A theory for prediction of the off design performance of kort nozzle ducted propellers

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2 DEC. 197!

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DATUM: DOCU.1HTATE

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A THEORY

FOR PREDICTION OF THE

OFF-DESIGN PERFORMANCE OF KORT

NOZZLE DUCTED PROPELLERS

by

R. I. LEWIS

(Professor of Fluid Mechanics and Thermodynamics,

Department of Mechanical Engineering, the

University of Newcastle upon Tyne)

i.ab.

y.

Scheepsbouwkuncle

Technische Hogeschool

Deilt

The author would like to dedicate this paper to Professor J. Weissinger on the occasion of his sixtieth birthday and with respect to his many

(2)

of the characteristic curves for Kort nozzle ducted

propellers. The only initial data required are the

values of the thrust coefficient, the thrust ratio and the advance coefficient at one central design point

for a given ducted propeller. _{Predicted curves of}

thrust ratio, advance coefficient and propulsive efficiency,

all versus thrust coefficient, compare cïosely with

published experimental test results for N.S.M.B. propeller

(3)

1.0 INTR0DUCTI0

For more than forty years accelerating Kort nozzles have been fitted to propellers to provide additional thrust for small heavily loaded vessels

such as tugs and trawlers. More recently these devices

have attracted interest with regard to supertankers and bulk carriers in view of the increased propulsive

efficiency which may be obtained with promise of

significant

savings in fuel cost.

Several British research publications have appeared recently relating to advanced fluid dynamic

ana1yses1 (2) or design performance prediction3 mostly

directed towards the application of Kort nozzles to very

large vessels. The present paper complements these by

providing a simple method for predicting the _performance

characteristics of a ducted propeller in open water, the

only prior information required being the values of the

thrust coefficient _{CT ,} thrust ratio 1- and advance

coefficient J at one central design point. _{The method}

involves the assumption that the fluid deflection

through the propeller at the r.m.s. radius is _constant

for all values of J. _{If cascade data are available to}

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the designer, then this assumption _{can be dropped.}

Satisfactory predictions are however obtained retaining

the assumption.

The propulsive efficiency has been shown by the present author3 _{to depend largely upon seven} principal dimensionless groups and two frictional

coefficients as follows,

17 Propulsive power

Shaft power

f(c , J, C, h, i /D , t/l ,

'd

I _CJJp)

(1) System Machine design _Frictional

variables variables coefficients

where the system variables _{are the most influential.}

Since these are pertinent to the present analysis _their

definition will be included here

c T _{(Thrust coefficient)} ₍₂₎ i 2ir2 PVaID j = _{(Advance coefficient)} (3) i:

-Propeller thrust (Thrust ratio) ₍₄₎

(5)

Reference 3 is concerned primarily with the reduction of equation (1) to analytic form to simplify

prediction of propulsive efficiency. On the other hand

the relationships between CT -c and J are not known a

priori. The burden of this present paper is indeed to

supply these relationships in the form of two characteristic curves,

-c ₌ (5)

J = fc '

(6)

for which closed analytic expressions are

derived. The first characteristic is dealt with in

section (3.0) and the second in section (4.0) following some general considerations of duct thrust and wake

vorticity outlined in section (2.0)

2.0 WAKE VORTICITY AND ORIGIN OF DUCT THRUST

The forward thrust upon the duct _{Td ,} Figure

(1), originates from the interaction between the

contracting flow produced by the propeller and the _Kort

nozzle behaving as an annular aerofoil. _{The simplest}

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case to consider is that of a free vortex propeller with

zero tip clearance. The effect of the propeller can

then be represented by the radial inflow or 'downwash'

induced by the trailing ring vortex sheet wake y. which divides the jet flow from the outer flow downstream of

the propulsion unit. This vortex sheet in fact comprises

spiralling vortex filaments, although the tangential component of vorticity is alone responsible for the

radially inward 'downwash'.

An expression for the value of this wake tangential

vorticity at infinity downstream,

e

ignoring the

real effects of viscous or turbulent diffusion, can be derived directly from the jet velocity V. , Figure (1),

Joo

y.

=

V. -V

Joo a

Following the assumption implicit in simple one dimensional theories such as that by Van Manen and

oosterveit, that for all streamlines

p2 - p1 p02 - Pol = constant,

the propeller thrust is then

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T =

(p2-p1)(l-h2)

p 2 2 2 liD 2

p(V.-V )(1-h

)o a 8

Also from equations (2) and (4) we have

i 2im

T

=1-T

CCpV

p T

Hence eliminating T

and finally from equation (7) p

- v/i

ICT

1 ₍₁₁₎ V - l-h2 a

In view of the rather unsatisfactory assumptions leading to this simple expression which is independent of J, a more rigorous analysis is presented in the Appendix resulting in the following improved equations for

je

fi+Jii

V A a (9) (lo) (12) 6. (Ba) (8b)

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where C

32

A = 2CT B

= 1-A

C = - (l+C Po

(

J2JCT

lnl ) C = po J 'klnl/hi ¿ i -(1_h2)2

Computations reveal however that equation (12)

gives surprisingly little improvement over the approximate

equation (11) which is therefore adopted in the following section concerned with the 1 _{(CT) characteristic.}

3.0 PREDICTION OF THE _{C (CT)} _{CHARACTERISTIC}

Prediction formulae for the t (CT) characteristic follow from the hypothesis that the duct forward thrust

is proportional to the square of the radial downwash

velocity. This hypothesis is, for example, precisely

true for the rectilinear flat plate aerofoil, Figure (2)

located in a uniform stream W at negative incidence _a

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to the Magnus law the lift force normal to W is given by

L pwF

where, for the flat plate aerofoil the bound circulation is given theoretica11y'5 by

F

= vlWsina

Thus the forward thrust T along the line of the plate is given by

T = Lsina

= pT1(Wsin)2

2

oC (Downwash velocity)

For any other aerofoil with zero lift at zero

incidence the same result can be argued cpialitatively.

If one assumes that the duct circulation of a Kort nozzle with no propeller present is negligible then the same

hypothesis holds true. Furthermore, provided the vortex

wake remains reasonably cylindrical the radial downwash

is proportional to resulting in the postulate

8.

2

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But we have already shown from equations (9) and (4) that Td (l-t)T

i 22

= (1-C)CpVD

T2 a4

Substituting this expression and equation (11)

into equation (14) we thus have the proportionality

(1 - t) CT

£

5 1-(1-C)

o CCT ₂

1+

-1

1-h'

If this is applied also to the central design condition

('t1

_{CT) we have finally the identity}

CT + l-h2 o To l-h2 C To) CT

which has the same general form as the characteristic

equation (5) but has been written _{as an identity for the}

purpose of computation. For a specified value of CT

the computational procedure is to obtain successively improved estimates of 11 by substituting the latest

estimate into the right hand side of equation (15).

Convergence is quite rapid.

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As seen from Figure (3) surprisingly accurate predictions were obtained by this simple theory when compared with the pub1ished6 experimental _{tests for}

the N.S.M.B. propeller Ka 4-55 in duct 19a over the wide

range 1.0 <CT< 25.0.

From a formulation derived

in the appendix, equation A(4)a, the jet _diameter

contraction ratio D. /D has also been calculated _{over this}

Jc

range, Figure (4). _{The earlier assumption that the}

jet remains reasonably cylindrical is certainly valid

for this case and would be for most _{Kort nozzles of this}

type although largely by accident. If _{values depart}

greatly from 0.7 at the central design point, _{as can be}

seen from equation A(4)a, D./D can differ considerably

from unity. _{A typical example would be the} _{pump jet}

designed for CT = 3.0, t 1.05 . In this case with

h = 0.2 equation A(4)a yields D. /D _{= 0.823, a considerable}

Jc

jet contraction.

4.0 PREDICTION 0F THE _{J(CT ,)} _{CHARACTERISTIC}

The J(CT ,1) characteristic _{represents the}

relationship between speed of rotation and _thrust.

Since thrust is dependent upon blade loading _{and velocity}

triangles which in general differ for _{each radius, we}

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shall assume that conditions at the r.m.s. radius r m are representative of the integrated effect of the whole propeller, where

r =

m 2\Á1 -h2) (16)

Also at this radius, for most radial

distributions of blade loading, it is reasonable to

assume that

vÇV

p p

At this radius the cascade static _pressure

rise coefficient follows directly from _{velocity triangle}

geometry as follows C_pm = = tan - tan'

12

1m 2m 2 (17)

¡pvp

But the propeller thrust is given approximately by

T ₌ (1-h2)

122 =-Cc

pv 'nID

(13)

so that alternatively we have

c V 2

T(a

pm - 1-h2 'p

But from derivations given in the appendix

2(1-h2)

-[ /1

CCT

V _CT

l-h2

whereupon equation (18) becomes

follows.

C 4(1-h2)1

pm _CT

Finally we may eliminate

1m from equation (17) to produce a system of equations _{involving J,}

CT t and . Thus 2m

tan1

U 2irnr V m m a V V V p a p )2 +

-1

1-h

and with substitutions from equations (3), (16)

and (19),

J

(1+h2) / C tane = 1-h2) -

/1+ T

1m _JCT V _1-ha

The computational procedure is then as

12.

(20)

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For the central design condition substitute

CT and J0 into equations (20) and (21) to

derive C and . Hence solve equation (17)

pm 1m

for and evaluate the fluid deflection 2m

1m - 2m

For each off design condition

CT and

t

are already known and we wish to calculate the value

of J.

Substitute _{CT and} c _{into equation (20)}

to obtain Cpm and into equation (21) to obtain

Jtan . Estimate J and thus and solve

1m _1m

equation (17) for Check

1m2m

against

the design fluid deflection and continue _{to make new}

improved estimates of J until the correct fluid

deflection is obtained.

The assumption implicit in the above proposed computational procedure is that the fluid deflection

lm - 2m is constant for all points on the characteristic. For low solidity high stagger cascades this is _{a good}

assumption usually but if cascade deflection _{data is}

available the assumption can in any case be lifted.

The J(CT) characteristic computed by this

technique is compared in Figure (3) with a curve derived

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14.

for the Ka 4-55 N.S.M.B. propeller in duct 19a.

Extremely good agreement was obtained particularly for values of CT greater than design but for very low _CT

values below about 2.25,J was overestimated by the theory. The same regions of error occur for the 1 characteristic

previously mentioned.

A possible explanation for this behaviour may be forthcoming from Figure (5) which shows the fluid relative inlet angle

lm at the r.m.s. radius for the

total range of CT values considered. At high speeds

of rotation and large thrust coefficients little change in occurs largely due to the increase in Vp/Va in

rough proportion to n . Only at the lower

CT values

does

lm decrease significantly. Thus at CT = 2.25

1m has decreased by 2 compared with the design value whereas at _CT 25.0,

lm has increased by only 10

5.0 PREDICTION OF PROPULSIVE EFFICIENCY

Using the method outlined by the author in

reference 3. the propulsive efficiency has been estimated neglecting tip leakage losses and compared with the

published experimental tests in Figure (6). Close

agreement was obtained over most of the range although,

as before, there is room for improvement at low CT

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values. _{Since, as explained in the introduction,}

is highly dependent upon _{CT ,}

I

_{and J,}

greater accuracy would be obtained if the _{1. (CT)}

j (1

, CT) curves could be improved in this region.

This would be possible to some extent by the _introduction

of cascade deflection data, although it should be

mentioned also that constant values of propeller and

duct drag coefficients of 0.006 and 0.04 _respectively

have been assumed for all conditions. _{At low} _C

values this assumption may tend to break down.

COMCLUS IONS

A simple theory has been presented for _prediction

of the _(CT) _{J(CT) open water characteristics of a}

Kort nozzle ducted propeller. _{Calculations have been}

compared with published

_N.S.M.B.

_{experimental tests for} the Ka 4-55 propeller in duct l9a which _{show extremely}

close agreement over the very wide range 2.25 < _{CT < 25.0}

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REFERENCES

P. G. Ryan and E. J. Glover

A ducted propeller design method: a new

approach using surface vorticity distribution

techniques and lifting line theory. R.I.N.A., 1972.

R. I. Lewis and P. G. Ryan

Surface vorticity theory for axisymmetric potential flow past annular aerofoils and bodies of revolution with application to

ducted propellers and cowls.

Jrn.Mech.Eng.Sci., Vol. 14, No. 4, 1972.

R. I. Lewis

Fluid dynamic design and performance analysis

of ducted propellers.

North East Coast Institution of Engineers and

Shipbuilders, Trans., Vol. 88, No. 3, 1972.

J. D. Van Manen and M. W. C. Oostervelt Analysis of ducted propeller design.

Society of Naval Architects and Marine Engineers, Annual meeting, Paper No. 13, 1966.

G. K. Batchelor

An introduction to fluid dynamics. Cambridge University Press, 1970.

J. D. Van Manen

Effect of radial load distribution on the

performance of shrouded propellers.

Netherlands Ship Model Basin Publication

No. 209, reprinted from International Shipbuilding

Progress, Vol. 9, No. 93, 1962.

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NOTAT ION

A)

B Functions of 1, CT and J given by

C ) equation (13).

Duct drag coefficient

Propeller drag coefficient

Cascade static pressure rise coefficient

12

2

C Stagnation pressure rise coefficient

Po

2

L

_Po/PVa

Thrust coefficient

Design thrust coefficient

D Propeller diameter

Jet diameter at infinity downstream

h Hub/tip ratio

j Advance coefficient

J Design advance coefficient

CDd C Dp C pm CT C To D. Jo Duct length

(19)

i Propeller chord at tip radius

L Lift force

n Revolutions per second

p Static pressure

Po Stagnation pressure

r Radius

Root mean square radius

Circumferential pitch between blades at

tip radius T Thrust Duct thrust T Propeller thrust p U Blade speed

VP Axial velocity in propeller plane

Ve Swirl velocity Va Advance velocity 18. r m t

(20)

V. J V p w p 11

¿po

Jet velocity at infinity downstream

Mean velocity in propeller plane

Uniform stream

a Negative angle of incidence of W

Relative fluid inlet angle at r m

Relative fluid outlet angle at r

m

Jet vorticity

Tangential component of jet vorticity

at infinity downstream.

Bound circulation of flat plate aerofoil

Tip clearance

- pl) Static pressure rise

- Pol) Stagnation pressure rise

Propulsive efficiency

Density 1m

2m

(21)

20.

'C Thrust ratio

to

Design thrust ratio

Suffixes

i Upstream of propeller

2 Downstream of propeller

c'o Infinity downstream of propeller

m Root mean square radius

(22)

APPENDIX Derivation of V

_ap

/v

T 2

-h2)v

(V.

-v

p

Joo a 2 i

2D

= CTPVT

From overall momentum considerations

But from equation (lo) we have that

thus upon substitution into A(i)

V

_2 (y.

2(1 h))

j

v

-

C

1v

'' a p T 2(l_h2)

_/

tCT - CT V/

_l-h2

Derivation of D/D. Jc

The jet contraction ratio D/D. _{can be}

Jc

derived as follows.

From mass flow continuity considerations applied to the jet flow

A(i)

A(2)

A(3)a

(23)

2 11D. 7TD 2 Jc4

V (l-h )

= V.

p4

joo 4 whence V. V _J.:2 _Ä a p l-h

After substitutions from A(2) and A(3) we have

finally 2 = - 1) _{A (4)} D

D.

Joo Derivation of

fJ=

-

l-h2 CCT + l-h2

An approximate analysis led to the derivation of equation (ll) for ry.e/V . The more accurate

equation (12) may be derived as follows.

The stagnation pressure rise through the propeller at radius r follows from the Euler pump

equation,

Po = P02 - Pol = p21vnrve2

Thus from Bernoulli's equation the static

22.

A(4)

a

(24)

pressure rise is p = p2 - p1

12

= po - DV

A2

A

LPQ

= - 2 2 8pr r

The

propeller thrust then follows directly from radial integration

D T = T = 2 2 7rrdr

--i-= 27r ¡ p0rdr _47r ED 2 2 po - dr r

If we now consider a free vortex _{design in}

which _{= constant, the integrals may be evaulated} resulting in T =

9(l-h2)Lp

mi/h

h2

° 4pwn2 2 A (6) A(7)

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The solution of this quadratic in

Lp

may be expressed as a stagnation pressure coefficient C which is a function of C , C and j.

Po _T 2 2 o

í1Tì1l

h C = = po 1 2 f

pv

₂ _a

fi

1 2 Jvjot2

Vê2t2

(DÌ

2 = pV 2

aìkV

'k. V i

<kiD?

-a a

2J2CT

_lnl/h 2 (l-h2)2 7T If

¿p

is expanded as a series in r to

represent more complex vortex flows, solutions of greater generality may be obtained

involving

CT C and J.

An expression relating _V. _{and hence} _y.

Jo,

to C_po can now be derived. At the edge of the jet

a long way downstream of the ducted propeller the jump in stagnation pressure from within to without, is, from