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