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AdvancedResearch, Inc. 100 HUOSON CiRCLE S ITHACA. N Cw YORK

THE GENERALIZED ACTUATOR DISK

by

G. R. Hough & D. E. Ordway

TAR-TR 6401 January 1964

This is a paper to be presented at the SecondSoutheastern Conference on Theoretical andApplied Mechanics to be held in Atlanta,Georgia on March 5-6, 1964. This researchwac supported by Air Programs, Office of NavalResearch, under Contract Nonr-2859(00).

Therm Advanced Research, Inc.Ithaca, N. Y.

ABSTRACT

Based upon the classical vortex system representation,the induced velocities of a finite-bladed propeller with

arbitrary circulation distribution are derived and Fourier

analyzed. The zeroth harmonic, or steady component, of the

induced velocities is considered in detail and tormulas involv-

ing only an integration over the blade radius are found. The

mathematical equivalence between these results and the conven-

tional actuator disk representation of the propeller is demon-

strated. Sample calculations of both the axial and radial

velocity components for an arbitrary representative circulation

distribution closely approximating the Goldstein optimum are

presented. For the special case of a uniformly loaded pro-

peller, expressions for the induced velocity components are

given in simple closed form and velocity profiles ar,. compAred

with the representative results.

1.i

2

INTRODUCTION

The problem of developing suitably simple, yet accurate,

expressions for the induced velocity components of a free

propeller has long occupied the attention of both the aErody-

namicist and the nFval architect. Detailed knr.-ledge of the

velocity field is extremely important for many applicationssuch as the design of ducted propellers for VTOL aircraft, the

evaluation of the interaction between a propeller and a wing

i-mersed in its slipstream, the prediction of the loading on

nearby appendages, and the determination of the free-space

sound field.

Early approaches to the development of airscrew theory

followed two lines: one, the momentum theory of R. E. Froude [1]and W. J. M. Rankine [2] and the related actuator disk con-

cept; and the other, the blade-element theory of W. Froude (3]and S. Drzewiecki (4]. While able to preaict propeller

performance with some reasonable accuracy, the utility of each

theory is limited by basic deficiencies. On the one hand,

momentum theory yields neither a relationship between the

propeller geometry and the thrust and torque nor any detailed

knowledge of the induced velocity field. On the other hand,

blade-element theory suffers from the uncertainties involved

in assuming the aerodynamic characteristics of the blade

sections. It also gives no information regarding the induced

velocity field.

Prediction of the induced velocity field and in partic-ular the local inflow at the propeller plane was finally

achieved with the now-classical vortex theory of the propell-er. Analyzed in detail by A. Betz and L. Prandtl (5] andS. Goldstein [6] and refined by others [7] & r8 1 , it hasenabled, in conjunction with blade-element theory, theaccurate determination of propeller performance for theforward flight regime. Still, most calculations of induced

velocities using vortex theory are either so complex as to

require extensive time on high-speed electronic computers,

can not be readily carried out for arbitrary distributions of

blade circulation, or art based on the oversimplifying assump-

tion that the number of blades is very large and the flow

periodicity may ba naglected.

The purpose of this paper is to remove some of theundesirable features of these theories, the results being

essentially an extension of part of an earlier analysis for

the ducted propeller by D. E. Ordway, M. M. Sluyter and

B. U. 0. Sonnerup [9]. In outline, we consider a lightly-loaded propeller of arbitrary blade number and circulation

distribution operating at zero incidence in a %mifornt free

stream. The propeller is represented by the conventional

vortex system and the induced velocities at any field point

are determined from the Biot-Savart law. These velocities

are then Fourier analyzed. The zeroth haimonic, or steady

component, is considered in detail and identified with the

actuator disk, or infinite-blade-number solutiou. Simplified

fo-mulas involving only an integration over the blade radius

are derived and sample calculations for an arbitrary represen-tative circulation distribution are carried out, For the

special case of uniform propeller loading, the integrations

are performed analytically yielding simple closed-form express-

ions for the velocity field. Several profiles are calculated

and compared with those for the representative case.

BASIC FORMULATION

Consider a propeller operating at zero incidence in a

uniform, inviscid, incompressible stream of speed U . We

assume the propeller to be lightly loaded with N equally

spaced blades of radius R which are rotating about their

axis at a constant angular velocity n . In addition, we

assume that both the blade thickness-to-chord and chord-to-

radius ratios are negligibly small and, for simplicity, dis-regard the hub.

A cylindrical propeller-fixed coordinate system (x,r,e)is chosen such that the propeller disk is normal to the

x-axis and is located at x = 0 , see Fig. 1. In accordance

with our assumptions, we can ncw represent the propeller by

r\ / /g01

dr

FIGURE 1

COORDINATE SYSTEM AND SCHEMATIC VORTEX REPRESENTATION

q0

the classical model of a bound radial vortex line of strength

P(rv) for each blade accompanied by a helical sheet of vor-

tices of strength - dr/drv trailing from each line, where

rv is the radial distance to any element of the vortex line.

Also consistent with our assumptions, the helical path of the

trailing vortex system is determined solely by the incoming

free stream with translation U and rotation .

In order to express the induced velocity field it is

convenient to introduce three elementary vector velcoity

fields [9]. These fields or influence functions I , I1rand 4 are due to vortices of unit strength and unit length

which lie in the axial, radial and circumferential directions

respectively. From the Biot-Savart law we have,

1, "" X - D / 4rIpI3 ()

where i is the unit vector in the poritive direction of our

desired element and D is the vector from the element at

(xv,rv,0v) to the field point (xr,e) . with ix , ir and

o corresponding to the unit vectors in the x, r and e

directions at the field point, the influence functiona reduce

to

-rvSin(0-0v)ir + [r-rvcos(e-ev) l4LI4,rD 3

7r ,in(O-O)K - (x-v)sin( e.)i, - :x-Yv)co,.-.v)q

[rv-r cos(e8o v ) ]i4 + CX-xv)COsCeov)i - (x-x.')sin(e4v)q

(2)where

D [(X-xv)2 + r2 + rv2 - 2rrvcos(e-Ov)1 (3)

With these results, the total velocity field for both the

bound and trailing vorticeg may be simply expressed. For * orthe velocity induced by the N bound blade vortex lines, we

integrate over each blade and sum the result over all blades, or

N R

Sr Z f r (rv) IO(, rv, WI/N) drv (41=1 0

Here, the argument of I is the location of the vortex element,

the field point remaining arbitrary for the present, and theindex I denotes the ord.tnal number of the blade.

For *, or the velocity induced by the trailing vortices,we first combine an axial and a tangential element appropriately

to form an arbitrary helical element lying along the free stream.

Then we integrate over the vortex trailing from each radial

location, carrying out a second integration over the radius and

8summing over the propeller blades, we arrive at

N R

Xr= f -r (rv) f u , av e) dr drv (5)Z=1 0 0

where we have set r" a dr/dr v and the arguments of Ix and

4 are

I = IX(UT,rv,2 /N + nT)

= 4(UT,rv,27rt/N + nT) (6)

The dummy variable r represents the time for a trailingelement convected by the free stream to travel from the blade

to its position downstream and the corresponding integral over

T is equal to the velocity induced by a semi-infinite helical

vortex of unit strength, radius rv and pitch U/ir v .Superposition of the velocity fields given by Eqs. 4 anu 5

determine then, within the limits of our formulation, the total

induced velocity at any field point. Within the propeller

slipstream these integrals are to be interpreted in the Cauchy

Principal Value sense.

FOURIER ANALYSIS OF VELOCI'ii FIELD

In the coordinate system translating but not rotating with

the propeller, the flow is periodic with time. That is, the

velocity at any field point will consist of a steady or time-

dependent component plus a superimposed fluctuating componentwhich has a period of 2w/N2 . Accordingly, we can express

and in the complex Fourier series

U c ~(xr) eimNGT

=U c (5x,r) ejMNO (7)m----c

in our propeller-fixed coordinates in which the angular varia-tion is equivalent to the time dependency. The complex Fourier

velocity coefficients c and c are

7rj, e-JmNO dO

inn

7r1 (' -mN-fli 2w do (8)

from orthogonality, U being introduced for dimensional pur-

poses.

At this point it is convenient to decompose _, ,

m and c into their axial, radial and tangential compo-

nents respectively, or

10

qr = Ur& + vri + wr

6 + Vrpr + Wr,+ (9)

and the components of the complex Fourier velocity coeffi-

cients associated with u will be denoted by cu

umTo find, say, the axial component c m , we replace I

by I:r.ix in Eq. 4 using the second of Eqs. 2 . From thefirst of Eqs. 9 the resulting expression represents ul . If,in turn, we replace S by this ur in the first of Eqs. 8and reverse the order of the r and e integrations, we cancarry out the e integration in terms of Legendre functions(10]. Noting that each blade contributes equally to any

harmonic for N identical blades, we obtain

cu -JN R r(r V) [Q+ 11 2 (aI) _ Q_ 3 2(wi)j drv"in 8ir2u3 0

(10)

where QmN+1/2 and mN-3/2 are the Legendre functtons of

the second kind and half-integer order with argument c01,

x2+(r-r ()

2I - 1 + 2rr (i

v

and the prime (') denotes differentiation with respect tothis indicated argument.

The remaining coefficients C"'.. are obtained inrm

similar fashion and are given by

R

= ~ f r~ 1/1 N-/2 drv0 rr) r.

[S+1 2 ( 1 N r x. V) dr~

cr= 8f~r/ f r'(~ T/ t r;N+1/2 + 32w) v0 v 0

vu -Nn r '(r~) f v ~ T87 f f3 1 [r;12mzq+1rr /22 )

o _v 0

2rx-T Qo;N 1 / 2 (cOO]e-INII ciT drv

rm= Nflr f Tr)f(. fx u')QN+ 1,2 (tD0 47-

+ ( t X+iUT)Q 3 2 a 2 _ 2u oN- 1 /2 (0w2 )eaiflNnW ciT dr~

(12)

12

where

(x-U ()2+r-r13)

For r > R , the coefficients for the radial components may

be identified with previous results [9].In general, further analytical simplification of these

coefficients to a simple, closed form involving no integra-

tions is not possible. Still they represent an essential

simplification. That is, for any harmonic m only 0'+I/2

%N-1/2 and N-3/2 are required, or equivalently%N+1/2 ' QmN-I/2 ' QmN-3/2 and OmN-5/2 from recurrenceformulas (10]. This is in contrast to results in terms ofelliptic integrals [11] in which the number of necessary func-tions to be computed rises in geometric progression with m

The Legendre function of second kind and half order hasbeen examined in detail [10] in connction with recent work onthe ducted propeller and extensive tables for orlers of -1/2through 21/2 have been computed (121. For a fixed value ofn , 0n1 /2 (o) has a simple logarithmic singularity at ) = 1and decays monotonically to zero as 0) increases to infinity.

In addition, 0n-1/2 (w) decays monotonically to zero withincreasing n for a given ) .

Altogether, it appears that the Legendre function of

second kind and half-integer order is the "natural" function

13

for the Ptudy of the velocity field of a propeller.

STEADY VELOCITY FIELD

If we examine that part of the velocity field correspond-

ing to m = 0 , considerable simplification in the complex

Fourier velocity coefficients is pe'ssible. This part corre-

sponds to the steady, or time-independent part of the veloc4ty

field. Using Ur ... to designate these steady velocity

components, we obtain from Eq.. 7, 9, 10 and 12 together with

the identity /'' 3 /e(=) , that

r=0

yr 0

RNr/ I r (rv)wI = NI'Q0(o0) drv

0 -47r2w 2 r3/ fV"()) dr dr2

0 v-N f f"(rv) f tr, (0)2) - drr" '4,2r3/2o 0 rv o0 %=) vW(2]d r

N R r'(r V) 00v' 47r32 Jo 157 f (X-UT) Qh(c2 ) dT drv

0 v 0-UN r'(r v )

0Wr"/ = 232 rv(2) -(=) dT drv

(14)

As a result, we find that the bound blade vortices contributeto the tangential velocity only.

The integrations over T occurriY.; in the expressions forUp, , o ad P, can be carried out analytically. Since it

is the simplest, we first consider r' and note the

relationship

Q rrv )(x-UT) (2) = - ' (15)

to obtain

R-= r2U f r'(r ) v T ,r (l)l dry (16)

0

Since x appears only as a squared quantity in the argument

M, given by Eq. 11, we see immediately that v r is symmetricin x . From appropriate expansions of Qh for small andlarge arguments [12], it can be shown that -r is logarith.mically infinite over the propeller disk and vanishes on the

propeller axis. Because vr = 0 , the swaie conclusions are

true for the total steady induced radial velocity=' (r +qr,)

The 5 rI and w r components are reduced by rewriting

Q'(w) and Q' (w) in their integral form [10], or

S = /2 sna (17)Qn_( )= - [2(m-l)+4sin a]3/2d 17

15

The order of the a and T integrations is then reversed,

after which the integrations over T and then over a may be

carried out. The final results are

RNn f ,'(rv) Kl(x,r;rv) drv

R

f N '(rv) (x,rlrv) drv (18)47r0

where

7 + x Q_ (I ) + M ) IF rC

(19)and K2 is identical to K with the radial inequality

signs reversed. Ao(Olfkl) is the Heuman Lambda Function [13]

with argument 01 and modulus k,

4rr1- sin- kI = t rv)_ (20)

x2+(r-r_) 2 x , (rTrv)

16

Eqs. 18 can be still further simplified if we integrateby parts with r'drv as the differential and use the relation-ship which we have found,

Ao 1 x (r Q,(Ml) + , + IF r

17

For wr" we get

- R r(r )IF r>R ,-oo

18

T he results of Eqs. 22 and 24 as regards to the inflow atthe propeller plane are especially noteworthy. In particular,

5A = nr/U and so from the velocity diagram at a blade section,the resultant induced velocity vector is perpendicular to theresultant free stream vector (uA.-nrh) . This is the sameas Moriya (7] found for the total inflow, or steady plushigher harmonics. Therefore, we conclude it must be true forthese harmonics as well.

In summary, the total axial component of the steady in-

duced velocity field is determined by Eqs. 22 and r = 0 , the

radial component by Eq. 16 and = 0 , and the tangentialcomponent by Eqs. 24. We see that these relations if expressed

in non-dimensional form would be independent of the bladenumber for a fixed advance ratio J E U/R and disk loading

dCT 2N r r (25)= - - (5

d(r/R) WYJ R UR

where CT is the propeller thrust coefficient. Cunsequently,on a physical basis they must be equal to the respective com-

limponents of the ,,m (o+a,) Of Eqs. 7 subject to the sameconditions. This limit corresponds to the velocity fieldassociated with what we now define as the GENERALIZEE ACTUATOR

DISK, i. e., the precise mathematical definition of the actua-

tor disk as opposed to a model with certain characteristics

assumed a priori.

19

REPRESENTATIVE BLADE CIRCULATION DISTRIBUTIONVELOCITY PROFILES

To illustrate the theory, we have calculated several i

and 7 profiles for the representative blade circulation dis-

tribution given by

r =(26)UK R v

This particular r was contrived because it simplifies the

calculations and approximates the familiar Goldstein optimum

distribution [6] quite well, particularly the proper squareroot behavior at the tip, see Fig. 2. The constant A is

proportional to the propeller thrust coefficient CT " Bysubstituting Eq. 26 into Eq. 25 and integrating, we have

A = 15 7ri(27)32N I CT

2 2where CT - T/fRR hpU . T is the pr,peller thrust and p is

the fluid density.The axial and radial velocities were calculated on a

CDC 1604 digital computer from Eqs. 22 and Eq. 16. The results

in terms of CT from Eq. 27 are tabulated in Tables 1 and 2

and sketched in Figs. 3 and 4 respectively. Several featuresare prominent in addition to those observed previously. From

Fig. 3 the steady induced axial velocity off the propeller disk

20

1.0 V \o

0.5 /

0

/ N=4, J= 1/50 N=3 , Jz 1/2

0

00 0.5 1.0

r/R

FIGURE 2

COMPARISON OF REPRESENTATIVE CIRCULATION DISTRIBUTIONWITH GOLDSTEIN OPTIMUM

'kD 0 co %D0 c( tn co %D0 -t un H i 0 w tq n N0 % 0 n MOOC t~ 0'0 k J CJN H #A 4 0 0* . . * . . . * 0 . . * *90 0 00 0 00 00 CI 0I 0)99

0 - N~4 N 00 tr4 VH4 H'C. m 00Dn* 0 w JY wA --tLA 0000m-- rmviH00

0t m. fV 4 a% (3' a% ON N 7 0 t- '.0 M m7' Cn Htn M N o c Ln0Ho r Ct n _:rN0C0

* 4 H0H 0 Ln .ncn 400 0 0 0C0

N 0 co i LA -NLAC~- ~ -1 c0NJ-- N -I 0 DoH qc)( OCCHOft.- co 0 o 0(N0 0 0 0H 0 iCjNMc o

-t- 0 n a r4 4 M 1 r0 0 0H O J C U0 (0 0O JC00 000 H

cli10 c\t- -r'li0 NU) OJH00 0 N 0 l 0Ni I n \0MCi4u fr Dc - ~

0A 0 HHr4m w (A0 w n 0 t-%D G\m

0 C\J CIJ C% oC 04 0 U\0 \Ot\Mz

00000!9 900 00 C) 0 00 00 0

kok o% ' l4 c \% H LAC'Jk

0 0 0 0 0 0 0o o o oo o,8 8 ; C

-JC tU\kot 0a '

22

4 m1 --.t n %0 t.- w O0 m' m' 0 0 0 CO -0 0 0 0 0 0 0 0 0 0 0,4 -I 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0ci . . . . . .

wU co 0 Hq (i -- Un ko t- t-.. t- Lt 0 -tooo 0o o0 o

0A o 0 0 0 ,- ,-i ,-4 ,4 ,4 H- "- ,-4 r- ,-i H-0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0r4 . . . ... . . . . . . . . .+1 00000000 0000

m M" t,- 0 - m r4 N- NJ NJ CN N" Nl aLo0 0 ,-4 ,-q N. N. NJ N. 0"n 0n 0n ( ,n N N A0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

+ 0 00000 0 00000o0

Ifn ,-4 t- 0.I t:- OJ t- 0 CI "O M" Csj WO U% M Lfo - '4 r4 1 01 Oj m "Y -* -.t -" -. " :* m," 0 -1I 0a,.000000000000900000o

. ooooo oooooooooooo

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0I I I I I I I I I I I I I I I 14

E-4.0 o 0 0 03 00 0 0l 0 00 0--" 0 , 0 ", -0 "0 0"

w. m 0 m %o 0 No m mo m o-o od% o

'.1 CO u'0 0 0 0 O 2- 0 .0 0 10 . 0t 0 00 1+1 OOC;CCCCCC 0 ,

z 0 co n* co 0 n cn 0 0 -- N k'.%0 -t -t .4 tf

0 0 0 (m 0 Nt00 , C %.--- -I 00

0 9 9 9 C0

+1 000 0 0 0 00 0 C 0 0 00

1 0 0 0 H 0 0 0 0 00... ...-............................OOOOn0OOt-N oOHHmHH0J(Vil 0

23

4- cc

C 4-U)0

O P4

0. 0C) 0

_ _ _ _ _ __i

L6

241

a, 0('J

- Ir~H

toLoI -cr4-

25

vanishes in the propeller plane, leaving the free stream unper-

turbed except for the radial velocity. Inside the slipstream

quickly approaches within a blade radius twice its value at

the propeller plane and decays just as fast upstream. FromFig. 4 we see a rapid change in the radial velocity in theimmediate vicinity of the propeller plane and a smoother vari-ation farther away. Within a certain radius from the propeller

axis and near the propeller plane, - reverses and the radialflow is outward. This reversal is due to the influence of the

trailing vortices inboard of the maximum value of r whichare of opposite sign to those outboard. At axial stations nearthe propeller plane the velocity is still appreciable for large

radial distances, whereas the profile quickly decays with large

axial distance away from the propeller in the vicinity of thepropeller axis.

CONSTANT BLADE CIRCULATION DISTRIBUTIONVELOCITY PROFILES

For the special case in which the circulation distributionover the propeller blade is constant, the results for tho

steady induced velocities can be put in simple, closed form

because fundamentally they are the "building Ltocks" for the

case of arbitrary circulation.

With r(r) = rc , it is easiest to reduce u by returningto the intermediate form of ur or the first of Eqs. 18.

26

This yields only a contribution from the tip which, together

with i = 0 , gives

NX ( FIF r

27

As opposed to the representative case, now is finite over

the propeller disk except at the tip. Furthermore, it does not

reverse sign as the only trailing vortices of opposite sign to

the tip vortices lie along the propeller axis and do not induce

any radial flow.

These velocities have been calculated and are tabulated in

Tables 3 and 4. They are compared back with the results forthe representative circulation distribution in Figs. 3 and 4.

From the figures we see that for radial stations less than the

propeller radius the results are not in close agreement, the

disparity being most pronounced within the propeller slipstream.

However, there is no significant difference between the two

results for r/R > 1.5 While the condition of constant circulation along the

entire blade is physically impossible [14], we can concludethat the determination of the propeller Induced velocities out-

side the slipstream may be satisfactorily approximated by

assuming such a circulation distribution. On the other hand,prediction of the flow field within the slipstream as well as

in the immediate vicinity ahead of the propeller plane will be

in serious error if a constant blade circulation is assumed.

IDENTIFICATION WITCH ACTUATOR DISK SOLUTION

As we said earlier, the assumption is often made simplya priori that the blade number N is infinite and so, the

28

4 ** l ni 'ko t- wx m~ mi 0 mi o H LtI N~41 **7***4***( 0 00 l l0 0

00 w0000000 0 ko r- Ln kC?%m tF

ci 0 m i Lc ' m U M:3: L 0 mO~ Lri 0Y 0

4 4 44 4 cli 0 00 0

* % H coY n H0 0 00

0 * * * . . . * . . * * * . . *

0 N000 00 0 N0 0 N0 0 0 0j C? 0? E

00

_0 000000( 300- k k 40 00 o4C\Mm m 8 l

* l -1 1 l 1 %' -! 9 0 004i0 00 0

0 N 4r

0

0 00 00 00 00 00 00 00 00

tCU0 0 0 0H H #Y4. r4 r H 4 r l00 0 000 00 00 00 0 0000 0

+1 9,,,,99999,,,,o999

0 0' 0~ I-0 0 0 '0 0' 0 H 0 0 ' 0 0 0

co 0 0HH04040404j mcfl (n4t r 4T ;lm NP-10 0 00 0 0000C)00 00 000 0

00

* 000%D00 l00 n0 n o0\000D000\ t e0 4 N m *n t n n *n *n *n *n -* N r4

0 E-0 C ' H C 0. 0. C..

01 0 0H0401 01 01A40JH

0 00 00 00000 0O 00 00+1 81 q 81 81 o 9,99,99,981c

04 0 4Cl 0 ~ 0- 0CT00 \ H 4 0 0 0f HO

0q 4 N~ (n n CO- 0OO ( C 40 0 0 00 0OOHH Y r rA on-4HH O O

0+1 9999999999999999

m p0C~C m O H (Y a o pt- 0pnV0 00 00 0 I 4 HHOOOO 00

0!C U ot oG qc -o o o o o 6 8 6 9, 19, ; 49,z 0; ,

13C,

propeller is replaced by an "imaginary" disk across which the

axial velocity is continuous while the fluid pressure is

suddenly increased in passinq from one side to the other. Inthe absence of swirl this permits the use of appropriate dis-tributions of ring vortices or sources to represent the

flow (16]. These representations can be related to our solu-tion for the steady induced flow field of the finite-bladed

propeller.

First consider the ring vortex representation for the

axial component of the induced velocity. The strecim functionT for a ring vortex of unit strength is

CD 1 + (X-Xv)2+(r-rv)2 (31)4 = 2rrv

where xv is the axial location of the ring and rv is the

ring radius. The corresponding induced axial velocity becomes

after simplification,

1 r 1 (rQ ( 4 ) - rvQ (c)D)] (32)v

With T replaced by xV/U , we see that 5 from Eqs. 14 is

equal to the slipstream integration of Eq. 32 weighted by the

31

vortex strength -NS1dP(r v)dx v/27rU . But this agrees exactlywith the usual exprcssion for the induced axial velocity

associated with the actuator disk in terms of ring vortices:

Take for simplicity the case of uniform loading which requires

only a single semi..infinite solenoid of such vortices of radius

R . From momentum theory, the ratio of the total velocity in

the ultimate slipstream to U is ./l+CT , or linearized forlight loading (I+ CT) ; cf. Table 3, u/UCT = 0.25 at x=0 ,

r/R

32

For the rcpresentation of the actuator disk by the distri-

bution of ring sources on the propeller disk, we can proceed

formally to prove the equivalence as for the vortex rings. In-

asmuch as the equivalence between the ring sources and vortex

rings has already been established (16] & (17], this is notnecessary. If desired, though, Eqs. 22 are best for the

axial velocity, For the radial velocity, integrate Eq. 16 by

parts. The respective velocities for a ring source follow from

the potential 0 per unit source strength,

1 1 (l)(33)v

where x = 0 is the axial location of the ring and rv , theradius. We find that outside the propeller slipstream the

velocities are equal and the required strength of the ring

source is -Nflrvr(rv)dr/U , i. e., a ring sink. Inside theslipstream the radial velocities are the same but the axial

velocities di.ffer by the constant Nnfl(r)/2tU , cf. Eq. 22,since 6 for the ring source is antisymmotric in x every-

whore.

CONCLUSIONS

From our study of the induced velocity field of a finite-

bladed propeller with arbitrary circulation distribution, we

have concluded the following:

33

Relatively simple forms for the Fourier coefficients ofthe axial, radial and tangential velocities can be obtainedinvolving only Legondre functions of the second kind and halfinteger order. These appear to be the "natural" functions foruse in propeller theory.

A comparison of the steady velocity profiles for both arepresentative and a constant circulation distribution revealsthat the two results in general agree closely outside the pro-peller slipstream. However, calculation of the field immedi-ately ahead of the propeller as well as inside the slipstreamunder the assumption of uniform loading may lead to seriouserrors.

For the special case of constant circulation, the steadyinduced velocity components can be expressed in closed forminvolving no integration.

The axial and radial induced velocities for the conven-tional actuator disk are established as exactly equal to theirsteady counterparts for a finite-bladed propeller, providedthe advance ratio and the disk loading arc the same.

REFERENCES

1. R. E. Froude, "On the Part Played in Propulsion byDifferences of Fluid Pressure", Trans. Inst. Nay. Arch.,Vol. 30, 1889, p. 390.

2. W. J. M. Rankine, "On the Mechanical Principles of theAction of Propellers", Trans. Inst. Nay. Arch., Vol. 6,1865, p. 13.

3. W. Froude, "On the Elementary Relation between Pitch Slip,and Propulsive Efficiency", Trans. Inst. Nay. Arch.,Vol. 19, 1878, p. 47.

4. S. Drzewiecki, "Th6orie Gen6rale de l'H61ice", Paris, 1920.

5. A. Betz, "Schraubenpropeller mit geringstem Enerciever-lust", Appendix by L. Prandtl, G~ttinger Nachr., 1919,P. 193.

6. S. Goldstein, "On the Vortex Theory of Screw Propellers",Proceedings of the Royal Society of London, A, Vol. 123,1929, p. 4140.

-;4

7. T. Moriya, "Selected Scientific and Technical Papers",Moriya Mcmorial Coarmittee, University of Tokyo, Tokyo,August 1959.

8. M. Iwasaki, "Diagrams for Use in Calculation of InducedVelocity by Propeller", Reports of Research Institute forApplied Mechanics, Kyushu University, Fukuoka City,Vol. VI, No. 23, 1958.

9. D. E. Ordway, M. M. Sluyter and B. U. 0. Sonnerup, "Three-Dimensional Theory of Ducted Propellers", T1ERM, Incorpo-rated, TAR-TR 602, August 1960.

10. B. U. 0. Sonnerup, "Expression as a Legendre Function of anElliptic Integral Occurring in Wing Theory", THERM, Incor-porated, TAR-TN 59-1, November 1960.

11. J. P. Breslin and S. Tsakonas, "Marine Propeller PressureField including Effects of Loading and Thickness", SNAMETransactions, Vol. 67, 1959, P. 396.

12. M. M. Sluyter, "A Computational Program and ExtendedTabulation of Legendre Functions of Second Kind and HalfOrder", THERM, Incorporated, TAR-TR 601, October 1960.

13. P. F. Byrd and M. D. Friedman, "Handbook of EllipticIntegrals for Engineers and Physicista3", Springer-Verlag,Berlin, 1954.

14. H. Glauert, "Airplane Propellers", Division L of "Aerody-nanic Theory", Edited by W. F. Durand, Dover Publications,New York, 1963,

15. R. H. Miller, "Rotor Blade Harmonic Air Loading", IAS PaperNo. 62-82, January 1962.

16. L. Meyerhoff and A. B. Finkelstein, "On Theories of theDuct Shape for a Ducted Propeller", Polytechnic Instituteof Brooklyn, PIBAL Report No. 484, August 1958.

17. D. KUchemann and J. Weber, "Aerodynamics of Propulsion",McGraw-Hill, New York, 1953.

35

PRINCIPAL NOMENCLATURE

T^ propeller thrust coefficient, T/iR2kU 2

, complex Fourier velocity coefficients form m * and Sr

J propeller advance ratio, U/SIR

m harmonic number

N number of propeller blades- %

Wn_Ia) Lcgendc'e tunction of second kind and half-integer order of argument to

vector velocity induced by bound blade vortices

Sr, vector velocity induced by blade trailingvortices

R propeller radius

U forward flight velocity

u,v,w axial, radial and tangential componentsrespectively of induced velocity

x,r,6 cylindrical propeller-fixed coordinate system

r(r) propeller blade circulation

A o(,k) Heuman's Lambda function of argumentand modulus k

36

propeller angular velocity

( )" differentiation of a function with respectto its indicated argument

(-) zeroth harmonic or steady part of inducedvelocity

.1 '

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