A note on the theory of ducted propellers working in open water

(1)

NORWEGIAN SHIP

MODEL EXPERIMENT

TANK THE TECHNICAL UNIVERSITY OF NORWAY

A NOTE ON THE THEORY OF

DUCTED PROPELLERS

WORKING IN OPEN WATER

BY

KNUT J. MINSAAS

NORWEGIAN

_{SHIP MODEL EXPERIMENT}

_TANK _{PUBLICATION N2 93}

JANUARY 1967

(2)

Contents.

Page

Introduction 1

The Thrust of the Ducted System i

Estimation of Axial and Radial Velocities Induced on Nozzle Profile by the Propeller

L The Thrust of the Nozzle 5

5. Propeller Thrust and Torque 8

6

Comparison between Values Found by

Experiment and Calculation 10

Nozzle Wake and Propeller Thrust 11

(3)

Definitions:

Velocities (roeller)

VA Intake velocity of the water into the nozzle system

V1 Total axial velocity in the propeller slip stream

UT Tangential induced velocity in the slip stream

UA Axial induced velocity in the slip stream

UN VUT2

V Velocity of the water at the propeller in the nozzle Velocity at the profile

w Angular velocity of the propeller

(Fig. 3)

V (Fig. 3)

Velocities (nozzle)

U_rm The mean of the radial induced velocity on the nozzle by the propeller

0xm The mean of the axial induced velocity on the nozzle

by the propeller

U_am The mean of the axial velocity increase in the slip stream

Constants and Coefficients c

_°

K

T

; Thrust

p/2

_VA 2 constant of the nozzle system

Thrustcoefficient of the nozzle system

pn D

T K

p 2 Thrustcoefficient of the propeller

pn D

KQ _{Torque coefficient of the propeller}

pn

D

T

Thrustcoeffjcjent of the nozzle

(4)

CL Liftcoefficient of the nozzle profile

V

- _{Nozzle wake}

d1

VA

K _{Masscoefficient or total loss factor}

P/D Pitch ratio of the propeller (hydrodynamical) VA

J _{Advance coefficient of the propeller + nozzle}

V

J

-

_{Velocity coefficient in the slipstream.}

x nD

E Axial loss factor and drag-lift coefficient

k1 Constant of circulation k2 Radial constant of velocity k3 Axial constant of velocity

Ao/AE Bladearearatio of the propeller

K Goldstein factor

g

l/D Length/Diameter ratio Geometrical dimensions

i Length of the nozzle profile

D Diameter of the propeller

D Diameter of the equivalent vortex ring

r Radius of the propeller

P Pitch of the propeller (hydrodynamical) Rd = Radius of the equivalent vortex ring

X _{Distance from the leading edge to the point of max. camber}

c Chord of the profile

t Max. thickness of the profile

Different definitions

n Number of revolutions of the propeller

T Thrust of the nozzle system (T _{+ Td)}

T Thrust of the propeller

(5)

Q Torque on the propeller

Total circulation of the nozzle profile

L Lift in general

Z Number of propeller blades

A Area of the equivalent foil

2

A Area of the propeller disk

AE Expanded blade area of the propeller Hydrodynamical angle of pitch

Angle of zero lift

Angle of attack measured from the propelleraxis a - a Effective angle of attack

1 o

a1 Angle between the propeller axis and the chord-line of the nozzle profile.

(6)

Summary.

A simple theory for ducted propellers is developed, which makes possible a rough estimation and analysis of different nozzle systems. It takes into consideration the

number of propeller blades and the nozzle shape. The development is based on simple and well-known principles from the _vortex

theory. _{A comparison between experiment and theory shows}

good agreement.

Introduction.

The author knows no method of calculation for ducted propellers which gives sufficient agreement with experimental

results. _{The method of Dickmann and 1eissinger (6) is valid}

only for propellers with an infinite number of blades and with no rotation in the slipstream.

The following theory takes into consideration the number of blades and the characteristics of the nozzle profile. The slipstream contraction is not taken into _{account and it is} assumed that the slipstream diameter is equal to the propeller diameter. _{The diameter of the boss is not considered.}

Dxpressions for the thrust of the system, the propeller and

the nozzle are developed.

The Thrust of the Ducted System.

Considering the slipstream some distance from the propeller, the volume streaming through an annular element of

the slipstream will be

(7)

which gives

U

VA UT

-

UT UA

This means that

UN /UA + UT2 and the energy is

(U 2 + U 2)

2

Equivalence of the energy in the system gives

dQ w

_{dTVA +}

(U2

+ UT2).

2

By means of the equations for dT, dQ and dm we get

U

2 2

wr.UT=VAUA+(A

T),

2

which gives the thrust

dT 2 r V1 UA dr.

The tangential induced., velocity is UT and the tangential force is

dK = 2 yr V1 UT dr, which gives the torque

dQ 2

_p

r2 V1 _{UT dr.}

The velocity diagram of the propeller is shown in Fig. 1. It

is seen that

(8)

Frg. 2 Fig. i VP

n

cor A

Fig. 8

B I 'A Vr

(9)

and

V

=V

₊₀₀

It is assumed that the free vortices being shed from the nozzle

and the propeller are forming regular helicoidal surfaces between

I and II in Fig. 2 with the pitch angle

_.

The 1nduce. velocity

will then be normal to the surface.

The velocity diagram is

shown in Fig. 3.

Constant V

along the radius is now assumed.

From dT

2ip r V1 UA dr

it follows that

dT

p2iî r(V

+ U)(V

_{- VA + U)dr}

R

(V2

- V V

+ (2V

- V )U

+ U

2)r dr.

x A x

Ax

X 3

The equations

o R

2iri

2 U

rdrw

2 itD2 14

and

JX

o (R 2ir U

r dr

KW 2 lTD 24

where

X o V

wV (P/D---)

give

X nD T

I.

K0 2 14

pn D

2

j

+ (2Jx - J )(P/D_Jx)K+c(P/D_Jx2)).

24 X X O O

(10)

60

0 L LL L E

6.d

9 !.d 80 a 90 7O

(11)

3) Estimation of Axial and Radial Velocities Induced on Nozzle Profile b the Propeller.

A propeller with an infinite number of blades and

no rotation in the slipstream is considered. It is assumed that the velocity V + U VA + UA is uniformly distributed over the propeller disk. Far ahead of the propeller the pressure is P0. The propeller causes a sudden pressure increase p. Behind the propeller the pressure is Po + Ap. The velocity in the slip-stream just inside the contour of the slip-stream, is VA + _UA. Just

outside the contour the velocity is VA. Applying Bernoullis equation for any point on the contour, we get

l/2((VA+UA)2_VA2)

= l/2P(VA+ UA + VA)(VA + - VA)

where

112(VA + + VA) VA +

is the velocity just on the contour and where

VA + UA - VA

is the strength of the vortex sheet replacing the discontinuity

of

the velocity. The equation of the vortex sheet is

(VA +

The propeller is now replaced by _{a semi-infinite vortex cylinder} (sheet) inducing a velocity UA uniformly distributed along the

radius. _{The induced velocities from this cylinder} _{are given by}

Küchemann and Weber in "Aerodynamics of Propulsion'.

₍₈₎

A single cylinder can only replace propellers with constant circulation along the radius. If variable circulation

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0.2 0.4 0.6 0.8 YR 1.0

Fig 9.

dC 120 110 100 go 1.1 tfl. 1.0 0.9 0.8 0.7

lo

20 30 40 Fig 12c 50 60

Xyc

10 A UA

(13)

1.0 Jo 0.8 0.6 0.4 0.2 0. E 0.6 0.4 0.2 O 0.8

_Jx

1.0 0.2 0.4 0.6

Fig. 4

0.8 0.6 1.0 12

F,g. 5a

1.6 1.4 I

(14)

1.1 'Zd 1.0 0.9 0.8 0.7 0.6 0.5 .7

d

t 2 1.4 0.8 0.6 0.4 0.2 O 0.8 1.0 1.2

Fig.5b

1.0 1.2 P/D 1.4

Fig.5c

0.2 0.4 0.6

Fig 30

o 0.8 J0

(15)

G(x)

2

X

+ (x tg )2 g

Kg is the Goldstein factor.

) The Thrust of the Nozzle.

-5

occurs, the propeller is replaced by an infinite number of coaxial cylinders each having a vortex strength equal to

d y dr.

(Kobylinski: mt. Shipb. Progress No.

88 Vol. 8, 1961). (7)

The estimation of the constants k2 and k3 requires the knowledge of the UA distribution. In Fig. 9 two

distri-butions: one for the usual propeller with elliptical blade area and one for the Kaplan type propeller are given. The distri-butions are based on very rough assumptions. Several vortex cylinders placed inside each other will give the wanted

distribution if the vortex strengths are correctly balanced.

Using the table of Küchemann and Weber

(8)

it will now be possible to estimate the axial and the radial induced velocities on every single point of the nozzle contour. Fig. ! shows the connection

between

x and J0 for different valaes of P/ID. K and c are given

in Fig. ₅ as functions of P/D x ir

tg ß.

K is given by the

equation

K

G(x)xdx,

where

For a foil of chord length i and of infinite length the lift L is given by

L L

12 V2 i

(16)

k

VA

where k2 is a constant.

If _{is the mean of the axial induced velocity} from the propeller on the nozzle the circulation will be

-6--We may also write L = pVr, where r is the total circulation. From this

CL V i CL a

i

2 2

where a is the angle of attack.

The same equations are valid for the annular wing but CL is greater because of the self-induced velocities. The ratio CL ran an k -1 CL' a

F1011

is shown in Fig. 6 as a function of lID. If i is the length of the nozzle the thrust will be

where

Td 2rrp R i- U

d d rrn

RL Radius of the equivalent vortex ring

Total nozzle circulation

0rm The mean of the radial Induced velocity on the nozzle from the propeller.

From (V - VA +

U)r

dr

we have

Uam

(17)

which gives II. where o

+k

2 Dd i

CLt

(1 ₃ _VA .

Kd=k2 Jo

;i-;

2 _VA

The constants k2 and k3 are given later. _{In Fig. 8 the velocity} diagram B is replaced by diagram A. This will have an influence on V only for the highest values of J0. It gives

Kt

0.5 + O.5K.

1)

C1'a

(VA

+ U)i

1d

2

where O is given by a constant k3 through

mx

o o

- k

VA

3VA

The induced angle of attack is given by U am o V rm

tga1

k2-Uam mx., VA(l

+ V-,

i + k3 VA

The effective angle of attack is

ct a1 - cx0

where a is the angle of zero lift (Fig.

7).

The nozzle is

replaced by a vortex ring with diameter Dd. The nozzle thrust is expressed non dimensional as

Td

Kd

(18)

The velocity ratios

i

U1p

11U

U i xm I rx and

-- dx

VA

iJ

VA VA

li

VA o o am as functions of give VA O B propeller Ka propeller

-8

n d

k2

_0.303

_0.305

k3

0.2L2

0.260

The drag of the nozzle will reduce the nozzle thrust:

CL sin

-= 1

C1)

Kdj CL sin 0L tg

1

where CL is the liftcoefficient of the profile and 01) the drag-coeffisient. 01) 0.01 gives

d 0.9 - 1.0 for P11) 1.0 - l.1 and J0 < 0.50. Fig. 30 shOws as a function of J0 for different numbers of PID.

5) Propeller Thrust and Torque.

The torque and the thrust of the propeller can be estimated in the same way as for an ordinary proneller, if this has a velocity of advance equal to V. If the propeller thrust is found as the difference between the tota' thrust and the nozzle thrust, the torque can be roughly estimated by regarding a foil which is equivalent to the propeller. The foil has an area A _{and profile characteristics equal to those} _{of' the} propeller at a certain value of x. The exact value of x is

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0.10 E 0.08 0.06 0.04 0.02 o Fig. 10

B 4-55 Yc_=oo53

t/c_

FÇ4-55

0.048 VC

'.4

C 0.06 0 01 _8.2 _0.3

0.4 CL 05

Fig. 11

(20)

x 0.66 to x 0.78. Usually x 0.70 is chosen as a standard

value. At any section of the foil the velocity w and- the angle of attack are identical to the values that occur at the station x on the propeller blade (Fig. 10).

The llftcoeffisient of the foil is decomposed into the components CQ in the tangential and CTI in the axial direction

(Fig. 10). We get the following equations:

and and 2 t c T n D

_KT

nD 2 K T /2 w2 A Q

-

p

fl'DKQ

(2)i2

2 KQ CQ -

-p12 w2A.0.35D

p12 w2A .035D w A 0.35 X X CT

CLcos

_CDsin

. CQ _{CL sin} ₊ _CD CO5 CD O gives KQ1 _{0.35 tg} If CD >0 we get CT KT CL cos _{- CD} sin

1 - 0»55 P/D

- C

cos1

CT! _Ti _L CQ KQ

_CLsin

_{CD cos}

C.

K Ql Ql CL sin P/D n Q

T'

pI2w2A

w A X X

(21)

1.5 . A IA

_Eo

where AE/AQ is the blade area ratio of the propeller.

c is a function of CL, the camber and the thickness of the

profile. In Fig. 11 has been drawn as f(CL)for an actual NACA pröfile. If a more exact expression for CL is wanted, the one on page 13 can be applied.

6) Com2arison between Values Found by Experiment and Calculation. The thrust of the system and the nozzle as well as the torque has been calculated for two fourblacled propellers in nozzle

(l/D z 0.50,

j

z 10.2°). One of the propellers is of the Kaplan type, ('k) the other is a B u-55 propeller (van Manen) (2.3.). Both have the same blade area ratio and the same pitch and pitch distribution (constant). The connection between the hydrodynamical

and the geometrical pitch is indicated by the equation 0.7 tg z

0.96 0.7 tg + 0.055 developed under the assumption of the velocities V + Ux and UT!2 at the propeller disk. is the geometrical angle of pitch. It is assumed that P/D z x tg 6.

is kept constant for different values _{of x.}

For nozzle 6 and 19 (Wageningen, van Manen) (2.'4) Dd/D is put equal to 1.06 and 1.09. _{The slope of the lift is given by}

CL

lo

-If KT, and are known, KQ can be estimated. is a function of CL. Gutsche has analysed a number of propellers calculated

by vortex theory and gives the following formula for a rough estimation of CL K p d C1 z 0.11 L k1, dOE

where k1 is shown in Fig. _{6 as function of l/D.}

d CL

da 21T k1

(22)

120 ,.

a.-'

.h

0.12

d110

100__

90

lo. -

_{100_ a}

90

110g

100

L'-90 80

II

lO 20 30 40 50 60 xf/c

Figl2a

1.0 0.8 0.6 0.4 0.2 120 d C1 dcx. ₁₁₀ 100 90 110 100 90 110 100 110 loo 10 20 30 40 50 60

Fg12b

0 0.6 0.8 1.0 1.2

Fig 27

1.4

/D

is

(23)

4., 0,2 41 44' 43 4 47 ¿3 4. 48

A

-Fig .1 3. 43 48 47 45 0,5 44'ç 43 o

8-/nOiscN,'6

m//N4C4-Prv/7/4'75 K

pDn

-

742e H

-A il

L

,

t

AiI1U1

I

UI

1I1I

IiL

ß'#-55 Scrie ni/I N4C4-PrcI)i 0,50 102° thOúc4'r.6 4'4'/5 ¿34' 45 O,6'O,7 0,8 45 1,0 7,1 2 1,3 O 41 42 43 1,3 1' ç' 45 46 47 48 ¿3 0

(24)

f1-I-O

4 0-5

o

_'n

np

Fg.14,

SCREW SERIES IN NOZZLE No.19

H,, 11/)-I.4 2 08

!FILIIIIIk

)l 0-4 17

1/

-' rL rA -%

-

-o OS

(25)

-dO

(21T k1).

In Fig. 12a and b we have given dL as function of

the distance between the leading edge and the point of max. camber.

in Fig. 12e is the mean of values measured by MACA and CØttingen. For the comparison with B 4-55 the K-values for z _{4 are employed directly.} _{In the case of the Ka 4-55 propeller}

K for z 5 is employed. _{By the calculation of KQi}

ktg jKTj,

k is chosen as 0.35 for B 4-55 and a bit higher for Ka 4-55.

A comparison between experimental and calculated values shows good agreement (Fig. 13 and 14). The conclusion ìs: The

theory seems applicable for a rough calculation and as base for a simple analysis of the influence of different parameters.

7) Mozzle Wake and Pro2eller Thrust.

It is important to know the velocity through the

propeller disk. This velocity VP is given in percent of VA by the following equation

V

-VA

where

'a is called the nozzle wake. ta iias been estimated by Dickmann and Weissinger for propellers in nozzles with contra systems (no rotation in the slipstream, UT 0). It may also be computed by the formula

0.5 (/1 + 2 C - 1), o where T C ° p12

V2

A o

The formula has been developed from simple momentum consideration. It gives values of the same order of magnitude as those for nozzles without contra systems but is too rough for an accurate calculation.

It does not take into consideration the influence of pitch, as will be seen from Fig. 23 where the formula has been compared to some

(26)

r--5.5 4) -5.0 -4.5

-4.0

-3.5

-3.0

- 2.5 -2.0

-1.5

-1.0

-0.5

o -5.0 -4.5

-4.0

-3.5

-3.0 -2.5 -2.0 -1.5

-1.0

-0.5

0 5 10 15 20 25 30 35 40 45

Fig 23

5 10 15 20 25 30 35 40 45 co

Fig 24

NACA 415

VD OES

B 4-55

«1 =12.70 SI/o 1.4

¿I.

,P

D.W.

/

/ì,

(27)

-5.0 'V -4.5 -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 - 0.5

-5.0

4)

-4.5

-4.0 -3.5 -3.0

-2.5

- 2.0 -1.5 -1.0 -0.5 o o

u

-u

IFA

r'

Alu

IUA

-u

O 5 10 15 20 25 30 35

40,-50

Fig.25.

10 15 20 25 30 35

_40c45

o

Fig.26.

(28)

12

-experimental results obtained by van ianen. (2,

3,

_,

5).

VA

0.5 + 0.5

<. This was assumed to be correct some distance behind

the nozzle but at the propeller disk Kt will often have a higher value. (Especially in the case of the Kaplan type propeller.)

For Ka

_a-55

and nozzle No. 19 it is assumed that

K + K1

\z

I _(Fig.

_27),

2

and for B

_a-55

and nozzle No.3 that K

= K(z=4).

The equation

U

- 1) gives good agreement with

the Wageningen tests as shown in Fig. 23, 224, ₂₅ and Fig. 2G.

At radius r we get the following circulation around the propeller blades

F 2 _{r UT}

According to Kutta Joukowski the propeller thrust will be

d T p 2it r U K (wr - U /2)dr. P T g T Integration gives T 2 _{UT Kg (wr - UT/2)r dr.} o and nondimerisional K _{UT (wr - U /2)K} _{x dx.} t g

2(R)2

U ar

was put equal to

K'(P/D/J0 -

1) and K' equal to

(29)

0.2 0.1 0.2 0.1 O

B 4-55

NACA 4415

o(10.2 WJ0.5

0.1 0.2

03Kp(dia'ì4

Fig 28

K 4-55

NACA 250

102 Íf)0.5

/0'

e O 0.1 0.2 0.3 Kp(dici)0.4

Fig 29

0.4 Kp 0.3 0.4 Kp 0.3

(30)

13

-As to Kg values are chosen which give good agreement between calculated and measured wake for the case considered. A distribution of circulation equal to optimal distribution for propellers working in open water (without shrouding) has been

chosen. K has been calculated for different pitch ratios and coefficients of advance for the Ka 'k-55 and B u-55 cases. A comparison is made in Fig. 28 and

29.

The agreement is not to bad.

8)

Blade Area-Ratio. Fig. 8.3 gives 2ir x K D CT g (cos - cos

zC

where z Number of blades

D Diameter of the propeller

C Chord of the profile K Goldstein factor

g

V + (wr)2

For x 0.7 and different values of . and

, CL has been

calculated. Then the angle of attack at x 0.7 has been estimated for AL/A 0.0, 0.55, and 0.70.

To obtain as a function of and t/C, the slope of the CL, curve has to be corrected for the curvature of the flow which in the case of the propeller is different from the wellknown 2 dimensional flow of the given section. In

2 dimensional flow the induced velocity in any direction on the section is U2 2 (x/C, x) CL and ur cos

(31)

0.8

K0

0.7

06

05

0.4 0.3 0.2

04

0.6 0.8

10

1.2 ₁₄ ₁₆

Fig.30.

0. K0 0. 0.

o

0.2 0.1

/

0.40

0.55

0.70

Fig.31. J

JO.2

06 08101214

(32)

0. K

0

0. 0. 0.1

o

0.4

06

0.8 1.0 1.2 14 16

Fig32.

I I

°.

/

(33)

1.0 K o 0.9

0.8

0.7 0.6

05

04

0.3 0.2

JQ

À

r

'VA

À

r

lì

7

055

070

0.4

06

0.8 10 12 14 1.6

Fi933,

(34)

If the same section is placed on the nropeller the corresponding velocity is

U3 f3 (x/C x)CL.

The difference U U3 - U2 C (f3 - f2) gives the change in effective camber. If CL

O,U =

O and no correction is

necessary. This means that the angle of zero lift remains unchanged for a given camber.

If the lift is different from zero and a certain is wanted for a given section and propeller, the slope of the CL o. curve can be reduced by a value equal to the usual camber correction. There are different methods of calculating the camber correction which all give different values. Ilost of them are only valid for certain distributions of circulation and

for the ideal angle of attack. Further no camber correction has been calculated for the ducted propeller. 'Je therefore have

used a slope reduction of 30 as a mean. Ludwieg Ginzel's camber correction gives a reduction having the same order of magnitude.

By drawing the CL curves for AE/A O.0, 0.55, and

0.70

a has been found for different values of _CL, =x Trtg ß and

J0.

Then for AE,A _{= O.O, 0.55, and 0.70 has been}

estimated by setting

a =

-where is given by

X

tg

.

From the simple geometrical relation between _{and sj'} K0 has been calculated for different values AE/A and

J0.

(35)

Fig.35.

-3

-2

-1 i 2

3

4 5 67

Fig.34.

(36)

- 15

This ha been donc for nozzle No. 7, 3 14_40, B 4-55 and B '4-70. In Fig. 30, 31, 32 the calculations arc represented by the

points and show good correlation with the curves obtained

from experiment. Concerning the Kaplan type propeller results are available from tests with Ka '4-55 and Ka '4-70 in nozzle No. 19A (Nageningen). At x 0,7 the tic ratios are 0,0370 and 0,0482. For circular sections the angle of zero lift is equal to tic. As C0 of the Kaplan propeller is greater than C0

.

of the B type, CL/B will be reduced. At x 0,7 the slope reduction is assumed to be about 14Q corresponding to the

reduction for a B '4-75 propeller. If C0 for the B propeller is 55 it will be 75 for the Kaplan type of equal blade arca

ratio. K0 has also been calculated for K '4-55 and Ka '470

for different values of AE/Ao and J0. As shown in Fig. 33

there is a good correlation between calculated and experimental

(37)

References

1) Horn Arntsberg: Entwurf von Schiffsdüsensystemen (Kort-Dysen), Jahrb. S.T.G. Bd. (1950).

Manen, J.D. van:

) Manen, J.D. van:

5) Manen-Superina:

Ergebnisse systematischer Versuche mit Schiffsdüsensystemen. Jahrb. S.T.G. Bd. L7 (1953), s. 216 - 2Ll.

Open Water Test Series With Propellers in Nozzles. mt. Shipb. Progress No. 36

(1957), Vol. 4..

Effect of Radial Load Distribution on the Performance of Shrouded Propellers.

Publication No. 209 of the N.S.M.B.

The Design of Screw-Propellers in Nozzles. mt. Shipb. Progress No. 55, Vol. 6 _(1959).

Dickmann-Weissinger: Beitrag zur Theorie optimaler Düsenschrauben. Jahrb. S.T.C. Bd. 9 (1955), s. 253-305.

Kobylinski: The Calculation of Nozzle Propeller Systems based on the Theory of thin annular Airfoils with arbitrary Circulation Distribution.

mt. Shipb. Progress No. 88, Vol. 8, (1961). KUchemann-Weber: Aero-Dynamics of Propulsion. Mc. Grw-Hi11.