1E~k~icA LFOZApi'

~ STL"DY TWO-DIMENSIONAL PRESSURE DISTRIBUTIONS

- ON ARBITRARY PROFILES

by DDC

Terry Brocket

TlSiA

Dhtribution of this document is unlimited.

MATHMATIS ~HYDROMECHANICS LABORATORY

RESEARCH AND DEVELOPMENT REPORT

0t

ACOUSTICS ANDVI~iATQNOctober 1965 Report 1821

JLf -.

STEADY TWO-DIMENSIONAL PRESSURE DISTRIBUTIONS

ON ARBITRARY PROFILES

by

Terry Brockett

K Distribution of this document in unlimited.

October 1965 Report 1821wo S-R009-0101

K (

TABLE OF CONTENTS

Pcge

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . I

ADMINISTRATIVE INFORMATION . .......... .... I

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . 2

POTENTIAL PRESSURE DISTRIBUTION FROM APPROXIMATECONFORMAL MAPPING ... . . . . .. .. . . 4

SUMMARY OF DERIVATION ......... ....... 4

COEFFICIENTS OF THE TRANSFORMATION . . . . . . . . . . 5

LIFT COEFFICIENT. ........... . ....... 9

VELOCITY AND PRESSURE DISTRIBUTION . . . . . . . . . 10

EMPIRICAL MODIFICATION TO ACCOUNT FOR VISCOUS EFFECTS . .. . 13

NUMERICAL METHOD . ... .... ........... 15

TRIGONOMETRIC SERIES TO REPRESENT THE ORDINATES. . .*. . . 15

EVALUATION AT SPECIFIED POINTS 18

EVALUATION AT AN ARBITRARY POINT . . . . .*. . . ... . 20

COMPARISON WITH ANALYTICAL RESULTS . . . . . .... . 22

COMPUTER PROGRAM .......... ....... . .... 24

COMPARISON OF POTENTIAL THEORY WITH OTHER METHODS . . . . . 36

COMPARISON WITH EXPERIMENTAL RESULTS . . . . . . . . . . 44

COMPARISON WITH MEASURED PRESSURE DISTRIBUTION .. .. . 44

COMPARISON WITH MEASURED CAVITATION INCEPTION . . . 48

U ii

Page

SUMMARY AND CONCLUSIONSa . . . . . . .. .. . . . .53

ACKNOWLEDGMENTS ........ ............... 54

APPENDIX A - EXAMPLES OF EXACT CONFORMAL TRANSFORMATION a 55

APPENDIX B - DISCUSSION OF AIRFOIL GEOMETRY . . . . . . 62

APPENDIX C - LISTING OF FORTRAN STATEMENTS FOR THE PRESSUREPROGRAM . . . . . . . . . . . .66

REFERENCES . . . . . . . . . . . . . . . . . . . . . . 72

ii

LIST OF FIGURES

Page

Figure 1 - Chordwise Stations x m =12(1tCOS ). . . . . . . . . . 21m '1

Figure 2 - Comparison of Analytical and Numerical So!itions forthe Location and Value of the Maximum Velocity onE II ipses . . . . . . . . . . . . . . . . . . . . . . 25

Figure 3 - Data Input to Computer at Required Stations .. ... . 28

Figure 4 - Data Input to Computer at Arbitrary Stations ......... 30

Figure 5 - Typical Output from Computer, Profile Constants ....... 33

Figure 6 - Typical Output from Computer, Pressure Distribution . . . . . . 34

Figure 7 - Comparison of Theoretical Pressure Distributions forNACA 16-009, Zero Incidence , a&0

Figure 8 - Comparison of Theoretical Pressure Distributions forNACA 0010, Zero Incidence . . . .......... 41

Figure 9 - Comparison of Theoretical Pressure Distributions forNACA 65AO06, Zero Incidence ... . . . . . . . . . . . 43

Figure 10- Ordinate versus Angular Variable, Showing Bump near theLeading Edge, NACA 65AO06 . .. . * 0 # 0 . *. .. .43

Figure 11 - Comparison of Theoretical Pressure Distributions forNACA 4412, c = 6.4 Degrees . . . . . . . . . . . . .46

Figure 12- Measured and Predicted Pressures on the NACA 4412,CL = 1.024, a= 6.4 Degrees. . ... . . . . . . . 46

Figure 13 - Measured and Predicted Pressures on the RAE 101 Section,10-Percent Thick, C L = 0, a = 0 .. . .. . .&. 47

Figure 14- Measured and Predicted Pressures on the RAE 101 Section,10-Percent Thick, CL =0.218, a,= 2.05 Degrees . . . . . . 47

iv

Page

Figure 15 - Measured and Predicted Pressures on the RAE 101 Section,10-Percent Thick, C L= 0.430, c = 4 .09 Degrees . . . . . . .47

Figure 16 - Comparison of Incipient Cavitation Number and MinimumPressure Coefficient, NACA 4412 . 4 * . ... . . . 50

Figure 17 - Comparison of Incipient Cavitation Number and MinimumPressure Coefficient, Elliptic-Parabolic Section,NACA a = 1.0 Meanline .. . . .. . . . . .* # * 0 0 .52

Figure A-1 -Comparison of Exact and Approximate Velocity Distributionfor Foils at 0 and 5-Degree Incidence . . . ..... 61

LIST OF TABLES

Table I - Values of xm = 1/2 (1 + cos MITT ). . .. . . ..... 2118

Table 2- intermediate Points . . ..... .. ....... 25

Table A-1 - Comparison of Exact und Approximate Velocity Distributionfor Two Foils . . . . ... . . . . . . . . 57

3

6

6

7

V

PRINCIPAL NOC)ATION

A Real part of complex hansformation coefficient Cnn

a Radius of circle which is transformed to profile

a Coefficient in Fourier series for ordinatesn

an Multiple of i in the complex transformation coefficient Cn

b Coefficient in Fourier series for ordinatesn

C (6) Theoretical velocity on profile at zero angle of attack

Cc Coefficient of force parallel to chord

C L Lift Coefficient of force normal to direction of motion,1/2PU c "lift coefficient"

C Moment Moment coefficient, positive clockwisem 1/2P U 2 c2

CN Coefficient of force normal to the chordline

C Complex transformation coefficientn

Cp= 1/2, 2 Pressure coefficient

c Chord length

D (SP) Theoret;,al velocity on profile at c = 90*

e Base of natural logarithms

F ( ) Complex potential in the plane of the circular cylinder

Imaginary constant, il = -1

K Modulus of complex transformation coefficient C_1,;equals 1/2 theoretical lift-slope coefficient

m, n Index of summation

N Number of angular intervals between 0 and n innumerical method (e.g., N = 18 in Figure 1)

vi

P Local static pressure

Pv Vapor pressure of the liquid

P Pressure For from the profile

q Magnitude of the velocity on the profile

R Exponent of distortion variation

Re Reynolds numbers, based on chord length

S (0) Combination of transformation coefficients

T(() Combination of transformation coefficients

+ Dummy integration variable

U Asymptotic velocity in profile plane

V Asymptotic velocity in circle plane

XN x-coordinate of profile nose

x Abscissa in plane of profile

Y Ordinate of profile

¥ C Camberline ordinate

Yeven Interpolation ordinate for 1/2 (Yu + YL ) ' i.e.,camber contribution

YL Lower surface ordinate

Y N Ordinate of profile nose

Y odd Interpolation ordinate for 1/2 (Y - YL ), i.e.,thickness contribution

YT Thickness distribution ordinate

Y U Upper surface ordinate

vii

z Complex variable in plane of profile

a Angle of incidence in plane of profile

a' Empirical function to approximate boundary layer effects,chordwise variable

ao0 e Experimental angle of zero lift

c Angle of incidence in plane of circular cylinder

r Circulation, lift/PU

A CL Decrease in geometrical angle of attack to obtain actual lift

Complex variable in circle plane

Lift-slope coefficient, (dCL/da) experimen / 2

x - a , equals theoretical angle of zero.lift

X. Index of summation

v Index of point at which velocity is obtained

tDummy integration variable

P Fluid mass density

P LE Leading edge radius of curvature, nondimensional_P -Pv

-'2 V Cavitation index at inception or desinence of cavitation

r Thickness ratio, maximum distance from upper to dowersurface of the thickness distribution divided by the chord

0 Angular variable in cirrcle plane, for the approximationit equals arc cos (2x - 1)

w Angle of rotation to put profile nose at (0,0)

viii

ABSTRACT

A method of determining two-dimensional pressure distributions on

arbitrary foils is explained. The development is based on an approximate

potential theory suggested by Moriya (1941) which is empirically modified in a

manner suggested by Pinkerton (1936) to give an arbitrary lift for a set

incidence and at the same time satisfy the Kutta condition. Interpola-

tion functions for the ordinates are used to reduce the calculationz to

a straightforward numerical procedure which is easily programmed for

machine calculations. (FORTRAN statements for such a computer

program are included.) The results are those of Riegels (1943).

Comparisons are made with othei theoretical methods for poten-

.ia! flow and with experimental results. Good agreement with both

calculated and measured pressure distributions is found when the lift

coefficients are matched. The assumption that cavitation occurs

when the local pressure falls to the vapor pressure is not upheld for

the cases considered.

ADMINISTRATIVE INFORMATION

This report is essentially a duplication of a Master's Thesis submitted to the

Graduate School of Cornell University in May 1965. The work was performed at the

Taylor Model Basin under Bureau of Ships Subproject S-R009 01 01.

INTRODUCTION

To obtain an accurate estimate of the actual pressure distribution on two-

dimensional foa.s, we have two approaches: (1) to solve the cumbersome Navier-

Stokes equations governing viscous flow or (2) to use a potential theory with empirical

modifications to approximate real fluid effects. The first approach may be simplified

to solving the boundary-layer equations for a two-dimensional curved surface, but

the simplification is nominal and the task remains formidable in application.9 2

Moreover, as yet, this method is not sufficiently advanced to give accurate results

unless experimental boundary-layer data are known. 3 The second possibility is based on

potential theory which means simpler develcpment and shorter computation time. As

normally used, this method makes use of the experimental lift which is either available

or can be estimated for conventional foils.

The first satisfactory method of using the experimental lift in potential theory

was developed by Pinkerton in 1936.4 He made use of the Betz (1915) observation'

that empioying the experimentally determined circulation in pace of the theoretical

value gives a pressure distribution that agrees well with measured results over most of

the chord. However, merely inserting the experimentally determined circulation

into potential theory will not, in general, satisfy the Kutta condition that the flow

leave the trailing edge smoothly. Pinkerton4 was able to retain the measured lift

and also satisfy the smooth-flow condition by introducing an empirical modification

in the profile shape. He applied the modification to a rigorous potential theory

'References are listed on page 72.

2

formulated by Theodorsen in 1932." Although Theodorsen's theory is easier to apply

than the bounday layer equations, the calculations are still lengthy.

Riegels' removed this lost complication in 1943. He adopted Pinkerton's idea

of an empirical distortion in the flow field to an approximate potential theory

developed by Moriya in 1938.- Riegels made a further simplification by linearizing

the equations on a small pa:ameter introduced in the modification. in addition, he

used interpolation functions for the ordinates which reduce the calculations to a

straightforward numerical procedure, requiring only offsets at fixed fractions of the

chord. The linearized numerical procedure of Riegels has been in use for some time*

at the David Taylor Model Basin where it was programmed' for the computers at the

Applied Mathematics Laboratory. In addition to giving the chordwise uressure

distribution, the results of this program con be used to predict cavitation inception

if the assumption is made that the cavitation starts when the local pressure falls

to the vapor pressure of the surrounding liquid. Unfortunately, output from this

program consists of the pressure distribution at only the input points so there is no

assurance that the minimum is obtained. During the analysis necessary to correct

this omission, il became obvious that the Riegels derivation .as quite obscure. A

detailed derivation of the theory was accordingly undertaken, and the results of the

in./estigation are outlined in this paper. The derivation closely follows later work

of Moriya1 (1941) which is an approximate conformal transformation of the circle

*In 1955, a Model Basin Memorandum (Aero 28) described a method of calculating

pressure distributions over profiles of arbitrary shapes. This memorandum included atranslation of the Riegels paper (Reference 7).

3

to an airfoil profile and gives the same equations for the velocity distribution as his

earlier work.

The present contribution to the development is a unified presentation of the

theory, simplification and extension of the numerical method, and extensive com-

parisons with experiment and heory. In the comparisons with theory, some new

results of considerable simpliclty are derived from an exact conformal transformation

of the circular cylinder (see Aopendix A). Airfoil geometry is discussed in Appendix B.

The FORTRAN statements of a new computer program for calculating the pressure

distribution on an arbitrary profile are given in the paper together with a description

of the program and input instructions. The program statements are listed in Appendix C.

This program will accept foil ordinates at arbitrary stations and interpolate between

them to obtain the ordinates at the required stations.

POTENTIAL PRESSURE DISTRIBUTION FROMAPPROXIMATE CONFORMAL MAPPING

SUMMARY OF DERIVATION

Using the method of conformal transformation, we can easily transform the known

flow about a simple body to that about an arbitrary profile. As is customary, we take

the circular cylinder for the simple body since both its geometry and complex potential

are simple in form.

This development retains the two simplifying approximations made by Moriya":

(1) a group of transformation coefficients is ignored in the calculations and (2) the

circulation is determined by placing a stagnation point at the trailing edge of this

approximate transformation rather than the exact one. For symmetrical foils, the second

approximation is exact. 4

'COEFFICIENTS OF THE TRANSFORMATION

A difficulty with all transformations of the circle into an airfoil is that of matching

the coefficients of the transformation with those of the airfoil. Except for special

cases, iteration is required. In the work that follows, we neglect the iteration and

instead evaluate the coefficients with a first approximation to the transformation.

The mcst general mapping function which maintains the type (but not necessarily

magnitude and direction) of the flow at large distances from the body is given by

z= C C + = n= E- C=' x+- iy r "

- a 0 1 n Cn

where z is the complex coordinate in the profile plane,

C =A +iB ,n n n

n = - 1 , 0 , 1 , 2, . . .

is the complex variable in the circle plane, and

= re ic" I ,as shown in the figure below which also shows the orientation of the

flow velocity.

imaginary

(a

realI

V

5

Substituting C = ae, '.,e obtain ana on the profile o3

A_ i cost- B_ siry - - -c , A cos r,(: - 3n sin n,,21

Y=B coso-A sirv -3 8 (B cosrr- A sinrv) -3--1 -1 o n n

where the capital letter Y is used to denote points on the profile. To ensure that

these expressions actually give the ordinates of the profile, the coefficients of the

series must match a similar expansion for the profile coo,'dinates. Usually %.ve have

0 : x < 1 and Y = Y(x). In Appendix A, several fci!s are derived from exact theory.

For foils which are slender in the horizontal direct ion, it is shown that x is

x = 1, 2 ( -cos CY-- O(7,f)

where T and fare the thickness ratio and camber ratio, respectively. Fcr such

slender foils, a slight change in the x value makes an even smaller change in the

ordinate (except in regions of large curvature, e.g., the nose). With these facts in

mind, we make the approximation* that

x = 1/2 (1 + cos p) r47

(note that this approximation reduces V0 to a parameter) and Y = Y((0) where Y(O)

is understood to be an implicit Fourier expansion, i.e.,

Y(p)=a + ' (a cosno+b sin n(p) F3a ,o n n1

*Alternatively, this could be the starting point for an iteration leading to the exact

transformation. However, the results show this approximation to be sufficiently accuratefor the shapes considered here (see Appendix A).

6

Matching coefficients with Equation r3", we obtain

2-B° = a = 1/2- .Ydq,

0

2- 2-

B +B a ,/- Ycospdp, A -A =b =1- F Ysin:pdci-1 1 1 -1

B1 = - Y cos ordp, ,A b Ysi rYsinnr(pn n 0, n n "o

n> 2

Also, from Equation -4-, we know that A c '2 :-ence the coefficients are

determiner, with tre. exceorc. :,: ,N cr ~ b -ing that the infin tre series in

Eauctiorr -)- on 7- cr'r -.urer conjugates ana n ot the A and 'D terms are no",

we can im'vc, ., write an exoression to- dererminina these co-n!. ,o-.cvu

tne resuirinc exoressions would invoive A ana . Tne depenoence Qn rneseC C

coefficients may be eliminated by considering differentials of Equations r21 and r3l.

Therefore, expressions which involve only A_ and B-, may be obtained by constructing

the quantities: 2-

T (,p) = dx/d- -1/2- .? Y'(t) c-t ((o-t),'2 dt02rT

S (p) = dYidp + 1/2- . x' (t) cot ((p - t)/2 dt F510

where P indicates thot the Cauchy Principie Value of the improper integral is to be

taken and the primes denote differentiation with respect to the argument. Moriya's

notation', '- - differs from ours since he used the Fourier development which is more

cumbersome fnan the integral and different al form piesented hete.

Substitution of the transformation equations (Equations 121 and [31) into

Equation [51 yields

T()= - 2 rA sin(p4 B coscpl-1 -1

S (tp) = 2 [AI cos ep - B_I sino" rSa'I

With the given airfoil ordinates, the expresskons for T and S are2Tr

T ( ) -1/2 sin V - 1-2- .P Y' (t) cot (rp - t)/2 dt

0

S(p) = 1/2 cos o + Y (c) [6]

Equating Equations f5a] and [6] fo; O= 0, we obtain

A_ = 1/4 + 2Y' (0)]

2nr

BI =-1/4,Tf Y'(t) cot t/2 dt F710

Ns ,: Y (U) = (dY/d p) = -1/2 sin op (dY/dx) is zero, for example,=0 TE

with ,rofiles having sharp trailing edges. Since not all profiles are of this form,

Y"'() will be retained in the development.

Although the Fourier conjugates method is only one possible way of obtaining

the .coefficients, it is shown later that the group of coefficients T(P) and S(,p) do

appear in the expression for the velocity distribution. Even more important, we have

found that other methods of obtaining only A I nd B do not give results which! -1

compare as favorably with exact results as do the T(p) and S(rp) expressions.

8

LIFT COEFFICIENT

Arguments based on physical concepts lead to the conclusion that there is no

flow around the trailing edge (Kutta condition). Thus, the numerical value of the

circulation is just enough to place a stagnation point at the location on the circle

which transforms to the trailing edge of the profile. In the exact transformation,

the value of'p which makes x a maximum describes this location. In the approxima-

tion, the maximum value of x is at p = 0.

The velocity in the circle plane is found by differentiating the complex potential

F(C) with respect to . For a circular cylinder inserted in a uniform stream such

that ihe stream makes an angle "x with the real axis, the complex potential is

F(C) = V LCe- +a e / + i(r /2-) Pn iC/a

where r is the circulation.

With (o as the independent variable, the velocity on the circular cylinder

C =ae becomes

dF/dC I = V e-i -e ° + i(r/2ra) e [81

1 C~ae I -aC- (L 4

Applying the Kutta condition, we put dF/dC = 0 at C = a, which sets

r = 4 -.a V sin (cLC) [91

Before the circulation can be converted to a lft coefficient, the effect

of the mapping function on the free stream must be investigated. Ini the circle

plane, the free stream velocity is given as

dF/dC V--Ve-'€k

9

and ir, the profile plane as

d F/dz I z (d F/dC) /(dz/dC) I.Ve iaca/C 1

We define dF/dz 1--Ue'iaZ

and let C =eI = ics+ si-1 %,e K(o + si

But c-i = A-1 + iB_

Thus K= -A 12 +B8- 2

arctan B F01

Equating the two descriptions for the freestream, we find

V= KU/a

a C- X rilli

With the above relations, the lift coefficient is found to be

CL fl /2P U'c = 2TT I(4 A_)2 +t4 12 i~c-)[2

where c is the chord and p the fluid mass density. Hence, x is the angle of zero

lift in potential flow.

VELOCITY AND PRESSURE DISTRIBUTION

The velocity on the profile q is found by evaluating the expression

IdF (C (z)) -bd /(I~ dz) =Ie;qdz lbddC a(i 1r ;

10

Performing the operations, we find

2V a I sin( -;)) - sint a

q = " 131

2jd' (dY)

Substituting Equation r I ] and expanding the right-hand side, we obtain

q 2 I (B B cosp -A sine) cosz

U F/dX\2 ldV\2 1 -1 -1

+(A cosp- B sinp- A )sinj! F141S-1 -i -1

The combination of transformation coefficients enclosed by parentheses is the

previously defined S ((p) and T (0), Equation F51. Hence, the expression for the

nondimensional velocity can be writtenI

q I [T() - T(0) ] cos a+ IS ()-S(0) sinci

U 1/4 sin2 o + Y'(.P)2

I C (p) cos + D (0) sk, c 1 F151

-where T (o) and S ((p) are given by Equations F61.

Since this method is for steady irrotational flow, Bernoulli's equation

holds and gives the pressure distribuation in coefficient form as

C - _ _P - l

P 1/2PU2

I

11

EMPIRICAL MODIFICATION TO ACCOUNT FOR VISCOUS EFFECTS

in Reference 4, Pinkerton shows that an empirical modification to Theodorsen's

exact potential theoryP gives results which agree reasonably well with experimental

measurements. In this chapter, we will show how Pikellon's idea can be extended

to the approximate potential theory just outlined.

Theodorsen -" noted thai conventional profiles are similar in shape and that

application of the Jourkowski transformation to an arbitrary profile will yield a

nearly circulcr shape of variable radius 1/4 e; and polar angle eo. The nearly

circular shape can then be tranrsformed into a circle of constant radius

1/4 e and polar angle o + E (,o). With our notation and the circulation as yet

unspecified, Theodorsen's expression for the velocity on the profile can be put in

the form:

... = f (0o) 1 sin (,a - ,p - E) -rU eoU

= ~ ~ ~ f fIO +snc--) z de~

where f (0) r =1 + snnJ

and

x = 1/2 [cosh LE+ cosh 0 cos i]

Y = 1/2 sinh p sin V

(Note that this V is different from that used in the previous section.) The expressions

for x and Y are sufficient to determine 4 and p, and from these we can compute

12

2TT 0 .1 EGOM - 7-t + EWt di,E0 = )" ( ) cot dt

2-,

12 = /2 - .-r ;(t) I + d dt0 0: dt

Pinkerton's modification is to add an arbitrary function to E (c0). Arguing that the

modification is a result of viscous effects and 1har the cumulative effect of the

viscous forces are toward the trailing edge, he kckes the addition to E as

S (1+ cos o) R= L 2

where La is to be determined from the requirement that the lift be the experimental

value ana R is some constant > 1/2 (if R < 1/2 the velocity goes to infinity at the

nose). Pinkerton considered only the case R = 1. Since C. is the Fourier conjugate

of E, a change in cE would also cause a change in i,. However, Pinkerton has

fauna 6hat the effect of this change in 0 is negligible in f (p).

For slender foils, both E and z are small and we can approximate

x 1/2 (1 + cos 0)

Y 1/2 Osino

Thus the new expression for velocity is

I= f( + dc ),sin (a-c')-o- c!-sin(,-x- 'a)whrU ) d

where x = E(0) when the Kutta condition is satisfied. Comparing this expression

with Equation 13], we can immediately write a similar modification to the

i3

approximate theory given previousiy:

q 2K(1 + T Isin(a - a' x- sin (a-x- 'SVdx-'

.dY "

T (e) cos ( x- a ') + S ( 'p)sin (a - ')+

or +d ' [T (0)cos (c-b+ S() sin (a - ) r16q GO - - 16

U ./ s n 'P+ dY2

4

where da-l - sino R (I +-Cos p2 R- 1

Except for notition, this is ths soluticn obtained by Riegels. 7

ImplieJ in t!,e -tbove equation is that he lift coefficient is

CL = 2 A 1[( + (4 Bi 2 sin (a.-x-La.) [171

We sp. at once that LCL is a ficlitious decrease in the geometrical angle of attack

to give the desired lift. ThiFs equation con be rearraned to give t'a in terms

of an experimental lift coeffic:entCL

LcL 'arcsin - (exp)

2TT (4 A_) 2 +(4B_)2

With known values of the I;ft coefficient, this expression determines LcL. Since

we rarely have specific test data, a representative actual lift curve slope

coefficient and angle of zero lift can be used to find the lift. Assuming these

14

values are known, we can determine the itt coefficient from the expression

CL)exp = 2 -7(a- ) 119-

where -n is equal to dC L /da I / 2- and '. is the experimental angle ofexp' o

zero lift. (na.,d -oe are discussed further later in the paper.)

In order to more clearly shiow the effect of -.- , the velocity distribution will

be linearized on ja. This will be p,;. "n the form of the potential velocity plus

a term multiplitd by _i . The linearization is performed by replacing sin a

by .y ' and cos -' by 1. Ignoring all small quantities multiplied by a', we find*

where E (,p) = (1/2) -R-x R D(,r)11/4 sin'°o+ Y (

and (30 ) is the potential velocity. In addition to showing the effect

of -a on the potential velocity, this expression is easier for hand calculations

with the potential theory known.

NUMERICAL METHOD

TRIGONOMETRIC SERIES TO REPRESENT T-E ORDINATES

To use Equation r1 6 1, we need an analytical expression for the ordinates. For

tabulated offsets, this means either constructing a Fourier series or curve fitting a

polynomial to the ordinotes. As will be shown in this section, a trigonometric

*With R = 1, the linearized expression for velocity would be the same as that

obtained by Riegels' in 1943 if he had recognized that C (o) sin 'X in our E ('p)term is small in the normal incidence angle.

15

series may be fitted analytically to ordinates given at certain specif;c stations along

the chord. The slope and cotangent integral then become a summation of the product

of constant coefficients and the ordinates at these fixed stations.

The formulas used to approximate the ordinates can be developed by either the

least-square-error approach' 1 or by requiring the series to pass through the given

points.' Both methods are equivalent so we will outline the more direct method of

curve fitting through the given points.

Values for the ordinates are required at 2N -- 1 equally spaced values of o

between 0 and 2-in Equation A4-, i.e., at the fixed stations

1+cost~ I = +cos-

xm + Cos Om =-_ _ , 0<.m<2N 21'2 2

Althcugh we have 2N + I values of the ordinates, only 2N are independent since

!he periodicity requires Y(0) = Y(2-), or the mean value if there is a discontinuity

at the trailing edge. Noting that sin Nmprr is zero at all m, we see that Equation -3a'

should be truncated to the 2N termsN-1

Y(, 0=ao + . (an cos nep+ bn sin np) +aN cos No

Setting this expression equal to the 2N independent ordinates, we obtain a system of

linear equations to be solved for the coefficients:

N-1Y = a + 7 (an cosn rm +bn sin n pm) + aN cos N0 mm o m

These equations are solved by multiplying both sides of the above eq-ention by the

multiple of the desired coefficient and summing from zero to 2N - 1. Using this

16

approach we obtain the constants

1 2N-1a = -Y0 2N m

1 2N-1 mn- 2N-1 tun-a N Y Cos N b - Y sin N

0n N 0

i 2N-1

aN - Y cos m--N 2N 0 m

(Note that for n <- N, these expressions are identical with results of evaluating integral

expressions for the Fourier -onstants by the trapezoidal rule with equal angular spacing

as above.) With Y(0) = 0 = Y(-), the expression for the ordinates, Equation -3a- ,

becomes

y (0) = Yodd(,P) Yeven (P) 22

whereN-1 N-i N-i

Ydd('p)= bn sin nq=rmN Z (Ym-Y 2N-m) - sin ni sinnT -231 m=l n=1

N 1 N-1 FN-IYee )=- an cos n p -(Y 2N-m cos npmcosnp

0 m=l Ln=l0 m

+ 1+ cos N Pm cos N4rr2

2 N/t p247-

und Ym and Y2N-m are the upper and lower surface ordinates, respectively, measuredm-

from the nose-tail line at p =,". = N "

The odd part of the series arises from thickness and the even part from camber.

It should be noted that the odd function given above assumes a rounded leading and

17

trailing edge and the even function assumes finite slope d(Ym + Y2N-m) at thedx

leading and trailing edges.

EVALUATION AT SPECIFIED POINTS

The troublesome expressions in T (,p) and S (,p) (Equation 6]) are the singular

corangent'ntegral and the slope, respectively. Using the numerical expressions for

Y ((p), we can easily perform the necessary operations. The expressions will involve

a double summation as in the Y (o) expression. Since the summation involving

the trigonometric terms is to only a finite limit, the sum may be calculatedV-r

analytically.* At a point corresponding to an input point (p = -- , v one of theV N

m), the cotangent integral reduces to

1/2 P Y'(t) cot dt 1/2NZ -Yj 2N, t-

Y

+Cos Mt'V 1

(for m = , the multiple of Y is easily shown to equal N/2).S m

Similarly, the slope (in the (p plane) reduces to

dY I N-i I +M+ v , (m-V)~ 2N 2N+V)T=1/2 7 (-1)1 Y cot 2Y mco tCk = 'pT M1Lm 2N Nm 2P L m

(for m = v, the multiple of Ym is zero).

The numerical expressions of Riegels' may be added together to obtain these results.

However, these equations require computation of a N + 1 by N - 1 array of coefficients;

*See, for example, References 13 and 14.

18

this is a lengthy calculation eithe, by hand or on a high-speed computer. Further

simplification is possible if we define = m - v in the first term of summotion and

= m + v in the second. The equations then reduce to

,0 o-t N I N-1l[y . y1/ I -1) 25-11/2- PY'(t) cotR dt Y +- "- -

22 2 1,r-cos-

dYl =1/N-i FV 1--1/2 NY (-1) cot -i26-

where we understand that Y = Y2 NmandY 2 N+Y

These expressions are now quite simple and can easily be set up for hand or

computer calculations with a minimum of calculations. It is necessary that only the

ordinates be cyclicaiy rearanged before each computation.

The expressions for T (,p) and S (p) at o = p V are now

V N V N = -1/2 in N Y - 1 (Y + Y H) 2Nd/s n - " k N 1 1 -cos .I

S /2o) c1/2os - + "" ( Y -Y )(1)X cot A28s (O I X =1 V- X , k" ct2N-

and the modified velocity at'p= (o is1'

T cos(c-a')+S sin(a-t')

-r o cos (,-x- ,a ) + S, sin (¢ ] X

(PV 1/sin2 !Z + -~ o -R -

1 - sin-'- R _, 16a'

19

EVALUATION AT AN ARBITRARY POINT

As mentioned in the Introduction, the motivation for this research is to obtain

the minimum piessure. Unfortunately, there is no guarantee that the prebsure

distribution computed at the fixed points will yield the minimum value. If %Ve note

that only when the minimum is near the nose does the pressure distribution ho',e a

steep gradient, then our problem can be reduced to obtaining more data ;ust near

the nose.

The procedure outlined for obtaining the slope and cotangent integral at

specified points can be followed for an arbitrary ;p value. The expressions obtained

are more complicated but still Jive a single summation of the ordinates at the input

points and triconometric expressions w i:h can be evaluated once and for all at any

desired point. Although 'the resu'ts appec:r to be new, the derivation is not sufficiently

involved to warrant presentation and we give only the final results:

2-1/2-.? Y'(t) cot--2"dt

0O N-1 I (_I) m N -4 sin0m sin~p

= ' (Yrn - Y 2 N - m kIco1)rl L 2N (cos(p- cos Pm

(-1 sin sin' m2 cosp - Cos 'P

N-1 1) 1 - coSpmcoso+ E (Ym + Y 2N-m ) (

m=l(cos- cOSm)2

+ (-1) sin Nq) • sinD j [291

2 cos - cos (PM

20

70 8

4

5.5

, I ,\\ '//"

26 27 2 "

Fiue- Chrdis Sttosx S/ 1+csm-,l

-" ",6.-'i '. S D '

I

TABLE 1

Values of xm 1,'2(1 cos mrw '18)

_ m o. deg . -.

1 1 I 0.992404

220 I 0 .969846

3 30 0,S1301322 0.88 22

5 50 0.821394

6 6 0 0.75

7 I 70 0.671010

880 0.586824

10 100437

11 100.328990

F2 120 0 25130 0178606

14 140 0 116978

S * 150 0066987

S160 0 030154

7 070 00075106

i g 180 0

x36-mn m

21

dY N-1 - P- ) m + 1 sin .siem sintpdY _ N-i _ sin Np* ____

0 = (Yi m- y 2N-m ) 2N (coso - coSrpm) -'

do m 1

+ (-1) cos Nip . m I2 cosv - cosrpJ

N-1 r(-I)m+i 1 -cosp Cos+ 2~ Lsin Nqo. m

+ (Y + Y2NmLm 2N (cos 'p - cosm=l - m

(-1) cos N, . sin 1 [3012 Cos V - Cos mP

In order to obtain a better estimate of the minimum pressure, the velocity is

evaluated at additional points near the nose using these expressions. Choice of

these additional points depends upon the choice of N. In the pressure program, N

will be 18, so that o increases in increments of 10 degrees; see Figure I and Table 1.

The intermediate points selected are listed in Table 2. They are each integer

degrees between the nose (o=TT) and the input point on each side of it

((o = TT± -L ) and three additional points 2 1/2 degrees apart between this point18

((o=TT± ' ) and its neighbor (,o ± ).18 9

COMPARISONS WITH ANALYTICAL RESULTS

The numerical expressions for the ordinates are obtained by forcing the

expression to pass through specified points. Thus there is no assurance that the

computations will be reasonable between the points. Also, we have assumed that a

22

finer spacing of computed data near the nose is sufficient to determine the minimum

pressure. We cannot check these problems in a general sense, but we can make

comparisons in specific cases.

In order to check the accuracy of the numerical solution, the computations

were performed both analytically and numerically for the linearized symmetrical

Joukowski profile1 3

x = 1/2 (1 + cos D)

Y - (sin co - 1/2 sin 2D)

added to the parabolic camberline"

Y = f sin2 (,

where T is the thickness ratio and f is the camber ratio. The analytical and numerical

solutions were accurate to at least four sigiificant figures when the input was specified

to five significant figures. This accuracy is more than adequate for any practical use.

In order to make a comparison with the minimun, pressure computed anc.ytically

and numerically at discrete points, the results for a symmetrical ellipse were used.

For this simple shape, the location and the value of the minimum pressure are

easily obtained analytically. From the exact potential velocity distribution

(see Appendix A), we obtain the location of maximum velocity d_ = )

from the foi lowing equation

tan -

sin ,0(1 - F )- tarnp

I23

where T is the thickness ratio. Particular ,, values can be selected and the angle of

incidence for maximum velocity calculated from this equation. A graphical

comparison of both the location and value of the maximum velocity as computed

analytically and numerically at the fixed points is given in Figure 2 for thickness

ratios of 0.05 and 0.1. This comparison shows that even though the location in

the nose region may be off somewhat, the velocity is calculated with sufficient accuracy.

COMPUTER PROGRAM

A computer prograrn has been written for calculating the pressure distribution

by the numerical method outlined in the previous section. The FORTRAN statements

of the source program are listed in Appendix C. These statements are for Applied

Mathematics Laboratory Problem 840-041-F.

The order of the computer operations is as follows: First, the constant coefficients

in Equations r271 through r301 and other constants are calculated and stored. Then the

input data are read and the summed products of the ordinates and the stored constants

obtained. Finally, for each angle of attack, the pressure distribution at the input

points and at the intermediate points is determined from Equation '116, (actually Equation

[16a] of the numerical method) with t' = cux (i.e,, R = 1). Since the last operation

performed is for he angle of attack, variations in angle of attack take least machine time.

The data needed to compute the pressure distribution consist of the angles, in

degrees, for which the pressure distribution is desired; estimates of the angle of zerodC

lift,* in degrees, and an average lift-curve slope ccefficient* v- d ex

*Reference 15 finds that 7 is primarily a function of the thickness form and that %

is primarily a function of the meanline. Tabulated vgalues for many sections can ebe found in Reference 16. As shown by the data in Reference 16, both 17 and cxOeare functions of the Reynolds number.

24

TABLE 2

Intermedi ate Pointsdek,

1 0. 006-4

3 0 Wt.bs

4 M6X1218

I '(0. 062739

0.l5

12.5 or)' 1-IS2

15 0J.0170i7

17 5 6.023142

20

LOCATION OFMAXIMUM1

I VELOCITh4 MAXl!M1LMVELOCITY.1

16 - I - 4 2.10

U 15

-4

12 --- --

10 -3 l0

0hion 0~Zion.

8 2

6 .05 J

me. t2

1 0 NUMERICAL ANSWFt.

7~ 05 10

2- 0 j

0

0 2 4 6 8 0 12 14

uDG. H

Figure 2 - Comiparison or Analytical and Numerical Solutions for the Locationand Value of th;e Maximum Velocity on Ellipses

25

for use in Equation '19 ; and, of course, the cirfoil ordinates, in fractions of the

chord. The nondimensional ordinates should be accurate to about one-tenth of I

percent of the maximum ordinate to give a srrooth pressure curve. (Accuracy to four

dec;mal places is usual!y sufficient.)

Th e stations for whicl the offsets are .equired in the numerical method are

given by Equation r21], shown in Figure 1, and tabulated in Table 1. Since these

stations are often inconvenient to use with tabulated data, provision fo, input at

arbitrary x values is also provided. The input at the required stations is obtained

from a third-degree polynomial fitted through four ordinates, two on each side of

the required station. The interpolation is performed between angular stations since

the ordinates do not then have an infinite slope at the leading edge (see, e.g.,

Figure 10). The arbitrary stations should be as near the required values as possible.

This is especially important at the leading and trailing edges.

As mentioned, several input options are provided. As with all machine

computations, input must be given in a ronvarying manner; hence the input

descriptions must be rigidly followed.

Option I: Ordinates at Required Stations

The shortest method of data input is to specify ordinates at the required stations.

This type of input naturally results in the shortest running time since no interpolation

for the ordinates is required. The necessary order of input is shown in Figure 3 a

where each horizontal block lepresents a different card starting from Column I.

However, not all the caras indicated for the angles of attack and the ordinates

26

should be filled; only as many boxes as are needed should be used. (Ser- *he sample

input, Figure 3b.) When several profiles are given for the same anges of attack,

it is important that only similar types of profiles be grouped together ( i.e., either

symmetric thickness forms or nonsymmetric cambered foils, not both). When changing

types of foils, a new set of input, starting with the initial control card for the angles,

is necessary. Additional members of a group of profiles require only the cards

indicated under PROFILE INFORMATION in Figure 3 a.

Option 2: Ordinates at Arbitrary Stations

As mentioned, input may also be given at arbitrary stations. Two sub-

options are considered in this case.

(a) Points on Profile

The first of these sub-options is to specify points (x,Y) around the foil

in the direction of increasing m (see Figure 1,. The upper surface trailing edge

ordinate must be the first point specified and the lower surface trailing edge ordinate

must be the last point given.* The nose (see Equation 'B2' itn Appendix B) must

also be given before the input points. In addition, the nose point should be given

again, in orde;, midway in the listing of coordinates. The program shrinks and

rotates the coordinatbs to put the nose at (0,0) since this orientation has been found

to be the most accurate. (This point is further discussed in Appendix B.) The rotation

angle is added to each of the input angles so that cx is measured from the original

reference line. A maximum of 53 points is permitted.

*Fven though there are provisions for inserting ordinates at the trailing edge, aid though

they are printed out in the pressure program, no use is made of them when computing thepressure distribution. They are used in the interpolation and hence sLIld not be omittedwhen using the arbitrary input option.

27

COWMTpL CARD. ?UM7RAM F~rua 234

Number I J.49L,-4 t =~4

0000 -. Lamt Jo7f _-~00 Eme 'W amw*e Jobs RAO.

ANGLES -0 Dftes Mauma. -1 24 Pl.-td Fom 47 F2 6

COrROL CARD Forma 214

M r0001 - No-y meta

PRCMIE L"FORIAATION Repea for ?.actf Fo.3 m the Group

prsme,,tw Daa Form.at 27F32 6

- - Aro; ~Zem Ll Deirtt. - hecautoen~n Needed

- _ LAft CC . e Co.fextee Fr3.1&oa.)422

Ordmaes tri.aij, edge t,,L i~d,; #Cd~e AL Wt upper surgue h~~ed.W CtraL~o ~eaie ~ ,.!~ce Onot the upper surface yOWYu 79

s=-z icafds ALl Ctws Y0 *.. Fo~sm=6F32 6

Figure 3a -Format for Data Input to Cornputer Program at Required Stations

* *1~~ 4 30 5 4 '

4. ... ... .... .. ... . f4:.::{ 4 1 4 -_______ ......... ...........#...... . . .

.......................4 ......... .............. S.. .....

. ... . .. . . . . O f i *4 , 4-

Figure~~~~~~ 3b - apeIpta eurdSain

Fig+r 3 + /at npu toCmue tReurdSain

f, o; M.28

(b) Values of Thickne s, Camber, and Comberline Slope

The second sub-option is to give values of the station, the thickness, the

camber, and the camberline slope from trailing edge to leading edge. These quantities

are combined in the NACA manner' - (thickness applied perpendicular to the camber)

to obtain the coordinates of the foil (see page 113 of Reference 15 and also Equation

B I in Appendix B of this report). The coordinates are rotated and shrunk as above.

If the surface is to be computed by Y = Y z Y , the camberline slope should be enteredc t

as zero in the input and the nose radius is not needed. Input at 27 stations is permitted.

The format for data input in both cases above is shown in Figure 4a, and

sample input in Figure 4b.

Other Options

Although not shown in Figures 3 a and 4a, if a 1 is placed in Column 16 of the

control card just before the PROFILE INFORMATION cards, values of the lift

coefficient (as many as the angles and the same format) may be given in place of the

angle of zero lift and lift-curve slope coefficient. (See the sample input, Figure 4b.)

Note that lift coefficients must be given for each profile grouped under this control

card. Also, if a 1 is placed in Column 12 of the initial control card, only values

of the angle of attack, lift coefficient, minimum pressure coefficient, maximum

velocity, x location, and integrated moment coefficient about the line x = 0.25

are printed out instead of the complete pressure distribution around the foil, There

is a considerable saving in paper and computer time when using this option.

29

Figure 4 - Data Input to Computer at Arbitrary StationsCONTROL CARD. FORTRAN Format 214

l -Number of Angles t 24)

SEither 10000 Li.st Jobe0001 "More jobs Fallow

ANGLES, in Degrees. Maximum of 24 Permittea. Fornat 6 F 12.6

!+

CON'TROL CARD. Format 314

• ' 0OO - Total Number of the Following Group of Similar Profiles10007~ ~ ~ ~~~00 Tyeo rfls ihr OD Symmetrical

,~~~~~ ~ ~~ ....TpODPofls zhr 001 =Non-symmetrical

PROFILE INFORMATION, Repeat for Each Foil in the Group

Identification, any. printable statement, centered

Experimental Data, Forn~t 2 F 12.6

r ........ Angle of Zero Lilt, Degrfees. with Negative Sign if Needed

-" ..... Lift Curve b,'lope Coeffic-ent. Fraction of 2 r

CONTROL CARD, Format 214

--- Number of Stations

' --" (A) ONO0 z input Is Points around Profile

(%umber of Sta:tons t' 0053)...Either (B) 0001 -Input Is *:ea C.mber. and Camberhine Slope

I (,'amiber of Stations ft; 0027)

ORDINATES - for (A) Points. Format 2F 12.6

Talub Nose Ascissa: XN - - PLE (r -co-"LE) Do not give this card

Type Offor symmetrcaPROFILE INFORMA. O, Repeat " or E.. Nose ordinate- YN G PLE s' foils

- -'- "UpperNeentcrfoCoordinates (. Y). Give same number on

-A. . . . . . . . . ,LA A ..... , 4

.. pper surface as on lower. For symmetricalS l foils g ie only the upper surface. The last

..... Lowe point :*uid De (0. 6;.

As eeedSurface

ORDINATES- for (B), Thickness. Camber, etc. Format 4 F 12.6

.. Fracton of Given Thckness-A 0 adng Edge Radius for Given Thickness

t heraction of Gen Camber and Slope

Th(ckness Camber S.pe ofStation Ordinate Ordinate Camborlne

As Needed Leading Edge

', ,.,os a a o l rr.......o . ...... pe srasn....sy m ic

Figure 4a -f Format for Data Input to Computer at Arbitrary Stations

0 30

5 4 3 1. 1c

1 - - - -- - - --

73*,AA

3@3

324 S72

A 046

005

.. 9124v4 .. 6

027

.10012633

f 13. ... S .. . . f .; 0 0 s 8

.02,39 01381 *2.0 984

.0&* 001021'7 A

0340 0.211 062

413. YA of l il , 22

8760 1/116 02.0404

*o: 0 o2AA 0*1 5 1812

001AC S 44.119 ?o 1,0T t$202C*4

@931

Sample output for the RAE 101 (10-percent thickness) is shown in Figures 5 and 6.

This is the data plotted in Figure 15. The first two pages of output (Figure 5) for each

set of ordinates are titled Profile Constants. The first page includes the stations (column

headed X) and the ordinates (column headed Y). The column headed C is the velocity

distribution on the profile at zero angle of attack and the column headed D is the

velocity distribution due to the angle of attack (Equation '15'). The colt,mn headed

E is a factor necessary to find the -,,odified, linearized pressure distribution (see

Equation r20.3). These three columns permit rapid evaluation by hand of the

velocity at other than the input angles without rerunning the data. (At the bottom

of the page is an expression which shows how to combine Columns C, D, and EdY dY

to obtain the linearized pressure distribution.) The last column is d - 1/2 sin -dx

dCL dx

The theoretical slope factor / 2-1 is printed out at the bottom of theda, C L = 0

page as well as the theoretical angle of zero lift.

Additional information may be obtained from the data on the Profile Constants

page of output. The angle of attack a, which a stagnation point lies at the nose,*

(q) L 0 = 0, may be calculated for potential flow from Equation F151:

C() coso+ D(-) sina. = 0

i.e.,

= arctan - D(rr)

For "ai 0, the nose stagnation angle may be similarly obtained from the linearized

*This angle is not the same as Theodorsen's ideal angle of attack.' The ideal angle

(for thick foils) is defined as the angle of which a stagnation point lies at the forwardend of the camberline. As noted by Theodorsen, this angle is of limited practicalimportance.

32

1= U8L1E c4. 419 3 3.4R8 ,.. A I O I P.t S- f 7,IbT.16%t.I I .-

PQcF ILE k _,,TAI5

RAE - 101 - A

0.is 00~o'0 O9C,9 . 243 -Z..3 0.8

0.~Rt C0)2y~. 0 .3347 1 C968-1 .30374-' 0

.0 e i" 0 4.5,c ?.1 le 0.160.,. -0. 025 4.81&

0.6733 134 3.e,84 0."1 b8I -0.753192 3.540515

0.3868c4 0 0 67C 1.3.2834 0'Z 7 -. 504 C..3Ito.'soo. .:7) 1.Ce0 . .3, -3. s7oC 0.',. 1o

0.1333. 4..047 1 i .444 1.350847 0 v713 0. 16.,.. 89Th~ 0.049A~rC P19 ,8? .471C c .I820's 0..04".5A

M12000 0.041360 1.34 "1 3.7,7f4, I "o.323 0 -0. oI55Abe76ob 0. C,4 470 3.419 2723 345740 -0.0 3"-.60.398 .. SRO. 33394 1. 8 594 143 6966d?? -0..,416.,

. 19" 1130t 3 .1~ 3. ~97 .:08?953 -0.0504 7609"35 0.0219o" I.874 t,.7 e -2.63 4 1 -0. 5071'6

1 0 0110 o~a~0.01010 .... o. .Qees3 -4 .7 73448 -0.0603of0. 0 . -0. 37.344 9 3 -8.056399 -0.0t206.

0.007-,Ye -0.0;0710 -0.945092 9.9836ft3 -4:773448 -~~nC.0301-.4 -0.0 093. -3 .087b3 S. 7067.e -,?.bb3 4"3 -0.Os674 3Q.8667 -0.0.31140 -3354b 3. 16 _k.087.53 -0.v 0476a.316978 -0.5 3*40 1 -. 394 2 . b59 31, -a696tiz -0. ?65.

178608 -0.0 4'7-11 -1.344 379 2 . Z2439 1 4657#Q -0.0 30 S-60.50 -0048850 - It491 3.773k4 -3.309325l -0.03S546

0.3Z89,30 -0.4Q850 -3.33557s 3.22630 1.82065 0.0045,681.31376 -0045 -31IS3 330647 -, .0779 003i

0.500000a -0.042670 -1.0va804 0.9379 -0.96760 3 0 .032 369

0.506824 -0.036360 -1.062834 0.640? -. 65047 0.03 075,

0 .7*010 -0 .029345 -3.0364Y4 O6.~I5677 -0 :.927 0:040635

0 .750000 -0 .022360 -3.036276 0.49033' -0.655257 0.61388340,21344 -0.03S970 -0.9 q2321 0.038 -0545076 0:034k450.883077 -0.00460 -0.978'4J 2776 -0.4er845 0.026"630.91 83033 -0.00,5990 -0.958660 0.1664b? -0.S0,426 0.a223850.98984 6 -0 .002700 -0.937 0,0968t8 -0:.109740 0.0309SZ.9Q24 4 -0.000680 -0.8390693 0.026483 -0.069a73 0.007z8?

I3. 010000 -0. -0. -0. 0). 0.001998

NON-Ot-kNS36NAL VELOCITYVZCCS4LFA).DSIN IALFA) 3*13-/$OELTAgSu9TIZ-Z**23 3*OELTA*E

OCL/DI'LPIAI/2Pl (THEORY)- t.003996A N0LE.CL-0 (THEORY) * 0. GE 11

NT3TE.tE OROINATF NOT USED INTERNALLY

AML PROMTEM 840-041,AR8ITRAPY AIRFOIL PRESSURF 035T8181.,T309

WITH EMOIRICAL CORRECTION FOR VISCOSItY

PROF ILE CONSTANTS

RAE-1303-010

3NTERMELIAE VALU.ES

UPPX34 SURIFACE

x C 0 E OY/08H30.000076 0.163276 36.982900 -7.983673 -0.062038O.C00305 0.31176? .6.522598 -7,7612 -0.0619660.000685 0.450839 35.633353 -1.45t068 -0.0618480:0037;6 0: S69054 . 92645 -1.065237 -0.0616890.00 403 0.667997 "340.35385 -b6.58289 -0:06:4920 .002719 0 .749i35s 33.3694 52 -6.22b603 -0.06326S0.003727 0.834890 32.284467 -U.0Z653 -0.0610320.004866 0.679 3:452906 -5.442S67 -0.060140.006356 .158 30. 1685003 -,042397 -0.060456

00315 3.e005922 8.503872r -4.3103034 -0.0593970.0""037 T.063992 7 145886 -3.588.95 -0.0586050 .023342 3.069452 6 .43 6098 -3.181521 -0.057745

LOW4ER SURIFACE

0.000076 -0.161276 16.982900 -7.98t671 -0.062038

0:000305 -0.353375? 36.522s98 -7.764532 -0.06:9660.000685 -0 4 08391 35.833353 -7.1106. -0.0638480 .003210 -0 .56905 34 .. 9 92645, -7 .06 ,237 -0.0636890.001903 -0.667997 340853e5 -6:6408Q -0.0614920.00.71F9 -0.749135 3.642 -6.22,800 3 -0.0612650 .003777 .0.814890 32.284467 -5.82e653 -0.0610320.004866 -:1.867892 11.452406 -r,.44.!567 -0.0607.00.006356 -0.4;0680 10.685003 -5.092397 -0.0604560.0338, -a ~5922 8.50387? -4.3034 -0:kb43970.03703? - 3.0 92 .356 -. 8 795 -. 0600.0.1144? 3.069457 6.436098 -J131321 -0.057745

Figure .5 - T,"\pica! Oulput from ('ompulor, Profile (on-iant-

33

A. 980wL. 640-041. A0813862 A3IRFOIL PRESSNf OI~tRICUff1O8

0:1" E.P92..AL CO.33.1033 0. 08 VISCOSITY

QAE-1302-010

Sy-E383(AL PROFIL(

At.t6 CL o FA03 S34.I.LFA Lift SLOPE ALrA.CL.0

A. 00,0 0.430329 0.003346 0.0723Z) 0,9t-9000 -0.

P37... 2t. S35 3 fist Vk350 POO- P-00 VISC P-fQPLo3., 0.0 .00000 1.000 000

0.60 0.29333A -0.OC.2b4 0.8900-Z 0.10.7266 0.207025

0.9A686 0.20 -0.00.,7 0.912 0.123 0312Is 3013 0.2'q69Sid -0.00315? 0.966006 0.05927 .962

110a 3012 0 .,a96316 -0.002.3 1 0.20 1 0 .0 07263 0. 510 t ... I C./8 57 .0.02 3.00A -0...12 .. 03:5

0 f..O'O 1.0486'. -0.0011,8 3.06563 -0 .0996a6 -0.392738-0.671010 3.07976, -0.003:.5 1. 0759'. .0.385823 -0.3s62 7

0588'),,-1t 0.045 1. 10126 0.2.3960 -0 . 32386,S600) * .3268 -80.006'12 3.3462, -0.329338 -. 325

0..3O7 0 . 334 '05,3 1.03 -0.6 2 -0.38s,

0.41892 ' .2ft5 -0.026 3.2126203 - .533o36* -0.5322Z

0.70032 1.1732?4.0016 . .l6 5 -0.6 36285 -0.6016220.370-16 1. 3.2996 -0 .0066 3.32li86 "J0.62 , -0.6 Z.6

0.697 1.396?33 -0 .0 076". 0.390ss7 -0..9 3 -0..33:68

0.013, 1.- "2030 -0 .009112 3.682226 -3.?26171 -. 1969890.007S526 3.624753 -" .0360 , .6392 -3.1 208 -. 6.7391

0,. 3.222838 -0.O15307 3.3;97453 -11:438t333 -0.6139010011,16 0.e10637 S .510 0.26717 0.636 .998

0.669.7 0..6711 .:6 '0.629? .82 0 .27 038a5

0. 36976 0.4 33335 0.006319 0.93t-34 0.1302, 0 .. .270.378636 0.929 1 0.000.850 0.988803 0.030) 022.9

0.11000 3.018663 0.005512 3.026236 -0.03 7636 -0.0620 220 . 3za940 3.0352.10 0.00336 3.060558 0O.0 1 680 -0.08271630.63 337?6 3 $Or5 0.005073 3 03C576 -0.01. 885 -0.' 02

,.00030 3098 0.30.7 1.02392 008 -0.0.067

0.58~ 3 . 00.582 3.6062I 3. 0?: -0.03,675 -0.02058

0.673030l 0.963943 0.006020 0.925963 0.016053 0.00620.750000 0.278Fit 0.0"3565 0.962282 0.00ZI233 .33

:.ale314 0.267517 00362 0.97052 0 0.063823s 0.;3579%t

0.883012 0. q265z8 0.00" ' 4 0.9590 12 0.085055 0.080 2580.2303 0.962939 0.003906 0.9668A 0.3 6 0.307269

0.9698!6 0.24130, 0.033Ma 0.92565314 0.682 0..31037

0.99e,6l6 0.8a065 36 0. 1^656b 0 .88'71 0 03058 0 ." I8..

1.000000 0. 0. 0. 13.00000 0 1..00000

INTtCGRATED CN:G .0. 122

NYCEAAE CRA83 0.0020652 .49l''1

3c2ls6fE 0(2. -0.00.200 C~ .8 A,,

AWL. PhOU33.0 60-063.AC8II.AQY AIRFOIL 'IlESSUEIiE 35R3U?30f4

I. E :I IAL COURECIIOft FOR VISCOS IT

PRESSURE OIO.TRIBUTIO.

RAC-1301-03 0

ALFA CL CELIA SINIALFA) LIFTSLOPE ALFA.CL-04.090000 0.430329 0.003346 0.0713a3 0.922000 - 0.

INTE2330036TE VALVES

UPPER SURFACE60

A P03'3L V0100 VI V 36006 VISC 503.00 P038. P/O VESt PlO0.000076 1.3 72143 -0058 2.3465 -0827 -0.8362850.0335 3.4230 -0026 366 -:.222:8 -33 230:00068S ... 803 .0026 -1.439 -3623 -3 862

0.031 .. 6 693; -0.'0"?7;0 3.6!49490 -1.619642 -1.6065780003:903 1.67?0932 -0 .02337S 364937 3736 -1.721632

.07321 3.68653? -0389 6.996610 -3.866338 -3.77763600037 Z7 3.688284 -0 .038667 1.61 037 -3.fi6?6 37:

0 .0 C.4866 1.682-542 -0. 037533 1.66S028 -1.830946 -3.772333

006306 1.6:0352 -0.03662 .. 66390 -. 790M1 -2.735386'4

'.302 1.60974d -0 .0313469 1. 596 273 -t.591269 -3."68087

0.F033 3.SO5266 -0.013W.) 3.55331 3 -3.650059 .3.!437640.02?3362 3.52S772 -0.03075 3.53503? -1.327902 -3933

LOVER SURFACE

I POINL 20100 V;St 383.333 235 VELOC P33.00 5 P/O S0.000076 0:050.12 -0.02532Z3 1.025282 -0.3336 -00 or20:000305 .86S489 -0.024637 0.903 0.5028 0.29526320.000605S 0679460 -0.023Z5 0656035 0.38336 0.562638:

0.033 0.5073 -0 .022 0.67953 0.r627 0.77007000. 1 S,100393 03363 -0.0290 0.7 03 01885539 0.6992486

0 .00 ,739 0.22062 -009597 0.172465 0.96332 0.9 70256

0.00 3727? 0.0633 5 -0.0383 0.005 0.99S96 0.29797

0.036866 0.. 82 0.017365 0.085987 0 .997616 0 .995b660061356 0.16121 0.03608? 0.362258 0.978636 0 .9736?0.082 0.3969 8 0.03 3070 0-13008 0.820 0.83306600.0:7037 0.517600 a.356 082 0.7 9 0.7205

0 .0 23342 0.607684 0.03O0323 0.638006 0.630720 0.620. 0.'?

Figure 6 - Typical Output from Computer, Pressure Distribution

34

velocity given by Equation '20 >

C(-) cos- - D(-) sin, - ',E(-) = 0

Usually two iterations are necessary to solve this equation.

Also, if the nose radius is desired, it may be found from

r dY 1PLE = j

If the nose radius is known, this relation provides a check of the computed slope.

Two pages titled "Pressure Distribution" are printed (Figure 6) for each input

angle. The first is the pressure distribution at the specified points and the second

is the pressure distribution at the intermediate points near the nose. Constants printed

out at the top of the page are for the input conditions. "Delta" stands for .a ,

calculated from Equation 18, and is given in radians. A measure of the accuracy

dYof the input data is the smoothness of the curve - since small errors in the ordinates

are magnified when taking the slope.

At the bottom of the page with the pressure distribution at the input points

(Figure 6) are integrated values for:2-

CN = - 1/2 0 C sin pdo The coefficient of force normal0 P to the chordline, (positive upward)

2-

C - C Y"' (0) dp The coefficient of force parallelo to the chordline (positive in the

positive x-directior)2-

CM - 1/2 0 C sin'p (x-. 25) dip Moment about the line x = 0.25x=.25 o P (clockwise positive)2-

C - C Y Y' (,p) dO Moment obout the line Y 0y =0 o P (clockwise positive)

35

The lift coefficient is given by CL = CN cosa.- C sinca, and the moment coefficients

about the point (0.25, 0) are given by C m(0.25,0) = C m + C I -mx-'0. 2 5 m =0

The integrals are evaluated with the trapezoidal rule at equal angular intervals.

The numerically integrated values should be reasonably accurate, except Cc since

there is not a smoothing factor (e.g., sin o, Y) under the integral sign. In fact,

the pressure program gives a nonzero chord force for foils at zero incidence.

(Potential theory gives zero drag.)

In all cases considered so far, the integrated and set lift coefficients have agreed

within 2 percent.

COMPARISON OF POTENTIAL THEORY WITH OTHER METHODS

Here, comparisons will be made with other methods for calculating two-dimensional

pressure distributions in potential flow. Substituting Equation r67- into Equation r151,

we find for the potential velocity on the profile

q I sin(l cos. L1/2sin pV i2 + Y ' (0o)2

4

-1/24I Y' (t) (cot-p cot t/2)dt

V 0 c oC I

+ sin 2 - (0)+Y (0) F31I

This is equivalent to the solution obtained by Moriya.8 , °

36

*5 Converting this expression to x dependence, we obtain

q _ 1 cos- [1 1 1 YU' Q YL dU NF+Y' (x) - 0 x -

1L- YUi~ YL1

± 7/2 P -x dJ0

s- sin - L-- x X) Y x)) 4Tx-) r3 2"

L x -I _ X

where 0 < x - 1, and Y and Y are the upper and lower surface ordinates, respectively,U L

measured from the nose-tail line, and primes denote differentiation with respect to

the argument.

Riegels and Wittich " obtained this same equation by noting that linear airfoil

theory, including the thickness correction term in sin a, g;ies a velocity on the

chordline. They obtained the velocity on the profile from the linearized velocity

q(x,O) by equating integrands of two expressions for the circulation:

r = .q(x, Y) ds =1 q(x, 0) dx

q(x,0) _ q(x,0)i.e., q(x,Y) ds/dx f !+ Y (x)2

since the singularities enclosed by both paths are the same. This neat trick can

also be used to improve three-dimensional linear theory.13 ' 14

Lquuhon 321j clearly shows the cr - :s-coupling effects of thickness and camberiin the velocity distribution. If the thickness and camber are added and subtiacted to

obtain the ordinates, then the only coupling at zero incidence is the term

1 + Y' (x)- which is negligible except near the nose. Hence the NACA

method' of combining velocity Increments computed separately for thickness and

camber can be expected to be accurate everywhere except near the nose. Comp'ratlve

calculations have shown that using the NACA tabulated velocity increments' is

sufficiently accurate for most sections but that there is simply not enough data near

the nose even for moderate incidence.

in References 13 and 14, Weber proposes a method of calculating two-dimensiona!

pressure distributions which is based on other work of Riegels. Her expressions for

C ((p) (Equation q5') is the same as ours, but for D (,p) she obtains

[y YT (I ) - T d

=1 Hx 1 4(10 JD (1p( D (x) X T, -y'(×)

where YT = 1/2 (Yu - Y L)"

This expression gives results which are closer to exact potential theory than does

the method presented earlier in this paper. Howevet the derivation is somewhat

loose when camber is included c.nd the method does not easily cllow the insertion

of the experimental lift while satisfying the Kutta condition.

The data tabulated in Reference 15 will b'. used to compare the approximate

potential solution, Equation '32', with the exact solution for the more usua! foils in

use today. Theorel ical velocity contributions for a !arge number of arbitrary thickness

forms, calculated numerically from Theodorsen's exact method for potential flow.'

are tabulated there. Pressure distributions at zero angle of attack calculated by that

38

method and by the method of this report are compared for three foilF--the NACA

16-009,* the NACA 0010, and the NACA 65A-006. As shown in Figures 7 and 8,

the two methods are in close agreement for both the NACA 16-009 and the NACA

0010. However, for the 65 form (Figure 9) the method of this report predicts a small

pressure peak near the nose. Plotting the ordinates at the angular variable

o-= or: cos (2 x - 1) instead of the usual x, we find a hump near the nose (Figure 10).

This hump causes the pressure peak since fairing it out was found to give a smoot-

pressure curve. The numerical method of Hess and Smith' predicts a similar

peek on this foil (see Figure 42 of R ,ference 18). They note ;he important point

that experiments fail to show the peak. The experimental dis:repancy is thought

by the authors to be the result of inaccurate machining near the nose.

To explain the theoretical differences, note the discussion on page 3 of Reference 19

in connection wth the design of the NACA 16 series:

"The Theodorsen method as ordinarily used for calculating the

pressure distributions about airfoils, was not sufficiently accurate near

the leading edge for prediction of the local pressure gradients."

Because of this deficiency, Theodorsen's numerical evaluation of a nonlinear integral

equation was not used for the design of the NACA 6 series foils, but was replaced by

another numerical procedure employing equations similar to the interpolation functions

*The equations

YT = T F.98883-x- .23702 x - .04398 x2 - .5576 2 x3 1, < x < .5

Y = r.01 + 2 .3 25(1-x)-3.42 (1-x) 2+ 1.46(1-x) 3 1 ,.5 <x< 1

give a good fit to the tabulated ordinates for the NACA 16 form. Th; nondimensionalnose radius is 0.4889 , , or 0.003960 for the original 9-percent thick foil.

39

m 0 0 0~C- 0

co~~ ~ ~ W 0wID 0 t

0~~~~ -t C04 O .? 0 ~ Cli C 0 01 CO S - wN - 0 -

z U.- 0 00 0 4 0O

04 01 ~N- 0 - O 0 0

04

- c-

w c-m CD

Ile-IN:

o

o "oese

00 0

0 /

V) C4u/

41I41~

for the ordinates (Equations r23] and r241) used in this report.'. " For these foils,

it is believed that the complicated behavior of the "design parameters" near the nose

(see Figure 4 on page 9 of Reference 19) is not adequately represented by such a

finite term approximation for the Fourier series unless a large number of terms is used.

Note, in ccatrast, the smoothness of the curves Y (o); see, e.g. , Figure 10. Accurate

integration of an irregular curve would depend upon the number of points taken.

Theodorsen6 recommended five equal divisions between 0 ani - with his method.

Reference 15 recommends 40 with the improved method, surely more than enough.

No indication of the number used in the design process is given .n Reference 19,

but it is believed to have been insufficient.

So far, comparisons have been made only for thickness distributions at zero

angle of attack. For cambered foils at an angle of attack, a potential flneory

t a = 0) comparison can be made or Lamay be "idjuered to give the same lift. Both

comparisons are shown in Figure 11 for the 4412 at c 6.4 degrees. (Pinkerton4

gives the potential lift, computed numerically from exact theory, as CL = 6.915

sin (a - ao) , a o = -0.0706 radians.) It is easily seen that the computed pressure

distribution with the exact lift (CL = 1.254) is not in agreement with the results

of the exact potential theory. It should be noted that the results of the exact theory

were taken from the small figure on page 62 of Reference 15. Even allowing for errors

in the tranfe, of data, the agreement is only fair at the same lifts. Moreover, the

compited minimum pressure is about 10 percent lowier than the "exact, " which

follows the trend in Appendix A. Integration of the curves in Figure 11 gives

42

, - - . - - - .- . -,.

-------- -------

L

2. N - -- 2-

100

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

0 I -

-- -

/ ""

43

CL = 1.274 for the "exact" pressure distribution and C 1.242 for the approximate

case ( - - 0), compared to the set C, of 1. 254. The diffe rence between the set

and integrated value for the pressure program (approximately . percent) is thought to

be caused by the approximations in the theory and possibly by inaccuracies in the

numerical method. The disagreement between the "exact" results and the integrated

vaue (approximately 4 percent) is obviously . ir prt by the small size of the

original C ,rve of & xact p,,..:. '-c- ,:h ch the data were taken. Some of the

discrepancy could akl. be a result of Pinkertfjn's use of approximate expressions

for E and L . That thesc are approximations is apparent if one compares the expressionsO

on page 13 of Reference 4 with those in Reference 6 and in the section on

empirical modification given in the present paper.

COMPARISON WITH EXPERIMENTAL RESULTS

iCvi,-;ARISON WITH MEASURED PRESSURE DISTRIBUTIONS

There are few measured pressure distributions tabulated at a sufficient

number of points to accurately define the pressure curve near the nose. The most

comprehensive tests are those of Pinkerton',' on the NACA 4412 section. Unhappily,

Pinkerton's measurements are actually for three-dimensional flow over a rectangular

foil with a 30-inch span and a 5-inch chord. The equivalent two-dimensional flow

is found by subtracting the theoretically calculated induced angle of attack from

the geometric angle of incidence. Nevertheless, these tests are often used in

two-dimensional comparisons because of a general iack of expetimental data, and

for that reason, we do so also,

44

Comparison of pressure distributions on the NACA 4412 section at an equivalent

two-dimensional flow angle of 6.4 degrees (geometric incidence of 8 degrees) is

shown in Figure 12. Agreement on the upper surface is excellent, but the lower

surface shows sl.ght differences. As expected, there is d;sagreemen. near the

trailing edge caused by the thickening boundary layer. Also shown ;n Figure 12 is

the pressure distribution computed from ordinates interpolated (in the pressure program)

from the tabulated measured' offsets of the foil. Some of the humps and hollows in the

measured pressure distribution are better predicted, but a new .redicted hump is also

obtained at quarter chord. (The measured nondimensional ordinate at this point is

0.0012 greater than the computed value.) Essentially though, the pressure distribution

is the same as for the mathematical ordinates.

Reference 3 contains one of the few tabulated sets of measured two-dimensional

pressure distributions with a sufficient number of experimental points near the nose.

The model tested was a symmetric RAE 101 section with a 30-inch chord and 10-percent

thickness ratio. The large model (surfaces were accurate to . 0.0003 c) and

accurately measured angle of incidence (to the nearest 0.01 degree) makes these

tests valuable for comparative purposes.

Two Reynolds numbers were considered in the experiments. For comparison,

we have selected the lower (Re = 1 .6 x 10' ) since it corresponds to a lower Mach

number (U = 100 ft/sec). (It is not completely clear whether or not compressibilty

corrections were applied to the data.) Measured and predicted pressure distributions

are shown in Figures 13, 14, and 15 for three angles of attack: 0, 2.05, and 4.09

45

2,424 NACA 4412

6 4

-2 J 6

1 6 Exact Potential Theory,( Page 62 o! Rel CL1.2}

Program Res%.tlL

n 1095 oe -404

-1 2 - - - Program Resauts-A- 0 CL 1.184

Upper S%:rface

-0 8

CI,

.0i 4

o I 'N.[]

0

FRACTION OF CHORD

Figure 11 - Comparison of Theoretical Pressure Distributions for NACA 4412,= 6.4 Degrees

-2C

NACA 4412

-! CL 1 024 . 6 4

r, L 3 x 106

C 5"

-12 2

- Program Resuttb, Mathematical Ordinatebx 0 Experimental Points. from Refererce 4

-0 8- x Program Results Measured Ordinates

Uptr Surfacs"

-0 4-

Cp

0

1 2 3 4 s '. lFPACTION Of CHIORD

Figurv 12 -Measured and~ 'r icte( Prvsure0- on the N - - 1 , C 1.02 1,. 6i. De-res

L

46

0 4

rRc wI% of C140R1

f ~~RAE - 10 R IC

IV' Thurk 10011I -

.0 CL 9- Pr.oera. Re41

0 Eeri*rt, i1 f r.., Rtltre-4c 5

Figure 13 -- "Measured and Predicted Pressures on the RAE 101 Section. l0-Percent

Thick. CL L 0 . o= 0

-1 2

RAE - 101

-0ThickS CL =021, V 2.0 0 Degrec

* 99C 30 -

0*

Figure15 -easurd andPrediced Prssur peoritenRaE 101 SectionR10-Percent

.0447

degrees. Agreement is excellent except at the trailing edge. This deviation is

again due to the rapid thickening of the boundary layer at the trailing edge.

COMPARISON WITH MEASURED CAV!TATION INCEPTION

Although measured pressure distributions are hard to find, incipient cavitation

tests have been conducted in several laboratories. With the classical assumptions

that cavitation starts when the local pressure falls to the vapor pressure of the flowing

liquid, cavitation inception may be predicted from the pressure distribution since the

cavitation index m. P is equal to minus the minimum pressure coefficientP mi I //2pU-2

-. Pmin

Pmin 11/2p U It is important to note that unsteady effects are not considered

when computing the minimum pressure. That ;s, the testing is actually done for either

varying P or - ' whereas the predictions are for fixed flow conditions. This means

that there *s some doubt that the predicted miminum coincides with the actual test

minimum. Assuming that they are equal is the usual "quasi-steady" approach, generally

the only reasonable banis of solution. For the (assumed) slow variations in flow

conditions here, it is reasonable to expect they would be equal.

Cavitation inception data from two different laboratories will be considered:

(1) tests by the Cal ifornia 'institute of Technology (CIT) on the NACA 4412 section

for various Reynolds numbers and (2) tests by Vosper Limited' on elliptic-parabolic

sections of various thickness and camber ratios (NACA a = 1 camberline) at constant

Reynclds number (Re = 1)" 10 ).

Measured values-' of o-i for (visual) inception and disappearance of ccvitation

at two different Reynolds numbers are compared with measured " values of -Cpin

48

for the NACA 4412 in Figures 16a and 16b. Predicted minimum pressures were

computed for values of 7T and - e interpolated from subcavitating tests' of t1'e

foil used in the "nception tests.

The good agreement of predicted and measured -CPmin is surprising since the

angles of zero lift and lift-curve slopes are not the same in the various tests. Also,

er negative angles of attack, the measured pressure distribution was not taken at a

sufficient number of points to ensure obtaining the minimum pressure.

Cavitation prediction from computed m'nimum pressure is not in such good agreement

(the piedictiun is conservative) although agreement improves with increasing test

free-stream velocity. Some of the discrepancy may be attributed to nonstecdy effects

(resulting in a poor predction of the minimum pressure) although, as already

discussed, such effects should be small. Again the fault seems to be with the assumption

that cavitation begin.- when the local pressure falls to the vapor pressure.* Perhaps

equally important in accounting for the inact-urate prediction is the small size of the

model (3-inch chord). For such a small model, slight machining errors could result

in iarge changes from the computed pressure distribution. It would thus be of con-

siderable interest to have measured ordinates rather than ordinates for only the

mathematical foil.

Machining should be most inaccurate for the rapidly changing geometry of the

nose, and at negative angles of attack, the minimum pressure is close to the nose.

*This has been known for a considerable time although it is generally assumed that

for "engineering problems," conditions are such that the assumption is adequate.(See References 23 and 24.)

49

C 3' *.2

Figu0 16a w CPpio at3 x. le= 1.1 X 10

-4

to 3 4.

x cairt" 1Sot Rk O e Refrnte2o 1Ud4MOS to 64.2 I S i 6

40 ..0 61 01 2.6 3.6 2.0 2.2 2.4 21 28 3

-h -z1 ENDn - Cp . a t, 32 x O. Refehc. 20

Fiar t16bvue R, Co.parso atRe 1. x

50 L 11

it is possible, then, that machining inaccuracies also contribute to the large differences

at negative incidence.

Since the incipient cavitation number for two-dimensional hydrofoils increases

with increasing free-stream velocity and since, in "certain instances," increasing

the body size also increases the incipient cavitation number, 2" it is imperative that,

with our present lack of knowledge of the proper scaling laws, laboratory testing

should be done for environments approaching full-scale conditions if meaningful data

are to be taken. A better alternative is to develop realistic scaling laws so that

models of small size may be tested at low-Reynolds numbers.

The cambered 9-percent thick foil was selected for comparison with the Vosper

test data. in these tests, only (visual) cavitation disappearance was recorded,

Experimental and predicted vnilues are shown in Figure 17. Figure 17a gives the

raw data points and Figure 17b the faired data corrected (by Vosper) for tunnel

interference. The raw data are included to show the experimental scatter. In both

cases, the prediction is based on the experimental lift-curve slope corrected for

tunnel interference. The raw data show that the agreement for angles greater than

zero is generally good except for the one point at 1/2 degree. For negative angles

of attack, the prediction is again too conservative.

Although the model chord was relatively large (8 inches) in these tests, the

machining was accurate to only ± 0.005 inch22 or ± 0.0006c. From actual numerical

j test cases run using the pressure program, ,xndom differences of this size in the

ordinates could account for the unusual result that >- C near zero anglePmi

51

0 01 0 0I 0

0

4 - I 0

0 13 Vosper s 9% Thick, Cambered Fol!* Elliptic ftrab~Iic Thickris

0 NACA io=I Mean$-e, C~jO.282

2 oltz-2.1r, -9=0.762F0 YT _LYC

Uj17 "Isec, C=8", Re lXIO'

0

0000

0 Predicted

~ [ nMwssured, Ref 22 0 0

Figure 17a - Comparison with Uncorrected Data PointsCr

06

05-

04r

o-td

.-

-Jd

Predcled-01 Measured

Figure 17b - Comparison %-with Faired Data, Corrected for Tunnel Interference

Figure 17 - Comparison of Incipient Cavitation Number and Minimumi Pressure Coefficient,Elliptic-Parabolic Section, NACA a = 1.0 Meanline

52

of atiack. (However, q. > - C at small incidences was also found for other foils

mintested by Vosper. It is interesting to nole that the air content during these tests was

only 50 percent of saturation2 whereas air content for the CIT tests was about

5) percent of saturation. 2 -)

Although measured ordinates would be useful in determining more precisely the

differences between and - Cp , the differences are too great to be attributed

solely to inaccurate machining. Most of the differences seem a result of assuming

that cavitation occurs when the local pressure falls to the vapor pressure.

SUMMARY AND CONCLUSIONS

A numerical method for calculating the two-dimensional pressure distribution

on arbitrary profiles with arbitrary lift has been explained. The development is-based

on an empirically modified, approximate, conformal transformation of the circle and

is limited to flow conditions befcre stall. In all cases considered so far, the numerical

approximate method gives integrated lift coefficients that differ by less than 2 percent

from the assigned value.

For the examples tested, when the pressure distribution about a foil at a given

incidence is computed with the appropriate lift, agreement is good with both exact

potential theory and experimental results. Comparisons with measured pressure

distributions show considerable disagreement near the trailing edge caused by the thicken-

ing boundary layer. The assumed empirical modification introduced in the potential

theory cannot adequately represent the flow conditions near the trailing edge since

it assumes a stagnation point there in contrast with the experimental result of almost

free-stream pressure.

53

Although predicted pressure distributions agree well with measurement, comparisons

c,f the minimum pressure coefficient and the incipient cavitation index show differences.

In the example considered, the differences decrease with increasing test Reynolds number

(i.e., free-stream velocity). Also, cavitation prediction on the upper surface is better

than on the lower and is generally conservative on both. Some of the discrepancy

may be explained by machining inaccuracies, especially near the nose. Hence, it would

be of interest to have measured ordinates to use in the pressure program when making

comparisons with the test results. If we assume that nonsteady effects in the testing

were small, most of the differences seem a result of cavitation not occuring when

the minimum pressure is equal to the vapor pressure. This points out the need for a

better scaling law if laboratory data are to be useful in predicting full-scale results.

This investigation has called attention to two areas of deficiency in experimental

results. First, t!ere is a lack of pressure distributions on two-dimensional cambered

foils. Second, cavitation inception tests have not been carried to the point of

constant ar. for increasing Reynolds number. Further investigation in both of these

areas would be valuable to further confirm the calculation method of this paper.

ACKNOWLEDGMENTS

SpeciaI thanks are due Professor Geoffrey Ludford of Cornell University for

his helpful suggestions, in particular, for catching inconsistencies in a previous

development of the empirical modification.

54

APPENDIX A

EXAMPLES OF EXACT CONFORMAL TRANSFORMATION

If specific numbers are substituted for the transformation coefficients in Equations

[2] and 133, we will obtain foil shapes from exact theory which can be used to check

the cpprcimate method where x has been approximated by Equation F4 1

SYMMETRICAL FOILS

Consider first symmetrical shapes. These forms will be generated if all Bn are

zero. For our purposes, sufficient generality is obtained by setting all An, n > 2,

to zero. Then we have

x A+ (A_1 +A )cos 0+A 2 cos2V

Y = (AI - A1 ) sinj - A 2 sin 2,p

Letting Y = E (sin (D - 6 sin 2P) and requiring x to lie between 0 and 1, we obtain

x = 1/2 (1 + cosp) + E6 (cos 2 (D- 1)

From Equation f141, the exact velocity is obtained:

q 1/ 2 + sn O cosL+ (1 -cosq,) sincal

Also, from Equation [12] the lift coefficient is

C L 2T (I+ 2c)sincaSL

55

First, consider the case 6 = 0, then

Y = E sin

x = / 2 (1 + cosCO)

which is the ellipse of thickness ratio Tr = 2 E.

The velocity and lift coefficient are

q (0 T) sin(Pcosa + (1 - cos p) sina

U Fsin' p + I cos2 (p

CL 2 (+r) sin.ct

(Note that this includes the special case of the flat plate, i- = 0.)

Since x is precisely the form of the approximation, the velocity from the

approximate met'hod is-also exact.

For the case of nonzero E and 6, we can find (P in terms of x:

/1/4- 8 6,E (1/2 - 2 E 6- x) - 1/2arc cos 4 6E

and obtain (P values which give x, Y, and q/U at the x values used in the numerical

method. This has been done for four cases:

6 = 1/2 T 0.

6 = 1/4 T 0 . 2

For 6 = 1/2, the foil has a cusped tail similar to the Joukowski foil. Maximum27r

Y value occurs at q ;- 120 degrees, from which E r . A comparison of exact

and approximate values is given in Part (A) of the following table.

56

TABLE A-I

Comparison of Exact and Approximate Velocity Distribution for Two Foils

X = 10iv oso0) (cos 20 - 1)

A) For the Foil:-1

=27 (si0- sin 20j

i =.20 7 = .10

X exact I1 Dexact Darox YT Cexact approx D exact j rox

0 0 0 0 1.4952 6.4036 0 0 0 113.9904 12.9480

007596 .031655 .7370 .7388 7.0136 5.9965 .014381 - .9340 .9343 9.8212 9.0912

030154 .059631 1.2296 1.2325 5.8179 4.9847 .027447 1.1659 1.1662 6.0849 5.6346

.066987 .081097 1.4283 1.4316 4.4618 3.8352 .038080 1.2116 1.2119 4.1636 3.8577

.116978 .094556 1.4422 1.4454 3.330 2.8741 .045505 1.2072 1.2075 3.0565 2.8341

.178606 .099855 1.3799 1.3828 2.4984 2.1657 .049373 1.1846 1.1849 2.3431 2.1746

.25 .097881 1.2979 1.3004 1.9067 1.6603 .049755 1.1547 1.1549 1.8465 1.7154

.328990 .090159 1.2182 1.2204 1.4824 1.2965 .047083 1.1225 1.1228 1.4816 1.3779

.413176 .078485 1.1479 1.1499 1.1708 1.0282 .042038 1.0908 1.0910 1.2027 1.1197

.5 .064652 1.0882 1.0900 .9351 .8244 .035440 1.0609 1.0612 .9826 .9157

.586824 .050267 1.0386 1.0402 .7516 .6649 .028131 1.0339 1.0341 .8043 .7502

.671010 .036642 .9977 .9992 .6045 .5364 .020881 1.0099 1.0101 .6562 .6127

.75 .024738 .9645 .9659 .4832 .4299 .014317 .9893 .9894 .5304 .4956

S.82194 .015149 .9379 .9392 .3804 .3392 .008881 .9719 .9721 .4212 .3939

.883022 .008111 .9171 .9184 .2909 .2598 .004805 .9579 .9581 .3242 .3033

.933013 .003540 .9016 .9028 .2108 .1886 .002114 .9471 .9472 .2361 .2210.968461 00174 .0918 899 .22 .0064 .394 -996 15236 1 4432

.992404 000136 .8844 85 .0676 .0306 .000082 .9348 .9350 .0761 .07131.0 0 1 0 0 01 0 0 0 0 0 0

Fr x = (1 cos9). (cos 2o -,)B) For the Foil: s ) c .45417t

IY mc (sin 9 - 1 sin 20)

_ .20 T = .10

X YT Cexact 1Capprox Dexact Dapprox YT Cexact Capprox Dexact Dapprox

0 0 0 8.6728 7.9168 0 0 0 16.0122 15.3005

.007596 .025944 .7397 .140". 7.6502 6.9847 .012328 .9377 .9378 10.2197 9.7659

.030154 .049890 1.1375 1.1384 5.8403 5.3350 .023823 1.1206 1.1207 6.0605 5.7921

.066987 .070132 1.2895 1.2904 4.3603 3.9865 .033746 1.1585 1.1586 4.1239 3.9420

.116978 1 .065472 1.3270 1.3280 3.3070 3.0269 .041532 1.1621 1.1622 3.0462 2.9125

.178606 .095321 1.3143 1.3152 2.5600 2.3461 .046838 1.1532 1.1534 2.3602 2.2572

.25 .099687 1.2803 1.2812 2.0171 1.8513 .049562 1.1387 1.1388 1.8827 1.8012

.328990 .099066 1.2388 1.2397 1.6117 1.4813 .049827 1.1214 1.1215 1.5294 1.4637

.413176 .094293 1.1964 1.1972 1.3008 1.1973 .047942 1.1032 1.1033 1.2560 1.2024

.5 .086389 1.1563 1.1571 1.0565 .9738 .044342 1.0850 1.0851 1.0369 .9931

.586824 .076410 1.1201 1.1208 .8598 I.7936 .039526 1.0678 1.0678 .8565 .8205

.671010 .065341 1.0881 1.0888 .6979 .6449 .033997 1.0518 1.0519 .7043 .6750

.75 .054009 1.0605 1.0611 .5615 .5194 .028205 1.0375 1.0375 .5730 .5493

.821394 .043034 1.0369 1.0375 .4438 .4109 .022512 1.0249 1.0250 .4573 .4385

.883022 .032809 1.0166 1.0171 .3399 .3149 .017162 1.0141 1.0142 .3533 .3388

.933013 .023505 .9976 .9982 .2457 .2278 .012279 1.0047 1.0048 .2577 .2472

.969846 .015089 .9733 .9739 .1578 .1464 .007867 .9.148 .9949 .1679 .1611

.992404 .007356 .8984 .8988 .0723 .0670 .003830 .9695 .9696 .0812 .0779

1.0 0 0 0 0 0 0 0 0 0 0

57

For 6 = 1/4, the trailing edge is rounded and the general foil shape is similar

to c6nventional foils. The maximum value of Y occurs at p= (ir - arc cos - ,2

for which c 0.45417 r. A comparison with the approximate solution is given in

Part (B) of the table.

These examples show negligible differences in the C (co) component of velocity

for the 10-percent thick foils. For the 20-percent thickness forms, the differences

are larger but still slight. It is interesting to note that the approximate velocity is

greater than the exact in this case. The D(0) component of velocity shows large

-differences, but the approximate velocity is lower than the exact in this case, which

is the trend indicated by viscous effects. Also this is the component of velocity

which is multiplied by sin so that its contribution to the total velocity is small

in the normal incidence range.

In general, these comparisons indicated that there are larger errors in the

approximate computations for foils which show large departures from an ellipse.

Most foils in use today have their maximum thickness near midchord and do not have

cusped tails; buth of these properties contribute to accurate calculations.

CAMBERED FOILS

For foils with camber, we consider terms in Equation '21 and r31 to the

second harmonic, i.e.,

x =A+ (AI + A cos + (B -B_ Isiro+ A 2 cos 2p+ B2 sip 2p

Y=B + (A 1 -A 1) sin p+(B 1 +B 1) cosV -A 2 sin2P+B 2 cos2P

58

Ie

We take

Y =E(sin -6 sin 2p)+- ( - cos 2 9) + Y (I - osV)2

and set - 0 x() = 1, x() = 0 to obtain

fx =1/2(l+cosp)+E 6(cos2,p- 1)+ fsincP-- sin2,p

dx 2

Unfortunately,* these foils do not have = 0 at the leading edge which means

there is some overhang there. The constantY is included to raise the nose to lie

close to the x axis. The velocity distribution on these foils is

cos a [(1/2 + E) sin,) - (f+ Y) (1 - cosp)]

q + sin a[(1/2 + E) (I - cos(p) - (f+ Y) sinp]I

dx 2 dy 2

dxwhere -- 1/2 sinP+ f cosp - 2 E6 sin 2V- f cos 2P

dy~

dy (cosep- 26 cos 2t/p) +y sinq+ f sin 2 Vo

The lift coefficient is C = 2 Tr I +[ sin(co-x),L C+2

2(f+ Y)y = - arctan 1 + 2 E

The complexity of the x ((p) term does not permit easy inversion to obtain

P = cP (x). Instecd, we simply calculated x, Y, and q/J at many (p values and put

*Nor do these expressions permit us to put E = 0 and obtain simply a camberline

shape since the x relation would then give a curve which crosses itself and henceis meaningless.

59.

the (x,Y) relation into the arbitrary input section of the pressure program. Since we

were unable to calculate the exact velocity at the points of the numerical method,

the comparisons must be made graphic~al Iy.

The thickness portion of the foil was tcken as YT 2 - (sino - 1/2 sin 20)3r-with r = 0. 10. The constants f and 7 were first taken as 0.03 and 0.01, respectively,

which gives a camber ratio of about 0.0369 and then f and Y were taken as 0.015 and

0.0075 which gives a camber ratio of about 0.0185. The foil shapes--rotated and

shrunk to put the nose at (0,0)--and velocity distribution computed from exact theory

and the approximate theory are shown in the following figures for both cases at two

angles: 0 and 5 degrees (angles referenced to the unrotated foil shape). The

calculations show the approximate method is sufficiently accurate for practical use

at 0 degrees. At 5 degrees, the approximation is showing enough error to be

questiobable fromt a potential standpoint. However, the error is aguha in the direction

indicated for viscous effects.

Also shown in these figures is the velocity distribution computed with Ac

adjusted to give the same lift as the exact theory. These comparisons indicate that the

approximate method is sufficiently accurate for most work when the lift coefficient

is known.

60

tor JO;

ID61 .06

0 0~

4 - / N

0 T J~.O, f =O5, .00750

z z02

j .2 3 .4, .5 _. S_ U J.2 .4A. .6 7 B S L

o FRACTION OF CHORD - RC-UNO HR-.02 1

-1FOIL SHAPES

-- EAC TEOY C,-.282 - ~ 1 -EXACT THEORY. CL=.5OI1.4t- APPROXIMATE METHOD, &2=O 1. 0- APPROX. METHODL* 0@~

- -- aAPPROXIMATE METHOD, CL ,282 - APPROX. METHD t.

12 1.2 .

FRACTION CF CHORDFRCINOCOD

i rz10, z.01, Y=0075T:IQ,. f=.03, Y:OI

VELOCITY DISTRBUTION AT (r INCIDENCE

I.8 * --- EXACT THEORY, cZ~j.870 LB 450 - EXACT THEORY, CLO086-APPROXIMATE METHOD, Av:O APPROXIMATE METHOD, Ai.'OoAPPROXIMATE METHOD, C0'.870 aAPPROXIMATE METHOD, C1 'I.086

161.6 -

00 0~

S1.4[~ L41

1.2j .2-

VELOCITY DITOBTO AT5'INIDNC

Figure A-i -. Comparison of Exact and Approximate Velocity Distribution for, Foilsat 0 and 5-Degree Incidence

61

APPENDIX B

DISCUSSION OF AIRFOIL GEOMETRY

In many cases a thickness distribution is combined with a camberline by laying

off the thickness ordinate perperdicular to the camberline,' 5 One result of this

combination method is a nonzero ordinate at the leading edge (the center of the nose

radius lies along the camberline tangent at the leading edge ). A tesm may be

added to Equation -241 to include the nonzero ordinate at the nose. However, when

that vas done for the NACA 4412, the computed pressure distributions gave a greater

negative peak for positive incidences than either the experimental or exact potential

values. This result could be anticipated from a consideration of the approximations

in the theory. At the nose, the velocity due to thickness is zero and the only

velocity is that of the camberline. Since the velocity at a point is considerably

influenced by the ordinate at that point, a large ordinate at the nose means a large

camberline velocity and correspondingly large errors in the computed velocity. To

handle this nose ordinate, the profile can be rotated to put the nose at (0,0). When

the pressure distribution was computed for a rotated foil, again the 4412 section,

reasonable results were obtained aothough the predictions for potential flow

(ACL = 0) were somewhat lower than the exact results (see Figure 11), as expected

from the comparisons in Appendix A.

In order to rotate the foil, it is first necessary to determine the nose point

accurately. Points along the profile are determined from the expressions* 15

*These expressions would follow more logically if the abscissa for the thickness ordinate

were measured along the camberline. Presently, this is not done." 5

62

upper surface: (ix - YT sin ,YC+ YT co e)

lower surface: (x + YT sin e, YC- YT cos8) [B-1]

where

Y T is the thickness ordinate,

YC is the camberline ordinate, anddY

a is the camberline inclination, = arctan dC)x

However, these expressions do not give a value for the nose point. The nose can

be found by recalling that the center for the nose radius lies along the camberline

tangent at the leading edge,* as below

Y

dYC

I 0.Y,- arc tan(X Nr YN ) ! ""

LLE x

Hence, the nose (the point of minimum x) is given by

-PL (0 - Cos eLE)N LE 0 cSLEI

YN PLEsin 8LE [B-21

The nose can be put at (0,0) by the coordinate system translation (maintaining a

chord length of unity):

*See footnote on preceding page.

63

x-XNX -X1-X N

Y-Y Ny - [B-3 1

I -XN

Then the coordinates can be rotated through the angle

Y N.w = arctan -B-4]

I - XN

to measure the ordinates from the nose-tail line. The new coordinates with a

chordlength of unity areX cos-,.V - Y Sir. wx y 2

I -XN)

Y cos i + X sin uB-5

R I _ _ _

In the pressure program, the rotated ordinates are found from Equation L-5].

APhough the above expressions are exact, they do require interpolation between

the computed values, or an iteration, to find the ordinates for a fixed ration.

An alternative is an approximate equation for combining a camberline and thickness

distribution for fixed x. Such an expression may be obtained by expanding the

difference between the known ordinate in Equation M-1] and the unknown ordinate

64

at x in a Taylor series. The results, retaining only the lowest order terms, are

[1-u Yc+YT [V+YC2 +T

YL C T /IY YTYcJ FB-6

where primes denote differentiation with respect to x.

With YT = 2 PLE ... these expressions give the nose as (0,YN) where

Y N= P LE Y C (0).

The rct." ion to put the nose at (0,0) is approximately

XR -x

YR z Y - YN 0 -x) rB-71

These expressions aie quite accurate for small thickness and camber ratios.

However, they are not used in the pressure program and are included only to show

a method of combining a thickness distribution perpendicular to a camberline at

fixed x so that ordinates may be obtained at the stations required in the pressure

program. As already mentioned, there is a considerable saving of machine time when

the ordinates are given at the required stations.

65

APPENDIX C

LISTING OF THE FORTRAN STATEMENTSFOR THE PRESSURE PROGRAM

66

0

0

Ni V

.4 0

ON a ai 00N

N Og -0 UU

z 42 .N0w w 0 ! 0 0 E

;0 .5 Z C- 0 00UIL I z

4 -5 0 LOraU4 M -W 0 W--owi t) ~

*pqS N c ac 0 fl' -9 (X - )-C 0 C

o2. -. 0 0 00 C cc 0 -0

0 0 01 * N -

44 0 . Iz. - - 4 -ZS U C N 0 C -. 0 -

A02I . *J.. C . ~ - - -02-2-- 0 440 -000 -~0 ) U ~ - - N U -E 0 -0

M19 1 0. uZt0 - ) - - ~ - C J , U

0 - . - S .- ~ . 2In0E-~~ -N fJ 0 - * N - 4 1 N O .J W zE m l-~ I1-

a..-uaox00-. na -~ -n-. 4 40-N.I-0 0 - O W 0 - .. 0 E 2O.

O 0 0; 0: al 0 - C In

U o

000 0

Nm

v Nm

w *t4 0' 0- -- u .3 z ( I

"V N- Z M"PIG

w0 -. 4-

0 m 11 i IA v0

20 MN WS0 mN 0- )- >4

%L? N-. 4c z .W 9 L P Z F- 0.

C X *0 2 CYe cj.""; 5 z 2 N UZ. In'. W2a0U aNN Or 2

00 -0.0 0 !; 0

00 -InU -. 00

I-C 04 N In

IL WN. r- m0 .SU :NU . NUaN "-Z . z z

L*'0 N 0 0

Cit. ... j. Uv P2u P.

~2- .4224J Onn 221 -67

a

2N

04

09

0~ ftx 0 9

Li .3 A1

C - -M 0 N N

in 0 0 u-o * 0 ze

- - 09 0 N -C N 0 Js - - p. 1 ;

4 _ 0 -. 7-M %

I nC . . .3 .. .. u0 0.E

N 4 I I 3

44.W= 4o m~m Ca X : *o wa - 0- LCcaa% --.1-- 4c- .. u- 0U4i 4

14 C3 4;v N N Ml 1% t.

0 0 2 0 0 0 0 o P. 0 P. 00 p

005 n N tv Nl 1) N ** 0 0 0* - *0 * -= . C.~- - 0 ~P U

LU..1E i VI 0 0 N 40 N... P4 0.

NN2Z3~- .4 0 .0 "0 fi~04 .30

00022J - 0 0 2 - * 0 4N .0-. uS~U.*0O>~0 - 44 *0 N - - LL NC~t 0J

*.-~.P---4.X 4~~Vt'P - - .. .. * N J30-444 I-

P P* PEV. *0d-VN * U'- ON.Ed-O .0ON4 ~ e -2*NL2

0 0 - -- 0-0! 4 ~dNflN~NC Z~N~2.. 0L. CZ0

0 '. N-N~0 .3I01 P 4.0 ZOI 0* oo 40--O00U 40aP--P--oNNNNU 0.0w-C-- 0

*UO 2~4N.N~ 122-4 3Z.p 0 ZZZZ..22-Z2JE.0-O00-ZPx

N - N N N N fiN 0 040% .20 U3

P. In> 0 2

w I!

04 0 + 4

w

0C I

z 0Z

0

U9 O x3 -U. "n 0 vu

Zo N Z: j- 0 .4 NJNi

04VO <' CcM M A 0 4Vo C - o m

-- 5 - 0 4k

CL C-

.4'.J

4 -4

CY -. n

1.4 2 0 r M 0 t4-

>S z 03

.5- C4<c 0 -w . 2 -Cz I

> ot a v . I, Zi-j >41 0S-4 4-

I,-J -. 0 Z 1A.f A0 M1 J Z =.4 -u L

z- D-n-T- ;!,

22 4-) A1 w x- - 1 IS- > M sI Q O ! c

*~ i ;;x = a . ~ o A cw - ~ .a 42 0 0z o

In in In 0 In - 1-0132 .34 2 3-

V J'U .5 -0 4 ~ 0 0

* ... X 5* S.Z2 *- 0 . 2. O 31, "5 1I0- *0 UUS 4* 0 2 .00 0

- .44.4 C. 5 S ** NO . *5 00 2 *.~aU * 4 .5 V - JU . 5 A -- 13 0. 0\ 913 2 3 4 4~J 4.13 i 0 ... 410

> J- . Z 4 - . .. 3 40 2 '

00 z.JC.a 0. 40 % .43 .k

5-'- 013 04 M. 404 - 3 3 3 N o3 04 0 0. -0 4 4*3.I'04. W UZ 0'VU*.I1J4 1 00 -0, J n

P. 4-. 0 1 2 a 1" aL .3 4J. 4-0.5 . .. .. % z4-~- ;" 0 4A --00322 .U--1z440 -1S 4 4 J r 0i'

*S3.0-.49 i 00. 2 0- 40 3 ; -..C-9 u0- o 0 - C - - .5 -

vW<A.. - 0 - Z 33..4 0, 0-. -2-- 02-. A1 04 0 I

4C.0 Of1 UV3)0C. 1 '.1 . 4U1V33.). 40..- 3. 1 - 00 Z. NN -4 am .-- 0.2 0. )0

0- O a M0 IO Z 4 0 -6.

--,. 0> J - - -- - - 00 -u 00 - !!1 0 -ISISoM 4w ISO ~ l 05 IS IS . ccO aS ofa 4 a cac '3c L X 0 C C- ~

4 M4 %, 'D . f * 0 .

000 00 0In I.I n nb n%9 69

I I

Nw

N -c

T NW !::! - : t00w -- x *C .0 - z C

0. CL VZI A- o- w40m 4 Ia 4l v - ; 4 ! - - .E. .N ::; S!W I: :; : IIo3 0. ul. .- I- I N) > ) -- - "I S

o J IV C-. ii- ZO 0- 1 mtu - w1L r% WUW1. m *

- 0 Z) - Z W I -~ Z Z' 41-WW

fff0N w co00 0- US Z..C) U - -- fl-0l ~ > ) I10 4C%)K .-- U a~ -S 4-V% 'Uq -0 10-C - I

0.9 .. 0.W.N S*00 U*. ~ - L l a - O 0.)

.a- -C v J A~ U 5 M ~ E ~ l0 Ulf

0)- 00 00 0 -X 0 - 0-CL %C 11 C4 Z0 I0 '0 z C

v Z Z Z

NMN

z S

4 4 I

N UU U 0 C-

0 I- Sad 0al-...coa 0 N m mO( C--2

00 00 IL 0 0 -0 00co5 00

0C)) 1-S 10 0. 0 )(X 04o -0 -00C - -1

0* 5S 4 .I - 51 I Z - . S. N IZ ~ 4 *Z -Il70. I

- 4t -16 .4~ w in-

o2 46 J4 4' w

o% s. = U 0 - 0

in 0 m~ w Co :.0 0 J-~: 0 ;. z0 N0 =~I~O 2 I .- L.. Z 0

-0 in Ik 3P 2 A M 4 O

1~0 K91 0.0 UZC 3C ~ 4-u -z. pI

zi A. = 45 DC *0 0~~.10 V*4mz * 2 * - X .

00Ow - .0- 0 cC .M. *20 - 41

"Oko0 400.0 w0 ) '- EOz.10v ;J-I'-1 J z -~ 2 z . OU, X43-2 1.

asoac 24 .0. . 22 a a- N 2

2200U00 V1) X- I Zo z 0 w 0,Ia-b. 0 L*49.1L01 Z2 a-2 vU m

wri ccno o a- WU C32 ;;)..m CU!O2A1 XC004040WJC

C. -.j . 0 J - w22 N-urtI 0* lv.

XZXXXXIZA~kXX Z N2E2 On222 MX X XX - X-XQN.

N - 0 fl- 90 ml n . al0 0 z N 4 .Ji z 0 4 ; 0 A1 m1 z 0 z a I.. '! I .

- -0- 4C-------

051000-0 0 0000C0C06Za0I'O00000 000J000004ZU. w.LLIL UUw IL I U. w" L L.U.U 40 .1& U.u.U.. U. U. w & L 1 L . L u

N -~n -mm o

00

z a

N0

14 I -JQ

cy y w*1 0 -

M l t x U. M

m 0x

In a 0

?y ry - - -

X ~~~ 0 -a -F, ~~~~~~ ~0 X 4Wi U- .t

A Of !Z -, 1 -2 - .J, D

x amx l DA 4 ~ W- + wz 0-04

N Z x x I a )

N Y -. t 4 F. 'c .9. NN C 4 0 4PK~b In in bl Nn 00I

Kt to 'n 11 11 ON1 ' 22

a 00 -Z 4 0 410.2.

4-. .~ W00~I* * I N *.471l

II

REFERENCES

1. Preston, J.H., "The Calcu'at, on of Lift Taking Account of the Boundary Layer,"

Aeronautical Research Council R&M 2725 (1949).

2. Spence, D.A., "Prediction of the Characteristics of Two-Oimensional Airfoils,"

Journal of the Aeronautical Sciences, Vol. 21 (1954), p. 577°

3. Brebner, G.G. and Bagley, J.A., "Pressure and Boundary Layer Measurements

on a Two-Dimensional Wing at Low Speed," Aeronautical Research Council R&M 2886

(1952).

4. Pinkerton, R.M., "Calculated and Measured Pressure Distributions over the Mid-

span Section of the NACA 4412 Airfoil," National Advisory Committee for Aeronautics

Report 563 (1936).

5. Betz, A., "Untersuchungen einer Schukowskyschen Tragflache, " ZFM VI,

(1915), p. 173.

6. Theodorsen, T., "Theory of Wing Sections of Arbitrary Shape," National Advisory

,ommb.tee for Aeronautics Report 411 (1932).

7. Riegels, F., "Uber Die Berechnung De~r Druckverteilung von Profilen," Technische

Brichte, Vol. 10 (1943).

8. Moriya, T., "A Method of Calculating Aerodynamic Characteristics of an

Arbitrary Wing Section," Journal of the Society of Aeronautical Science, Japan, Vol. 5,

No. 33 (1938), p. 7.

9. Hecker, R., "Ma3nual for Preparing and Interpreting Data of Propeller Problems

which are Programmed for the High-Speed Computers at the David Taylor Model Basin,"

David Taylor Model Bcsin Report 1244 (1959).

72

1

10. Moriya, T., "On the Aerodynamic Theory of an Arbitrary Wing :,t.ction,"

Journal of the Society of Aerona.tical Science, Japan, Vol. 8, No. 78 (1941),

p. 054, ralso published in"Selected Scientific and Technical Papers."the Moriya

Memorial Committee, Department of Aeronautics, University of Tokyo (Aug 1959) 1.

11. Hildebrand, F.B., "Introduction to Numerical Analysis," McGraw-Hill 1956.

12. Sokolnikoff, I.S. and Redheffer, R.M., "Mathematics of Physics and Modern

Engineering," McGraw-Hill (1958).

13. Weber, J., "The Calculation of the Pressure Distribution over the Surface of

Two-Dimensional and Swept Wings with Symmetrical Aerofoil Sections," Aeronautical

Research Council R&M 2913 (1953).

14. Weber, J., "fhie Calculation of the P,-essure Distribution on the Surface of

Thick Cambered Wings and the Design of Wings with Given Pressure Distribution,"

Aeronautical Research Council R&M 3026 (1955).

15. Abbott, 1.H. and von Doenhoff, A.E., "Theory of Wing Sections," Dover

Publications (1959).

,16. Riegels, F., "Aerofoil Sections," Butterworths, London (1961).

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filen," Jahrbuch 1942 der dtsch. Luftfahrtg., i, p. 120.

18. Hess, J.L. and Smith, A.M.O., "Calculation of the Non-Lifting Potential

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ES-40622 (1962).

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19. Abbott, I.H. et al., "Summary of Airfoil Data," National Advisory Committee

for Aeronautics Report 824 (1945).

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over an Airfoil Section," National Advisory Committee for Aeronautics Report 613

(1937).

2i. Kermeen, R.W., "Water Tunnel Tests of NACA 4412 and Walchner Profile 7

Hydrofoils in Noncavitating and Cavitating Flows," Cali '3rnia Institute of Technology,

Hydrodynamics Laboratory Report 47-5 (1956).

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23. Eisenberg, P., "On the Mechanism and Prevention of Cavitation, ' David

Taylor Model Basin Report 712 (1950); see also the addendum to this report, David

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24. Breslin, J.P. and Landweber, L., "A Manual for Calculation of Inception

of Cavitation on Two and Three-Dimensional Forms," Society of Naval Architects

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of Technology, Hydrodynamics Laboratory Report 21-8 (1952).

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