PREDICTION OF SPAN LOADING OF STRAIGHT-WING ...

N A S A C O N T R A C T O R

R E P O R T "

N 0 *o cv pe: U

I

-LOAN COPY: RETURN TO AFWL TECHNICAL LrSRARY

KIRTLAND AFB, N. M.

PREDICTION OF SPAN LOADING OF STRAIGHT-WING/PROPELLER COMBINATIONS UP TO STALL

M . A. McVeigh, L . Gmy, and E. Kisielowski

Prepared by UNITED TECHNOLOGY, INC. Blue Bell, Pa. 19422

for Langley Research Center

N A T I O N A L A E R O N A U T I C S A N D S P A C E A D M I N I S T R A T I O N W A S H I N G T O N , D. C. 0 , OCTOBER '1975

. I

. .

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https://ntrs.nasa.gov/search.jsp?R=19760004918 2020-03-22T17:45:48+00:00Z

17 - TECH LIBRARY KAFB. NM ~~ ~.

'1. Report No. 2. Govcmnwnt Accession No.

NASA CR-2602 . l lnl l lmlHll l r l rrnl l l~~~~ 4. Title and Subtitle Prediction of span loading of ObbLSLL

straight-wing/propeller combinations up to stall

October IY13 6. Performing organization Coda

- 7. AuthoM M . A. McVeigh 8. Performing Organization Report No.

L. Gray 10. Work Unit No. E. Kisielowski UTR-004

United Techhology , Inc . 11. Contract or Grant No. 1777 Walton Road NAS1-12238 Blue Beil, Pa. 19422

9. Performing Organization Name and Address

13. Type of Report and Period Covered r12: Sponsoring Agency Name and Address

National Aeronautics and Space Administration Washington, D.C. 20546

14. Sponsoring Agency code

I .__." 15. NotesThe N A G technical representatives were Mr. Robert T. Taylor and Mr. Louis P. Tosti. The contributions of the NASA technical personnel to this work are gratefully acknowledged.

16. Abstract

distribution on straight-wing/propeller combinations. The method combines a modified form of the Prandtl wing theory with a realistic representation of the propeller slipstream distribution. The slipstream analysis permits calculations of the non-uniform axial and rotational slipstream velocity field of propeller/nacelle-combinations. This non-uniform field is then used to calculate the wing lift distribution by means of the modified Prandtl wing theory.

for both the wing.and the propeller blade airfoil sections and is applicable up to stall. The theory is developed for any number of non-overlapping propellers, on a wing with partial or full-span flaps, and is applicable throughout the aspect ratio range from 2.0 and higher

digital computer. The computer program is used to calculate slipstream characteristics and wing span load distributions for a number of configurations for which experimental data are available. Favorable comparisons are demonstrated between the theoretical predictions and the existing data.

The method utilizes non-linear aerodynamic section data

The analysis is programmed for use on the CDC 6600 series

17. Key Words (Suggested by Author(s)I Load distribution Propeller Distribution unlimited

Wing Angle of attack

18. Distribution Statement

Aircraft I Twist Stall Taper ratio Subject Category 01

1 I 19. Sccurity Classif. (of this report)

. . . - . . . 20. Security Classif. (of this page)

. ~. . ~~~

21. NO. of Pages 22. Rice'

Unclassified Unclassified $7.25 208 . - ... .. . . -

For sale by the National Technical Information Service, Springfield, Virginia 22161

SUMMARY

A method is presented for calculating the spanwise lift distribution on straight-wing/propeller combinations. The method combines a modified form of the Prandtl wing .theory with a realistic representation of the propeller slipstream distribution. The slipstream analysis pennits calculations of the non-uniform axial and rotational slipstream velocity field of propeller/nacelle combinations. This non-uniform field is then used to calculate the wing lift distribution by means of the modified Prandtl wing theory.

The method utilizes non-linear aerodynamic section data for both the wing and the propeller blade airfoil sections and is applicable up to stall. The theory is developed for any number of non-overlapping propellers, on a wing with partial or full-span flaps, and is applicable throughout the aspect ratio range from 2 .O and higher .

The analysis is programmed for use on the CDC 6600 series digital computer. The computer program is used to calculate slipstream characteristics and wing span load distributions for a number of configurations for which experimental data are available. Favorable comparisons are demonstrated between the theoretical predictions and the existing data.

iii

CONTENTS Paqe

S U M M A R Y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

LIST OF ILLUSTRATIONS. . . . . . . . . . . . . . . . . . viii

SECTION 1 INTRODUCTION........................... 1

SECTION 2

2.1 2.2 2.3

3

3 4 6

SECTION 3 TiIEORETICAL ANALYSIS................... 8

3.1 PROPELLER SLIPSTREAM ANALYSIS. . . . . . . . . . 8 3.1.1 General Propeller Solution............. 8 3.1.2 I n i t i a l Calcu la t ion of Inf low Angle.... 13

Solution....o....o...~................. 15

D i s t r i b u t i o n s ......................... 18

,' 3.1.3 Convergence of the Iterative P r o p e l l e r

3.1.4 Analys is for Slipstream Velocity

3.2 WING-IN-SLIPSTREAM ANALYSIS. . 21

o r W i t h Full-Span Deflected Flaps...... 21

Deflected Flaps. .~. . . . . . . . . ......,..... 29

Small Aspe'ct Ratios.................... 39

3.2.1 Analys is for a Wing W i t h no Flaps

3.2.2 Analys is for a Wing With Part-Span

3.2.3 Extension of. the Wing Analysis t o

SECTION 4 DIGITAL COMPUmR PROGRAM............... 42

4.1 PROPEZLER SLIP-S'I'REAM COMPUTATIONS. . . . . 42

Sl ips t ream Veloc i ty Dis t r ibu t ions . . . . . 42 4.1.1 Computational Procedures for Propeller

V

. % , . 4.x.2

4.1.3 - . . . .

1

4.2 ,.. 4'.2.1

4.2.2

4.2.3 4.2.4

4.3

4.4

SECTION 5

5.1

5.2 5.2.1

5.2.2

5.2.3

5.2.4 5.2.5

SECTION 6

SECTION 7

Propeller: B l a d e Section C h a r a c t e r - istics......................... ...... Table Look-Up Procedures fo r Pro- peller A i r f o i l C h a r a c t e r i s t i c s . . . . . .

- . .

WING-IN-SLIPSTREAM COMPUTATIONS. . . . . C o m p u t a t i o n a l Pracedures f o r Span- w i s e Loading on a Wing With no Flaps o r w i t h Full-Span D e f l e c t e d Flaps.. . C o m p u t a t i o n a l Procedures for Span- w i s e Loading on a Wing With P a r t - Span D e f l e c t e d Flaps ................ Wing Section Characteristics........ T a b l e L o o k - U p Procedures fo r Wing Section Characteristics.............

DESCRIPTION OF THE: COMPUTER PROGRAM LOGIC...............................

SAMPLE OUTPUT.......................

VERIFICATION O F THE DEVELOPED THEORY

CORRELATIONS EQR AN ISOLATED !

PROPELLER........................'...

CORFSLATIONS FOR WING-IN-SLIPSTREAM C o r r e l a t i o n s f o r Low A s p e c t R a t i o Wings C o r r e l a t i o n ?or C e n & 3 r a l l y - M o u n t e d Propellers and Jets................. C o r r e l a t i o n fo r Twin Propeller C o n f i g u r a t i o n s ...................... Effect of Propeller R o t a t i o n . . . . . . . . Effect of Flap Deflection.. . . . . . . . . . CONCLUSIONS AND RECOMMENDATIONS.. . . . REFERENCES..........................

vi

48

52

54

54

57 61

61

61

65

68

68

77

\ 78

80

84 88 96

98

99

APPENDIX A PROPELLER TIP LOSS CORRECTION TABLES 104

APPENDIX B PROPELLER AIRFOIL TABLES............ 108

APPENDIX C PROGRAM USER INSTRUCTI0,NS. . . 121

APPENDIX D INTERNAL LISTING O F THE COMPUTER PROGfZIIM............................. 146

V i i

LIST OF ILLUSTRATIONS

Figure 1

2

3

4

5

6

7

8

9

10

11

12

Notation for a Propel le r Opera t ing i n the Presence of a Wing ......... Blade Element Velocity Diagram .... Analy t i ca l Model for Sl ipstream Contraction.......................

Notation for Wing-In-Slipstream Model.............................

Mathematical Representat ion of Flap Discont inui ty ................ Method for Superpos i t ion of Solutions.........................

Computer Program Flow Diagram. .... Logic Diagram for Propeller Slip- stream Subroutine.................

Paqe

9

12

20

24

30

35

62

64

Cor re l a t ion Between Predicted and Measured Elemental Thrust and Torque Loadings a t 75 Percent Radius..... 69

Cor re l a t ion Between P red ic t ed and Measured Elemental T h r u s t and Torque Loadings a t 52 Percent Radius... . . 70

Cor re l a t ion Between P r e d i c t e d and Measured Elemental Tnrust and Torque Loadings a t 25 Percent Radius... . . 7 1

Comparison Between P r e d i c t e d and Msasured Distributions of Slipstream Axial Velocity and Swirl Angle f o r t h e P-2 Propel ler of Reference 1 7 , a t J = 0.12....................... 73

v i i i

Figure 13

14

15

16

17

l a

19

20

21

, .

Paqe Comparison Between Predicted and Measured D i s t t i b u t i o n s . of Slipstream . . ,

Axial Ve.locity and Swirl Angle for the P-1 Propeller of Reference 17, a t J = 0.26.. ...................... 74

Comparison Between Predicted and Measured Distr ibut ions of Sl ipstream Swirl Angle for Typical Test Conditions of Reference 42........" 76

V e r i f i c a t i o n of Low Aspect Ratio ~ a l y s i s .......................... 79

Comparison Between Predic ted Span- w i s e Loading and.Measurements of Reference 6 f o r a Rectangular Wing With End Plates Subjected t o a Uniform Je t ; Vs/Vo = 1.36.........

Predicted Versus Measured Spanwise Loadings for the Rectangular Wing of Reference 29 With a Central ly- Mounted P rope l l e r ; AR = 6.........

Predicted Versus Measured Spanwise Loadings for the Rectangular Wing of Reference 29 With a Central ly- Mounted P r o p e l l e r ; AR = 3.. ....... P r e d i c t e d Versus Measured S,panwise Loadings f o r the Twin-Propeller Configuration of Reference 42; AR = 3-01 C T s = o.................

Predicted Versus Measured S,panwise Loadings for the Twin-Propeller Configuration of Reference 42; AR = 3.0, C T ~ = 0.36, p 75 = 25O..

Predicted Versus Measured Spanwise Loadings for the Twin-Propeller Configuration of Reference 42; AR = 3.0, CT = 0.64, p75 = 25O.o

S

81

0 2

8 3

85

86

8 7

ix

Figure 22

23

24

25

24

27

2 8

29

30

Paqe Predicted Versus Measured Spanwise Loadings for the Twin-Propeller Configurat ion of Reference 44; AR = 4.7, c = o................. 89

Predicted Versus Measured Spanwise Loadings for the Twin-Propeller Configurat ion of Reference 44; AR = 3.26-, C = O..................... 90

Predicted Versus Measured Spanwise Loadings for the Twin-Propeller Configuration of Reference 44; AR = 2.28, C = O................ 91

Predicted Versus Measured Spanwise Loadings for the Twin-Propeller Configuration of Reference 44;

TS

TS

TS

AR = 4-07, CT = 0.4............... S 92

P red ic t ed Versus Measured Spanwise Loadings f o r the Twin-Propeller Configuration of Reference 44: AR = 3.26, C T ~ = 0.4.............. 93

P r e d i c t e d Versus Measured Spanwise Loadings for the Twin-Propeller Configuration of Reference 44; AR = 2.28, CTs = 0.4.............. 94

Effect of Propeller Rotat ion on Span Loading for the Configurat ion of Reference 42; AR = 3.0, C T ~ = 0.64, 0 = 10 Degrees. ....... 95

Pred ic t ed Spanwise Loadings f o r t h e Twin-Propeller Configuration of Reference 44 t o Show the Effect of Flap Deflect ion; AR = 4.7, Q = 10 Degrees... ........................ 97

Assembly of Computer Program Inpu t Data Card Deck.................... 123

X

LIST OF TABLES

Table I

- I1

I11

IV

V

Paqe

Typical. Propeller Blade Sections . . 49

. .

. Summary .of Propeller Airfoil Sections Tabulated for use in the Computer Program............. ..... 51

Sample' Output for Lift. Distribution on a Wing-In-Slipstre am........... 66

Sample Output. for Propeller Velocity Distribution............'.......... 67

Card Format for Wing Section Airfoil Tables............................ 130

xi

LIST O F . SYMBOLS

AR wing aspect ratio

a lift curve slope for finite aspect ratio, per degree . .

B

B, b

CD

cd

CL

C 1

CQ

CT

C

CR

D

E

F

. .

speed of sound, m/sec

section lift curve slope, per degree

number of blades per propeller

coefficients in trigonometric series

wing span, m

total wing drag coefficient

section drag coefficient

total wing lift coefficient

section lift coefficient

propeller torque coefficient , Q/, n2 D5

propeller thrust coefficient, T/pn2 D4

propeller thrust coefficient , T/qs T R2

wing local chord, m

wing root chord, m

propeller tip diameter, m

edge velocity factor

propeller tip loss factor

inclination of the propeller axis to the fuselage centerline, degrees

x i i

J

1

43

4 7

n

Q

qs ,

R

Re

r

rS

T

u

V

Va

Vn

vo

VS

propeller advance ratio, Vo/ nD

wing section lift, per unit span, N

freesteam Mach number, V d a s

local Mach number for propeller blade element, V/as

rotational speed, rev/sec

propeller shaft torque, Nom.

average slipstream dynamic pressure, N/m2

propeller tip radius, D/2, m

Reynolds number

local radius in propeller disk plane, m

local radius in slipstream, m

propeller thrust, N

axial component of velocity induced by a blade element in the propeller disk plane, m/sec.

local velocity, m/sec.

component of freestream velocity along the propeller axis, m/sec.

component of local slipstream velocity normal to the zero-lift line, m/sec.

freestream velocity, m/sec.

component of local slipstream velocity parallel to the zero-lift line, m/sec.

axial component of local velocity in the fully-developed

I I slipstream, m/sec.

X i i P

V St

vW

Y

Y*

yP

Q

at3

“P

momentum-we&gh,ted . . mean . . . , axia l - , . >veloci$y in,,the fully-:. developed sl ips. tream, m/sec.

t a n g e n t i a l component of locai.. veloci.ty’ in‘, the f u l l y - developed s l ipstream, m/sec.

I

j - . . , _ . . . , . . : . . L

- . .. . . _ . . : I ._ ~ I . . . . . . . ~ .

. I . . ., . . . ‘ 4

. . , , .

.. . % . . ~ , . ~ i I, ’. ; ’

I .

. upwash v e l o c i t y component a c t i n g ‘ i n the p r o p e l l e r d i s k p lane . . due t o , t h e . , . . . . presence, .of . :a l i f t ing. wing, m/se,c,.j;

spanwise co-ordinate of local wing element, m

spanwise co-ordinate of f lap end, ,m . .

spanwise co-ordiante of p r o p e l l e r . a x i s , . m,,

angle o f attack r e l a t i v e . to. a i r f o i l s e c t i o n chord- l i n e , d e g r e e s

ang le of attack r e 1 a t i v e ” t o f u s e l a g e c e n t e r l i n e , ‘ degrees . I . . . .

, . _. . .

corrected sec t ion ang le o f attack, , deg-rees

e f f e c t i v e a n g l e o f a t tack of wing s e c t b n , d e g r e e s . .I

. . . I : .. l L , . .,_ . .

geometr ic angle of at tack of w.ing sec t ion , deg rees

induced angle o f a t tack .of wing sec t ion , degrees

ang le of attack o f a i r f o i l s e c t i o n a t ze ro lift, ;

degrees

sec t ion ang le 0.f attack for two-dimensional air- ~

f o i l , degrees . . . . .

. . . . .. . 2.-

p r o p e l l e r a x i s a n g l e o f , a t t a c k , d e g r e e s ~.

x iv

geometr ic angle of a t t ack o f .w ing roo t , deg rees -.

i n c l i n a t i o n o f the slipstream a x i s to t h e f l i g h t path, degrees

pitch ang le fo r p rope l l e r b l ade e l emen t , deg rees

mul t ip l i e r fo r i nduced ang le o f a t t ack , deg rees

magnitude of discont inui ty in absolute and induced a n g l e s of a t t ack , deg rees

geometric t w i s t a t any wing section, degrees

c i rcu la t ion about any wing sec t ion , m2/sec.

wing t a p e r r a t i o

b l a d e s p e e d r a t i o f o r p r o p e l l e r blade element

blade t i p speed r a t i o

k inemat i c v i scos i ty , m2/sec.

inf low angle for p rope l le r b lade e lement , degrees

i n f l o w a n g l e f o r p r o p e l l e r blade element, excluding c o n t r i b u t i o n from induced velocity in d i s k p l ane , degrees

ambient density , kg/m3

s o l i d i t y f o r p r o p e l l e r blade element

wing spanwise coordinate, COS"( 2y/b)

spanwise co-ordinate for

r o t a t i o n a l s p e e d , 2 a n

angular veloci ty induced

* f l a p end , cos" ( ylb )

radians/sec.

by a blade element behind the p rope l l e r d i sk p l ane , r ad ians / sec .

factor to accoun t fo r low aspect r a t i o e f f e c t s

xv

PREDICTION OF SPAN LOADING O F STRAIGHT-WING/PROPELLER COMBINATIONS

UP To STALL

M. A. McVeigh, L. Gray, and E. Kisielowski

UNITED TECHNOLOGY, I N C . SECTION 1

INTRODUCTION

The propel le r s l ips t ream exer t s an impor tan t in f luence on w ing l oad d i s t r ibu t ion , wh ich i n t u rn a f f ec t s t he . a i r c ra f t s t a l l c h a r a c t e r i s t i c s . T h i s e f f e c t i s introduced through a n increase in local veloci ty over the s l ipstream-immersed por t ion o f the wing and a change of wing loca l angle o f attack due t o s l i p s t r e a m r o t a t i o n . While the i nc reased ve loc i ty t e n d s t o s t a b i l i z e the flow over that wing po r t ion , t he s l i p s t r e a m r o t a t i o n may g ive r i se to an asymmetr ic stall condition due to inc reased l oca l ang le s o f a t t ack of the wing sec t ions behind the up-going p r o p e l l e r b l a d e s , and reduced a n g l e s of a t t a c k o f t h e wing sec t ions behind the down-going b l ades .

A rev iew of the ava i lab le t echnica l l i terature indicates t h a t there are no r e l i ab le t heo re t i ca l o r s emi -empi r i ca l methods which can adequately predict the e f f e c t s of a p r o p e l l e r s l i p s t r eam on the spanwise load d i s t r ibu t ion of an en t i re wing. Many of the e x i s t i n g methods a r e s u i t a b l e o n l y f o r computing t o t a l wing forces s ince they are of ten based on gross s implifying assumptions. Thus, for example, an assumption tha t the slipstream-immersed portions of the wing can be t r e a t e d as i so la ted p lanforms neglec ts the s t rong in f luence o f the s l ip s t r eam on ad jacen t wing regions. Other theoret ical methods are gene ra l ly classed a s rigorous mathematical approaches w h i c h are usua l ly very complex and a re f requent ly no t in suf f ic ien t agreement with experimental data t o w a r r a n t t h e i r use as a design tool. Furthermore, most of the above theories use l i n e a r l i f t c u r v e s and as a r e s u l t can not be expected t o y i e l d sa t i s fac tory agreement w i t h tes t data near wing s t a l l .

The l im- i t a t ion imposed by the use of l i n e a r l i f t curves f d r the wing has been successful ly removed i n the work r epor t ed i n Reference 1. This re fe rence p re sen t s a computerized method for predic t ing spanwise load d i s t r ibu t ions of straight-wing/ fuselage combinations a t ang le s of attack up t o stall. This method, which i s based on the Prand t l wing or " l i f t i n g l i n e ' ! theory as formulated by S ive l l s i n Re fe rence 2, provides a reliable a n a l y t i c a l tool for p red ic t ing w ing s t a l l i ng characteristics of gene ra l av i a t ion t ype a i rcraf t , bu t is only a p p l i c a b l e t o power-off f l i gh t cond i t ions , such as might be encountered during landing.

The cu r ren t i nves t iga t ion ex tends the ana lys i s o f Reference 1 to p e r m i t c a l c u l a t i o n s of span loading and s ta l l ing c h a r a c t e r i s t i c s under power-on condi t ions (e .g . take-off) f o r wings w i t h o r wi thout f laps and having any . number of non- ove r l app ing p rope l l e r s . The p r e s e n t method i s based on employ- i n g n o n - l i n e a r a i r f o i l s e c t i o n c h a r a c t e r i s t i c s ' f o r b o t h the propel le r and the wing. The bas i c ana ly t i ca l app roach of this method i s t o r e t a i n the inhe ren t s imp l i c i ty o f the Prandt l wing theory, modify the t h e o r y as r e q u i r e d t o a c c e p t non-uniform I

s l i p s t r e a m v e l o c i t i e s , a n d e f f e c t i v e l y combine this modified l i f t i n g l i n e t h e o r y w i t h a realist ic p r o p e l l e r t h e o r y t o form a u n i f i e d a n a l y t i c a l t o o l .

A detailed d e s c r i p t i o n of th i s analyt ical method, toge ther w i t h the specially developed digital computer program i s p resen ted i n the fol lowing pages.

2

SECTION 2

GENERAL REVIEW OF THE ANALYTICAL METHODS

The prime o b j e c t i v e of the current development is t o provide a . practical ana ly t i ca l so lu t ion fo r de t e rmin ing the l i f t d i s t r i b u t i o n a n d s t a l l i n g characteristics of wings p a r t i a l l y or t o t a l l y immersed i n a p rope l l e r s l i p s t r eam. I:n order t o depict some of the h i g h l i g h t s o f the c u r r e n t w o r k r e l a t i v e t o o t h e r a p p r o a c h e s , t h i s s e c t i o n p r e s e n t s a brief review of the ex i s t ing ana ly t i ca l and expe r imen ta l i n v e s t i g a t i o n s that a t t e m p t s o l u t i o n s of the wing/propeller problem

2.1 STATEMENT OF THE PROBLEM

The basic l i m i t a t i o n i n p r o v i d i n g reliable s o l u t i o n s t o the wing/propeller problem is re l a t e ' d t o a lack of complete understanding of the flow f ie ld generated ''by the wing/propeller i n t e r a c t i o n u n d e r p r a c t i c a l o p e r a t i n g c o n d i t i o n s . The problem is further compounded by the d i f f i c u l t y ok developing realistic a n a l y t i c a l r e p r e s e n t a t i o n s o f this complex flow f i e l d environ- ment so as t o a c c o u n t f o r the ma jo r i n t e rac t ion effects a c t i n g .on a wing/propeller combination. A complete solut ion t o the problem must therefore account for a l l these effects, which as a minimum should include the fol lowing

(a) Local wing angle-of-attack changes due t o the mean i n c l i n a t i o n a n d r o t a t i o n of the s l i p s t r eam f l o w .

(b) Non-uniform s , p n w i s e d i s t r i b u t i o n o f v e l o c i t y o v e r those p o r t i o n s of the wing w i t h i n the s l ip s t r eam.

(c) Non-uniform v e r t i c a l d i s t r i b u t i o n of v e l o c i t y w i t h - i n the slipstream-immersed regions of the wing.

( d ) Viscous mixing between the s l ips t ream and freestream f l o w along the s l ipstream boundary.

I n view of the real f lu id f l ow effects involved it is un l ike ly that eve ry a spec t of the problem can be treated adequately, us ing the established ana ly t ica l approaches . H i s t o r i c a l l y , the approach has been t o in t roduce a series of s implifying assumptions in order t o arr ive a t a so lu t ion . These approaches are discussed below.

3

2.2 REVIEW O F EXISTING SOLUTIONS

The earliest t r ea tmen t of the p r o p e l l e r slipstream problem i s con ta ined i n the pioneering work of Koning (Reference 3) who treated the, s i m p l i f i e d case of a wing c e n t r a l l y immersed i n a c i r c u l a r u n i n c l i n e d slipstream of uniform axial ve loc i ty wi thoyt ro ta t ion . Koning applied the methods of l i f t i n g l i n e t h e o g y t o ob ta in a s o l u t i o n when the ratio of free stream v e l o c i t y t o slipstream v e l o c i t y is close t o uni ty . This work was'extended by Glauert (Reference 4) and by Franke and Weinig. (Reference 5) t o a wider range of forward speeds.

Stuper (Reference 6 ) conducted a series of experiments t o v e r i f y the p r e d i c t i o n s of Koning's theory by measuring the l i f t d i s t r i b u t i o n o n a r ec t angu la r wing w i t h end p l a t e s unde r the a c t i o n of a c i r c u l a r j e t . Stuper used a spec ia l ly des igned f an t o produce a j e t wi thou t ro t a t ion and w i t h a v e l o c i t y c ross -sec t ion which w a s approximately uniform. While the results of those experiments are somew3at impaired by the

1 p a r t i c u l a r t e s t arrangement used by Stuper , there is s u f f i c i e n t evidence t o show that the Koning theory over-predicts the l i f t increase due t o the j e t .

Because of the i n a b i l i t y of the l i f t i n g l i n e a p p r o a c h , as formulated by Koning, Glauert e t . a l . , t o s a t i s f a c t o r i l y pred ic t exper imenta l measurements , subsequent inves t iga tors assumed that the f a i l u r e o f the l i f t i n g - l i n e t h e o r y w a s a s s o c i a t e d w i t h the fac t t ha t the por t ion of the wing immersed i n the slipstream w a s u sua l ly o f small aspect ra t io . Graham, e t . a l . , (Refereace 7 ) therefore approached a s o l u t i o n via s l ende r body theory and the approximate l i f t i n g s u r f a c e t h e o r y of Weissinger (Reference 8 ) . Ca lcu la t ions made by Graham showed improved agreement w i t h Stuper ' s experimental data.

Ribner and E l l i s (Reference 9 ) genera l ized the Weissinger l i f t i n g s u r f a c e f o r m u l a t i o n t o mul t ip le , un inc l ined slipstreams. Their r e s u l t s showed reasonable agreement w i t h the experimental data obtained by Brenckmann (Reference 10 ) f o r the o v e r a l l l i f t increase due t o the p rope l l e r s l i p s t r eam.

The t e s t r e s u l t s o b t a i n e d by Brenckmann rep resen t an improvement over the experimental data o f S tupe r i n t h a t the former e x p e r i m e n t s u t i l i z e d a n i n f i n i t e aspect ra t io wing, thus avoiding the use of end-plates which introduce uncertain-

4

t i es as t o the e f f e c t i v e v a l u e of wing aspect ratio. Since Brenckmann employed a free p r o p e l l e r y i e l d i n g a non-uniform slipstream v e l o c i t y p r o f i l e , the Ribner -El l i s theory , which assumes uni form ve loc i ty d i s t r ibu t ions , would no t be expected t o y ie ld adequate p red ic t ions of the spanwise l i f t distri- bu t ions as compared w i t h Brenckmann' s measurements.

Another series of tests of i n t e r e s t are those of Gobetz (Reference 11) and Snedeker (Reference 1 2 ) who employed a similar experimental arrangement t o that of S tupe r , i n tha t a je t of a i r of qpproximately uniform velocity p rof i le w a s used t o s i m u l a t e the p r o p e l l e r slipstream. These test's w e r e designed to determine the basic effects of both w'ng aspect r a t i o and wing chord/slipstream diameter. The r J s u l t s were compared w i t h theoretical ca l cu la t ions u s ing the modi f i ed l i f t i n g - l i n e t h e o r y o f Rethorst (Reference 13) and it is shown that t h i s theory is a t least capable o f p red ic t ing the t r e n d s of the tes t da ta .

Goland, e t . a l . , (Reference 14) formulated a mathematical model based on poten t ia l theory approach t o p p e d i c t o v e r a l l performance and s t a b i l i t y characterist ics of small aspec t r a t i o wing spanning a s l ipstream of uniform vdloci ty . Al though this work e f f e c t i v e l y combined t h e R. T. Jones small a s p e c t r a t i o t h e o r y w i t h t h e p o t e n t i a l f l o w theory t o y i e l d good c o r r e l a t i o n w i t h t e s t data, no attempt w a s made t o p r e d i c t and correlate t h e wing spanwise load dis t r ibut ions. This work w a s extended in References 15 and 16 to p rovide equat ions and charts f o r e s t i m a t i n g l i f t and longi tudinal force c o e f f i c i e n t s of STOL a i r c ra f t wings immersed i n p r o p e l l e r s l i p s t r eams . t '

George and Kisielowski (Reference 1 7 ) modified the work of Reference 16 to account for non-uniformity of the prop- e l l e r s l i p s t r eam. In this a n a l y s i s the p r o p e l l e r s l i p s t r e a m v e l o c i t y w a s represented by a number of concentric zones of uniform veloci ty (s ta i rcase func t ions ) w i t h the wing spanning the s l ip s t r eam. A l though s a t i s f ac to ry co r re l a t ions w e r e obtained between the theore t ica l and exper imenta l t es t data fo r l o w and moderate wing angles of attack , the theory of Reference 1 7 d i d not adequately predict l i f t d i s t r i b u t i o n s c l o s e t o the wing s t a l l .

In reviewing t h e a b o v e a n a l y t i c a l a t t e m p t s t o s o l v e the wing-slipstream problem, it is apparent that none of these

5

approaches i s s u i t a b l e fo r d i r e c t a p p l i c a t i o n t o the p r e s e n t problem of predict ing the effects of propeller s l i p s t r e a m on the s t a l l characteristics of s t r a i g h t wing a i r p l a n e s . Either the e x i s t i n g theoretical models are too s impl i f ied and d i s r ega rd effects which are known t o be c r i t i ca l , (e.g. Reference 6 and 13), or the ana lyses are too complex and do n o t y i e l d practical and reliable so lu t ions (e .g . Reference 9 ) . Therefore , there exis ts a requirement t o develop an improved mathematical model capable of providing practical and reliable a n a l y t i c a l s o l u t i o n s t o the wing/propeller problem.

"he analyt ical methods developed under the current program potent ia l ly re ,present an answer t o t h i s problem. Although this optimism i s based on a f e w isolated c o r r e l a t i o n s w i t h the a v a i l a b l e t es t data, s u f f i c i e n t i n d i c a t i o n o f the e f f e c t i v e n e s s of the developed methodology has already been obtained, as confirmed by compara t ive r e su l t s p re sen ted l a t e r i n the text. The basis f o r this improved mathematical model i s described be low.

- 2.3 BAS1 S FOR THE PRESENT ANALYSIS

A commm approach of p a s t i n v e s t i g a t i o n s i n v o l v e s a n i d e a l i z e d r e p r e s e n t a t i o n of the p r o p e l l e r s l i p s t r e a m i n which the v e l o c i t y i s d iscont inuous across the s l i p s t r e a m boundary. This model gene ra l ly r equ i r e s complex so lu t ions t o the boundary condi t ions associated with the d i s c o n t i n u i t y ,

The basis of the c u r r e n t a n a l y s i s l i e s i n t h e o b s e r v a t i o n tha t i n a real s l i p s t r eam the v e l o c i t y d i s t r i b u t i o n r e m a i n s continuous throughout the slipstream boundary.

An examination of experimeatal data obtained on the v e l o c i t y d i s t r i b u t i o n s i n the wakes of propel lers shows tha t there is no sudden jump i n v e l o c i t y a c r o s s the s l ip s t r eam boundary. There i s , however, a rapid i n c r e a s e i n v e l o c i t y as the boundary i s c rossed bu t the cofi t inui ty of veloci ty i s s t i l l preserved. Since the l i f t d i s t r i b u t i o n m u s t be continuous and the v e l o c i t y d i s t r i b u t i o n is cont inuous, . then the a s s o c i a t e d c i r c u l a t i o n d i s t r i b u t i o n m u s t a l s o be continuous. Therefore, the s t r e n g t h of t h e s h e d v o r t i c i t y may be obtained by d i f f e r e n t i a t i n g the s p a n w i s e d i s t r i b u t i o n o f c i r c u l a t i o n i n the usual manner, without the complication of accounting f o r d i s c o n t i n u i t i e s i n c i r c u l a t i o n .

6

The approach presented in the fo l lowing pages u t i l i ze s a comprehens ive p rope l le r ana lys i s to compute t h e s l i p s t r e a m f low f i e ld i nc lud ing s w i r l components of velocity. The wing- nacel le combinat ion is t h e n i n t r o d u c e d i n t o t h i s f l o w f i e l d and t he e f f ec t s o f t he non-un i fo rm p rope l l e r f l ow f i e ld on the wing l i f t d i s t r i b u t i o n i s computed using a modif icat ' ion o f l i f t i n g l i n e t h e o r y which permits the calculat ion of the l o w a s p e c t r a t i o e f f e c t s a s s o c i a t e d w i t h the s l ips t ream- immersed p o s i t i o n s of the wing.

The v a l i d i t y of t h e s i m p l e l i f t i n g l i n e t h e o r y u t i l i z e d he re in fo r t r ea t ing w ings w i t h non-uniform spanwise velocity d i s t r i b u t i o n s h a s b e e n v e r i f i e d by applying it t o a problem of l inear ly vary ing spanwise ve loc i ty g rad ien ts t rea ted in a more general and complex manner by Fe jer in Reference 18. The implementation of th i s l i f t i n g l i n e t h e o r y t o p r a c t i c a l wing/propeller combinations i s presented in Sec t ions 3 and 4 below.

7

SECTION 3

THEORETICAL ANALYSIS

This s e c t i o n p r e s e n t s a summary of the a n a l y t i c a l methods developed for ,p red ic t ing the p r o p e l l e r slipstream e f f e c t s on. the spanwise load d i s t r ibu t ion of wings operating a t ang le s o f a t t ack up t o s t a l l . The analyt ical a ,pproach ,presented h e r e i n i s based upon f irst determining the v e l o c i t y d i s t r i b u t i o n i n the p r o p e l l e r wake and thenca lcu la t ing i t s e f f e c t on the wing l i f t d i s t r i b u t i o n . The ana lys is ,p rovides for the use of non-linear l i f t curves for both the p rope l l e r and the winq i n o r d e r t o r e a l i s t i c a l l y r e p r e s e n t t h e p r o p e l l e r s l i p s t r e a m d i s t r i b u t i o n a n d i t s e f f e c t on wing loading a t ang le s of a t t a c k u,p t o s t a l l .

Accordingly, the first p a r t o f th i s s e c t i o n d e a l s w i t h the p rope l le r S l ips t ream ca lcu la t ions , and the second ,part pre- s e n t s the implementation of the s l ip s t r eam pa rame te r s i n the modified wing theory.

". 3,1 PROPELLER SLIPSmAM ANALYSIS

The f i r s t p a r t o f the a n a l y s i s d e a l s w i t h the p r o p e l l e r s l i p s t r eam r ep resen ta t ion , i nc lud ing the r e q u i r e d i t e r a t i v e solution and convergence procedures.

3 1.1 Genera l Propel le r - . - Solut ion ""

Consider a p r o p e l l e r o p e r a t i n g a t an ang le of a t t a c k aP t o the remote f r ees t r eam o f ve loc i ty Vo , as shown in F igu re 1. The presence of a l i f t i n g wing behind the p rope l l e r mod i f i e s the i n f l o w t o the p r o p e l l e r d i s k through an induced upwash v e l o c i t y V, . An an approximation this upwash v e l o c i t y is assumed t o be uniform across t h e p r o p e l l e r d i s k , a n d t o l i e w i t h h . the d isk p lane . The method used f o r c a l c u l a t i n g t h i s upwash v e l o c i t y i s ,presented in Sec t ion 4.2.

For the purpose of analyzing the wing l i f t d i s t r i b u t i o n it i s assumed t h a t the s l i p s t r eam can be considered as being ful ly developed. W i t h th is assumption the ave rage i nc l ina t ion of the contracted s l ipstream can be r ead i ly ca l cu la t ed u s ing

8

s imple ac tua tor d i sk theory (e.9, Reference 19) and , in the nota t ion of F igure 1, i s obtained from

where u i s the axial induced veloci ty increment a t the p r o p e l l e r d i sk . From momentum theory , u is r e l a t e d t o p r o p e l l e r t h r u s t , T, by

where VI is the r e s u l t a n t v e l o c i t y a t the d i s k and i s given by

On combining equations ( 2 ) and ( 3 ) a q u a r t i c i n u is obtained and i s genera l ly so lved by i t e r a t i o n . However, s i n c e the p r e s e n t a p p l i c a t i o n i s to convent iona l a i rc raf t where (I p is small and V, 5 Vo this q u a r t i c may be reduced to a q u a d r a t i c whose s o l u t i o n i s

With the above value of u ,equat ion (1) can be solved t o y i e l d the mean s l i p s t r e a m i n c l i n a t i o n as , r e l a t i v e t o the freestream.

To ob ta in the detailed v e l o c i t y d i s t r i b u t i o n w i t h i n the i n c l i n e d s l i p s t r e a m it i s assumed t h a t , t o a good approximation, this can be o b t a i n e d d i r e c t l y from the s o l u t i o n fo r an i so l a t ed p rope l l e r ope ra t ing i n ax i a l f l ow a t speed

The c a l c u l a t i o n of non-uniform sl ipstream velocity

LO

d i s t r ibu t ions beh ind a p rope l l e r o f a rb i t r a ry geomet ry is based upon es tabl ished blade element-momentum theory as presented in Reference 20. While the s o l u t i o n t o the genera l theory is very complex, a r e l a t i v e l y simple and practical s o l u t i o n i s obtained on the assumptions that the r o t a t i o n a l e n e r g y i n the slipstream is small compared t o the axia l energy and that the radial v a r i a t i o n o f static p res su re i n the slipstream can be .neglec ted .

Standard blade element-momentum theory assumes that the f l o w i s both incompressible and inviscid. Thus the f l o w i n annu la r stream tube e lements , i s treated in an independent manner. For any annular stream tube element, the slipstream v e l o c i t i e s are related t o b o t h the i nduced ve loc i t i e s a t the p r o p e l l e r d i sk and the r a d i u s i n the f u l l y c o n t r a c t e d slipstream,

Following the a n a l y s i s of Reference 20, the induced axial and r o t a t i o n a l v e l o c i t y components, u , 1/2wr, a t any rad ius , T, i n the propeller disk can be obta ined by a n i t e r a t i v e s o l u t i o n t o the equa t ions

and

where, from Figure 2 , the inflow angle, +, is given by

and the blade s e c t i o n list and drag coef f ic ien ts are known i n terns of ang le o f attack, a, given by

a = p - + (9 )

Figure 2. Blade Element Velocity Diagram

12

To accoun t fo r t he s ign i f i can t and w e l l known loss of l i f t toward the b l a d e t i p e q u a t i o n s ( 6 ) and ( 7 ) are r e w r i t t e n i n the f o r m

i

and

where F is the t ip - los s co r rec t ion f ac to r g iven by Lock i n . Reference 21 and u*, 1/2(w*r) are modif ied values of the induced v e l o c i t y components.

Equations (10) and (11) yield improved values of the inf low angle C#I and sec t ion characteristics C$ and

Cd , as a f f e c t e d by the t i p - l o s s c o r r e c t i o n f a c t o r F . These va lues are then used t o ob ta in 'a better approximation for slipstream-induced velocity components u and w r / 2 , using equat ions (6) and ( 7 ) . 3.1.2 I n i t i a l C a l c u l a t i o n o f I n f l o w Anqle

The i t e r a t i v e s o l u t i o n f o r the system of equat ions (8) through (11) be obta ined . t o t a l t h r u s t is gene ra l ly s a t i s f a c t o r y

r e q u i r e s that an i n i t i a l app rox ima te va lue of 4 Equation (4) can be used i f the propeller

i s known. However, s i n c e the p r o p e l l e r t h r u s t n o t known in advance, a method that y i e l d s a s t a r t i n g v a l u e for $ is developed as follows:

From Figure 2 the inf low angle C#I can be expressed a s

+ = U*

where the resu l tan t induced ve loc i ty increment is assumed to- be normal t o the l o c a l b l a d e v e l o c i t y .

On making the assumption that cd <, c&' and %*I equat ion (11) can be w r i t t e n as

13

By solving equat ion (13) f o r u * / a r and subs t i t u t ing this va lue i n equa t ion (12) a n i n i t i a l v a l u e f o r + is obtained. However, the r i g h t hand side of equation (13) must first be reduced to a tractable form. This i s accomplished by applying the fo l lowing re la t ionships :

(a) A l i n e a r i z e d e x p r e s s i o n f o r the blade s e c t i o n l i f t curve, g iven by

cL~ = oo ( Q -Q.) (14)

where a 0 is a r e p r e s e n t a t i v e l i f t s lope Q is given by equat ion (9 ) , and Q~ i s the angle of attack a t z e r o l i f t .

(b) An express ion for V, obtained from Figure 2 as ,

v = J v c 2 + ( s l r ) 2

(c) P r a n d t l ' s e x p r e s s i o n f o r the t i p loss f a c t o r F obtained from Reference 2 1 as P

Fp = 7 c o s ' [exp{-+ ( , - + ) J q y } ] where B is the number of blades

Combining equat ions (12) through (15) and sub- s t i t u t i n g f o r the t i p loss fac tor , Fp, given by equat ion (16) leads t o the fol lowing expression:

from which the s o l a t i o n f o r "*/a, i s obtained as

14

* VQ (18)

nr u= 112 [ . (~+x)2+. . (p -+o- .o ) - (z.x)]

where

I

3.1.3 Convergence of the I t e r a t i v e P r o p e l l e r S o l u t i o n

The i t e r a t i v e s o l u t i o n t o e q u a t i o n s (8) through (11) is n a t u r a l l y d i v e r g e n t w i t h i n the normal- range of the b l a d e s e c t i o n l i f t curves. Therefore, convergence of the so lu t ion mus t be forced by applying a c o r r e c t i o n t o each new computed va lue of #I . A comect ion procedure which yields rapid convergence i s derived by the method presented below

L e t the e x a c t s o l u t i o n f o r i n f l o w a n g l e , $I , be expressed as

where i s the value used as i n p u t t o the nth i t e r a t i o n a n d SI is a small unknown increment.

I n the gene ra l i t e r a t ion p rocedure , is first used i n e q u a t i o n (9) t o o b t a i n a v a l u e of Q from which C$ and Cd may be determined knowing the b l a d e a i r f o i l s e c t i o n characteristics. Next, equat ions (10) and (11) are u s e d t o s o l v e for u * and u* and these va lues are t h e n s u b s t i t u t e d i n equat ion (8) t o o b t a i n a new va lue of inf low angle , denoted by + I 1 . It is this new va lue of in f low ang le w h i c h must be co r rec t ed be fo re p roceed ing t o the n + l t h i t e r a t i o n .

m e r e f o r e , l e t the exact s o l u t i o n for inf low angle , $I also be expressed as

15

where 82 is a second small unknown increment.

Combining equat ions (19) a n d ( 2 0 ) , t o e l i m i n a t e + , y i e l d s

Subs t i tu t ing equat ion ( 2 1 ) in to equat ion (19), there fol lows:

Equation (22) forms the basis of a method f o r c a l c u l a t i n g an improved value of inf low angle for input to the n e x t i t e r a t i o n c y c l e by using the guessed and calculated values from the previous cyc le . The r a t i o , 82/81, remains to be determined from an approximate error analysis in the following manner:

From equat ion (20 ) the va lue of tan 4 is expressed t o first o r d e r i n S2 by

From equat ions (9 ) and , 419) the exact s o l u t i o n f o r blade l i f t c o e f f i c i e n t , - "Cd , is expres sed i n terms of the va lue C& c a l c u l a t e d i n the nth i t e r a t i o n c y c l e from

where a, is a mean va lue of l i f t - c u r v e s l o p e . Equation (10) w r i t t e n i n terms of v a l u e s f o r the exact

16

s o l u t i o n b u t w i t h the assumption that cd < - c.l 8 reduces to

Subs t i t u t ing equa t ions (19) aril (24) i n t o equat ion (25) , and re ta ining only f i rs t o rde r terms i n 8 , , there f 01 lows :

where

and where small changes i n the t i p loss f a c t o r are neglec ted .

S imi la r ly , equa t ion (11) reduces to

k y = k: [ I - 8, (s + cot $91 w h e r e

Now, using equat ions (25) and ( 2 7 ) , equat ion (8) can be expressed as

tan # = p (1 + kx) + k y

where p i s the local forward speed ra t io ( V a / n r ) . . .

Rewri t ing equat ion (29 ) i n terms of the va lues k,,ky I I

c a l c u l a t e d i n the n t h i t e r a t i o n c y c l e y i e l d s I

tan # ' I = p ( I t k:) + ky I

Combining equations (29 ) and (30 ) l eads t o the following r e l a t i o n s h i p

F ina l ly u s ing equa t ions (23 ) , (26 ) , (28 ) and (31) a solution f o r the r a t i o 8,/8, i s obtained as fol lows:

Equation ( 3 2 ) thus provides the e s s e n t i a l r e l a t i o n s h i p by which equat ion ( 2 2 ) i s appl ied to ob ta in an improved va lue of i n f l o w a n g l e f o r i n p u t t o the n e x t i t e r a t i o n c y c l e . I n p r a c t i c e , t h e i t e r a t i o n p r o c e d u r e i s terminated when the d i f f e r e n c e . (r$'-#) for e a c h s u c c e s s i v e i t e r a t i o n c y c l e h a s converged to within a prescr ibed margin of error.

3.1.4 Analysis fo r S l ip s t r eam Ve loc i ty D i s t r ibu t ions ~ ~ ~~ ~- ""

Upon reaching a converged so lu t ion for the inf low angle # , the f i n a l v a l u e s of (#I , CIQ and cd are then sub-

s t i t u t e d i n e q u a t i o n ( 6 ) and ( 7 ) t o 'solve f o r t h e t r u e i n - duced veloci ty components in the p r o p e l l e r d i s k plane,^ and 1/2 w r .

The local a x i a l v e l o c i t y component VSa i n the f u l l y developed s l ipstream i s obtained from Figure 1 as

vo COS ap + 2u Vsa =

18

( 3 3 )

The local r o t a t i o n a l v e l o c i t y component, Vst, i n the ful ly developed slipstream i s obtained from conservation of angular momentum and i s given by

vst = w r (+-) ( 3 4 )

where rs is the local r a d i u s i n the s l i p s t r e a m f o r the streamtube element which has a local r a d i u s r i n the propeller disk plane.

The local r a d i u s rs f o r each flow element in the s l ip s t r eam is der ived from a s i m p l i f i e d a p p l i c a t i o n o f the cont inui ty express ion to success ive s t reamtube e lements . For the nth blade e lement s ta t ion a t r a d i u s rn the corresponding radius rsn i n the s l i p s t r e a m is given by

where, i n the notation of .Figure 3 , r l and r ~ l are the va lues of hub and nace l le rad ius , res ,pc t ive ly .

,' The value of rsn given by equat ion (35) is based upon r ep resen t ing the s l i p s t r e a m by a series of concen t r i c annular streamtubes w i t h uniform velocity between each element s t a t i o n . This s o l u t i o n , while approximate, i s found to be more than adequate for a l l reasonable var ia t ions between u r l and Un .

19

r

-t

k E

R l

k E

\'

!i

~-

Propeller and Nacelle Axis

Figure 3 . Analytical Model For Slipstream Contraction.

- 3.2 ." . " WING-IN-SLIPSTREAM - - ANALYSIS

I n the second pa r t of the a n a l y s i s a modified form of l i f t i n g l i n e t h e o r y is presented w h i c h uses the nonuniform s l ip s t r eam ve loc i ty d i s t r ibu t ion , a sde te rmined above, to c a l c u l a t e the l i f t d i s t r i b u t i o n on wings w i t h p r o p e l l e r s . The approadh presented re l ies on the use of a simple phys ica l model to ob ta in a s o l u t i o n for wing-in-slipstream l o a d i n g s f o r a wide range of real a i r c ra f t p rope l l e r -wing combinations.

F i r s t the method is ,presented for an unflapped wing immersed in one or more non-overlapping slipstreams. Follow- ing this, the modi f ica t ions requi red to i n c l u d e f l a p s are descr ibed . F ina l ly , an ex tens ion of the a n a l y s i s t o inc lude l o w aspect-ratio propeller-wing combinations i s discussed.

3.2.1 Analys is for a Winq w i t h no Flaps or w i t h Full-Span Deflected Flaps

Consider the basic case of a wing i n a uniform stream of v e l o c i t y , vo . If the l o c a l c i r c u l a t i o n is ro , then the span l oad d i s t r ibu t ion a t any spanwise s ta t ion i s given by

90 = P vo ro The supe rpos i t i on of a propel le r s l ips t ream f low g ives rise t o a n i n c r e a s e d l o c a l v e l o c i t y vd and an i nc reased c i r cu la t ion r; , which can be expressed, r e s p e c t i v e l y , as

v i = vo -t n v (37)

where n V and Ar are the incremental changes in local ve loc i ty and c i r cu la t ion , r e spec t ive ly , due t o p r o p e l l e r slip- strew. Now the corresponding spanwise load dis t r ibut ion for the b a s i c wing immersed i n the p r o p e l l e r slipstream can be w r i t t e n as

1 = v,' rsl (39) 21

Subs t i tu t ing equat ions (37) and (38) in to equat ion . (39) . .y ie lds : a genera l express ion for the spanwise load d i s t r i b u t i o n of , . , -. a wing immersed i n the p r o p e l l e r s l i p s t r e a m , as follows:

where r2 E A r i s the change i n wing c i r c u l a t i o n due . t o . . . p r o p e l l e r slipstream a lone .

I t can be noted from equat ion (40) t ha t the first com- , .

ponent dl of the t o t a l l i f t d i s t r i b u t i o n i s that which would .

be obtained i f the local ve loc i ty i nc reased while the cir- c u l a t i o n ro remained unchanged. If the c i r c u l a t i o n is unchanged, then there i s no change i n the t r a i l i n g v o r t i c i t y ' and therefore no change in the wing downwash f i e ld . The second t e r m r e p r e s e n t s the change in spanwise l i f t d i s t r i b u t i o n due t o the c i r c u l a t i o n I-2 and is t h e r e f o r e associated w i t h wing downwash changes caused by the p r o p e l l e r s l i p s t r e a m .

, .

The problem o f a wing immersed i n the p r o p e l l e r s l i p s t r e a m i s now reduced t o proper determination of local va lues for the r e s u l t a n t v e l o c i t y V,' and the c i r c u l a t i o n r2 f o r the e n t i r e wing. This a n a l y s i s i s developed below.

For typ ica l p rope l le r /wing conf igura t ions , the r e s u l t a n t l o c a l v e l o c i t y V,' can be equated (within the small ang le assumpt ion) to the combined freestream and s l ips t ream component along the wing s e c t i o n z e r o - l i f t l i n e , t h u s

v, s= v:

22

Using the nomenclature of Figure 4, t h i s v e l o c i t y component can be expressed as

Also, the corresponding component of the to ta l f low normal to the w i n g s e c t i o n z e r o - l i f t l i n e is given by

I

The q u a n t i t i t e s "ha and Vst i n the above equat ions represent the ax ia l and s w i r k veloci ty components of the combined freestrean and ' s l ips t ream f lob and are given by equat ions (33) and (34 ) r e spec t ive ly . AlsD, t h e a n g l e s a s and Q e are known q u a n t i t i e s which r e p r e s e n t i n c l i n a t i o n s o f the s l i p s t r e a m and t h e z e r o - l i f t l i n e r e l a t i v e t o the r emote f r ees t r eam ve loc i ty , r e spec t ive ly .

The e x t r a J , i n g c i r c u l a t i o n r2 , caused by the a c t i o n of the propel ler s , l ipstream, i s determined by equa t ing t he resul t ing change i p wing u,pwash t o the downwash change a s s o c i a t e d w i t h . This upwash change, in non-dimensional form, i s def ined as

Subs t i t u t ing equa t ion (43 ) i n to equa t ion (44 ) y i e lds t h e extra dpwash due to t h e s l i p s t r e a m as

v = - sin (as + Qe) + - COS 6 s -t Qe) -sin Qe (45) VO VO

I n o r d e r t o s a t i s f y the wing boundary condition of no f l o w through the s u r f a c e , this e x t r a upwash or crossflow must be balanced by the combined in f luence of the e x t r a bound v o r t i c i t y , l-2 , and the a s s o c i a t e d streamwise ( i . e . chordwise and t r a i l i n g ) v o r t i c i t y , - d r2

dY dY.

23

Figure 4 . Notat ion for Wing-in-Slipstream Model

24

Observat ion of the lift on wings with s l ipstreams shows t h a t t h e major p o r t i o n of the ex t ra loading caused by the slipstream i s concentrated on and near that p o r t i o n a c t u a l l y immersed i n the s l ip s t r eam and , fo r mos t con f igu ra t ions , the "aspec t ratio" of this immersed po r t ion i s small', usua l ly about 1.0. I t is h e r e i n p o s t u l a t e d t h a t t h e d i s t r i b u t i o n of extra vor t i c i ty caused by the s l i p s t r e a m e x h i b i t s the c h a r a c t e r i s t i c s of that found on l o w aspec t - ra t io wings . Kuchemann, in Reference 22, shows that f o r 'low aspect-rat io wings only the chordwise and t ra i l ing v o r t i c e s are r e q u i r e d t o f u l f i l l the boundary conditions. In fact, as the a s p e c t r a t i o t e n d s t o z e r o , the t r a i l i n g v o r t i c e s c a n c e l h a l f the upwash, and the chordwise vort ices cancel the remainder.

I n the p r e s e n t c a s e , 'the n e t e x t r a upwash is Vov and t h e downwash change due t o t h e . t r a i l i n g v o r t i c e s a s s o c i a t e d wi th the e x t r a c i r c u l a t i o n , r 2 , i s given b y

J - b/2 Y , - Y

Therefore, from the above considerat ions it f o l l o w s t h a t

wj2 = 1/2 vov (47 1

and hence

Equation (48) i s supported by the ana lys i s o f Reference 2 3 , which deals with the determinat ion of the l i f t on a wing pass ing close t o a l i n e vortex. This r e f e r e n c e s t a t e s t h a t l i f t i n g l i n e t h e o r y a lways overes t imates the l i f t induced by r a p i d l y changing upwash f i e lds (such as f rom p rope l l e r s and l i ne vo r t i ce s ) by a f a c t o r of 2 , due t o a corresponding underestimation of the wing downwash.

Now, express ing the c i r c u l a t i o n d i s t r i b u t i o n , r2, as a Four i e r s ine series i n t e r m s of the wing spanwise angular coord ina te , 8 = cos"l (2y) y i e l d s

2 5

Combining equat ions (48) w i t h (49) and performing the requ i r ed ma themat i ca l ope ra t ions , t he Four i e r coe f f i c i en t s

Bn can be expressed as

By l i m i t i n g t h e s e r i e s f o r r2 t o r-l terms, equat ion (49 ) be come s

r- I r Z m = 112 b V o x Bn sin n - m r r

r n=l

and equation (50) i s reduced t o t h e summation

m = l

From equat ion (40) t h e l i f t d 2 a s s o c i a t e d w i t h t h e s l i p s t r e a m may be ex.pressed a s

where the l i f t c o e f f i c i e n t i s based on V, . Therefore

26

Comparing equat ions (51 and (54) there resul ts

If the r e l a t i o n s h i p f o r the c o e f f i c i e n t s , 8, is sub- s t i t u t e d i n t o e q u a t i o n ( 5 5 ) , t h e l i f t d i s t r i b u t i o n a s s o c i a t e d w i t h the slipstream i s o b t a i n e d i n t h e form

(56) Having determined the l i f t a s s o c i a t e d w i t h the s l i p s t r e a m upwash the o v e r a l l wing l i f t is c a l c u l a t e d as fol lows.

L e t ail and ai2 be the induced angles of attached assoc- i a t ed w i th the l i f t d i s t r i b u t i o n s dl and 4 r e s p e c t i v e l y , as given by equation ( 4 0 ) . Then the to ta l induced angle o f attack a t any po in t k i s given by

Also, r e - e q r e s s i n g e q u a t i o n (40) i n terms of the lift coe f f i c - ients based on V, y i e l d s

using the m u l t i p l i e r s P m k from Reference 1 i n equa t ion (58 ) there fol lows

Now, by d e f i n i t i o n

27

Adding Qi2k t o b o t h s i d e s of equation (59). and using equations ( 5 7 ) and (60) l e a d s t o

I - ... rn = I - I I ...

Equation (61) g ives t he t o t a l i nduced ang le of a t t a c k i n terms of, . the unknown wing l i f t d i s t r i b u t i o n , CdC/b, t he known induced angle o f a t tack , Qi2’Vo V/VS , and t h e known s l i p - stream l i f t d i s t r i b u t i o n , c,& c / b , given by equat ion (56) . The s o l u t i o n is obtained by i t e r a t i o n a s f o l l o w s .

&I approximation t o the o v e r a l l l i f t d i s t r i b u t i o n i s ca lcu la ted and equat ion (61) i s used t o o b t a i n a f i r s t approximat ion to the induced angle o f a t tack . The e f f e c t i v e a n g l e o f a t t a c k a t t h e wing s e c t i o n i s obtained from

Where Qgk is the sec t ion geomet r i c ang le o f a t t ack , add( , i s t h e s e c t i o n z e r o - l i f t a n g l e a n d E is the edge-ve loc i ty fac tor o f Reference (1) . This va lue o f e f fec t ive angle o f a t tack i s then used w i t h the two-dimensional section l i f t c u r v e s a t t h e e f f ec t ive s ec t ion Reyno lds number Rek t o o b t a i n t h e l i f t c o e f f - i c i e n t CL . The va lue o f C d d b t h u s c a l c u l a t e d i s compared t o the i n i t i a l a p p r o x i m a t i o n a n d , i f s u f f i c i e n t a g r e e m e n t is n o t o b t a i n e d , a new va lue i s computed using the method given in Reference 1. This i t e r a t i o n p r o c e s s i s t h e n r e p e a t e d u n t i l guessed and ca lcu la ted va lues agree to w i t h i n p re sc r ibed to l e rance .

I t should be no ted tha t the above ana lys i s i s a l s o a p p l i c a b l e to a wing wi th fu l l - span de f l ec t ed f l aps , p rov ided that a p p r o p r i a t e a i r f o i l c h a r a c t e r i s t i c s are employed f o r wing s e c t i o n s w i t h f l a p s .

28

3.2.2 Analysis ~~ . f o r a Winq w i t h Part-Span Deflected Flaps

The d e f l e c t i o n of a par t -span f lap causes a d i scon t i - n u i t y 8 i n the d i s t r i b u t i o n of a b s o l u t e a n g l e of attack a t the end of the flap, and produces a corresponding discont inui ty i n the slipstream-induced crossflow. The effect of these dis- c o n t i n u i t i e s on the span load d i s t r i b u t i o n i s treated below.

The a n a l y s i s is developed for a wing having a deflected part-span f lap extending f rom y=-b/2 t o y = y * . The most general case is that of a f lap whose end l ies wi th in the slipstream, as i l l u s t r a t e d i n F i g u r e 5.

Following the preceding t reatment of a wing w i t h no f l a p s or w i t h fu l l - span deflected f l a p s , the t o t a l wing l i f t distri- but ion given by equat ion (40) can be d iv ided i n to two po r t ions and can be expressed i n non-dimensional form a s

where C & . c / b is the l i f t d i s t r i b u t i o n a s s o c i a t e d w i t h s l i p - . stream-induced upwash and CJ, .c/b i s the remainder of the d i s t r i b u t i o n .

I n the p r e s e n t case, however, the slipstream-induced upwash VO v , given by equat ion (48), is discont inuous a t t h e end of the f l a p a s shown i n F igu re 5. The n e t d i s c o n t i n u i t y i n c r o s s f l o w a t the edge of the flap, y = y * , can be obta ined from equat ion (45) by consider ing the upwash on both sides of the f lap end. Thus, using equation (45) for the flapped side of the wing, a t y = y*-o , this e x t r a upwash can be expressed a s fo l lows

vo.v = vS% sin (a,*+ a$ +8) + vtt. cos (Y "0 ) (a*, +.:,ss, ( 4)

- V, sin ( ~ g + 8)

29

I

Propeller Slipstream Diameter \ / Diameter

? . . ,

-I-

T I

Distxibution (a) = (b) + (c)

Fig.ure 5. Mathematical Representation of Flap Discontinuity

30

Simi la r ly , for the unfla,pped side a t y = y + o , the crossf low is given by

*

The n e t d i s c o n t i n u i t y i n c r o s s f l o w is obtained as the d i f f e rence between equat ions (64) and (65), thus

vo v (y "0) - vo v(y*+o) = V, A v * (Ci6)

Because of the d i scon t inu i ty i n c ros s f low, Vo nv * , given by equat ion (66), the s o l u t i o n f o r the l i f t d i s t r i b u t i o n

Cd2 can n o t be obta ined from a simple Fourier ser ies f o r

bution VO v i s s p l i t i n t o two por t ions , one a continuous dis- t r i b u t i o n V,.vI and the o t h e r a s t e p f u n c t i o n d i s t r i b u t i o n VO.vl1 8 where -

r2, as w a s poss ib l e i n equa t ion (48) . Therefore, the distri-

Now, it i s necessary t o relate ~e v e l o c i t y d i s t r i b u t i o n s given by equat ion (67) t o their cor responding c i rcu la t ion d i s t r i b u t i o n s . S i n c e r 2 is the t o t a l c i r cu la t ion co r re spond ing t o Vo v , as given by equat ion (48) , it can a l so be mlit i n t o two d i s t r i b u t i o n s , i .e. , I-2' , corresponding to V, V I and r2" corresponding to V, v l l , w h e r e

31

r2 = r2' + r2 II

Thusr using equations (48) and ( 6 8 ) , the v e l o c i t y d i s t r i b u t i o n s g i v e n by equat ion (67)can now be re-expressed i n terms of the cor re spond ing c i r cu la t ion d i s t r ibu t ions

and r2I1 as fol lows :

v,v = v,v + VOV 1

where

and

Now the problem reduces to determining r2' and r2I1 and the corresponding l i f t d i s t r i b u t i o n Cd2. c/b . This i s accomplished as outl ined below.

32

! '/ I .

Since V 0 . v ' i s continuous then, following the a n a l y s i s of subsec t ion 32.1, r2' can be expressed as the simple Fourier series

r- I

rzl = - b Vo 1 8, sin n e I 2

n = I

where the c o e f f i c i e n t s Bn are given by:

r- I

B, = - 1 ( v - v i l ) sin - sin n - 4 m l r m l r n.r r r

m = I m ( 7 3 )

The r e l a t i o n s h i p f o r r2" is obtained from t h e a n a l y s i s of Reference 24 i n the form

The d i s t r i b u t i o n r2I' given by equat ion (74) satisfies the r e q u i r e d d i s c o n t i n u i t y i n the crossf low, Vo.v1l = VO. OV i n e q u a t i o n ( 7 1 ) .

x

The c i r c u l a t i o n d i s t r i b u t i o n s r2I and r2I' , determined i n e q u a t i o n s ( 7 2 ) and (74) respect ively, can now be used to ob ta in . the corresponding l i f t d i s t r i b u t i o n C d 2 c/b assoc ia t ed w i t h the slipstream-induced upwash. This i s accomplished by rearranging equat ion (54) and using equat ion (68), thus

3 3

I . I . , ...-... .... .”._. . ”””-.-

F i n a l l y , s u b s t i t u t i n g e q u a t i o n s ( 7 2 ) , (73) and (74) i n t o equaiiion (75) y i e l d s the l i f t d i s t r i b u t i o n CJ2 c / b a t any point k on the wing, in the form

The above ana lys i s g ives the s o l u t i o n f o r t h e c a s e where the f l ap ex tends from t h e l e f t wing t i p t o a p o i n t Y = Y on the*r ight wipg . ?he s o l u t i o n f o r a f lap extending between

obta ined by supe rpos i t i on of s o l u t i o n s as shown in Figure 6 .

*

- Y S Y l Y , or any o ther combina t ion of f lap pos i t ians , i s

I t should be no ted t ha t equa t ion ( 7 6 ) represents on ly one p a r t o f the s o l u t i o n f o r the t o t a l l i f t d i s t r i b u t i o n C d . c / b as g iven i n equa t ion (63 ) . I t is now necessary to obta in an a p p r o p r i a t e s o l u t i o n f o r the d i s t r i b u t i o n CJI c/b . This is accomplished as outl ined below.

The d i s c o n t i n u i t y 8 i n a b s o l u t e a n g l e o f attack caused by the f l a p d e f l e c t i o n a l s o a f f e c t s t h e d i s t r i b u t i o n C d , d b , This dis t r ibu t ion , , a l though cont inuous , p o s s e s s e s a n i n f i n i t e d e r i v a t i v e a t y = y . Therefore , the m u l t i p l i e r s Pmk developed i n Reference 1 can no t be used d i r ec t ly t o ob ta in t he i nduced ang le o f a t t ack due t o t h i s d i s t r i b u t i o n . This r e s t r i c t i o n i s removed by the fo l lowing ana ly t ica l approach .

The d i s t r i b u t i o n C,J$ c/b can a l s o b e d i v i d e d i n t o two p o r t i o n s , t h u s I

where c$’ c / b s i s the l i f t d i s t r i b u t i o n due t o a u n i t I

34

Propeller S1 i p s tream .er

I

Distribution (a) = (b) + (c) + (d ) + (e)

Figure 6. Method for Superposition ,of S o l u t i o n s

35

I I I I I I II I Ill 11111 II 111111 I I I 'I I 'I I I I I I I I II I I I II I I

d i s c o n t i n u i t y 8 , and Cd,' c/b , is the remainder.

Applying the m u l t i p l i e r s P m k t o e q u a t i o n ( 7 7 ) y i e l d s the fol lowing

/

A l s o , applying the m u l t i p l e r s Prnk t o equat ion (63) y i e l d s the t o t a l l i f t d i s t r i b u t i o n C d c/b as

Subs t i t u t ing equa t ion ( 7 8 ) i n t o e q u a t i o n ( 7 9 ) , y i e l d s \

rn = I "

I n order t o o b t a i n the i t e r a t i v e s o l u t i o n t o equat ion ( 8 0 ) , it is now necessary t o relate the lift d i s t r i b u t i o n s g i v e n i n equat ions (63) and ( 7 7 ) t o the i r corresponding induced angle of a t t a c k d i s t r i b u t i o n s . If Q i is the induced angle of a t t a c k d i s t r i b u t i o n c o r r r e s p o n d i n g t o t h e t o t a l l i f t d i s t r i b u t i o n ,

C& d b and Qii and Q i 2 are the induced angle of a t t a c k d is t r ibu t ions cor responding t o t h e l i f t components C d l c /b and

C & c / b then from equation ( 6 3 ) there fol lows

A l s o , applying similar cons ide ra t ions t o equat ion ( 7 7 ) theze

36

r e s u l t s

Subs t i tu t ing equat ion (82) i n to equa t ion (81) y i e l d s the r e l a t i o n s h i p f o r the induced t o t a l ang le of a t t a c k as follows:

where ail' = 8 over t he f l ap span o o u t s i d e the f l a p span

I t can be no ted i n equa t ion ( 8 3 ) that the induced ,ang le of attack d i s t r i b u t i o n ail' must be cont inuous, s ince i ts corresponding l i f t d i s t r i b u t i o n Cdl' c/b as g iven i n equa t ion ( 7 7 ) i s continuous. Therefore, this induced angle of a t t a c k d i s t r i b u t i o n i s obta ined d i r e c t l y u s ing t he mu l t ip l i e r s , t hus

The above re la t ionship can be re-expressed in terms o f t h e t o t a l induced angle of a t t a c k d i s t r i b u t i o n ai by using equat ion (83), t hus

Now, equat ion ( 8 5 ) can be s u b s t i t u t e d i n t o e q u a t i o n (80) t o e l imina te the C$/ c/b d i s t r i b u t i o n a n d t o y i e l d the t o t a l l i f t d i s t r i b u t i o n CJ c/b i n t h e d e s i r e d m u l t i p l i e r form as follows

37

I

Fina l ly , r ea r r ang ing the equat ion (86), the t o t a l ' i n d u c e d angle of at tack a t any po in t k on the wing can be related t o _. t he co r re spond ing t o t a l l i f t d i s t r i b u t i o n a n d the know d i s t r i b u - t i o n s of induced angle of attack ailt1 and Qi2 and their corresponding l i f t d i s t r i b u t i o n s Cd,llc/b and Cd2 c/b , respec t - t i v e l y . The r e s u l t i n g r e l a t i o n , s h i p i s \

> I

.. .

.._ .:

. . ..

where from the a n a l y s i s of Reference 1

and Cd2c/bhas been a l ready determined in equat ion ( 7 6 ) . 1

Equation ( 8 7 ) i s ana logous to equa t ion (61) deve loped in subsec t ion 3.2.1 for no f l a p d e f l e c t i o n . This equat ion is a l s o solved by an i t e r a t ion p rocedure , similar t o that used for so1vi .q equation .(.61). Thus, upon obta in ing the required convergence of t h e i t e r a t i v e s o l u t i o n , e q u a t i o n ( 8 7 ) y i e l d s the t o t a l lift d i s t r i b u t i o n C$ c/b f o r a wing w l t h a deflected f l a p w i t h i n the p rope l l e r s l i p s t r eam. . .

38

3.2,3 Extension of the Winq Ana lys i s t o Small Aspect Rat ios

To prov ide added f l ex ib i l i t y t o the methodology developed herein, the wing a n a l y s i s t r e a t e d i n S e c t i o n s 3.2.1 and 3.2.2 is extended to include wings of s m a l l a s p e c t r a t i o . This a n a l y s i s is p a r t i c u l a r l y u s e f u l f o r the cu r ren t app l i ca - t i o n , s i n c e much o f t h e a v a i l a b l e t es t data on spanwise load- i n g s for w i n g s i n s l i p s t r e a m f a l l s w i t h i n t h e low a s p e c t r a t i o range. The c o r r e l a t i o n s o f this extended ana lys i s w i t h the corresponding t e s t data where appropriate i s shown i n S e c t i o n 5.0.

The modif ica t ion o f t h e p r e s e n t a n a l y s i s t o s m a l l a s p e c t r a t io wings i s based on t h e wing theory of Kuchemann (Reference 22), as outlined below.

In equa t ions (61) and ( 8 7 ) a se t o f m u l t i p l i e r s w a s u s e d t o o b t a i n the induced angle of at tack d i s t r i b u t i o n s for a wing wi th no f l a p s and w i t h par t -span deflected f l a p s , r e s p e c t i v e l y . These multipliers were obtained from the fund- amental equation of the h i g h - a s p e c t - r a t i o , l i f t i n g - l i n e t h e o r y which expresses the induced angle of a t tack in terms of the span loading,

b/2 d (CJ c/b)

d Y l Y, - Y

dy I Qi -

- b/2

Kuchemann, Reference 22 , h a s shown t h a t t h i s e q u a t i o n may be gene ra l i zed t o w ings o f any a spec t r a t io by w r i t i n g

where w' i s a f a c t o r which varies between 1 f o r h i g h a s p e c t r a t i o ( A R + a ) , and 2 f o r low aspect ra t io(AR+O) .

39

Kuchemann obtained the fol lowing equat ion for ‘w’ . ’ ’ . - .

I 4 - 1/4 0 = 2 - [ I +x2]

If the m u l t i p l i e r s P m k are reder ived us ing equat ion (91), a new set of m u l t i p l i e r s , Phk , is o b t a i n e d r e l a t e d t o the o l d se t by

I P m k = Prnk (92)

I

This new set of m u l t i p l i e r s P m k may then be used t o calc .-

l a t e induced angle of attack throughout the e n t i r e aspect i d t i o range.

The second equation that must be mod i f i ed is tha t which d e f i n e s the edge-velocity factor E. In Reference 1, this q u a n t i t y is given by

= J I + G 2 4 ( 9 3 )

where a. is the two-dimensional l i f t curve s lope (AR-oo)and a i s the corresponding value fo r f i n i t e aspect ratio.

Reference 22 p re sen t s an expres s ion for the r a t i o of the l i f t curve slopes as

2 - 7r w’ cot (+) “ a0 - (94)

where w ’ is given by equat ion (91 ) .

40

Thus , subs t i tu t ing equat ion (94) in to equat ion (93) y i e l d s the edge-ve loc i ty fac tor E. a p p l i c a b l e t o a l l va lues of a s p e c t r a t i o as

I

2 - T W I cot(?) E =

2 w' (95)

I n the extended ana lys i s equa t ion (95) is used i n place of equat ion (93) .

F i n a l l y , t h e e x p r e s s i o n f o r t h e , l i f t d i s t r i b u t i o n a s s o - ciated w i t h a d i scon t inu i ty i n i nduced ang le o f a t t ack , as given by ,equation (88),must be mod i f i ed i n the following form

Equation (96) i s now a p p l i c a b l e t o any value of aspect ratio. This equat ion i s implemented i n the computer program and extends the program c a p a b i l i t i e s t o wings w i t h low and high a s p e c t r a t i o s ranging from about 2.0 t o i n f i n i t y .

41

SECTION 4

DIGITAL COMPUTER P R O G R m

The theo re t i ca l ana lys i s p re sen ted i n Sec t ion 3 w a s programmed f o r u s e on the CDC 6600 series d i g i t a l computer. This was accomplished by extensively modifying the computer

.program of Reference 1 t o i n c l u d e the p r o p e l l e r s l i p s t r e a m and the wing in-s l ipstream analysis .

This s e c t i o n p r e s e n t s a d e s c r i p t i o n o f t h e combined computer program logic, the selection and assembly of the p e r t i n e n t a i r f o i l s e c t i o n c h a r a c t e r i s t i c s , and a sample computer output. Wherever appropriate, the d i scuss ion is directed towards those features of the modif ied program that are d i r e c t l y r e l e v a n t t o t h e t r e a t m e n t o f t h e p r o p e l l e r s l i p - stream and i t s e f f e c t on t h e wing spanwise loading. Addition- a l information per ta ining to computat ions of the wing loading f o r a basic wing/fuselage combination can be obtained from Reference 1 . 4 1 PROPELLER SLIPSTREAM COMPUTATION-S

This subsect ion presents the methodology and the a s s o c i a t e d a i r fo i l s ec t ion da t a u sed i n computa t ions o f the p r o p e l l e r s l i p s t r e a m v e l o c i t y d i s t r i b u t i o n s , w h i c h a r e la ter implemented i n t h e o v e r a l l s o l u t i o n f o r t h e wing spanwise loading of a general wing/propeller combination. The basic computat ional s t eps fo r imp lemen t ing t he s l i p s t r eam ve loc i ty d i s t r ibu t ions i n t o the wing ana lys i s are summarized in subsec t ion 4.2

4.1.1 Computat ional Procedures for Propel ler Slipstream Veloc i ty Dis t r ibu t ions

( a ) C a l c u l a t e the p r o p e l l e r a n g l e o f a t t a c k a n d t i p speed r a t i o from

42

J COS U p P.LT = 7r

J COS a p P = 7 r r -

(c) obtain an approximate solut ion for the t i p loss f ac to r u s ing

where, by d e f i n i t i o n

k = FP

and ao, Q&) a r e the lif t-curve slope and angle of attack a t ze ro l i f t , - r e spec t ive ly , fo r a linearized approximation t o the tabula ted a i r fo i l sec t ion charac te r i s t ics .

( e ) :Compute an ini t ia l inf low angle a t each blade element s ta t ion from

U * (p' = A + x ( f ) Obtain an ini t ia l value for the quant i ty def ined

as

(9) As the first s tep i n the basic i teration routine, calculate a b e t t e r approximation for t h e t i p loss factor from

where F /Fp is obtained by interpolat ing the resul ts from the t i p loss correction tables for specified values of B , T and sin$ . A l i s t i n g of the t ip loss correction tables stored and u t i l i zed by the computer program i s presented i n Appendix A.

(h) Calculate the blade section angle of attack and the blade section Mach number from

44

Then o b t a i n the sec t ion characteristics C& and cd by i n t e r - po la t ion and /o r ex t r apo la t ion o f the data p resen ted i n the propeller a i r f o i l tables, f o r the s p e c i f i e d a i r f o i l s e c t i o n geometry and values of Q b and M, .

(i) Compute the

k, = L [ 4F

fo l lowing quan t i t i e s de f ined as

and then c a l c u l a t e a new value of 4 from

(j) If the absolute magnitude of (t#)"-~#)')>O.l degrees , then the s o l u t i o n for # r e q u i r e s r e i t e r a t i o n . I n this case the va lue of + t o be s u b s t i t u t e d f o r #I i n s t e p s (9) through (i) is obtained from

45

. . . . . . . . .

where .kc is given by

(k ) If the absolute magnitude of ( #I1- #I) 5 0.1 degrees then the f ina l s l ips t ream ve loc i ty components f o r the streamtube element passing through the .specified blade element s t a t ion a r e de t e rmined a s fo l lows . First, c a l c u l a t e the t r u e i n d u c e d a x i a l v e l o c i t y r a t i o i n the p r o p e l l e r d i sk plane using

U 4 F ky kz S l r 2 i- ( I -I- k x ) 2 -4

and then obtain the a x i a l v e l o c i t y r a t i o i n the f u l l y contracted s l ip s t r eam from

(1) Obtain the l o c a l r a d i u s i n the f u l l y c o n t r a c t e d s l ip s t r eam which corresponds t o the s p e c i f i e d blade e l emen t s t a t i o n from

- where rsp , rp and VSap/va are the values corresponding t o the immediately preceding inboard blade s l emen t s t a t ion . The v e l o c i t y r a t i o a t the outer s l ipstream boundary i s taken as u n i t y , as is that a t fhe hub/nacelle boundary unless a blade e lement s ta t ion is s p e c i f i e d a t the hub.

-

46

( m ) Compute the t a n g e n t i a l v e l o c i t y r a t i o i n the f u l l y c o n t r a c t e d s l i p s t r e a m as

(n ) Having obtained solut ions for the f l o w corresponding t o all propel le r b lade e lement s ta t ions m = l ( a t the hub) through m = M ( a t the b l a d e t i p ) , c a l c u l a t e the value of the i n t e g r a t e d p r o p e l l e r t h r u s t c o e f f i c i e n t from

1

where 3 F ky k z

- cT = (7r T) d 7 ( I +k)"

(0) Obtain the momentum value of p r o p e l l e r t h r u s t c o e f f i c i e n t from

(p ) Compute the in t eg ra t ed p rope l l e r t o rque coe f f i c i en t u s ing

47

~ ~

where

(9) Calcu la t e the value of the momentum-weighted average axial v e l o c i t y r a t i o i n the f u l l y c o n t r a c t e d sl ipstream from

4.1.2 P rope l l e r B lade Sec t ion-Charac te r i s t ics

The ana ly t ica l methods deve loped here in requi re that su i t ab le ae rodynamic cha rac t e r i s t i c s be employed f o r the blade sec t ions o f p rope l l e r s u sed on genera l av ia t ion- type a i r c r a f t . The information on t y p i c a l blade s e c t i o n s was obtained from the a v a i l a b l e t e c h n i c a l l i t e r a t u r e a n d is summarized i n Table I .

A s can be noted from this t ab le , e a r l y blade .sections used i n t yp ica l p rope l l e r s are of the USNPS and Clark Y a i r f o i l series. These sec t ions have very similar p r o f i l e s and members of each series a re un ique ly i den t i f i ed by the va lue o f th ickness /chord ra t io a lone .

Later blade s e c t i o n s a re of the NACA 16-ser ies fami ly , which have a w i d e r a ,p ,pl icat ion in modern p rope l l e r des ign because of the i r supe r io r low-drag characteristics (see Reference 25) . These cons ide ra t ions a l so app ly t o the use of NACA 64 and 65 a i r f o i l series. All of the latter air- f o i l s a r e s p e c i f i e d i n terms of both a design l i f t c o e f f i c i e n t and a th ickness /chord ra t io .

Based on a review of published experimental measurements o f p r o p e l l e r a i r f o i l s e c t i o n c h a r a c t e r i s t i c s , it is evident t h a t the most re l iable data f o r t h e c u r r e n t a p p l i c a t i o n can be obtained from tes ts conducted in three wind tunnel

48

'Table I. Typical Propeller Blade Sections.

A i r f o i l Series

~~

USNPS

Clark Y

NACA 16XXX

NACA 64-XXX

NACA 65-XXX

Design Lift Coeff ic ients

- -

0.2 to 0.7

0 to 0.2

0 to 0.2

Thickness/Chord R a t i o s

0.05 to 0.35

0.07 to 0.50

0.04 to 0.40

0.07 to 0.26

0.04 to 0.40

29

17,30,31

17,30,32,33,34

35

35,36

fac i l i t i es only. These are the Langley Low Turbulence Pressure Tunnel (Reference 261, for s e c t i o n d a t a a t l o w speed condi t ions ( M e 0.15) 8 and both the Langley and Ames High Speed Wind Tunnels (References 27 and 2 8 , respect- i v e l y ) for s e c t i o n data a t high speeds (0.3 < M < 0 . 8 5 ) . Exper imenta l da ta ava i lab le from tests i n these fac i l i t i es were therefore used as the basis f o r p r e p a r a t i o n of the required s e c t i o n characteristics f o r a l l s e l e c t e d a i r foi ls w i t h the except ion of the USNPS and Clark Y series. The s e c t i o n data f o r the l a t t e r two a i r f o i l s w a s generated from the measurements obtained in the Langley Variable-Density Tunnel .

"

Application of the present ana ly t ica l methods requi res information on the two-dimensional behavior of both l i f t and d rag fo r the s p e c i f i e d blade a i r f o i l s . However, an import- an t s imp l i f i ca t ion i n p re ,pa r ing these a i r fo i l c h a r a c t e r i s t i c s i s rea l ized th rough the use of a cons t an t va lue for drag coe f f i c i en t on the basis of the following approximation.

From the p r o p e l l e r a n a l y s i s it can be noted that the con t r ibu t ions o f the blade s e c t i o n d r a g c o e f f i c i e n t c d t o the a x i a l and s w i r l v e l o c i t y components i n the s l i p s t r e a m are given, approximately, by ( Cd/Cd) tan d and (Cd/Cd) cot $ r e s p e c t i v e l y , where c# i s the inflow angle. For low speed f l i g h t c o n d i t i o n s a p p r o p r i a t e t o g e n e r a l a v i a t i o n t y p e a i r - c r a f t , the con t r ibu t ions o f blade s e c t i o n d r a g t o the l o c a l a x i a l v e l o c i t y component i n the slipstream are found to be negl ig ib le , whereas the c o n t r i b u t i o n s t o t h e l o c a l s w i r l v e l o c i t y a re t y p i c a l l y n o t more than a f e w pe rcen t . Thus it i s considered a j u s t i f i a b l e s i m p l i f i c a t i o n i n the computer program t o s u b s t i t u t e a r e p r e s e n t a t i v e c o n s t a n t v a l u e f o r Cd i n p l a c e o f the a c t u a l v a r i a t i o n s as a func t ion of angle of attack and Mach number.

I t is t h u s e v i d e n t t h a t rea l i s t ic a p p l i c a t i o n of the p rope l l e r - s l ip s t r eam ana lys i s demands t h a t selected data on b l a d e s e c t i o n l i f t characterist ics be accura t e ly de f ined a s a funct ion of local angle-of-at tack and Mach number f o r t h o s e t y p i c a l a i r f o i l s e c t i o n s i d e n t i f i e d a b o v e .

Table I1 summarizes the a i r f o i l s e c t i o n s for which aerodynamic charac te r i s t ics have been ob ta ined and ident i f ies the source re fe rences . I n genera l it i s apparent t h a t insuf- f i c i e n t d a t a e x i s t t o e n a b l e a thorough coverage of a l l the poss ib l e va r i a t ions i n s ec t ion geomet ry , ang le -o f -a t t ack and

50

T a b l e 11. Summary of P r o p e l l e r A i r f o i l Sections T a b u l a t e d for U s e i n the Computer P r o g r a m .

Airfoil Series . T h i c k n e s s / C h o r d Ratios i n Percent ' . ' .' I'

. .

. . . . . e

USNPS

Clark Y -4 6 8 1 0 11.7. 1 4 1 8 2 2

NACA 1 6 1 X X

6 9 1 2 1 5 .2l - 163XX

6 9 - 1 5 ' " - ' 30

165XX 6 9 1 2 15 2 1 3 0

167XX : - 9 1 2 15 - -

-4 j -1 4 6 8 1 0 . 1 2 1 4 1 6 1 8 20

___.. " ~ -..- " - - .- "

, .

. ;I . "

..

.- -

NACA 6 4 - OXX

- 9 1 2 1 5 18 2 1 64-4XX

6 9 1 2 15 1 8 2 1 64-2XX

6 9 1 2 15 1 8 2 1

NACA 65-OXX 6 9 1 2 15 18 2 1

65-2XX 6 9 1 2 1 5 1 8 2 1

65-4XX - 1 0 1 2 1 5 18 2 1

"_______" - -_______

Mach No. R a n g e

-0.07- '

0,.07 j ___-

So,urce :References

"" . -

0 .3 t o ' i 0 . 8 1 , 39 '

. .

. r . .

'. . .- .

- - _" . .

0 . 1 5 40,4L

Mach number range. Accordingly a number of s imple empir ical techniques have been developed to permit a reasonable ex t ra - p o l a t i o n o f the a v a i l a b l e data, as w i l l be d iscussed la ter i n the text.

Fur thermore , in p repara t ion of the f i n a l s e c t i o n characteristics, faired curves o f . the experimental cd versus

Q were u t i l i z e d . The da ta w a s c a r e f u l l y selected so as t o best d e f i n e the n o n - l i n e a r i t i e s i n the faired curves . In gene ra l the data r e p r e s e n t s the f u l l r a n g e o f the experimental measurements extending from the zero l i f t c o n d i t i o n t o a p o i n t c l o s e t o s t a l l and in most cases through the s t a l l .

A complete computer l i s t i n g o f the t a b u l a t e d s e c t i o n c h a r a c t e r i s t i c s f o r the p r o p e l l e r a i r f o i l s l i s ted i n Table 11, is presented in Appendix B. The a i r f o i l tables are arranged so as t o provide the maximum f l e x i b i l i t y i n their use i n the computer program. These tables can be easi ly extended o r deleted t o i n c l u d e o t h e r a i r f o i l families o r s p e c i a l l y modified aerodynamic characteristics of the selected sec t ions .

These t a b l e s form the basis for look-up procedures which th rough i n t e rpo la t ion and ex t r apo la t ion o f the s t o r e d data provide the requi red va lues o f Cd f o r s p e c i f i e d blade s e c t i o n s . These table look-up procedures are descr ibed i n d e t a i l i n the next subsec t ion .

4.1.3 Table Look-Up P r o c e d u r e s f o r P r o p e l l e r A i r f o i l Characteristics

?he p r o p e l l e r a i r f o i l data tables a r e read i n and s t o r e d by the computer immediately p r io r t o execu t ion o f the p r o p e l l e r - s l i p s t r e a m c a l c u l a t i o n s . The computer program provides data tables f o r up t o 9 a i r f o i l families, i d e n t i f i e d by a n a i r f o i l series code between 1 and 9 i n c l u s i v e , b u t is capable of accept ing a maximum of 150 tables. This s to rage capac i ty i s considered more than adequate under most cir- cumstances but could be extended, i f r e q u i r e d , by an i n t e rna l program change. A s a r u l e the only tables read i n w i l l be those sets corresponding to the blade s e c t i o n s of the prop- e l ler-wing configurat ion being evaluated.

Each table, as it is r e a d i n by the computer, is indexed consecu t ive ly i n o rde r t o pe rmi t e f f i c i en t ope ra t ion of the look-up procedure. For proper uti l ization of these

52

d a t a t a b l e s , it is essential that t h e y be assembled i n a s p e c i a l o r d e r . The assembly of a l l t a b l e s f o r each given a i r f o i l f a m i l y must be in ascending order of Mach number, th ickness /chord ra t io and design l i f t c o e f f i c i e n t . However, t h e sets of tables f o r any a i r f o i l f a m i l y may be assembled i n any order.

As a n i n i t i a l s t e p i n the table look-up procedure, the computer program f i r s t s e a r c h e s t h r o u g h t h e t a b l e s to locate and index those par t icu lar tables r e q u i r e d f o r i n t e r - p o l a t i o n as each p rope l l e r b l ade e l emen t s t a t ion i s s p e c i f i e d . The ac tua l l ook-up p rocedure u t i l i ze s l i nea r i n t e rpo la t ion throughout and is performed f i r s t f o r the requi red va lue of

a , secondly for the value of Mach number, t h i r d l y f o r the sec t ion t h i ckness / chord r a t io and f i n a l l y f o r the design l i f t c o e f f i c i e n t o f the a i r f o i l f a m i l y s p e c i f i e d .

To permi t sa t i s fac tory opera t ion of the computer program f o r c o n d i t i o n s o u t s i d e the r ange o f t he da t a t ab l e s a series of s imple extrapolat ion procedures have been developed empi r i ca l ly from the ava i l ab le expe r imen ta l da t a . These procedures are out l ined below.

For ang le s of a t t a c k o u t s i d e the t abu la t ed r ange i n each table it i s assumed t h a t the value of CJ remains cons tan t , and for a Mach number outs ide the g iven range the extrapolat ion procedure determines a c o r r e c t i o n t o the requ i r ed value of Q , d e f i n e d a s ac , thus

where the s u b s c r i p t T denotes values for the table t o be ex t r apo la t ed .

This method i s based on an appl icat ion of the s tandard Prandt l -Glauer t ru le for the change i n l i f t - c u r v e s l o p e w i t h Mach number and assumes t h a t t h e e x t r a p o l a t e d f a m i l y o f l i f t curves can be represented by a simple adjustment of the ang le of attack scale about ~1~ po in t .

For s ec t ion t h i ckness / chord r a t io s ou t s ide the given r ange o f t ab l e s a t each value of design l i f t c o e f f i c i e n t it i s assumed t h a t t h e a i r f o i l c h a r a c t e r i s t i c s w i l l be i n v a r i a n t . While t h i s a s s u m p t i o n d o e s n o t s a t i s f a c t o r i l y ' r e p r e s e n t the

53

g e n e r a l r e d u c t i o n i n l i f t - c u r v e s l o p e . f o r - t h i c k sections., , .-..-.

the exis t ing da , ta . does .no t - p rovide . :a b,,ase .. . . . I-

for a better approximation. ( ' t i c ,> 0.2 1

. . . .- ,

.. , . . . . , . 8 .. 2

. . For a sec t ion des ign l i f t c o e f f i c i e n t " c&i , o u t s i d e

. ,. . . . . . - .

the tabulated range; an extrapolat ion proc;edure is .'used t o o b t a i n a corrected v a l u e - o f C& defined as :-C& , t h u s

- I.

. kc& - , 8

. . . . . , , ,

. ' , .

. . ccec = CLeT + (c9 - CLeiTj :.. -: (126 1

. . _ . ,

where the subscript T denotes va lues ' fo r the table t o be ex t rapola ted and kCdi is an empiFica1 constant which g e n e r a l l y v a r i e s far each a i r fo i l ' f ak i iy and t h i ckness / chord ra t io . This cons tan t has been determined for each a i r f o i l f a m i l y u s e d h e r e i n , a n d c o n s t i t u t e s a n i n h e r e n t p a r t of the computer program table look-up subroutine. . .

.~ .. .. , . .

4.2 WING . . IN-SLIPSTREAM -COMPUTATIONS . . . ,

This subsec t ion p re sen t s t he method of implementation of the p r o p e l l e r slipstream d i s t r i b u t i o n s o b t a i n e d a,bove i n t o the spanwise load ca lcu la t ions of a pro,peller/wing combination.,.., The es sen t i a l computa t iona l s t eps a re described below.'

4.2.1 Computational Procedures for Spanwise Loadinq on a Winq w i t h no Flaps , o r w i t h Full-Span Deflected Flaps

~ . - ..

( a ) Obtain the wing basic geometric ,parameters namely,, . .

sect ion chord ra t io c / C R , t w i s t d i s t r i b u t i o n . ' E 8

th ickness-chord ra t io t/c , and camber d i s t r i b u t i o n . Then c a l c u l a t e the wing section Reynolds number Re based on the local chord c and the l o c a l r e s u l t a n t v e l o c i t y V , thys. .

. . c'

, , . . .

. .Pi , , . . : , . :' .. . ~.. . . ' - .. ." .

where V i s the combined freestream and slipstr'eam v e l o c i t y g iven i n equa t ion ( 3 ) and Y is the k inemat ic , v i scos i ty . A l s o , ob ta in the s e c t i o n z e r o - l i f t a n g l e ado . . . . .

, . . .I

. .

%. .

(b) Compute the wing-induced upwash func t ion , f , . .

.s 4

from the fo l lowing .eqyat ion which is based on a simple horse- shoe model of t h e wake (Reference 1 9 )

J (7?4-$2+ xp - xp - 2 -

are the non-dimensional spanwise and chordwise locations of the r igh t -hand propel le r hub.

(c) Calcula te the geometr ic angle of attack a t each wing s t a t i o n from

where ag is the fuse lage angle o f attack aR is the wing / fuse l age roo t s e t t i ng E i s the loca l geomet r i c t w i s t

T [E$-,] is the c o r r e c t i o n f a c t o r f o r f u s e l a g e upwash given in Reference (1) and A E n is the s e t t i n g of the equiva len t chord l ine , of the nacel le above the wing chord l i n e a t the n a c e l l e s t a t i o n . The quan t i ty A E n is o n l y t o be included when a computa t ion s ta t ion co inc ides w i t h ' the n a c e l l e l o c a t i o n .

(a) Calcu la t e the f o l l o w i n g i n i t i a l a p p r o x i m a t i o n t o the o v e r a l l wing l i f t c o e f f i c i e n t

- I CLAPPROX- (I + %j ("B "R -0.4 ado TIP -0.6 "do ROOT )(130)

5-5

( e ) Compute the wing-induced upwash a t the p r o p e l l e r d i sc us ing equat ion (128) as follows :

CL APPROX. f v, = ~ - 2 A R

( f ) Calculate the p r o p e l l e r t h r u s t - l i n e a n g l e o f attack and average inc l ina t ion of the p r o p e l l e r s l i p s t r e a m t o the freestream from

ap = + ~ T L

as = tan” { - Vo sin QP + V,

VStl 1 where iTL is the p r o p e l l e r t h r u s t - l i n e a n g l e r e l a t i v e t o the f u s e l a g e c e n t e r l i n e .

( 9 ) ’ A t each Qing s t a t i o n calculate the e f f e c t i v e a n g l e s of a t t a c k , the r e s u l t a n t local slipstream v e l o c i t y , V , and the non-dimensional slipstream upwash, v , from the fol luwing equat ions:

(h) Calcu la t e the d i s t r i b u t i o n of l i f t due t o s l i p - stream upwash C& c/b using equat ion (56) .

(i) Using the e f f e c t i v e a n g l e s of attack, ae , computed from equation (134) f i n d the va lues of s e c t i o n l i f t c o e f f i c i e n t , C& , f rm the two-dimensional section data a t the proper va lues of Reynolds number, thickness-chord ratio and camber l e v e l .

56

( j ) Calcu la te an i n i t i a l a p p r o x i m a t i o n t o the spanwise I

loading d i s t r ibu t ion us ing

"

b AR + 1.8 (137)

where X i s the wing t a p e r r a t i o .

(k) Compute the values of induced angle of attack f o r this load d i s t r ibu t ion u s ing equa t ion (61) and determine the r e su l t an t s ec t ion ang le s o f a t t ack f rom equa t ion (62) .

(1) From the s e c t i o n data obta in the va lues of l i f t coe f f i c i en t co r re spond ing t o the r e s u l t a n t a n g l e s of attack from step (k) and ca lcu la te the new values of me span loading , c(e c/b .

(m) Compare the approximate values of span loading w i t h the ca l cu la t ed va lues . If these are n o t i n s u f f i c i e n t l y close agreement, compute a new set of appro.ximate values of

of Reference 1. Repeat the i t e r a t i o n p r o c e s s u n t i l the requ i r ed convergence i s achieved.

CCe c / b using the procedures p resented in subsec t ion 3.2.2

( n ) I n t e g r a t e the new s p a n l o a d d i s t r i b u t i o n t o o b t a i n the o v e r a l l wing l i f t c o e f f i c i e n t CL and ca l cu la t e a new va lue of wing-induced upwash a t the p r o p e l l e r d i s c using equat ions (128) and (131).

(0) Repeat s teps (f), (9 ) , ( h ) , ( i) , (k), (11, (m) , ( n ) u n t i l the approximate and calculated values of span loading are in sa t i s fac tory agreement .

(p) Having determined the l i f t d i s t r i b u t i o n o b t a i n the sec t ion p ro f i l e d rag and p i t ch ing moment va lues from the sec t ion da t a and ca l cu la t e the overa l l wing l i f t , drag, and p i t c h i n g moment c o e f f i c i e n t s .

4.2.2 Computational Procedures for Spanwise Loadinq on a W i n q w i t h Part-Span Deflected Flaps

(a) Calculate a n i n i t i a l a p p r o x i m a t i o n t o the f lapped wing l i f t d i s t r ibu t ion f rom the fol lowing equat ions

57

where C ~ R is the value of the l i f t c o e f f i c i e n t a t the r o o t obtained from the f lapped sec t ion data a t the angle of attack

(b) Determine the uncorrected values of l i f t coe f f i - c i e n t c>, a t each f lap end as fol lows

where FF is the co r rec t ion f ac to r wh ich accoun t s fo r the change i n the two-dimensional section data a t the f l a p end. The c a l c u l a t i o n p r o c e d u r e f o r o b t a i n i n g these c o r r e c t i o n f a c t o r s is descr ibed in detail in subsec t ion 4.1.3 of Reference 1, and w i l l n o t be dupl ica ted here.

(c) For the va lues o f c 2 , o b t a i n e d i n step (b) above obtain the corresponding angles of attack Qo from t h e data f o r flapped sec t ions , Calcu la te the corresponding corrected ang les of attack Qc8 a t each end of the flap from

(d) Using the same procedure as i n s t e p (c) above, calculate the values o f angle of a t t a c k a c S=O on the unflapped s i d e s o f the wing. Then ob ta in the first approx- ima t ion fo r the values of the d i s c o n t i n u i t i e s i n a n g l e of attack 8 , t h u s

58

(e) I n t e g r a t e the l i f t d i s t r i b u t i o n g i v e n by equat ion (138) t o obtain an approximate value of the ove ra l l f l apped l i f t c o e f f i c i e n t , CL , and using equation ( 3 ) , determine the wing-induced upwash a t the p r o p e l l e r disc. Then c a l c u l a t e the value of slipstream i n c l i n a t i o n , a s , using equat ion (5)

I ' (E )~ Using the va lues of 8 and as f rom s teps (d ) ,

and *Fej , r e s p e c t i v e l y , c a l c u l a t e the d i s t r i b u t i o n of slipstream crossflow from the fol lowing equat ion:

+ (%) COS (as + .e) - sin Qe

and use equations (64) and (65) to determine the d iscont in- u i t i e s i n c r o s s f l o w Av*= vII . . . " .

- NOTE: - I n the most general case of a wing having t w o : p r o p e l l e r s , ' (one mounted on' each wing panel) , r o t a t i n g i n &e . . same d i r e c t i o n , the s l ips t ream-induced c ross f low d is t r ibu t ion ..

w i l l be d i f f e r e n t a t ' the same spanwise s ta t ion Y on each s i d e of the f u s e l a g e c e n t e r l i n e . This d i f f e r e n c e is caused by upward s l i p s t r e a m s w i r l v e l o c i t i e s on one wing.pane1 and . . downward on the other, occurr ing a t the' same spanwise s ta t ions on each side of the fuse l age , i.e. v (y) # v (-y) . I n the case of two propellers r o t a t i n g i n o p p o s i t e d i r e c t i o n s , each sl ipstream-induced crossf low is symmetrical about the fuse lage centerline and equation (143) need only be appl ied once, s i n c e v'(y)= v(-y) .

" . .

~. -

(9) Using*ihe appropriate values of the discont in- u i t i e s 8 and nv = v i r from steps ( d ) and ( f ) , r e spec t i , ve ly , .. compute the l i f t d i s t r i b u t i o n 'Cd2.c/b using equat ion (76 ) . "

- NOTE: For the most general case, as d i s c u s s e d i n step . . - ( f ) above, this l i f t d i s t r ibu t ion mus t be c a l c u l a t e d ' s e p a r a t e l y for each wing panel.

(h) Determine the l i f t d i s t r i b u t i o n CdZ,lI-c/b 8

corresponding to the l e f t and r i gh t spanwise d i scon t inu i t i e s from equation (88).

_ _

59

(i) Calcula te the overal l induced angle-of-at tack d i s t r i b u t i o n Q i from equation (87), using the approximate span load d i s t r ibu t ion computed i n step (a) above.

(j) Compute the e f f e c t i v e r e s u l t a n t s e c t i o n a n g l e of d i s t r ibu t ion f rom the fol lowing equat ion

Qe

where Qg i s the geometric angle of attack, Cdmax is the value of CLemax obtained from the co r rec t ed s ec t ion data and

(Cd?max)o i s the uncorrec ted va lue o f CJma . (k) Using the va lues of Qe from s t e p ( j) above,

o b t a i n the corresponding values ,of l i f t c o e f f i c i e n t c& from the uncorrected two-dimensional section l i f t data. Then determine the c o r r e c t v a l u e s of l i f t c o e f f i c i e n t C& by s c a l i n g , as fol lows:

(1) Ca lcu la t e the d i s t r i b u t i o n C&c/b from (145) and compare this calculated d i s t r i b u t i o n w i t h the approximate d i s t r i b u t i o n . If agreement between the d i s t r i b u t i o n s i s n o t s u f f i c i e n t l y c l o s e , c a l c u l a t e a new and better approximation using the procedures presented in subsect ion 3,2.2 of Reference 1..

(m) Repeat steps (b) through (1) above, u n t i l a g r e e - ment i s reached between the approximate and ca lcu la ted va lues of the span l oad d i s t r ibu t ion .

(n) Having determined the l i f t d i s t r i b u t i o n i n s t e p (m), c a l c u l a t e the corresponding value of the o v e r a l l i n t e - g r a t e d wing l i f t c o e f f i c i e n t CL .

60

4.2.3 Winq Sec t ion Characteristics

The wing a i r f o i l s e c t i o n characteristics f o r t y p i c a l g e n e r a l a v i a t i o n aircraft are p resen ted i n Sec t ion 4.2 of Reference 1, and w i l l n o t be d u p l i c a t e d i n t h i s r e p o r t . These characteristics are u s e d d i r e c t l y i n t h e c u r r e n t computer program and constitute a p a r t of the o v e r a l l t o o l f o r p r e d i c t i o n o f s t a l l i n g characteristics of general wing/ propel ler combinat ions . 4.2.4 Table Look-Up Prdcedures for Winq Section

C h a r a c t e r i s t i c s '

The tab le look-up subrout ine for wing sec t ion characteristics used i n the current program is i d e n t i c a l t o that descr ibed in Sec t ion 4.2 of Reference 1.

4.3 DESCRIPTION OF THE COMPUTER PROGRAM LOGIC

The computat ional procedures descr ibed in Sect ion 3.0 have been programmed fo r u se on a CDC 6600 series d i g i t a l computer. The program user ins t ruc t ions are g i v e n i n Appendix C. The flow diagram for the program is shown in F igu re 7 and a l i s t i n g o f the program is p resen ted i n Appendix D. The program was accomplished by an ex tens ive r e s t ruc tu r ing and enlargement of the basic power-off wing s ta l l ana lys i s p rogram contained in Reference 1.

The program i s i n i t i a t e d by r e a d i n g i n the basic wing-fuselage configurat ion parameters . In this inpu t format , p rovis ion has been made to inc lude an increment r ep resen t ing the d rag coe f f i c i en t o f the n a c e l l e s . I f the c a l c u l a t i o n s are t o be performed for the power-on case this is ind ica t ed t o the program by s e t t i n g the parameter NSLIP e q u a l t o 1. If NSLIP=O, the s l i p s t r e a m c a l c u l a t i o n l o o p s are bypassed and the program only computes the power-off characteristics.

The computer program arrays are dimensioned t o e n a b l e c a l c u l a t i o n s o f the span loading to be made using 10 con t ro l points per semispan.

For tw in p rope l l e r aircraft computations where t h e p r o p e l l e r s are s i t u a t e d n e a r the center of each wing panel or

61

62

. . CALCULATE

PARAMETERS

. .- . I . . .. AIRFOILTAQLES - FUSELAGE .

,_ *. .. . ~

.) -! , .

c

- :, . NO CALCULATE READ LIST

- . I .I

WING GEOMETRY - OF

OF ATTACK

. , I . .., . . ..I d

CALCULATE CONVERGENCE .C- MATRIX Ki j Cd',c/b 6

CALCULATE at , / 8, CALCULATE

AND STORE AND STORE

I t 4

I I NO I

PRINT OUT

INFORMATION

APPROXIMATE CALCULATE CALCULATE

L IFT DlSTRlQUTlON THERE A UPWASH SLIPSTREAM

FUNCTION, f SUBROUTINE - .

. .

SLIPSTREAM UPWASH, v

I No I

Y APPROXIMATION TO CALCULATE FIRST

POWER-OFF LIFT DlSTRlQUilON I

- .

. . . , .

CALCULATE INDUCED ANGLES OF ATTACK AND NEW LIFT DISTRIBUTION

*

I I . .. . . I . * . NO . ,

* , - . . I " ,

AND PRINT OUT

SELECT VALUES

DEFINE STALL . OF a8 TO

-

(e) Figure 7. Computer Program Flow Diagram

a t the wing t i p s , this number of wing cont ro l s ta t ions i s adequate . However, f o r s i n g l e p r o p e l l e r c o n f i g u r a t i o n s , a better d e f i n i t i o n o f the span loading in the s l i p s t r e a m region i s obtained i f the number o f c o n t r o l s t a t i o n s i s doubled t o 20 per semispan. This is readi ly ach ieved by redimensioning the requ i r ed a r r ays .

Having i n p u t the basic data, the r e q u i r e d wing s e c t i o n da ta tables are read i n a n d s t o r e d on tape . If the case is f o r a wing w i t h fu se l age the requi red t ransformat ion parameters are computed. The l is t of fuse lage angles of attack is now read in and the first v a l u e i n the l i s t is s e l e c t e d . If the computation i s t o be performed for a power-on case the p rope l l e r s l i p s t r eam sub rou t ine i s then called.

Execution of the s l ip s t r eam sub rou t ine shown i n Figure 8 i s i n i t i a t e d w i t h input and s torage a t the p r o p e l l e r t i p l o s s c o r r e c t i o n f a c t o r t a b l e s . This i s followed by r ead ing and . s to r ing the r e q u i r e d b l a d e s e c t i o n data tables, toge the r w i t h the data spec i fy ing the basic propel ler geometry and operat ing condi t ion. ?he program then proceeds w i t h the main computations as the parameters for each success ive blade element are read in. For each b l a d e s t a t i o n , the solu- t i o n f o r blade s e c t i o n a n g l e o f attack and lift is i terated t o convergence. The v e l o c i t y components f o r the corresponding s t reamtube e lement in the con t r ac t ed s l i p s t r eam are then computed. Final ly , having obtained the complete veloci ty d i s t r i b u t i o n f o r t h e s l i p s t r e a m , the s l i p s t r e a m v e l o c i t i e s a t the wing c o n t r o l s t a t i o n s are determined by i n t e r p o l a t i o n b e f o r e r e t u r n i n g t o the main program logic.

Having calculated the s l i p s t r e a m v e l o c i t y d i s t r i b u t i o n s the wing upwash funct ion and the induced angle-of-attack m u l t i p l i e r s are now computed. If a p a r t s p a n d e f l e c t e d f l a p - is p r e s e n t the parameters a s s o c i a t e d w i t h the spanwise discon- t i n u i t i e s are c a l c u l a t e d t o g e t h e r w i t h the f a c t o r s u s e d t o c o r r e c t the two-dimensional section data.

The matrix of c o e f f i c i e n t s K i j used i n the i t e r a t i o n procedure i s now computed and stored. If the c a l c u l a t i o n s are t o i n c l u d e slipstream effects, the s l i p s t r e a m i n c l i n a t i o n t o the freestream, the s l i p s t r e a m upwash func t ion , v , and the load ing a s soc ia t ed w i t h this upwash func t ion , C& c/b, are computed .

63

START

I. np LOSS CORRECTION FACTOR TABLES

2 AIRFOIL TABLES I 1

1

1

1

READ BASIC INPUT DATA FOR TEST CASE

CALCULATE BASIC CASE PARAMETERS

PRINT OUT CASE HEADING AND BASIC INPUT DATA

SET INITIALVALUES FOR SLIPSTREAM SOLUTION AT HUB AND NACELLE

READ INPUT DATA FOR LOCAL BLADE STATION

1 YES

[OF 9 FOR ITERATION I CALCULATE INITIALVALUE

LOOK UP TIP LOSS CORRECTION FACTOR

LOOK UP CL FROM AIRFOILTABLES r"l CALCULATE NEXT VALUE

OF # FOR ITERATION

I O F O , I CALCULATE NEW VALUE

CALCULATE SLIPSTREAM ELEMENT VELOCITIES AND INTEGRAL FUNCTIONS

SET BOUNDARY VALUES FOR SLIPSTREAM SOLUTION

PRINT OUT SOLUTIONS FOR BLADE STATION AND SLIPSTREAM ELEMENT

CALCULATE AND PRINT OUT INTEGRATED VALUES FOR SLIPSTREAM

OUT SLIPSTREAM VELOCITIES AT WING SPAN STATIONS

RETURN TO MAIN PROGRAM

Figure 8. Logic D i a j r a m f o r P r o p e l l e r S l i p s t r e a m Subroutine

64

The cen t r a l p rog ram i t e r a t ion l oop i s now entered . A new l i f t d i s t r i b u t i o n i s computed and compared t o an approximate inpu t va lue . If convergence is not ach ieved a new and bet ter approximate value i s computed. If slipstream effects are being considered, this new l i f t d i s t r i b u t i o n is used t o r e c a l c u l a t e - t h e upwash a t the p rope l l e r d i sc s and a modified s l i p s t r e a m i n c l i n a t i o n is obtained. A new upwash d i s t r i b u t i o n is ca lcu la ted and the basic i t e r a t i o n l o ~ p r e e n t e r e d .

Once convergence is obtained, the program computes and p r in t s ou t the o v e r a l l wing in tegra ted va lues o f CL ,

CD , etc. toge ther w i t h the d i s t r i b u t i o n s . If s ta l l i s de tec t ed a t any wing s t a t i o n , the program enters a r o u t i n e t o select values of fuselage angle of at tack t h a t w i l l de f ine the exac t s t a l l ang le more c lose ly .

4.4 SAMPLE OUTPUT

A t yp ica l ou tpu t ob ta ined f r m the computer program, as described above, i s presented in Tables I11 and I V . Table I11 shows sample computations for the spanwise l i f t d i s t r i - bu t ion on a wing-in-slipstream, whereas Table I V p re sen t s a sample output for the s l ip s t r eam ve loc i ty d i s t r ibu t ion u sed i n the wing computations.

6 5

Dis t r ibu t ion I . , . _ . . , . . . . . . . . ,. . , ; . , . , . .

P R O P E L L E R S L I P S T K E A K A N A L Y S I S ' z . N R C LR-284 F I G ,26 CT1'NOM=.0.64~, BETA75=25!, Jr0.605

P R O P E L C E R - k I i \ r G G € O M E T R Y P R O P E L L E R - N A C E L L E G E O M E T R Y . P R O P E L ~ E R OPERAT-ING CONDITION

. , . . . r . . . .

N U M B E R OF P R O P S

P R O P G I A / W l N G S P A N = O i 4 7 5 3 P R O P A X I S R E L BOOY A X I S = 0.000 D E G - PROP A N G L E OF A T T A C K - . * :4.250 D E G

P R O P FkD COURU 2.XP/B = 0.5667 P H O P S P A N C C O R D 2.YP/.8 = .0,6179

= 2 N O OF 8 L A D E S P E R P R O P = 4 L E F T / R I G H T P R O P R O T N RH, LH = 0.6050 * 0.0581

HUtl D I A / P R O P D I A N A C E L L E D I A / P R O P D I A x 0.1673 .

0.1673 . P R O P A D V A N C E R A T I O , F L I G H T M A C H N U M B E R

P L A G P E L E V E N T G E O . P E T R Y b L A D E E L E M E N T S O L U T I O N '.. S L I P S T R E A M E L E M . E N T S O L U T I O N

R $ / R P C B / R P P I T C H A / f S E R C L I T / C F M A C H A L P H A C L c o RSIRP U S A / U A * USTIUA PHIS

0.2000 0.3000 O..400G 0.50@C, 0.6600 0.7000 0.8000 G.9000 0.9500 1.0000

0.2500 0.2500 0.2500 0.2500 0.2500 C;. 2500 0.25Cti 0 .2500 0.2500 0.2500

A9 200 '43.350

55.350

37.760 32.350

22.850 27.300

ZB.900 17 L O O 1 5 - 3 6 0

N A C A 16 N A C A 16

N A C A . 16 N A C A 16 N A C A 16 N A C A 16 N A C A 16 N A C A 16 N A C A 16 N A C A 16

0.500 0 500 0.500 0.500 0.500

0.500 0.5C@

0.500 0.SOD 0.500

0..994

0.956

1 0 4 0

Oe.972

0.933

0.811 0.893

0.473 0.638

o.:ooo

0.082 .a. 105 0.131 0.159

0.217 0.246 0.275 0.289 0.307

a. 188

3.461 6.981 7.968 7.645

4.722 6.403

2.739

-0.458 0.699

0.000

0.582 0.. 848

0.957 0..970

0.764 0.878

0.655 0.487 0.367 0.000

0.1983 0.2878 0.3750 0.4610

0.6326 0.7190 0.8067 0.8518. 0.9012

0.~5467

P R O P E L L E R T H R U S T C C I E F F I C I E N T ~ ' C T " = 0.6327 P R O P E L L E R T H R U S T C O E F F I C I . E N T , I C T = 0.2462 P R O P E L L E R T G R U U E C O E F F I . C € E N T V * C Q = 0.0374 M G M E N T U k W G T U S L I P S T K E A M V E L R A r I O = 1.6455

1 e2525

1.61.PO 1 7270

1.7986 1.7895 1.6853

1 .oooo 1 5 6 7 9

1.4486

1.7843

S L I P S T R f A Y V A L U E S A T W I N G C O N T K O L S T A T I O N S

K 2Y / D WS/RP I ;SA/UO UST/UO U S T / U S A

1 0.9876 2 0.9510 3 0.8910 4 0.8090 5 0.7071 6 0.5677 7 G.4579 8

1 2 0.3G90

13 -0.3090

1 4 -0.4539 -0.5b77

1 5 -0 m 107 1 1 6 -0 8G90 L7 -0.89 10 l b -0.9510

-0.9876 1 9 .

0 .77BO 0.7009 0.5745

O. la76 0.4C20

0.3448 C.0633

0.0490

0.3448 0.6498

C.0633

0.4020 0.5745

0.1876

0 7009 0.7780

1.7148 1.7865

1.6470 1.2490

1.5544

1:7e40

1 24YO

1.7918 1 7918

1 2490 1 5544

1 - 2 4 9 0 1.6470

1.7865 1.7148

1.7840

0.2903

0.4136

0.31 75 0.4730

-0.1071 -0.4613 -0.3712 -0.3712 -0.4613 -0.1071

0.4730

0.3452

0.3175

0.4136 0 3452 0.2903

0.1693

0.2318 0.1932

0.2541 0.2872

-0e.2968 -0.0858

-0.2071 -0.2071

-0.0858 0.2872 0.2318 0.1932 0.1693

'-0.2968

0.2541

SECTION 5

VERIFICATION O F THE DEVELOPED ~ . " THEORY

Th i s s ec t ion p re sen t s a series of correlat ions between the p red ic t ed r e su l t s , ob ta ined f rom the computer program described in Sec t ion 4 and the avai lable ex,per imental data . A discussion of these co r re l a t ions has been separated i n t o t w o na tu ra l ca t egor i e s . The f i rs t p a r t d e a l s w i t h a ve r i - f i c a t i o n of the s o l u t i o n f o r a n i s o l a t e d p r o p e l l e r - n a c e l l e conf igu ra t ion , w h i l e the second p a r t c o n s i d e r s the combined wing-in-slipstream case.

5.1 CORRELATIONS FOR AN ISOLATED PROPELLER

A major i ty of the ava i lab le exper imenta l data on iso- l a t e d p r o p e l l e r s is l i m i t e d to measu remen t s o f t o t a l t h rus t and torque. Even i n t h e f e w repor ted s tud ies where the prop- e l l e r s l i p s t r e a m v e l o c i t i e s were measured, the data ,presented i s gene ra l ly i ncom,p le t e and i n su f f i c i en t t o pe rmi t a comprehens ive eva lua t ion of the p r o p e l l e r a n a l y s i s . I t w a s t h e r e f o r e necessa ry t o es tabl ish the overal l adequacy of the a n a l y t i c a l p r e d i c t i o n s by present ing a series of p a r t i a l c o r r e l a t i o n s w i t h the app l i cab le data from each experimental source.

Cor re l a t ions of the elemental loading on a p r o p e l l e r blade are l i m i t e d t o the experimental data r e p o r t e d i n Reference 30. This data i s presented for two 2.8-foot diameter model p rope l l e r s o f similar d e s i g n , b u t d i f f e r e n t t w i s t d i s t r i b u t i o n s . The experimental loadings were obta ined d i r - e c t l y from measurements of the sl ipstream velocity and s w i r l angle in a plam immediately behind the propeller disc. This t e s t information forms the basis f o r t h e c o r r e l a t i o n s shown i n F i g u r e s 9 , 10, and 11.

Figure 9 presents comparisons between the predicted and measured elemental thrust and torque loadings, expressed as ra t ios o f p red ic ted over measured va lues , versus p red ic ted l o c a l l i f t c o e f f i c i e n t a t a blade r ad ius o f 75 .2 percent . As can be noted from this f igu re , t he t h rus t l oad ing p re - dict ions, employing the t a b u l a t e d a i r f o i l c h a r a c t e r i s t i c s , are i n s a t i s f a c t o r y t o good agreement w i t h the t e s t data throughout the range of the l i f t c a r v e . A l s o , t h e

68

a k Q)

73 m

a LI Q)

a

1 . 5

1.0

0.5

0

F i g u r e

4

~~

.L

"

" -

I c o n d i t i o n

."" "- 1 L" -I----~

Sym Prop. L i f t C u r v e "

0 U - 2 4 Tables x II L i n e a r El 0.4E Tables + II L i n e a r

T a s t Data from R e f e r e n c e 30 . _ " ~

0 0.5 1.0 1.5 2 .0 2.5 c1

predicted '

9. C o r r e l a t i o n b e t w e e n P r e d i c t e d a n d M e a s u r e d E l e m e n t a l T h r u s t a n d T o r q u e Lrsadings a t 75 P e r c e n t R a d i u s .

1 . 5 . :

I I

I

1 . 0 , ;

I

I

0.5 i

0 '

F i g u r e

1 . . , .. . ' - 1

T e s t Data f r o m R e f e r e n c e 30

c

.L _... "-.-. . "..- .... u.,.."nC.- ., ..., I".>.->: " 0 0 .5 . ., . . .-.. 2 .'o ;i i 1.0

xl... 5 . . . , :. Y . 2.5

. , .

predicted 10. C o r r e l a t i o n b e t w e e n P r e d i c t e d a n d M e a s u r e d E l e m e n t a l

Thrus t and Torque Load1ngs : : a t 5 2 Percent Ra,dips,.,,.:. .,.': . . . , I

>. ,. . . . .

, ,.. I , .. : . i . ,:. i , - , ~.

' : :?

7 0

. .

. .

1.5

1.0:

. .

, .

0 . 5.

0

.1.5

1.0

0.5

0 0

-

. . . . . .

+ + o

b " 5

+ I-

" Sym Prop. L i f t C u r v e

0 U-24 Tab le s x II L i n e a r I3 0.4E Tab les 4- I I L i n e a r

Test D a t a from Reference 3(

0.5 1 .0 1 . 5 2 . 0

p r e d i c t e d

2.5

11 .. C o r r e l a t i o n b e t w e e n P r e d i c t e d and Measured Elemental . . Thrust and, Torque Loadings at 25 Percent Radius .

. .

71

I

~" ~ ~

c o r r e s p o n d i n g t o r q u e l o a d i n g s are i n r e a s o n a b l e a g r e e m e n t , e x c e p t a t c o n d i t i o n s n e a r the s t a l l , where the d i s c r e p a n c i e s may be a t t r i b u t e d t o a n u n d e r e s t i m a t i o n o f the d r a g c o e f f i c i e n t , assumed cons tan t a t a nominal va lue o f 0.01. However, it shou ld be n o t e d tha t the e x p e r i m e n t a l t o r q u e l o a d i n g . i s p a r t i c u l a r l y s e n s i t i v e t o t h e m e a s u r e m e n t o f s l i p s t r e a m s w i r l a n g l e , a n d t h e r e f o r e may be s u b j e c t t o a p p r e c i a b l e e x p e r i m e n t a l e r r o r . F i g u r e 9 a l s o i n c l u d e s a compar ison of the p r e d i c t e d r e s u l t s o b t a i n e d by u s i n g a n unstallable l i n e a r l i f t ' cu rve to , app rox ima te the a i r f o i l character is t ics . The l i m i t a t i o n of the l i n e a r i z e d r e p r e s e n t a t i o n i s r e f l e c t e d by a n i n f e r i o r p r e d i c t i o n o f t h r u s t l o a d i n g s n e a r a n d a b o v e the s t a l l p o i n t .

F i g u r e 10 shows similar c o r r e l a t i o n s , t o t h o s e p r e - s e n t e d i n F i g u r e 9 b u t f o r a blade s t a t i o n f u r t h e r i n b o a r d a t 52 p e r c e n t r a d i u s . I n t h i s c a s e , s a t i s f a c t o r y t o good c o r r e l a t i o n s are a l s o i n d i c a t e d .

F i g u r e 11 shows similar c o m p a r i s o n s t o t h o s e shown i n F i g u r e s 9 and 10 , b u t a t a blade r a d i u s n e a r the h u b , a t 25.3 p e r c e n t . W n i l e a n i n c r e a s e d s ca t t e r i n the c o r r e l a t i o n s may be p a r t l y a t t r i b u t e d t o the smaller magni tude o f the measured q u a n t i t i e s , it i s e v i d e n t t h a t t h e a s s u m p t i o n o f a n u n s t a l l a b l e l i n e a r l i f t c u r v e o f f e r s a bet ter c o r r e l a t i o n for b o t h the t h r u s t a n d t o r q u e l o a d i n g s . A s u g g e s t i o n t h a t t h e s t a l l p o i n t f o r t h i s a i r f o i l s l ? o u l d be ex tended t o a h i g h e r a n g ' l e - o f - a t t a c k , i s c o n s i s t e n t w i t h the p robab le e x i s t e n c e of a favorable boundary layer deve1opr;Ient caused by c e n t r i f u g a l pumping near the hub r eg ion .

I n r ev iewing the c o r r e l a t i o n s shown i n F i g u r e s 9 through 11, it is a p p a r e n t t ha t there may be a r e s t r i c t ed r e g i o n o f the blade c l o s e t o the hub where s t a l l d e l a y e f f e c t s are present . However , there i s c l ea r ly a n i n s u f f - i c i e n t s u b s t a n t i a t i o n o f t h i s phenomenon t o p e r m i t a n y r a t i o n a l e m p i r i c a l t r e a t m e n t .

F i g u r e s 1 2 and 13 p r e s e n t c a r r e l a t i o n s b e t w e e n the p red ic t ed and measu red va lues o f the a x i a l a n d s w i r i v e l o c i t y d i s t r i b u t i o n s w i t h i n the s l i p s t r e a m o f p r o p e l l e r s o p e r a t i n g a t r e l a t i v e l y low advance r a t io s . The e x p e r i m e n t a l data shown w a s obta ined f rom Reference 1 7 , which p r e s e n t s s l i p - stream ve loc i ty measu remen t s fo r two 39-inch diameter prop- e l l e r - n a c e l l e m o d e l s , i n a p l ane approx ima te ly 0.44 diameters

72

30

. Q) d

0

0 0 1 P r e d i c t e d

.2 . 4 .6 08 1.0 r,/R

F i g u r e 12. COmpariSOn Between P red ic t ed and Measured .-

D i s t r i b u t i o n s o f S l i p s t r e a m Axial V e l o c i t y a n d S w i r l Angle f o r the P-2 P r o p e l l e r 015 Refe rence 1 7 a t J = 0.12.

7 3

30

20

10

0

.2 4 rs/R

.6 1.0

Figure 1 3 . Compari-son Between Predicted and Measured D i s t r i b u t i o n s o f S l i p s t r e a m Axial Veloc i ty and .

Swirl Angle for the P-1 P r o p e l l e r of Reference 17 a t J = 0.26.

7 4

@ownstream..of---the- propeller disc.. The. slipstream v e l o c i t y 1

9 ',<., : .; symetr~&&~~y:< "a3jout '~e"',p$ope~l.er -axis . easur&&$:%&*& -abt&;n?e&:'.deirig ,an eight-probe rake mounted

I i

deasured s !ipstream ." " .- -. , . , . . v e l o c i t i e s . " .., . for- a propel ler designed w i t h high tape$and t w i s t :so as t o produce an axial veloci ty peak qe11 inboard. As can be seen 5rom this f i g u r e , the p red ic t ed Gxial v e l o c i t y d i s t r - i b u t i o n . ... with in the s l ip s t r eam i s i n good

agreement q t h [tihe- corrgkponding Swirl angl? predict . ibp. ,can not be properly assessed because df the -exc$.s3PVe '&itter ".oaf ' the . experimenfal data points . It dhould be .!kioted that jtlagged and unflagged t e s t p o i n t s shown 4n Figure , : l2 represent image pos i t ions on"each s ide of the 3 : propeller; . I

I : r I..- . .--E-ig&-.l&-shows ~~a--shilar degree of ,cor re la t ion be tween the

i Figure 12 shows :a comparison between the predicted and

-. ..' - t es t data; However, the

' .

I I .

predicted and measured s l i p s t r e a m v e l o c i t i e s for a p r o p e l l e r Qavirig.a'€rtCYre- convent ional plan .form and t w i s t d i s t r i b u t i o n . I n this caGe, the p r e d i c t e d r e s u l t s are p resen ted fo r a

This co r rec t ion w a s introduced . to overcome an apparent discrepancy i n the tes t 'measurements , as suggested by the authors

. I prope l l e r Speed reduced t o 80 percent of the reported value.

df. Reference .:1;7 -*.. . _ _ '.'.:, . . - . . .

1 . <*" 9 a " " - , ~ ,

"_l

Figure,-.14 preser&s'.twQ a d d i t i o n a l c o r r e l a t i o n s f o r s l i p - Stream swi$ l angle di i t r ibut ions kqsed on the t e s t data of Reference $2. The exper imenta l dasurements were obtained a t a distance! .. - . .. :of. 2 d iameters , downstream of an isolated p rope l l e r . $ t can be :Seen from Figure 14 that the ana lys i s gene ra l ly Bredicts Brof i les o f the exper imenta l d i s t r ibu t ions , a l though an i nc remknta l sh i f t , i n the s w i r l angle of up t o 4 degrees i s Qvident. i Sowever, the absence of the corresponding t e s t d a t a dn the a$ ia l ve loc i ty d i s t r ibu t ions p rec ludes a proper eyplan- ~Ui i i f i 'T€$&Ts- ' s h i f t in the s l ip s t r eam s w i r l angle .

". I --- ~ -

.Ir.-, i -2.. , '{ One aspec t o f t l i e ana lys i s no t cons ide red i n the above

Gorre la t ions i s an assumption %hat the s l ip s t r eam may be considered \as fully developed or f u l l y c o n t r a c t e d , f o r the ~ ~ ~ n s e . . - o f - , p r a d i c t i n g . . the-wing. span loading. While the e f f e c t

:bg s l ip s t seam con t r i c t ion on wing loading can only be properly assessed by comprehensive- measurements, some observat ions on the ra t e o f s l i p s t r eam con t r ac t ion can be made from the a v a i l a b l e 'lie*% 'data .'

, 1 -:. " : .: _. . . ' ,'" *. 7 . 1

. . _ . ~ . . . A . ' . .

. , : I . I . . ..

?' : ,.-.',:I.,.: , . . . : : -. : . . "

, . -... .: , , . . , .. . - - .- . .. - . . . .

7 5

c

I

/

I \ \ I +"------- I I

\

0 . 2 . 4 .6 . 8 1.0 r s /R

F igu re 14. Comparison Between Predicted and Measured D i s t r i b u t i o n s o f S l i p s t r e a m Swirl Ang le fo r Typ ica l Test Cond i t ions Repor t ed i n Re fe rence 4 2 .

76

I n practical aircraft conf igura t ions the p r o p e l l e r d i sc p l ane i s generally located between one-half and one diameter fonmrd of the wing quarter-chord l ine. From the c o r r e l a t i o n s shown in Figures 1 2 and 13, it may be in fe r r ed that slipstream contract ion could be ful ly developed at d i s t ances w i th in 0.44 (D) behind the p rope l l e r . If this is the case, then the rate of s l ipstream contract ion i s s ign i f i can t ly h ighe r t han that p red ic t ed by po ten t i a l t heo ry .

From the foregoing discussion and t h e c o r r e l a t i o n s presented above, it can be concluded that the computerized a n a l y t i c a l method developed he re in y i e l d s more than adequate s o l u t i o n for the non-uniform p rope l l e r s l i p s t r eam ve loc i ty d i s t r i b u t i o n s , which can be conf ident ly used for p red ic t ion of wing spanwise loadings.

5.2 CORRELATIONS FOR WING-IN-SLIPSTREAM

Th i s subsec t ion p re sen t s t he co r re l a t ions o f the theory with experimental data on wing spanwise loadings with slipstream effects. In s e l ec t ing expe r imen ta l da t a t o thoroughly tes t t h e t h e o r e t i c a l model the following c r i te r ia were used :

Complete information on the geometric parameters of wings, nacel les and p rope l l e r s .

Adequate d e f i n i t i o n of the wing and p r o p e l l e r a i r f o i l sect ions used i n the tes ts .

A thorough description of the propel le r opera t ing condi t ions i n terms of blade angle , advance ra t io , and rotat ional speed.

An accurate determination of the spanwise l i f t d i s t r ibu t ion ob ta ined by chordwise pressure surveys.

I t was found that the amount of ava i lab le t e s t data that meets a l l of the above c r i t e r i a is extremely l i m i t e d . However, s u f f i c i e n t d a t a w a s obtained from the t echn ica l l i t e r a t u r e t o p r o v i d e a fa i r ly adequate basis f o r v e r i f i c a t i o n of the t h e o r e t i c a l model. Some t e s t data was obtained on wings immersed i n j e t s , wings having centrally mounted

77

propellers, and wings w i t h propellers p laced a t d i f f e r e n t spanwise s t a t ions up t o and inc luding the wing t ips .

- ,

W i , t h f e w e x c e ~ t i o n s , t h e m a j o r i t y :of the available .. . .

tes t data w a s ob ta ined a t wing angles of a t t a c k below stall. Therefore ,I the adequacy of the IdeveXoped :methods t o pregict the s p a n " l o a d d i s t r i b u t i o n a t f i e o n s e t of s t a l l cou ld no t be thoroughly ver i f ied. However, based on the c o r r e l a t i o n s p r e s e n t e d h e r e i n , a t a n g l e s of a t t a c k c$,ose t o stall, it can be i n f e r r e d tha t the -span ' loading a t stdll can be reasonably w e l l p red ic t ed u s ing ;the p r e s e n t a n a l y s i s .:. Unfor tuna te ly , no t es t data i s available on spanwise l i f t d i s t r i b u t i o n s for wings w i t h par t - span def lec ted flaps. Therefore , c o r r e l a t i o n s fo r this case can not be p resen ted a t the p r e s e n t t i m e .

The c o r r e l a t i o n s that are presented -below show the a p p l i c a b i l i t y of the a n a l y s i s t o low aspec t - r a t io w ings ; the c a p a b i l i t y t o p r e d i c t w i n g - i n - j e t effects , t h e p r e d i c t i o n of span loading for s ing le @propel le r conf igura t ions and , f i n a l l y , the a b i l i t y t o predict the l i f t d i s t r i b u t i o n s on ._

twin engine a i rc raf t including those having t ip-mounted'prdp- e l lers .

. .

5 .2 .1 Cor re l a t ions fo r Low Aspect-Ratio Winqs

The a p p l i c a b i l i t y of t h e p r e s e n t method t o l o w aspect r a t i o w i n g s , (see subsect ion 3 .2 .3) , was" Y e r i f ied by performing c o r r e l a t i o n s of the span loading on a rec tangular wing of a s p e c t r a t i o e q u a l t o 1.0. These co r re l a t ions which are shown i n F i g u r e 15, are based on the a n a l y t i c a l r e s u l t s of Reference 2 2 and the a v a i l a b l e t e s t data obta ined from a number of sources .

Figure 15 ( a ) shows a comparison of the p r e d i c t e d span loading (expressed as CJ/a )., w i t h the a n a l y t i c a l d a t a of the two selected References. A s can be noted f r o m this f i g u r e , the predicted r e s u l t s . are i n s a t i s f a c t o r y a g r e e m e n t w i t h t h e r e s u l t s o f Referedces 22 and 43. A l s o F igure 15 ( b ) shows a comparison between the predicted and measured var i - a t i o n s o f l i f t - c u r v e ,slope fo r . a rec tangular wing versus aspect r a t i o . A g a i n , the computed r e s u l t s match those o f References 22 and 43, and agree with the corresponding experimental values .

78

5

4

3

cLa 2 P e r

r a d i a n 1

0

. . . . . .

Reference 2 2

"" Reference 4 3 . .

0

A

0 1 2 3 4 5 6

AR

(b ) V a r i a t i o n of L i f t Curve Slope wi th Aspec t Rat io

Computer Program

Test Data

2.0

1.6

1.2

%/a Per

r a d i a n

. 8

. 4

0 0 .2 ..4 . .6 - . 8 1.0 -

2Y/b

(a) Spanwise L i f t D i s t r i b u t i o n for Rectangular Wing, AR = 1

Figure 15 . V e r i f i c a t i o n of Low Aspect Rat io Analys is

5.2.2 C o r r e l a t i o n for Centrally-Mounted Propellers and Jets

In Re fe rence 6 Stuper measured the l i f t d i s t r i b u t i o n on a rectangular wing having a u n i f o r m c i r c u l a r j e t of a i r blowing over the center span . The je t w a s produced by a s p e c i a l l y d e s i g n e d f a n g e n e r a t i n g a uniform j e t f low without ro t a t ion . Th i s t e s t d a t a was chosen for comparison because it provides a check of the wing-in-s l ipstream theory, wi thout r e f e r e n c e t o t h e propeller a n a l y s i s .

Figure 16 shows a comparison between the p r e d i c t e d and measured spanwise l i f t d i s t r i b u t i o n s f rom S tuper ' s test; and aga in sa t i s fac tory agreement be tween the theory and the experimental data i s obta ined .

Measurements of l i f t d i s t r i b u t i o n s on wings with c e n t r a l l y mounted p r o p e l l e r s are presented in Reference 29. I n this series of tes ts , data was o b t a i n e d f o r a f u l l - s c a l e wing/propel le r combina t ion in the la r g e 30' X 60 ' wind tunne l a t NASA, Langley. The p r o p e l l e r had a diameter o f 4 feet and the wing w a s r e c t a n g u l a r w i t h a 5 foot chord . A s p e c t r a t i o w a s v a r i e d by changing the wing span.

F igure 1 7 p r e s e n t s a comparison between the t h e o r e t i c a l p r e d i c t i o n s a n d the experimental data o b t a i n e d f o r a s p e c t r a t i o of 6.0 f o r wing a lone, wing and nacel le , and wing, n a c e l l e a n d p r o p e l l e r . Similar comparisons for a - w i n g a s p e c t r a t i o o f 3.0 a re shown i n F i g u r e 18. I t can be noted f rom these f igures that t h e combined wing/propeller t h e o r y p r e d i c t s the span loading very w e l l , except near the t i p s where the experimental data shows the characterist ic squa re - t ip l oad ing wh ich canno t be p red ic t ed u s ing l i f t i ng 1 ine theory .

I t should be noted t ha t f o r b o t h the Stuper j e t case (Figure 16 ) and the c e n t r a l p r o p e l l e r cases (F igu res 117 and 18), twenty s ta t ions per semispan were used in the comput- a t i o n s i n o r d e r t o o b t a i n a d e q u a t e d e f i n i t i o n of the Load d i s t r i b u t i o n w i t h i n t h e p r o p e l l e r slipstream region . - -If the p r o p e l l e r slipstream is n o t p r e s e n t , s u f f i c i e n t d e f i n i t i o n i s gene ra l ly ach ieved w i t h the s t anda rd 10 poin ts per semispan .

80

P

1 .o

Conf igura t ion Tes't P red ic t ed

Wing Alone a "-

0.8

0.6

0.4

0.2

0

Jing With Je t 0

-5 -4 -3 -2 -1 0 1 2 3 4 5 Y/R

Figure 16. Comparison Between P r e d i c t e d Spanwise Loading and Measurements

OD of Reference 6 f o r a Rectangular Wing With End Plates Subjected

r t o a Uniform Jet : V,/Vo = 1.36.

03 h)

1.2

1.0

0.8

C1

0.6

0.4

0.2

0

J = 0.42 (Climb Condition)

I

-1.0 -0.8 -0.6 -0.4 -0.2 0 0.4 0.6 0.8 1.0

Figure 17. Predicted Versus Measured Spanwise Loadings for the'Rectangular Wing of Reference 29 With a Centrally-Mounted Propeller; AR = 6.

J = 0.42 ( C l i m b Condi t ion)

1.2

1.0

0.8

C 1

0.6

0.4

0.2

0

Conf igu ra t ion Tes t P red ic t ed

Wing Alone - -___ --- Wing With P

Nacelle Only *-e-. I i

Figure 18. Predic ted Versus Measured Spanwise Loadings for the Rectangular Wing of Reference 29 With a Central ly-Mounted Propel ler ; AR = 3 .

03 w

5 .2 .3 Cor re l a t ion fo r Twin P rope l l e r Co.nfiquratitons

Reference 42 p r e s e n t s the r e s u l t s o f wind tunnel tests on a re f lec t ion-p lane model of a twin-engined tilt- wing VTOL conf igura t ion . The model tested consis ted of a low a s p e c t r a t i o r e c t a n g u l a r (18" X 26") unswept wing w i t h a n a c e l l e and p r o p e l l e r s i t u a t e d a t 62 pe rcen t of the semispan. The wing a i r f o i l s e c t i o n w a s a NACA 0015, the p r o p e l l e r blade sec t ions were of the NACA 16 series and the p r o p e l l e r diameter w a s 26". The tes t r e p o r t p r e s e n t s measured spanwise l o a d d i s t r i b u t i o n s a t v a r i o u s wing angles of attack f o r a l i m i t e d r ange o f p rope l l e r t h rus t coe f f i c i en t s .

Figures 19, 20 , and 2 1 show comparisons of the pre- d i c t e d and measured span loadings for power-off and power- on c o n d i t i o n s , f o r p r o p e l l e r t h r u s t c o e f f i c i e n t s o f CTI=O , 0.36 and 0.64, r e spec t ive ly . I t w a s found t h a t i n o r d e r t o match the measured l i f t d i s t r ibu t ions power-of f , the t h e o r e t i c a l c a l c u l a t i o n s had t o be performed a t angles of attack s l i g h t l y below those values quoted in Reference 42. For example, i n o r d e r t o match the C d i s t r i b u t i o n f o r 5O angle of a t tack, the c a l c u l a t i o n s h a d t o be made a t 4.25O. The reason for t h i s discrepancy is n o t clear s ince , a s is shoim elsewhere, ,predictions for other wings, power-off, agree w i t h the ex,perimental data. The discrepancy could be a t t r i b u t a b l e t o t u n n e l f l o w i n c l i n a t i o n effects. -In the comparisons shown f o r power-on condi t ions, the angle of a t tack values used are those that match the power-off loading.

Despite the differences noted above, the t h e o r e t i c a l p red ic t ions o f the spanwise l i f t d i s t r i b u t i o n a g r e e w e l l w i t h the experimental data of Reference 42 except near the wing r o o t . This discrepancy i s a t t r i b u t e d t o the presence of tunnel w a l l boundary l a y e r effects , a s mentioned in Reference 42 .

In Reference 44, a series o.f tests are repor ted that were made on rec tangular wingsor aspec t ra t io 2.28, 3.26, and 4.7 w i t h an underslung nacelle and propeller placed a t 83 percent , 58 percent and 40 percent of the semispan, respec t ive ly . The p r o p e l l e r was the same p r o p e l l e r used i n the tests of Reference 42. The wing a i r f o i l s e c t i o n was a NACA 4415 series. The tests were conducted f o r wing angles of attack of Oo through 120° a t var ious values of propel ler t h r u s t c o e f f i c i e n t .

84

~

- SYm T e s t Predicted A a = loo a = 8.75O B a = 50 Q = 4.25O

v a = -loo a = - 8 . 7 ~ ~

a = oo a = o0 $ Q = "50 a = -4.250

0.6

1 , . .

0.4

. _

0.2

C1

0

-0.2

-0.4

-0.6

W Q -

0 w

pa-

0 0.2 0.4 0.6 0.8 1.0 2Y/b

Figture 19. P r e d i c t e d Versus Measured spanwise Loadings f o r the Twin-Propeller Configuration of Reference 42; AR = 3.0, CT = 0. S

/

a5

". . . .. ... . . ". _" . . . . .. . .. .. . . -. .

0.4

0.2

0

-0.2

-0.4

-0 .6

P r e d i c t e d a = 8.75O a = 4.25O a = oo

a = -8.750 a = -4.25O

0 0.2 0.4 0.6 0.8 1.0 I 2y/b

I I \

~~ ~.

I \ I 11 I

F i g u r e 20. P r e d i c t e d Versus Measured S,panwise Loadings f o r the Twin-Propel le r Conf igura t ion of Refe rence 42 ; AR = 3.0, C T ~ = 0.36, p75 = 250.

86

0.4

0.2

0

-0.2

-0.4

-0.6

I

\ I

Predicted a = 8.x0 Q = 4.25O a = 00 Q =I -4.25O

= -8.75O

A h A

V V

0 0.2 0.4 0.6 0.8 1.0 I 2 Y / h

I I

I I

0

Figure 21. Predicted Versus Measured Spanwise Loadings f o r the Twin-Propeller Configuration of Reference 42; AR = 3.0, C T ~ 5: 0.64, p75 =-25O.

/

I

Figures 22, 23, and 24 show power-off c o r r e l a t i o n s of the spanwise l i f t d i s t r ibu t ion ob ta ined us ing the p r e s e n t t h e o r e t i c a l a n a l y s i s f o r the three wing a s p e c t r a t i o s a t wing ang le s o f a t t ack below s t a l l . The agreement between the theory and tes t i s good throughout a l l values of wing aspect r a t i o except near the wing r o o t where s u b s t a n t i a l w a l l e f f e c t s are evident . Figures 25 through 27 show the theore t ica l span loading versus the measured loading for a p r o p e l l e r t h r u s t c o e f f i c i e n t C T ~ = 0.4. I n a l l cases, excel lent agreement i s obtained between the predic t ions and the t e s t d i s t r i b u t i o n s .

Although tes t da ta w a s obtained a t angles o f a t tack up t o 120°, the angles o f a t tack were either below s t a l l or w e l l above s t a l l . Thus no data w a s obtained a t the point of i n i t i a l s t a l l o n s e t . N o check of t h e t h e o r y c l o s e t o t h e s t a l l p o i n t is , t h e r e f o r e , a v a i l a b l e from t h i s t e s t s e r i e s .

5.2.4 E f f e c t of Propel le r Rota t ion

The d i r e c t i o n o f r o t a t i o n of propel le rs o f mul t i - p rope l l e r con f igu ra t ions may introduce appreciable changes in the wing span loading. For example, rotation of p r o p e l l e r s i n the same d i r e c t i o n of a twin-propel ler configurat ion causes asymmetry i n the span loading, which in turn gives rise t o t h e a i r c r a f t r o l l i n g moment.

-. ,

Although no t es t data e x i s t s t o v e r i f y t h i s a s p e c t of the present theory, computat ions were performed for the conf igura t ion of Reference 42 to demonst ra te the a b i l i t y of the computer program t o h a n d l e d i f f e r e n t p r o p e l l e r r o t a t i o n s . The p r e d i c t e d r e s u l t s showing the e f f e c t o f p rope l le r ro ta - t i o n s on t h e wing span loading are presented in F igure 28. The r e s u l t s are a p p l i c a b l e t o wing a s p e c t r a t i o of 3.0, wing ang le o f a t t ack o f loo and p r o p e l l e r t h r u s t c o e f f i c i e n t of

cTS = 0.64.

A s can be noted from Figure 28 counterclockwise r o t a t i o n o f both p r o p e l l e r s (as viewed from the rear) r e s u l t s in the asymmetric span loading, which could be i n t e g r a t e d t o y i e l d the a i rc raf t r o l l i n g moment due t o power e f f e c t s .

88 -- .

1.0

0.8

C 1

0.6

0.4

0.2

0

Sym Test - a = 0: a = 10

"g

F i g u r e 22. Predicted Versus Measured S,panwise Loadings f o r the Twin-Prope l l e r Conf igu ra t ion of Reference 44: AR = 4 . 7 , C T ~ = 0.

89

1.0

0.8

C1

0.6

0.4

0.2

0 0 0.2 0.4 0.6 0.8 1.0

2y/b

I

I

Figure 23 . Pred ic ted Versus Measured .Spanwise Loadings f o r the Twin-Propel le r Conf igura t ion of Reference 44; AR = 3.26, C T ~ = 0 .

90

0.8

0.6

C1

0.4

0.2

0 0 0.2 0.4 0.6 0.8 1.0

2 Y / b

Figure 24. Predicted Versus Measured Spanwise Loadings f o r the Twin-Propeller Configuration of Reference 44; AR = 2.28 , C T ~ = 0 .

91

1.0

0.8

0.6

0.4

0.2

0

- Sym Test

o a = 0' EJ Q = 10'

0 0.2 0.4 0.6 0.8 1.0 2Y/b

, \

I

I

, /

Figure, 25. Predicted Versus .Measured Spanwise Loadings f o r the Twin-Propeller Configuration of Reference 44; AR = 4.7, C T ~ = 0.4.

92

0.8

0.6

0.4

0 .2

0

- Sym Test

0 . a = 00 Q = loo

0 0.2 0.4 0.6 0.8 1.0 2Y /b

Figure 26. Predicted Versus Measured Spanwise Loadings f o r the Twin-Propeller Configuration of Reference 44; AR = 3.26, CTs = 0.4.

93.

J

0.6

0.4

C l S

0.2

0

-0.2

Sym- T e s t

0 0.2 0.4 0.6 0.8 1.0 2Y/b

I /

F i g u r e 27. P r e d i c t e d Versus Measured S,panwise Loadings f o r the Twin-Propel le r Conf igura t ion of Reference 44; AR = 2.28, C = 0.4. TS

94

0.4

0.2

0

-0.2 I -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1.0

2Y/b I \ t \

\

Figure 28. Effect of Propel ler Rotat ion on Span Loading for the Configuration of Reference 42; AR = 3.0, C T ~ = 0.64, Q = 10 Degrees. 1.

1

5.2.5 Effect of Flap Deflect ion

One of the prime concerns in the design of modern genera l av ia t ion t ype a i rc raf t i s the e f f e c t of f l a p d e f l e c t i o n ( p a r t - span and fu l l - span) o r b r ing s ta l l ing charac te r i s t ics dur ing take-off and landing, i.e. power-on and power-off conditions r e spec t ive ly . This e f f e c t can be r e a d i l y predicted by the computer program developed herein, however the adequacy of the ana lys i s can no t be ver i f ied because o f the l ack o f su i t ab le experimental data.

Figure 29 demonst ra tes the capabi l i ty o f the cur ren t computer program t o p r e d i c t power-on span load d i s t r ibu t ions assoc ia ted wi th the def lec t ion of par t - span f laps . This f igu re p re sen t s the computed r e s u l t s f o r the twin-propeller configuration of Reference 44, w i t h a n a r b i t r a r y f l a p o f 60 percent span. The p red ic t ed power-off span loadings, with and wi thou t f l ap de f l ec t ion , a r e a l so shown for comparison.

Based on t h e c o r r e l a t i o n s p r e s e n t e d i n t h i s s e c t i o n it i s concluded that the wing-in-slipstream theory developed herein p rov ides an e f f ec t ive ana ly t i ca l tool f o r p r e d i c t i n g t h e e f f e c t s of p rope l le r s l ips t ream on wing spanwise loadings. Unfor tuna te ly , due to the l ack o f su i tab le exper imenta l da ta , t hese co r re l a t ions had t o be l i m i t e d to unflapped wings operat ing a t cond i t ions be low s t a l l . I t is expected, however, t h a t i f t h e p r e s s u r e d a t a was a v a i l a b l e f o r w i n g s a t t h e o n s e t of s t a l l and w i t h par t - span def lec ted f laps , the p resent theory would a l so p rove s a t i s f ac to ry fo r t hese cond i t ions . I t i s the re fo re recommended t h a t this p a r t of the theory be v e r i f i e d by wind tunnel tes ts which should include pressure measurements for bo th the wing and the s l ips t ream for typ ica l wing/propel le r combinat ions operat ing c lose to s t a l l .

96

I

i 1.0

0.8

0.6

C1

0.4

0.2

0

""

"""

"

---"" " -""

Sym Flaps C T ~ - "" rn 0 --- Down 0

0.4

0 0.2 0.4 0.6 0.8 1.0 2y/b

/ \

1 I L ~

1 "_

I .

\ I

Figure 29. Predicted Spanwise Loadings for the Twin- Propel le r Conf igura t ion of Reference 44 to show the Ef fec t o f F l ap Def l ec t ion : AR = 4.7, Q = 10 Degrees.

9 7

,' / SECTION 6 i

'CO@CLUSIONS AND RECOI@ENDATIONS / /

, I I

, I

1. This report p r e s e n t s a n a l y t i c a l m e t h o d s fo r p r e d i c t i n g spanwise lopd d i s t r ibu t ions o f s t ra ight -wing/propel le r combinat ions operat ing up t o s t a l l , f o r a range of a s p e c t r a t i o , f r o m a b o u t 2.0 and higher .

\

2. The ana ly t ic41 methods deve loped here in employ non- l i n e a r l i f t c u r v e s , i n the form of computerized table look-u,p sub rou t ines , fo r a v a r i e t y of wing a i r fo i l s and an e x t e n s i v e s e l e c t i o n of t y p i c a l p r o p e l l e r blade s e c t i o n s . These methods are therefore a p p l i c a b l e t o a wide range of wing/ ,propel le r conf igura t ions and opera t ing condi t ions .

3 . The p r e d i c t e d r e s u l t s f o r both p r o p e l l e r slipstream v e l o c i t y d i s t r i b u t i o n s a n d f o r wing spanwise loadings are g e n e r a l l y i n good agreement w i t h the l i m i t e d test da ta . However, due to the l a c k of s u i t a b l e experimental d a t a involving pressure measurements on wings c lose to s t a l l and with park-span deflected f l a p s , t h e f u l l capabili ty of the a n a l y s i s c o u l d n o t be v e r i f i e d .

4. Based o n t h e c o r r e l a t i o n s shown in Sec t ion 5, it is concluded that the computer ized methods developed herein r e p r e s e n t a n e f f e c t i v e a n a l y t i c a l t o o l f o r p r e d i c t i n g the ,power-on a n d p o w e r - o f f s t a l l i n g c h a r a c t e r i s t i c s of g e n e r a l a v i a t i o n a i rc raf t .

5. I n view of the p r o m i s i n g r e s u l t s o b t a i n e d i n this s tudy , it i s s t rong ly recommended that a comprehensive wind tunnel program be under taken to p rovide the necessary ex,perimental data base t o comple t e ve r i f i ca t ion of the a n a l y s i s . I

6. I t i s f u r t h e r recommended that the wind tunnel t e s t , program must include detailed pressure measurements for both the wing and the slipstream f o r t y p i c a l w i n g / i

p rope l l e r mmbina t ions ope ra t ing t h roughou t the e n t i r e range of angle o f a t tack up to and inc luding s t a l l .

I

98

SECTION 7

REFERENCES

1,. McVeigh, M. A.: and Kisielowski, E.: Design Summary of I Stall Characteristics of S t r a i g h t Wing Aircraft. NASA 1

i CR 1646, June 1971.

2. S i v e l l s , James C.: and Westrick, Gertrude, C.: Method for Calcu la t ing L i f t D i s t r ibu t ions for Unswept Wings w i t h

I , Flaps or Ailerons by use of Nonlinear Section L i f t D a t a . I NACA Rep. 1090, 1952.

: 3. Koning, C. : Inf luence of the P rope l l e r on O t h e r Parts of the Airplane Structure . Aerodynamic Theory (Durand, F. W., Ed i to r ) V o l . 4. Divis ion W. Jul ius Springer , Berl in , 1935.

4. Glauert , H. : "he L i f t and Drag of a Wing Spanning a Free Jet . R & M 1603.

5. Franke, A.: and Weinig, F.: The E f f e c t of the Sl ipstream on an Airplane Wing. NACA TM 920.

6. Stuper, J.: Effect of Propel ler Sl ipstream on Wing and T a i l . NACA m874, 1938.

7. Graham, E. W.: Lagerstrom, P. A.: Licher , R. M.: and Beane, B. J.: A Preliminary Theoretical Inves t iga t ion of the Ef fec ts o f Propel le r S l ips t ream on Wing L i f t . Douglas Aircraf t Co. Report SM-14991, 1953.

8 . Weissinger, J.: The L i f t D i s t r ibu t ion o f Swept-Back Wings. NACA TM 1120, 1947 .

9. Ribner, H. s.: and E l l i s , N. D.: Theory and Computer Study of a Wing i n a Slipstream. AIM Paper No. 66-466, 1966.

10. Brenckmann, J.: Experimental Invest igat ion of t h e A e r o - I dynamics of a Wing i n a Slipstream, J. Aeron. Sci. 8 Vo1.25,

NO. 5 May 1958, pp. 324-328

11, Gobetz, F, W,: A Review of the Wing-Slipstream Problem w i t h Experiments on a Wing Spanning a C i rcu la r Jet. Prince-

, ton University . Department of Aeronautical Engineering 8

Report 489, 1960.

99

12. Snedeker, Richard S.: Experimental Determination of Span- w i s e L i f t Effects on a Wing of I n f i n i t e A s p e c t Ratio Spanning a C i rcu la r Jet. Pr inceton Universi ty , Depart- ment of Aeronautical Engineering, Report 525, 1961.

13,. Rethorst, S.: Royce, W.: and wu8 T. Yao-tsu: L i f t C h a r - acteristics of Wings Extending Through Propeller Slip-. streams. Vehicle Research Corporation Report No. 1, 1958.

14. Goland, L.: Miller, N.: But le r , L.: Effects of P rope l l e r Slipstream on V/STOL Aircraft Performance and Stabil i ty. USAAVLABS Technical Report 64-47, 1964.

15. Huang, K. P.: Goland, L.: and Balin, J.: Charts for Estima- t i n g Aerodynamic Forces on SWL Aircraft Wings Immersed in Propeller Slipstream. Dynasciences Report No. DCR-161, Bureau of Naval Weapons, Department of the Navy, Washing- ton, D O C . # November 1965.

16. Butler, L. : HUand, K. P. : and Goland, L. : An Inves t iga t ion o f P rope l l e r Slipstream Effects on V/STOL Aircraft Perfor- mance and S t ab i l i t y , USAAVLABS Technical Report 65-81, February 1966.

17 . George, M.: and Kisielowski, E.: I nves t iga t ion of P rope l l e r Slipstream Effects on Wing Performance. USAAVLABS Tech- nical Report 67-67, 1967.

18. Fejer,: A. : Lift ing Line Theory for a Wing in L inear ly Varying Flow. Ph.D. Thesis, C a l i f o r n i a I n s t i t u t e of Technology, Pasadena, California, 1945.

19. Glauert , H.: The Elements of A i r fo i l and A i r s c r e w Theory. Cambridge Universi ty Press, 1959.

20. Durand, W. F. ( E d i t o r i n Chief) : Aerodynamic Theory, Volume I V . Dover Publ icat ions, Inc. , New York, 1963.

21. Lock, C. N. H.: and Yeatman, D.: Tables for' Use. i n a n 'Improved Method of A i r s c r e w S t r i p Theory Calculat ion. Aeronauti-ca1 Research Council, Reports and Memoranda 1674, October 1934.

100

22 .

23.

24 .

25.

26 .

27.

28 .

29.

30.

31.

32 .

Kuchemann, D.: A Simple Method for Calculating the Span and Chordwise Loading on Straight and Swept Wings of any Aspect Ratio. R & M 2935, August 1952.

F i l o t a s , L. T.: F i n i t e Chord Effects on Vortex Induced Wing Loads. AIAA Journal Volume 11, No. 6 , June 1973.

I

,

I

&Young, J.: Spanwise Loading f o r Wings and Control I Surfaces of Low Aspect Rat io , NACA TN 2011, August , 1940. I

Abbott, I . H. and Von Doenhoff, A. E.: Theory of Wing I !

Abbott, Jr., F. T. and Von Doenhoff, A. E.: The Langley

I

,

, Sections. Dover, New York 1958.

TWO-Dimensional Low-Turbulence Pressure Tunnel. I

NACA TN-1283 I

Stack, J.: Lindsey, W. F. and L i t t e l l , R. E.: The Com- p re s s ib i l i t y Burb le and the Ef fec t o f Compress ib i l i ty on Pressures and Forces Acting on an A i r f o i l . NACA Rept. N o . 646, 1938.

Summers, J. L. and Treon, S. L.: The Ef fec t s of Amounh and Type of Camber on The Var ia t ion wi th Mach N u m b e r of the Aerodynamic Characteristics of a 10-Percent-Thick, NACA 64A-Series Airfoil Section. NACA TN-2096.

Robinson, R. G. and H e r r n s t e i n , W. H.: Wing-Nacelle- P r o p e l l e r I n t e r f e r e n c e f o r Wings of Various Spans; Force and Pressure Dis t r ibu t ion Tests. NACA RidpfXt No . 569, 1936.

Reid, E. G.: Wake Studies of Eight Model P rope l l e r s . NACA TN-1040, J u l y , 1946.

Draper, J. W. and Kuhn, R. E.: I nves t iga t ion o f the Aerodynamic C h a r a c t e r i s t i c s o f a Model Wing-Propeller Combination and of the Wing and Propeller Separately a t Angles of Attack up t o 90°, NACA TN-3304.

Hammach, J. and Bogeley, A. W.: Prope l l e r F l igh t I n v e s t i g a t i o n t o Determine t h e E f f e c t s of Blade Loading, NACA TN-2022 .

101

33. McLemore, H. C. and Cannon, M. D.: Aerodynamic Invebti- g a t i o n of a Four-Blade Propel ler Operating through an Angle-of- A t t a c k Range from Oo t o 180°, NACA TN-3228

34. C u r r i e , M. M. and Dunsby, J. A.: P re s su re D i s t r ibu t ions and Force Measurements on a VTOL Ti l t ing Wing-Propel ler Model. P a r t 11: Analysis of Results. National R e s - earch Council of Canada, NRC A e r o . R e p t . LR-284, June 1960 .

35. Borst , H. V. and Ladden, R. M.: Propeller Tes t ing a t Zero Veloc i ty , C&/AVLABS Symposium Proceedings: V o l 1, June 1966.

36. Wickens, R. H.: Aerodynamic Force and Moment Charact- erist ics of a Four-Bladed Propel ler Yawed Through 120 Degrees. National Research Council of Canada. NRC A e r o . Rept. LR-454, May 1966.

37. Weick, F. E.: Aircraft Propel ler Design. McGraw H i l l , 1960.

38 . Pinker ton , R. M. and Greenberg, H. : Aerodynamic Characteristics. of a Large Number of A i r f o i l s Tested i n the Var iab le-Dens i ty Wind Tunnel, NACA Report No. 628, 1938.

39. Lindsey, W. F,: Stevenson, D. B. and Daley, B. N.: Aerodynamic Characteristics of 24 NACA 16-Series Ai r fo i l s a t Mach Numbers Between 0.3 and 0.8. NACA TN-1546, September 1948 .

40. Abbott, I. H.: Von Doenhoff, A. E. and S t ive r s , L. S.: Sumnary of A i r fo i l Data. NACA TR-824, 1945.

41. Loft in , L. K.8 Jr. and Sni th , H. A,: Aerodynamic Characteristics of 15 NACA Air fo i l Sec t ions a t Seven Reynolds N u m b e r s from 0.7 x lo6 to 9.0 x 10 . 6

NACA TN-1945, 1949.

42 . Cur r i e , M. M., and Dunsby, J. A.: P r e s s u r e D i s t r i b u t i o n and Force Measurements on a V m L Ti l t ing Wing-Propel ler Model Pa r t 11; Analysis of Resul t s . NRC LR-284, June 1960.

102

43. Falkner, V. M.: The Solu t ion of L i f t ing P lane Problems by Vortex Lattice Theory. R & M 2591, September 1947.

44. N i s h i m u r a , Y.: An Experimental Investigation by Force and Surface Pressure Measurements of a Wing Irmnersed i n a Propel le r S l ips t ream Part 11; Surface Pressure Measurements. NRC LR-525, June 1969.

103

APPENDIX A

PROPELLER TIP LOSS CORRECTION TABLES

This appendix p resents a computer -genera ted l i s t ing of the propeller t i p loss c o r r e c t i o n factors u t i l i z e d by the computer p rogram descr ibed here in . These cor rec t ion fac tors are app l i ed as d e s c r i b e d i n S e c t i o n 4.1 t o o b t a i n a n im,prove- ment t o the a p p r o x i m a t e t i p loss f a c t o r g i v e n by equat ion ( 1 6 ) .

T h e s e c o r r e c t i o n f a c t o r s are based d i r ec t ly on the t abu la t ed va lues gene ra t ed by Lock, as g iven in Reference 21. However, the o r i g i n a l t ab les given by Lock have been modified and en larged to p rovide f o r more uniform increments i n the two parametric v a r i a b l e s , r and s in $ . These changes permit a more e f f i c i e n t table look-u,p in te rpola t ion procedure and pro- v i d e for an improved def in i t ion o f the c o r r e c t i o n f a c t o r s . The add i t iona l and i n t e rmed ia t e va lues o f these f a c t o r s were o b t a i n e d t h r o u g h c r o s s p l o t t i n g o f the o r i g i n a l t a b u l a t e d da ta and by u s i n g s u i t a b l y f a i r e d c u r v e s .

The f o l l o w i n g t a b l e s are ,p re sen ted fo r , p rope l l e r s having either 2 , 3 , o r 4 blades. For ,p rope l le rs wi th more than 4 b lades , the compuber program assumes a c o r r e c t i o n factor of u n i t y f o r a l l va lues o f 2 and s i n # .

104

I - T I P LC.SS CCKRECTICNS - T A B U L A T I O N CF k / F P FOR 2 I ~ L A ~ E C PREPELLEKS

R / R P c 0: 3 0.4 c.5 C.6 0.7 ~ -0 .8 ~” 0.9 1.0 -~ ~

S € N ( P l - I ) /

0.co 1.ooc 1..000 1 . o o c 1.CCO 1.000 1.000 1.G00 $ A 0 0

0.. 0 5 1 . O O C 1 . C O C 1 . C C C 1 . C C O 1 .000 0 .998 Ot9~k”’C.990

(2.10 1.000 1.COO 0.999 0.998 (3.996 G.990 C.983 G.976 ,

0.0 15 0.999 0 .99b 0 .997 0 .994 0 .987 0.97,8, 0.966. ’ ’ 0 .955 1

0.20 0.995 C.994 O.99C 0.984 C.972 (3.959 C,.,$’40 p.923 ,I ,

/

Q.45

0.50

0.55

0 060

0 0.6 5

0.70

O . i ! 5

0.80

0.85

0.90

0 - 9 5

r l .356

0.954

C.953

0.955

0.Y63

0 . 9 7 5

C.992

1.015

1 .056

1.128

1.240

C.341

C.929

G.91@

Go907

0-0 9C 0

C.897

c.E!99

c.. 9 11

0 . 4 4 2

C.990

1 .C6C

0.913

C.894

0 .876

0.851:

0 . 8 4 3

0 .835

0.833

0 . 8 4 0

0.855

0.877

0.906

(2.879

0 .852

0 .827

0 . 8 0 6

6 .791

0 0-700

0 ..777

0.777

0 .i780

0 .784

0 .791

0.831

0 . 8 0 1

c . 7 7 7

0 .759

c . 7 4 4

c .735

C.726

C.718

0 .710

0.7U4

C.638

C. i ’86; ’C‘ .746 0 . 7 1 1

0.757 $.717 0.681

0.734 I 0.694 0 .656 /

/.

Go658 C.6G7 0 . 5 6 4

C.646 0.592 0.547

C.634 0.577 c . 5 3 1

C.623 0.563 0.515

1.00 1,512 1.170 0.940 Cot798 0.632 (3.612 0.5SO 0.500

105

T1.P L C S S CCRKECTICNS - T A B l j L A T I O h i CF F / F P FOR 3 B L A U f O PROP€LLEHS

R'/ R P

0.1c

0.15

0.2c

u.25

0.3C

0.35

0.4C

0 .45

5.50

u.55

0.60

0.65

0 . 7 0

0.75

0.80

0.85

0.90

0 095

1.co

o . 3

1 .ooo

2.000

1 .oco

(3.999

c.997

C . Y 9 5

C.992

c .9e9

C . 9 8 5

C.980

C.976

(2.975

C.978

G.986

c . 9 9 q

1.02c

1.051

1.099

1.163

1.255

1 . 5 3 5

0.4

1 .coo

1.coo

1.OOC

c . 9 9 9

C.996

c.9c33

C.930

G.985

G.975

0 . 9 7 3

C.567

C.964

C.965

C.568

C.976

(2.986

1.co1

1.C28

1 . C 7 3

1,145

1.275

c . 5

1.ccc

1.ccc

0 . S S S

0.997

O.cj94

C.931

C.986

0.975

0.97c

(3.961

0.953

0.947

C.341

0.935

0.938

0 0 9 3 s

(2 .344

0.957

(3.9e4

1.c25

1 . c o c

C.6

1.GOcj

1.oco

G.4S9

0.957

(3.993

C.587

C.980

C.971

(3.553

(3.945

C O S 3 1

C.318

C.906

C.897

0,890

c . 8 ~ 3 5

C.1182

c.886

C.897

Go914

0.935

c.7

1.ooc 1.000

0.999

0.995

C 990

C . Y 8 1

(2.968

c.954

0.Y38

G.919

G.900

G.882

C.865

C. 8 4 9

0.835

C.823

C.815

0.812

0.814

C.815

C.817

0.8

1.ocii c. 999

0.995

0.990

C. 982

0.9-/1

0.955

6.935

G.91L'

C.886

0.861

C.837

Go816

C. 796

c. 779

c;. P63

c. 7 5 0

0.739

C. 7 3 0

C, 722

(2,715

0.9

1.coo G.997

C.992

C.985

6 .972

c.953

0.931

C.905

Go876

(2.845

0.815

C.789

0.765

c.744

G o 7 2 5

0.706

0.689

0.675

C.663

(2.652

0.642

1.0

1.ooci

0.995

0.987

0 . 9 7 6

0.96C

0.938

0.910

6.879

ci.844

0.803

3077.7

lJ.749

0.724

0.701

(3.680

0.661

0.644

0.627

0.611

0.597

0.583

106

T I - P L E S S C C H R E C T I C N S - TABCILbTION CF F / F P FOR 4 BLADED P R O P E L L E R S

0.25

0.30

0.35

0 .4i)

0.45

0.50

0.55

0.6C

0 - 6 5

0.70

0.75

0.. 8 0

0.85

0 . 9.0

0.95

1.00

0.998 C.998 0.596

C.996 c.995 0.593

‘2.994 C.992 C.9139

0 .991 C.588 0.984

C.987 C..Y84 0.97s

C.985 0 .981 0.$74

c . 3 8 5 0.980 0 . 9 7 ~

C.987 (2.981 0.966

c.993

1.004

1.022

1.049

1.090

1 .155

1.25C

1.493

0.994

C.591

0.98s

0.977

0.967

0.958

0.949

0.942

0.991

0.984

0.974

C . 9.6 2

0.9411

c.933

Go919

(3.905

Q.985 0.965 OoG32 0.892 0.848 Co8Cl

C.991 0.965 0.926 C.8UO G o 8 3 3 (2.783

1.004 0 . 5 6 5 0.922 0.870 G o 8 1 8 0.765

1.021 0.976 0.921 0.862 0.806 (3.748

1.C48 01989 0.923 0.857 0.795 0.733

1 . ~ 9 0 1 . ~ 0 ~ 0.929 0.855 (2.7~6 c .720

1.156 1.043 0.943 C.857 0.779 0.709

1.267 1.095 0.965 0.862 0.773 0.699

(3.756

0.737

0.719

0.701

0.684

0.667

0.652

0.637

107

I APPENDIX B

PROPELLER AIREXIIL T.ABLE-S Y 3 . -

."

[ This ap.pndi-x presents a, computer-generated l i s t i n g of the pro-peller blade s e c t i o n data t a b l e s that are a v a i l a b l e for use by the computer program.

E&h table con ta ins the va lues o f l i f t c o e f f i c i e n t verpus angle of attack f o r a range of Mach number condi t ions f o r : o n e s p e c i f i e d a i r f o i l s ec t ion . A t the head of each table i s ' desc r ip t ive i n fo rma t ion on the a i r f o i l name and d a t a source. This is follows by the a i r f o i l series code identi- f ica t ion and the main geometric and t e s t ,parameters.

The fo l lowing tab les conta in the selected p rope l l e r s tatim characteristics f o r a i r f o i l s of the U.S.N.P.S., Clark Y, NACA 16, NACA 64 and NACA 65 families. The sets of tables f o r each a i r f o i l series are a r r anged i n the s p e c i f i c order of increasing Mach number, thickness/chord ratio and design l i f t c o e f f i c i e n t , as necessa ry fo r p rope r u t i l i za t ion by the computer program.

108

PROPELLER BLADE SECTION A IRFOIL TABLES

A I R F O I L SECTIOIY

TASLB DATA SOURCE

A I R F O I L SER CODE

O E S I C N L I F T C O E F F THICKNESS / CHORO MACH NUMBER ZERO L I F T ALPHA EXTRAP COEFF K C L I

ALPHA, CL VALUES

A l R F O I L S E C T I O N

TABLE DATA SUUKCC


OESICN L I F T COEFF THlCKNESS / CHORO MACH NUMBER Z E R O L I F T ALPHA EXTRAP COEFF K C L I

ALPHA, CL VALUES

USNPS -1104

F E WEICK

1

0.000 0.040 0.070

-1.900 0 . 0 0 0

-1.900 0 .000 0 . 0 0 0 0.200 2.000 0.420 4.000 0.615 6.000 0.760 8.000 0.860

10.000 0.915 12.000 0.930 14.000 0.925 16.000 0.885 0.000 0.000 0.000 0.000

USNPS ”16

F E WEICK

1

0 .000 0.160 0.070

-7.400 0.000

-7.400 0 .000

-4.000 0.270 -2.000 0.450

2.000 0.795 4.000 0.950 6 .000 1.095 8.000 1.210

10.000 1.305

13.000 1.365 12.000 1.375

16.000 1.195 14.000 1.305

0 .000 0 .000

-6.000 0.110

USNPS -Mob

F E WEICK

1

0.000 0.060 0.070

-2.600 0.000

-2,600 0.000

4.000 0.660 2.000 0.465

6.000 0.850 8.000 1.020

10.000 1.075 12.000 1.040 14.000 1.010 16.000 0.990 0.000 0.000 0.000 0.000 0.000 0 .000

USNPS “18

F E WEICK

1

0 .000 0.180 0.070

-8.700 0 .000

-8.700 0.000 -4.000 0.320 -2.000 0.490

0.000 0.650 2.000 0.795 4.000 0.950 6.000 1.070 8.000 1.165

10.000 1.240 12.000 1.285 14.000 1.220 16.000 1.135 0.000 0.000 0.000 0.000

USNPS -1100

F E Y E I C K

1

0.000 0.080 0.070

-3.350 0.000

-3.350 0.000 4.000 0.735 6 .000 0.925

10.000 1.200

12.000 1.160 11.000 1.215

16 .000 1.060 14.000 1.105

0.000 0.000 0.000 0.000 0.000 0.000

8.000 1.105

USNPS -HZ0

F E WEICK

1

0.000 0.200 0.070

-10.200 0 .000

-10.200 0.000

-4 .000 0.345 -6 .000 0.215

-2.000 0.490 0 .000 0.630

4.000 0.890 2.000 0.760

6 .000 1.010

10.000 1.165 8.000 1.110

11.000 1.1.80 12.000 1.170 14.000 1.135 16.000 1.060

USNPS “10

F E Y E I C K

1

0.000 0.100 0.070

-4.300 0.000

-4.300 0.000 4.000 0.800 6.000 0.985 8.000 1.145

10.000 1.310 11.000 1.395 12.000 1.435 14.000 1.235 16.000 1.095 0.000 0.000 0.000 0.000 0.000 0.000

0

0.000 0.000

0 .ooo 0.000

0.000

0.000 0.000

0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0-000

0.000 0.000 0.000 0.000

0.000 0.000

USNPS -1112

F E WEICK

1

0.000 0.120 0.070

-5 250 0.000

-5.250 0.000

4.000 0.880 2.000 0.700

8.000 1.220 6.000 1.055

10.000 1.370 12.000 1.470 13.200 1.490 14.000 1.480 16.000 1.240 0.000 0.000 0.000 0.000

0

0.000 0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000 .o.ooo 0.000 0.000

0.000 0.000 0.000 0.000

0.000 0.000

USNPS -)I14

F E Y E I C K

1

0.000 00140 0.070

-6.300 0.000

-6.300 0.000

-2.000 0.390 -4.000 0.200

2.000 0.765 4.000 0.935 6.000 1.090 8.000 1.230

10.000 1.355 12.000 1.440 13.000 1.430 14.000 1.385 16.000 1.230

0

0.000 0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000 .o.ooo 0.000 0.000 0.000 0.000 0.000 0.000 ,’ 0.000 0.000 0.000 0.000 o..ooo 0.000 ’ 0.000 0.000 0.000 0;ooo 0.000 0.000

r r 0

PROPELLER BLADE SECTION AIRFOIL TABLES

AIRFOlL SECTION

TA8LE D A T A SOURCE

AlRFOIL SER C O O €

DESIGN L IFT COEFF THICKNESS / CHORD NACH NUMBER Z E R O L IFT ALPHA E X T R A P COEFF KCLI

ALPHA, CL: VALUES

AIRFOIL SECTION

TABLE DATA SOURCE

AIRFOIL SER CODE

DESIGN L IFT COEW THICKNESS / CHORD HACH NUMBER ZERO LIFT ALPHA E X T R A P COEFF KCLI

ALPHA, CL VALUES

CLARK.Y-MO6

NACA TR-628

2

0 .000 0.060 0 .060

-2.950 0 . 0 0 0

-6.000 -0.300 -2.950 0 .000

6.000 0.865 8.000 0.985

10.000 1.060 12.000 1.070 14.000 1.050 16.000 1.020

0 . 0 0 0 0 . 0 0 0 0.000 0.000 0.000 0 . 0 0 0 0 . 0 0 0 0 . 0 0 0

4.000 0.680

CLARK.Y-MZ2

NACA TR-628

2

0.000

0.060 0.220

-9.290 0 .000

-9.290 0.000

-4.000 0.480 -6.000 0.300

0 . 0 0 0 0.800 2.000 0.950 4.000 1.085 6.000 1.185 8.000 1.265

10.000 1.320 12.000 1.350 13.000 1.360

20.000 1.240 16.000 1.300 14.000 1.340

CLARK-Y-HOB

NACA TR-628

2

0.000 0.080 0.060

-3.560 0.000

-6.000 -0.235 -3.560 0.000

6.000 0.925 8.000 1.110

10.000 1.280 11.500 1.370 12.000 1.290 14.000 1.170 16 .000 1.100 0 .000 0.000 0.000 0 . 0 0 0 0 .000 0 .000 0 .000 0 .000

NACA 16106

NACA TN-1546

3

0.100 0.060 0 .300

-1.050 0.760

-2.000 -0.085 -1.050 0.000

0.000 0.100 2.000 0.305 4.000 0.510 6.000 0.670 8.000 0.8.15 9.000 0.850

10.000 0.855 .o.arro- 0.000 0 .000 0.000

0.000 0.000 0.000 0.000

0.000 0.000

CLARK-Y-MI0

NACA TR-628

2

0.000 0.100 0.060

-4.560 0 .000

-6.000 -0.130

2.000 0.640 4.000 0.830 6.000 1.015

-4.560 0.000

8.000 1.195

12.000 1.500 14.000 1.635 14.800 1.680 16.000 1.420 20.000 1.220

0 .000 0 .000

1o.aoo 1.365

NACA 16106

NACA TN-1546

3

0.100 0.060 0.450

-1.100 0.760

-2.000 -0.090 0.000 0.105

4.000 0.540 2.000 0.305

6.000 0.705 8.000 0.820

10.000 0.810 9.000 0.835

0.000 0.000

0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000 0.000

CLARK-Y

NACA TR-628

0.000 0.117

-5 .000 0 060

0.000

-6.000 -0.095 -5.000 0.000 -2.000 0.285

6.000 1.030

10.000 1.360 8.000 1.200

12.000 1.485 14.000 1.610

16.000 1.500 20.000 1.300

0 .000 0.000 0 .000 0.000

15.300 1.680

NACA 16106

NACA TN-1546

3

0.100 0 060

-1.000 0.600

0.760

-2.000 -0.105 0.000 0.110 2.000 0.340 4.000 0.600 5.000 0.705 6.000 0.780 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

CLARK-Y-Ill4

NACA TR-628

2

0.000 0.140 0.060

-6.200 0.000

-6.200 0.000

4.000 0.985 2.000 0.800

6.000 1.160

10.000 1.465 8.000 1.315

12.000 1.590 14.000 1.720

20.000 1.430 16.000 1.550

0.000 0.000 0.000 0.000 0.000 0.000

NACA 16106

NACA TN-1546

3

0.100 0.060 0 e 700

-1.000 0.160

-2.000 -0.130 0.000 0.120 2.000 0.37s 3.000 0.540 3.770 0.725 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.000’ 0.000 0.000 0.000

0.~000 0.000

CLARKeY-MlB

NACA TR-628

2

0.000 0.180

-7.600 0.060

- 0 . 0 0 0

-7.600 0.000 -6.000 0.160 -4.000 0.350

2.000 0.890 4.000 1.040 6.000 1.185

10.000 1.420 8.000 1.315

12.000 1.470 13.000 1.480 14.000 1.470

20.000 1.360 16.000 1.430

NACA 16106

3

0.100 0.060 0.750

-1.000 0.760

-2.000 -0.145 2.000 0.405 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000.

0.000 0.000 0.000 0.000 0.000 0.000


AIRFOIL SECTION

TABLE DATA SOURCE

AIRFOIL SER COOE

DESIGN L IFT COEFF THICKNESS I CHORO MACH NUWER LfRO LIFT ALPHA EXTRAP COEFF KCLI

ALPHA, C'L VALUES

AIRFOIL SECTION

TABLE DATA SOURCE

AIRFOIL SER COOE

DESIGN LIFT COEFP THICKNESS / CHORO FACH NUMBER Z E R O LIFT ALPHA E X T R A P C O E F F KCLI,

9LPHAq C C VALUES

NACA 16106

NACA TN-1546

3

0.100 0.060 0.800

-0.950 0.760

-2.000 -0.175 -0.950 0.000

0.000 0.000 1.770 0.450

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

N A C A 16109

NACA TN-1546

3

0.100 0.090

-1.000 0.800

0.730

-2.000 -0.130 -1.000 0.000

0.000 0.000 1.770 0.360

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

NACA 16109

NACA TN-1546

3

0.100 0.090 0.300

-1.000 0.730

-2.000 -0.085 -1.000 0.000

0.000 0.090 2.000 0.295 4.000 0.465 6.000 0.660

10.000 0.835 8.000 0.790

11.000 0.850 0 . 0 0 0 0 .000

NACA 16115

NACA TN-1546

3

0.100 0.150 0.300

-0 .eoo 0.600

-2.000 -0.110 -0.800 0.000

0 .000 0.080 2.000 0.260 4.000 0.350

6.000 0.445 5.000 0.390

7.000 0.515 8.000 0.620

10.000 0.785 11.770 0.855

NACA

NACA

, 16109

TN-1546

3

0.100 0 090 0.450

-1.100 0.730

-2.000 -0.080 0.000 0.095

4.000 0.470 2.000 0.295

6.000 0.665

9.000 0.805 8.000 0.785

10.000 0.795 11.000 0.775 12.000 0.755

NACA 16115

NACA TN-1546

3

0.100 0.150

-0.800 0.450

0.600

-2.000 -0.110 0.000 0.080 2.000 0.260 4.000 0.330 5.000 0.380 6 .000 0.445 8.000 0 .620

10.000 0.790 11.000 0.800 11.770 0.780 0.000 0.000

NACA

NACA

, 16109

TN-1546

3

0.100 0.090 0.600

-1.000 0.730

-2.000 -0.090 0.000 0.100 2.000 0.320 4.000 0.500 6.000 0.700 7.000 0.750

10.000 0.790 12.000 0.785

0.000 0.000

8.000 0.780

NACA 16115

NACA TN-1546

3

0.100 O.lS&-J" 0.600

-0.800 0.600

-2.000 -0.115 0.000 0.085 2.000 0.280 4.000 0.355 5.000 0.405 6.000 0.470 8.000 0.620

10.000 0.860 11.000 0.830 11.770 0.690

0.000 0.000

NACA 16109

NACA TN-1546

' 3

0.100 0.090 0.700

-1.000 0.730

-2.000 -0.100 0.000 0.115 2.000 0.365 4.000 0.520 0.000 0.000 0.000 q.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

NACA 16115

NACA TN-1546

3

0.100 . 0.150

-0 . BOO 0 600

0.70-0 . -

-0.800 0.000 -2.000 -0.130

0.000 0.085 2.000 0.295

0.000 0.000 3.770 0.400

0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000

0.000 0.000

NACA 16109

NACA TN-1546

3

0.100 0.090 0.750

-0.850 0.730 I

~

-2.000 -1.000

0.000 2.000 3.770 0.000 0.000

0.000 0.000

0.000

-0.110 I

-0.025 i '0.125 0.420

0.000 0.630

0.000. 0.000

0.000 0.000

NACA 16130

NACA TN-1546

3

0.100

0.300 0.500

' 0.300 . . . "

-0.190

-2.000 -0.110 0.000 -0.025 0.500 0.000

4.000 0.195 2.000 0.100

6.000 0.245

10.000 0.295 8.000 0.260

11.770 0.345 0.000 0.000 0.000 0.000

r r I-J


AIRFOIL SECTION

TABLE O A T A SOURCE

AIRFOIL SER CODE

DESIGN L IFT COEFF THICKNESS / CHORO MACH NUMBER L E K 0 LIFT ALPHA E X T R A P COEFF KCLI

ALPHA, CL VALUES

AIRFOIL SECTION


AIRFOIL SER CODE

DESIGN L IFT COEFF THICKNESS / CHORO MACH NUMBER Z E R O L I F T ALPHA: E X T R A P COEFF KCLI

ALPHA, CL VALUES

NACA 16130

NACA TN-1546

3

0.100 0.300 0.450 0.500

-0.190

-2.000 -0.140 0.000 -0.030 2.000 0.080 4.000 0.160 6.000 0.180 8.000 0.185

10.000 0.235 11.770 0.290 0.000 0.000 0.000 0.000

NACA 16306

N A C A TN-1546

3

0.300 0.060 0.750

-2.200 0.760

-4.000 -0.340 -2.200 0.000

0.000 0.360 1.770 0.730 0.000 0.000 0.000 0.000 0.000 0.000 0.000 o.oc0 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

NACA 16130

NACA TN-1546

3

0.100 0.300 0.600

-0.500 -0.190

-2.000 -0.155

0.000 0.030

4.000 0.085 2.000 0.030

6.000 0.105 8.000 '2.135 9.770 0.190 0.000 0.000 0.000 0.000

-0.500 0.000

NACA 16309

NACA TN-1546

3

0.300 0.090 0.300

-2.450 0.730

-4.000 -0.155 -2.450 0.000 -2.000 0.045

0.000 0.240 2.000 0.450 4.000 0.610 6.000 0.750 8.000 0.905

10.000 0.975 9.000 0.950

11.000 0.980 11.770 0.960

N A C A 16306

NACA TN-1546

3

0.300

0.300 0.060

-2.400 0.760

-4.000 -0.190 -2.400 0.000 -2.000 0.045

0.000 0.265 2.000 0.490 4.000 0.640 6.000 0.800 8.000 0.960 9.770 1.010 0.000 0.000

N A C A 16309

NACA TN-1546

3

0.300 0.090 0.450

-2.600 0.730

-4.000 -0.160 -2.000 0.055

2.000 0.455 4.000 0.620 6.000 0.800 8.000 0.930 9.000 0.955

10.UOO 0.950 11.770 0.920 0.000 0.000 0.000 0 . 0 0 0

0.000 0.250

NACA 16306

NACA TN-1546

3

0.300 0.060 0.450

-2.400 0.760

-4.000 -0.190

0.000 0.270 2.000 0.490 4.000 0.675 6.000 0.865 8.000 0.965 9.000 0.980 9.770 0.980 0.000 0.000

-2.000 0.045

NACA 16309

NACA TN-1546

3

0.300 0.090 0.600

-2.500 0.730

-4.000 -0.190 -2 .000 0.050

0.000 0.270

4.000 0.690 2.000 0.490

6.000 0.900 8.000 1.080 9.000 1.070 9.770 0.950

0.000 0.000 0.000 0.000

0.000 0.000

NACA 16306

NACA TN-1546

3

0.300

0.600 -2.300

0.760

0.060

-4.000 -0.220

0.000 0.295

2.000 0.540 1.000 0.420

-2.000 0.035

3.000 0.650 4.000 0.780

6.000 0.970 5.000 0.895

7.770 1.010

NACA 16309

NACA TN-1546

3

0 300 0.090

-2.500 0.700

0 730

-4.000 -0.190 0.000 0.320 3.770 0.890 0.000 0 .000 O.ObO 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0 .000 0.000 0.000 0.000 0 .000 0.000 0.000

NACA 16306

NACA TN-1546

3

0.300 0.060 01700

-2 300 0.760

-4.000 -0.260 ,3.770 0.920 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0 .000 0.000 0.000 0 .000 0.000 0.000

NACA 16309

NACA TN-1546

3

0.300 0.090 0.750

-2 350 0.730

-4.000 -0.205 -2.350 0.000

0.000 0.000 1.770 0.650

0.000 0.000 0.000 0.000 0 .000 0.000 0 .000 0.000 0.000 0.000 0.000 0.000 0 .000 . 0 .000 0.000 0.000


AIRFOIL SECTION

TABLE OATA SOURCE

AIRFOIL SER CODE

DESIGN LIFT COEFf THICKNESS / CHORO Y A C H NUMBER Z E R O LIFT ALPHA E X T R A P COEFF KCLI

ALPHA, CL VALUES

AIRFOIL SECTION


AIRFOIL SER CODE

DESIGN LIFT COEFF THICKNESS / CPORO MACH NUMBER Z E R O LIFT ALPHA E X T R A P COEFF KCLI

ALPHA, CL VALUES

NACA 16312

NACA TN-1546

3

0 . 1 2 0 0.300

-2 .600 0.300

0.690

-4.000 -0.110 -2.600 0 .000 0.000 0.215 2 .000 0.410

4.000 0.540 3.000 0.485

6.000 0.700 8.000 0 .835

10.000 0.935 11.000 0.965 11.170 0.960 0.000 0.000 0.000 0.000

NACA 16315

N A C A TN-1546

3

0.300 0.150

-3.600 0.600

0 . 6 0 0

-4.000 -0 .020 -2 .000 0.075 -1.000 0.115 0.000 0 . 2 0 0 2.000 0.415 3.000 0.515 4.000 0.570 6.000 0.620

10.000 0.940 8.000 0.790

11.770 1.050

NACA 16312

NACA TN-1546

3

0.300 0.120

-2.800 0.450

0.690

-4.000 -0.100 -2.000 0.065 0.000 0.230

4.000 0.550

6 .000 0.715 8.000 0.900 9.000 0.940 9.770 0.950 0.000 0.000 0.000 0.000 0.000 0.000

2.000 0.420

5.000 0.620

NACA 16315

NACA TN-1546

3

0.300 0.150

-3.750 0.600

0.700

-3.750 0.000 -2.000 0.085 -1 . .OOO 0.130

0 . 0 0 0 0 . 1 9 0

2.000 0.440 3.000 0.580 4.000 0.725 7.000 0.725 7.770 0.740 0.000 0 .000

1 .000 0 .290

NACA 16312

NACA TN-1546

3

0.300

0 .600 0.120

-2.800 0.690

-4.000 -0.105 -2.000 0.080

0.000 0.265 2.000 0.490 3.000 0.570

5.000 0.680 6.000 0.770

0 .000 0 . 0 0 0 0 . 0 0 0 0.000 0.000 0.000 0 .000 0.000

4.000 0.615

7.770 0.940

NACA 16321

NACA TN-1546

3

0.300 0.210

-1,300 0.300

0.370

-4.000 -0.100 -3.000 -0.090 -2.000 -0.060 -1.300 0.000

4 .000 0.425 5.000 0.460 6 .000 0.485

11.710 0.720 8.000 0.500

0.000 0.000

2.000 0.270

NACA 16312

NACA TN-1546

3

0 300 0.120

-3.000 0.690

0.700

-4.000 -0.080 -3.000 0.000

0.000 0.285 2.000 0.505 3.770 0.705 0.000 0.000 0.000 0 .000 0.000 0.000 0.000 0.000 0.000 0.000 0 .000 0.000 0 .~000 0.000 0 .000 0.000

NACA 16321

NACA TN-1546

3

0.300 0.210

-1 300 0.450

0.370

-4.000 -0.030 -3.000 -0.070 -2.000 -0.060

2.000 0.255 -3.000 0.330 4.000 0.400

11.770 0.680 8.000 0.495

0.000 0.000 0.000 0.000 0 .000 0.000

NACA 16315

NACA TN-1546

3

0 300 0.150

-2.100 0.300

0.600

-4.000 -0.070

-2.100 0.000 -3.000 -0.040

0.000 0.195 2.000 0.370 4.000 0.515

6.000 0.570 5.000 0.540

7.000 0.655 8.000 0.760

10.000 0.885

11.770 0.930 11.000 0.915

NACA 16321

NACA TN-1546

3

0.300 0.210

-1.100 0.600

0.370

-4.000 0.000 -2.340 -0.100 -1.100 0.000

4.000 0.410 3.000 0.345

6.000 0.430 8.000 0.480

11.770 0.790 10.000 0.580

0.000 0.000 0.000 0.000

NACA 16315

NACA TN-1546

3

0.300 0.150

-3.000 0.450

0.600 I

-4.000 -0.045 I -2.000 0.050 -1.000 0.110

0.000. 0 .200 2.000 0.400 4.000 0.520 6.000 0.580 7.000 0.650 8.000 0.750

10.000 0.890 11.770 0.960

0.000 0.000 0.000 0.000

N K A 16506

NACA TN-1546

3

0.500 0.060

-3.900 0.760

0.300

-3.900 0.000

2.000 0.625 1.000 0.525

4.000 0.740 6.000 0.895

9.000 1.090 8.000 1.050

0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000

PROPELLER 8LAOE SECTION AIRFOIL TABLES

AIRFOIL SECTION

TABLE OATA SOURCE

AIRFOIL SER CODE

DESIGN LIFT COEFF THICKNESS / CHORO MACH NUMBER ZERO LIFT ALPHA E X T R A P COEFF KCLI

ALPHA, CL VALUES

AIRFOIL SECTION


AIRFOIL SER CODE

DESIGN LIFT COEFF THICKNESS / CHORO MACH NUMBER LERO LIFT ALPHA E X T R A P COEFF KCL€

ALPHA, CL VALUES

NACA 16506

NACA TN-1546

3

0.500 0.060 0.450

-3.900 0.160

-4.000 -0.010 2.000 0.660 3.000 0.720 4.000 0.795 6.000 0.970 7.770. 1.085

NACA 16509

NACA TN-1546

3

0.500 0.090 0.600

0.730 -4.200

-4.000 0.020

2.000 0.720 1.000 0 .605

4.000 0.810

0.000 0.000 5.770 0.930

0.000 0.000

NACA 16506

NACA TN-1546

3

0.500 0.060 0.600

-3.700 0.760

-4.000 -0.040

2.000 0.730 1.000 0.605

0.000 0.000 3.770 0.765

0.000 0.000

NACA 16509

NACA TN-1546

3

0.500 0.090 0.700

-4.200 0.730

-5.000 -0.085 -4 .000 0.020

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

1.770 0.700

NACA 16506

NACA TN-1546

3

0.500 0.060 0. 700

-3.500 0.760

-4.000 -0.080 -2.000 0.230

0.000 0.000 1.770 0.830

0.000 0.000 0.000 0.000

NACA 16509

NACA TN-1546

3

0.500 0.090 0.750

-4.100 0.730

-4.100 0.000 0.000 0.560 1.710 0.750 0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000

NACA 16506

NACA TN-1546

3

0.500 0.060 0.750

-3.400 0.760

-3.400 0.000 -2.000 0.240

0.000 0.595 1.770 0.050 0.000 0.000 0.000 0.000

NACA 16512

NACA TN-1546

3

0.500 0.120

-4.200 0.300

0.690

-4.200 0.000 2.000 0.545 4.000 0.100 6.000 0.170

10.000 1.010 8.000 0.905

11.770 1.070

NACA U S 0 9

WACA TN-1546

i

0.500 0.090 0.300

-4.200 0.730

-4.200 0.000

4.000 0.110 2.000 0.615

7.170 0.990 0.000 0.000 0.000 0.000

NACA 16512

NACA TN-1546

3

0.120 0 500

-4;250 0.450

0 690

-41000 0.02s 2.000 0.510 3.000 0.645 4.000 0.700 6.000 0.790

0.000 0.000 1.770 0.910

NACA 16512

NACA TN-1546

3

0.500 0.120

-4.200 0.600

0.690

-4.000 0.02s ob000 0.420 2.000 0.620

4.000 0.175 3.000 0.705

0.000 0.000 5.710 0.840


AIRFOIL SECTIOY

TABLE DATA SOURCE

AIRFOIL SER CODE

D E S I G N LIFT COEFP THICKNESS I CHORO MACH NUMBER Z € R O LIFT ALPHA E X T R A P COEFF KCLI

ALPHA, CL' VALUES

AIRFOIL SECTION

TABLE DATA SOURCE

AIRFOlL SER CODE

DESIGN L€FT COEFF THICKNESS I CHOKO MACH NUMBER Z E R O LIFT ALPliA EXTKAP COEFF KCLI

ALPI~AI CL VALUES

NACA 16512

NACA TN-1546

3

0.500 0.120 0.700

-4.100 0.690

-4.000 0.005 -2.000 0.225

0.000 0.440 2.000 0.675 3.000 0.780

0.000 0.000 3.770 0 . 8 2 5

0.000 0.000 0.000 0.000 0 .ooo 0 .ooo 0.000 0.000

NACA 16515

NACA TN-1546

3

0.500

0.750 0.150

-1.600 0.600

-2.000 -0.045 -1.600 0.000

0.000 0.145 1.000 0.210 2.000 0.295 3.770 0.480 0.000 0.000

0.000 0.000 0.000 0.000

0.000 0.000

NACA 16512 NACA 16515 NACA 16515

NACA TN-1546 NACA TN-1546 NACA TN-1546

3 3 3

0.500 0.120

-3.600 0.750

0.690

-3.600 -2.000

0.000 2.000 3.770 0.000 0.000 0.000 0.000 0.000 0.000

0.000 0.200 0.380 0.530 0.680 0.000 0.000 0.000 0.000 0.000 0.000

0.150 0.500

-4.400 0.300

0.600

-4.400 -4 .OOO -2.000

0.000

4 .ooo 2.000

6.000 8.000

10.000 12.000 13.770

0.000 0.025 0.155 0.305 0.495 0.660 0.710 0.815 0.920 1.010 1.030

0.500 0.150

'4 500 0.450

0.600

-4.000 -2.000

0.000 2 .ooo

4.000 3.000

6 000 8.000

10.000 12.000 13.770

0.040 0.195 0.325 0.520 0.600 0.655 0.715 0.825 0.940

1.010 1.025.

NACA 16521 NACA 16521 NACA 16521

NACA TN-1546 NACA TN-1546 NACA TN-1546

3

0.500

0.300 0.210

-2.300 0.370

-2.000

4.000 3.000

6.000 5.000

8. 000 10.000 11.770 0.000 0.000

0.030

0.540 0.455

0.605 0.650

0.750 0.685

0.000 0.860

0.000

3

0.500

0.450 0.210

-2.000 0.370

-2.000 -1.000

0.000 2.000 4.000 6.000

11.770 8.000

0.000 0.000

0.070

0.180 0.000

0.360 0.520 0.620 0.670 0.840 0.000 0.000

3

0.500

0.600 0.210

-1.800 0.370

-2.000 -1.000

0.000

4.000 3.000

5.000 6.000

8.000 7.000

9.770

0.120

0.170 0.120

0.530 0.440

0.590 0.610

0.750 0.910

0.660

NACA 16515

NACA TN-1546

3

0.500 0.150

-4.600 0.600

0.600

-4.000 0.045 -2.000 0.220

0.000 0.325

4.000 0.720 2.000 0.525

6.000 0.780 7.000 0.840 7.770 0.915 0.000 0.000 0.000 0 .000 0.000 0.000

NACA 16521

NACA TN-1546

3

0.500 0.210 0 700

-0.400 0.370

-2.000 -0.210 -1.000 -0.080 -0.400 0.000

0.000 0.035 2.000 0.020 3.000 0.190

0.000 0.000 3.770 0.270

0.000 0.000 0.000 0.000

NACA 16515

NACA TN-1546

3

0.150 0.500

-4 700 0.700

0.600

-4.000 0.065 -2.000 0.235 -1.000 0.265

0.000 0.315 2.000 0.530

0.000 0 .000 3.770 0.710

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

NACA 16530

NACA TN-1546

3

0.500

0.300 0.300

1 500 -0.190

0.000 -0.100 6.000 0.300 8.000 0.405 3.770 0.470 0.000 0 .000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

r r ul

PROPELLER BLADE SECTION A I R F O I L T A B L E S

A I R F O I L S E C T I O Y

TABLE DATA SOURCE


D E S I G N L I F T C O E F F 1 H I C K N E S S / CHORO k A C H NUMBER ZERO L I F T ALPHA E X T R A P C O E F F K C L I

ALPHA, CL VALUES

A I R F O I L S E C T I O N

TABLE DATA SOURCE

A I R F O I L SER COO€

D E S I G N L I F T C O E F F THICKNESS I CHORO MACH hlUM8ER ZERO L I F T ALPHA t X T R A P C O E F F K C L I

ALPHA, C L V A L U E S

NACA 16530

NACA TN-1546

3

0 . 3 0 0 0 .500

0.450 2.200

-0.190

0.000 -0.120

4.000 0.115 2.000 -0.015

6.000 0.245 8.000 0.355 9 .770 0.415 0.000 0.000 0 .000 0.000 0.000 0 .000 0.000 0.000 0 .000 0.000

NACA 16709

NACA TN-1546

3

0.700 0.090 0.750

-5.400 0.130

-4.000 0.150 -2.000 0.375

0.000 0.585 1.000 0.620 2.000 0.670

0 .000 0 .000 3.770 0.820

0 .000 0 .000 0 .000 0.000

NACA 16530

NACA TN-1546

3

0.500 0.300 0.600 3.200

-0.190

0.000 -0 .280

4.000 0.070 3.200 0.000

6.000 0.120 8.000 0 .260 9.773) 9.430 0 .000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0 .000

NACA 16709

NACA TN-1546

3

0.700 0.090 0.775

-3.500 0.730

-4.000 -0.100 -3.500 0.000 -2.000 0.205

0.000 0.460

0.000 0.000 1.170 0.535

0.000 0.000 0.000 0.000 0.000 0.000

NACA 16109

NACA TN-1546

3

0.100 0.090 0.300

-5.400 0 .730

-6.000 -0.065 -5.400 0.000 -4.000 0.140 -2.000 0.345

2 .000 0.755 3.000 0.815 4.000 0.850 5.000 0.895 6.000 6.965 1.770 1.080

0 .000 0.550

NACA 16712

NACA TN-1546

3

0.700 0.120 0.300

-5.500 0.690

-6.000 -0.040 -4.000 0.120

0.000 0.495 2.000 0.685

6.000 0.930

10.000 1.140 8 .000 1.025

4.000 0.885

11.770 1.210

NACA 16109

NACA TN-1546

3

0.700 0.090

-5.500 0.450

0.730

-6.000 -0.060 -4.000 0.145

0.000 0.580 1.000 0.690 2.000 0.780 4.000 0.880 6.000 1.005 8.000 1.140 9.000 1.200

0.000 0.000 9.170 1.210

NACA 16712

NACA TN-1546

3

0.700 0.120 0.450

-5 600 0 s 690

-6.000 -0.035 -4.000 0.130 -2.000 0.340

2.000 0.740 4.000 0.910

8.000 1.085 9.770 1.115 0.000 0.000

6.000 0.960

NACA 1b709

NACA TN-1546

3

0.700 0.090

-5 500 0.600

0.730

-6.000 -0.050 -40000 0.150 -2.000 0.395

0.000 0.645 2.000 0.855 4.000 1.010 5.770 1.110 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

NACA 16712

NACA TN-1546

3

0. 700 0.120

-6.000 0.600

0 690

-6.000 0.000

-4.000 0.150 -5.000 0.060

2.000 0.825 3.770 1.040 0.000 0.000 \. 0.000 0.000 0.000 0.000 0.000 0.000

NACA 16709

NACA TN-1546

3

0.090 0.100

0.700 -5.600

0.730

-6.000 -0.020

-4.000 0.150 -5.000 0.040

-2.000 0.445 0.000 0.750

0.000 0.000 2.000 0.915

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

NACA 16712

NACA TN-1546

3

0.700 0.120

-6.000 0.700

0.690

4 .000 0.150 -2.000 0.365

0.000 0.575 1.770 0.760 0.000 0.000 0.000 0.000

I 0.000 0.000 0.000. 0.000 Ow000 0.000

AIRFOIL SECTION

TABLE DATA SOURCP

AIRFOIL SER COO€

DESIGN L IFT COEFC THICKNESS / CHORO hACH NUMBER Z E R O L I F T ALPHA E X T R A P COEFF KCLI

ALPHA, CL: VALUES

NACA 16715

NACA TN-1546

3

0.700

0.300 0.150

- 5 . 4 0 0 0.600

-6.000 - 0 . 0 4 5

-4.000 0.110 -5.400 0.000

-2.000 0.295 0.000 0.445

4.000 0.800 2.000 0.625

5.000 0.865 6.000 0.905 8.000 0.950

11.770 1.110 10.000 1.050

PROPELLER BLADE SECTION AIRFOIL

NACA 16715 NACA 16715

NACA TN-1546 NACA TN-1546

3 3

0.150 0.700 0.700

0.450 -5.500

0.600 -5 A 0 0

0.600 0.600

0.150

-6.000 -4.000 -2.000 0.000

4.000 2.000

5.000 6.000 8.000 9.770 0.000 0.000

-0 .050 0.130

0.460 0.660

0.895 0.915 0.975 1.090 0.000 0.000

0.300

0.850

-6.000 - -5.400 -4 000 -2.000 -1.000

0.000 2.000 3.000 4.000 6.000 7.770 9.770

m0.040

0.120 0.000

0.325 0.390 0.470 0.720 0.840 0 .930 1.050 1.150 1.200

TABLES

0

0.000 0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

0

0.000 0.000

0.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.wo 0.000

0

0.000 0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.000 0.~000 0.000 0.000

0.000 0.000 0.000 0.000


AIRFOIL SECTION

TABLE D A T A SOURCE

AIRFOIL SER CODE

O E S I G V L IFT COEFF THICKNESS I CHORO PlACH NUMBER ZEHO L I F T ALPHA E X T R A P C O E t F KCLI

ALPHA, CL VALUES

AIRFOIL SECTION

TABLE D A T A SOURCE

AIRFOIL SER C O O €

DESIGN L IFT COEFF THICKNESS I CHORO MACH NUMBER Z E R O L IFT ALPHA EXTRAP COEFF KCLI

ALPHA, CL VALUES

NACA 64-006

N A C A TR-824

4

0 . 0 0 0 0.060 0.150 0.000 0.740

-4.000 -0.430 4.000 0. '430 6.000 0.620 8.COO 0.790

10.000 0.810 12.000 0.770 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

NACA 64-206

NACA TR-824

4

0.060 0.200

-1 330 0.150

0 740

-6.000 -0.490 -1.330 0.000

4.000 0.560 6.000 0.770

10.000 1.000 8.000 0.900

12.000 1.010

16.000 0.910 14.000 0.970

0.000 0 .000 0.000 0.000

NACA 64-009

NACA TR-824

4

0.000 0.090 0.150 0.000 0.740

-6.000 -0 .700 0.000 0.000 6.000 0 . 6 2 0 8.000 0.770 10.000 0.870 12.000 0.890 14.000 0.870 16.000 0 . 8 0 0 0.000 0.000 0.000 0.000 0.000 0 . 0 0 0

NACA 64-209

NACA TR-824

4

0.200 0.090

-1.400 0.150

0.740

-6.000 -0.490

6.000 0.790 8.000 0.970

12.000 1.040 10.000 1.050

14.000 1.010

18.000 0.860 16.000 0.9bO

0.000 0.000 0.000 0.000

-1.400 0.000

NACA 64-012

NACA TR-824

4

0 .000 0.120 0.150 0.000 0.740

-6.000 -0.650 6 .000 0.650

10.000 0.930 8.000 0.830

12.000 0.910 14.000 0.850 16.000 0.800 0.000 0.000 0 . 0 0 0 0 .000 0.000 0 .000 0.000 0.000

NACA 64-212

NACA TR-824

4

0.120 0.200

0.150 -1.240

0.740

-6.000 -0.520 '

-1.240 0.000 6.000 0.790 8.000 0.980

10.000 1.125 11.000 1.160 12.000 1.130 14.000 1.010

0.000 0.000 0.000 0.000 0.000 0.000

NACA 64-015

NACA TR-824

4

0.000 0.150 0.150 0 .000 0 .740

-6.000 -0.670 6 .000 0.670

10.000 1.000 8 .000 0.870

12.000 1.100 14.000 0.900 0.000 0.000 0.000 0.000 0 . 0 0 0 0.000 0.000 0 .000 0 .000 0.000

NACA 64-215

NACA TR-824

4

0.150 0.200

0.150 -1.440

0.740

-6.000 -0.490 -1.440 0.000

6.000 0.800 8.000 0.990

12.000 1.200 10.000 1.120

13.000 1.200 14.000 1.150 16.000 1.020

0.000 0.000 0.000 0.000

NACA 64-018

N4CA TR-824

4

0.000 0.180 0.150 0 .ooo 0.740

-6.000 -0.650 6.000 0.650

10.000 0.980 8.000 0.840

12.000 1.040 14.000 1.070 16.000 1.040 17.200 0.640

0.000 0.000 0.000 0.000 0.000 0.000

NACA 64-218

NACA TR-824

4

0.180 0.200

-1.220 0.150

0 740

-6.000 -0.530 -1.220 0.000

6.000 0.000 8.000 0.930

10.000 1.060 12.000 1.135 14.000 1.180

17.000 1.200 16.000 1.200

18.000 0.790 20.000 0.760

NACA 64-021

NACA TR-824

4

0.000

0.150 0.000 0.740

0.210

-6.000 -0.630 6.000 0.630

10.000 0.900 8.000 0.800

14.000 1.010 16.000 1.030 17.000 1.030 18.000 0.900 19.000 0.700 20.000 0.680

12.000 0.980

NACA 64-221

N4CA TR-824

4

(* 0.200 0.210 0.150

-1.260 0.740

-6.000 -0.540 -1.260 0.000

4.000 0.600 6.000 0.800

10.000 1.030 12.000 1.090

16.000 1.140 14.000 1.130

18.000 1.000

8.000 0.940

20.000 0.920


AIRFOIL SECTION


AIRFOIL SER CODE

OESICN LIFT COEFF THICKNESS / CHORD MACH IWMBER 1 € R O LIFT ALPHA E X T R A P COEFF KCLl

ALPHA, CL VALUES

AIRFOIL SECTION


AIRFOIL SER CODE

OESICN LIFT COEFF THICKNESS / CHORD HACH NUMBER Z E R O LIFT ALPHA E X T R A P COEFF KCLI,

ALPHA, CL VALUES

NACA 64-409

NACA TN-1945

4

0.400 0.090

-2.540 0.150

0.740

-6.000 -0.360 -2.540 0.000

6.000 0.890 8.000 1.040

10.000 1.110 11.000 1.120 12.000 1.090

15.000 0 .960 14.000 1.020

0.000 0.000 O.OGO 0.000 0.000 0.000 0.000 0.000

NACA 65-006

NACA TU-824

5

0.000 0.060 0.150 0.000 0.740

-6.000 -0.640

4.000 0.420 0.000 0.000

6.000 0.620

10.000 0.870 8.000 0.770

14.000 0.880 16.000 0.790 0.000 0 .060 0 . 0 0 0 0.000 0.000 0.000

12.000 0.920

NACA 64-412

NACA TR-824

4

0 .400

0.150 0.120

-2.860 0.740

-6.000 -0.340 -2.860 0.000

6.000 0.960 8.000 1.150

10.000 1.290 11.000 1.340 12.000 1.340 14.000 1.220 16.000 1.100 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

NACA 65-009

NACA TR-824

5

0.090 0.000

0.150 0.000 0.740

-6.000 -0.630 6.000 0.630

10.000 0.900 8.000 0.820

11.000 0.920 12.000 0.910 14.000 0.850 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

NACA 64-415

NACA TR-824

4

0.400 0.150 0.150

-2.820 0.740

6.000 -0.360

6.000 0.950 0.000 0.320

8.000 1.110

12.000 1.290 10.000 1.220

13.000 1.300 14.000 1.290 16.000 1.240 18.000 1.050 0.000 0.000 0.000 0.000 0.000 0.000

NACA 64-418

NACA TR-826

4

0.180 0.400

0.150 -2.800

0.740

-6.000 -0.360 -2.800 0.000

2.000 0.540 4.000 0.740 6 .000 0.930 8.000 1.080

10.000 1.170 12.000 1.230 14.000 1.240 16.000 1.230 18.000 1.170 0.000 0.000 0.000 0.000

N A C A 65-012

NACA TR-024

5

0.120 0.000

0.150 0 .000 0 .740

-6.000 -0.650 0.000 0.000 6.000 0.670 8.000 0.870

12.000 0.950 14.000 0.900 16.000 0.840

0.000 0.000 0 .000 0.000 0.000 0.000 0.000 0.000

10.000 0.970

NACA 65-015

NACA TR-824

5

0.000

0.150 0.000 0 740

' i 0.150

-6.000 -0.630 -4.000 -0.440

6.000 0.660 8.000 0.840

10.000 1.000 11.000 1.030 12.000 1.020 14.000 0.880 16.000 0.790 18.000 0.680 0.000 0.000 0.000 0.000

NACA 64-421

NACA TR-824

4

0.400 0.210 0.150

-2.570 0.740

-6.000 -0.400 -2.570 0.000

0.000 0.300

4.000 0.720 2.000 0.530

6.000 0.890 8.000 1.000

12.000 1.120 10.000 1.070

16.000 1.180 14.000 1.160

20.000 I . 00.0 18.000 1.180

NACA 65-018

NACA TR-824

5

0.000 0.180 0.150 0.000 0 740

-6.000 -0.600 -4.000 -0.420

0.000 0.000 6.000 0.620

10.000 0.930 8.000 0.800

12.000 1.020 14.000 1.070 16.000 0.900

0.000 0.000 0.000 0.000

0.000 0.000

0

0.000 .

0.0.00 0.000 0.000 0.000

0.000 0.000

0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

NACA 65-021

NACA TR-824

5

0.000 0.210 0.150 0.000 0.740

-6.000 -0.580 -4.000 -0.390

0.000 0.000 4.000 0.400 6.000 0.590

10.000 0.850 8.000 0.140

12.000 0.930 14.000 1.020 16.000 1.070 18.000 1.080 20.000 0.860

P R O P E L L E R B L A O E S E C T I O N A I R F O I L T A B L E S


TABLE DATA SOURCE


D E S I G N L I F T C O E F F THICKNESS / CHORO RACH NUMBER ZERO L I F T ALPHA EXTRAP COEFF KCL I

ALPHA, CL VALUES


TABLE DATA SOURCE


DESIGN L I F T COEFF THICKNESS / CHOHO MACH NUMBER Z E R O L I F T ALPHA EXTRAP COEFF KCLI

ALPHA, CL VALUES

NACA 65-206

NACA TR-824

5

0.200 0 060 0.150

-1.330 0.740

-6 .000 -0.490 -1.330 0.000

4.000 0 .560 6.000 0.760

10.000 0.990 8.000 0.900

12.000 1.000 14.000 0.960 16.000 0.900 0.000 0.000 0.000 0.000 0.000 0.000

IJACA 65-410

NACA TR-824

5

0.400

0.150 0.100

-2.370 0.740

-6.000 -0.390 -2.370 0.000

8.000 1.090 10.000 1.260

14.000 1.100 12.000 1.180

0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000

6.000 0.900

NACA 65-209

NACA TR-824

5

0.200 0.090 0.150

-1.280 0.740

-6 .000 -0.500 -1.280 0.000

4.000 0.560 6.000 0.770

10.000 0.990 8.000 0.900

11.000 1.000 12.000 0.980 14.000 0.930 0.000 0.000 0.000 0.000 0.000 0.000

NACA 65-412

NACA TR-824

5

0.400 0.120 0.150

-2.660 0.740

-6 .000 -0.360 4.000 0.720 6.000 0.920 8.000 1.100

10.000 1.250

12.000 1.300 11.000 1.310

14.000 1.220 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

NACA 65-212

NACA TR-824

5

0.200 0.120 0.150

-1.180 0.740

-6.000 -0 .500 -4.000 -0.310 -1.180 0.000

6 .000 0.790

10.000 1.070 8.000 0.960

12.000 1.060 14.000 1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

NACA 64-415

NACA TR-824

5

0.400 0.150 0.150

-2.640 0.740

-6 .000 -0.360 4.000 0.710 6.000 0.890 8.000 1.060

10.000 1.170 12.000 1.220 14.000 1.240 15.200 1.070 0.000 0.000

0.000 0.000 0 .000 0.000

0.000 0.000 0.000 0.000

NACA 65-215

NACA TR-E24

5

0.200 0.150 0.150

-1.250 0.740

-6.000 -0.500 -4 .000 -0.300 -1.250 0.000

4.000 0.570 6.000 0.770 8.000 0.960

10.000 1.070

14.000 1.090 16.000 0.990 0.000 0.000 0.000 0.000

12.000 1.130

NACA 65-418

NACA TR-824

5

0.400 0.180 0.150

-2.450 0.740

-6.000 -0.370 4.000 0.670 6.000 0.820 8.000 0.970

10.000 1.070

14.000 1.200 12.000 1.140

15.000 1.220 16.000 1.200 18.000 1.120 22.000 1.000 0.000 0 . 0 0 0 0.000 0.000

NACA 65-218

NACA TR-B24

5

0.200 O e I B O 0.150

0.740 -1 270

-6.000 -0.470 -4.000 -0.280 -1.270 0.000

4.000 0.540 6.000 0.730 8.000 0.870

10.000 0.970 12.000 1.060 14.000 1.110 15.000 1.130 16.000 1.120 18.000 0.720

NACA 65-421

NACA TR-824

5

0 A 0 0 0.210

-2.490 0.150

0 .740

-6.000 -0.360 -2.490 0.000

2.000 0.460 4.000 0.630 6.000 0.770 8.000 0.890

10.000 0.990

14.000 1.150 12.000 1.080

16.000 1.180 18.000 1.190 20.000 1.180 22.000 0.850

NACA 65-221

NACA T R - 8 2 4

5

0.200 0.210 0.150

-1 530 0.740

-6.000 -0.440 -4.000 -0.250 -1.530 0.000

4.000 0.560 6.000 0.710

10.000 0.930 12.000 1.040 14.000 1.100

. l6.000 1.130 17.000 1.140 18.000 1.130 20.000 0.900

0

0.000 0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000

0 .000 0.000 0.000 0.000

0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000

- AqPENDIX PROGRAM USER INSTRUCTIONS

1.0 INTRODUCTION

This appendix conta ins a g u i d e t o t h e s e t t i n g up and running of the computer program. The es sen t i a l computa t iona l s t e p s are desc r ibed i n Sec t ion 4, and the underlying theory i n S e c t i o n 3 of this report, and Section 3 of Reference.1. It should be noted that whereas this program i s a n extension and modi f ica t ion of the computer program developed i n Refer- ence I, c o n s i d e r a b l e d i f f e r e n c e s e x i s t between the two programs. U s e r s of the o ld program are , the re fo re , cau t ioned aga ins t a t t empt ing mod i f i ca t ion of the former program without a ca re fu l s tudy o f the present p rogram layout and da ta format requirements .

2 .O PROGRAM LANGUAGE

The program i s w r i t t e n e n t i r e l y i n F o r t r a n I V , Version 2.3 for a Scope 3.1 opera t ing sys tem and l ib rary t ape .

3.0 MACHINE REQLJIREMENTS

The program i s designed to run on a CDC-6600 computer. The program makes use of the o v e r l a y c a p a b i l i t y a n d r e q u i r e s 15 d i s c f i l e s and a t a p e u n i t f o r p e r i p h e r a l s t o r a g e .

4.0 INPUT DATA

The i n p u t data r equ i r ed by the program t o compute a s e r i e s o f so lu t ions a t s p e c i f i e d a n g l e s o f at tack up t o s t a l l f o r each wing- fuse l age -p rope l l e r con f igu ra t ion cons t i t u t e s one input case. This iqpu- t da ta mus t inc lude tabula t ions o f appl icable wing-sec t ion and propel le r b lade-sec t ion aerodynamic characteristics, p r o p e l l e r t i p loss c o r r e c t i o n f a c t o r s , and a r ange o f geomet r i c and f l i gh t cond i t ion parameters toge the r w i t h spec i f i ca t ion o f s eve ra l cmnpu ta t iona l con t ro l and sequence opt ions .

121

The input data f o r each case i s i n the f o r m of punched cards arranged in sequent ia l groups as shown i n F igu re 30. The pr inc ipa l : card groups are i d e n t i f i e d as follows:

CARD GROUP DESCRIPTION

A Wing-fuselage geometry and control and sequencing options

B Wing-section aerodynamic data

C Wing geometry i f no t s t ra ight - tapered

D Fuselage angles of attack

E P r o p e l l e r t i p loss factors

F Pro,peller blade-section aerodynamic data

G Propeller geometry and opera t ing condi t ions

I n general, the first case r e q u i r e s a f u l l s p e c i f i c a t i o n of the inpu t data conta ined in each group. However, f o r the second and subsequent cases, card groqps specifying tabulated data (Groups B, C , E , and F) may be omitted where there i s no change i n the inpu t data requirements. See Figure 30.

4.1 Winq Fuselaqe Geometry (Card Group A )

Wing-fuselage geometry i s en tered on the first three ca rds as follows:

VARIABLE PROGRAM - CARD COL .LOC TYPE NAME: VARIABLE DE SCRI PTION

1 1-10 R e a l ASPEC AR wing a spec t r a t io

11-20 R e a l TAUT ( t / c )T Wing t i p t h i d c n e s s / c h o r d r a t i o

21-30 R e a l TAUR ( t /c)R Wing root thickness/ chord r a t i o

1 2 2

Program termination

Second and subsequent input cases each consist of card groups A through G as i n - t h e first case, but card group elements indicated by * should be omitted unless new data is required. Note that blank cards i n groups E and F must, however, be retained

" -

Blade element geometry

Propeller-nacelle geometry, Propeller operating condition

Propeller case descr ipt ion <L-\\y tJ Blank card -rxA\)w

*Propel le r a i r fo i l da ta t ab les , Blank card

*Propel le r a i r fo i l sec t ion t i t les Blank card

*Pro.peller t i p loss correct ion tables

Wing geomeitryt omitbed f o r straight-tapered config,

Fuselage angles of a t t ack

*Air fo i l da ta t ab les for

Figure 30, Assembly of Computer Program Input Data Card Deck

Y H

VARIABLE PROGRAM - CARD COL.LOC' TYPE NAME VARIABLE DE SCRIPTION

1 31-40

41-50

51-60

61-70

71-80

2 1-10

11-20

21-30

31-40

41-50

60

' R e a l

Real

R e a1

R e a l

Real

Real

R e a l

R e a l

R e a l

R e a l

I n t ege r

TAPER

TWIST

R

BF

REYND

D I SCR

A

B

H

ALPHR

NFLAP

r

b f / b

Rd

A

A

B

H

QR

Wing t a p e r ra t io

Wing geometric t w i s t (If geometric t w i s t i s specified, TWISA on card 2 must be set t o 100.0)

N u m b e r of spanwise s t a t i o n s

Flap span/wing span

Reynolds number i n m i l l i o n s based on wing mean aerodynamic chord

Cri ter ion for convergence o f i t e ra t ion loop

Fuselage semi-height/ wing semispan

Fuselage s e m i - w i d t h / wing semispan

Height of wing above fuse l age cen te r l ine / wing semispan

Wingbody incidence - degrees

F lap ind ica tor : i f NFLAP = 1, Flap i s def lec ted : i f NFLAP = 0 , Flap i s undeflected

124

VARIABLE PROGRAM - CARD COLmLOc TYPE NAME VARIABLE DE SCRIPTION

2 61-65 Real FLAP Sf Flap setting - degrees: If 8f is zero, ie, f laps not deflected, then BF on card 1 should be s e t t o 1.0

66-70 Real X X X-coordinate of moment reference point

I !

' I

71-75 Real Z Z Z-coordinate of moment reference point

76-80 Real TWI SA ea Aerodynamic t w i s t - degrees ( I f aerodynamic t w i s t i s speci- f ied, TWIST on card 1 must be se t to 100.0)

3 1-10 Re a1 CAMBT KT T i p a i r fo i l s e r i e s camber level

11-20 Real CAMBR KR Root a i r fo i l se r ies camber level

21-30 Integer N S L I P " NSLIP = 0, power off case, no propeller data required; NSLIP = 1, power on case, propeller data required

31-40 Real CDNAC QN Total drag coefficient of a l l nace l les based on wing area, set to zero i f no nacelles

125

4.2 Control and Sequencinq Options -~ (Card Group A)

Card 4 of Group A don t ro l s data read-in and optional p r i n t o u t features, in add i t ion t o s t anda rd ou tpu t . ' The card layout i s shown below.

VARIABLE PROGRAM - CARD COL.LOC TYPE NAME VARIABLE DESCRIPTION

4 1 In tege r NLVL

2 In teger ISWIT( 1)

3 In t ege r ISWIT(2)

4 In teger I SWI T ( 3 )

5 In t ege r I G

6-55 Alphanumeric --

The opt ions are as follows:

ICSWIT(1)

This con t ro l op t ion i s used

N u m b e r of va lues of th i ckness cho rd r a t io (limit 5)

Option for reading i n wing geometric parameters

Option t o p r i n t o u t intermediate calcu- l a t i o n s as they are performed

Option t o p r i n t o u t matrices

Switch used t o set up d a t a t a b l e s

Run number, date, etc.

f o r wings having nonlinear t ape r . The loca l va lues of Reynolds number, geometric t w i s t , and r a t i o s of thickness/chord and local chord/root chord must be key-punched f o r each spanwise s t a t i o n t o g e t h e r w i t h the va lues of edge ve loc i ty fac tor and the r a t i o s of mean aerodynamic chord/root chord and root chord/span. By s e t t i n g ISWIT(1) = 1, these values (Card Group C) are read i n the following order using Format (16F5.0) :

126

PROGRAM VARIABLE NAME

TAU

BEY

C

EP S

EDGE

CRB

ACC

DESCRIPTION

Thickness/chord ra t ios

Reynolds numbers

Chord/root chord ratios

Geometric t w i s t

Edge v e l o c i t y f a c t o r

Root chord/span ra t io

Mean aerodynamic chord

- TYPE

a r r a y

a r r a y

a r r a y

a r r a y

s ing le va lue

s ing le va lue

s ing le va lue

NUMBERS OF

VALUES

R-1

R-1

R- 1

R - l

1

1

1

If ISWIT(1) = 0, the program assumes a s t r a igh t - t ape red wing and ca lcu la tes the values. Normally this is the case.

ISWIT(2L

This op t ion a l lows fo r d i f f e ren t types o f p r in tou t r e q u i r e d i n the computation.

Se t t ing ISWIT(2) = 1, causes the program t o p r i n t out in te rmedia te ca lcu la t ions as they are performed. This p r i n t o u t is very lengthy and should be used only when abso lu te ly necessary to a id in debugging.

If I’SWIT(2) = 0, in te rmedia te ca lcu la t ions are n o t p r in t ed .

IswIT(3)

If this con t ro l op t ion is set t o 1, the two major matrices of the program w i l l be listed. F i r s t , fimk (BETA), the matrix of mul t ip l ie rs used to obtain induced angles of attack is p r in t ed ou t . Then this i s followed by the ma t r ix Ki j (TRIX) which is used in the i t e r a t i o n c y c l e .

I f ISWIT(3) is se t to zero , no p r in tout o f the matrices w i l l be obtained.

127

I 1 I I

This control option i s used t o s e t up the wing a i r f o i l data tables. For the case of a wing having the root airfoil se r ies the same a s t h e t i p a i r f o i l s e r i e s , o r for a wing with a deflected part-span flap, set:

I G = 3 , i f a i r f o i l d a t a is being read f o r the f irst time from cards

o r I G = 1, i f the a i r foi l data has a l ready been read i n and stored on tape

I f the r o o t a i r f o i l s e r i e s is d i f fe ren t from the t i p se r ies then s e t :

I G = 4, i f a i r f o i l data i s being read for the f i r s t time from cards

o r I G = 2, if the airfoil data has already been read and stored on tape

Note that for a wing w i t h a deflected part-span flap, the a i r f o i l s e r i e s from r o o t t o t i p must be the same. The value of I G causes the computer t o perform the following operations.

VALUE O F I G OPERATION

1 Read a i r f o i l d a t a from peripheral Storage device (PsD) t o cube 1 on disc then copy cube 1 t o cube 2 (See Figure 7 ) .

4

Read roo t s e r i e s a i r fo i l da t a from PSD t o cube 1 on disc, then read t i p series data from P S D t o cube 2 on disc.

Read a i r fo i l da t a from cards , load t o PSD, then read from PSD t o cube 1 on disc. Copy cube 1 t o cube 2 .

Read r o o t s e r i e s a i r fo i l da t a from cards,load t o PSD, then t o cube 1 on disc. Read t i p ser ies a i r fo i l da t a from cards, then load t o PSD, then t o cube 2 on disc.

128

The aerodynamic section characteristics of the a i r fo i l are read i n t o the computer i n the form of tables of l i f t c o e f f i c i e n t ( C 1 ) versus angle o f a t tack ( Q ) , drag coeffi- c i e n t (cd) versus C 1 , p i t ch ing moment c o e f f i c i e n t (G) versus C 1 and l i f t coe f f i c i en t w i th f l ap de f l ec t ed ve r sus a . The tables must be selected t o cover the range. of values of thickness/chord ratio, Reynolds number, and camber as soc ia t ed with the wing under consideration.

For each value of thickness/chord ra t io , ' the d a t a tables are arranged as ind ica t ed i n Tab le V. The first card (punched i n columns 1 through 7 ) i n d i c a t e s the number of rows i n the t a b l e (columns 1 and 2 ) , t he number of columns i n the t a b l e (columns 3 and 4 ) and the a i r f o i l t h i c k n e s s / c h o r d r a t i o (columns 5, 6 , and 7 ) . The second card (in alphanumeric format ) ind ica tes the a i r f o i l t y p e and the type of d a t a e.g. NACA 230xX, C1.

A l l cards in the table have format (F7.3,9F8.3) . The f i r s t card contains the values of Reynolds number ( i n m i l l i o n s ) and begins w i t h a b lank in columns 1 throush 7. I f t h e table i s t o c o n t a i n C 1 values , the next card reads - 90.0 i n columns 1 through 7, and -9.0 i n columns 8 through 80. If t h e t a b l e i s for va lues o f Cd , t h e card reads -10.0 i n columns 1 through 7, and 2.0 i n columns 8 th rough 80 . I f the t ab le i s to con ta in Cm va lues , the card reads -10.0 i n columns 1 through 7 , and z e r o s i n columns 8 through 80.

The remaining c a r d s c o n t a i n e i t h e r : (1) a value of angle of attack (columns 1 through 7 ) followed by the va lues of C 1 corresponding to each Reynolds number, o r ( 2 ) a value of C 1 (columns 1 through 7 ) followed by the values of cd, o r (3 ) a value of C1 (columns 1 through 7 ) followed by the values of Cm, depending on whether the table conta ins C1 , C d , or Cm data. The las t th ree ca rds i n each t ab l e are :

129

Table V. Card Format for ,Winq Sec t ion Ai r fo i l Tables

Col. LOC. $ 1 8 16

Header Cards

Reynolds Numbers 1 0.5 1.0

$311-12 NACq 23012 L i f t C o e f f i c i e n t CL

-9oio I -9.0 -9.0 -1410 -0.5 -0 . 45

a Values C& Values . . +90.0

Cd Max Values Q Max Values I

. . 4.0 9.0 1.51 1.54 17 .2 17.0

1511.12 NACA 23012 Drag C o e f f i c i e n t CD

0.5 1 .o -10 2.0 2.0 -0.3 0.004 0 . 0042

C d Values cd Values

a . 10.0

8

2.0 2.0 Blank card Blank card

1611.12 NACA 23012 P i t ch ing Moment C o e f f i c i e n t

-10.0 0.0 0.0 -0.36 -0.40 -0.41

CM1/4 Chord

C& Values

10.0 0.0 Blank card Blank card

C, Values . 0.0

130

r 3 Table

Card - No. Columns 1 throuqh 7 Remain.inq F i e l d s

1 90.0 9.0 9.0 e tc .

2*

3*

blank

blank

Values of C 1 max

Values of Q a t C 1 max i.e. Qmax

* I n the C 1 t a b l e s the v a l u e s of CLmax and Q max appear ing on cards 2 and 3 r e s p e c t i v e l y , m u s t a l s o a p p e a r i n the main body of the t a b l e .

3

C, Table

1

10.0

blank card

blank card

10 .o

2.0 2.0 e tc .

0.0 0.0 e tc .

2 blank card

3 blank card

U,p t o 5 " l e v e l s " of t h i c k n e s s / c h o r d r a t i o may be used. Within each l e v e l the table s i z e i s l i m i t e d t o 8 columns ( 7 values of Reynolds number) and 25 r o w s ( 2 2 d i f f e r e n t v a l u e s of Q or C 1 ) .

The a i r f o i l data cards are assembled as shown i n F i g u r e 30.

131

4.4 Fuselage Anqles of Attack (Card Group D)

The fuselage angles-of-attack a t which calculations are t o be made are read i n on 2 cards, Format (10F8.0). These are used sequentially as punched unt i l e i ther 99.0 i s encountered o r s t a l l i s reached. If the former condition i s encountered, the program w i l l automatically proceed to the next case; i f s t a l l i s reached, the program w i l l search for an angle-of- attack close to the value of anglerof-attack a t which s t a l l j u s t occurs. The accuracy of this search depends on how closely the angles-of-attack are chosen’ near s t a l l .

Note t h a t a l l t h e l i f t t ab les a re a r t i f ic ia l ly extended beyond the C 1 max point w i t h a positive slope. For example, i n a l l t a b l e s C1 a t 01 = f 90 i s se t t o k 9.0. This is done t o ensure convergence. The e f fec t of this i s that the overall wing Cb versus a curve predicted by the program w i l l be correct up t o ;the angle-of-attack a t which s t a l l f i rs t occurs on the wing. ’ Thereafter, the predicted CL i s incorrect. This i s consistent w i t h the purpose of the program which i s to predict the point of stall onset only.

4.5 Propeller T i p Loss Correction Factors (Card Group E )

The propel ler t ip loss correction tables are defined by a three-dimensional array of size ( 3 , 21,8) . The data values are read i n on a to t a l of 63 data cards, each card containing 8 values and having the following format:

VARIABLE PROGRAM ALGEBRAIC COL .LOC TYPE SYMBOL SYMBOL DE SCRIPTION

1 Integer I B (B-1) Array index identifying number of blades per propel ler , less one

2- 3 Integer I P (1+20 sin#) Array index identifying value of s i n #

4-32 ” ” ” Blank

33-38 Real moss( 1) F/FP Data value a t r / R = 0.3

39-44 Real TLOSS ( 2 ) F/FP Data value a t r /R = 0.4

132

VARIABLE PROGRAM ALGEBRAIC COL. LOC. TYPE SYMBOL SYMBOL DESCRIPTION

t

I I

I

75-80 Real TLOSS(8) F/FP Data value a t r / R = 1.0

These 63 data cards may be assembled i n any order.

4.6 Propell-er .Blade Sec t ion Aerodynamic D a t a (Card Group F)

I Up t o 150 propel ler b lade-sect ion ddta tables*can be

accepted and stored by t he computer program. Each t a b l e conta ' ins an array of up t o 20 p a i r s of 0 add C1 values for one a i r f o i l s e c t i o n a t one Mach number conditjion. Up t o 25 air- f o i l s e c t i o n s may be spec i f i ed fo r each 4 i r fo i l f ami ly . A maximum of 9 families can be stored, eacq being assigned an a r b i t r a r y s i n g l e - d i g i t a i r f o i l series code between 1 and 9 inc lus ive .

The s tandard b lade-sec t ion da ta t ab les ava i lab le wi th the program use preassignod a i r f o i l s e r i e s c o d e s (as given below) for which the computer program stores the following t i t les and values of two cons tan ts , k l and k2.

Code A i r f o i l S e r i e s N a m e -1 k -2 k

1 USNP s 0 . -40.0

2 Clark, Y 0. -40 .O

3 NACA 16XXX -7.3 0 .

4 NACA 64-XXX -6.9 0.

5 NACA 65-XXX -6.9 0 .

6-9 Blank 0 . 0 .

*See s e c t i o n 4.1.3 of main report for order of assembly.

133

The cons tan ts k l , k2 are empirical va lues t ha t permi t a n i n i t i a l v a l u e f o r a, , angle-of-at tack a t z e r o l i f t used in equa t ion 104 , to be obta ined from a l i n e a r i z e d ap,prD.xi- mation to the a i r f o i l characteristics given by the express ion

a, = k l X (des ign l i f t c o e f f i c i e n t ) + k2 x ( t h i c k n e s s / c h o r d r a t i o )

The a i r f o i l series t i t l e ca rd i s u s e d t o r e a s s i g n a i r f o i l series names and k l , k? v a l u e s as r equ i r ed . One card i s used f o r each reassignment grid has the fo l lowing format :

VARIABGE PROGRAl4 ALGEBRAIC ” COL .Loc . TYPE Wi” SYivl30L DESCRIPTION

P I n t e g e r I “ A i r f o i l series code

11-18 Alphanumeric AFSER(1) ” A i r f o i l series name

21-30 R e a 1 AK(I,l) Em,pirical constants de f ined i n

31-40 R e a 1 =(It21 preceeding t e x t

Each a i r f o i l table i s de f ined on a sequence of from 2 t o 5 ca rds . The f irst card i n the sequence contains header in format ion spec i fy ing the a i r f o i l s e c t i o n a n d o t h e r pe r t inen t pa rame te r s as fo l lows:

VARIABLE PROGRAM ALGEBRAIC COL .Loc . TYPE NAME: SYMBOL DE SCRIPTION

1-2 I n t e g e r IHEDD ( 1 ) ” N u m b e r of p a i r s o f a , C 1 v a l u e s i n . table

5 I n t e g e r IHEDD( 2 ) “ A i r f o i l series code

9- 16 R e a 1 THEDD( 1) C 1 i Design l i f t c o e f f i - c i e n t

17-24 R e a l THEDD ( 2 ) t /c Thickness/chord r a t io

134

VARIABLE TYPE

PROGRAM ALGEBRAIC NAME SYMBOL COL .Loc .

25-32

33-40

DE SCRI P TI ON

M E D D ( 3) Mo R e a l Mach number

R e a 1 M E D D ( 4) a0 ;P;rrgle-of-attack for ze ro l i f t

41-48 Real Extrapolation c o e f f i c i e n t g i v e n by. equat ion ( 126)

57-68* Alphanumeric

69-80* Alphanumeric

Table da ta sou rce " "

" " I A 5 r f n ; i l section name

* For reference only; these columns not read by program.

The second and subsequent c a r d s i n each a i r f o i l table sequence contain the pairs of Q and C1 va lues . These va lues must be ar ranged i n order o f a i nc reas ing . The format i s a s follows:

VARIABLE PROGRAM TYPE NAME

ALGEBRAIC SYMBOL COL .LOC .

1-8

DE SCRI PTION

R e a l TALFA(1) 1st p a i r of va lues

9-16 Real TLIFT(1)

R e a l TALFA ( 2 ) 17-24 2nd p a i r o f values

I I I I I I I I

5th pa i r of values

25-32 I I I I I I I

65-72

R e a 1 I I I

I I

I I

R e a l

TLIFT( 2) I I I

I I I I

TALFA(5)

R e a l TLIFT( 5) 73-80

135

I

4.7 P r o p e l l e r Geometry and Operatinq Conditions (Card Group: G )

Information, t o s p e c i f y the propel le r geometr ic and ope ra t ing parameters is ,provided on a series of cards o f four types. These cards must be a r r a n g e d i n the o r d e r desc r ibed below. Each c a r d type i s ass igned a numerical code value i n column 80. ?he va lue i s read by the program t o ensure tha t the c a r d s are in proper sequence.

?he f i rs t card ,p rovides for the i n c l u s i o n of a r b i t r a r y t i t l e informat ion as fo l lows :

VARIABLE PROGRAM COL .mc i TYPE NAME DE SCRI PTION

1- 76 Alphanumeric TITLE P r o p e l l e r i d e n t i f i c a t i o n t i t l e

80 I n t e g e r IDENT Card i den t i f i ca t ion code = 1

The second card provides information specifying the propeller and nacel le geometry as fo l lows:

VARIABLE PROGRAM COL .Loc . TYPE NAME

1-2 I n t e g e r NP

11-12 I n t e g e r NB

21-30 Real DPB

31-40 R e a 1 RHBR

41-50 Real: RNBR

80 I n t e g e r I DENT

WGEBRAIC SYMBOL

"

B

rhub/R

l n a c e l l e /R

DESCRIPTION

N u m b e r o f p r o p e l l e r s

N u m b e r o f b l ades pe r p r o p e l l e r

Propeller diameter/ wing s , p n

Hub r a d i u s / t i p r a d i u s

Nacelle r a d i u s / t i p r a d i u s

C a r d i d e n t i f i c a t i o n code = 2

136

The t h i r d card p rovides in format ion spec i fy ing the ope ra t ing cond i t ions for the p r o p e l l e r ( s ) as ,follows: I

VARIABLE PROGRAM ALGEBRAIC COL .Loc . TYPE NAME SYMBOL DE SCRIPTION

1-2 I n t e g e r NROT( 1) " L .H. p r o p e l l e r ro ta t ion index

11-12 I n t e g e r NROT(2) " R .H. propeller r o t a t i o n i n d e x

21-30 Rea 1 AJ J Propeller advance ra t io

31-40 R e a l AMCHU MO F l i g h t Mach number

80 I n t e g e r I DENT " C a r d i d e n t i f i c a t i o n code = 3

The p o s i t i v e v a l u e (01) f o r the p r o p e l l e r r o t a t i o n i n d e x is used t o s p e c i f y a r igh t -hand ro t a t ion fo r e i ther p r o p e l l e r . The nega t ive va lue (-1) is used t o spec i fy a le f t -hand r o t a t i o n . Where a s ing le -p rope l l e r con f igu ra t ion i s con- s i d e r e d , the r o t a t i o n i n d e x f o r the L.H. p r o p e l l e r must be set t o z e r o (00 ) while the R.H. p rope l l e r i ndex has the a p p r o p r i a t e v a l u e f o r the s i n g l e p r o p e l l e r r o t a t i o n s e n s e .

The four th- through- las t cards o f the series provide informat ion spec i fy ing the b lade sec t ion geometry . One ca rd i s r e q u i r e d f o r each selected b lade s ta t ion be tween hub a n d t i p , up t o a maximum of 1 2 ca rds . These cards must be assembled in order o f increas ing blade s t a t i o n r a d i u s . The f irst s e l e c t e d s t a t i o n n e e d n o t b e a t t he hub , bu t t he l a s t ca rd must spec i fy the s t a t i o n a t the t i p w i t h a va lue o f 1.0 i n columns 1-10.

VARIABLE PROGRAM ALGEBRAIC COL .LOC . TYPE NAME SYMBOL DE SCRIPTION

1-10 R e a l RPBR r / R Blade s t a t i o n r ad ius / t i p r a d i u s

11-20 R e a 1 CPBR C/R Blade-section chord/ t i p r a d i u s

137

VARIABLE PROGRAM ALGEBRAIC MOL .Loc . TYPE NAME SYMBOL I3E SCRIPTION

21-30 R e a 1 BETA P Blade s ec t ion pitch ang le I degrees

31 In t ege r NA " Blade sec t ion a i r f o i l series code

41-50 R e a 1 CLI C1 i Blade sec t ion des ign l i f t c o e f f i c i e n t

51-60 R e a 1 TOC t/c Blade sec t ion th ick- nes s / chord r a t io

80 I n t e g e r I DENT " Card i d e n t i f i c a t i o n code = 4

4.8 Proqram Termination (Card Group H )

The program operation i s terminated by three i n p u t data c a r d s ' l o c a t e d a t the end of the input da ta deck . The f i r s t card uses the format of card number 1 of Group A w i t h the value of ASPEC set t o 99.0. The second and th i rd cards must be blank.

4.9 Data R e s t r i c t i o n s

The fol lowing i s a l i s t of i n p u t q u a n t i t i e s t o g e t h e r w i t h the , r e s t r i c t i o n s and normal range of values.

Q3ANTI TY

ASPEC

SIGN-RESTRICTIONS-NORMAL RANGE

+ # 2 2.0

TAUT, TAUR, 1 TAPER +, 1.0

TWIST, TWISA between +150 and -15O

R +20 only*

138

QUANTITY SIGN-RESTFUCTIONS-NORMAL RAWE

REYND + D I SCR +, suggested value -001

ALPHR between +loo and -loo

NFLAP 0 or 1

FLAP +, between Oo and 90°

x, z + o r - N S L I P 0 or 1

CDNAC +, 0 5 1.0

* R , which must be an even integer, may be changed t o a l low ca l cu la t ion a t a g r e a t e r number of spanwise s t a t i o n s . This requires changing the DIMENSION statements.

5.0 OUTPUT

5.1 Printout Options

All output i s from a standard 120-characters-,per-line p r i n t e r . The amount and type of data output depends on the opt ions exercised and on whether the computation i s f o r a wing w i t h o r w i thou t a de f l ec t ed f l ap / w i t h or without sl ipstream. For the s tandard run (without the debug p r i n t - o u t ) , the output data is self-explanatory. When the opt ion fo r p r in t ing i n t e rmed ia t e ca l cu la t ions i s exercised, the output conta ins the fol lowing addi t ional parameters whose meaning i s given below.

139

I

'QUANTITY

Al

DESCRIPTION (see also Reference 1) TYPE

s i n g l e v a l u e Angle of a t t a c k c o r r e s p o n d i n g t o C 1 a t f lap end, from flapped s e c t i o n data

A2 s i n g l e v a l u e Angle of a t t ack co r re spond ing t o C 1 a t f lap end, from unflapped s e c t i o n data

Zero-l i f t a n g l e a t f l ap end - f l a p p e d s e c t i o n d a t a

A3 s i n g l e v a l u e

A 4 s i n g l e v a l u e Z e r o - l i f t a n g l e a t f l ap end - unf lapped sec t ion data

ALPC Angle of attack c o r r e c t i o n s f o r f l a p e f fec t

a r r a y

ALPH Angles of a t tack c o r r e c t e d f o r downwash, f l a p e f fec ts and body upwa sh

a r r a y

ALPHE

ALPG

E f f e c t i v e a n g l e s o f a t t a c k a r r a y

a r r a y Wing sec t ion geomet r i c ang le s of a t t a c k

ALPHU Sect ion downwash a n g l e s corrected f o r f u s e l a g e e f f e c t s

a r r a y

S e c t i o n z e r o - l i f t a n g l e s ALPHZ

ALFPR

a r r a y

s i n g l e v a l u e P r o p e l l e r s h a f t a n g l e o f a t t a c k i n r a d i a n s

s i n g l e v a l u e

a r r a y

a r r a y

a r r ay

s i n g l e v a l u e

Average slipstream a n g l e ASBAR

CBC Calcula ted va lues o f C l c / b

Approximate values of C l c / b CBG

Values of Cdc/b CDOC

CL I n t e g r a t e d l i f t c o e f f i c i e n t u s e d to normalize CLADD

240

QUANTITY

CLADD

CLAD1

CLAD2

CL2CB

CLDEL

CLMAX

CL STA

CL STU

cvffi

DELTA

DDCLMA

EDGE

F

TYPE

a r r a y

a r ray

array

a r r ay

a r r a y

a r r a y

s ingle va lue

s ingle va lue

a r r a y

a r r a y

s ingle va lue

s ingle va lue

a r r a y

DESCRIPTION (see a l s o Reference 1)

Additional l i f t c o e f f i c i e n t d i s t r i b u t i o n

Modif ied dis t r ibut ion of add- i t i o n a l l i f t c o e f f i c i e n t s from t i p t o f l a p end

Modif ied dis t r ibut ion of add- i t i o n a l l i f t c o e f f i c i e n t s f r o m . f lapend to wing/fuselage junct ion

Di s t r ibu t ion of l i f t associated with equation (56)

Di s t r ibu t ion of l i f t c o e f f i c i e n t due t o f l a p d e f l e c t i o n o n l y

Values of section maximum l i f t c o e f f i c i e n t s

Section l i f t c o e f f i c i e n t a t f l a p end

Uncorrected section lift co- e f f i c i e n t a t f l a p end

L i f t coe f f i c i en t s co r re spond ing t o ALPG

Differences between guessed and ca l cu la t ed l i f t d i s t r i b u t i o n s

Increment i n s e c t i o n maximum l i f t c o e f f i c i e n t a t f lap end due t o f l a p d e f l e c t i o n

Edge ve loc i ty f ac to r

Fac tors used in a l te r ing two- dimensional section d a t a t o three-dimensional data

141

QUANTITY

FF

F1

F2

FUNC

GE l?F,

HOPP

RDUBAR/DU

SGENE

SmNY

SIGMA

sv

TONY

vw

TYPE

s ingle va lue

s ing le va lue

s ing le va lue

s ing le va lue

a r r a y

a r r a y

s ing le va lue

a r r a y

a r r a y

a r r a y

a r r ay

a r r a y

s ingle va lue

5.2 Er ror Messaqes

~~

DESCRIPTION (see a l s o Referanqe 1)

Factors used a t flapside of f l a p end t o a l t e r 2-dimensional da t a

Factor used t o scale a d d i t i o n a l l i f t d i s t r i b u t i o n CLADD

Factor used to scale CLDEL

Wing-on-propeller upwash funct ion, equation (128)

Values of equat ion (88) f o r e* = K - 8"

Values of Q c K 8 - see Reference 1 equat ion ( 3 8 )

R e a l p a r t of the der iva t ive o f the conformal transformation funct ion, equat ion ( 9 ) of R e f . 1

Function associated w i t h equation (76)

Funct ion associated w i t h equat ion ( 7 6 )

Function associated w i t h HOPP

S l ips t ream c ross f low d is t r ibu t ion

Values of equat ion (88)

Wing-induced upwash i n p r o p e l l e r disc plane

In developing the program it was found that the most common source o f e r ror w a s the a i r f o i l da ta tables. I n p a r t i c u l a r , the t a b l e s of C1 versus a are most c r i t i ca l . The v a r i a t i o n of C 1 w i t h increas ing Q should be smooth

142

without sudden breaks, especially for values higher than Q max- A sharp break i n the slope of 'C1 versus Q a f t e r =?X may cause the i teration procedure to I diverge. If this occurs a message i s printed as follows:

UNABLE TO CONVERGE AFTER 30 I T E R A T I O N S ABORTED

The l a s t values of DELTA and the values of l i f t co- e f f ic ien t a re then listed together with a dump of the a i r fo i l tables i n core a t t h a t time.

'

A second error message associated with \table interpolation is a s follows:

ERROR CODE ( N )

I F 1 CVAL GTR THAN MAX VALUE L I S T E D

I F 2 CVAL GTR THAN TABLE VALUE

I F 3 ALPHA VALUE GTR THAN TABLE VALUE

I F 6 " I I C K N E S S CHORD RATIO VALUE CANNOT BE FOUND

If these errors occur, the tables should be examined for mistakes i n key punching, etc.

Several error messages are associated with the propeller slipstream analysis subroutine and operate as follows:

If an invalid number of tip loss correction cards are read, i.e. other than 0 o r 63, t h i s i s indicated by the message:

XX T I P LOSS CORRECTION TABLE DATA CARDS READ I S I N V A L I D - SLIPSTREAM COMPUTATIONS ABORTED

If the propeller geometry and operating condition data cards are read out of sequence, th i s i s indicated by the message:

CARD IDENT X X HAS BEEN READ OUT OF SEQUENCE, SLIPSTREAM COMPUTATIONS ABORTElD

143

I

If propeller airfoil tables are not stored for the a i r fo i l s e r i e s code specified on a blade station data card ( I D E N T 4), this is indicated by the following message :

A I R F O I L T A B L E S NOT STORED FOR A I R F O I L S E R I E S )(x

SPECIFIED AT RB/RP = 2 C X 0 X X X X - THIS ELEMENT I S DELETED FROM THE ANALYSIS

If the propeller-slipstream analysis fails to converge within 9 iterations, then the following message appears after printing out the solution for the las t i terat ion:

S3LUTION FOR PRECEEDING ELEMENT FAILED TO CONVERGE I N 9 I T E R A T I O N S AND I S DELETED FROM THE SLIPSTREAM LNALYSI s

6.0 PROGRAM STRUCTURE

The program i s written to operate i n OVERLAY mode. The central controlling portion of the overall program i s called STALL. The remainder of t h e program i s sp l i t in to 3 parts, ONE, TWO, THREE:, which are overlaid. STAI;L ca l l s ONE which then ca l l s e i ther TWO or THREE depending on whether the cal- culation i s f o r a flapped wing or not. The major subroutines called by each overlay are as follows:

ONE ca l l s MAIN, MAINA, MAIN1 TWO ca l l s MAIN2, MAN2A, MAIN4 THREE ca l l s MAIN2, MAIN3, MAIN5

These major subroutines call the remaining subroutines i n the program as follows:

MAIN c a l l s MAINA, SETSW, AERDA, DATSW, ZZZ MAINA Calls SLIP, AAA, D A T m , Z Z Z , BRIDG, MAIN1 MAIN1 c a l l s DATSW, IVLA, MINV, GRIDG, SSS MAN2A c a l l s DATSW, AAA, BRIDG, Z Z Z , MAIN4 MAIN3 c a l l s DAGET, DATSW, AAA, ARC, Z Z Z , MAIN5 MAIN4 c a l l s BRIDG, AAA, DATSW, MAN2A MAIN5 c a l l s DAGET, ARC, DATSW, AAA, ZZZ BRIDG c a l l s DAGET, ARC ARC c a l l s LOOK

144

7 .O OPERATING PROCEDURE

Logical TAPE5 i s named as the working (scratch) tape. The program deck and data deck are loaded in the following sequence: job card, system control cards, end-of-record card, program deck, end-of-record card, data deck,' end-of- f i l e card.

8 .O PROGRAM TIMING

Central processor unit t i m e for an average run of six (6) angles-of-attack is approximately 60 seconds-

145

APPmDIX Q

INTERNAL LIS'PIING OF THE CDMPUTER PROGRAM

Presented in this appendix is an internal listing of the computer program developed under the present contract.

146

5 0 30

10 2c 4c

1c 2 0

30

l i P R D E * d L P l - e * K C C P P C h & S L I P *

S

A t

T 1 * 1 * I 1

= O U T P U T , T A P t l O l r T A P E 1 5 = 2 0 1

t R E C A . L L 1

* R E C A L L 1

P R E C A L L 1

TC 2C

. I GC TC 1 c C O TLi 3C

9 ) , A L P G i 19

0 * C D N A C . .

147

I

10

20 C

C C

30

C 4 8

h T C P

C 50

00

7 3 t C C C

80 9s

1co 110

120 C t C

130 1 4 C

1 5 C C C C

I t 0 1 7 0

C C

C

C C

C

C C C

180

1 9 C

20c

213

2 3C 2 2 0

240

IhTERCEANGE C O L L H N S

~ K = h t i + ~ ~. .I Fs . IP+ . I I

K J = I J - I * K A C I J ) = F C L C * C C h T I h C t

h

h.

I 1501 14C 1 1 5 C 1 1 3 C

P ( K J ) + A ( I J )

O I V I D E ROW B Y P I V O T

K J=.K-F; C C 170 J ~ 1 . h K J = K J + k

b (KJ )=A(KJ ) /€ ! IGA € F ( J - i o 1 6 C 9 1 7 0 1 1 6 C

CCNT INUE

- 0 4 *B.I GA

ACKKl * l .O ' / f lEGA C C h T I h U t

K =N

0 I 2co 230r210

J=P (.K I € F ( J - K ) 1 K €=!(-!I I IC 25c I= H C . L C = A ( K I K €=K €:+k

JI=KI-K+J

PRODUCT OF PIVOTS

R E P L A C E P I V O T t i Y R E C I P R O C A L

F I K A L R G k AND CCLLMN I N T E H C k A N G E

149

C c C C

C C C C C C

C c C

C C

C C

C

C C L C C

C

256 A L J € )=l-CLI: G C T C ,190

200 R E T L R h

S U B R C L T I V E C 4 6 E T ( A R K A Y . t I F I L E t I I I t h C

SLBRCbTINE T G GET T A B L E FHCY UISK AND PUT I T I h T C C O R E

r

T P e L E P R E S E b i T L Y I K C O R E IS IhNCjW

I F hGT T H E N GET 1 A t i L . E F R C R C I S K I f - T I B L E I S A L R t A C Y IN C C R E T H t N R E T U R N

I . F ' ( I F I L E - I h h C h ) 1 0 t 9 0 r 1 0 1c €Is1

R E k I R C I F I L E H E L C ( 1 F I L E I A H t l A Y R E i I h C I F I L E

3c

40 50 60

7 c

9c 8 0

A R R P Y ~ J F C X G C T C S C € F ( I F I L F - 1

R E S E T T A B L E P R E S t K T L Y I N C C R E I N U I C A T C R

T C O U l P U T 4 OhE C I C E N S I O N A L

S U B R C U T I N E TI? O U T P U T A .SINGLE V A L U E

C L T P L T V A L L E I N S E T H F AFIlC E FORKAT

€ P = 6 W R I T E ( I P t ~ 1 0 1 . V A L U E * V A L U E R E T L R h

10 F C ~ C A T ( l X . t F 1 5 . 8 r E 1 6 . 8 / 1 X ~

150

C C

t C

C 10

20

C C C

10

20

C

C E

10

20 3c 4 c 5 0

6s 70 8C

r

EhC S L B H C U T I Y E S S S C A r k P )

S L B R C b T I N F T C CLTPUT A SCLARE TkC C I P E k S I C I \ i A L AHHAY

K = C . k W I T E ( I P * 9 0 ) I P = 6

I . F ' ( I S h I T ( 1 ) ) 2 C r 2 0 r 1 0

k R I T E ( 1 P r l C C ) € F ( I S h I T ( 2 ) ) 4 0 r 4 G 1 3 1 i

K = K + 1

K = K + 1

C C

C

10 10 16 40

50

t C

C C

V b K E S I C L N C

O T SE i T = C .

3 )

50 50

50

C E C I S I C C H C

V L r k h

A S E D Clc S E T T I k G CF ThE S k - I T C k I S S E T r I C O ~ ~ O = L

X C U L r C B X X r I F I L E r K E E P )

S b t I l C H , If

C A R C S ANL' PLT T A B L E S ON UlSK S L f ! R C U T I N E TC REAL A E R O D Y K 4 Y I C TABLES FRCH

151

l A L P t - V ( 16)

c ZhTECER.2 KEEP C C P P C h I N h ' C k

c" C C C C

C C C

Ea=@ 1 P*O

CC 10 J = l r h ; L V L

Y A X X t C ) C O h T A l N S K U M B E R C F RChS IK A T A B L E

V b L L t I h f i C k IS TkE T A B L E P H E S E h T L Y IN C O K E

h k E i ? E ( . 2 r b ) C C N T A l h S I t i 6LE N L P B E K AN0 T A U

I h I T I P L I Z C L E V t L S I N Y H E R E

LCCP rc R E A C I N L E V E L S GF LL S E T CF EITHER L I F T , C R A G , CR P I l C H I N G E C K E h T

DC 40 L V L = 1 r h L V L C C C C

C C C S T C R € kLMDER OF C C L U F N S

C C C

C C KEbC A h C l n R I l E T I T L E OF T A B L E C

C

R E A L A h D P R I F j T NUMI3ER OF K C k S r C C L U M P i S , T A L V A L L E S FUR A G I V E h L t V E L

R i b C t I Q , 5 C ) ~ C , ' h C C L , ~ ~ E ~ ~ E ( Z , L V L ) W R I T E ( I P v 6 C I ~ c , h C C L T h t E i ~ E ( 2 r L V L )

PX.CCLf LVL J = h C C L

S T C R E . N L K B E R O F R C k S

M A X X L L V L 1 s t ~ ~

R E b C ( . I R , 7 0 ) h A C E W R I T E ( I P , E O ) h A F E , L V L

E R E A L V A L U E S FOR T A B L E

W R I T E C O P P L E T E T A B L E UN D I S K C

R E.T LR K

'C, YO r C D N A C t

OLPY?=C.@ DL;CVl=C.O € f - ( P E Y C h - S S S . ) 2 0 * l U I 2 C

GC T C 3C LO K E Y = l

- 2 G K E Y = 2 : S T C R E C V A L b E C

3C C L T = C l L C C C C

hEk I S THE F I L E N L K b E R OF T k € O T h C R CUeE P R I C A H Y C L B E S A R E FiUK:bEKEC l r 2 t 3 r 4 S E C C N C O H Y C L b E S AHk k U K t i E H E C 5*lLr15120

c hrEh=LCY*S

C C

C

C C C C

C

C

C

S T C R E X C A X

XIVX=XPbX

C E T E R F I N E IF S I h G L t V A L U E I S T O B E USED CR L I S T L F V A L L E S IS lfl 6 E LSEC I N L C O K UP

4C K C t = L I . F ( IS-IP) 5 C 9 4 0 r 5 0

C S E T U P FljK C G N S T A \ T V A L U E OF X

A T P P I L ) = X X ( l ) GC TC 7C

E S E T b P F O R V A R I A B L E V A L U E GF X L

5 0 K C C = 2

60 A T Y F L J ) = X X ( J ) DC C C J = N S * h P

t P U T P R C P € R T A B L E IN C O R E C

C C S E T U P A L P I - A E I T H c R V A i 3 I A 6 L E OH C C V S T A N T

7C CALL C P C E T ( . ~ R R A Y I ~ E ~ ~ I Z X Y )

C cc 1sc I ( = K S * h P GC T C leO?SClc K G C

8 3 A L P F A = A I Y P ( I S ) GC TC 1CC

9C A L P I - P = A T F P ( K )

:: C L T T A U V A L U E FRnP A R A A Y C

C C G E T P R C P E R R E Y h C L C S h L K B E R E I T H E r ! R E G L L A H

L C C T P L X = T P L ( K ) GC T C ( 1 2 C , l l ? ) ) r K E Y

5 C R P A X b i ~ P b E 1 3 L

110 RkYC%=.9EY CK 1

. 1 2 C R E Y h = H E Y I K ) GC T C 1 2 0

1 3 C C L L = C L T X C ~ X = X C X

R E Y h t S S 9 .

r

i: P F R F O R P LCCK FCR L I F T ? ORAGI P I T C H I N G 4 PCIUCNT C k FLAP C A S t L

CjC T C 1 1 4 0 r 1 5 t r 1 6 C r 1 7 C ) r L C K 1 4 G . C A L L J ~ ~ C C A H H P Y * T A ~ X ? ~ P P A r P ~ C C L * I E I A ~ . E K E ? ~ L V L )

1 5 C C A L L A ? C I P R R A Y * T A L X * M A @ O r ~ ~ C C l * I ~ * d ~ E R E I N L V L I

160 . C A L L A R C ( A R R A Y ~ T A U X ~ C A C C I P C C C L * I E P C P E R E ~ N L V L L

CC T C l e 0

GC T C l e 0

GC TC 1bO

153

C 17C . C A L L d R C ( ~ R R A Y , T A ~ X , ~ A C D , P ~ C C L t I ~ ~ O ~ E R E I N L V L L

4. S T C R E V A L U E S FCUNC I N L O C K LP L

1 H G C C Z Z C K I = Y Y

C A L R 2 ( k I = C L P Y 2 C C L R 2 ( K ) = C U P Y 1

C C R E P E A T F O R hLMBER CF V A L U E S T O B E FCUND l- L

C C C

C

190 C C A T € I \ b E

C t T N E X T T A B L E I N C O R E

C A L L C & C E T ( b R R A Y * L C E * I Y l

5 S E T U P A L P P A V A L U E L

OC 310 K=hS,hP G C T C ( 2 0 C , 2 1 C l r K C 0

Z O C A L P k A = P T P P ( I S l LC T C 2 2 0

210 . A L P k b = A T P P ( K l e L 5 S t T UP R E Y R C L O S UUFtlER L

225 GC T C C 2 4 C ~ 2 3 C ) r K E Y 23C R E Y C h = ! ? E Y ( K )

l ? f Y k = S S S . G C T C 250

240 R E Y h = K E Y (K 1

t S E T LP C V A L , T A U , A N C X P A X V A L U E C

25C C L . L = C L T T L L X = T L L ( X I X P b X = X P X

L C C K LP FCH SECCNC L I F T , DRAG, P I T C H I N G P C C E N T , CR F L A P C A S E

3 C O C C Z l ( . K I = Y Y

310 . C C l i T I h U E

CCLRL(K)=CLWYl CLLR 1 ( Y )=CLPYZ

153

39C C C h T I h U E HE T L R h

C c R E C C L A R I N T E R P C L A T I O t d B E T k E E h TAGLES

P W R A Y I 5

i 6 1

h F!

r: i. C f

A G I V E & T A B L E SLt3RCbTIbiE T C I N T t R P C L A T E B E T k E E X L E V E L S OF

C

T P L J A L L t I C T 4 b L E S S T H A N C R E Q U A L T C L C i h E S T L E V E L

Y E S , S E T J E C U A L T C P A X L E V E L

3C J = \ L V . L S .XTRPU=l. GC T C 80

hCt S E A 2 C k L I S T G F L C V E L S ( . T A U V A L U E S ) U h T I L C h i G I U L N I S G R E A T E R T I - A h OhE T @ BC FCLhU

ERRCR I F NCNE C A h f3E FOUhC

60 7 6

9 5 ao

1oc 11s 12c 13C 143 150 160 173 l e 0 190

YON X M A X

zcc 2 10

GC 1C 210 I F L P = 2

2 C T C 2 7 0 I .S=5

G C T C 270 I S - 4

GC T C 2 7 0 I s = l

b C T C 270 FS=? G C TC 270

R E T L R h I . t = t

C L L L L C C K ( P R R P Y V L C L 2 (.2 ) = c u P Y z P L R (.2 ) = C U V Y 1 GC TC ( Z H C w 2 S C 1 3 C F l h L ' = A L P H A A L P k A = S S 9 . G C T C 3 5 0

AEYr \=SS9. F I h C = R E Y N

GC TC 3 5 0 F I A C = C V P L C V P L = 9 S S . GC TC 3 5 0 F I h C = R E Y @ N GC TC 350 F € \ L = X P b X GC T C ( 3 3 C 9 3 4 C ) r X P P X = C . GC TC 3 5 0

,YdX2=h2ChS ( J - 1 ) L V L = k k F ? E ( l r J - l )

P P X C = h C C L S ( . J - l )

I. 5 = 2

R E Y C h = S S 9 ,

X P A X = l C $ .

2 2 0

230

24C

25C

26C

2 7C V L , F A X R I C A X C ~ I E ) .

C , 3 1 C T 3 2 C ) * IS

290

3cc

310

320 3 3 c

3 5 c 340

1 F L ?

r

hEICkbLRhCbC O f C L M A X S P E C I A L I J R G C E O b r l E FOR I N T C R P C L A T I C K IN T F E

L

36C

J )

5 3 c 1 T IS 3 7 3 3 6 2 39c

C 1 5 1 0

4 1 2 4 c 2

4 2 C

4 3 0

4 4 0 4 5 c

4 7 c 4 c o

c" A L P t A

4 8 0

490

59c

510

156

C

C C C

C C C

C C C C C

C C C C C C C C

C C C C C C

t C C C C C

t C

C C

520

5 3s

. C V A L = T € R P ( R I (ETLR)\ C l = R E Y C \ Q E Y C k = T E R P (

C l = X C A X R E T L R h

X P A X = T E d P ( R R E T L H h

EhC S L E R C U T I N E . L C C X ( A ~ ~ L V L I C P X R ~ P A X C , I E )

SLBRCUTIhk F C R T H C D I M E N S I O k A L T A h L E LCOKUP

Y L 5

V 2

i

i L

* f I!

Y P B JPI

' N A C

w a x x r C L P A X * C

A X * E€IGE. AClK * E K E *

S E C T I C h T O L C C A T L H E Y h O L C S h L M o E R 1N T A B L E

4 0 E F ( R E Y h - A ( L V L * l r Z ) ) 5 C t 6 C ~ t C

PLW k t h a N T T O USE T h E TI-E T A B L E

5 0 L C C b 3 XTHAL,l.

157-

#- GC TC 7C

. A R C t . T A B L E U h T I L kE AND G R E . A T E R T H A R

R V R E A T E H T P A N 1HE

C i r d L . A V A L L r E C.F.599 SPkCIFIES T H A T THIS S E C T I C N F@R N O R M A L L O C K UP CF A L P H A A N D

V A R I A P L E I S T H E OhE hHCSE V A L U E IS T C BE F C L k C I N T I - € T A d L E .

C c S E C T I C N T O L C U K UP A L P H A FCR G I V L A C V A L BY I h T E K P C L u T I R G d E T h € E h C C L L Y k S hHERE R E Y h C L C S K U V r j E R S B R A C K E T G I V E N RkYN

L C C K L f ' . C V A L FOK G I V E N ALPkA. C k e C K T G SEE I F h E t-.uvC P L I F T C R C R A G T 4 H L E l i \ A T THIS P 4 X I V L E V 4 L L E S ( . L A S 1 TWU FiChS) . T l r E . LKAG T A i 3 L i S h A V E L E H C ' d A L L E S F C H A L L

C C P L

210

22r 230

2 4 0

2c;C 2 5 c

270

2 d G 290 300 3 1 0 3 2 0

330

A ) 24

,158

340

s5ii 360 3 7 0

39c 40C 4 19

42c

430

440 453

4bi!

4 70

2% 509

51c

520 53c

54C

E 550 C

560

57c

59s

590

600 510

620

E 159

S L E R G L T I NE T C S T O R t T . A U C L A T E C PKCPELLER T I P L C S S C U H K E C T I C f \ C A T A CH CISK

C C C C

R E P L T L C S S ( 8 1 K E A C L l l ' I T K C ) ' T L O S S R t T L R h EhC .sc:BRcuTIr:E G C : S K C I T I ~ C . T A L F I . T L I F T )

C C

C C C C C

S L B R C L ; T I r \ r € T C S T C R E T A B L J L A T E C P R O P E L L k R A I K F C I L S t C T I O N G A T A ON U I S K - A L P l - A A K L ' C L V A L U L S C N L Y

R E A L T 3 L F P ( 2 @ ) r T L I F T ( 2 C )

REAC: l.13 ' I . T r \ C 1 T L I F T R L A C C I Z ' I T h C ) T A L f A

S L B R C L T I N E S L I P EhC R € . T L R h

c L C C C c C

S L R R C U T I N E T C C A L C L L A T E P R C P E L L E R S L I P S T R E A K V i L C C I T Y C I S T K I B L T I L J N b S I N G

L I F l C l - A K A C T t K I S T I C S k c r d - LINEAR ULAL;E ~IRIXIL SECTIGX

L 2

q 4 4 4

DbTP DAT P D A 1 A D 4 T P

C P T P C P T i

D b l P C P T G

D P T P D A T A

4

C F S E P i A F S € R ( P R C T i

I 4 4 . t

C . C / 4

F S E k F S E K F S t k

I h I T I

e / 3 l - Ll-

A L I Z E

t- us. I- C L A T l- F. I c .

C A T A A R R A Y S

41-NPS /

41-A 16/ 4t -A 64/ 4 h A 65/

4 P R X . Y /

t3H RH/

Ct-ECK V A L U t GF I P R A B

t C

E C 1 - h e h C A S E , k E A 3 C A H C I N P L ' T hE 1 - h t k A L P H C Oi\lLY, S K I P C A R D I N P U T

L

2000 C C h T I hLE I F ( € P Q b B - l ) 2 3 5 0 , 2 C O C . 2 3 5 C

t S E T F I X E D C C K S T A N T S K P = 6 KR=€! P 1 = 3 . : 4 1 5 $ 3 7 . u ~ ~ = 5 7 . 2 9 5 7 e

C S E T A I R F O I L C O h S T A K T S t C R I h I T I A L S G L L T I C N A K ( 1 . 2 ) = - 4 C . C A K ( 5 , 2 ) = - 4 C , C AK(3,1)=-7.3 A K ( 5 * 1 ) = - 6 . 4 A K ( 4 , 1 ) = - C . 4

C ..*............ e.... R E P C . S T C k i : ANC I h C t X T I P L C S S C G R R T A B L t S C C C c 4 + G U T r.LST NLFdER E I T k E R C OK 6 3

4 4 4 4 C A T A C A R D S A L C E P T E O I Y 4NY CHDEr l . *+

.4

4 4 4 +

4. *+ *+ 4 4 hCME H E G U I H L C I F T I P L C S S T A b L E S

4 4 P R C - A L R k A C Y S l O K E D C K CISK F I L E

4 4 F C L L O I U I I \ G C A R C PUS1 B E B L A N K .t te 4 4

160

2 c 2 100

2110

2120

213C 2140

5 k C C C C C

1: C

t C C C C C C C C 4 L 2200

2 2 C 1

2202 C

22C3 t c 2204 2205

2 2 C 8

C C 22 10

C 2 2 2 0 .-

3c s 214C

R E A C

I R

*

) *

s1

IK=

'CR t

1 r 8

ANiJ I R C E X A I R F C I L T A B L E S 11 .. -1

++ T I T L E C A R C S P C S T BE R E A D F I R S T - 4. NCT R t i Q C f R E D FOH S T A l l D A H O T A & L E S **

tt

4. 4 4 F C L L O k l K G C A R L P U S 1 E € B L A N K - I

4. e a ..

4 4 C A T A C A R D S FC,K E A C H A I H F G I L S t T 4 4 K L S T C E A S S € P B L c L : I n C E S C E N O I N G 4 4 C R C E R C F - C L I I f i C R E A S I N G 4 4 - T / C I N C w E A S € N G *it 4 4 - P A C k I N C R E A S I N G * 4

"

.4 4 4 +.

+* a + NCNE R E E U I H E C F C R S t C C h C C A S E

et F L L L O h I N G C A R D F'UST B E B L A N K 4 4 4 4 4.

4. 4 4 4 4 4 4

R E A C A I R F O I L S E R I E S 1 I . T L E C A R D S

1 1 IICr r Z i L 2 r E F S P . 1 A F S R 2

FSR 2 2 9

C . € A D

l r A F S R f r A K 1 1

S € T T A L L

A K 2

E I K C E X Ah F I R S T T A B L E

I h I T I A L I L E A I R F C I L S E C T I C h I h D E X

TI-E € A =

T k E

REA I T =

( I

I T

CL!* T / C p AkL 4 I R F L i L S E C T I C h I I L D E X S E T L A S T V A L L E S OF A I R F O I L C C D f c

i: C l - E C K F C H L A S T A I H f - O I L T A 8 L E C C

€ F t f l - E A D ( I T . 2 1 ) 2 2 2 5 ~ 2 2 3 C s 2 2 2 5

T A B L E

c C C h T I K i L E * R E A 0 C A T A CARDS F C R N E X T T A B L E 2225 € B = I t f A C ( I T 1)

R f A C C K R t 2 9 1 3 ) C T A L F A ( I ) r T L I F T ( I ) ~ I = l ~ I B )

161

. . . . . . . . . . ...... . . . . I ,I I

2230

2 2 4 C

2 2 5 G 2 2 6 C

C 220C

229C C

2 2 9 5 C

2296

C C

c C C

C

C

C C

2330 C C C 234C

2 3 4 1

2342

2343

2 3 4 5 2344

C C

2350

C

C C

C

162

................... P L F P = A L P H B + E Y E T L

C P = C C S ( P L F P P I A L F P Q = P L F F / P T C

A F L T = G J * C P / F I A C C t - T = F l * P C C t L / A J DhP=CP@*RhE!*

C

C C

C C 2360

2365

2370

C C

C C z3ac

2400 C C

C C

2 4 C 5

24 10

2 4 1 1

2415

24212 2425

2430 2435

2440

2445 2 4 5 C

24.5 5 2460

1 4 .+ .* 4 t * 4 L A S T D A T A . C A R D Y U S l E A V t R P U A = l . D *+

C t E C K

I F GHPER-loC 1 2 4 C C r 2 3 f ! C 9 2380

= 1 , 3 l r k G A T A L I S ) r

FCk L A S T B L A C E S T A T I C h D A T A CARD

.................... C b L C b L A T E B C U N O A R h € T = C F=C .O

.ALFI-A=C 00 A C P C i - = 4 C C l - T + S C R T ( l . C + A ~ U 1 4 4 2 )

LC=C.C CL=C.C

~ S ~ R C I S ) = S G R T ( H S ~ H X 4 + 2 + 1 1 . C - K P B R X 1 + 2 ) + ( 0 R S T C Q = R S ! ? R ( IS1 C E P E i i ( I S I = l . G b S T E K ( I S ) = C . C

C T = C T + C 5 * C C T X * ( 1 .C-HPHRX I PI- 1 s=c .c C C = C G + C . S * C . C C X I ( l . C - R ~ ~ R X I GC TC 275C C C h T € k U E .................... L C C P T E 4\13 I R D E X

4kC G F C P t l k I C O A T

A I R F C I L S E C T I O N G TC L C C h UP C L F 0 ; i

I . T = I T h ( k P , l ) I.F ( . I T ) 2 4 1 C 9 2 4 C 5 , 2 4 1 0 W R I T E ( K P v Z S S 4 ) h A s R P t l R I S = I S - l GC TC 2 3 6 C T t E C l = T l - E A C ( E T s l I D C 5 4 1 1 I = 1 , 4 I T C ( I t l ) = C I , T C C I I ~ ) = C I P = l I.@=i I-c = 1

Y

d k

' . VALUES FC

5 + 1 . 0 / ( 1 . C

H S L I P S T R E A P

+USt lRX I 1 I

L E S E D

16 3

2 4 6 5

2 4 7 5 24 7 0

I F I ' ( ' 4

- T I - E D 1 1 7 5 * 2 4 8 0 i i i

T C L 1 € T Z (,

T T C C T X C L IC=1 F2= I IC)

I T - 1 GC T

T T C C ZTG (

T X C L

Iici 2 4 8 0

T T

9 5 9 2 4 9 0

1)

2 4 6 5 2 4 9 0

1e=4 I C = C 1. E.= C GC T C 2 4 5 5 . C C r \ T l i ~ b €

I3 2495

C L . A T E I N I T I A L

I T E R A T I O N R C L A L U E S C F UT I NE

P H I R A N D C X FOR - ...... . . . . . . . .e ..*.e

C

Y- FP X = 'V = L ' * 2 ) / (

. C - Y * * Z

e L t 4 . C .

= ( Z . C S C L * b ( F t T J

P i - . Ih=Pk PI- I =PI- I . c x z x / ( 1 X=L 'XCZK

h i F T = l

U X C Z k = C

C 25CO

c ..*................. B A S I C I T E I I A T I O N R C U T I N E C P = C t S ( P Y I R ) S P = S I h ( P l - I P )

L C C K CP TIP L O S S C G R R E C T I C N FACT~IR

I.F ( . h e - 4 ) ~ 5 2 ~ ~ 2 5 ~ ~ ~ ~ 5 1 0 GC rc 256c

2510 F C f F = l . C

2520 l@=hC-l C CI-ECK I F R P B R L T C.3 - P S S L k E R P B R = G.3

AH=lC.S.f!PBR I . R = P R 1.F ( I R - 3 ) : 2 5 3 C * 2 5 4 C * 2 5 4 C

2530 k H = 3

C C C h T I h b E - C C M P b T t : I N T E H P C L A T I G N F R A C C I O k S C C V P L L E S T L , b f I N T E Q P O L A T E O 2 5 4 0 D R = P R - I R

AR=3.C

Ah2 A H H A Y I h C I C E S FUK T I P LCSS C G R Z L C T I O K

€ C * I R-2

A P ~ L C . C ~ S P € P = A P D F = P P - I P

I C = I C + l

r I h T E H P C L A T E FOR R P B R A T E A C F S I Y I P H I ) L DC 255C 111 r 2

164

^.L F C R € A C k A I H F C I L T A B L E

2686 .CLP ( I C ) = T L F T l GC TC 269C

F

EXTRAPCLATE FOR ALPHa

I h T E R P C L A T E FOR ALPHA b 2687 C L P f I C l ~ T L F T l + L T L F T 2 - T L F T l ~ * ~ A L F L ) / ( T A L F Z ~ ~ ~ L F l ~ 2690 . C C h T I h L E

C C 2691 . C L C t I P ) = C L P ! l )

GC T I : ( 2 6 5 1 * 2 t 9 2 ) r I 8

EXTRAPCLATE FOR APACh

165

GO TC 2695 F IhTERPCLATE FUR APACH L 2692

C 2695

27C5

. L Y ( l ) l *

hTERPCL

( AE:ACH-

A T € CL

(.TPCHZ-

AhrC CL

,TKCHL i I-

, 2 7 0 5 t 2710

L C IhTERPCLATE FOR T C C C 271G L

T

T 1

G ( I A ) CC( I k - 7 XCL C C ( I A

C 2715

E X 1 2 APCLA T E FOR TGC

2 7 2 0 c

2 7 2 5 r 2 7 3 C ,2725

k IhTERPCLATE FOR C L I

F

C 2725 C L = C L C ( 1 ) + L C L C L Z ) - C L C ( l ~ ~ + ( C L I - T C L I ( l ~ ~ / ( T C L I ( Z ~ - T C L I ( l ) )

G C T C 2 7 3 5 EXTRAPCLATE FOX C L I ObTSIGE TABLE L I P I T

L 2730

C 2 7 3 5

A C t i * *

A G C

2740

t C C 2745

01 SK PLANE ELEF ElrtT

c

L 2750 W R I

1FvP I F

2755 I F 2760 k R I

IS= 28CO .CCh

GC

c

h A , 2 ) r C L I ~ T C C I , PI- IS

I P S T R E A K ELtMENT

C 1

I s-1 T C 2 3 6 0 T kHLE

SLP INTEGRAL

I

C 281C

C

E E C

2831

2812 2813

211 15 2816 28 17

2820 2625

2830

2e35 2840

2845

2850

2b55

2060

286 1 2862

C

2865

,~"..,............. CALCI

"

VALU6.S OF IIJTtGRAL TERMS

1~....-............. PRIhT FI!VAL SLIPSTREAM IhlEGRAL

HRITE(KPrs295@) C T S r C T r C C r V S B P R .................... CALCCLATE SLIPSTRiAM VALbES FOR

EF ChPCT( 11 1 2812r2811.,2@12

IRPLT TC; h lNt i ANALYSIS

GC TC 2E13 N R = - l + I A H S ( h R C T ( 2 ) - h K C f ( l ) ) KCP=C

kR=- 1

*ISIGh

S

STATIGNS

16 7

28 7C .CCh T I hL E 2897 WRITE(KPr2SSl) I D E R T

C 2901 2911 EY 1 2 2913 292 1 2923 2924 2925

.e. R E L I C

. C . C ) 0 )

FGRKATS

b C e.. 0 "0." . 0 . . e e .o 0 . . . PR 1 i \T FCRaYATS

168

C C

R E A C E H ( I R ) A N D P H I N T E K I I P ) L O G I C A L LNIT FiLPBEHS

lR=8 € P = 6

. R c h I h C 2 R E C E K C . 3 R E h I h C 4 .RE.h I h C 5 REhIhC 10 .KEA IkC 15 REhIhC 20 R E k I S C 7 K E E F x . 1

RGk1h.C 1

L I h P L T C A T A S E C T I O h ;

C C C c C C C C C C 4 z C C C C C C C C

C C

C c E C C c

L P Y C U T CF F C L ' R T H C A T A C A R 0

F l E L C 1 I1 N L Y B t H C F T A L V A L U E S PER T A B L E

F I E L C 2 11 1 FOR K E A D - . I h C F T A L I R E Y t E T C . LJ F U R ItL X c A C I h

F I E L D 3 I 1 1 F C R O b ~ P C F C C C P U T E O A K R A Y S 0 FOR NC DUKP

G k(1h NU UUPP F I E L C 4 I 1 1 F@R I)UrlP C F B E T A A R R A Y

F I E L C 5 I1 WOK TAPE LGAO CUtiE 1 f G CUBE

C C C * L

30

CbLL SE

Ul--ERZ( 1 DC ? C J

HI-ERE (2 P A k h t J ) M P C C L ( J I Y = C €F ( EG-3

S h I T C k L C K C CN L l L L READ I N VALUES FRCR C A R C S FCk TAL, R E Y : C I E P S , E C G E t CRL Ah0 ACC. F C R F ' A T I S L ~ F ~ . C . A R A A Y S A R ~ REA^ I N R C h n l S k

t 40

169

I

40

50

6C

70 00

100 90

110

12c

130

l K = l

r F A C C L r P A Y k r 7 r K E E P )

r Y A C C L * P A k k r 7 r K E E P )

170

150

- 160 L 5 I F hO F U S E ( F U S E L A G E 1 L

€ F . ( 4 1 4 1 0 e 4 1 0 e 1 7 0 170 Y C = e r S C R T ( l . - ~ + * Z / A * * 2 ~

E C . C = S C R l ( A * + 2 - 8 * + 2 ) C A L L C A T S h ( C e I ) GC T C ( 1 9 C e l e C I r I

183 C R O = 2 0 * ~ 1 . - Y C 1 / ~ A S P f C . ( 1 . + T A P E R - 2 . r Y O + T A P ~ R - 2 ~ * Y U ~ T A P E R 1 ~ 190 ~ F P C = l o - ( . Y C r T P U H + C R ~ / ( ~ . 1 4 l 5 9 *

Yo..( I I C S E E CK1646 F C R E X P L A R A T I O h S L

Ct-ECK IF FOSE E L L I P T I C A L OR C I R C U L A R C

C € E ( A - F ! ) 2 3 0 e 2 3 0 e 2 1 C

F E L L I P T I C A L FUSE

Y B A R P R I F E ( I . ! 2 2 0 Y C 4 X ~ . I l ~ ~ Y C ~ ~ I l / ~ A - D ) . . ~ ~ - b + C I S T ~ I l / S Q ~ l ~ C I S T ~ . I l ~ * 2 - ~ C C * ~ Z l l l / 8 W X

C

C C Y B A R P R I M E 1.11 FCR C I R C L L A R FUSE C

GC T C 2 5 0

230 B k X = l . - . A * * 2 / ( 1 . + ~ + + 2 1

24r) Y C b X ~ ~ I ~ = Y C A ~ I ~ * ~ 1 . - A r r Z / l Y C A ~ I l ~ ~ Z + ~ ~ + Z l l / ~ ~ X 3C 2 4 C I = l r J P P

C C C

C

CCPPCIS TI1 t L L I P T I C A K O C I k C L L A R FUSE

250 DC 260 I P l e J P P

5 Y BAR . ( I 1 L

A [ = I - 1 260 Y X ( € I = C C S ( 4 I * P I E i 3 1

I

171

t C

360

37P

391 380

40C 390

4 lC

42G

C C C

C C

430 44 0

4 5 C 4 6 0

410

4 8 0

49c

50L:

51L 52C 53c

54 1

k I h 6 ON TOP GR 8OTTUP OF FLSELAGE

2 + l - + + 2

* + 2 l+SC ( X I S F + +

CC k C T k.A!qT TU CllPPUTk C R O . S P k L I I L CASE I F V I L U E S HAVE BEEN READ I N

k I L L I \OT COCPbTE VALLES ALREADY K E A O I N

,172

A L P k P ” S 9 9 - .CL.L=C. R E Y C h r S 9 9 . XCPX=O.

r

C A L L t ? R l C G €F( € E l 5 6 C

G L

LCCK LP ZERO L I F T A N G L E F C R .

LCCK LP ZEKC LIFT ANGLE FCR H c o r SECTIUN

t

C C

5ec 570

bOC 59c

6 1 L 62G 610

L C

T h I S T = T ’ n I S IF ( T h l S A - 1

GC T C t C O

C b L . L C P T S k T h I S P = T h I S

GC 1C (-i3C cc 6 S b 1 = 1 E P S L I . ) = T h l

.LC.C E R = 1 € F ( h F L b F . h

C L L = 9 4 9 .

R E Y h Z S S 9 o A L P k / ; = S 9 9 0

.REYCh‘=C.

C6LCL ‘LATE AERODY!JAP! IC ANC G E C M E T R I C T l v I S T

ALPI -Z 5sc

ALPI-Z

1 I Y ( I )

FCR X P A X = l C C . A R C h I L L L C C K LP A L P H A F A X F C R XI’AX=C. ARC t. I L L L C O K U P C L F A X

320 F C R P P T r e F 1 C . C ) 6 b C F C R P b T ( 1 X ~ l l k t R R G H CCCE ,12,13F -IF1 S E C T I F N r I 3 , S H A B C R T E D I

E ~ , T :

C S C B R C L T I I J € P A I N 1

2 P A I h l - - - - C C h T I N b A T I . O h CF S U B R C L T I h E P & I N L

C C C

3.74

c .12c 13C

1 4 C

150

-60

17C

L80

190

2CC

2 1.n Z Z C 2 3C

240

250

27G 26C

280 C E

. I 13c1 I = 1 1 hP

P €ER T S T I X )

t c = 1 . / 4 5 .

, ( 1.-cc = I . / s c . = l . / S C . (l.+CCS 1=20*SX )=2.*SI = T C h l ( I = G E h E ( I E

3 10

14C115 AX*Slh

C S 1 TI-.F t - T b T A

+ 1.5 I A X € 1 - S C . E ) - 9 0 . E G A E G A

c s ( T I - €

STbX 1 1 * T S T A X S T A X 1 (3.141 9 * T C h Y cj*.C-EKE

CI-ECK FGR CLWP

CkECK FCR O W P CALL CAlSW( 3.r.I')

175

29C

3 C O 31C

C

C z C C C

C C C

320

330 C C C

C c

S T C H E B E T A T E M P G R A R I L ’ Y Oh DISK SO WE C A N C C C F U T E THk TRAlriSPCSF OF T R I X

R E h I h C 44

R E h I h C 44 ~ W I l E ( 4 4 1 e E T 4

REPC 1.44 1 E E T A

S T C R E T R I X I N B E T A

hCk TRAKSPCJSE b E T A ( C L D T R I X )

R € S T C H E D E T A

I h V E R T T R I X

C

c C C

$20

S E T tip FOR C L P A X L O O K UP

lrCK CP C L P A X V A L U E S

XM A X I

176

490 4 'IC

5 1 C 52C !a32

5 5 2 563 5 IC)

sou

54c

590 6C 0

L

FCklvAT I

F C R C A T I F C P P b T (

F r P F A T I F C j u C i i

l o x 1 x / 1 ll- 1/

l o x . l o x * 1l)X * l r r X * l o x * l o x * l o x * L O A * 1 3 Y *

' I N E FA I R 2

PA€1\2----- G

/ 1 X ) S E S T A T I G N S - Z Y / B )

E Y h C L C S h b K e E R S , M I L L I O N S ) / C P C R C D I S T R € B U T I U N )

T R I Z U T l C i J ) T R I B G T I O K ) , S T R I t L T l O h )

EhETIPL P R I N T S U R i t C U T I N E - - - - - - - - -

Ci-CHC. K k E S S C f l C . 2 / 1 I C T b I S F l C . L + 1 I h G I LE

A T t Ut: F l C . 2 / 1

/ l X 1

T ~ P ~ R

4 2 t 1 C X 9 3 U H G E C C E T H I hhlSt S T A T I O ' V S . = Z / 1 5 X r 3 C h F L A P S P A

L CI C €

Z * T 1 H C

4 Y l - 3 A L

C € LC

11.; cc

1 A L cc 1 E P 2 * P k Tk 1 1 s

5Wk

, ~ a z z it

K C L * v S T A L * C P ( 1

Pl-A * K E Y t E R E r k I - TAPEt3, U * I Y * I Z * P u Y Y v F L RICACBT

177

C 70

C C C C

80

90

66

47 101 135 136 1c2

103

104

105

1 C6 109

1003 55

lOiC

101 1

5 1 1 G 3 1

1 C 7

1071 52

1072

OC 7 C K = l r h P . A L P G L K ) P A L P ~ B + P L P ~ R + E ~ S ~ K ) + A L P ~ ~ ~ T F A C . C ~ ~ & N $ ~ K ) . - ~ ~ ~

CLWP CF A R H A Y S COPPUTED CLWIhG I T k R A C I O h : S h I T C l - KUVEER 3 IS U S E D F C R 4N I N T E R N A L

PPCCESS

178

53 DC 120 Kz1~h.P AK=F SLVl=C.C qc 121 h=lrhP Ah=& SLtJZ=C.C 3 c 122 C=l.hP

122 121 3 2c

SI ) I 'AN

l': (. AN* .AM*P 1 ER

130 L 13G 13 L 1311

54

/ E D G E

(.K 1 +CRB+ 60 1 ) )

I Y T I E c R E Y c T A C

100 11c PSPECt1.E) 1*C

/C1.14159*CIK

.L 20 14 0

159

6.0 1 /EDGE L C C c

170

c 18C

G Cf-ECK FCR CUPR L

8

1 9 C

4 I

4 I

A I

d I

A I

I C E Cl-ECK TCLERAhCE

179

C A L L C P T S k ( 3 9 JUhK ) GC 'IC (26Z127C)r JIJKK

C P L L b A A ( C 8 G T h P I 26C W R I T E ( I P , 3 9 C )

t F R E P E A T C Y C L E +.

27C € T R = € T R + l €F.( €TI?-30 1 14 1914 1928C .- 1 4 1 IF ( A S L I P - 1 ) 1 4 C , l C 5 r l C 5

280

3 0 C

3 10

c

CLFP C E L T A V A L L E S , C V P L U E S , i4ND T A B L E I F L N A H L E T C C O N V E R G E A F T E R 30 I T E R A T I O N S

P R E S E h T L Y I-N C C R E BEIhG LSEC F O R L C C K UP

68 f C R r ~ T ( l O X , l t l - Z E R C - L I F T A N G L E S ) 37C FCRf'A.T I 1 O X 4 M L P G 1 383 F ~ R C A T ( I G X * 4 F C V A L ) 330 I-LRPA r O 1 G X v 2 k C E G I 40C F C R C b T L l O X * S l - A L P l ' U I 4 1 C F C R P A T ( l C X , 5 l " b L P h E ) 420 F C R f J P T C l O X , 3 P C H C ) 430 F C R C L T L l G X , S t - C E L T A l 440 F C R t ' A T ( l X t 4 E t L X A B L E TC C C h V E R G E A F T E R 30 I T E T R A T I G N S A B O A T E D ) 470 FCRCbT( i O ! p I 2 , 6 3 F I T E ? A T I C h S R E Q C I R E O T O C C N V E K G E FOR AhGLE CF AT'

l F i l C K E6L.A- T C v F 8 . 2 / 1 X ) 480 F C R I A T ( l X i l C l - . E R R C K C C G E , € Z , l X ~ l C P A T S E C T I O N r € 3 , l X o 3 2 H I ~ PROGRAKT E

1 X E . C L T I C h . T E H Y I h A T t C 1 E h C SU@RCL;T I R E P P I K 4

C c I s A I k 4 - - - - - CGKrINUAT€CN CF S U B R C L I T I N E ).'.PIN2 L

C C C

1

J.80

b

CCPPLTE QUARTER Cl-CRD P € T C H I N G WOPENT C C E F F I C I E k T S

4 Cl-ECK I F S E C T I O N S T A L L E D

C C P P U T E S E C T i O h PITChZNG CCPENT

40

50 .R€!/ELrX

L. 5 Cl-ECK FGR CLiPP L.

53

70

80

80 1

- z c

9C

StiP2-COO

'2

CCP t CLPP CCNAC, 'ZKrCC ;ch!pp r C O P P

C ~ F I ( K S T P L . F C o C ) . A N C . ( K C ~ ~ T o ~ ~ o C ) ) GC TO 250

4 CEFIhrE E X A C T S T A L L A h G L E CF A T T A C K L

10oc

2GCC

2 102

30c0 31CO

323C

3 3 0-c

9 9 C O

170

25C

1'7 1 c

IH(-101 R E T C R K

GC TO 9 S C O

GC TO 171

C C"- K P I h 3 - - S U B R C b T I h E FGR TI-E C A S E hITl-! P A R T - S P A N FLAPS---------------

1.82

I C t AXX CGE , t

I t

P U T F L A P S - L P CL D A T A I N T C CGRE CbLL LAGETLPRRAY t 1 9 1Y 1

CC 6C K = l r h P A L P G C K l = 3 -

. L C L E R = i

LCCK LP C L V A L U E S

ALPkU=ALPE(Al C L L = S 9 9 .

R E Y h = P E Y ( K )

REYCh-SS9, T L L X = T J ~ L [ K I

X P A X = C . .CP.LL P H C ( A R H A Y I T A U X ~ ~ ~ X X , ~ X C C L ~ I E T ~ ~ E R E , ~ L V L )

Cl-ECK FCR ERROR STGP

83

9G

LCCK LP A L P F A F U R LERC L I F T

LCCER=3 ALPkh=SS9. CL.L=C.C K€Yh=HkY(KI T A L X = T A L ( K ) R E Y C h 5 S S 9 -

L

4 LCCK L P C L V A L L E S L

L C C E R d 4 .CLL=SS9. ALPI-P.rALPkE.1X4

183

t C

C C

C 220

O W P A L L

C

C C

2 C

C A L C L L b T E A C C I T I C h A L L I F T D I S T K € J U T I C N I C L 1

I h B C A R D CF

C c . L E F = G ~ S C r J

. C L A C C ( K ) = & V A L ( K ) / C C L I F CLL=O. AL.PkP=S99. R E Y h = R E Y ( I S T A R ) T A b X = T A L ( I S T A R )

C A L C L L A T E Z E H C - L I F L A P E h C

D C Z 6 C K = l # h P

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PREDICTION OF SPAN LOADING OF STRAIGHT-WING ...

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