A transonic propeller of triangular plan form (an extension)

A TRANSONIC PROPELLER OF TRIANGULAR PLAN FORM

(AN EXTENSION)*

TECHNISCHE

HOGESCHOOL DElFT

WCHTVAART- EN

RUJMTEVAARTTECH~JIEK

BIBLIOTHEEK

Kluyverweg 1 -

DELFT

by

Herbert S. Ribner

Hay,

1978

UTIAS Technica1 Note No. 218

..

A TRANSONIC PROPELLER OF TRIANGULAR PLAN FORM

CAN

EXTENSION)

by

•

Foreword

There has been a resurgence of interest in high speed propellers after two decades of daminance by jet propulsion. Thus it has seemed worth-while to put into print a hitherto unpublished design study of a high speed propeller concept with several nevel features. The design, carried out at

NACA Langley in 1949, was intended as a basis for model tests in a wind tunnel.

Summary

An isosceles triangle twisted into a screw surface about its axis

was proposed in NACA TN 1303 as a propeller for transonic flight speeds. The mathematical analysis is extended herein to cover the aerodynamic and structural effects of a structurally feasible camber distribution. The basic camber is localized near the leading edge; the camber is determined in such a way that

the theoretical infinite pressure peak along the leading edge is eliminated. An

auxiliary camber or rake is derived for the alleviation of the deflection caused by the centrifugal forces.

The earlier formulas we re derived on the assumption that the triangle is narrow compared with the Mach cone from its vertex. A factor to correct approximately for the breadth of the triangle is introduced herein. The selec-tion of the advance-diameter coefficient and pitch-diameter ratio for maximum

efficiency is 'treated. Finally, the design of a 3.5-foot diameter model

propeller for wind-tunnel tests at a stream Mach nuniber of 0.9 is carried through in a numeri cal example.

1-2.

3·

4.

5.

6.

CONTENTS Foreword Su:mma.ry INffiODUmION

CALCULATION OF THE LEADING-EDGE CAMBER CALCULATION OF THE RAKE

CORRECTION FOR ASPECT RATIO .AND MACH NUMBER SELEmION OF J .AND p/D

NUMERICAL EX:AMPLE

APPENDIX A - EVALUATION OF I(BC) FOR EXAMPLE REFERENCES FIGURES ~ i i i i i 1 1

5

8

9

11

..

1. INTRODUCTION

In Ref. 1 an isosceles triangle twisted into a screw surface about its axis was proposed as a propeller for transonic flight speeds. The purpose was to attain the drag reduction associated with large sweep-back. in a structurally practicable configuration . The desirabili ty for introducing camber was touched upon, but the mathematical analysis was limited to an exact helicoid surface without camber. In the present paper an investigation is made of the determination and behaviour of suitable

distributions of camber. \

An exact helicoid surface has the property that sections cut by planes perpendicular to the axis (x = constant, see Fig. 1) are straight

and radial. Thus the centrifugal forces on a helicoid in screw motion give rise to no primary bending moments. The, si tuation is alter,ed by camber. The resulting bending moments for an aerodynamically desirable design may easily be prohibitive. After laborious calculations, this was found to be the case for a shape cambered so that the lift would be distri-buted uniformly from leading edge to trailing edge. The spanwise sections exhibited marked curvature and, inclination fram a radial direction.

A compromise would be to limit the camber to a narrow region near the leading edge. The surface is then essentially a helicoid again, and sections cut by planes (x

=

constant) are straight radial lineswith a slight hook at the end (see Fig. 2). For such a surface the centrifugal stresses need not be excessive.

From the aerodynamic aspect the-leading .. edge camber is eapal:>le of being adjusted to bring the stagnation point of each section to the 'leading edge. This is the condition for smooth flow entry: it is an

optimum situation in which the theoretical infinite pressure peak is elim-inated, but the thrust and efficiency are not thereby impaired. There is a forward component of the lift force on the drooped leading edge that takes the place of the leading-edge' suction. There is good reason, fram consideration of boundary-layer behaviour, to believe that most of the predicted forward component will be obtained in practi ce. There' is good reason, on the other hand, to doubt that the leading-edge suction theore-tically predicted for an uncambered design will be obtained in practice.

2 • CAICUL.A!rION OF THE LEADING-EDGE CAMBER

The calculation employs the narrow triangle theory of Ref. las extended to Ref. 2 by means of Fourier series. (A correction is introduced in the numerical example later to account for the finite vertex angle.) The analysis makes use of Fig. 2. The figure has been drawn as though there were no twist in the propeller. The effect of the twist will be brought in later.

Let the arbitrary sec·tion AA be .defined by

z

=

0 -Ta.$ y ~ Ta

k(y + 2

=

Ta) a

=s

y ~ -Ta (1)

=

-k(y - Ta) 2 _a~y~ Ta

where

a

=

Cx

C

=

constant T = constant

<

1

The component of velocity, w, normal to the surface of the propeller may be obtained by an extension ofthe reasoning leading to Eq. 10 of Ref. 1. The result is

(2)

The evaluation of dz/dx from Eq. 1 gives

- a

=s

y

=s

-Ta

This distribution of normal velocity is shown in the bottom part of Fig. 2. A surface distribution of vortieity that will induee (approximately) the normal velocity (3) is to be determined. The calculation will be eoneerned with thesurfaee· potential of this distribution. The potential will be expressed

(Ref. 2, Eq.

16)

by the trigonometrie series

A..- 00

A

cp = aV { - Aosin \l

+

.

~

sin 2l)

+

I

2n

l

S~~i~~l)!)_ sini~i~)

_

!}

J}

(4)

2

where

'1

= cos -l(y/a) and the constants

Ac,

Al, .•• An are to be determined from the boundary condition equation

(3).

Equations 17 of Ref. 2 give

'TT

IJ

w

Aa

= - ;

V

dl) o

_

_~J'TT

_w ~

=

"

V

cos nl) dl)

(5)

o

Figure 3(c) shows that w is antisymmetrie. Hence all the even-sul;>scripted coefficients are zero. Upon carryine through the integrations the odd-stibscripted coefficients are found to be

~

=---V

g

kTCa(2e + sin 2e) + 8k T2Ca sin e

7r 7r

= -

~

kTCa [ sin(n-l)e + sin(n+l)e ] + 8kT2Ca sin ne

An 7r n-l n+l 7rn

where

-1 e E cos T

( 6)

The surface potential (Eq.

4)

is now known.. The pressure coeffi-cient is proportional to the x-derivative of this potential:

=

4,c

cot 1'1 {- AO cos 1'1 o(B-Aa)

4 {-

sin T)+

dx

~

=

4cp

Iv

q x 00 A Al 21'1'+

I

cos(n-l)

~J}+

.

'

+ - cos ₂ 2n [cos(n+l) 1'1 -2

O(aA

_l) 00

Si~

2T) +

I

ex

₂ sin(n-l)T) ] } n-l o(aA n) 1

l

sin(n+l)T)

2

-dx:

n+l

Condi tions at the edges (T)

=

0 or 7r) are of primary concern. In particular it is desired to specify the leading edge camber such that the infinite pressure peak at the edges is made to vanish. The infini ty comes from the

cot 1'1; thus i ts factor in brackets must vanish to eliminate the infini ty . Inserting 1'1

=

0 or 7r in this factor, discarding

Aa,

whicll is zero, and equating to zero yields simply

~ = 0 (8)

By insertion of the value, of Al from Eq.

6,

Eq. 8 may be solved for the camber parameter k. The result simplifies by use of the identity T

=

cos e to

k

=

or k

=

where

nw~-P7n)

2VC COS €(2€ - sin 2€) 7T 2

~

-

PID)

CDJ cos . €(2€ - sin 2€) V J = -nD

(9)

Equation

9,

when used with Eq.

1,

determines the optimum camber for the edges of propeller cross sections like

AA,

Fig. 2. A practical case is worked out in the later numerical example.

Pressure Distribution

The pressure coefficient is given by Eq.

7.

Addi tional simplifica-tion is needed. By differentiasimplifica-tion of Eq.

6,

Also, An = 0 for even values of n, and the camber constant k has been chosen to make Al =

o.

The remaining coefficients An may be expressed in the form

where a == 2cos € sin € n n sin(n-l)€ n-l sin(n+l) € n+l

When these relations are introduced into Eq.

7

there results

00

~

= Bl {

co~

Tl

I

a_n [ cos(n+l)T) - cos(n-l) T)

J

2

sin(n+l)n _ sin(n-l)n

n+l n-l

J}

4

A curve or ~/q versus y

=

a cos ~ for the case T

=

cos €

=

0.925 is shown in

Fig. 3 along with the uncambered case (,.

=

1.00) for comparison. Note how the

leading-edge camber has modified the infinite pressure peak at the edge into a finite peak slightly inboard of the edge. Ca.ni:>ering more or the edge (reducing ,.) wiU lower and broaden the peak and move i tinboard.

60

rque

The torque can be obtained from the equation at the bottom of page

15

of Ref.

2:

(13)

For the optimum camber i t was shown that Al

=

O. Tbe constant A3 is gi ven

by Eq.

6

with n

=

3; also, at the trailing edge (T.E.) a

=

R. Thus

7T

4 { (

J)

2kCV,.

r

4 .

1 . ) , .

J}

Q

= -

'8'

pVR w 1 - p/D - 7T 2€ -

3

s~n 2€ +

b

s~n ~€ _

(14)

Insertion of the value of the camber constant k corresponding to the optimum

camber, Eq.

9,

gives

2_{€ -}

₃

4 .

_s~n 2 _€

₊

1

_b

_s~n .

4

_€

J

; 2€ - sin 2€ -1 €

=

cos T

(15)

where 7T

4 (

J)

%

= - 128 pVD w 1 - p/D

(16)

is the' torque for a (narrow) triangular propeller without camber.

Equation

15

shows how the torque falls off as the cambered region

1 - cos €

=

1 - T is increased. A curve of the variation of

Q/%

wi th 1 - ,.

is presented in Fig.

4.

3. CALCtJLArION OF THE RAKE

Examination of section AA, Fig. 2, shows that the cambered tips give rise to a counterclockwise bending moment under the influence of the centrifugal force field. Strip theory calculations indicate relatively large deflections, increasing as the uncambered fraction ,. is decreased. As a correcti ve measure an opposing mo:nient may be obtained by raking the section

upward as in Fig.

5.

Ir this curvature is identical from section to section

as x is varied, the curvature will cause no change in the distribution of

angle of attack, and hence in the aerodynamic loading. It should be possible

to select the rake so that the tips or section AA have zero deflection. Under this circumstance the deflections elsewhere along the blade will be at a

practical minimum. Tbe following analysis provides a determination of this

The right half of the section AA of Fig.

5

is shown in more detail,

with curvature exaggerated, in Fig.

6.

A110wance is made for a hub or spinner

whose radius is a fraction ( j of the tip radius. A uniform thickness

distri-bution ,is assumed, and the cross- sectional area of the strip is taken as A.

The desired shape bf the strip is specified by the subscript 1, and is defined by

2

=

cr 2

= cr Ta

<

r

<

1 (17)

where c and k are constants • The rake term cr2 approximates a parabola concave

upward with vertex at the origin~ The tip-camber term -k(r - -ra)2

approx-imates another parabola concave downward wi th vertex at r :;: Ta touching the

first parabola. The centrifugal force on an element between r' and r' + dr'. is

approximately

2

pAw r'dr'

This force acts along r' which has the polar angle e(r'). The moment of this force about an arbitrary point of the strip located by r and the polar angle e.(r) is given by

r[e(r) - e(r')]pA

w

2r'dr'

The (positive counterclockwise) bending moment about point r of the entire portion of the strip outboard of r is given by

a

M( r)

= -

pA ,W 2 r

J

[e (r') - e (r ) lr' dr' (18)

r

The equilibrium shape of the strip AA has been specified by Eq. 17. It is desired to calculate the "prebending" or initial shape the strip must

have in order that i t sha11 deform to the specified shape 'under load. The

distribution of bending moment for the specified shape will be needed. By the substitute of Eq. 17 in Eq. 18 this distribution is

a a M.L(r) , 2 [ c

J

(r' - r)r'dr' - k J (r' - Ta)2dr ,

J

=

- pA w r r Ta 2 [ c

(;3

ra 2 r

3 )

a

3

₍₁ -T)3

J

.

(19)

=

- pA w r

_{- T+"b}

_-

k -3

for ca

<

r

<

Ta. A different equation is obtained for the tip region

Ta

<

r :( 1.- In what fOllows, however, the single equation (19) will be

applied-as though it were valid over the entire region Ta

<

r

<

1.

Exper-ience with a closely similar problem has shown that if T ~-O.9-such a procedure

gives surprisingly good agreement with the exact calculation of the tip defle cti on , and is very much simpler. It is felt that the error from this source will be less than those resulting from the basic imperfections of

strip theory, as well as fram the use of an idealized, uniform thickness

distribution.

The curvature of the strip has been assumed small and therefore

r may be replaced by y as a good approximation • An appropriate form of the

beam equation is then

1\(y)

= (20)

EI

where Zl(y) is the prescribed equilibrium shape of the strip and zO(y) is the initial shape to be determined. Referring to Eq. 19 and making the

substitution Il

=

y/a yields

2 d

(z - zJ

3

1

=

B

[2k(1 -

T)

-

c(2 - 31l

+

1l2)] (21)

<%12

1

where Integrating twice (22) (23)

The constants of integration Cl and C2 may be determined from the boundary condi'tions:

for Il

=

cr

(24)

Inserting these relations in Eqs. 22 and 23 and sOlving yields

(25)

2

3~

(2~

34 1 6 )

The constant k' in these equations specif'ies the tip droop and the

constant c specifies the general upward curvature or rake: it is desired to

determine the relationship between the two constants that wiU make the tip'

deflection equal to zero. The tip deflection, which will be designated Zl -

Zo,

is obtained by setting J:L = 1 in Eq. 23. This gi ves, by use of Eq. 25,

(26)

The solution for zero tip deflection, Zl -

Zo,

gives for the required rake

constant,

(27)

Numerical values are worked out in an example in a later section.

4.

CORRECT ION FOR ASPECT RATIO ANI) MACH NUMBER

The pressure distribution on a triangular propeller wi thout camber bears a close relationship to th at on a flat triangle in rOlling mot i on • The

formulas of Ref • .1 apply, in both cases, to relatively narrow triangles such

that the ratio of the tangent of the semivertex angle to the tangent of the

Mach angle, tan €/tan j..l == BC, is small compared with unity. The pressure

distribution on a rOlling triangle for an arbitrary value of BC is deri ved

in Ref.

3.

For BC

<

1 the formula camprises the expression for a narrow

triangle (BC -+ 0) mÜltiplied by an additional factor that is a function of,

BC alone. This factor has been termed I(BC) in Ref.

4

and its variation wi th

BC is shown in Fig.

4

therein.

A simple extension of the development in Ref. 1 will .show that the

pressure distribution on a triangular propeller may be corrected to apply to an arbitrary value of BC

<

1 by multiplication by the same factor :t(BC).

This generalization is subj ect to two conditions, however: the twist of the

propeller must be smaU, and the angular velocity must be sufficiently small so that the resultant velocity at any point is not appreciably greater than

the stream velocity • The presemce of camber should cause li ttle error.

In a practical design, however, neither of these two conditions

will be met. The tip sections may exhibit of the order of

45

ó _{twist relative}

to the root sections, and the tip velocity may exceed the stream velocity by

4CYfo

or more. The formulas become crude approximations at best. The factor

I (BC) may still afford some improvement, however, if C

=

cot A is replaced by

an effective sweep parameter Cé and the average resultant Mach number, instead

of the stream Mach number, is used to calculate B

=

.

..;}#. -

1. If subsonic veloeities are experienced over portions of the blade, itappears reasonable '

to use

11 - M2

for B in these portions.

The effective sweep parameter Cr

=

eet lIê is obtained from the following considerations. For a flat triàngle or swept wing. the component of velocity normal to the edge Vn is related to the resultant velocity VR (in this case the same as the stream velocity V) by

V_n __

cos .A

V:R

The parameter C

=

cot A is thus given by

V 2 n

(28)

(29)

The edge velo city and resultant velocity for a triangular propeller bear different relations to the geometrie sweep angle from the foregoing. However, an effective sweep angle

1le

may be defined that satisfies the same relations. Thus

Ce

V n

=

Jv

2 _ V 2

R

n

~~nl

(30)

=

jCRJ -

Cn)2

Introduction of Eqs.

n8

and

D9

of Appendix

n,

Ref. 1, plus simplification, leads to

(31)

An average value of V IVR over the blade I;Ihould be used.

5 . SELECT ION OF J AND

pin

The selection of the advance diameter coefficient J

=

VinD

and the pitch-diameter ratio

pin

will probably be made on the basis of propulsive efficiency. The following analysis gives the optimum value of

pin

for a given value of J and provides a general guide to the selection of J.

It is useful, as in the examp1e of Ref. 1, to define a weighted b1ade area Swt such that the product 1/2 pV3

en

Swt wi11 give the amount of power consumed in overcoming profile drag. Then the propulsi ve efficiency is approximate1y given by

Qw

where Ti is the ideal thrust and Q is the torque. This may be written

~ p~

CD Swt lli

T.

J.

(32)

(33)

where 'li

=

TiV/QJJJ is the ideal efficiency (efficiency neg1ecting profile drag). Equations 16 and 15 of Ref .. 1 give expressions for 1li and Ti' respective1y. The expression for Ti should be multiplied by I(BC) to

correct for aspect ratio and Mach number and by

QjQ{)

to correct for 1eading-edge camber (see earlier sections). Thus '

. The required condition for maximum efficiency is that the derivative with' respect to J!(P!D) for fixed J should vanish:

o

1 16

~

CD Swt

~

= 0 =

2 - -3

: : 2 : :

-o (

P/D ) 7T

~

- P/D) D 2 I(BC)

(~o

)

(35)

The soiution for J/p/D gives the optimum value

( _p/D J ) _opt

₌

_{1 -} 4.J2J _{; / 2} (36) The maxinrum efficiency is obtained by substi tuting Eq. 36 in Eq. 34. The result is simp1y

Equations

37

and

36

predict thai the maximum efficiency increases toward uni ty as J is reduced toward zero. The vali di ty of the formulas becomes progressively impaired, however, as J is reduced below perhaps

5.

It is like1y that there is an optimum value for J and that it lies within the range ordinarily employed by propellers, sa:y 1 to

3.

.

6.

NUMERICAL EXAMPLE Genera]. Specifications

The design of a model. propeller for wind-tunnel tests at a stream Mach number of

0.9

will be carried through. The basic specifications are

n

=

3.5

ft

V

=

1000

ft/sec (38)

p =

0.00145

slugs/cu ft

The propeller is to be mounted on a cylindrical hub

13

inches in diameter which houses the electric drive motors and dynamometer. The hub and a projected plan view of a blade are shown in Fig.

7.

Because of the twist the leading edges of a triangular propeller lie on a cone rather than a triangle. For the propeller of the exlmIple the

semi-vertex angle of

39°

is chosen for this cone (sweep angle

51°).

A greatersweep would decrease the wave drag, but the resulting increase in area wou).d increase the friction drag. . Tl1e value selected is in the natur.e of a compromise • The tips of the twisted triangle are cut off at the required diameter of

3.5

feet. The depth of the triangle is such that the projected tip chord is 2.1 inches (see Fig.

7).

The finite tip chord is intended to provide strength to resist the local bending loads resulting fram the aero-dynamic and centrifugal forces.

A thickness distribution with the line of maximum thickness well forward (25% C, confer Ref. 5) is selected for low wave drag. The NACA

00035

~ection .. (streamwise) is adopted. The sec"tions are

3.5"/0

thick based on the projected chord.

Values of J and pin

It was concluded in an earlier seetion that the advance-diameter ratio J should be of the order of 1 to

3

for best efficiency. The value

J

=

TT is specified as corresponding to a practical minimum for a particular

installation. This corresponds to an angular velocity

w

=

572 rad/ sec. The optimum value for the pitch-diameter ratio is obtained fram Eq.

36

as

J

=

1 _ 1.021 J

n

wt ~

(

)

~

s CL

CD is the average profile drag coefficient; it is estimated to be 0.0086. The uncertainty is large and' the basis of the estimate is not worth e1abo-rating. Swt is a weighted b1ade area (exc1usive of the htib) computed as in Tab1e I, Ref. 1; the va1ue is 4.16 square feet. D is the diameter, 3.5 feet. I(BC) is a factor intended to correct roughly for aspect ratio and compress-ibili ty effects . This factor was discussed in an earlier section; i t is evaluated in the Appendix as 0.96. Upon insertion of these numerical values there is obtained (

~)

_{Pt D opt -}- 1 - 1.0217T

=

0.811 Leading-Edge Camber (0.0086)(4.16) 1 (3.5)2(0.96) 0.879 (40)

Values of all the quanti ties necessary for the determination of the camber parameter k are at hand, with the exception af E. The quant i ty 1 - cos E

=

1 - T is the fraction of the semispan devoted to 1eading-edge

camber. A curve of the pressure distribution for 7.5% semispan cambered

(1 - T = 0.075) was presented in Fig. 3. Cambering more of the edge (reducin~

T)

wi11 10wer and broaden the peak and move it inboard. This is aerodynaJllical1y desirab1e. The increased extent of the cambered region wi11, however, result in increased def1ections as a consequence of the centrifugal forces. By means of Eqs. 23 wi th 27 the def1ections have been found to be sufficient1y large so that the ca;mbered region must be held to a minimum. The pressure peak shown in Fig. 3 is about as large as may be to1erated. The corresponding value of 1 - T, 0.075, is thus a suitab1e campromise extent of camber.

The selection 1 - cos E

=

0.075 yie1ds E

=

0.390 radians. The sub-stitution for E and the other parameters in Eq. 9 then give for the camber

parameter 2 . _ 7T (1 - 0.811) . -1 k - tan 39° (42)(7T)(0.925)(2

x

0.390 _ sin 0.780)

~n

0.2466 in-1 (41) Rake

Equation 27 specifies the rake parameter C in terms of k, T, and (J. From the last section 1 - T

=

0.075 and k

=

0.2466 in-1 . The ratio (J

of the spinner diameter to the propeller diameter is 13/42 = 0.31. The rake parameter is thus

4

-4

= .26 x 10 (42)

r---~

Thrust, Torque and Efficiency

Several factors render a precise estimate of the performance very difficult. One is the marked variation of resultant speed along the b1ade:

no present aerodynamic theory can handle a variation fram M

=

0.94 at the

hub . to M

=

1.27 at the tip. Another is the presence of the large cylindrical

hub: the theory of the effect of such a hub has not yet been published. A

third factor is the rectangular trai1ing-edge extension: its contribution

can be evaluated by integration. The 1eading-edge camber is a further com-p1ication: the torque has already been calculated in an earlier section, but a precise evaluation of the thrust would require further manipulations of the unwie1dy Fourier series.

several of the mentioned difficulties arise from uncertainties in the theory; the remainder represent elements for which the theory is c1ear enough, but for which the computation is tedious and invo1ved. The uncer-tainties in the theory appear to justify simp1e estimates in p1ace of the more tedious computations. Thus, same 10ss in thrust and torque may be expected from the presence of the large hub and same gain from the trai1ing-edge extension. The 10sses and gains wi11 be assumed to cancel for the

chosen proportions • (More e1aborate estimates and calculations lead to very

near1y thi s result.) Again, the use of 1eading-edge camber may be expected to impair the ideal efficiency only slight1y. According1y, the torque and

ideal thrust bear the same proportions specified by the e·quations of Ref. 1,

but both are reduced by the factor Q(

T) / Qo

as a consequence of the

leading-edge camber. There i s a further reduction for aspect ratio and compressibi1i ty embodied very approximate1y in the factor I(Be).

The expression for the torque is thus (cf Ref. 1)

(43)

From the curve, Fig. 4, the value of

Q(T)/Qo

for 1 -

T =

0.075 is 0.879. In

the Appendix I(Be) is estimated as 0.96. Inserting the values of the several

variables,

Q

=

~(0.00145)(3.5)4(572)(1000)(1

- 0.811)(0.879)(0.96) 128

= 485 1b-ft

The ideal thrust is (cf Ref. 1): 2 4 2

Ti

=

~6D

[ 1 (

P/D)

J

.

Qá2

I(Be)

=

p{0.00145)(572)2(3.5)4 r1 - (0.811)2

1

(0.879)(0.96)

256

The e~~ective drag is (Re~. 1):

=

(0.0086)(1/2)(0.00145)(1000)2(4.16)

=

26 lb The netthrust is The e~~iciency is T

=

T. - D _{J. . e} = 226 lb TV TI

=

Qw _ (226 ) ~ 1000 ~ - (485 (572

=

0.815 or 81.5 percent Calculation o~ the Ordinates

(46)

(47)

(48)

The mean surface is most conveniently defined in terms o~ cylindrical coordinates r, 9 and x (see Fig. 8). The sur~ace does not deviate greatly

~rom the ~asic helicoid determined by the pitch-diameter ratio p/D. This basic helicoid is defined by

57.3n{x - xO)

90

=

R(p!D) degrees

where XC is arbitrary. In the present case XC will be taken as 28 inches, which makes 90 = 0 at the trailing. edge. The remaining constants are R =

21 in,

.r;tP/D.)

=

0.811, J

=

7T. The helicoid equation is thus _; 9

0 = 2.213(x - 28) degrees ' (6.5::S r

::s

0.810x) (50) The propeller mean sur~ace is obtained by adding to 90 an incremental value 91 given by Eq. 17. upon stibstitution

o~

c

=

4.26 x 10-

4

in-1 , k

=

0.2466 in-1 , T

=

0.925, a =0.810x, and multip1ication by 57.3 ~or conversion to degrees,

9

=

0.0244r

1

(51)

2

=

0.0244r _ 14.14 (r - 0.749x)

r

(0.749x

~

r

~

0.810x)

The equation of the propeller mean surf ace is therefore

= 2.213(x - 28)

+

O.0244r

(52)

=

2.213(x - 28)

+

O.0244r _ 14.14 (r - 0.749x)

r

(0.749x

~

r

~

O.810x)

where 9 is in degrees and x and r are in inches.

The slope ~ of the mean surf ace in the radial direction will be

needed (see Fig.

8).

This slope enters into the calcu1.ation of the coordinates of the upper and lower surfaces when the thiCkness is introduced. The slope ~

differs but slightly from 90, the slope of the basic helicoid, except nearthe tip. It is approximately

~ =

9 +

0

(r9)

o

d'r

1

=

2.213(x - 28)

+

2(0.0244)r

=

2.213(x - 28)

+

2(0.0244)r -2(14.14)(r-

"

0.749x)

(6.5

<

r

<

0.749x)

-

-

(53)

(0.749x

~

r

~

0.810x)

The Cartesian coordinates of the:mean surface, uppersurface, and lower surface are then given by

y

=

r cos 9

}

mean surface z

=

r sin 9 (54) y "= Y - t sin ~

}

u upper surface z

=

z + t cos _~ u

(55)

y~

=

y + t sin ~

}

lower surf ace

=

z - 't cos

respectively, where t=t(x, r) isthe 10caJ.. semi-thickness.

Tbe . semi-thickness distribution t is now to be determined. The

basic data are a table of ordinatesof the NACA 0006 profile (Ref. 6). From

tliis a table of ordinates 'of the NACA 00035 profile is obtained upon .

multi-plication by 3.5/6. The notation is changed so that tl now represents

semi-thickness as ·a fraction of 'the chord (not percent) and sl represents the

di stance· from the leading edge as a fraction of proj ected chord. Tbe proj ected

chord length is taken as s; this is the chord length that would be obtained

ifthe ·twist were ·neglected

(9

set eqtiaJ.. to zero). The following relations

are· then obtained from Fig. 7 by geometry:

s

=

28 - r ·tan 51 cl

R R( =21)

Solving for r, with R

=

21 inches,

r

=

Also 0.8l0x - 22.70sl 1 - 'sl (57) (58) (59)

Equations 58 and 59 ma,y be 'used wi th the table of profile ordinates

tl versus sl to obtain a table of correspondingvalues of r, x and t 0 The .

. specified values of sl are those of the .table ·of tl versus Alo Values of x

are specified, such as x = 1, 2, 3 ••. 28 inches. Correspondingvalues of

rare computed from Eq. 58. These values of r, together wi th the 'values of

t l corresponding to the specified sl' are · substituted in Eq. 59 for the

deter-mination of the 'corresponding values of t.

The calculation of the ordinates has been carried through

f.o

.

r

the

propeller design of the example. The shape of the propeller is indicated in Fig. 9. Note how the camber of the righthand sedion is very slightly

con-cave upward in the·inboard region and concave downward near the tip.

APPENDIlC A

. EVALUATION OF I(BC) FOR EXAMPLE

A

weighted average value ofthe Mach nuni>er·is required for 'the

evaluation. The Mach mmi>er 'varies from

0.94

at the hub to

1.27

at the tips.

The pressure distribution in the general vicinity of the ,tips makes the

greatest contribution to the torque and thrust. Thus the weighting should

favour 'the' tip region . The value

M

==

1.2

is estimated to be a suitable

weighted average. The corresponding Yalue of B is

==

0.663

(Al)

i

I

The Mach number of th!e stream is

0.9.

The weighted ave rage ratio

of V to VR is therefore

(

_V

ij ) =

0·9

=

0.75

R

1.2

(A2)

The average effectivesweep parameter according to Eq.

31

is

=

(0.75)( 0 .810 )

==

0.608

(A3)

The argument of I(BC) is

BC

e =

(0.663)(0.608)

==

0.403

(A4)

The value of I(BC) corresponding to BC

=

0.403

may be read off from Fig.

4,

Ref.

4.

The value is

1. Ribner, Herbert S.

2. Ribner, Herbert S.

3. Brown, C1inton E. Adams, Mac C.

REFERENCES

A Transonic Propeller of' Triangular Plan Form. NACA TN 1303, 1947.

The Stabi1ity Derivations of' Low-Aspect-Ratio Triangular Wings at Subsonic and Supersonic

Speeds. NACA TN 1423, 1947.

Damping in Pitch and Ro11 of' Triangular Wings

at Supersonic Speeds. NACA TN 1566, 1948.

4. Ribner, Herbert S. Stability Derivatives of' Triangular Wings at

Mal vestuto, Frank S. Jr. Supersoni c Speeds. NACA TN 1572, 1948.

5 • Puckett, Allen E.

6. Abbott, Ira H.

von Doenhof'f', Albert E. Stivers, Louis S. Jr.

Supersonic Wave Drag of' Thin Airf'oils. Jour. Aero. Sci., Vol. 13, No. 9, Sept. 1946, pp. 475-484.

Sumnary of' Airf'oil Data. NACA ACR L5C05 (War-time Report L-560) (Now available as TR 824), p. s4 in ACR version.

Flight

Speed

1

Ratafian

r - - - -....

y

PLAN VIEW

x

z ... - - - .

SIDE VIEW

x

FIG. 1 TRANSONIC PROPELLER OF TRIANGULAR PLANFORM

_---I~

Y

(a)

A

A

x

z

w

( b)

A

A

w

(c)

A

'~'-~t.-.::::::::::::-~"-

_A

FIG. 2 (a) PLAN VIEW OF UNTWISTED TRIANGULAR PROPELLER WITH

LEADING-EDGE CAMBER, (b) VIEW OF SECTION AA OF (a),

(c) DISTRIBurION OF NORMAL VELOCITY ALONG AA.

INTRO-DUCTION OF TWIST CHANGES THE RELATIVE ORIENTATION OF SUCCESSIVE SECTIONS AA AND THE MAGNITUDE OF THE W DISTRIBtJrION.

Ap

q

o

T =.925

- - T= 1.00

y/a

I

I

I

I

I

1.0

FIG.

3

SPANWISE LOADING WITH CAMBERED TIP

(T

=

.925)

AND WTIH UNCAMBERED TIP

(T

= 1.00) •

. 8

Q

.6

-

_Qo

.4

.2

0

0

.4

.6

.8

I-T

A

First Parabola

--Second Para bola

Center

FIG.

5

MODIFICATION OF SECTION AA, FIG. 2, TO INCORPORATE UPWARD RAKE (lst PARABOLA) FOR RELIEF OF CEN.rRIFUGAL STRESSES.

FIG.

6

RIGHT HALF OF FIG.

5

IN MORE DETAIL, WITH CURV ATURE

28"

"""-_ _ :==:,It~c:,,-::;;;-- -

f - -

---;.~

r

/ " 5/0

,.

\'.~

,.

\ -/

"

,.

\' /

"

,

~,

,.

"

/

"

/

"

,

_,

R=2/" _ _

~~IT

x

FIG.

7

HUB AND PROJECTED PLAN FORM OF PROPELLER BLADE SPECIFIED

FOR EXAMPLE.

z

"-point

(X,r,B)

~---~~--~---~y

HUB CENTER LINE

Y=O, z=o

HUB RAD/US

=

6.5

inches

Z,

inches

I

FIG.

9

TRANSONIC PROPELLER: SECTIONS TRANSVERSE TO HUB AXIS.

25

24

_----'-_ _ _ _ _ --'-_ _ _ _ _ -'--_ _ _ _ _

L----'x

27

26

6

8

/0

12

/4

16

18

20

22

Y,

inches

UTIAS TechnicaJ. !lote 110. 218

Institute for Aerospace studies, University of Toronto (UTrAS) 4925 Dufferin street, Downsview, Ontario, Canada, l(3H 5T6 A TRANSONIC PROPELLER OF TRIAH:lULAR PLAN FORM (AN EXTENSION)* Rlbner, Herbert s.

1. Propellers, high speed

r. Ribner, H. S.

29 pages 9 tigures

2. Propeller design 3. Wing theory

II. UTIAS TechnicaJ. Note No. 218

~

(Tbere has been a resurgence of interest in high speed propellers af ter two decades ot dom1nance by jet propulsion. Thus it has seemed wortbwhile to put into print a hitherto unpubl1shed design study ot a high speed propeller concept with BeveraJ. navel features. Tbe design, carried out at NACA Langley in 1949, was"intended as a basis tor model tests in a wind t,unne1. OWing to speciaJ. circumstances the tests were never undertaken.)

All isosceles triangle twisted lnto a screw surf'ace about lts e.x1s. was proposed in NACA TN 1303 as a

propeller f'or transonic flight speeds. The ma.thematlca.l. an&1.ysls ls extended berein to cover the

aero-dynamic and structural effects of' a structurally f'easlble camber distribution. The basic camber ls localized near the leading edge; the camber is detennined in such a way that the theoretical infinite presBure peak aJ.ong the leading edge iB eliminated. An awdl1ary cBl!iler or rake is derived for the aJ.leviation of the deflection caused by the centrifUgaJ. forces.

The earl1er formulas were derived on the assumption tbat the triangle is narrow campared with the

Mach cone from its vertex. A factor to correct approximately for tbe breadth of the triangle is

introduced herein. Tbe seleotion of the advance-d1"8lIleter coeffic1ent and pitch-di8llleter ratio for

maximum efficiency is treated. Finally, tbe design ot a 3.5-foot diameter model propeller for wind-twmel tests at a atream Mach nUIlber of 0.9 is carried through in a numeri cal example.

*Prepared under NACA auspices in 1949 and hitherto unpubl1shed.

UTIAS TechnicaJ. Note Ilo. 218

Institute for Aerospace studies, University of Toronto (UTIAS) 4925 Dufferin street, Downsview, Ontario, Canada, l(3H 5T6 A TRANSONIC PROPELLER OF TRIANlULAR PLAN FORM (AN EXTENSION)* Ribner, Herbert s. 29 pages 9 tigures 1. Propellers, high speed 2. Propeller design 3. wing theory

I. Ribner, H. S. II. UXIAS TechnicaJ. Note No. 218

~

(There has been a resurgence of interest in high speed propellers af ter ho decades or dom1nance by jet propulsion. Tbus it has seemed worthwhile to put into print a hitherto unpubl1shed design study of a high speed propeller concept with severaJ. novel features. Tbe design, carried out at NACA Langley in

1949, was intended as a basis for model tests in a wind t,unnel. Owing to spec1aJ. circumstances the

tests ~ere never tUldertaken.)

AA lsosceles triangle twlsted lnto a screw surface &bout its a.x1s was proposed in NACA TH 1303 as a

propeller for transonic flight speeds. The mathematical analysls ls extended berein to cover the aero-dynamic and structural. effects of a structurally feasible canber d1stributlon. The basic camber ia

localized near the leading edge; the camber is detennined in such a way that the theoretica! infinite

pressure peak aJ.ong the 1eading edge is eliminated. An awdl1ary cBl!iler or rake is derived for the alleviation of the deflection caused by tbe centrifUgaJ. forces.

Tbe earl1er formulas were derived on the assumption tbat the triangle is narrow campared with the

Mach cone frm its vertex. A factor to correct approximately for tbe breadth ot the triangle is

introduced herein. The seleotion of the advance-di"8lIleter coefficient and pitch-di8llleter ratio tor

maximum efficiency is treated. Finally, tbe design of a 3.5-foot di·ameter model propeller for

wind-tunnel tests at a streem Mach nlllliler of 0,9 is carried through in a numeri cal enmple.

*Prepared under NACA auspices in 1949 and hitherto unpublished.

Available copies of this report: are limited. Return this card to UTIAS, if you require a copy. Available copies of this report: are limited. Return this cllrd to UTIAS, if you require..a -copy.

UTIAS TechnicaJ. Note No. 218

Institute for Aeroepace Studies, University of Toronto (UTIAS) 4925 Dufferin street, Downsview, Ontario, Canada, l(3H 5T6 A TRANSONIC PROPELLER OF TRIANlULAR PLAN FORM (AN EXTENSION)* Ribner, Herbert S.

1. Propellers, high speed

I. Ribner, H. S.

29 pages 9 figures

2. Propeller design 3. Wing theory

II. UTIAS Technical Note No. 218

~

(There has been a resurgence of interest in high speed propellers af ter bo decades ot dominanee by

jet propUlsion. Thus it has seemed worthwhile to put into print a hitherto unpubl1shed design study ot

a high speed propeller concept with several novel features, The design, carried out at NACA Langley in

1949, was intended as a basis for model tests in a wind t,unne1. OWing to speciaJ. circumstances the

tests were never tUldertaken.)

An isosceles triangle twisted into a screw surface about its axis was proposed in NACA TN 1303 as a

propeller for transonic flight speeds, The mathematical analysis is extended herein to cover the

aere-dynamic and structural effects of a structurally feasible cBl!iler distribution. The basic camber is

locaJ.ized near the leading edge; the camber is detennined in such a way that the theoretical infinite

pressure peak along the leading edge is eliminated. An auxiliary camber or rake is derived for the

alleviation of the deflection caused by the centrifUgal forces.

The earlier formUlas were derived on the assumption that the tr1angle is narrqw compared ",ith the ~ch cone frcun its vertex. A factor to correct approximately for the breadth of the triangle is

introduced herein. Tbe selection of the advance-di"8lIleter coefficient and pitch-di8llleter ratio for

maxilnum effiCiency is treated. Fina.lly, tbe design of a 3.5-foot diameter IOOdel propeller for

wind-tunnel tests at a stre8lll Mach nlllliler of 0.9 is carried through in a numeri caJ. enmple.

*Prepu.rod under NACA auspices in 1949 and hi therto unpubl1shed.

UTrAS Technical !'Iote No. 218

Institute for Aerospace Studies, University ot Toronto (UTIAS) 4925 Dufferin street, Downsview, Ontario, Canada, l(3H 5T6 A TRANSONIC PROPELLER OF TRIANlULAR PLAN FORM (AN EXTENSION)* Ribner, Herbert S.

1, Propellers, high speed

I. Ribner, H. S.

29 "ages 9 figures

2. Propeller design 3. Wing theory

Ir. UXIAS Technical Note No. 218

~

(There has been a resurgence ot interest in high speed propellers af ter two decades of dom1nance by jet propulsion. Thus it has seemed worthwhile to put into print a hitherto unpubl1shed design study ot

a high speed propeller concept with severaJ. novel features. The design, carried out at NACA Langley in

1949, was intended as a basis for model tests in a wind ~unne1. OWing to speciaJ. circumstances the

tests were never Wldertaken.)

An isosceles triangle twisted into a screw surface about its axis" was proposed in NACA TN 1303 as a

propeller for transonic flight speeds. The mathematical analysis is extended herein to cover the aero-dynamic and structuraJ. effeots of a structuraJ.1y fea.ible camber distribution. The basic camber is

localized near the leading edge; the camber is detennined in such a way that the theoretical infinite pressure peak along the leading edge is eliminated. An awdliary camber or rake rs derived for the

alleviation of the deflection eau Bed by the centrifUgal force •.

The earlier formUlas were derived on the assumption that the trle.ngle is narrow compared with the

Mach cone fran it. vertex. A factor to correct approximately for tbe breadtb of the triangle is

introduced herein. The seleotion of the advance-di"8lIleter coefficient and pitèh-diameter ratio for

maximum efficiency is treated. Finally, the design of a 3.5-foot diameter model propeller for \/ind-tunnel tests at a stre8lll Mach nlllliler of 0.9 is carried through in a numericaJ. enmple.