.AT1O wJB-DIPiIETER RATIO

DISTRIBUTIO

0.200

;1(;/R):uAUA n,eanref (e/O)

--

U max VA where: CQ Q p VA2 4- D3

The relation to the mean Induced tangential velocity ¡s:

U-r

3/4U

mean Tmax

The radial distribution of the tangential ve!oclties can be calculated with Eq. (2.2).

The ¡hduced velocities have been obtained neglecting the effect of slipstream contraction. In order to àllow or this, we assume the râdiai coordinate to be expressed as a fration 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 ¡n the linearized cóncèpt so after allowing för contraction. As mentioned before, the sllpstieäm contraction ¡s determined by usiñg

the law of continûlty.

919

Having established the basis of cóunter-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 ¡h Fig. 12.

35

2CQ

1 + a

(2.14)

36

APRE ¡UD IIITEAFEREIICE EFFECTS WITII

EQ. (2.21)

B OES lUll FORO/IDO I'ROPELLTR

advanc

C7/-IPINE TI-DUSTS

ROD P.DEIIAI. CIRCULATIIOI DISTRIBUTI7/IS AITH

ASSU'1E0 VALUES

-10 IOESIGI

CORRECTICII TI pri. OF AFT PrTrCLLRT

FOR SLIPSTREVE ROTATIVES AlTrI

CQS. (2.22) 31.0 (2.23)

APT PROPELLER

P.501811. DISTRIRUTIOI CF TAQETITIAI. VELOCITIES AT URO PROPELLER IIIDUCFD CV FORi IRRT PROPELLER F1303 ET. (224)

Fig. 12. Flowchart of counterrotáting propeller design procedure

At first the basic design parameters have to be established (block i

of F-ig-. 12): .

velocity and. wake distribution,

diameter of forward propellerr

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

axial distance between the propellers,

singlerpropel ICr design parameters, such-as number of blades, staticpressure at shaft height, margins against cavitation, admissible static stresses, etc..

ESTARA.I5S SASIC DCSIC.PARNIETLRS

2 EST TISTE -TI DUSTS,

ITDUSTS 7/ID CIRCULATIDO OISTRIBDTIDN SET ACCOROIID

10 TO LILXKS 8 NEI 3 EE/S ADIAL VELOCITIES rIUTUALLYI DUCES 811TH EQS.. (DIE) NS (2.17) DT

SLIPSTRCNI carTr.AcTITII RITTI [0. (2.13)

1IUTUALLY EDUCED PRESSURES VITI-i

EQS. (2.19) AID (2.20)

RADIAL 2ISTF.IDUIIOIIS UF SAIAL IIUTUALLY

IITOUCEO VELOCITIES (EQ. (IlE) OR

FIES. II. A/ID IS I

T 15175E VELOCITIES COPPECTED FOP SATT SrACrop.v

12 CORRECTTCRO OC PITO-O TI500I001IOC OF

AFT PROPELLER FOR RADIAL DISTRITDTICRI OF TAIDENTIUL TELEICITITS 011TH El. (2.28)

Advance velocity and wake distribution are determinèd as in thé case. of a single propeller. If the rotational speed is not prescribed, thé diameter of the forward propeller should be chosen as large as can conveniently be fitted In the aperture. According to moméntum theory the rotational losses are zero If the torque of both prOpellers is the same. Then thé propellers wi!l also have approximately the same optimum rotational speed. The usual practice of sèlecting equal power

and r.p.m. for the two propellers is therefore jutifled 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. [16] that more power should be assigned to the forward propeller for optimum efficiency in the béhind condition. Apparently, hull-propeller interäction

Is Important in this respect.

I

75 70 .65 60 .55 .50 45 .5 .6 .7

.8.

.9 "A.

Jr..

nD1

Fig. 13. Diagram for selecting optimum r.pm. and estimating efficiency

of counter-rotating propellers .37 6, 5 SDS E.

PR

R.P.M.

-A

r,

KQ/J3 r. - n 2.s D12VA3 s PtVA3

38

Optimum diarneter or optimum r.p.m. can most conveniently be determined with a systematic propeller series, because thé. optimum design points are to only a slight extent dependent on blade area ratio, number of blades, eta.. Diagrams for this are given in Refs. [21.] and [26). Usually the r.p.m. are selected slightly above the optimüm value fòr which the diagram given in Fig. 13 can be helpful. The diagram is based on the ser es given in Ref.. [21] supplemented by some ûnpublished 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 añd its interactiön with shafting and gearing. lt 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 propel 1er (block .2 of Fig. 12); this can be done with the aid of Fig. 13. As an error of

even 15 appears to have no noticeable impact on the design, this estimate need only be rough. The factors F1 (SIR1) and F1 (- SIR2) relating the mean axial mutually induced velocities to the reference velocities are obtained from Fig. 11, assuming at first R2 = R1 (block 3 of Fig. 12). The mean axial mutually induced velocities are calculated using momentum theory relations:

UA ₂

-

SIR1

)

VA a1

Vi + (SIR1)2

where:

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

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

continuity (block L1 of Fig. 12). Taking into account first order effects only, we obtain:

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 thç aft propeller ai,d conversely the aft propeller operates in a region of positive

differential pressure due to the forward propeller. The static pressure used for thé cavitation calculation has therefore to be corrected with the mutually Induced pressures. The correction can be derived from the velocity field. 1h general, the static pressure is of interest, only at

O.8R. At this radius the mutually induced velocities are approximately equal to the mean mutually induced velocities. We therefore derive the corréctions on the static pressure from the meän induced velocities.

From Bernoulli's law and taking into account first order effects only, we obtain: F1 ( S/R2)

(2.19)

and:

/

SIR2 = p VA2 a2 + p2 = p VA2 a1

Vi + (SIR2) 2/

/

SIR1

2 -

i + ) F1 (SIR1)

\

Vi + (SIR1)2!

(2.18)

(2.20)

39 (SIR1)

(2.17)

where P0 e denotes the static pressure without allowance

fr

mutually induced éffècts.

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 decreasingeffect on that of the aft propeller, both of approximately 5. The radial distributions of

the axial mutually Induced velocities UA (r)12/U

1,2 are obtained with Eqs. (2.9) and (2.11) from the giveÌ or assumed circulation

distributions r (r)121r

mean 1,2 (block 6 öf Fig. 12). DIstributions associated with an elliptical circulati6n distributiön which can be

used as a first approximation are given In Figs. 14 and 15. =

- e+ p12

.2 .6 .8 1. 1 2

UA (r)/UA r (r)Irmean

Fig. 11+. Distribution of axial velocity induced at forward propeller by

aft propeller with elliptic circulation distribution

1 .4 dID n 0.2

n/R.6

\

-r (-r)/-r CELLIPTI() mean - /U mean 9 f .8 .4

VA (r) 1,2 VA UA_{mean 1,2} = 1 - w (r)1,2 + VA LJA (r)12 u,' mean 1,2 (2.21)

Now the forward propeller can be designed using any available design procedure for single propellers (block 8 of Fig. 12). The aft propeller operates in the rotating screw racè öf the forward propeller, which ImpI les that the rotational speed with respect to the incoming flow is higher thañ the rated r.p.m.. The correction on rotational speed,

corr' Is ôbtained using momentum theory (block 9 of Fly. 12):

- -..--..--d/DO.2 (r)IT (ELLIPTÌC) -a, UA (r)IUA n,ea .4 .6 r8 12 U (r)/U6 an r

Fig. 15. Distribution of axia! velocity induced at aft propeller by

forward propeller with elliptic circulation distribution

The mean mutually induced velocities, the radial distributions as calcúlatéd In block 6 of Fig. 12 and thé given wàke distribution, w (r), determihe the axial infläw velocity (blôck 7of Fig. 12):

.9 f .7 .6 .5 .4 .3

UT

P Vp - D13

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

corr

-n2

1+

n2 D2,

n2

where deñotés the nominal rotational speed of the aft propeller.

The aft prOpeller can then be designed according to a procedure for single propellers, using the increased rotational speed n2 (block 10 of Fig. 12).

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 distrWution of the tangential velocities follows from Eqs. (2.2) and (2.10) (block 11 of

Ql

2 CQ t + a1 VA

(2.23)

rh

If the circulation distribution of the forward propeller is not known a priori, an elliptical distribution can be assumed as a firs.t approximation (Fig. 16). Fig. 12): T (r) F (r)1 rh R r2 d r (2.2+) UT mea n r -R

r(r)1rdr

The radial distribution of tangential velocities induced by the forward

UT n max (2.22) cor r 01

.9 .6 .5 .2 dio O.2 Uf (r)/U

Fig. 16. Distribution of tàngential vàlocity induced at aft propeller

by forwàrd propeller with elliptic circulation distribution

The actuál distribution of the tangential velocities ¡s taken into accouflt by correcting the radial. pitch distribution of te aft propeller (block 12 f F1. 12): P (r)2 U1(r) 1 + 2'2 - 1) _UT -- mean where:

(r)2 denotes the pitch distribution of the.áft propeller as resulting fróm the single propeller design procedure.

The thrusts and circulation distributions are theñ compared with the assuÑed values. If the agreement is not satisfactory we repeat the procedure starting with block 3 of Fig. 12. Usually no iteratiöri ¡s necessary when the design procedure for singlë propellers is based on

.2 - .8

- 1.2 1 C 1.6

UT (r)/UT

r (r)iran

1.8

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

2.3. NLZ'4ERICAL RESULTS

The method was applied in a computer program for thè design of counter-rotating propellers in which the distribution of power and

r.p.m. may be arbitrary. Lips's single propel lér design procedure was used. This is a lifting line method with còrrections oñ induced veloci ties

number

stream

for the following effects:

of blades (and thickness effect),

curvature (lifting line - lifting surface correction),

propeller loading (slipstream contraction).

The correctiòns have been derived from a regression analysis of the Wageningen B-series, with the principal advantage that the mean pitch accordiñg to this method ¡s ¡n agreement with the value obtained from

the propel er series. This feature enabled us to check whether the present thethod yields corréct power absorption. Fôr this purpose we redesighed the systematic series of counter-rotating propellers that are g ven in Ref. [26]. The results of the calculation are

compared w th the data from the experiments in Table 1. lt appears that the agreement asto power absorption and efficiency is satisfactory for the higher advance ratios, which is the range of interest for counter-rotating propel 1ers.

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

of

coúnter-rotating propelilers ISet no of counter- rotating All propellers KTmeas 0 38 -contractIon D /D 2-I pItch forward P /0 1.7R pitch aft P2 ID .7R effIciency ri propellers .1 Q21Q1 Tcalc meas calc dcv meas calc dey meas calc dcv meas caic dey Ref. [26] -(n1n2) in 8 In O in % fl 0.373 0.960 0.376 0891 -0.852 -4.4 0.893 -0.941 5.4. 1.192 F204 1.0 0.392 0.389 -0.8 -2 0.523 0.941 0.379 0912 0:895 -1:9 0.972 1.052 8.2 1.199 1.189 0.8 0.500 0.502 0.3 3 0.708 O.949 0.366 0.935 0.924 -1.2 l.tIOO 1.-122 1.7 1-.275 1.227 -3.1 0.615 0.598 -2.8 4 1-.027-0.958 --0368 0.958 0.953 -0.5 1.360 LOCO 2:9 1.465 1.425 -2.7 0.713 0.691 -3.1

Since counter-rotatiñg propellers äre most attràctivè 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 powercan be regarded to be not far fri 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 rotatiönal 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 désign parameters were the same as for the design of the 60 000 hip. single propeller.

Fig.. 17. Efficiency - r.p.m. relationship for 60 000 h.p. counter-rotating propeller arrangement

CALCUI.ATEDITH D 7.00 M

STA6CRAD RADIAL D1STRIBJT1OS V 33 KOTS

OF MJTUAU.Y IIDXED VELOCITIES w 0.112

78 I 76 .72 80 go 100 110 R.P.M.

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

A comparison of the particulars of the counter-rotating propellers with those of the single propeller is made ¡n Table 2. After correction for mutually induced pressures, all propellers have the same cavitation margin lt can be seen that a significant reduction in blade area per propeller is obtainéd b9 distributing the power over two propel 1ers. This shows that the. cavitation problem, which is the limiting factor as to pwer and speed for fast contaiher ships and naval vessels, can be relieved to a great extent by adoption of counter-rotating propellers.

47

All propellers: Couhter-rotàting propellers Single

pro-Vs = 33 knots

w = 0.1l2 Forward Aft peller

Power [h.p.] 30 oob 30 000 60 000 R.p.m. 100 100 135 Number of blades 5 ¿4 ₅ Diameter [m] 7.000 6.778 7.000 Mean pitch [m] 10.916 11.155 8.436 Pitch at 0.7R Em] - 11.077 _11.1492 8.547

Blade area' ratio 0.539 0.1485 0.909

Weight of blades [kgfJ 16 067 140 28 31k (Ni-Al-Bronze) Distance between propellers [mV] .7

.9 2 A 5IILE PROP. B F-1MD PROP. C AFTROP.

BWA

C -: FIlCH +46X6

AB

.6 .8 1.2 1.1

-

P (r/R)/Pmn - w (r)

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

The wake distribution and the pitch distributions are given in Fig. 18. The pattern of the velocities ¡nduced by the forward propeller leads to a pitch distribution of the aft propeller which is increased at 0.8 radius and reduced at, the very tip and the root of the blades. The effect of the velocities induced by the aft propeller on the pitch distribution of the forwàrd propeller is a slight decrease at the tip and a slight increase at the root. Thé circulation distributions,

together with the associated distributions of mutually induced vélocities are given in Figs. 19 and 20.

.7

6

L

g .8 .6 .5 k .3

f

Fig. 20., Circulation

distribution of aft propeller

of 60 000 h.p.

counter-rotating propeller arrangement

and associated distributions

ofinduced velocities at forward propeller a 6 11A (r)1 r 'UA _an k

/

2 2 1 .2 .4 .6 .3 1. 1.2 14 1.6

la

rr)1 UA(r)2 __5. L msan Up fl2 U

Fig. 19. Circulation distribution of forward propeller of 60 000 h.p.

counter-rotating propeller arrangement and associated diStrIbutions

of induced velocities at aft propeller

1 .4

12

2 .4 .6 .8

F(r)2 UA (r)1

Ta6e 3. Impact of assignment of blade numbers on efficiency of 60 000 h.p. counter-rotating propeller arrangement

Table ii. Impact of distribution of torque and r.p.m. on efficiency of

16 000 h.p. counter-rotating propeller arrangement

50

Number of blades Efficiency

(calculated with standard radial distrbutions of mutually induced velocities) Forward Aft 1 5 3 6 5 4 6 3 0.7575 0.7530 0.7479 0.7483 Design data: =

20.5

knots w = 0.25 For D1 =

5.22

m Z1 = 4 r.p.m.1 = 105 Z2 = 5

wake distribution see 'cargo liner' of Ref.

[24]

r.p.m.

aft/forward

torque

aft/forward

Efficiency

(calculated with standard radial distributions of mutually induced velocities)

1. 1.1

0.9

1. 1. 1. 1. 1. 1.1

0.9

0.6398 0.6387 0.6347 0.6381 0.6366

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 réasons of efficiency as stated in Ref. [21*]. 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 ± 1O 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.

2.4. CONCLUSIONS

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

f èa tu res

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

Vortex theory (according to the variable load actuator disk model) s used for the calculation of the radi-al distributions of the mutually induced velocities, taking intO account the actual load distribution.

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

52

The theory

design

4. Miitually induced effects are separated from self-induced effects in such a way that each propeller can be designed according to an e5tablished procedure for single propellers.

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

1. Comparison with experiments shows that the method yields the

correct power absorption, provided a reliable method for the. of single propellers is used.

The pattern of the vlocities 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 bladés. 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 increâse at thè root.

The mutually induced pressures call for an increase in the blade area of the fórward propeller by about 5 and a decrease, by the same amount, in that of the aft propeller.

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

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

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

3. CAVITATION INCEPTION OF SHAFT BRACKETS

The straightfòrward 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 ¡t, ¡s of interest. Cases have been encountered in pratice, 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).

5

The pröpeller radial, load distribütion is elliptical; the hub-diameter ratio d/D 0.2.

The maximum propeller ìnduced velocity at the bracket is then:

UA (xIR) VA where:

S=

From Fig. p rac t ica I p C- SIR) ¿4 / S/R - F1 (- S/R) a

(i

-\

V1 +

(SIR)2

distance between mid-chord of bracket and propeller disk.

11 it is apparent that for axial clearances in the range O.k < (S/R) <1.0, one may put F1 (SIR) = 1.00. This implies that the referencevelocity can be used for calculating

the inception of cavitation. According to Eq. (2.19) the propeller induced pressure is:

-pVA2a

(i-

S/R.

As in the previous chapter, the section at an height of 0.8R above the shaft line is taken as the determinant section, when calculating the inception of cavitation. The cavitation number of the bracket is then:

P0 -

e - 0.8 R y G

p VA2 {

1 + a

Vi +

(S/R) SIR

V i

+ (S/R)2

2e

-}

S/R + (sIR) (3.1) ) (3.2) 2 S/R

Vi +

(S/R)2

To show the effect of the propeller loading upon the inception of cavitation on a shaft bracket, an example relàting to a twin-screw container ship with 60 000 h.p. per shaft is given in Fig. 21.

1.5 lo 0.5 0.0 EUER LOADIIAI 55 OF PRO EFFECT J J oc 1.13 ' 1.35

PO.\

0.2 O O :02

LE. 0.0 0.2 O'. 06 0.8 .0 I.E.

0/c

STATIC PRESSURE P - e - OIR 1301.0 kgf 2 INFLRE VELOCITy VA 15.6 e.

OE1ISITY o 101.5 kgl n. ecc2

TIPJST CXFFICIE1IT CT 0.6

DISTAICE REThEER SF*IFT BRACRET AID PROPELLER SIR 0.8

Fig. 21. Profile of shaft bracket and pressure distribution at varying incidence

The cavitation number without allowance for the propeller loading is: P0 - e - 0.8 R y

- 1.35

-p VA2

Taking into account the propeller loading according to Eq. (3.3) the cavitation number becomes:

a = 1.13

In view of varying incidence due to ship motions and manoeuvring, and also because of uncertainties ¡n estimating the streamline direction, the bracket should remain free from cavitation at non

56

zéro angles of attack. The pressure distribution on the bracket was therefore calculated for various angles of attack (using Theodorsen's conformal mapping method),. The results (Fig. 21) show that the unfavourable effect of the propeller loading corresponds to more than i degree in angle of attack.

In bracket design, therefore, the selection of the thickness distribution, and particularly the thickness-chord ratio, should be based on calculations of cavitation Inception which include:

(i) anti ipated angles of attack,

pressure distributions at these angles of attack,

cavitation numbers of anticipated operating conditions, including thé negative effect of the propeller loading (according to the method outlined in this chapter)..

It is only when there Is no conflict with the requirement of freedom from cavitation, that strength considerations may prevail.