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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
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Research Council R&M 2913 (1953).
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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,
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