4O. I-W

CASE FILE APR 14 o - =' 15

NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

11111k ItI' I illE RE PORT ORIGINALLY ISSUED

October 1944 as Advance Restricted Report L1G15

THE FLOW OF A COMPRESSIBLE FLUID

PAST A CIRCULAR ARC PROFILE

By Carl Kaplan

Langley Memorial Aeronautical Laboratory Langley Field, Va.

FILE COPY To be returned to

th'i fles of the National Advisory Comnittas

for Aeronautics Washington, 0. C.

NACA WASHINGTON

NACA WARTIME REPORTS are reprints of papers originally issued to provide rapid distribution of advance research results to an authorized group requiring them for the war effort. They were pre-viously held under a security status but are now unclassified. Some of these reports were not tech-nically edited. All have been reproduced without change In order to expedite general distribution.

L - 216

NACA ARR No. L4G15

NATIONAL ADVISORY COMMITTEE FORERONi.UTICS

ADVANCE RESTRICTED REPORT

THE FLOW OF A COMPRESSIBLE FLUID

PAST A CIRCULAR ARC PROFILE

By Carl Kaplan

SUMMARY

The Ackeret iteration process is utilized to obtain higher approximations than that of Prandtl and Giausrt for the flow of a compressible fluid past a circular are profile. The procedure is to expand the velocity poten- tial in a power series of the caiber coefficient. The first two terms of the devslo'pment correspond to the Prandti-Glauert approximation and yield the well-known correction to the crcuistion, abbut the profile. The second approximation, involving the square of the camber coefficient, improves the velocity and pressure fields but yields no new results with ragard to the circulation, since the circulation about the profile is an odd junc-tion of the camber coefficient. The third approximation, involving the cube of the camber coefficient, permits the use of higher values of the camber coefficient and furthermore yields an improvement to the Prandtl-Glauert rule with regard to the effect of compressibility on the circulation of the circular arc profile Numerical examples with tables and graphs illustrate the results of the analysis.

INTRODUCTI ON

The calculation of the two-dimensional steady flow of a compressible fluid past aprescribed body can be performed by a method independently discovered by Janzen (reference 1) and by Rayleigh (reference 2), which con- sists in developing the velocity potential or the stream function according to powers of the stream Mach number. The first approximation IS the incompressible case and the succeeding approximations represent the effect of compressibility. The method has in recent years been

2 NACA ARR No. L14G15

successively improved by Poggi (reference 3), by Imai and Aihara (reference 14), and by the present author (refer-ence 5). Although the method can be applied to an arbi- trary profile, it suffers from the practical restriction to small stream Mach numbers, because approximations beyond the second or third entail a prohibitive amount of labor.

For the flow past a profile of small thickness, camber, and angle of attack, Frandtl (reference 6), Glauert (reference 7), and Ackeret (reference 8) obtained by various means an approximation that applies to the entire subsonic range of velocity. The present author (reference 9) extended the method of Ackeret by an itera-tion process that takes into account the effect of thick-ness and applied the method to a particular family of symmetrical profiles. In the present paper, the effect of camber is investigated. by a similar application of the method of reference C to a circular arc profile. In the appicat1on of the metnoa, it is desirable to avold stagnation points so t,iat:thevariation of tiiC local velocity from that of the undisturbed stream can he made small. For this reason the direction of the undisturbed stream is chosen parallel to the chord ofthe circular arc (ideal angle of attack) and the circulation, about the profile is determined in accordance with the K.utta condi- tion; namely, that the flow past the profile leave the trailing edge tangentially. The flow is symmetrical iore and. aft and the velocity remains finite at all points. The circulation in a compressible flow will be seen to be an odd function of the camber coefficient. In order, then, to obtain an improvement of the Prandtl-Glauent rule, it is necessary to carry the iteration process through three approximations

THE ITERATION PROCESS

The velocity potential Ø(x, Y) of the two-dimensional, steady , irrotaitional flow of a compressible fluid satisfies the following differential equation of the second order:

(c2 - u2) . - 2uv + ( 2 - v2) (1) axa y ay

NACA ARR No. LG15

where

X, Y rectangular Cartesian coordinates in plane of flow

u= —, v = - fluid velocity components along X and Y aX axes, respectively

c local velocity of sound

The local vaiocfty of sound c is expressed in terms of the fluid velocity q by means of Bernoulli t a equation

'p

/+_q2= (2)

U- 1

the equation defining the velocity of sound

(3) U.

and the adiabatic relation between the pressure and the density

p -(p \Y Pi - P1)

(L1)

In equations (2), (3), and (4),

p static pressure in fluid

P1 static pressure in undisturbed s

P density of fluid

P1 density of undisturbed sbream at

q magnitude of velocity of fluid

Y adiabatic index (approx. 1.4 for

tream at infinity

infinity

air)

j . NACA ARR No. L)Xi5

For the adiabatic case, equation () yields

0 P c- =

By means of equations (Lb ) and (5) Bernoulli's equation, equation (2), yields the following relations:

2 2 t-1 (q2 C = C1 -

M1Tj

P Pl-

( 6)

=- 2

Ivii2 ('_ i)

where

U velocity of undisturbed stream at infinity

C1 velocity of sound in undisturbed stream at infinity

Mi Mach nunber of undisturbed stream at infinity

Now, if the profile is held fixed in the uniform stream of velocity U and if a characteristic length a is assumed to be the unit of length and the stream velocity U is assumed to he the unit of velcciy, the fundamental differential equation (1) and the first of equations (6) become

('-4-2M12uv 122 (7)

àXÔY

and

2- Y jq12(q2 i) (8)

Cl

NACA ARR No.L)4-Gl5

5

where X, Y, U, v, q, and ' now denote, respectively, the nondimensional quantities x/s, Y/s, u/U, v/U, q/ /. and 0/us.

The iteration process consists in developing the velocity potential 0 in powers of a parameter h, the camber of the circular arc profile. Thus

OX h0i- h202- h O ••• (9)

WITNI

°i h2ÔØ

U = -1 - hax

-a--

--

(10)

h2 h V =-h -

-...

6Y6.Y

When these expressions for /, u , and v, together With the expression for c 2/c 12 given by equation (8),

are introduced into the fundamental dif w i

ferential equa- tion (7) end hen the coefficients of the various powers of h are equated to zero, the following differential equations :L, 02' 0, . . . result:

9

(1 - ') + = o M1 (11) \ I

ax2

9

- M1 àØ + a02 =

/ 6x 6Y

601 ^21 62

(12) ox ax ax 6 y2 ax 6x 6-1r 1

6 NACA AFR No L4G15

2 2.-, M 2 ) + =

/6X 2 10

2 1 l Ml 6X_+ (1)1J 6X2 y2

+ l(a Ø 2 r a2+ (+ll

2Ôy/ L 6x

aør ___ ___

+ (y + 1) - - + (y - 1) - aX L ax

aØ1 [ a20 a20' + - (y + 1) + (y - 1)

ax. aYc___.

ø

601+ 602 a20 + a2ø)l

(13) +2 ( 671 ày àxay ày axay ày àxôy ( .-1

These differential equations may be put into more familiar forms by the introduction of a new set of independent variables x and y, where

X 1 y = 3YJ

and

2)

1/2 [3 = - M1

For M1 < 1, equation (11) then becomes a Laplace equa-tion and equations (12) and (13) become Poisson equations. Equation (11) replaces the fundamental differential equation (7) for flows that differ only slightly from the

NACA APR No. L4Gl5 7

undisturbed stream, and its solution yields the well- I Prandtl-Glauert result. The solutions of eque- Lions (12) and (13) provide successive improvements in the approximation to the solution of a compressible-flow problem.

For the present problem, the procedure to be fol-lowed in solving equations (11) to (13) is first to obtain the velocity potential for the incompressible case in the form of a power series in the camber coeffi-cient h of the circular arc profile. The solution for the first approximation '- of the compressible flow is then obtained by analogy from the form of the coefficient of h of the incompressible velocity potential. The solutions of equations .,

(12) and (13) for the second and third approximations 2 and 03 follow by a straight-

forward procedure. The boundary conditions - that the. flow be tangential to the profile and that the disturb- ance to the main stream vanish at infinity - are satisfied to the same power of the camber coefficient h. that is involved in the approximation for the velocity poten -tial Ø The calculations are laborious when more than two steps in the iteration process are involved but the third step is necessary to obtain results that extend present-day knowledge. Most of the details of calculation are given in appendixes in orde not to obscure the presentation of the main r-esults.

IIESULTS OF THE ANALYSIS

Expression for the velocity potential.- The choice of the circular arc as the solid boundary was made for two reasons: (1) the solution of the incompressible flow can he easiLy expressed in a closed, form, and (2) when the circular arc is fixed in a uniform stream flowing in a direction parallel to the chord and when the Kutta condition - that the flow leave the trailing edge tangen -tially - is applied, the velocities at the nose and the tail are finite and different from zero. No stagnation points occur, therefore, on the boundary or in the field of flow and a greater degree of accuracy J r, the iteration process is assured. Atpendix A contains the calculation of the incompressible flow past the circular arc profile and appendixes B, C, and D contain the detailed calcula-tions for 01 , 12,and Ø, respectively.The final

8 NhCA ARR

expression for the velocity potential 0 takes the following form:

Ø=-cosh cos '-hØ1 -h212 ..-hØ3 (15)

where, from e q uation (39),

(e __22 i sln 2 - 2)

from equation (Cl),

(21) + 41 ^-.,e + 2De' + 2 ^3 - D+ D 1) D

1) + 12D- D + 1) D+ 41 39 cos

and from equation (DIB),

G 1 () sin 2r + G2() sin 4j1

/ . (1 sin 2i

+ G()(-- --e sin -e sin Lrj, \ cosh 2 - cos /

- [2G.1 (o) +

In these eauations

D =

G1 (), G2 (), and G() functions of tions (P12 (Dl)), re

e1liitic coordinates relatc•ci to Cartesian coordinates N, Y trans Co rm at ion:

01 =

, given by equa-), (D].), and pectively

re c tongul ar by equations of

NACA ARR No. Ltj.G15

x = X = cosh cos

Y = ç y = sinh , Sin Y)

The circulaticn correction formula,- Equation (15) re presents the s5Fu OT -. , e fundamental differential equation (1) that s&ti.1iCS the boundary conditions at the surface of the circular are profile and at infinity insofar as the terms Inclusive of the third power of the camber coefficient h are concerned. Each of the expressions l' ø2, and are obtained in closed

form and are finite for all values of the stream Mach numb er Y1 , from zero up to but not including unit y . The

Kutta condition, which determines the circulation uniquely by stipulating a finite velocity at the sharp trailing edge of the circular arc, yields the following circulation correction formula (see equation (D6):

[Lc-)+

+ 1) J-LL (e + p1 3 p3 p5

± ( + 1 )2i + p2)1 h6 (16)

2 P7

where I' and F1 are, respectively, the circulations

in the compressible and incompressible flows. The incorn-pi'essible circulation 171 is proportional to the first

power of h so that the compressible circulation '

is an odd function of h. The second approximation of

is therefore identical with the first approximation and no departure from the P.,andtl-Glauert rule is obtained until the third power of h is included. This result explains why the simple Pranitl-Glauert

rule for the effect of compressibility on the circulation or lift of an airfoil has been very satisfactory.

For comp arison, a formula analogouc toequation (16) has been obtained by applying the von Krmdn-Tsien velocity correction formula to the circular arc profile.

10

NACA ARR No. LLC-l5

From reference 10

= q

where

velocity of compressible fluid

qi velocity of incompressible fluid

- 12 MI.

/)

1/2 l±(1-M I L I -

By an e1emntary integration around the circle, corre-sponding conformally, to the circular arc, the following relation is then obtained:

1 - 1-p1/2

C08 2ö

-l/2 sin26 In - 21/2 (i + s26) + p. cosö 1/2

/2 1 2 + L COS 5

- [i + 21/2 ( + +ccs)45]i/21

where the angle 5. (see fig. 1) is related to the camber coefficient h by means of the equation

tan S = 2h

Table I gives values of the ratio for various

values of the stream Mach number and the camber coeffi-cient h, calculated by means of equations (16) and (17). Figure 2 shows the graphs of as functions of for various values of h. The curves based on the von Karmn-Tsien velocity correction formula lie between the Prandtl-G-lauert curve and the curves based

NACA ARR No. L4G1511

03 H

(/)a)

0+

I)

o VII •r- -P o

-

c-co

4' C\j

• 4' --P 1)

0 r-C' VIi—I (1)

Cl) 4'. LU- 0

-P r-C\ I N- + 4) 0

Cl) 0

"-c- 00

i— N

H 'H

0 'H H (\J

'H0

+r-e

0 H H,0

'H a) ci) t-- c--i cl. N — 0

H

Cl)

Cl) 0(1) HIr-C\ ci) (1)

4 3 (1)H N

U)rLil 4-

+4- ç

H + --' ,-. co 0 0— - _0 -S

Hfl. 00

Cl) cD- ci) +r-(\ (D•H

+

ci) 4)ç -1- N 0 •r1 (I) •c'J - - Cl 'ri -P

_QDL c- N N

iO-H I I N ci) 0+

tcOrd

(I) Ci))1) .-dHk CO

4 CO N I C!, H Ico..

COCI) -H (\j

"'-.I

+ '-__---' .-

(1) ()) H

El 0 N N H -P ,C)

0 CO -P ICci

c--i

'.U. + .srJ)

Cl) •H 4' 0 0r

Hi)) C\ H CO c- H

rd N Cl) • E H 'H + C-- 'H c- c-jr-I c-o Cl) ci)

-• HIC\J-P0 00

fl0c-

-H_ I V I I ' - + ci)

H VM4

0 -' I 4'

0 c--i

0 H CO

Ici)

0 0 (1) - C6 -T .---,. 0) -P

('-I VII +

.0) Cl) Cl)

-P 0

" C0 U) C') 0

Cl)I'C'-)- RI + ' H

•r4 j U) H

Ci) c--i

c—i0 N

4) — —i H CL).

0 4) ())

')ci) H >

c--I 0 0)

I "-----. --s 0

'dc-c ) 0 .-d1 r-' +

\,/ OH c- 'H

co 0 0 -f - .15 C') C') H 0 ± _- 'do

coi. c-i ,c U) Id t 0 -P -r k'-1 -

ci. -c-- Ci. I 4-" 'H c- c--c

•H C)

'4' N I'd. I) H i-C\ ul Ci)

ci)c15 ci) >C H Ci) -I1 4' 0 + -zt I 4)

0) c--i c--i -P

'H )

__

0) 0 -j-IQi)-C')

H I'd. 0q--,

P, 0 L-_--_-J OH + 00

c-i CD-P .0 0

Cl) ,CC15 (I) U C))

$1 El 00 occi

H c--i 0 0 CH

H ' 40 H N -i 00 c--i o C-') C!) FbI

12 NACA ARR No. Lii5

0 H c\j H N N N ci)

a - — 0 -i

ci)

ci)N

'H cn CID

H00

0 0 H H

s:1 0 C) (J

cci bO -I- 0 ___

rC'. •H N Lf' Lr\

CO cl HI 0 U) 6) 6)

1 H I zt zt

4' (I) H N -•

0'H

N • N N 4) U)

0 -P 0 + N a

N•

RI -z1 ,--, U + CO CO

a

H I ' CO 4' N 1 4' N + "HO

lrH + H'TThN cli

0 ",-' -P H c,--,, +4) H RI (1) a

N 0 "—.—" 0^ ci 4)

0 ,si 0p u

U) H N\ --Zr H \ j 'H .-Zf 0

(1)I (1)

+

43)

> IO+-

cii + H N N H E H H ci) C

r-t 0 II H H

--I

o CIA

U s- 0.H 0104 •

0 ca•H co

.,-1

.'

0 •r4

•Ha Q C)

HP')

bQH cO ci)

C.) ro ci ci I' (1) 0 II 0 0 I _l 'H N 4) (\j 4) -P xi -P 00

0 •r-1 "H -r "H -p ® 0

01 0C Q1

'

2

.4.) 24

ci)

.-II +

c-

(3)—.J-

I

I-I ) Cl)

O ci) 0 ,S H H (1) a 0 -4 4.) -P H Oil (1) -P C) 'H -P 01 4) c ,J

0101 0110.4 E-----

NACA kRR No. LLGl5

13

shows the graphs of ( ••.Q- 1)r.,as functions of M1

for the three approximations for various values of the camber coefficient h.

The critical velocity q,, defined as the value

for which the velocity of the fluid equals the local velocity of sound, is obtained from the first of equa-tions (6) by putt

ing q c = q, Thus

+ M

q =1- -. (23) cL'

\ Y+i2 I - DA

N

The values of qcr are given in table V in the column for which the local Mach number is unity. The ratio

qcr/qj is easily calculated for the various approxima-

tions The graphs of only the third approximation of q, .

- 1 are induced in figure L. Table VI lists the qj

first, second, and third approximate values of the critical stream Mach number M1 , and figure 5 shows

cr the corresponding graphs as functions of the camber coefficient h.

The graphs of the third approximation of the maximum and minimum values of q, obtained from tables III and IV, are shown in figure 6 as functions of the stream Mach number M1 . The constant local Mach number lines

shown in figure 6 are obtained, from equation (3) by introducing the local Mach number M in place of the local velocity of sound c. Thus

a --__- J I (2)

+ LI

2 7/

Note that equation (24) becomes equation (23) when

MACA ARR No. L14.G15

M = 1. Table V contains values of q for various values of M and M1,

A. comparison of the results of reference 9 on the compressibility effect of thickness and the results of the present paper on the compressibility effect of camber is of interest. For this purpose, a symmetrical shape of reference 9 was compared with a circular arc' profile with the same incompressible maximum speed at the surface. Results of this comparison for several corresponding thickness and camber coefficients are given in table VII. The dashed curves in figure 6 are asso- ciated with the various symmetrical shapes. For moderate values of camber and thickness the difference may be seen to be negligible over the entire subsonic range. This observation indicates that, at least to a very good approximation, the effect of compressibility in the subsonic range can be considered to depend explicitly only on the incompressibie.fiuid velocity and thestream Mach number and to be independent of the shape of the profile This result therefore substantiates the use of velocity correction formulas such as the Prandtl-Glauert, the von Ka'rmn-Tsien, the Temple-Yarwood, and the Garrick-Kaplan (reference 11) formulas, which depend only on the incompressible fluid velocity and on the stream Mach number,

In general, the velocity q at the surface of the circular are profile may be written as follows:

ql+a1h sin +h2 (a2 +a cos 2 4)

+ h3( a

l, . sin + a5 sin + ... (25)

where, from equation (18),

a1 =

2 (__^:2\2 a2 = -2 + +

NACA ARR No. L4G15 15

/ a3 =---- 2(y -

a =+ G1 (0) ± 2G2(O

a7 = + 2 13 ( 2D + 3) + us(o)1.

Values of a, a2 . a3 , ai, and a,- . for various values

of the stream Mach number zdj are given in table VIII.

As an example of the behavior of the velocity distribu-tion over a circular arc profile as the stream Mach number is varied, the case cf h = 005 with N 1 = 0.3, 05, and 0.7 is calculated and compared with the incom-pressible case. The calculated values of the velocity at the upper and lower surfaces sf the circular arc profile, h = 005, for the v5rioUs values of M 1 are

given in table IX and tuhe corresponding velocity-distribution curves are shown in figure 7.

The pressure coefficient.- In the case of a uniform flow past a fixed boundery, the pressure coefficient is defined as

P -p1 0 r,Ivh =

L2L

From the third of e quations (6) it follo lks easily that

r

=

-1 + 1 + -(y l) M12 (1

(26)

For the incompressible case, M 1 = 0,

C. 0 =-

16 NCA ARR No. I4G1

For the sonic case, q =

C Yl I.-I

2 )2 + (y ) ivi-'- ç

(CMl)cr = yM1 + + 1 J

J

For the limiting case of, absolute vacuum, lvi =CO

___ 1\\1/2

and q(—.

(Cp,ji ) =

Table X gives associated values of the velocity and the

pressure coefficient C. icr various values of the

stream Mach number M1 • and figure 8 shows the corre - sronding graphs. By means of table X and flure 8, the velocity readings from figures b and 7 can be replaced by the corresponding pressure coefficients.

Langley Memorial Aeronautical Laboratory National Advisory Committee for Aeronautics

Langley Field, Va.,

NACA ARR No. LGl5 17

AP?SiWIX A

DETERMINATION OF TI COMPLEX POTNTlAL FUNCTION w

The Incompressible Flow past a Circular Arc Profile

Consider the mapping of a circle C' in the Z' -plLane into a circular arc C in the Z-plane. (See fig. 1.) If the center is at (O ,m) on the Y'-axis and the circle passes through the points (a,O) and (-a,O) on the X I -axis, then. the Joukowski transformation

Z = Z +

(Al)

maps the ciucle C' in the Z' -plane into a circular arc C in the Z-plane. Tiie e q uation of the circular arc is

X2 + ya2 - m2\2 JT12 + a2"\2 ! +

= (.) is in j

The parts of the circle C' lying above arid below the X'-axis correspond, respectively, to the uprer and lower surfaces of the circular arc C The end points A and B of the circular arc are the r)oints

X = ±2a

and the maximum ordinate is

7 2a tan ö

=

The comber coefficient h is defined as the ratio of the maximum ordinate to the chord, or

• 2m -

Lfc

= tan 6 (A)

NLCA ARR No. L'LjG15

The complex potential of the flow past a circular cylinder, of radius R fixed in a uniform flow of velocity U at zero angle of attack and with a circu-lation F is given by

W = _(z + - ii' z11 Z11 2ir J

log (ALt)

where

= Z' - Ia tan 5

For the purpose of the present paper the circulation F must be so chosen that the stagnation points on the circle C' lie at the points X' = ±a corresponding to the leading and trailing edges of the circular arc C; that is,

F = Birtjah

= LpUR sin 6 (A5)

With this value of the 'circulation inserted in equa-

tion (A4) and with Z replaced by Rele, the complex velocity at the surface of the circular are C becomes

2tJe8sin B + sin S

(e + i sin Ii 21B - 2ie sin S - e

The magnitude of the velocity is

/dw d\1/2 q = ( -

\dZ d

= u(i + 2 sin e sin ö + sin21)) (A6)

It is recalled that the upper surface of the circular arc is traversed in a clockwise sense as B goes from -o to iT + S and the lower surface, as 9 goes from -(ir - 5) to -S. The velocity at the nose or tail is then given by

NACI IARR No. L)..G15

nose = =(j cosô

The maximum and riinim'un velocities occur at 8 = and

at U = - res pectively, and are given by

q,max= u(i +(A7)

q = ii(l - sin ö)' mri

Ecuation of Circular Arc as Power Series in h

The e'quation of the circular arc, obtained from equation (A2) for the entire circle, is

(2 2

Y 21n - r ± - X) (A8)

2 where r =

is the radius of the circle. Expan- m

sion of the radical in e quation (8) according to powers of X/r yields

(L9) r 8 r-' 16 r-'

m By use of h=-

2a

Cl -

r - , + LLh2

= 2h - 8h + 2h5 -

Then"equatlon (A9) becomes

±--2)h5 + ()

20 NACA AFR No. L4G15

x Now, put -- = cos . and replace x y and - by X

-2a 2a 2a and Y, respectively. Equation (AlO) then becomes

y2h sin2 ±2h3 sin22+8h5 sin22 cos 2+... (All

and

cos -16h3 cos 4 cos 2-16h5 cos 4 (1+3 cos ... (Al2) dX

Equation of w as a Power Series in h

Consider equation. (AL) with F = 8iruah and = a2(i + 2) Then

/ 2\ W = _TJZ U + a2 ). - )4uah log (Al)

\ Z /

Now

Z' = -. ia tan 5

= Z I - 2iah

Then by expanding the right-hand side of equation (A13) according to powers of h and replacing Z,' I by

+ (2

4.a )__ obtained from the Joukowski transfor-

2 nation (Al), it follows that

w=-UZ+2iaLth7+(2a2)/2L +

4a2 2

If w/2aU and Z/2a are written, respectively, w and Z, then

1/2 1I

w= -Z+ih - lz_(z2_1) j -2 log EZ+ (z2_ 1) + ... (A1)

NACA ARR No. L)4G].5

21

From equation (.1L1 ) for w and a co'responding equation

for the complex conjugate w, the noridimensional velocity Potential becomes

ihI [ l/2]2

0 = -x + 1- Lz - (z2 -1)

± r(2 i)1/22 -2 log2 +

(Al')) 1/ 211

z^(z'_i)

22 NACAARR No. L1Gl5

APPENDIX B

DETERMINATION OF THE FIRST APPROXIMATION

By means of transformation (14), equation (11) or becomes

62Ø

2 + 2 (Bi) ax ày

A comparison of the expressions for 0 given by equa- tions (9) and (Al'-) suggests the assumption

2 11212 0i1 (z21) (z21)j

L+ ( 2 - i1/21

+ 2 log (R2)

z+z-1)j

where z x + ly, x - iy, and k is an arbitrary constant. Since this expression for 0. is the sum of a function of z only and a function of only, it satisfies Lapiace t s equation (Bi). The arbitrary con-stant k is to he determined from the boundary condition

a0 dy a0 ÔXdX 6

or

.2 (pz) Ox dx ày

The expression for 0, insofar as the first power in h is concerned, is

0 = -x - hØ1

NACA ARR No. LGl5 2

and, to the first power in h, from equation (Al2),

dy= -Lfl cos

The boundary condition, e quation (B), then becomes

cos -B2hy

= (ø

\ôz à/

= 2Ik2) J1 [(/2j2)

() (2_i2 \ 'I

By definition

the first powe first power in

z

z

Z2

(Z2

x = cos 4 1,, and from equation. (All), to in h, y = 2h sin2 . Hence, to the

= cos 4 + 241-1 siri2

= cos 4 - 21ph sin

= ccs + L4h sin2 cos

= cos 24 - 4iJi s Ln2 4 cos

1)112 = i sin 4 (1 - 2iph cos

(2 - = -i sin (1 + 2ih cos

Then

Lph cos . = -2iIth321 Slfl

2 sin L sin -

= -Sikh(.-,'2 Cos 4

2)4

NACA ARR No. L)4G15

or

The expression for the first approximation of 0 is then

ih 0 = -x - - (z 2 '/ (j72 i)'2J2

±( _i) L + 2 log1/2 (B5)

-+ (- - i)

This expression for 0 can be simplified considerably by introducing elliptic coordinates !, and 1). Thus, let

z = cosh S DO

where

Then

x + i cosh + ir,)

= cosh , cos 'r ± I sinh F. sin

so that

x = cosh cos(B'?)

y sinh , SIui 1

Equation ( B5) can then be written

Ø-(cosh +cosh ) log e) 2 2P

or

O = - cash cos i - (e sin 2 y - 2T) (E9)

IN

NACA ARR No. L1G15 25

From a comparison of equations (Al - ) and (B5) note that, if F and 1' denote the circulation in the incompressible case and the coZilpressjble case, then

pc

p . -.

= (1 w2)1T2(KO)

Equation (BlO) is the well-jcnown Frandtl-Qjauert rule connecting the circulations (or lifts) in the incom-pressible and comp:essjbe cases.

In order to utilize equation (f39) for the celcula- tions, the equations of transformation (B7) must be inverted. Thus,

+ -',----= 1 2 2 coshsirm

X2 2(B1.1)

y =1 cosr) sin2

From equations (Bfl),

2 si 2 = -b + (b2 2)1/2

2 sin2 = b + (b2 + 2) 112 ((l2)

where

b = 1 .- (v:? .. 2)

By means of transformation (iL),

2 sinh2 = - b + (b2 +

2 sin2r, = b + (b2 +

26 NACA JIRR No. L2.G15

where

b 1 - (x2 + P-2.^^2)

In terms of the complex variables t and f, the velocity components in the direction of the coordinate axes are

u-1----J—.c-ø-+-- aX - Siflh Siflh

(Bl)

v== ie(_'_- à y \Siflh sinh à

Let 0 be given by equation (B3); then,

U 1 sin

(Bl)

V = 005 fl

in h, at the boundery,

= 0

denote the magnitudes of the of the circular are profile for incompressible cases, respec-

q=1+sin c(5)

= 1 +11 4Ja sin .

or, when h sin

is eliminated,

qc1) (1316)

q 8

Now, to the first power

Hence, if q and q1 velocity at the surface the compressible and th tively, then

NJCA ARR No. L4G15

27

where

2 1/2 = (i - r1 )

Equation (B16) represents the velocity-correction formula for the Prandtl-Glauert approximation. Equations (B15) can also be written as £o11ovs:

qc(B17)

Since the Prandtl-Glauert approximation is strictly true for infinitesj pn1 disturbances bo the uniform stream; equation (B16) may he replaced by tiie di2ferentiai coef-ficient (from reference Ii)

Uqc - 1(BiS)

p

28 MACA LFR Ico. L.4ci5

APPENDIX C

DETERMINATION OF THE SECOND APPROXIMATION

By means of transformation (14), the symbolic relations,

ax àz OF

62 622 ____ a2

= + 2 0 +

2 2 6 ôzz à

0

6 Y 6 z 6 Z) - -

(Cl)

6 2 62 6 2 - —+2 ' — - àz2 aza a

a 2 - . (a 2 a2 - 11

axay \a z2 à-

and the equation of transformation (p6)

z = cosh

or

= cosh.

differential equation (12) for '2 can be expressed in

terms of the complex variables t and as follows:

NACA ;.PR No. LLjG15

.Tk\J

H

CO

tC\to

r-14

Cl) '0

'H Cl) 'U i

+ !l

N Ci)

rH01 .,-1 Id

(1) rj

'H Cl) rHj-

0.0) 0C\J

Ho CQU)

+ 'H j0), dr-1 'H -P L j

+ 'H ,

°Qj to

Li'° %1'°

'H II - N

IJ

CO-010) (\J ito 0

I I I ii C\JI--J) 0

N '-J) Co4.O

/0 /0 Co

0) cti ro

00) 'H

0)'Cir

Cl) —k

o —J '--

-P -P-P

'HO) 'H

(1)0)(1 CO

29

30

NACA ARR No. L4015

LX y^l)( y l(ee)(e sinh-e sinh ILr(e -t - 8 -e ' sinh +e sinh

Finally, by putting t = I + i7j . and = -

— ^! - E+i e+2e3j cos

+( + i)i e cos (03)

The right-hand side of equation (c3) suggests a solution of the form

7711() cos + () cos 3r) (04)

By substituting this expression for 02 into eque-

tion (03) and by equating the coefficients of cos Tj and cos 3r,, to zero, the folloving differential equations for F1() and F3 () are obtained

I -F1 )4 (05)

3.- 9F3 =

(c6) cl

02 =

The solutions of these equations are

F 1 2L .+') - (y-3) e +2e3+A1e (07)

/

NACA ARR No. L'-T15

31

F3 =- ( + l)(LLJ) (e + (c8)

where A l and A7 are arbitrary constants to be deter-

mmcd by the boundary condition at the surface of the profiie. The other two arbitrari constants are taken equal to zero since F1 and F-. must vanish at infinity.

In terms of the variables . and r, the boundary condition (B3) takes the form

Sifl TIO (sinhcos r

r

--cosh 0 )à, dx

+ siith cos rX (c9)

where the velocity potential 0 has the form

O=_coshcos_(e_2Sifl2T)_2fl)

- h2 F 1 cos r + F3 cos 3 + I'2) (010)

and where I'2 is an arbitrary circulation to he deter-

mmcd by tIm Kutta condition at the trailing edge of the circular are profile.

In order to make use of the boundary equation (c9), the various functions of and appearing in equ-

tion (ala) must be expressed as functions of 4 evalu-ated at the boundary. From equations (d1) and (Al2), the boundary and its slope are now given by

23h sin2 + 8ph3 smn2 cos 2 +

dy - = -L3h cos - dx

32

NACA ARR No. L4G15

At the boundary then, with x = cos 4, when powers of h above the second are neglected,

b = 1 - (2 + 2)

= - )4!32h2 Sifl4

Then, from equations (312)

Sin = i Sfl 1 + 4 h cos

.2 2( 22 2

Cos T)= (i - 2b2sin2)

sin sin + 2 2h2 cos2)

cos = cos - 2 2h2 sin 24)

sinh2, = 422

cosh2 1 + 14.f32h2 sth2..

sir

h = 2h sin .

cosh 1 + 2ph sin

e -9 = 1 - 2h sin 4 + 222 sin h sin.

= 23li sin

'When these expressions, with equations (C7), (CS), and (010), are utilized in the boundary equation ( C9), the

following results are obtained:

NACA ARR No. LG15

2 1-p ) 4

/1 — p2

1,

Ai +2(+l)( 2 j

(cii)

2 2 1 2 ±^ 1 A

+i p2 2 2+(32 1

'3/

The value of the arbitrary constant' 2 is deter-

mined in the following way. The magnitude of the velocity, when terms containing powers of h higher than the second re neglected, is given by

q 1 + h + h2{ 2(ø1)

+

^1^2] +

or, in the variables and r,

q = 1 +

2h inh cos cosh sin cosh 2-cos 2\

2 / + 2h (sinh cos cosh sin .a ) cosh 2-cos 2\ àj

+ 2p2h2 (Cosh sin 1)

+ sjrih eQS T LJ_ \ +... (c12)

(cosh 2- cos 2)2 \ afl,)

From equations (B9) and (do),

0i =(e_ 2 sin 2_2)

and

02 - F 1 cos •r + F cos + (c13)

NACA ARR No. L4cT15

o - H H a) r 0

4) 0 C4-1

4)

00 co I 10 __A._ 0

0 - • r-4 -P 0 rq N \/1I

+ 00 00 H C) 00 H H 0

_A_ iTh I) ç 1H

(1)a)

N -0 •

-P

C\J

rci(0

H 0 I -P-P 4) CL

W I CIH -1 0-P N

0.1 -PH -P H p-

CO EOH CD H H

O a)Q 0 O

H I

o

(J,.— Ha)H H H Q a)4

N-0 0- OEH

I ' 0

.0

-ci ON oj

1rn -i 0 a) P rH C) 'ti t')

I)-4' N 0

II H

C!)-PC) L_ I

U) H

0 II4-P-1 0-' 0 o cj Qi

H . 4' ra) C)

U)

co 0 NC)

(1) C)

CO + a)C Q) co bU -

Na)

C)-P tI ci

i 0 a) + N bQ0 4) C4.-i

ra) ,0 H CD (1L I 0 cL) N

O+ (DCi

Li 1iH H

•-_- 4' C) çj - /' U C) CO o- ra) • H N C) U) H H C) H + H

V)CL ,I'

H 0 H

H bf)

0) (1) ID

'H Cli

Cl)a)

coH

C) .Li ' Lr

ci 0 -p C)

(DO -P 0 +

CH 0 N -P ,00

(1) ICD a) oc

,- .C!1 0 bfl

O '-• C.H H r-4 0 'H 'H

ON S- r-1 N - +)

H 10 Id r.

NI--

4-:) 0

a)

ra) a)

C)

H

CC 0 • l - —1 4

ci CC H '-4 0 4i H -

N ç- (31 c- N C4-1 -PC)

C) (31

H 4

0 0 H CCC1

0 H

'r C -

•P:

,1i H

0•H ZO .P

• -P

ci

N 'H

a) (C 0C a) cC r

4'\itI C)

+ c-- o) 0 0 CIO (31 p (. 4' H Cj1

C) C) C Cl) Cl)

H -% -r1 CC C

C

(3 :)

\lI

E±4 >e -P H H C -i __ 0 CD CD

cC

--- ç

H

C1 -

0 C) 0 El

C) H

-3 +- C) -P

II C) C

0 U

C)

C)

C

H

C

0 0 'H

Ci)O

0 -1 C) 0 'H 0 01 0 4D

NACA ARR No. L14G15

35

For the position of maximum velocity, 4 = TT

\ 2 1 2 1+

qc I J

(l+2h)2

Tr For the position of minimum velocity, 4 = -

(cr7)

1 +h2[8+ +y(__]

qi - (1 - 2h)2

(c18)

NACA ARR No. L4G15

ci)IN

.11 El

r1'-I

N

o•

- i)H (N

E r) r4

Q) + .4) U) N

N

z•r4 H

(N o 'd N

(D H U) (1) +

N N N CI)

H -4 0 cli H

o H

a)

CI)

4:

('4.

+

N-

H

I C o

Es+

.'-1

ItTiCC\ qj

'

('4N I N

I '0 '0

a)(\J .C\J JI

a) El ('4

('4

o 4-1 H NN

C\J I N 0 N

- Nk

CD IN

,-1

H El

HO -

N N rI

,I N r4 N

(1) 4J

+•- r-.4 H

H O H r-;

N N

+N

+

a)

4-3r-4 r4

El Cl) +N o4 c

0 a)I— t C'J

NH (r. ML

N('4 .

+ N H 4.) 0

''.— N

N C!).

Na) H 0

Q) co

r4

+ > r.4

+ r.4

IN E

a) L_.J r-1('4 S.

a). 0

C.--4a) E

-+ t I)) H

ci)

C'J

+

> __

N ('4

Q . a) I.

IN

.'-. L__J N

- IN- 'I

E-4+ + +

H a) (1)

-dIN N a) C))

coEl

H

z c.i P-4 P-4

NACA ARR No. LLG15

01=-

(Z 2 /212

2 )1//2 + 2 log

+ (-Z2 -

Introduce new complex variables X and X, where

+ (z2 1/2 z 21)1/2

z Z( > (i)

= + (2 - 1 )1/2 1 = - 1)1/2

The relations between the complex variables X and and the complex variables and 1, respecti-vely, are

e t e(DLv)

Then

(D5)

Similarly, the expression for obtained from equa-

tions (c4), (c7), (c8), and (Cll2, is

(D2)

r 12 - (-2

1_ xx x'-x xx

+ E x + + ( D6)

OR

NACA ARR No. L4G15

where

2 1 P2 +2,

2 12 \\ ,J

1 - D =

22 -

and

C -D--E1

From equations (D5) and (D6) with the use of equa-tion (D), the following relations are obtained:

1

lz

_L2

ipx

-

02Z=(DE)(x21

C.I.lo x)-2D 2X2(2_1)

____6

1 + 2(C-D+5E)

1+2E x22(x2C

_i) - x2(x2i)

NACA ARR No. L4G15

2zz =8(D_E)[2-X 3 -

log Xx-

2X3( +

K(X2

-1

- i)-

1)2

+ -8-D

-T^ 7- (2 -

71 2- -

+ 2C•1

2 (X2 - 1 )

2i)

x + 8

X )2

02z8P()(21')(2]) ^X2

and expressions for the corresponding conjugate complex

quantities.

When the foregoing expressions are introduced into equation (Di), and when equations (14) are used to express the various quantities in terms of the variables and r,, the following differential equation for Ø is obtained:

40

i'UCA ARR No. LLGi5

c'J

r1

Cl r

0J0

+

a?U)

N

+ kfl '0

a)

+ Cu

'0

+

a)

Cu

+ (\J

C'J

1cu , I

AJ

+

ion I/o

U)

* tJ 0

co

8f

a) -'

.i'

çr) + (\J

-zt + '0

--1 9•4fl I Cu

a)

0

+

+.-1.

-t I

j LJ + +

cu

(I)

cj

U)

U '.' U) + ("J c'J + + C\J

iJi

co

(NU) N-

Il)

Th -'

^

(1) cij -

-

'0

a) +

+ jJ

a) -

a) + '0

C9ifl

U) C\J -1- I

-. oJ I

Cu +

)&i) Cu

LJ AUTh

+ +

co

Q

kI)

U)

+

U)

&f)

U)

+

+

U) If"

U)

Cu 0

th cx

U)

5q

a) +

U) Cu +

c\I'J

U) oJ

r1

C\JU)

0 IN

LJ +

ç N-0

cu

11)

U)

8L,\j

4J a

a) +

()

(j) cJ

Q) Cu +

a)

Ii)

+

CO a) +

'0

+

U) Cu

cJ

a) IN

+ 941) Cu

Cu

+

Cu

•r•I U)

rl cJ

a)

8J

-t

U)

j)

Il)

'-4

+

Cu

C" e cu

,---'

U) +

j) CJ

U) +

cu a)

-1

+

Co

+

0 N-

Cu

'-I

+

I +

0 U-'

+

Cu0

Co

0

'0

CuCu

U)

U)

NACA ARR No. LIG15

= 80D(D +1) + 8D2 (y + 1)(6D + 1) + +

A6 2 = -192 D2 - 2 PD3 ( + 1) + 12 ( + 1 ) 2

-6oD ( + 1) - 15 (y + 1)2

A 102 = 96D4 + 8D3 (y + 1) + 6D(y +

(y + )2 (15D + 8) + D2 (y + 1)(7D + 9)

ALL = 96 PD2 - i6 D3 (y + 1) - 10 p + 1)2

A6 = (y + 1) - 8D(y + 1)2

A8 = -221D2 - 16D3 (y + 1) + 22D (y +

Aio -8D (y + 1) - 21D(y + 1)2

= 128pD2 + 64D (y + 1) + 8&t (y + 1)2

B22 _ 8pD2 [( + 1) D +

= -12D ( + 1) ( y + 1) D +

= 16pD2 +1) D +

B2 12 D3 (y + 1) y + 1) D +

= -16D2 [(y + 1) D +

B6 L -20D(y + 1) [y + 1)D +

B8 = 2LD2 I(y + 1) c +

I'UACA ARR Nc'. L4G15 42

C 1 = +

C2 = 8pD2 r(y +

Di = +

= D3 ( + I

1) D + 211 + 1) D + 4]

1) D +

1) D + 21

)LY + 1)D +

8D

= 2D3 (y + 1)

Note that

B62 = _2322 -Bj = - B 2Al2 =

and

B2 _B2B6AloAB2

The right-hand side of equation (D7) suggests a solution of the form

co

sin 2,n+ G2( sin L+ G() sin 2n (D8)

n=

When this expression for 03 is inserted in the left-hand side of equation (D7) arid the coefficients of sin 2, sin L1 r, end sin 2n O are equated to zero, the following differential equations for G 1 (), G2(),

and G() result:

-= (A22 + + + + (A62 + B62)e6

+ A62e + A102e_10 (D9)

NACA ARR No. L4G15

0 4J)

P a) . p

kfl

I

P I

P '-U a)

+

Ui + Ui I +

-zr sJ J)p a' Ui

p b(1) Ui

+ Ui41)

0 I Ui

+

Ui ,'. _t P + Ui

+D 1J

+ 0I

+ + (74 P

C.) r-4 I UI C C)

'--'

I

r-;m-iUI

41)

----+

+

+

+

-zt+

UI.. I C)- P ,.-.'.' 0 + I

41) Ui

++ .+ ri

(;-4 + r4 "..-' Ui UII

C.)

+'-/ Ui

+

i0) +

-

CQ+

r-4 41) a)

N- + P:)

Ji

• I (\l ('I a) I Ui p

UI

+kfl

r4 P - + P

P +

e' p4 UI L—I

c\i L__J + I ::r

+I

('I

'.,J) + c +

_______ ________________ jI

P+

UI + 01 -z1

41) (\JP '.0 Ui

0 .-4 -'. Ui C.)

co P UI

"CliP +

P + ++

•U)

,-, J)

+

0

p I

+-i

.-4 p4 '.0

+

,- P

4 p4 Ui

+ rl

-Ui

-1 •

UI I_ Li 'j i_Li j

+a) () + + + + I + +

('I + II

LI UI c'_JUI

I I

('I kn

Ld

U, 0 -I .4.) 0 a)

C'-4 0 U) 0

4.) 0 -4 0 (4) a)

C-

NACA ARR No. r4c-15

(\J H H

,J) a) &fl

a) aD

a)

N IN

+ lo&j)

IrC\10

HJ) 0

J)+

a) N

J)

+ I

CD

a) &fl '-0 0 H

N Ia)

N H1 H! IOD I I+ +

+

&fl

kO

Nc't

I '.0J1

CC

N I +

CD Ia) ci)

9J N\0

N X4 -J CID

NN H

'.0

+

++

J) J1N

+

9J)

+

.-t'J)

NN

.d I

N I HIN

COci)N

CD ci)HH ci)

9J) H IGO

Q14 N, ,+

+ +

. +

H a) H

9J)

0 ti) J)

N H N I 4

H (\J I '.0 CD I ci) a)

N N N0

ci) + CDfl N

H

NH N C

H r-1 '.D HH r-1 1c0 HCC rH1N

H I + + I II

+ + I II

I

iI- 9j)

-

ti

r1 N cci

NACA ARR No. LIG15

45

kn '0

(\J I -zr

kf) r1

ci) ti)

I N i

(\Jci) r4

H +

- H

N I

N

4J)+ N

I - zJ(11

N +

9J) H

^Zn • N + N + N Qa.

CO

0 N

• H + C H H

+ H •1 i+ +

H

N - H

+ + N >-

+ + r"I

+ H L-±+

J)d CD

rc I

I '-1

ci ci

N

ci ci9J)

-±- kfl -4 H

fl N + + + HIN N

I rld CO NI0 N

-.N

— H

+ fl

ci)

N0 +I

(1) c(J)

____

1C'J H H

N N INI

•0.. + + > _

N -

H

+

D

cci .-

ciI

(D'-0 H N

'-0 H

H,i -zi -

+ CD ^+

cci ci

Ztcci

-Zt H + H H >- 0

II + + H

++

fl HN 0

N 0

Nci

ci CO

ci)+

0 N

+ ici a) +

NH 0

H 0 N °

+- -1 +

N H + H

+ N + N H + 0

H

-H+ H +

>- H Q

ci+

HLtci)

HI'4

I CCLL_-J I

(U1 r-11N a)

+ + I + - ii

NAC A ARR No. LOU

Dl - ID3 (y +1)

D2

D1+D2+D3+DPD(Y+1)+83(Y++16_(dl+c2)

Di + D2 - - D 4=

PD4 (y + 1)2+ D3 (y + 1) = - c2)

The arbitrary constants k1 , k2, and k(n 1 ) are

to be determined by the boundary condition at the sur-face (the boundary condition at infinity is taken care of . by putting equal to zero the other set of arbitary constants that normally appear in the solutions of linear second-order differential equations with constant coef-ficients). It is now anticipated that the arbitrary constants k are independent of n and equal, to k, say. Then

R

sin2sin 2n

CTccsh 2 - cos 2 n=)

- e sin Q - sin 4 1 (D15)

where

0

Cl) H

H C')

ci) 4.) 0

J.-) H

C) 0

r0)

0)

4.)

0

0 H

U) CO 'I)

ci)

ci)

C')

N

H

C!) 0 0

p4-n

(I) 0 C)

IX)

0 H 4) co

ci)

E 0

ci)

C)

NAG A APR No LLG1

L7

(\J

ci) C')

H

+

9cj) RI

ci)

H

+

+

RI

H

+.

02

Ii

±

iizt

ci) -.- 1

H

+

n cci

+

C')

H

+

i021

—I-. ci)

Ij

C')

H

+

01

+

C')

H

+

(a

LJ +

rci)

i---

+

Ci

+ C')

H

+

.4-(Li

LJ

9&J) Cci)

01

H

+

H

±

r'C\

ci Cci)

+

N

rH

±

+

N

R

H

+

H

±

>-

cci. co

+

N

H

+

(Li

+

c'&fl

02 -0 H

H

+

(Li 01

Co

+

C')

H

+

>-

-4-

cci KN L_J

H I Cci

+

4J) —4-

(D

f-i

cci '-U H

+

H +

Ci Co

+

C')

H +

r—C'J

+

C\ji

ci) kJ)

H + >-

IL)

+

C')

H

+

Li

'-U H IL)

C')

Ccl

'H C/)

C')

HIQi

H

L'!

NCA ARR No. LLG15

from equation (c3), (c6), (C7), and (do),

(2D -. Y + 1) D + + 2T)e L(

+2)3 -D+D[(Y+1)D+)4] e- Cos

(-1(y+1)D2e+i2+12D_D[(y+1)D+L1>e3) cos 3 (n17)

where

and, from equations (D8),

sin 2y

03 G() sin 2+G2 () sin L+G(\.si-i °2 -cos 2r

- e -2,,- sin 2-e sin L) +r, r, (DiS)

where G1 () and. G() are given by equations (D12)

and (Dr5), respeciveiy, and G() can be written

TT - 2(

+ K 2 - Je 2 -/ e2--2J-L )e

+ . Ke - VTK Ke8b+k (D19)

with

JPD4. (y + 1 ) 2

K D2[(y + 1)D + I2

D3 (y + i)[(y + 1)D +

-

NACA ARR No. L4G15

The arbitrary constants k k2, and k appearing in

the expressions for a1 , G2, and G, respectively, are determined by the boundary condition at the surface of the circular arc profile. The value of the arbitrary circulation r is determined by the Kutta condition at the trailing edge - that the velocity there be finite.

In order to evaluate the various terms appearing in the boundary condition, equation (c9), the following relations are necessary: From equations (All) and (Al2)

y = (2h sin 2 + 8h sin 2 cos 2 ) +

L -Lh cos . - 16ph3 cos 4 cos 2 - dx

From equation (B12)

b = sin2 - 2 (4h2 sin + 2h s in 4 C OS2 ) +

sinh 2h sin cos24 2h2(2 c032+sin)1+

cosh 1 + 2 2h2 sin 4 +

1 + 2h sin 4 + 2 2h2 sin

+ 2h sin 4 Cos 4 - p2(2s24 +sin)4 )l +

= 1 2ph sin 4 + 2p2h2 sin

- 2ph3 sin Cos 24- p2 (2 cos 2 4 + sin) + ...

= 2h sin 4 + 20 sin

- 2 (^- sin2 + 2 cos 2 + sink)] + •••

sin sin + 2 2h2 cos 2 ) +

cos i = COS (1 - 2p2h2 +

50 NACL ARR No. L4G15

When the expresSion t'or 0 given by equation (D16) is substituted into the boundary condition, equation (C9),

the coefficient of on the left-hand side is

Dky+i)D+ -+6p cos+ cos

+ DL(y+1)D++2 cos 5 (D20)

and the coefficient of h3 on the right-hand side is

'^-6-Lo D+.LD __ ^)2 (Y + 1 ) L

+2kg, k

+ (y+')cos 34

+ 3 7-D+D2(i+i)(Y1) +(^i)2+O(y+1)2

(' 2k2 + 2k\ - (D21) Cos

NACA ARR No. 14G15

In obtaining exp 'ession ( p21), the following rela- tions were utilized:

-. PD

G2(0)D22D(y+1)_PD(Y+l)

- 3D (+1)2 .j?. Cp-' +i\2 ±k2

i 2 D(y+]- +- D(y+l)-D+k

(p22)

+14D (+ 1) 2 + ?- pp4 (-+i -2k.

IdG -, = Rp2 +D2 ( y +1) - 1D) (y+I)

c /

+ . +i)+-- pD(y+l)2)4k

/ 1 --.- = U \ d

Fr 3cliatIng the. coef.Cictents of cos , cos 3, ani oo 4v al in exp:osions ( p20), and (D21), th3 I :LP)v-ng -o, for the arbitrary constants k1, k23, and k are cbained:

ci .1-

52 NLOA ARR No. LLG15

(k1_k)=[(Y4l)D+iiD- D2 + D4(y+l)

+-D)(y+l)+2D3(Y)+Y+1) (D2)

(-ki + 2k2 0 %y+1)D++]+i85D

r 2 29-- -, 2 - L D (y 1) - -* (y + 1) - LP (y+ 1)

2 0 2

-L c- - D (y + 1 ) (D2L.)

+ 2k) = + i) D + + + 0/ + 11

'n2 ( Y + 1)

-p+l)- D3(y+1)2 D(y+1)2 (D25)

Note that the arm of equations (P2)4) and (P25) yields

equation (D25 so that. these ecuationS for k1 , k2,

and k are j-irt indacendont. hence, one of the cotntS,

av k, La e rely arbitrary. Ibnill be sean in the

fo11ovin dis3usSiOfl that th e arbitrary disposal of k

is necessary in order that no infinite velocities occur

on the circular arc profile.

The velocity components along the X and Y axes

are given by

1 - (sixth , cos T1L- .. COSh, sin ,e_

aX cosh 2-cos 2eTj \> (D26)

cos)J

NACA ARR No. L4G15 53

Along the chord of the circular are profile, 0; equations (D26) therefore become

- sin)

7 (D27) a0'

V = -'---- -

sin rô

By means. of equation (D16) for 0 and the expressions for 01' arid it fo l.lows easily from equa-

tions (D27) tr?.t

+h2 i2 cos 2+Dy+l)D+.I(2 cos 2-i) p J ± G1(0) cos ?+LG2 (0) cos

• Sifl 1) L. - -

+ - G(0)[_2 cos 2 ,,j 4 cos 4-cos2 sin •q

Cos + ----- + ••• (D28) 2 siny,

v= cos r1 +pii2 (2+3) sin 2-2ph3 cosi

+ 2 +V----l-2 cos 2 - \d2 \dJ/d\J

+ 2G(0)(1+1L Cos 2r) + ... (D29)

At the trailing edg, i or sin rj 0. Hence, according to equations (D28) and (D29), an infinite velocity seems to occur there. The Kuttacondition at

I

514 ACA ARR No. L)4G15

the trailing edge, however, demands that the velocity be

finite. From equs.tioris (D22) it i t.at () = 0= o

so that the velocity component v is finite on the boundary. The velocity oomponent u can be rendered

finite by showing that the coefficiert; of in Sin fl

equation (D28) can be made to equal z.eno when r, = 0 or ir. Thus, since the constant k c)ccorrinig . 1-I n equa-tion (D15) is arbitrary, it. can be cIiosen so that G(0) = 0. Again, if the first coeff.cIent of

'-i

• - in equation (D26) van i shes io rj = ir, then, tile

sin fl circulation constant

F3 = -2G1 (0) — 14G2 ( 0 ) DO)

where G 1 (0) and G2 (0) are given y equations (D22).

The arbitrary constant k has .. )CSfl determined by the condition G(0) = 0. From equatJons (D22), therefore,

= 17 L ( +— - D5 (y + 1) + D (D31) 76

and frc'ni equations (D2) and (125), respectvely,

D 2ñ -) -=---l(y+l)D+iL ---+l2+-- fl D'+-J- 3) (y+]) T V.

+D3(y+l)+2D3(y+l)2^D(+1)2 (o32)

k2D[(+i)141

_ 2D+D2+2D2(+l)-D3(+1) 2 -

(D33)

I'. P H 133 3 L(i

p2)3

+

2)] 112

(ir2)2 (s +c@2)

+ (y+l)2

NACA ARR No. L)4015

55

Note that, had the incompressible flow past a circular arc profile been determined according to the methods of the present paper, a discussion similar to the foregoing would have been necessary, with the result that ki =

= -)4, Ic = 0, and = 0.

Substituting from equations (D21) for G 1 (0) and

G2 (0) into equation ( D3 0 ) gives

20 20 - 6 2

3 3 -- D-- D-- pD(y+l) - D(y+l)

-;

- pD(y+l)2_ p 3 (y+l) 2 (D3L)

The circulation F in the incompressible case, obtained

from equation (A5), is

F.2h

LTJa

The circulation F0 in the compressible case, inclusive of terms containing the third power of h, is obtained by adding the circulation term from equation (B9) to the value of F3 given by equatton (D3 L. ) and multiplying 2 the result by ).iiTJa. Thus, if P is replaced by -.L,

PC 2h+Hlv+ (+1)

8+sp2)

Lua P 3 p p

(' 2\ ) I ^ (y +1)2

( + p2) h (D3 5) 12

The circulation correction formula then becomes

(p36)

56 NAG A AR T Io. LLG15

The first term on the right-hand side is the familiar prandti-Glauert term so that the second term represents the first departure from the Prandtl-G1aUert rule.

The magnitude of the velocity at the surface of the

circular arc profile is calculated by the use of equa-

tions (D26). Thus

q1+h +h2() ++h)[4(

^6y

i)2

+ 602+ + .. * (D37)

ày oy oxi where, symbolically,

C) 2 (sinli cos r --cosh r sin r ax cosh 2-COS 2ri \.

2 - (cash sin —+sinh cos i

ày ' cash 2 -cc s 2r \

and the expressions for 0 1' 02'and Ø are given by

equations (B9), (D17), and (D18). When all the functions

of L and 1) are expressed as functions of at the

surface of the profile ar±d terms involving powers of h higher than the third are neglected, the expression for q

becomes

NACA ARR No L4G15

Co NC\

ç 4.) H H

• c_)

QA H

Riia)

H r-R

+

'ClH Cl)

H -.

Cli _______RI -P

0 _

(11 H CD•'

H / Ri —1 +

H C0. 0.-I 0 -P

'H

CL HO

-v -r

+ rcICli C/)

(-\J -S.-•----., ® '1)

RI -rCli

(.i

-r RIk.1-.-_- +

C) (I) Cl)

Ri -- (i) C

RI d0

co H I4) (1) Ri HI- H H

.-, () H RI RI -.- tJJ-.--I

ol NR fl ----- H

H H - I r-c'

+ ^RI

o I 0

- •-'

---.

' (/1. CO CO -I I RI(1) 0 -P C

RI RI 0 0 H ;- Ci RI H - -'- Cl

RI +Al

RI (Ili

(Dl))

RI N(-\IQi - (0 Cl) '-' H

-H 0- •HO H C) 0 Cl) '-._-." (31

RI - H -ftr\ - Cl . fl co ct

Irl H Ci

-P-r

C)

H0) H

Cl - 4-

Ri Cl u) RI I C0. ci) H -i •r1 C)

Cl l- o Cl o _:fl'il

r.:C\J Ci) (2W

0!- -- q

. I + Cl o -01 C- CH

+1-4 (3 cJ

(i '- C) ci) rH + 0 s C) H CD -

H0 Cl

-RI

Ci -PC

(157

Cl H

C')

co

RI

Cl)

0

('I

Cl

Cl)

-1-

H

C) H

58

N ACA ARR No. LL.G1.5

czr-1 j

C-) < Co 4-) Cij

E 4) C.) >

CDH 0 0 rc- CD

c-i -P >0

c. •) .H l: 0 c 4.)

VII CD CD

'C.i 0 'r.4 cz

VII CD

C) CD 4.)

0 1 CDH 0

rJ-1

CD £1) 0

,-I -0--

'ij '-0 0 II H

co +- C) 4. --

wo. I'C\ •'0

iVIICD---1 -_-

Di 1 0VII 01 H

Di I H

0' r4

Cl) ti CDCD

F4 CD-'-

o

c-i 0 (CS ç

rd LCO.-4 CD I:.4 .4 CD

1 CD • -I bU

H H -d Di

co iCD 4..) C) Cl) Cl) IC'J

IQ - r.4

II

H cI) + 01

4) C) a

01 001

0 H

(1) 0 C)

ci) • • r 0

ci) 'CI 0-P i-1 ,Cj Iri c-i CD ci)

rnCDc-i

.4.)

0

H

Cc)

C

0

4)

.1

CD 0

21

Ci)

4.)

c-I C)

cr.4

a

C'J

(D

0

(

+

LCT\

±

cx)-

LLJ

+

KN

4:1 a)

H

c'J

>-

+

+

RI

H

+

+

H C'J

+

cx)

L J

RI

+

ri CL

+

H

C) H

01 01

H

-

o

H

+

Lr\

+

(Cl

(\j

KN

CA

LJ

H

H1

-4-'

+

H

+

+

0.')

LJ....J

RI

ca

+

H

C) •r-4

0101

.41 Cr-)

+

C')

HZ-

H

NACA ARR . L1L&15

REFERENCES

1. Janzen, 0.: Beitrag zu elner Theorie der stationaren Str6mung koirrnressibler FlUssigkeiten. Phys. Zeitschr., lii Jahrg., Nr. iL, 15.July 1913, pp . 639-6L13.

2. Rayleigh, Lord: On the Flow of Com pressible Fluid Past an Obstacle. Phil. W1ag., ser. 6, vol. 32, no 187, July 1916, PP. 1-6.

3. Poggi, Lorenzo: Campo di velocit. in una corrente piana di fluidocompressibile. Parte II.- Caso del profili ottenuti con rap presentazione conforme dal cerchio ed inparticolare dci prOfili JoukOwski. L'Aerotecnica, vol. XIV, fasc. 5, May 1934, PP. 532-549.

L. Imai, Isao, and Aihara, Takasi: On the Subsonic Flow of a Compressible Fluid past an Elliptic Cylinder. Rep. No. 194 (vol. XV,8),Aero. Res. Inst., Tokyo Imperial Univ., Aug. 194O.

5. Kaplan, Carl: On the Use of Residue Theory for Treating the Subsonic Flow of a Compressible Fluid. I'ACA Rep. No. 728, 1922.

6. Prandtl, L.: Remarks on paper by A. Busemann entitled "Frofilmessungen bel Geschwindigkeiten nahe der Schallgeschwlndigkeit (im Hinblick auf Luftschrauben Jahrb. W.G.L., 1928, pp. 95_99.

7. Glauert, H.: The Effect of Compressibility on the Lift of an Aerofoil. H. & M. No. 1135, British A.R.C., 1927. (Also, Proc. Roy. Soc. (London), ser. A, vol. 118 1 no. 779, March 1, 1928, pp. 113-119).

8. Ackeret,J.: Uber Luftkrifte bel sehr grossen Geschwindigkeiten insbësondere bei ebenn Strmungen. Helvetica Physica Acta, vol. 1, fasc. 5, 1928, pp. 301-322.

9. Kaplan, Carl: The Flow of a Compressible Fluid past 3 a Curved Surface. NACA ARR No. 3K02 1 19L.

NIiCA MR No. LLG15

10. von Krnn, Tb.: Compressibility Effects in Aero-dynamics. Joir. Aero. Sc., vol. U. no. 91 July 1941 , pp. 7-356.

11. Garrick, I. E., and Kaplan, Carl: On the Flow of a Compre3sible Fluid by the Hodograph Method. I - Unification and Extension of Present-Day Results. NACA ACR No. L14.C2)4., l91.

NACA ARR No. L4G15

61

TABLE I

RATIO OF CIRCULATIONS FOR COMPRESSIBLE AND INCOMPRESSIBLE FLOWS

M1 Approxi-

mation..

0.10 0.20 0.30 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90

Qladert 1.0050 1.0206 1.0483 1.0911 1.1198 1.1547 1.19'741 1.2500 1.5159 1.4003 1.5119 1.666 1.8985 2.2942

5=0.010

Third 1.0050 von Krmn11.005011:020611.048311.091N

10206 1.0483 1.09121 1.1200 4 1199

1.1550 1.1548

1.197 1.19751

1.2508 1.2505

1.3172 1.3165

1.4029 1.4013

1.5170 1.5135

1.618 1.669

1.9348 1.9025

2.4603 2.3030

S = 0.015

Third. von Kim

1.0051 1.0050

1.0207 1.0207

1.0484 1.0484

1.09131 1.09131

1.1202 1.1202

1.1553 1.1553

1.1984! 1.19831

1.2517 1.2511

1.3189 1.3175

1.4059 1.4026

1.5234 1.5152

1.6940! 1.672(

1.9804 1.9078

2.6680 2.3142

5=0.020 - -

Third von KArtnn

1.0051 1.0051

1.0207 1.0207

1.0485 1.0485

1.0915 1.0915

1.1205 1.1205

1.' .1557

1.1992 1.1987 1.155; 1 1:1987 1 1.2520 1 1.3187 1 1:

41021.2530 1.2520

1.3212 1.3187

1.4102 1.4043

1.5323 1.5179

1.71531 1.6763!

2.6441 1.9153

2.9588 2.3301

S = 0.025

Third von Krmn

1.0051 1.0051

1.0207 1.020?

1.0486 1.0488

1.0918 1.0916

1.1209 1.3206

1.1565 1.1560

1.2002 1.1996

1.254? 1.2531

1.5242 1.5205

1.4158 1.4066

1.5438 1.5213

1.7427! 1.68]i

2.1262 1.9250

-

h. 0.050

Third von Krn

1.0061 1.0051

1.0206 1.0208

10487 1:0486 1 1.092011.2212

1.0921 1.1214 1.1572 1.1570 1.2006

1.2015.1.2568 1.2544

1.5278 1.5222

1.4226 1.4094

15578 1:5256 1 1.68

1.776 85!

2.2264 1.9370

-

5 = 0.035

Third von RArTnn

1.0051 1.0051

1.0208 1.0208

1.0488 1.0486

1.0925 1.0922

1.1220 1.1215

1. 1.1575 1.

1581 1.2030 2017

1.2593 1.2560

1.5321 1.3245

1.4307 1.4127

1.5744 1.5306

1.8157 1.6966

2.3449 1.9514

-

S = 0.040

ThIrdrm von KLn

10051 1 1:0051

1.0209 1.0209

1.0490 1.0488

1.0929 1.0928

1.1226 1.3220

1.1592 1.1588

1.2047 1.2031

1.2621 1.2579

1.5371 1.5271

1.4400 1.4166

15936 1. . 5364

1.86131.7060- -

5=0.045

Third von Krmn 1.0051

1.00511.0210 1.0210

1.0492 1.0490 1.0930

1.09341.32341.1604 1.3228 1.1599

1.2066 1.2048

1.2653 1.2600

1.3427 1.5301

1.4505 1.4209

1.6153 1.5430

1.91301 1.7161

-------

h 0.050

Third von K(rnin

1.0051 1.0051

1.0210 1.0210

1.049 1.0492

1.0939 1.0936

1.1242 1.1238

1.1617 1.1611

1.2087 1.2083

1.2689 12624

1.5490 1.3355

1.4625 1.4258

1.6396 1.5505

1.97081.7290- -

h0.060

Third von Krii.n

1.0052 1.0052

1.0212 1.0210

1.0499 1.0496

1.0952 1.0950

1.1282 1.1260

1.1648 1.1640

1.2137 1.2102

1.2775 1.2699

1.3636 1.3413

1.4605 1.4373

1.69581.5681- -

S = 0.070

Third von Krinn

1.0052 1.0052

1.0214 1.0212

1.0505 1.0500

1.0967 1.0960

1.1285 1.1281

1.1685 1.1693

1.2196 1.2148

1.2871 1.2744

1.3808 1.3507

1.5218 1.4511

1.76221.5895- -

5=0.080

- Third von KLrmn

1.0053 1.005?

1.0217 1.0215

1.0512 1.0510

1.0984 1.0976

1.1312 1.1289

1.1727 1.1711

1.2265 1.2202

1.2985 1.2820

1.4007 1.3618

1.5589 1.4673- I

5=0.090

Third von Krmn

1.0053 1.0052

1.0219 1.0217

1.0520 1.0515

1.1003 1.1001

1.1342 1.1540

1.1775 1.1755

1.2342 1.2263

1.3144 1.2907

1.42321.3741

F1161- -

5=0.100

Third von Arvin

1.0054 1.0053

1.022211 1.0220

.0528 1.0526

1.1025 1.1020

1.1576 1.1570

1.1828 1.1804

1.2428 1.2332

1.3258 1.3004

14484 1.3883

1.6482 1.5078-

NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

NACA ARR No. L4G15 , 62

TABLE II RATIO OF VELOCITIES AT TR ADING OR TRAILING EDGE FOR COMPRESSIBLE AND INCOMPRESSIBLE FLORS

\ ml Approxi-mation

0.10 0.20 0.30 0.40 0.43 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95

0.010; Qiey • 1/1.0004 -

First Third

1.0000 1.0000

1.0000 1.0000

1.0000 1.0000

1.000010 .9999

.9999

.90900.9999 1 0

.9998.9999 .9998

0.9999 .9997

0.9999 .9996

0.9998 .9994

0.9998 .9991

0.9997 .9965

0.9996 .9973

0.9995 .9939

0.9991 .9757

h • 0.015; 91exect 14.0009

First Third

1.0000 1.0000

1.0000 1.0000

1.0000 .9999

0.9999 .9998

0.9999 .999?

0.9999 .9996

0.9998 .9995 .9995

0.990810.9997 1 0 .0991

.9996

.99860.9995

.99800.9994

.99670.9992

.99400 9988

98630.9980

.9454

h 0.020; (q1) ex.c t 1/1.0016

First Third

1.0000 1.0000

1.0000 .9999

0.9999 .9998

0.9999 .9997

0.9998 .9995

0.9998 .9994

0.9997 .9991

0.9996 .9088

0.0995 .9083

0.9994 .9976

0.9992 .9964

0.9989 .9941

0.9986 .9893

0.9979 .9757

0.9963 .9028

h • 0.025; • 1/1.0025

First Third 1.0000 .9999 .9997

1.00001.00000.99990.99980.99970.9996 .9995 .9993 .9990

0.0993 .9986

0.9994 .9981

0.9992 1 0 .9074

.9990

.99620.9987

.99430.9983

.99080.9978

.98330.9968

.96200.9945

.8480

A 0.030; • 1/1.0036

First Third 1.0000 .9999 .9996 .9992

1.00000.99990.99980.99970.99960.99940.99030.90910.9969 .0900 .9986 .9980 .9975 .9962

0.9986 .9945 .

0.9982 .9910

0.9976 .9867

0.9068 .9759

0.9953 .9452

0.9921 .7808

h • 0.035; (qi) 50 1/1.0049

Fir it Third

1.0000 .0999

0.9999 .9998

0.9998 .9995

0.9996 1 0 .9990

9994 .9086

0.9992 .9980

0.9990 .9973

0.9988 .9063

0.9985 .9948

0.9980 .9925 0.99751 0.9967

.O$88j .98190.9956

.96720.9937

.92530.9892

.7013

h • 0.040; Oexact 1/1.0064

First Third

1.0000 .9990

0.9999 .9997

0.9997 .9993

0.9994 .9986

0.0902 .9081

0.9990 .9974

0.9987 .9965

0.9084 .9952 .9932

0.9980 1 0:9974 1 09903

.9967 .9853

0.9957 .9763

0.9943 .0570

0.9917 .9023

0.9859 .6092

S a 0.045; • 1/1.0081

First Third

1.0000 .9999

0.9998 .9997

0.9996 .9991

0.9993 .9985

0.9990 .9976

0.9988 .9967

0.9984 .9955

0.9980 .9939

0.0974 .9914

0.9968 .9876

0.9959 .9814

0.9946 .9700

0.9927 .9455

0.9895 .8761

0.9822 .5046

S = 0.050; • 1/1.01

First Third .9999j 1.000010.9998 .9996

0.9995 .9989

0.9991 .9979 1 0:9.70'3 1

0.9983 .9960

0.9980 .9945

0.9075 .9924

0.9968 .9894

0.9960 .9847

0.9949 .9770

0.9933 .9629

0.9910 .9326

0.9871 .8468

0.9780 .3872

S = 0.060; (q)exact = 1/1.0144 -

FirstThird

6.9999 .0999

0.9997 .9994

0.9993 .9985

0.9987 .9960

0.9083 .9957

0.9978 1 0 .9942

.9972

.99200.9964

.98900.9955

.98460.9942

.97790.9926 1 0

.9667.9904 .9463

0.9871 .9025

0.9814 .7784

0.0683 .1137

S = 0.070; (i)5= = 1/1.0196

FirstThird

b.0000 .9998

0.9996 1 0 .9991

.9991

.99790.9982

.9958 .9042 .9920 0.9277 1 0.9970 1 0.9961 1 0:9951 1 0:9938 1 0

.9891 9850 9790 .9922

.96970.9900

.95450.9869

.92660.9824

.86660.9746

.69680.9568 -

S - 0.080; (i)5 a 1/1.0256

First Third

0.9999 .9097

0.9995 .9989

0.9938 .9973

0.9977 .9944

0.0969 .0923 .9895 .9856 .9802 .9723 .9602

0.99600.99800.99360.99190.98980.98690.98290.97700.966909436 .9402 .9055 .8247 .6015 1- :----

S • 0.090 (i)00 1/1.0324 -

FirstThird

0.0998 .9997

0.9993 .9986

0.9984 .9965

0.9971 .9929

0.0961 .9902

0.9950 .9866

0.9936 .9817 0.991010.9898 .974B .9648

0.9870 .9493

0.9834 .9237

0.0784 .8770

0.9709 .7766

0.9581 .4921

0.9286

S • 0.100; (q ) = 1/1.04 - exact - First T)ird

0.9998 .9996

0.9992 .9982

0.9901 .9956

0.9964 0.9052 .9912 .9878

0.9930 .0633

0.9921 .9772

0.9900 .9887 .9561

0.9874 1 0 : 9840 1 0 : 9799369 90511

510.9733 .84690.9841

.72210.9482

.36600.9119

NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

NACA 'ARR No. L4G15

63

TABLE III

RATIO OF MAXIMUM VELOCITIES FOR COlIPRESSIBLE AND INCOMPRESSIBLE FLOES

4max/rnaxj

1 0.10 0.20 0.30 0.40 0.45 0.10 0.55 0.60 0.65 0.70 0.75 0.60 0.82 0.90 0.95 Appr3i- satiNri

h 0.010; (qj) 1 = 1.04; (41)3 = 1.0404; (q1) 3 1.0404; (4i)exact 1.0404

First 1.0002 1.0008 1.0019 1.0035 1.004611.0060 1.0076 1.0096 1 1.0122 1.015411.019711.025611.0346 1.0498 1.0847 Second 1.0002 1.0008 1.0020 1.003? 1.004911.0064 1.0082 1.0105 11.0134 1.017111.022311.029911.0422 1.0673 1.1547 Third 1.0002 1.0006 1.0020 1.0038 1.0050[1.0064 1.0083 1.0105 1.0135 1.0171L1.0227 1 j 1.0446_1.0779 1.2529

h = 0.015; 1.08; (q j ) 2 1.0609; )qj) 3 = 1.0609;(qi)exact 1.0609

First 1.0003 1.0012 1.0027 1.O0521.0066 1.0088 1.0112 1.0142 1.0179 1.0227 1.0290 1.0377 1.0509 1.0733 1.1247 Second 1.0003 1.0013 1.0030 1.00571.0075 1.0098 1.0126 1.0161 1.0205 1.0265 1.0347 1.0471 1.0678 1.1118 1.2790 Third 1.0003 1.0013 1.0030 1.005?jl.00?6 1.0098 1.0127 1.0162 1.0209 1.0271 1.0359 1.0498 1.0758 1.1469 1.7034

h 0.020; (q1) 1 1.08; = 1.0816; (4i) = 1.0815; ( i ) C = 1.0815

Flret I1.000411.0015 1.0036 1.0068 1.0089 1.0115 1.0146 1.0185 1.0234 1.0297 1.0379 1.0494 1.0665 1.0957 1.1631 Second 11.000411.0017 1.0040 1.0077 1.0102 1.0132 1.0170 1.0218 1.0280 1.0363 1.0480 1.0656 1.0981 1.1630 1.4321 Third _1 04[0O17_1.0041 1.007? 1.0103 1.0134 1.0173 1.0222 1.0288 1.0377 1.0507 1.0720 1.1146 1.2448 2.4191

h = 0.025; (qi) = 1.10; 4i2 1.1025; lq03 1.1024;(q,)exact 1.1024

Fire 1.0005 1.0019 .0083 1.0109 1.0141 1.0179 1.0227 1.0287 1.0364 1.0465 1.0606 1.081? 1.1177 1.2002 Second 1.0005 1.0022

1.004411 1.0051 1.0097 1.0129 1.01168 1.0216 1.0276 1.0358 1.0466 1.0619 1.0855 1.1269 1.2206 1.5308

Third 1.0005 1.0022 1.0052 1.0098 1.0130 1.0171 1.0221 1.0286 1.0372 1.0492 1.0671 1.0976 1.1624 1.3772 3.4221

h = 0.030; = 1.12; (q j ) 2 = 1.1236; (qj) 3 = 1.1234; 'exact = 1.1234

First 1.0005 1.0022 1.0052 1.0098 1.0128 1.0166 1.0212 1.0268 1.0339 1.0429 1.0548 1.0714 1.0963 1.1387 1.2360 Second 1.0006 1.0026 1.0062 1.0118 1.0156 1.0204 1.0263 1.0339 1.0439 1.0573 1.0766 1.1065 1.1601 1.2840 1.8183 Third 1.0006 1.0026 1.0063 1.0120 1.0159 1.0209 1.0272 1.0353 1.0462 1.0616 1.0854 1.1271 1.2203 1.5496 5.0253

S = 0.035;(q) 1 1.24; 1.1449; (45) 3 = 1.1446; 1.1446

Firet 1.0006 1.0025 1.0059 1.0112 1.0147 1.0190 1.0242 1.0307 1.0388 1.0492 1.0629 9 1.1103 1.1589 1.2705 Second 1.0007 1.0031 1.0073 1.0139 1.0184 1.0241 1.0312 1.0402 1.0522 1.0684 1.0918 F.12.7 1.1955 1.3530 2.0483 Third 1.0008 1.0031 1.0074 1.0142 1.0169 1.0248 1.0324 1.0423 1.0558 1.0752 1.1056 8 1.2894 1.7670 7.0467

- h = 0.040; (q j ) 1 = 1.16; (41)2 1.1664; (q 5 ) 3 = 1.1659;(q i ) exaet = 1.1659

Firet 1.000? 1.0028 1.0067 1.0126 1.0165 1.0213 1.0272 1.0345 1.0436 1.0552 1.0706 1.0920 1.1239 1.1785 1.I038 Second 1.0009 1.0035 1.0084 1.0160 1.0213 1.0278 1.0361 1.046? 1.0607 1.0798 1.1077 1.1519 1.2330 1.4272 2.3007 Third 1.0009 1.0036 1.0085 1.0165 1.0220 1.0290 1.0379 1.0498 1.0660 1.0898 1.1279 1.1990 1.3706 2.0339 9.6255

h = 0.045; (q 1 ) 1 C 1.18; (q j ) 2 c 1.1881; (qi)3 = 1.1874; 1.1874

First 1.0008 1.0032 1.0074 1.0139 1.0183 1.0236 1.0301 1.0381 1.0482 1.0611 1.0781 1.1017 1.1370 1.1974 1.3360 Second 1.0010 1.0040 1.0095 1.0182 1.0241 1.0316 1.0411 1.0533 1.0694 1.0916 1.1241 1.1762 1.2725 1.1063 2.5745 Third 1.0010 1.0041 1.0097 1.0188 1.0252 1.0332 1.0437 1.0576 1.0769 1.1055 1.1523 1.2420 1.4649 2.3545 12.8152

h 0.050; (q 1 ) 1 C 1.20; (q 1 ) 2 = 1.2100; (q j ) 3 = 1.2090; = 1.2089

First 1.0008 1.0Ô34 1.0081 1.0152 1.0200 1.0258 1.0329 1.0417 1.0527 1.0667 1.0853 1.1111 1.1497 1.2157 1.3671 Second 1.0011 1.0045 1.0106 1.0203 1.0270 1.0355 1.0462 1.0600 1.0783 1.1037 1.1411 1.2013 1.3139 1.5900 2.8682 Third 1.0011 1.0046 1.0109 1.0212 1.0284 1.0377 1.0497 1.0658 1.0884 1.1224 1.1791 1.2900 1.5731 2.7327 6.6647

S = 0.060; ( q i ) 1 = 1.24; 1.2544; (q 1 ) 3 5 1.2527; iexact 1.2525

Firs t 1.0010 1.0040 1.0094 1.0176 1.0232 1.0299 1.0382 1.0484 1.0611 1.0775 1.0991 1.1290 1.1739 1.2505 1.4263 Second 1.0013 1.0054 1.0129 1.0247 1.0330 1.0434 1.0566 1.0737 1.0967 1.1288 1.1764 1.2541 1.4016 1.7700 3.5106 Third 1.0014 1.0056 1.0135 1.0262 1.0353 1.0470 1.0625 1.0835 1.1136 1.1599 1.2396 1.4021 1.8341 3.676126.5210

- S = 0.070; (i)1 = 1.28; (45) 3 = 1.2696; (q 5 ) 3 1.2969; (q 0exact 1.2965

First 1.0011 1.0045 1.0106 1.0199 1.0262 1.0338 1.0432 1.0547 1.0691 1.0876 1.1120 1.1458 1.1965 1.281111.4618 Second 1.0016 1.0064 1.0151 1.0292 1.0391 1.0514 1.0673 1.0879 1.1157 1.1548 1.2133 1.3099 1.4954 1.965114.2193 Third 1.0016 1.0067 1.0161 1.031511 0426 1.0570 1.0763 1.1028 1.1416 1.2027 1.3106 1.5371 2.1589 4.8892139.5150

6 0.080; (q 5 ) 1 = 1.32; (qj)2 1.3456; (qj) 3 =1.3415; iexact = 1.3409

First 1.0012 1.0050 1.0117 1.0221 1.0290 1.0375 1.0479 1.0606 1.0766 1.0970 1.1241 1.1616 1.2178 1.2137 1.5339 Second 1.0018 1.cc073 1.0175 1.0338 1.0452 1.0596 1.0761 1.1024 1.1352 1.1616 1.2517 1.3682 1.5944 2.1735 4.9863 Third 1.0019 1.0078 1.0188 1.0371 1.0504 1.0678 1.0912 1.1240 1.1776 1.2508 1.3923 1.6963 2.5522 6.393755.9220

h = 0.090; (q 1 ) 1 = 1.36; (q 1 ) 2 1.3924; (45) 3 = 1.3866; (q i) 5 1.3857 -

First 1.0013 1.0055 1.0126 1:0241 1.031? 1.0409 1.0522 1.0662 1.0836 1.1060 1.1355 1.1765 1.2378 1.3426 1.5830 Second 1.0020 1.0083 1.0198 1.0384 1.0514 1.0679 1.0892 1.1172 1.1552 1.2092 1.2913 1.4288 1.6979 2.3934 5.6044 Third 1.00221.0090 1.0217 1.0430 1.0586 1.0792 1.1072 1.1470 1.2068 1.3047 1.4851 1.8810 3.0178 8.207975.974C

5 0.100; (4i)i 1.40; (q1) 2 1.4400; (qj) 3 1.4320; 4 i 1.4307 exact

First 1.0014'l.00Sg 1.0138 1.0260 1.034211 .0442 1.0564 1.0714 1.0903 1.1144 1.1462 1.1905 1.2567 1.3697 1.6293 Second 1.0022:1.0092 1.0221 1.0430 1.0577 1.0763 1.1004 1.1221 1.1755 1.2374 1.3320 1.4914 1.8054 2.6235 6.6672 Third i1.00241.0102 1.0247 1.0491 1.0672 1.09131.1244 1.1718 1.2442 1.3643 1.5696 2.0923 3.559110.348099.8730

NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

NACA ARR No. L4G15

64

TABLE IV

RATIO OF MINIMUM VELOCITIES FOR COMPRESSIBLE AND INCOMPRESSIBLE FLOWS

I-

Axi- 811010 0.20 0.30 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.71 0.80 0.85 0.90 0.95 mation -

h = 0.010; (q 1 ) 1 • 0.96; (q j ) 2 = 0.9604; (q 1 ) 3 r 0.9604; iexact = 0.9604

First 0.9998 0.9991 0.9980 0.9962 0.9950 0.9936 0.9918 0.9898 0.9868 0.9832 0.9787 0.9722 0.9826 0.9461 0.9082 Second .9998 .9992 .9981 .9965 .9954 .9941 .9925 .9904 .9882 .9852 .9815 .9768 .9709 .9651 .9841 Third .9998 .9992 .9981 .9985 .9954 .9940 .9924 .9904 .9881 .9850 .9812 .9759 .9683 .9535 .8451

h = 0.015; (g j ) 1 • 0.94; (q1) 2 0.9409; (qj ) 3 = 0.9409; 0exact = 0.9409

First 0.9997 0.9987 09969 0.9942 0.9924 0.9901 0.9874 0.9840 0.9798 0.9745 0.9673 0.9575 0.9427 0.9174 0.8594 Second .9997 .9988 .9972 .9948 .9932 .9913 .9900 .9860 .9829 .9788 .9739 .9680 .9619 .9605 1.0336 Third .9997 .9988 .9972 .9948 .9932 .9912 .9889 .9858 .9825 .9780 .9726 .9650 .9529 .9214 .5551

= 0.020; (q j ) 1 = 0.921 (q j ) 2 = 0.9216; (qj ) 3 = 0.9217; = 0.9217

First .9982 0.9958 0.9921 0.9896 .9828 0.9783 0.9725 0.9652 0.9555 0.9420 0.9219 0.8875 0.8085 Second

0.9996 1 0 .9996 .9984 .9964 .9952 911

0.9866 1 0 .9187 .9857 .9818 .9781 .9731 .9674 .9613 .9568 .9667

Third .9996 .9984 .9963 .9931. 99910 . .9885 .9854 .9815 .9772 .9716 .9643 .9538 .9350 .8708

-

h 0.025; • 0.90; (q 1 ) 2 5 0.9025; = 0.9026; (i) 5 = 0.9026

First 0.9994 0.9977 0.9946 0.9899 0.988? 0.9828 0.9781 0.9722 0.9649 0.9555 0.9431 0.9259 0.9002 0.8562 0.7553 Second .9995 .9981 .9955 .9917 .9892 .9862 .9827 .9779 .9738 .9682 .9622 .9567 .9559 .9827 Third .9995 .9981 .9955 .9915 .9889 .9858 .9821 .9769 .9721 .9651 .9559 .9418 .9126 .7914

- -

h = 0.030; (qi)1 • 0.881 (qi) 2 = 0.8836; (qj ) = 0.8838; = 0.8838 3

First 0.9993 0.99720.99340.98760.98370.97890.97310.96590.95890.94540.9302 0.9091 0.8775 0.8235 0.6997 Second .9994 .9977 .9947 .9902 .9875 .9839 .9799 .9142 .9700 .9642 .9583 .9544 .9595 1.0096 Third .9994 .9977 .9946 .9900 .9869 .9833 .9789 .9725 .9670 .9587 .9472 .9282 .8830 .6721

h = 0.035; • 0.86; (qj) 2 0.8649; (qj) 3 = 0.8652; 0.8652 -

First 0.9992 0.9966 0.9921 0.9852 0.9805 0.9748 0.9679 0.9593 0.9486 0.9348 0.9167 0.8915 0.8538 0.78930.6415 Second .9994 .9974 .9940 .9889 .9856 .9818 .9773 .9709 .9668 .9610 .9559 .9545 .9680 1.0483 Third .9994 .9973 .9938 .9885 .9849 .9607 .9757 .9681 .9620 .9521 .9377 .9121 .8439 .5008

- -

h 0.040; (qj) 1 0.84; (qi) 2 0.8464; (q0 3 = 0.8469; (i) 5 = 0.8469

First 0.9990 0.9961 0.9908 0.9827 0.9772 0.9705 96 0 24 0.24.95.9398.92380 0 0 0 535 0.5805 Second .9993 .9971 .9933 .9876 .9840 .9798 .9751 .9679 .9642 .9687 .9549

.9025.8^0:1211 .9994

.9993 .9970 .9930 .9870 .9830 .9783 8Third .9726 .9637 .9569 .9451 .9273 .645--

S = 0.045; (q j ) 1 0.821 (qj) 2 = 0.8281; (q j ) 3 0.8288; (qi) exact = 0.8288

First 0.9989 0.9955 0.9894 0.9800 0.9737 0.9660 0.9567 0.9451 0.9307 0.9121 0.8876 0.8537 0.8028 0.7159 0.5165 Second .9992 .9968 .9926 .9864 .9826 .9781 .9731 .9653 .9622 .9574 .9556 .9629 1.0005Third .9992 .9967 .9922 .9855 .9811 .9758 .9695 .9592 .9516 .9377 .9154 .8688 .7251

- -

5 0.050; (qj) 1 0.801 (q j ) 2 = 0.8100; 5 0.8110; = 0.8109

First 0.9987 0.9948 0.879 0.9772 0.9701 0.9613 0.9507 0.9375 0.9210 0.8999 0.8720 0.8333 0.7754 0.6765 0.4494 Second .9991 .9965 .9920 .9854 .9813 .9766 .9715 .9631 .9610 .9573 .9580 .9715 1.02 53Third .9991 .9963 .9914 .9841 .9195 .9734 .9664 .9545 .9481 .9296 .9016 .8395 .6392

- -

h = 0.060; (qi)1 0.76; (qi) 2 = 0.7744; (qj) 3 = 0.7761; (qi) 5 = 0.7759

First 0.9984 0.9935 0.9848 0.9712 0.9822 0.9512 0.9377 0.9211 736 0.8384 0.7895 0.7163 0.5913 0.3045 Second .9990 .9960 .9909 .9836 .9792 .9744 .9693 .9645

1 0:9002 9608

01 :89604 .9884 .9984 1.0937

Third .9989 .9957 .9900 .9813 .9756 .9686 .9601 .9491 .9339 .9105 .8866 .7600 .3964- -

h = 0.070; (qj) 1 0.72; (q5) 2 = 0.7396; (qj)3 = 0.7423; ( qi)ea = 0.7419

First 0.9980 0.9920 0.9812 0.9646 0.9534 0.9398 0.9232 0.9028 0.8772 0.8443 0.8010 0.7407 0.6507 0.4967 0.1435 Second .9989 .9956 .9901 .9824 .9779 .9732 .9687 .9652 .9641 .9689 .9873 1.0397 1.1903Third .9988 .9951 .9885 .9786 .9719 .9637 .9534 .9398 .9195 .8859 .8181 .6438 .0322

- -

S = 0.080; (q;) 1 = 0.68; ( qi) 2 = 0.7056; (q 1 ) 3 0.7097; texact = 0.7090

First 0.9976 0.9903 0.9773 0.9571 0.9436 0.9272 0.9071 0.8824 0.8513 0.8116 0.0365 Second .9988 .9953 .9895 .9818 .9775 .9734 .9700 .9685 .9713 .9833

0.7591 1 0:9761.0158 10

686310.577310.3910

Third .9986 .9945 .9871 .9758 .9681 .9585 .9459 .9285 .9016 .8537 .7515 .4790-

h 0.090; (q,) 1 • 0.64; (q 02 = 0.6724; (qj) 3 = 0.6782; ) exact = 0.6771

First 0.9972 0.9884 0.9728 0.9488 0.9326 0.9130 0.8890 0.8594 0.8223 0.7748 0.7121 0.6250 0.4947 0.2720 0.2389 Second .9988 .9951i .9893 .9819 .9782 .9750 .9733 .9749 .9831 1.0046 1.0552 1.1742

- Third .9985 .9939 .9857 .9730 .9642 .9528 .9375 .9152 .8790 .8113 .6611 .2520--

-

S = 1.000; 0.60; (qi)2 0.6400; (q1 ) 3 = 0.6480; ( q i ) exact = 0.6462 -

First 0.9966 0.9863 0.9678 0.9393 0.9202 0.8969 0.8684 0.8333 0.7894 0.7331 0.6588 0.5555 0.4011 0.1372 0.4684 Second .9988 .9951 .9895 .9829 .9800 .9783 .9791 .9848 1.0000 1.0338 1.1071 Third .9984 .9933 .9843 .9700 .9599 .9464 .9296 .8990 .8604 .7559 .5407

I

NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

NACA ARR No. L4G1'5 65

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NACA ARR No. L4G15 66 I

TABLE VI

VALUES OF CRITICAL STREAM MACH NUMBER FOR VARIOUS

VALUES OF CAMBER COEFFICIENT

- M1or

Approximation h

First Second Third

0 .02 0.848 0.832 0.825

.04 .770 .7146 .738

.o6 - .716 .682 .672

.08 .670 .628 .620

.10 .625 .585 574

TABLE VII

VALUES OF MAXIMUM VELOCITY FOR CORRESPONDING BUMP

AND CIRCULAR ARC PROFILE

qa

M Camber coefficient h • Thickness coefficient t

0.02 1 0 - 04 o.o6 0.08 0.10 0.052 0.100 0.1)4.5 0.186 0.226

0 1.0815 1.1659 1.2527 1.3415 1. 4520 1.0816 i.i66o 1.2527 1.3414 1.4320

.2 1.0834 1.1701 1.2597 1.3520 1.L466 1.0834 1.1701 1.2595 1.3513 1.141454

.3 1.0859 1.1759 1.2695 1.3668 1.14673 1.0859 1.1757 1.2689 1.3651 1.4641

.4 1.0899 1.1851 1.2855 1.3913 1.5024 1.0900 1.18147 1. 28140 1.3876 1.4950

.5 1.0960 1.1997 1.3116.1.4324 1.5627 1.0959 1.1988 1.3084 1.1.245 1.5467

.6 1.1056 1.2239 1.3572 1.5078 1.6780 1.1052 1.2217 1.3492 1.4879 1.6373

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