-- N A S A T E C H N I C A L N A S A T T F - 7 0 2 T R A N S L A T I O N el f

m0 = IN r l

0 6 h a 7

I U 81

41

I- Fi+ 0--5

4 LOAN COPY RETURN TO’ - P i -c/I AFWL (DOGL) -s ,

4 I

z KIRTLAND AFB, N.M,

THE CALCULATION OF THE PRESSURE DISTRIBUTION O N A CASCADE OF THICK AIRFOILS BY MEANS OF FREDHOLM INTEGRAL EQUATIONS OF THE SECOND KIND

by E. Murtensen

No. 23, Max-Planck-Institate for Flaid Research and

Aerodynamic Experimental Station, Gottingen, 195 9

N A T I O N A L AERONAUTICS A N D SPACE A D M I N I S T R A T I O N W A S H I N G T O N , D. C. JULY 1971

TECH LIBRARY KAFB, NM

IllllllIIIII1111Ill11IIIIIIIIIIIll11111MI 00b9050

NASA TT F-702

THE CALCULATION OF THE PRESSURE DISTRIBUTION ON A

CASCADE OF THICK AIRFOILS BY MEANS O F FREDHOLM

INTEGRAL EQUATIONS OF THE SECOND KIND

By E. Martensen

Aerodynamic Experimental Station, Gb'ttingen

Translation of "Die Berechnung der Druckverteilung an Dicken Gitterprofilen mit Hilfe von Fredholmschen Integralgleichungen Zweiter Art." Nr. 23,

Mitteilungen aus dem Max-Planck-Institut far Str6mungsforschung und de r Aerodynamischen Versuchsanstalt, Gb'ttingen, 1959

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION

For Sale by the National Technical Information Service, Springfield, Virginia 22151

$3.00

\

I

THE CALCULATION OF THE PRESSURE DISTRIBUTION ON A CASCADE OF THICK AIRFOILS EY MEANS OF

FREDHOLM INTEGRAL EQUATIONS OF THE SECOND KIND

E. Martensen

Aerodynamic Experimental Stat ion, Gottingen

ABSTRACT

Two independent l i nea r integral equations of the second kind w i t h

continuous kernels are derived f o r the exact potential theory f o r the

velocity d is t r ibu t ion on a cascade of thick a i r f o i l s . I t i s shown t h a t

the corresponding homogeneous integral equations possess one and only

one nontr ivial solut ion, so t h a t one knows the general results on the

basis of the Fredholm theorems. In the l i m i t i n g case of in f in i te separa­

ti on between a i r fo i 1s the equations reduce t o the fami 1i a r expressions

f o r s ing le a i r f o i l s . In the l i g h t of the per iodici ty properties which

a re present, one may develop a numerical calculat ion technique based on

the solution from a system of l inear equations. By select ing an ade­

quately arge number of unknowns, the des red accuracy i s obtained.

Examples are shown correlat ing the theory w i t h an exact known solut ion

and w i t h measurements.

iii

I

TABLE OF CONTENTS

Section Page

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . 1

2 GEOMETRY OF THE CASCADE . . . . . . . . . . . . . . . . 6

3 DERIVATION OF THE INTEGRAL EQUATION FOR A SINGLE AIRFOIL ( t = m) . . . . . . . . . . . . . . . . . . . . 6

4 THE PARAMETRIC REPRESENTATION OF io. . . . . . . . . 11

5 EXISTENCE AND MULTIPLICITY OF THE SOLUTION OF THE INTEGRAL EQUATION OF THE SECOND KIND . . . . . . . . . 14

6 TRANSFORMATION OF THE INTEGRAL EQUATION OF THE FIRST KIND TO AN EQUATION OF THE SECOND KIND. AND THE SOLUBILITY OF THE LATTER . . . . . . . . . . . . . . . 1 5

7 DERIVATION OF THE INTEGRAL EQUATION FOR A CASCADE OF AIRFOILS . . . . . . . . . . . . . . . . . . . . . . . 1 7

8 EXISTENCE AND MULTIPLICITY OF THE SOLUTIONS FOR THE INTEGRAL EQUATIONS OF THE SECOND KIND FOR A CASCADE OF AIRFOILS . . . . . . . . . . . . . . . . . . . . . . 2 2

9 NUMERICAL AUXIL IARIES . . . . . . . . . . . . . . . . . 26

10 PRACTICAL CALCULATION PROCEDURE . . . . . . . . . . . . 2 7

11 EXAMPLE FOR 2N = 24 CONTOUR POINTS . . . . . . . . . . 30

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . 34

TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

V

I

I

1. INTRODUCTION

The p o t e n t i a l f low about a two-dimensional body can be represented

by an arrangement o f s i n g u l a r i t i e s i n the enclosed reg ion and on i t s

boundaries i n numerous ways. These s i n g u l a r i t i e s can occur a t p o i n t s o r

l i n e s , o r t h e i r combinations. The ques t ion now i s which one o f these

u n l i m i t e d p o s s i b i l i t i e s i s the s implest . One i s the re fo re l e d t o a

f i e l d which vanishes i d e n t i c a l l y i n s i d e the reg ion and has a jump a t

t he w a l l , which g ives r i s e t o a vor tex d i s t r i b u t i o n corresponding t o the

l o c a l v e l o c i t y d i s t r i b u t i o n . U n t i l r e c e n t l y on l y approximate t r e a t ­

ments f o r these methods were pub l ished because o f the h igh c a l c u l a t i n g

costs. These s i n g u l a r i t i e s may be vo r t i ces , sources and/or vor tex-

source combinations a r b i t r a r i l y arranged i n po in ts and/or l i n e s . An

example o f the use o f t h i s method i s f o r t h e f l o w around a i r f o i l s which

has been accomplished by p l a c i n g these s i n g u l a r i t i e s over i t s chord.

However, a more s u i t a b l e , simple and e legan t s i n g u l a r i t y method t h a t

agrees w i t h the exac t s o l u t i o n of t he c l a s s i c a l f l o w problem f o r t h i c k

a i r f o i l s , which vanishes i d e n t i c a l l y i n the f i e l d , cons is ts o f cover ing

the boundary w i t h a vor tex d i s t r i b u t i o n , f o r t he f o l l o w i n g reasons:

1.

2.

The v e l o c i t y d i s t r i b u t i o n can be s u b s t i t u t e d f o r the vo r tex

d i s t r i b u t i o n s ince bo th views are i d e n t i c a l t o each o ther .

While f o r the prev ious case o f i n c l u d i n g s i n g u l a r i t i e s i n the

f i e l d t h e o r e t i c a l cons idera t ions and p r a c t i c a l ca l cu la t i ons

are r e q u i r e d i n o rder t o e s t a b l i s h the r e l a t i o n s h i p between the

vor tex d i s t r i b u t i o n and the v e l o c i t y d i s t r i b u t i o n .

Exis tence and c o n t i n u i t y o f the s o l u t i o n may be es tab l i shed on

II I1 l1llll111l11111l11llll I I I

the basis of Fredholm's theory, u t i l i z i n g continuity and

d i f f e r e n t i a b i l i t y properties. For in tegra l equations of the

f i r s t k i n d w i t h s ingular kernels, such conclusions are not

avai 1able.

3. The numerical procedure f o r solving the above integral equations

of the second k i n d w i t h continuous and, i n the present case,

periodic kernels, reduces t o a summation procedure. In

cont ras t , the s ingular i ty f o r the in tegra l equation of the

f i r s t k i n d gives r i s e t o a cha rac t e r i s t i c d i f f i cu l ty .

The method of covering the boundary w i t h vor t ices has been presented

previously by Korn i n a textbook [5]. Later Prandtl [6] expounded on

th i s idea as follows.

Imagine the i n t e r i o r of a body replaced by f l u i d a t r e s t , of a pressure p, + q l , as i s present a t the stagnation point. A t the boundary a vortex sheet produces a veloci ty j u m p of magnitude v . T h i s vortex sheet is the desired bound vortex system.

Prager [7] es tab l i shes , by the exclusive use of this method, a

Fredholm integral equation of the second k i n d f o r the velocity d is t r ibu t ion

around thick s ingle a i r f o i l s and obtains good numerical r e su l t s . He used

an integrat ion procedure of I'4ystram [8] and the Tschebyscheff quadrature

formula. The rapid development of computer mathematics s ince Prager 's

original work has increased the usefulness of his approach. The great ly

increased storage capaci t ies o f the presently used e lec t ronic computers

permit improving the accuracy of the calculat ions t o any desired extent.

In reference t o the work of Fottinger [9], Prager has demonstrated the

in fe r io r i ty of a source d is t r ibu t ion over the boundary as compared w i t h

a vortex d is t r ibu t ion . Goldstein and Jerison [lo] adapt this idea t o a

2

cascade of a i r f o i l s . They determined the a i r f o i l contour f o r a g iven

velocity d is t r ibu t ion ( ind i r ec t problem). T h i s point of view i s used

advantageously f o r the derivation of the fundamental equations, even

though the spec i f i c Fredholm theory is not needed. Finally Isay [ l l ]

has a l so t rea ted a cascade of thick a i r f o i l s by u s i n g vortex d is t r ibu t ions

over their boundaries. No use i s made i n th is work of the iden t i ty

of the vortex d is t r ibu t ion and the velocity d is t r ibu t ion s ince the l a t t e r

i s provided from the vortex d is t r ibu t ion by means of an integrat ion

formula. Nor i s the per t inent integral equation o f the f i r s t kind w i t h

a s ingular kernel converted t o an integral equation of the second kind

w i t h a continuous kernel. So Fredholm's theorems are not applicable.

Existence of the solution can be established only by insis tence on the

additional condition f o r the cascade geometry ( t equals the pi tch)

Inasmuch as this l imitat ion does not a r i s e i n the treatment of the flow

problem u s i n g Fredholm integral equations of the second k i n d , an example

u s i n g an e l l i p s e (thickness r a t i o 0 < E < 1 ) wil l be given f o r i l l u s t r a ­

t ion. T h i s example cannot be t reated by the Isay procedure. First , i n

u s i n g the parametric representation f o r the e l l ipse one obtains the

following formula [ l l ]

(1 - E ~ )sin (+ + $) H ( $ 3 $; m) = (1 + E Z ) - (1 - E Z ) cos ( $ + $ )

For the evaluation of this double integral we now examine the integral

equation

3

and seek n o n t r i v i a l s o l u t i o n s which are square i n t e g r a b l e from 0 t o 2 ~ .

Since

2a = -1

[ l n ( ( l t E 2

) - (1 - E 2

) cos (+ t $ ) ) I = o ,2Tr 0

the c o n d i t i o n

2a1 f ( $ ) d+ = 0 2a 0

must necessa r i l y be obeyed. By the i n t e g r a t i o n i n the complex plane

( res idue method) one can e a s i l y con f i rm t h a t

v = l , 2 , 3 , ... . There r e s u l t the eigenvalues

associated w i t h the e igenfunc t ions

f,(+) = cos v+ t s i n v+ , v = +1, +2 , k3, ... .

Since t h i s system o f f unc t i ons i s complete, i t f o l l o w s from the theory o f

i n t e g r a l equations o f t he second k i n d t h a t :

4

I I 1111 111 I1111I11111111111 I1II I I I I II I I I I ll11111111111

Hence the e a r l i e r s t a t e d i n e q u a l i t y reduces t o

Bu t t h i s i s equ iva len t t o say ing t h a t a l l e l l i p t i c a l contours w i t h a

th ickness r a t i o

0 < E -< 2 - 0.26795

( i n t e r e s t i n g cases) a re excluded by I s a y ' s cond i t ion .

I n the fo l l ow ing , the Prager i n t e g r a l equat ion f o r a s i n g l e a i r f o i l

[7] w i l l be der ived i n a s i m p l i f i e d fash ion and t h e t reatment f o r a

cascade o f a i r f o i l s w i l l be formulated us ing the work o f Goldste in and

Jer ison [lo]. Next, I say ' s s i n g u l a r i n t e g r a l equat ion o f the f i r s t k ind,

( t o be der ived by an a l t e r n a t e method), w i l l be converted t o a Fredholm

i n t e g r a l equat ion o f the second k i n d w i t h a continuous kerne l . Thus two

Fredholm equations o f the second k i n d w i l l be a v a i l a b l e f o r c a l c u l a t i n g

the pressure d i s t r i b u t i o n f o r a cascade o f t h i c k a i r f o i l s . Which o f the

two i s more s u i t a b l e f o r numerical c a l c u l a t i o n s cannot be determined i n

general. I t w i l l depend on the spec i f i cs o f the problem. P a r t i c u l a r

emphasis w i l l be p laced i n the degenerate case o f a s i n g l e eigenvalue

s ince the Fredholm theory i s app l i ed t o t h i s f i r s t and i t permi ts the

development o f a s u i t a b l e c a l c u l a t i o n scheme. This invo lves e s s e n t i a l l y

t he s o l u t i o n o f a system o f l i n e a r equations w i t h th ree inhomogeneous

terms corresponding t o the two main f l o w d i r e c t i o n s and the f r e e c i r c u l a ­

t i o n . A se r ies o f examples w i l l be c a r r i e d ou t f o r i l l u s t r a t i o n us ing the

Gott ingen e l e c t r o n i c computer 62.

5

2. GEOMETRY OF THE CASCADE

ifl, $-1,The cascade i s described by congruent contours io,... , which are arranged a t equal in te rva ls t i n the y-direction ( F i g . 1 ) . The

contour d o s a t i s f i e s (and so do i,,$-1, ...) continuity and d i f f e r ­

e n t i a b i l i t y conditions, t o be specified l a t e r . Essent ia l ly one prohibi ts

d i scont inui t ies i n the tangent, and curvature and the t h i r d and fourth

derivatives , as we1 1 as corners, cusps and double values .

Y

Figure 1

3. DERIVATION OF THE INTEGRAL EQUATION FOR A SINGLE AIRFOIL ( t = m)

I t i s desired t o o b t a i n the velocity d is t r ibu t ion ( a s well as the

pressure d i s t r i b u t i o n ) on the contour f o r a simply connected contour $o -f

w i t h no double points. The f r ee stream velocity i s given by wm i n an

incompressible potential flow (see Figure 2 ) . The components of this flow

are re la ted to the stream function Y by

wx = YY ' w

Y = - Y x ,

or equivalently

6

Y

Figure 2

holds; the f l ow d i r e c t i o n i s a t 90" counterclockwise t o the d i r e c t i o n o f

decreasing Y (F igure 3).

F igure 3

J, i t s e l f obeys Laplace 's equat ion

A Y = o , ( 3 )

i n accordance w i t h the vo r tex f r e e na ture o f the f i e l d (1). The problem

7

II I l l I I

i s t o determine a f l o w f i e l d t h a t goes t o G W a t i n f i n i t y and has ioas

a s t reaml ine, i.e., a f u n c t i o n Y t h a t has the f o l l o w i n g p roper t i es :

I. AY = 0 ou ts ide 3, ,

=11. [g]= vm cos a , -(E]vW s i n a , W m

111. Y = constant a long the ou te r boundary o f z0. One assumes the s o l u t i o n

~ ( x ,y ) = vm(y cos ~1 - x s i n a) + -' Ido v I n ds + constant2Tr

( 4 )

This s a t i s f i e s requirements I and I1 w i t h a surface d i s t r i b u t i o n along

$. equal t o v. When cross ing iothe values o f t he normal d e r i v a t i v e s

( n i s the ou te r normal) jump as fo l l ows :

($)o - [%Ix 0 = - s V

=[$]LO - [%Ii- 2-

These are proven i n Courant and H i l b e r t [4] s u b j e c t t o the c o n d i t i o n t h a t

the contour Lo i s f o u r times continuous y d i f f e r e n t i a b l e . However, t h i s

i n v e s t i g a t i o n w i l l n o t be concerned w i t h an e x p l o r a t i o n whether these

cond i t ions are a l s o necessary o r whether they may be m i t i ga ted . The

i n t e g r a l which appears i n (aYu/an)& i s n te rp re ted as a Cauchy p r i n c 0

value. The tangen t ia l d e r i v a t i v e remains constant i n c ross ing do;

[%lo - = o

8

- --

I

[x] - ($$Iiat Ro = 0 .

There a r i s e two p o s s i b i l i t i e s f o r the d i s t r i b u t i o n v t o conform t o

cond i t i on 111. I n one case, one i s lead t o an i n t e g r a l equat ion o f the

f i r s t k i n d f o r v, i n the second case t o one o f t he second kind.

The f i r s t i s obta ined by s e t t i n g

(51= 0 . (9) 0

Use o f (4) and (7) then es tab l i shes the f o l l o w i n g i n t e g r a l equation:

aIit Id v I n 1 ds = 2vm - (y cos ~1 - x s i n a) .a t 0

Before i n v e s t i g a t i n g t h i s i n t e g r a l equat ion o f the f i r s t k ind, we exp lo re

the meaning o f the sur face d i s t r i b u t i o n v. (7), (8) and (9) imp ly t h a t

[s)= 0 a i

i.e., t h a t Y i s constant (yi) a l s o along the i n s i d e o f io.Green's

theorem,

2( Y A Y + yx + Y 2 ) dx dy = Y

app l i ed t o the i n s i d e o f Logives

i.e., i n s i d e gothe v e l o c i t y f i e l d vanishes i d e n t i c a l l y ,

Y x = Y = o .Y

Therefore

9

(%IiO=

which combined w i t h (5) and (6) gives:

v = -[g]. 0

This means t h a t , by ( 2 ) t h a t v represents the magnitude o f the f l o w

v e l o c i t y around x0and, from F igure 3, t h a t v i s p o s i t i v e (negat ive)

f o r f l o w i n the counterclockwise (c lockwise) d i r e c t i o n . The i n t e g r a l

equation (10) i s a spec ia l case ( t = a) o f I s a y ' s work [11] which i s

p r i m a r i l y concerned w i t h a cascade o f a i r f o i l s , b u t n o t recogn iz ing the

s i g n i f i c a n c e o f v as shown i n (15).

The i n t e g r a l equat ion o f the second k i n d t h a t s a t i s f i e s requirement

I11 i s n o t found as immediately as i n the preceding case. One requ i res

t h a t

which leads, by (4 ) and (6 ) t o the f o l l o w i n g i n t e g r a l equat ion o f the

second k ind:

,+--I1 ds = -2v, - ( y cos a - x s i n a) .a v I n aan 71 an

x0

Again, apply ing Green's theorem (12) t o the i n s i d e o f bo,furn ishes, on

the bas is o f (16), t h e r e s u l t (13) f o r t he v e l o c i t y f i e l d i n s i d e go. As a r e s u l t , a l s o

10

and because o f (7) and ( 8 ) , a l s o

[$] = 0 9

0

i.e., i t i s es tab l i shed t h a t Y i s constant along the ou ts ide edge o f so i n accordance w i t h requirement 111. The sur face d i s t r i b u t i o n v represents

the v e l o c i t y o f t he f l o w around ioas i n the prev ious example because

(5 ) and (6 ) again imp ly (15) i n the l i g h t of (16). Th is i s a s i m p l i f i e d

d e r i v a t i o n o f t he i n t e g r a l equat ion (17) f o r the v e l o c i t y d i s t r i b u t i o n v,

which was f i r s t es tab l i shed by Prager [7].

4. THE PARAMETRIC REPRESENTATION OF to

L e t the contour go be represented pa ramet r i ca l l y by

x = X ( + L Y = Y ( + L 0 -<+zZlT,

w i t h + i nc reas ing i n the counterclockwise d i r e c t i o n . The f u n c t ons

x (+ ) and y ( + ) have per iods o f 2~ and are i n accordance w i t h Sec i o n 3,

supposed t o be f o u r t imes cont inuously d i f f e r e n t i a b l e . Furthermore , i n

o rder t h a t one may de f i ne a tangen t ia l vec tor a t each p o i n t on the contour

i n accordance w i t h Sec t ion 3, the subs id ia ry cond i t i on ( i n a d d i t i o n t o (20))

i s a l so imposed. With t h i s c o n d i t i o n i t w i l l be impossib le t o have cusps

on the contour, as was poss ib le w i t h (20) and mere r e g u l a r i t y requirements.

F i n a l l y , t o make the f o l l o w i n g d e r i v a t i o n eas ie r , the a u x i l i a r y v a r i a b l e

y (+) w i l l be in t roduced through the r e l a t i o n

11

II - l l l l l l l l 1 1 1 l 1 l l l - I

- - --

The i n t e g r a l equat ion o f the f i r s t k i n d (10) may be p u t i n the

f o l l o w i n g form, where x, y (i.e., 4) denotes a f i x e d p o i n t , and 5 , 0

(i.e., $) a v a r i a b l e p o i n t on go,and 2 i s a tangent vec tor :

;i:0

x - 5 V X x, Y) + (Y - d.t,(X, Y) 7r= ‘Iv(5, r l ) ( x - E;)2 + (y- +--~ ds;

d 0

I n the l i g h t o f ( 2 2 ) , (10) now becomes

= 2vm(y(+) cos ~1 - x (+ ) s i n a)

where

- c o t y. Since . _ _ . _.

H($, $) i s a continuous kerne l everywhere.

Correspondingly, one may modify the i n t e g r a l equat ion o f the second

12

, ,,.. , .._.._... .I

k i n d (17) , using the parametr ic representa t ion f o r ko as fo l l ows :

Because 2 represents the ou ts ide u n i t normal and 4 increases i n the

counterclockwise d i r e c t i o n ,

Using (22) and (26), the i n t e g r a l equat ion (17) takes the f o l l o w i n g form:

w i t h

K(9, $) i s an everywhere continuous kernel . The i n t e g r a l equat ion (27)

a r i ses i n Fredholm's t reatment o f t he Neuman problem i n p o t e n t i a l theory

[l;21 i n the form:

13

The transpose of equation (30) (replacing a / a + by a / a $ ) was likewise

considered by Fredholm f o r the solution of Di r ich le t ' s problem, and was

the s t a r t i n g point of Fredholm's theory of integral equations of the

second k i n d [l].

5. EXISTENCE AND MULTIPLICITY OF THE SOLUTION OF THE INTEGRAL EQUATION OF THE SECOND KIND

In order t o decide about the so lub i l i t y of the integral equation ( 2 7 )

one must invest igate , according t o the Fredholm theory, the existence of

nontrivi a1 sol utions of the associated transposed homogeneous integral

equation:

We now maintain tha t

%y ( + ) = constant # 0

i s a nontrivial solution of (31). To prove i t , we wri te

As shown i n Figure 4,

runs through a l l values from a cer ta in -c0 t o T0

+ IT as $I increases from

0 t o ZIT;each value i s assumed a t l e a s t once. Thereby one obtains the

relat ion

14

Y

Figure 4

'1 ZIT K(+, $1 d+ = 1 , (33)

0

which has a l ready been used by Fredholm [2]. By in terchanging + and

$ i n (33), one can see t h a t (32) i s a n o n t r i v i a l s o l u t i o n o f (31).*

There we e s t a b l i s h the ex is tence o f a n o n t r i v i a l s o l u t i o n y (+ ) o f (27),

when vm = 0, which we recognize as the pure c i r c u l a t i o n f l o w around the

p r o f i l e . Since i t i s known f rom t h e theory o f conformal mapping, t h a t no

two l i n e a r l y independent c i r c u l a t i o n f lows e x i s t i t fo l l ows , conversely,

t h a t the homogeneous equat ion associated w i t h (27) has n o t more than one

l i n e a r l y independent s o l u t i o n . Hence we have the degenerate case w i t h a

s i n g l e eigenvalue. The general s o l u t i o n f o r (27) may be w r i t t e n , t he re fo re ,

as

** Here y (+) i s the p a r t i c u l a r s o l u t i o n o f t h e inhomogeneous equat ion (27).

6. TRANSFORMATION OF THE INTEGRAL EQUATION OF THE FIRST KIND TO AN EQUATION OF THE SECOND KIND, AND THE SOLUBILITY OF THE LATTER

L e t f(+)be a cont inuously d i f f e r e n t i a b l e func t i on , def ined over

15

- -

- -

0 < + < ZIT, w i t h a period of 2n.

One c a l l s the function,

the harmonic conjugate function ("cotangent in tegra l" ) of f ( + ) .

(Cauchy principal value i s understood.) I t results from the Fourier

development of f ( + ) , by ignoring the constant term and replacing cos V +

by -sin V + and sin V + by cos v + , v = 1 , 2 , ... . Passing, once more,

from g (+) t o i t s harmonic conjugate, one obtains:

We now apply the cotangent integral operator t o the integral equation of

the f i rs t kind (23 ) and obtain the following integral equation o f the

second k i n d :

v, ,2IT .-J o

cy($) cos cx - ;($) sin a] cot d$ (37)

where

i s an everywhere continuous kernel. B u t the in tegra l over a conjugate

harmonic function vanishes because o f the missing constant temi i n the

Fourier development; t h u s (38 ) leads to:

(39)0

As i n Section 5 , i t is now established t h a t the general solution of (37 )

16

- - - -

i s i n the form of (34).

7. DERIVATION OF THE INTEGRAL EQUATION FOR A CASCADE OF AIRFOILS

S i m i l a r l y as above, two i n t e g r a l equations o f the second k i n d w i l l

be der ived f o r t he v e l o c i t y d i s t r i b u t i o n on a cascade o f t h i c k a i r f o i l s

(F igure 1) corresponding t o (27) and (37). We thus expect two equations

o f the form

and

V = LE &($) cos - x($) s i n CC]c o t 9 d$

TI

which f o r t -+ w i 11 reduce t o (27) and (37), respec t i ve l y . The

q u a n t i t i e s vm and cx s t i l l t o be c l a r i f i e d , s ince i n f i n i t y w i t h respect t o

the cascade i s n o t s imply r e l a t e d t o the "outs ide" o f the contours

do,sl, J-,,0 . . Because o f t h i s we r e l i n q u i s h f o r the t ime being

requirement I 1 o f Sec t ion 3 and t r e a t vm and a merely as f r e e parameters,

whereas we s h a l l r e t a i n requirements I and I11 f o r a l l contours do, z , ¶- c

-1, ... . Then the stream f u n c t i o n expression which s a t i s f i e s

requirement I i s g iven by

Y(X, y; t) = v,(y cos cx - x s i n a)

I+ z Jv ~ n ~

I _ _ ds

\/cosh--(x2TI - e) - cos (y - n)$0 t t

+ cons tan t . 17

This expression and formula (4 ) have a l ready been found by Go lds te in

and Jer ison [lo]. Furthermore, i t can be seen from (42) t h a t

yX(x, Y; t) = yx(x, Y + k t ; t )

yY(x, Y; t) = yY(x, Y + k t ; t)i , k = +1, +2, +3, ... , (43)

so t h a t i f requirement I 1 1 i s s a t i s f i e d f o r k alone, then i t w i l l be

s a t i s f i e d f o r a l l o f Xl, Lm1,... as w e l l . Therefore, we can conf ine

the i n v e s t i g a t i o n t o the cond i t ions a t x,. It w i l l be shown nex t t h a t

r e l a t i o n s (5 ) t o (8 ) are v a l i d a l s o f o r (42). For t h i s purpose, we w i l l

i n v e s t i g a t e the expression

F i r s t , one can e s t a b l i s h tha t , f o r f i x e d t, F(x, y; t) i s an a n a l y t i c

f u n c t i o n ou ts ide of v i o , xl, z-l, ... . This i s equa l l y v a l i d f o r the

Zl, Jm1,i n s i d e reg ions o f *io, ... . Next we show t h a t F(x, y; t) i s

a n a l y t i c on -ioi t s e l f ( n o t f o r X,, imlY...). For t h i s purpose we s e t

ZIT- ( x - E ) = x (45)t

2 " ( y - 11) = Y (46)t

and expand the argument f o r the l oga r i t hm i n (44) as fo l l ows :

18

.. . . . . ..

cash X - COS Y x2 + Y'

m v+l

x2 + Y2

= 2 y &*2 v+l 2 v+l

v =o

W 1 v = 2 2v + 2 ! p=o ( x2

)v-p ( - Y 2 )p .

v =o To show tha t the se r i e s converges fo r a l l values of X and Y we l e t

( w i t h M equal t o a real posi t ive number)

The estimate

W sinh M1 ( v + 1)M2v = -M ' v =o

i s thereby established. Thus

cash X - COS Y = 1 + x2 -12

Y2 + x4 - X2Y2 + Y4 + ... (47)x2 + Y2 360

i s convergent f o r a l l values of X and Y and hence i s an e n t i r e function

i n both var iables , X and Y. Therefore, the ser ies

(which follows from (47)) also converges i n a f i n i t e region from X = 0 ,

Y = 0, and therefore represents there an analyt ic function i n X and Y.

Finally, the integrand of (44) , w i t h f ixed 5 , I-, and t , i s a l so ana ly t ic

f o r a l l values of x and y on do (including the inside and outside of Lo,

19

excepting g1, i-l,...I. AS a result, ~ ( x ,y; t) experiences no

d iscont inui t ies as one crosses to.Final ly , s ince the form (42) f o r

Y(X,y; t ) differs from ( 4 ) f o r ~ ( x ,y) only by the expression,

-F(x, y; t ) + constant,

equations (5) t o (8) a re seen t o be val id a l so f o r expression (42) . One

can t h u s t r ans fe r the reasoning of Sections 3 and 4 , without additional

consi derat i ons , t o cascades. A1 so , the requi rement I I I can be thought

of as being fu l f i l l ed by the corresponding in tegra l equations (40) and

(41). As s t a t ed before, the corresponding flow conditions f o r sl,;eWl, ... are a l so automatically s a t i s f i e d .

Now, a l l t h a t remains i s the c l a r i f i c a t i o n of the s ignif icance o f

vm, ~1 and of the kernels K(+, $; t) and L(+, I); t ) which appear i n

equations (40) and (41). I t follows from (42) t h a t :

lim Y,(x, y; t) = -v, s i n ~1 - (49)X++m

ol I

lim YY

(x, y ; t) = v, cos ~1 I X-tt,

w i t h 2Tr

r = v ds = 1 y($) d+ 0;t

0

representing the c i rcu la t ion around each cascade p r o f i l e , one obtains

from (1):

20

= wwx ,-03 X ,+- = vm cos

WY 9-m = v, s i n a - -r 2 t

WY ¶+m

= vm s i n a + -r . J2 t

= vm cos u and wY 9-

= v, s i n CL are now recognized as componentswX,­

o f t he so-ca l led " t r a n s p o r t f l o w ve loc i ty , " G, ( v e c t o r average o f the

upstream v e l o c i t y $-, and downstream v e l o c i t y G+,) shown i n F igure 5.

Furthermore,

W y , b - Wyp

- 1 (52)-

i s the c h a r a c t e r i s t i c " d e f l e c t i o n " caused by t h e cascade.

To ob ta in the forms o f t h e kernels K(+, q ; t) and L(+, J I ; t),

we proceed as i n Sect ion 4 (be ing c a r e f u l n o t t o confuse a / a t and the

p i t c h t), and o b t a i n an i n t e g r a l equat ion of t he f i r s t k ind:

This d i f f e r s f rom (23) o n l y by the kernel

- c o t y (53)

and i n t h i s form was the bas i s o f I s a y ' s work. As can be e a s i l y shown,

I

the lirniting value ( 2 5 ) f o r H ( $ , $; t ) i s s t i l l valid. Correspondingly,

one establ ishes an integral equation of the second kind analogous t o

( 2 7 ) :

w i t h the kernel :

Also K ( $ , JI ; t) re ta ins i t s l imit ing value. Finally the kernel

L($, $; t) corresponding t o Section 6 assumes the expression:

W i t h t h i s , one can handle a l l the terms i n the integral equations (40)

and (41) f o r the treatment o f the cascade problem.

8. EXISTENCE AND MULTIPLICITY OF THE SOLUTIONS FOR THE INTEGRAL EQUATIONS OF THE SECOND KIND FOR A CASCADE OF AIRFOILS

One shows t h a t the homogeneous integral equations (40) and (41)

possess nontrivial solutions by verifying, as i n Sections 5 and 6 , t h a t

22

While the p r o o f o f (57) fo l l ows d i r e c t l y from the d e f i n i t i o n (55) f o r

L(+, I); t), the p roo f o f (56) i s n o t obta ined as e a s i l y . F i r s t o f a l l ,

we de f i ne

which leads, w i t h the a i d o f (54) , t o

IT

+ -27T [arc tan s i n

IT

( y - n ) j'" . s i n h 7 ( x - 5)

Now, by arguments s i m i l a r t o those p e r t a i n i n g t o F igure 4, one ob ta ins ,

on the bas is o f (54) and (58) :

23

I

Furthermore,

1 1 1J($; 8t ) = 8 J($; t) + 8 + 1+ 7

and J($; 2't) = 2-'J($; t) + 2-'(1 + 2 + 2' + ... + ~ " 1 )

= 1 + ZV[J($; 2't) - 11, v = 0, 1, 2, ... (60)

We now s e t

and i n v e s t i g a t e the a n a l y t i c i t y o f (54) w i t h respect t o z ( i n the neighbor­

hood o f z = 0) i n the form

f o r f i x e d + and $.

Expanding numerator and denominator o f the l a s t expression i n power

o f z one obta ins:

24

The constant term i n t h e denominator does n o t vanish when

J , f + + 2 k a 9 k = 0, +1, 22, ... . Thus there e x i s t s a neighborhood o f z = 0 where K ( + , J,; ")Z i s a n a l y t i c ,

and may be represented by a convergent power se r ies .

Since the l i m i t i n g value, as J, + + + 2ka e x i s t s on the l e f t s ide, i t

must a l so e x i s t on t h e r i g h t s de and must h o l d f o r each c o e f f i c i e n t o f

z2'. I n t h i s l i m i t t he re fo re , t he dependence on z disappears as seen

from (29). Thus K ( @ , @ + 2ka, $) i s a constant and hence a n a l y t i c . I t

27lfol lows t h a t K(+, J,; 7)i s a n a l y t i c f o r a l l values o f +, J, i n a c e r t a i n

v i c i n i t y o f z = 0 and admits a power se r ies rep resen ta t i on (64) .

Because o f un i fo rm convergence f o r f i x e d J, and z, we may i n t e g r a t e

(64) w i t h respect t o + term by term, and o b t a i n on the bas is o f ( 3 3 ) ,

(58) and (61) the expansion

which converges f o r s u f f i c i e n t l y l a r g e t. We s e l e c t v f o r a s p e c i f i c

cascade so l a r g e t h a t

converges f o r a l l values o f J, (such a v may always be found). Replace­

ment i n t o (60) now g ives:

25

I

which i s v a l i d f o r t he chosen v and a l l g r e a t e r values. Using v -f 0 3 ,

f o l l ows as w e l l as statement (56) .

9. NUMERICAL AUXILIARIES

The i n t e r p o l a t i o n formula f o r a f u n c t i o n f ($)s p e c i f i e d by

- V V fV = f($J Y $,--pv = 0, ... , (2N - 1 ) . (67)

w i t h p e r i od i c i ty 2~ is the trigonometri c polynomi a1

N N- 1 f ( $ ) = 1 a cos p$ + 1 bp s i n 14

p=O lJ p=1 where

1 2N-1 a o = 2 N f v v=o

The f o l l o w i n g formulas r e s u l t from t h i s representat ion.

I nt e g r a t i on :

26

- -

Cotangent Integral :

Differentiation :

. 2N-1 f ( @ = f P = 1 Bvfl-I+v1-I v=l

w i t h

2 @VB V

= ( - lp ) - l cot -2 , v = 1 , ... , (2N - 1 )

10. PRACTICAL CALCULATION PROCEDURE

Using the above, equations (40) and (41) a re converted t o the

following sys tem of l i nea r equations:

1 214-1 = 2vc4[x cos a + y s in a]

1-I 1J.

1 ~ . = 0, ... , (2N - 1 ) . ( 74b

Summing up a l l the above equations f o r both systems ( 7 4 ) , i t follows

from previous work tha t (74a) i s nearly l inear ly dependent, while i n

(74b) the l i nea r dependence i s exact. T h u s , one can omi t any equation

i n the second system (e.g., the 1-1 = 0 equation) while the f i r s t system

27

should be so lved i n a l e a s t square sense. To accomplish t h i s (74a) i s

rendered, on the bas is o f (56), which i s w r i t t e n as

2i5-1 2N-1 2N- 1 1 214-1 ( 1 - 2N e Kpv)Yv = 0 9

ll=O v=o v=o u=o

l i n e a r l y dependent. Again any equat ion, say, t h e p = 0 equation, may

be omit ted. Therefore, we have a system o f (2N - 1) equations f o r 2N

unknowns yo, yl, ... ’ ’2N-1. The (2N)th equat ion i s obta ined from

re1a t i o n (50) f o r t he t o t a l c i r c u l a t i o n , numer i ca l l y represented by:

The t o t a l i t y o f s o l u t i o n s can be obta ined by the superposi t ion o f

a l l poss ib le so lu t i ons . It s u f f i c e s t o so lve (74) f o r t h ree inhomogeneous

sides, au’ b 1J.

and c !J , where a

1J. and b

1J. correspond t o the perpendicu lar

f l ow (a = 0) and p a r a l l e l f l o w (a = IT/^), r e s p e c t i v e l y , t o the cascade

i n the absence o f c i r c u l a t i o n ( r = 0). The t h i r d inhomogeneous s ide c !J

s p e c i f i e s a f r e e c i r c u l a t i o n f l o w w i t h no incoming f l o w (v, = 0). Thus,

we have the f o l l o w i n g :

v, ,ZIT . $ - 4

r2vmx1J. and -IT J o Y($) c o t 2 d$ Y 1

aP = 1 1-1 = 1, ... , (2N - l ) ,

0 !J = 2N ,

b!J = 1 0

28

- -

0 !J = 1 , ... , (2N - l ) , c =

!J G p = 2 N . (79)

In the numerical calculations i t i s desirable t o make x!J

and y lJ

dimensionless, se t vm = 1 and r = 27l. In superimposing free circulat ion

one must take i n t o account, f o r cascades, the upstream and downstream

flow modifications according t o (51). Noting (22), we denote the three

(p) (p)above solutions v( ' ) ( (p) , v ( ~ ) ( and v ( ~ ) ( and refer t o them as the

fundamental solutions. Thus

V 2= av(') + bv52) + CV:~) , v = 0 , ... , (2N - 1) ,V V -m

where

a = cos a_,

1b = 7 (cos a-mtan a+- + sin a-m)

represents the solution w i t h a_, equal t o the upstream flow angle and

equal t o the downstream flow angle (Figure 5 ) . The accompanying

pressure d is t r ibu t ion is g i v e n by:

while the expression f o r the t o t a l c i rculat ion from (52) and (31)--see

Figure 5--is g iven by

I' - t (cos a-mtan a+m- sin a-m) = 27lc . '-m

I t i s t o be noted t h a t a and b are dimensionless constants whereas c has

the dimension of length.

29

In the following, the solut ions of the in tegra l equations (27) and

(40) will be referred t o as procedure A, while the solut ions of the

integral equations (37) and (41) will be referred t o as procedure B.

11. EXAMPLE FOR 2rj = 24 CONTOUR P o r r m

F i r s t of a l l , we shal l compare the r e su l t s of the foregoing analysis

w i t h the known solution for a s ing le e l l i p s e of thickness r a t i o 0 < E < 1

( c f . introduction o f this report) . The parametric representation is

g i ven by :

x ( + ) = cos + y ( + ) = E s in +

A t an angle of a t tack a s , the fundamental solut ions a re , as obtained by

the methods of conformal mapping:

(See Table 1 and Figure 6 delet ing one o f the two e l l i p ses . ) The resu l t s

are shown f o r E = 0.2 and as = 30" i n Table 2 (procedure A ) and Table 3

(procedure B). These agree th roughou t t o nearly four s ign i f i can t

The error is t h u s f a r belowfigures w i t h the exact solution (Table 1) .

the graphical accuracy. A cascade of these contours, w i t h pitch r a t i o

30

t / a , equal t o 1.0, will be i n v e s t i g a t e d n e x t (Table 4 and Figures 5 and

6) . Since, a t the present time, there i s no known e x a c t , c losed s o l u t i o n

t o this problem, one must conten t onese l f w i t h the f a c t t h a t both above

independent procedures lead t o results w h i c h agree t o near ly f o u r

s i g n i f i c a n t f i g u r e s . The pressure d i s t r i b u t i o n s f o r two contours w i t h sharp t r a i l i n g

edges, designated as No. 9 and No. 10, were c a l c u l a t e d f o r various

upstream flow condi t ions and f o r t / a = 1 and as = 40". The fundamental

s o l u t i o n s , c a l c u l a t e d by procedures A and 6 (Tables 6 and 7 ) , show some

disagreement near the t r a i l i n g edge. T h i s i s n o t s u r p r i s i n g , s i n c e cases

w i t h sharp t r a i l i n g edges were excluded from the beginning , and the

l a r g e curvature of the g i v e n contours make them appear as i f they had

cusps. However, th i s does not prevent a good agreement between theory

and measurements f o r contour No. 9 (Figures 9 t o 13) . The upstream

flow angle a_ , i s var ied from -15" t o 45" i n i n t e r v a l s of 15", while f o r

the downstream flow angle the experimental , near ly cons tan t , a+m = -69.6"

was chosen. The numerical values f o r these cases a r e c o l l e c t e d i n the

. .

a b c/ t r /av -m

CP ,+m ~ i : . . ; z : : . ; = , , . .~ - I _ _ ~ ~ -~. ~~ ~~ . - ~

-15" - 69.6" 0.9659 -1.4281 -0.3722 -2.3385 -6.6791

0" - 69.6" 1 .oooo -1 .3445 -0.4280 -2.6889 -7.2304

15" - 69.6" 0.9659 -1.1693 -0.4546 -2.8561 -6.6791

30" - 69.6" 0.8660 -0.9143 -0.4502 ~ -2.8287 -5.1753

45" - 69.6" 0.7071 -0.5971 -0.41 52 ~ -2.6085 -3.1152

31

- -

For s implici ty , contour No. 10 will be investigated next ( t / a = 1 ,

cxS

= 40") us ing procedure B only (Table 8, Figure 8). In order t o

evaluate the r e l a t ive merits o f contours No. 9 and No. 10 (see

Figure 8) we l e t , f o r both contours, the stagnation point coincide w i t h

the t r a i l i n g edge ( @= 0 and v = 0 , respectively) and se t the upstream

flow angle equal zero. For superposition o f the fundamental

solut ions, the following coef f ic ien ts are required (see formulas (81)) :

I _.­

c / t r /ac - W

. - _ - _ _ _ -1 5851 -0.5046 -3.1 703

-1.4470 -0.4606 -2.8940 - -~

~

Both resul tants a re displayed i n Figure 14.

Finally, a fu r the r contour, designated No. 12, wil l be investigated

(Table 9, Figures 15 and 16). An upstream flow angle C X - ~ = 0 i s assumed

and the rear stagnation point wil l again be placed a t the t r a i l i n g edge

( $ = 0 and v = 0 , respect ively) . The solut ion i s carr ied out w i t h :

The theoret i cal pressure d is t r ibu t ion , calculated by procedure B , i s shown

i n Figure 17.

The numerical calculat ions were carr ied out on the moderate speed

Gsttingen 62 computer (approximately 20 operations/sec) . The approximate

computation time f o r a s ing le calculation amounted t o the following:

32

-- --

-- --

Single a i r f o i l w i t h procedure A 1 hour 15 minutes

Single a i r f o i l w i t h procedure B 1 hour 45 minutes

Cascade of a i r f o i l s w i t h procedure A 3 hours 15 m nutes

Cascade of a i r f o i l s w i t h procedure B 3 hours 45 m nutes

The computer program i s so arranged t h a t only the angle of a t tack, pitch

r a t i o and contour coordinates must be s u p p l i e d t o the machine. The

program will then be automatically carr ied through a f t e r the machine is

s t a r t ed , and without further i n p u t , i t will end by automatically outputting

the expressions of the contour data, and the desired fundamental solutions.

The author thanks Dr. F. W . Riegels f o r his encouragement and

assis tance i n th i s work and Professor Dr. Biermann f o r providing the

62 computer. The support f o r this investigation was provided by the

I' De u t chen Fo rs ch un gsg emei n s haf t .'I

The theoret ical pa r t of th i s investigation was presented on 5 December

1957 in a Seminar on Instrumental Mathematics a t the Max Planck I n s t i t u t e

f o r Physics.

Translated f o r the National Aeronautics and Space Administration by R. J. Weetman Department of Mechanical and fierospace Engineering University of Massachusetts Amherst, Mass. 01003

33

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[2] I.Fredholm, "On an I n t e g r a l Equation w i th an A n a l y t i c a l Kernel," Acta Math. 45 (1924), pp. 11-27.

[3] W . Schmeidler, I n t e g r a l Equations w i t h App l i ca t i ons i n Physics and Engineering, Vol. I,Leipz ig, 1950.

[4] R. Courant and D. H i l b e r t , Methods o f Mathematics Physics, Vols. I and 11, B e r l i n , 1931 and 1937, respec t i ve l y .

[5] A. Korn, Textbook on P o t e n t i a l Theory, B e r l i n , 1899.

[6] L. Prandt l , A i r f o i l Theory, Vol. 1 , Report o f t he Imper ia l Associat ion f o r Science, Gatt ingen, Math-Phys. Class 1918, pp. 451-477.

[7] W . Prager, "The Pressure D i s t r i b u t i o n on Bodies i n Plane P o t e n t i a l Flow," Physik. Ze i tschr . 29 (1928) , pp. 865-869.

[8] E. J. Nystrom, " P r a c t i c a l S o l u t i o n o f L inea r I n t e g r a l Equations w i t h App l i ca t i ons t o Boundary Value Problems i n P o t e n t i a l Theory," Comm. Phys.-Math. SOC. Scient. Fennica -4, No. 15 (1928), pp. 1-25.

[9] H. Fo t t i nge r , "The Development o f V e c t o r i a l I n t e g r a t i o n t o Computer Solut ions f o r P o t e n t i a l and Vortex Problems , I ' l e i t sch r . f. Techn. Phys. , 1928.

[ l o ] A. W . Goldste in and M. Jer ison, " I s o l a t e d and Cascade A i r f o i l s w i t h Prescr ibed V e l o c i t y D i s t r i b u t i o n s ,I' NACA Techn. Rep. No. 869 (1957) , pp. 201-215.

[ll] W . H. Isay, "Comments on the P o t e n t i a l Flow through A x i a l Cascades," D isse r ta t i on , Z. f. Angew. Math. & Mech., 33 (1953), pp. 397-409.

[12] Hutte, The Engineer's Handbook, Vol. 1, 26. Auf l . , B e r l i n , 1936, p. 191.

[13] N. Scholz, Report 52/25, I n s t i t u t e f o r Flow Mechanics o f t he Technical Un ive rs i t y , Braunschweig (1952)(unpublished).

34

TABLE 1. Coordinates and the exact values o f the fundamental so lu t i ons f o r a 20% t h i c k s i n g l e e l l i p s e a t an angle o f a t tack as = 30".

V xV Y V

(3 ) "V

0 1 .ooooooo 0.0000000 0.20000 3.00000 5.19615 5.00000 1 0.9659258 0.051 7638 0.32297 0.961 65 3.58894 3.09629 2 0.8660254 0.1000000 0.5291 5 0.00000 2.26779 1.88982 3 0.7071 06 8 0.1 4142 14 0.721 11 -0.43070 1 .60740 1.38675 4 0.5000000 0.1732051 0.87178 -0.68825 1.19208 1.14708

5 0.25881 90 0.1931852 0.96731 -0.87720 0.87720 1.03379

6 0.0000000 0.2000000 1.00000 -1.03923 0.60000 1.ooooo 7 -0.25881 90 0.1931 852 0.96731 -1.19828 0.32108 1.03379

8 -0.5000000 0.1732051 0.87178 -1 .37649 0.00000 1.14708

9 -0.7071068 0.1414214 0.721 11 -1 .60740 -0.43070 1.38675 10 -0.8660254 0.1000000 0.5291 5 -1.96396 -1.13389 1.88982

11 -0.9659258 0.051 7638 0.32297 -2.62729 -2.62729 3.09629

12 -1.0000000 0.0000000 0.20000 -3.00000 -5.19615 5.00000

13 -0.9659258 -0.0517638 0.32297 -0.961 65 -3.58894 3.09629 14 -0.8660254 -0.1000000 0. 52915 0.00000 -2.26779 1.88982

15 -0.7071068 -0.141 4214 0.721 11 0.43070 -1.60740 1.38675 16 -0.5000000 -0.1732051 0.87 178 0.68825 -1.19208 1 .14708

17 -0.25881 90 -0.1931852 0.96731 0.87720 -0.87720 1.03379

18 0.0000000 -0.2000000 1 .ooooo 1.03923 -0.60000 1 .ooooo 19 0.25881 90 -0.1931 852 0.96731 1.19828 -0.32108 1.03379

20 0.5000000 -0.1732051 0.871 78 1.37649 0.00000 1 .14708

21 0.7071068 -0.1414214 0.72111 1.60740 0.43070 1.38675

22 0.8660254 -0.1000000 0.52915 1.96396 1.13389 1.88982

23 0.9659258 -0.0517638 0.32297 2.62729 2.62729 3.09629

35

TABLE 2. Coordinates and fundamental so lu t i ons us ing procedure A f o r a 20% t h i c k s i n g l e e l l i p s e a t an angle o f a t tack as = 30".

V xV Y V s V

(3) vV

0 1.0000000 0.0000000 0.20000 3.001 16 5.19816 5.00000

1 0.9659258 0.051 7638 0.32297 0.96241 3.5901 1 3.09629

2 0.8660254 0.1000000 0.52915 0.00046 2.26840 1.88982

3 0.7071068 0.1414214 0.721 11 -0.43039 1.60775 1.38675

4 0.5000000 0.1732051 0.87178 -0.68803 1.19226 1.14708

5 0.2588190 0.1931852 0.96731 -0.87706 0.87726 1.03379

6 0.0000000 0.2000000 1.00000 -1.03915 0.59995 1 .ooooo 7 -0.25881 90 0.1931 852 0.96731 -1.19826 0.32092 1.03379

8 -0.5000000 0.1732051 0.87178 -1.37655 -0.00028 1.14708

9 -0.7071068 0.1 414214 0.721 11 -1.60755 -0.431 14 1.38675

10 -0.8660254 0.1000000 0.5291 5 -1.96426 -1.13459 1.88982

11 -0.9659258 0.051 7638 0.32297 -2.62792 -2.62853 3.09629

12 -1.0000000 0.0000000 0.20000 -3.001 16 -5.1 9816 5.00000

13 -0.9659258 -0.0517638 0.32297 -0.96241 -3.5901 1 3.09629

14 -0.8660254 -0.1000000 0.5291 5 -0.00046 -2.26840 1.88982

15 -0.7071 068 -0.1414214 0.721 11 0.43039 -1 .60775 1.38675

16 -0.5000000 -0.1732051 0.87178 0.68803 -1.19226 1 .14708

17 -0.25881 90 -0.1931852 0.96731 0.87706 -0.87726 1.03379

18 0.0000000 -0.2000000 1 .ooooo 1.03915 -0.59995 1 .ooooo 19 0.25881 90 -0.1931 852 0.96731 1 .19826 -0.32092 1.03379

20 0.5000000 -0.1732051 0.87 178 1.37655 0.00028 1.14708

21 0.7071068 -0.141421 4 0.72111 1.60755 0.431 14 1.38675

22 0.8660254 -0.1000000 0.5291 5 1.96426 1.13459 1 .88982

23 0.9659258 -0.0517638 0.32297 2.62792 2.62853 3.09629

36

TABLE 3. Coordinates and fundamental s o l u t i o n s u s i n g procedure B f o r a 20% t h i c k s i n g l e ellipse a t an angle o f a t t a c k as = 30'.

yV

0 1.0000000 0.0000000 1 0.9659258 0.0517638 2 0.8660254 0.1000000 3 0.7071 068 0.1414214 4 0.5000000 0.1732051 5 0.25881 90 0.1931 852 6 0.0000000 0.2000000 7 -0.25881 90 0.1931 852 8 -0.5000000 0.1732051 9 -0.7071068 0.141 4214

10 -0.8660254 0.1000000 11 -0.9659258 0.0517638 12 -1.0000000 0.0000000 1 3 -0.9659258 -0.0517638 14 -0.8660254 -0.1000000 15 -0.7071068 -0.1414214 16 -0.5000000 -0.1732051 17 -0.25881 90 -0.1931852 1 8 0.0000000 -0.2000000 19 0.25881 90 -0.1931 852 20 0.5000000 -0.1732051 21 0.7071 068 -0.141 4214 22 0.8660254 -0.1000000 23 0.9659258 -0.0517638

S V (1) ( 3 ) V V vV

0.20000 3.00009 5.19631 5.00000 0.32297 0.96183 3.58896 3.09629 0.52915 0.00017 2.26775 1 .88982 0.72111 -0.43053 1.60734 1.38675 0.871 78 -0.68808 1.19201 1 .14708 0.96731 -0.87704 0.87712 1.03379 1 .ooooo -1.03908 0.59991 1 .ooooo 0.96731 -1.19813 0.32098 1 .03379 0.87178 -1 .37635 -0.00011 1 .14708 0.721 11 -1 -60727 -0.43082 1.38675 0.52915 -1.96384 -1.13403 1.88982 0.32297 -2.62722 -2.62745 3.09629 0.20000 -3.00009 -5.19631 5.00000 0.32297 -0.961 83 -3.58896 3.09629 0.52915 -0.0001 7 -2.26775 1 .88982 0.72111 0.43053 -1.60734 1.38675 0.87178 0.68808 -1.19201 1 .14708 0.96731 0.87704 -0.87712 1.03379 1 .ooooo 1.03908 -0.59991 1.00000 0.96731 1.19813 -0.32098 1.03379 0.87178 1.37635 0.00011 1.14708 0.721 11 1.60726 0.43082 1.38675 0.5291 5 1.96384 1.13403 1.88982 0.32297 2.62722 2.62745 3.09629

37

I

TABLE 4. Coordinates and fundamental s o l u t i o n s using procedure A f o r a cascade o f 20% thick e l l i p t i c a l contours w i t h the p i t c hr a t i o equal t o 1.0, a t an angle o f a t t a c k as = 30'.

V xV YV sV

( 3 ) vV

0 1.0000000 0.0000000 0.20000 2.65278 4.0321 3 6.56070 1 0.9659258 0.0517638 0.32297 0.61760 2.76939 4.46483 2 0.8660254 0.1000000 0.5291 5 -0.28272 1.71578 2.76477 3 0.7071068 0.1414214 0.72111 -0.69331 1.16664 1.89483 4 0.5000000 0.1732051 0.871 78 -0.95337 0.80452 1.33927 5 0.25881 90 0.1 931 852 0.96731 -1.15457 0.52683 0.94268 6 0.0000000 0.2000000 1.00000 -1.32438 0.29861 0.67333 7 -0.25881 90 0.1931852 0.96731 -1.47200 0.09875 0.54733 8 -0.5000000 0.1732051 0.87178 -1.61238 -0.10933 0.59990

9 -0.7071068 0.1414214 0.721 11 -1.78155 -0.39549 0.89822 10 -0.8660254 0.1000000 0.52915 -2.05236 -0.90329 1.62716 11 -0.9659258 0.0517638 0.32297 -2.56998 -2.04295 3.41 399 12 -1.0000000 0.0000000 0.20000 -2.65278 -4.032 13 6.56070

1 3 -0.9659258 -0.0517638 0.32297 -0.61760 -2.76939 4.46483 14 -0.8660254 -0.1000000 0.5291 5 0.28272 -1.71578 2.76477 15 -0.7071068 -0.1414214 0.721 11 0.69331 -1 .16664 1.89483 16 -0.5000000 -0.1732051 0.87178 0.95337 -0.80452 1 .33927 17 -0.25881 90 -0.1931 852 0.96731 1.15457 -0.52683 0.94268 18 0.0000000 -0.2000000 1 .ooooo 1 .32438 -0.29861 0.67333 19 0.25881 90 -0.1931 852 0.96731 1.47200 -0.09875 0.54733 20 0.5000000 -0.1732051 0.87178 1.61238 0.10933 0.59990 21 0.7071 068 -0.1 41421 4 0.721 11 1.78155 0.39549 0.89822 22 0.8660254 -0.1000000 0.5291 5 2.05236 0.90329 1.62716 23 0.9659258 -0.0517638 0.32297 2.56998 2.04295 3.41399

38

TABLE 5. Coordinates and fundamental s o l u t i o n s using procedure B for a cascade of 20% t h i c k e l l i p t i c a l contours w i t h the p i t c hr a t i o equal t o 1.0, a t an angle of a t t a c k as = 30".

V xV Y V s V

( 3 ) vV

0 1.0000000 0.0000000 0.20000 2.65199 4.03099 6.56037 1 0.9659258 0.051 7638 0.32297 0.61723 2.76872 4.46461 2 0.8660254 0.1000000 0.52915 -0.28286 1.71 543 2.76466 3 0.7071 068 0.141 421 4 0.721 11 -0.69333 1.16646 1.89479 4 0.5000000 0.1732051 0.87178 -0.95331 0.80444 1.33928 5 0.2588190 0.1931852 0.96731 -1.15445 0.52684 0.94273 6 0.0000000 0.2000000 1.ooooo -1.32421 0.29869 0.67341 7 -0.25881 90 0.1931852 0.96731 -1.47179 0.09888 0.54741 8 -0.5000000 0.1732051 0.87178 -1.61213 -0.1091 4 0.59997 9 -0.7071068 0.1414214 0.721 11 -1.73125 -0.39523 0.89825

10 -0.8660254 0.1000000 0.5291 5 -2.051 97 -0.90289 1.62714 11 -0.9659258 0.051 7638 0.32297 -2.56941 -2.04226 3.41384 12 -1 .0000000 0.0000000 0.20000 -2.65199 -4.03099 6.56037 1 3 -0.9659258 -0.0517638 0.32297 -0.61 723 -2.76872 4.46461 14 -0.8660254 -0.1000000 0.52915 0.28286 -1 .71543 2.76466 15 -0.7071068 -0.1414214 0.72111 0,69333 -1.16646 1.89479 16 -0.5000000 -0.1732051 0.87178 0.95331 -0.80444 1.33928 17 -0.25881 90 -0.1931852 0.96731 1.15445 -0.52684 0.94273 1 8 0.0000000 -0.2000000 1.ooooo 1.32421 -0.29869 0.67341 19 0.25881 90 -0.1931852 0.96731 1.471 79 -0.09688 0.54741 20 0.5000000 -0.1732051 0.87178 1.61213 0.10914 0.59997 21 0.7071 068 -0.1414214 0.721 11 1.78125 0.39523 0.89825 22 0.8660254 -0.1000000 0.52915 2.051 97 0.90289 1.6271 4 23 0.9659258 -0.0517638 0.32297 2.56941 2.04226 3.41 384

39

TABLE 6. Coordinates and fundamental so lu t i ons us ing procedure A f o r a cascade using contour No. 9 w i t h p i t c h r a t i o 1.0, a t an angle o f a t tack as = 40°.

V xV Y V sV

(1 1 vv

(2) vV

(3) vV

0 0.9992000 0.0011000 0.01772 13.12884 6.35601 21.44227 1 0.9877000 0.01 68000 0.14368 -9.19287 -2.68695 -9.0741 1 2 0.9480000 0.0540000 0.271 22 2.91 501 2.05747 6.68050 3 0.8806000 0.1090000 0.38955 3.41 342 1 A8176 5.07616 4 0.7889000 0.1769000 0.47799 0.59877 1.31449 4.22795 5 0.6780000 0.2529000 0.5441 2 0.01610 1 .17992 3.73079 6 0.5524000 0.3289000 0.57134 -0.48377 1.04014 3.30015 7 0.41 93000 0.3960000 0.55883 -1.15351 0.89762 2.91 561 8 0.2850000 0.4279000 0.50321 -2.21 244 0.54864 2.15021 9 0.1625000 0.3976000 0.46855 -2.58026 0.00501 1 .49058

10 0.0710000 0.3273000 0.421 69 -2.38235 -0.48263 2.12001

11 0.01 70000 0.2368000 0.37966 -1.87626 -1.01275 3.48586 12 0.0000000 0.151 0000 0.29398 -1.31 187 -1.60843 5.29746 13 0.01 55000 0.0821 000 0.25801 -0.50998 -2.00905 6.54315 14 0.0575000 0.031 0000 0.25956 0.40968 -2.01 880 6.56958 15 0.1231000 0.0024000 0.29878 1.26563 -1 .70812 5.61 329 16 0.21 45000 0.01 70000 0.43682 1.4421 1 -0.86805 2.97636 17 0.3347000 0.0682000 0.53395 1.20191 -0.36562 1.38960 18 0.471 3000 0.1009000 0.54022 1.27164 -0.20600 0.93765 19 0.61 19000 0.1087000 0.52872 1.48617 -0.08848 0.77 702 20 0.7424000 0.0945000 0.47033 1.81 352 0.03095 0.81334 21 0.8510000 0.0641 000 0.38726 2.57623 0.28866 1.45509 22 0.9323000 0.0330000 0.27299 3.91 447 0.83235 3.10944 23 0.981 2000 0.0089000 0.14284 -8.53086 -3.87085 -12.60546

40

TABLE 7. Coordinates and fundamental s o l u t i o n s using procedure B f o r cascade using 9 w i t h p i tcha contour No. r a t i o 1 .0 ,

angle of a t t a c k as = 40".

V xV Y V s V

0 0.9992000 0.0011000 0.01 772 37.69728 11.64376 1 0.9877000 0.01 68000 0.14368 3.56773 1.84881 2 0.9480000 0.0540000 0.27122 1.26393 1.28465 3 0.8806000 0.1090000 0.38955 0.33291 1.10863 4 0.7889000 0.1769000 0.47799 -0.16430 1.03142 5 0.6780000 0.2529000 0.5441 2 -0.621 47 0.94657 6 0.5524000 0.3289000 0.57134 -1 .KO35 0.81358 7 0.41 93000 0.3960000 0.55883 -2.08929 0.58726 8 0.2850000 0.4279000 0.50321 -3.23053 0.21872 9 0.1625000 0.39 76000 0.46855 -3.14004 -0.18241

10 0.071 0000 0.3273000 0.421 69 -2.63424 -0.56694 11 0.01 70000 0.2368000 0.37966 -2.02799 -1.06606 12 0.0000000 0.1510000 0.29398 -1 .43435 -1 .65006 1 3 0.01 55000 0.0821 000 0.25801 -0.63046 -2.04477 14 0.0575000 0.0310000 0.25956 0.29985 -2,05118 15 0.1231000 0.0024000 0.29878 1.15424 -1.73264 16 0.21 45000 0.01 70000 0.43682 1.33547 -0.89292 17 0.3347000 0.0682000 0.53395 1.11637 -0.37605 1 8 0.471 3000 0.1009000 0.54022 1.18476 -0.21 340 19 0.6119000 0.1 087000 0.52872 1.37230 -0.08893 20 0.7424000 0.0945OOO 0.47033 1.64536 0.02994 21 0.851 0000 0.0641 000 0.38726 2.20081 0.20389 22 0.9323000 0.0330000 0.27299 2.96024 0.45854 23 0.981 2000 0.0089000 0.14284 5.2401 0 1 .14526

a t an

( 3 ) vV

38.13269 5.92096

e 4.08919 3.50984 3.25466 2.98701 2.58061 1.93000 1.10587 0.90001 1.86857 3.31108 5.19100 6.41 553 6.50051 5.51 976 2.12402 1.34830 0.92670 0.7421 7 0.77741 1 .09959 1.77294 3.89806

41

TABLE 8. Coordinates and fundamental s o l u t i o n s us ing procedure B for a cascade using contour No. 10 w i t h p i t c h r a t i o 1.0, a t an angle o f a t t a c k as = 40".

'V xV Y V sV

(1) vV

(2) vV

(3) vV

0 0.9993000 0.001 3000 0.01632 38.75262 13.09774 42.98791 1 0.9870000 0.01 61000 0.14283 3.24950 1.79803 5.781 14 2 0.9462000 0.0520000 0.27065 1.14987 1.23524 3.94024 3 0.8797000 0.1070000 0.38547 0.34619 1.13135 3.59244 4 0.7878000 0.1738000 0.47886 -0.17430 1.03952 3.28696 5 0.6762000 0.2487000 0.54096 -0.63922 0.95147 3.01 047 6 0.5509000 0.3220000 0.56321 -1.23809 0.82549 2.6301 5

7 0.41 55000 0.3780000 0.54935 -2.08386 0.54699 1.84001 8 0.2808000 0.3994000 0.49560 -2.96166 0.181 19 1.05685 9 0.1607000 0.3699000 0.45859 -2.9 7790 -0.19227 1.00905

10 0.0701000 0.3030000 0.40250 -2.57497 -0.59296 1.95930 11 0.01 72000 0.2209000 0.34995 -2.07299 -1.11039 3.48879 12 0.0000000 0.1 390000 0.29170 -1.46601 -1 ,70277 5.35685

13 0.01 59000 0.0713000 0.25097 -0.61364 -2.12962 6.70415 14 0.0587000 0.0235000 0.25284 0.34280 -2.05679 6.64272 15 0.1259000 0.0004000 0.30446 1.12108 -1.64222 5.24750 16 0.21 85000 0.01 65000 0.431 50 1.26588 -0.85776 2.81777 17 0.3380000 0.0629000 0.52802 1.10984 -0.391 04 1.39926 18 0.4745000 0.0950000 0.53918 1.15905 -0.21 596 0.94097 19 0.6140000 0.1033000 0.52388 1.32910 -0.09451 0.75926 20 0.7440000 0.0907000 0.46843 1.56536 0.02991 0.78995 21 0.851 9000 0.0620000 0.38121 2.061 46 0.20363 1.12590 22 0.9321 000 0.0314000 0.27075 2.87368 0.50706 1.95443 23 0.981 4000 0.0087000 0. '14216 5.12538 1.27646 4.36522

42

TABLE 9. Coordinates and fundamental s o l u t i o n s using procedure B f o r a cascade using contour No. 12 w i t h p i tch r a t i o 1.0, a t an angle of a t t a c k as = 40".

V X S ( 1 ) ( 2 ) ( 3 ) V Y V V vV vV vV

0 0.9993000 0.0011000 0.01879 31.11854 10.231 28 34.08404 1 0.9886000 0.0175000 0.14406 3.12023 1.72565 5.58351 2 0.9507000 0.0570000 0.27376 1.30847 1.33220 4.24986 3 0.8879000 0.1 189000 0.3961 6 0.45421 1.20684 3.83439 4 0.8000000 0.1953000 0.489 34 -0.10702 1.16994 3.70265 5 0.6885000 0.2750000 0.55118 -0.71061 1.05164 3.34226 6 0.5573000 0.341 5000 0.56737 -1 .43778 0.85039 2.73334 7 0.4153000 0.3775000 0.54992 -2.24461 0.46136 1.66382 8 0.2767000 0.3723000 0.51347 -2.7101 5 0.07539 0.96885 9 0.1569000 0.3259000 0.46682 -2.5341 1 -0.26705 I . 16685

10 0.0688000 0.2580000 0.38448 -2.2971 3 -0.67450 2.24471 11 0.01 70000 0.1812000 0.33077 -1.92031 -1.23455 3.91729 12 0.0000000 0.1060000 0.26044 -1.36916 -1.91326 6.0481 3 13 0.01 57000 0.0456000 0.22977 -0.44097 -2.39772 7.57305 14 0.0581 000 0.0060000 0.23029 0.69703 -2.45255 7.76483 15 0.1275000 0.0083000 0.34313 1.30706 -1,43418 4.59496 16 0.2293000 0.0 747000 0.57771 0.91 568 -0.5031 3 1.66119 17 0.361 2000 0.1502000 0.54427 0.64737 -0.20736 0.77785 18 0.4956000 0.1526000 0.51 243 0.86079 -0.19748 0.77831 19 0.6269000 0.1287000 0.50085 1.16605 -0.14432 0.75168 20 0.7479000 0.0946000 0.45558 1 .49237 -0.06510 0.71 421 21 0.851 4000 0.0584000 0.37668 2.04776 0.0841 6 0.94807 22 0.931 0000 0.0278000 0.27057 2.75786 0.34500 1.57397 23 0.981 0000 0.0070000 0.14064 5.3361 8 1.20478 4.30404

43

IIIIII llllIIllllllII I I1 I1 I

V

t I

FIGURE 5. Labeling of the pertinent quant i t ies f a r upstream and downstream.

44

FIGURE 6. Cascade o f 20% thick e l l i p ses a t pitch r a t i o , t / a = 1 , and a t an angle o f at tack, as = 30".

45

I l l llIllIlllllIl I I I I I

Y

FIGURE 7. Contour 140. 9 ( ) and 10 (------- 1.

46

Y

t r FxI:

-0.5

F.WUKE 8. Contour 140s. 9 ( ) and 10 (------- ) w i t h p i t c h r a t i o t / a = 1 and as = 40".

47

-

I I l l 1 I llIllIlIllIlll I l l 1 I

FIGURE 9. Conpari son between the theoret ical and measured pressuredis t r ibu t ion f o r contour No. 9 with t / a = 1 , as = 40",

C1-m - -15", a+co = 4 9 . 6 " . Solid l i n e , method A ; dashed l i n e ,

method B ; ci rc l es , rtieas ured val ues according to [131.

48

0 0.2 0.4 0.6 0.8 1.0

FIGURE 10. Conpari son between the theoret ical and measured pressure dis t r ibu t ion f o r contour r40. 9 w i t h t/n. = 1 , as = 40",

= 0", = -69.6". Solid l i ne , method A; dashed l i n e , method B; c i r c l e s , measured values according t o [13].

49

I

FIGUIK 11. Comparison between the theoret ical and measured pressuredis t r ibut ion f o r contour !lo. 9 w i t h t/a. = 1 , as = 40",

cl-co = 15", = -69.6". Solid l i n e , method A; dashed l i n e ,

method 8; c i r c l e s , measured values according t o [13].

50

I "

0 0.2 0.4 0.G 0.8 1.0

FIGURE 12. Comparison between the theoret ical and measured pressuredis t r ibu t ion f o r contour No. 9 with t / a = 1 , as = 40°, cl-m = 30°, = -G9.Go. Solid l i n e , nettiod A; dashed l i n e , method 6; c i r c l e s , measured values according t o [13].

51

IIIIIII Ill Ill I1 I

FIGURE 13. Comparison between the t h e o r e t i c a l and measured pressured i s t r i b u t i o n f o r contour 140. 9 w i t h t / a = 1 , as = 40', a-m = 45', a+m = -69.6'. S o l i d l i n e , method A; dashed l i ne , method E ; c i r c l e s , measured values according t o [13].

52

t " I

2

O

-2

-4

-G

-8

-10

-12

-14

-16

-18 0 0.2 0.4 0.G 0.8 1.0

FIGURE 14. Theoretical pressure d is t r ibu t ion using rocedure B f o r the contour rlos. 9 ( ) and 10 (------- p , t / a = 1 , as = 40",

= O", and a+m = -72.5" f o r contour 9 and -70.9" f o r a-m

contour 10 (stagnation point a t t r a i l i n g edge).

53

I1 I I1 I1 Ill Ill1 IIIIIII I I

Y

FIGURE 15. Contour No. 12.

54

FIGURE 16. Contour No. 12, w i t h the p i t c h r a t i o t / a = 1.0, and a = 40".

S

55

0 0.2 0.4 0.6 0.8 1.0

FIGURE 17. T h e o r e t i c a l p r e s s u r e d i s t r i b u t i o n u s i n g p r o c e d u r e 13 for c o n t o u r No. 12 w i t h t / a = 1 , as = 40", a_, = 0", = -71.3'

( s t a g n a t i o n p o i n t a t t ra i 1i ng e d g e ) .

56 NASA-Langley, 1911 -1 F-702