Generalization of Joukowski formula to an airfoil of a cascade in compressible gas stream with subsonic velocities

MISSING IMAGE

Material Information

Title:
Generalization of Joukowski formula to an airfoil of a cascade in compressible gas stream with subsonic velocities
Series Title:
NACA TM
Physical Description:
16 p. : ill. ; 27 cm.
Language:
English
Creator:
Loĭt︠s︡i︠a︡nskiĭ, L. G ( Lev Gerasimovich )
United States -- National Advisory Committee for Aeronautics
Publisher:
NACA
Place of Publication:
Washington, D.C
Publication Date:

Subjects

Subjects / Keywords:
Aerodynamics   ( lcsh )
Aerofoils   ( lcsh )
Genre:
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Abstract:
It is shown that the ordinary Joukowski formula for lift force of cascade blades in incompressible flow can be applied to the case of subsonic compressible flow with sufficient accuracy, provided that the density in the formula is taken as the arithmetic mean of the densities far ahead of and behind the cascade.
Bibliography:
Includes bibliographic reference (p. 15).
Funding:
Sponsored by National Advisory Committee for Aeronautics
Statement of Responsibility:
by L.G. Loitsianskii.
General Note:
"Report date September 1951."
General Note:
"Translation of "Obobshchenie formuly zhukovskogo na sluchai profilia v reshetke obtekaemoi szhimaemym gazom pri dozvukovykh skorostiakh." Prikladnaia Matematika i Mekhanika. Vol. XIII; No. 2, 1949."

Record Information

Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003779967
oclc - 99056333
System ID:
AA00009231:00001


This item is only available as the following downloads:


Full Text
UpA(Tm- 130f







SC ? 37 7i1



NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS


TECHNICAL MEMORANDUM 1304


GENERALIZATION OF JOUKOWSKI FORMULA TO AN AIRFOIL OF A CASCADE

IN COMPRESSIBLE GAS STRFAM WITH SUBSONIC VELOCITIES*

By L. G. Loitsianskii


In the computation of the impellers of turbomachines, the Joukowski
formula is applied; according to the formula, the lift force of the blade
is equal to the product of the density of the gas, the mean vector of the
velocities ahead of and behind the cascade, and the circulation about the
blade.

The presence of high oncoming relative velocities of the gas flow at
the blade make it necessary to account for the effect of the compressi-
bility of the air on the lift force of the blade.

In 1934, Keldysh and Frankl (reference 1) showed that, in the case
of the isolated wing profile, the Joukowski formula retains its usual
form for both incompressible and compressible flow. As far as is known
to this author, no generalizations have yet been made to the case of the
airfoil in cascade.1

It is shown herein that, for subcritical values of the Mach number,
the Joukowski formula may, with an approximation entirely sufficient for
practical purposes, also be applied in its usual form to a cascade in a
compressible flow, provided that the density is taken as the arithmetical
mean of the densities of the gas at infinity ahead of and behind the
cascade.

This simple result may be useful for the computation of cascades,
especially in the solution of the problem of the resistance of a cascade
when it is necessary to separate the lift force from the total force deter-
mined by the momentum theorem (reference 2).


*"Obobshchenie Formuly Zhukovskogo na Sluchai Profilia v Reshetke
Obtekaemoi Szhimaemym Gazom pri Dozvukovykh Skorostiakh." Prikladnaia
Matematika i Mekhanika. Vol. XIII, No. 2, 1949, pp. 209-216.

Very recently, there had come to this author's attention the as yet
unpublished generalization of E. M. Berson, which differs from the one
given herein.







2 NACA TM 1504


1. Vector equation of lift force of an airfoil of a two-dimensional
infinite cascade in an ideal incompressible fluid. The flow about one
of the airfoils of a cascade situated in a tube of flow having at infin-
ity ahead of and behind the cascade the dimension equal to the pitch t
in the direction of the axis of the cascade is considered (fig. 1). The
density of the fluid is denoted by p, the velocities and pressures at
infinity ahead of and behind the cascade by wl, w2 and p1, P2,
respectively. When the momentum theorem is applied to the mass of fluid
enclosed in the flow tube between the infinitely removed sections al
and 02, the equation for the vector lift force may be written as


R = (pl-p2) T + p (F i) i P ( ." 2) W (1.1)


where the vector t is equal in magnitude to the pitch and is directed
at right angles to the axis of the cascade downstream of the flow.

The difference in pressures pl-p2 in equation (1.1) may be
expressed in terms of the velocity vectors at infinity by the equation


p p = (w22-w2) P- (W2+7wj) (_--l) (1.2)
1 2 2 '1 2 w1 2 1) (1.2)


The mean vector velocity 1m and the velocity wd characterizing
the vector change of velocity of the flow in passing through the cascade
are then introduced:




S- (1.3)
wd = v2 1Wj

Equation (1.2) then becomes


pl 2 = m wd (1.4)

The condition of the conservation of mass along the tube of flow
gives







NACA TM 1304


wl t w2 t = wm t





From equations (1.4) and (1.5), equation (1.1) takes the form


R = p (w wd) t ( ) w m ( ~) Px ( d) (1.6)


As is seen from equation (1.6), the vector R lies in the flow
plane and is directed perpendicular to wm toward the side determined
by the vector product on the right side of equation (1.6).

The magnitude of the vector R is given by


R = pwmwdt (1.7)


2. Possible transformation of Bernoulli formula for compressible
gas at subsonic velocities. In the flow of an ideal compressible gas
about a two-dimensional cascade, the condition of conservation of mass
flow along a flow tube yields


01 (t W) = P2 (t w2) (2.1)

and with the notation of equation (1.3), the equation for the vector
lift force may be given in the form


= (Pl-p2) t+ P (t w) v P2 ( 2 = 2 = 1(- P2 ( ;1) wd

(2.2)

where p1 and p2 are the densities of the gas far ahead of and behind
the cascade.

The first component of equation (2.2) is now considered. When it
is assumed that the motion is adiabatic, the known relations for the pres-
sure and the density are







NACA TM 1304


k





P k-i k-1
1


i k X2 k \4
SfO k+l 2(k+l)2



1 E2 2-k 4
0 k+l 2(k+1)2 J


where po and pO denote the pressure and the density, respectively,
of gas adiabatically brought to rest, k is the ratio of specific heats
of gas at constant pressure cp and constant volume c and X is the
ratio of the velocity of flow w to the critical velocity of gas a,,
expressed in terms of the velocity of sound aO in the adiabatic stag-
nation of the gas:


a,
(2.5)
F2 4 2k PO
a* ai 0 = k+ PO


From expansion (2.3), the difference of the pressures becomes


2 i p) 1 (112+22) + p] (2.6
p1 P2 = k-1 P0(%2-ro ) 2 %1 (- 1


where

1
p (X22_, 2)k/(k+1)


k
k-l k-1
1 k--i Xl


1 + 2(1 (X12+2) 2-k (2 14 + 122+X24) (2.7)
6(k+1)2

and X1 and X2 are the values of the parameter X ahead of and behind
the cascade.


k"
k-1 -1
-( k+1- ) 2







NACA TM 1304 5

From equation (2.4), the arithmetic mean pm of the densities pi
and P2 can be expressed as


(2.8)


Pm = (PliP2) = PO 2(k+l) b21 +


1
k- 2 k-1
k+l I


1 ( 2 2
2(k+71 1 2


k-i+ 1 1 k -1"
k+l 2 1


2-k
4(k+l) 2


(Xl4+X4)-


(2.9)


When equations (2.6) and (2.8) are compared,


k PO 2 2
P P2 = k+l 2 (2 1-
a*


Pm- Wd


(+m
VO


p P


0P

I ( I-


(2.10)


Use of equations (2.5) and (1.5) yield, respectively,


k P0
k+l POa*2


kpo 1
2 2
POO


2 2 1 -
w2 -1 W 2 m Wd

From equation (2.10) it follows that, with a certain error, the
order of which will subsequently be discussed, the differences in pres-
sures ahead of and behind the cascade may be determined by an equation
similar to equation (1.4) for the incompressible fluid if the density of
the gas is replaced by the arithmetic mean of the densities ahead of and
behind the cascade.


where


1p






NACA TM 1304


3. Generalization of Joukowski formula to flow of a compressible
gas at subsonic velocities about a cascade. When equation (2.1) is
used, the second component of equation (2.2) becomes


p(" .)d = [Pm m) + P+

= Pm( md +

-)
= Pm ^m P d +

/ m 12
( 2Pm


1T -- --
( m. w) m( m)] id
Lp- (T" m) Pm(i -m) Wd
- m J


/Pfl2
\ Pm


-)


% W1)Wdj

Pmc' Wm)Wd
PM 'w' Wd


In order to estimate the second component in the brackets, note
that, according to equation (2.4),

PI 2 2 2 12 )2 2(2 12 + 2)


(3.1)


P + P2 = 2p0 (1


= 1 (N _1 )2
4(k+1)


S1+


(12+X 2) +


k-i ( 2 ) + .]


Therefore,


pl1(t. &)wd = pnm(' w)d Ep'Pm(t Wm)d


Hence,


( P1-Pp2
pl- 2


(3.2)


(3.3)







NACA TM 1304


where, from what has just been shown and when, for convenience,
K = l/(k-l),


P(l + P2
? 1 2ii4 p


1 (P1-P2 2
4 p 2
Pm


S- (k-,)/(k+l)12 1 (k-)/(k+1X22


1[ --1)(k-h+i1)l2]K 1-(k-1)/(k+1)A2 K[


1 (2- 2_1)
4(k+l)2


1 + k-i
+1-+H


(hl 2 +2) .


(3.4)


The equation of the order of the
present; equation (2.2), on the basis
then be represented in the form


error c will be ignored for the
of equations (2.10) and (3.3), may


R = pm (wm'. ) Pm m) d + R Pjm (t xd) + R (3.5)


where ,R represents a certain small vector


PO
R p (P-) m (m" d)
m


t+ p'pm (t Wm) Wd


the order of magnitude of which will subsequently be discussed.

When this small vector ER is neglected, generalization of the
Joukowski formula to the flow of a compressible gas about an airfoil in
cascade yields


R = Pm m (x~ d)


(3.6)


(3.7)







8 NACA TM 1304

When equation (3.7) is compared with the previously derived equa-
tion (1.6) for the flow of an incompressible fluid about a cascade, the
following result is obtained: At subsonic velocities, the lift force of
an airfoil in cascade can be approximately determined by the Joukowski
formula for the incompressible fluid if the density of this fluid is
equated to the arithmetic mean of the densities of the gas at a great
distance ahead of and behind the cascade.
The absolute magnitude and the direction of the vector R with the
assumed approximation will now be determined and the difference from the
magnitude and the direction of the vector R determined for the incom-
pressible fluid by equation (1.6) will be shown.

The vector product t X wd is perpendicular to the plane of flow of
the gas and therefore to the vector wm' which yields


R = p m T X d (3.8)

In contrast to the incompressible case, in the compressible case the
vector wd is not perpendicular to the vector t, but is inclined by a
certain small angle to the axis of the cascade (fig. 2). Hence, in con-
trast to the condition of perpendicularity (equation (1.5)) in the case
of the incompressible gas, equation (2.1) gives

T- T P- 1 (1 1 1
tiwd = t *(w2) = 2 PA + >(T P^) (- 2 p2


P2 2 1 2


Hence,

t Pi P2 Pi P2
= 2 = = 2 (3.9)
t w PI + P2 Pm p

Whence, from equation (3.2),


t d 2 7' m 1 2 2))(E m) [1+ j (3.10)
"Wd= 'm k + 1""







NACA TM 1304


Denoting the angles between the vector t and the vectors wm and
wd by em and Gd yields the following expression for the modulus



[tX dl= td sin 9d = t2 d2 t2wd2 cos2 9= t2w (2 d)


= t2 d2 -


4p' (~t )2


Stwd 1l 4c \ cos2 em
(WM m


= tW 2c-' ) cos2 9


The modulus of the vector R is thus given by


R = PmWmwdt


1- 2c~p' (


The direction of the vector R for compressible gas also remains
perpendicular to the vector of the mean velocity wm (fig. 2).

4. Estimate of accuracy of generalized Joukowski formula. In order
to answer the question of the possibility of application of the previously
derived equations, it is necessary to estimate the order of the neglected
terms and also the interval of Mach numbers in which, with a preassigned
accuracy, the proposed approximate formula may be applied.

For this purpose, consider the estimate of the order of magnitude of
the vector CR. The simplest method of estimating is applied in writing
the evident inequality

PO
R : p C PmWmWdt- cos emd + (p PmWmWdt cos Gm


or, when the cosines are replaced by their maximum values,


R< C I + C m0 dt


(3.11)


cos2 em
C o s ) M


W PmWmWdt


(3.12)


(4.1)







NACA TM 1304


The ratio


p /p is readily estimated from the known equations:
0OM


k-1 I
+ 2 M, =P2 1


k- 2 )
2


Setting M1 = M2 = 1 yields


PO




PO <2 P(


fk+ l
= 2



=2 \2)


Assuming, however, that M1 = M2 = 0 yields


PO > P1


PO > P2


The result immediately follows:


1k
K-k-i


The estimate of the remaining magnitudes entering the parentheses of
equation (4.1) are now considered. For this purpose, the following
simple device is used: The values k = 4/3 = 1.33 and k = 3/2 = 1.5,
between which the region of most practical values of k is found (for
air and certain other gases), are considered and for these values of k,
the values of cp, C p and cp' are computed. Similar computations may
easily be made if desired for the interval 5/4 < k < 4/3, and so forth.


For the chosen values of
tions (2.7), (2.9), and (3.4)
is easily carried out. Then,


k, the exponents of the terms in equa-
become integers and the entire computation
for k = 4/3,


PO = p1 1


1
k-1


(4.2)


k-1 K
+ -


k-1
+
2


l< P Tm V-







NACA TM 1304


12)4 1 2 4
2 2 ( 1 7 2)
1 7 1 ~~ 2 '


- 1+ (x2 + X22)
14 1 2


7 (X22 12)



( 4 x2 2 + 2 4)
7 2


( 4 + \12 X 22 + 2 4)


1
4.73



p
P

s-






P 2
1
-7


. 2 )3
7 1


24 ) ( 3 6 26)


3 (\i4 + x
27


p 1 1 2
p- 4 P2


P02
2


1h2


(1 X2)


(h12 -_2)2


1 2
7 2


(i 1-
(1 -
7


(>66 + 4 X2 2 + 1224 + X26)



- + (3 2 + X2)
14 1 2


1 2)2
7


1 2
- 7X1


1
7T


221 2


~


1 P2 2
472 pM
-- f


+ 1-1\2
7 2







NACA TM 1304


and for k = 3/2,


1 2 1 2
(1 X12) (1 ( 1 ,22)
5 (52 2)
3 2 X12)
(\2 1


-1 ( + 2 2)


S1 (14 + X12 X22 + 24)
352


1 ? 2
-i\J


1 2
- ) _


+1 12+ X2)


= 2 (X4 + 24)
7-52 A2


1 1 Pl-P2 )2
p 4 2
m


Po2
pM2
49


1 P0
4-52 m/


Therefore,

cp p = -


Ep Cp
P p


1 2 -2
5 2) -2)


( 2 2 2 )
1x X2


1 1\ 2 2 2) ( 2 + 2
98 ( 1 X2 ) [1 14 1 + 2 )


1 ( 1a2 _2)2
150 2


for k = -
2


-2] 2
22)

S( 2
-1 ( 2 2 22
+ I\ 2


4
for k =
3
(4.3)
(4.4)


On the basis of the obtained formulas, an estimate of the expression


(4.5)


P-
P- Cp + Cp
Pm 01


Cp "


Ep =


gR
PmWmWdt






NACA TM 1304


for any values of k in the interval 4/3 S k S 3/2 is given: From
equations (5.2), (5.3) (Ed. note: equations (4.2) and (4.3)j] and so
forth, for k = 4/3,


+ 6 2 ( 12 22)2< P J + '


S9 12 2)2


0.033 ( 212 2)2 2 0


Ip + C' 0.133 ( X12 22)2


and for k = 3/2,


P1 + 1 (.- 2
iT^T-i^


12 -2)2< ,p Cp + Cp


2 1 41 (5 ) 4 4 2)2
[G 15) 2 4 12-X22


0.032 (\12 22~2< L P -CI 4 p'< 0.108 (12 2)2


Thus, for the magnitude ER, the estimate is


0.033 (K12 22)2


CR < 0.133 (Al X22)2
mmd


From the parameters %, and X2, the Mach numbers M1 = wl/a, and
M2 = w2/a2 are obtained, where al and a2 are the velocities ahead of
and behind the cascade. By the known equations of gas dynamics,


(4.6)


(4.7)


(4.8)







NACA TM 1304


2 k+l
1 2


2 2 k+l
2 2


M 2

1 + (k-1) M12

M2
1 + 2 (k- 2
1 (k-1) M2
i+


Whence,
2 2
2 k2 k+l M 2
2 T[I 1+ (k-1) M+ 1 (k-)) M22


Hence,


2l (M12 M2)2< (X12 22) 2
k+M1 -Mg)< A X2 )<- ^2 )


(MI2 M22)2


Thus, in the interval 4/3 < k < 3/2, the following estimate may be
used in place of equation (4.8):


0.03 (M12 M22)2 < R
mmm


<0.2 (M 2 M22)2


Equation (4.10) is a very rough estimate; in fact, the modulus of
the vector CR will be much less. From this simple estimate, however,
the possibility of practical application of the approximate equation (3.7)
is evident. Thus, for example, if at the entrance to a turbine cascade
the Mach number is equal to M1 = 0.2 and at the exit M2 = 0.7, the
relative error will not exceed 4 percent, even for the very rough estimate
assumed.


Translated by S. Reiss,
National Advisory Committee
for Aeronautics.


(4.9)


(4.10)








NACA TM 1304


REFERENCES


1. Keldysh, M. V., and Frankl, F. I.:
the Nonlinear Elliptic Equations
the Wing in a Compressible Gas.
est. nauk, 1934.


External Problem of Neumann for
with Application to the Theory of
Izvestia AN SSSR, Otd. mat. i.


2. Loitsianskii, L. G.: Resistance of Cascade of Airfoils in Viscous
Incompressible Fluid. PMM, vol. XI, no. 4, 1947. (Translation
available from NACA Headquarters.)







NACA TM 1304


Figure 1.


Figure 2.


NACA-Langley 9-4-51 -1000






L9 z
pf '"' d" 1n 'S
- a S2

C6 c 2 -.




2 i 3 B^ S
0 c



. 0 en _;Oen


14I.,


t


''* U1 se 1
- e.zi -: g 5



S :C W ;.

uommm
*oe- q. CV -^ l




E 5 .^cig
c4 v a
a o. I- -.vc
U U ) : ~

Ca C. ~U ~Z~ha-

- ; c'a d .z


0 o .


3a 0 m.-Ss ,





O,-U; o '

,. N .
sE P V"








g a in! o .


z w2O4OQ0. z* .4-:
z z a -.4na


L. 41
0 R ZE
al
a IUb ;11 _

ap
g~r-^
*2 2 w 5 *

E oS c
0 0" 0 ._*
s U.= wP


eon 0u i


C : m .





5 en- C-
0 C L V
ra ian r m





0 "1P aI
>, vi4fru

-*0 wQ m -
OCU ea,
( a 0
i" rw
a-rgvrSsS
0 C. Irl
0 w 961
'nSE'-
usn 2
rz r Mn
C: w
V


0 .
H N :=






Wd >

0 ol .>


4wEE-. d
in a C.


o Nc a ; 8


oo 2 d



< w < -C CL


.4 en



s, _
0 V





0 W0ca 0
1.0 g t ;
CJ CU :

L. a, 6
a en C
C a 0 e ,







-- U= u = U


5 0
0 i 0 w l
S' w c! cu w "O 3 '
d" W -3n

1 M ..e \ 4 ill m .4 ; N \i o ; '2 c
0.I r;Q) -nw >" H 42cvcJ0gs >M


U u CM1 -* ar s E aui' : """ i
o n IfvP 1

m o -C k V 6 Cd 26a
P. rs u -- z 7 U
00 r-i 1. V)
g .qS" V g 3 g
C U azO... BCa C u aza.. .

- a -ij cO r OS


Ea
0 0






0 v




z E a
ten Co.
< M 3 > -,



0 n0 -
o o0o




z S z a aE'n A ;.
filssjiisf5?~


Ut

-.a



5 0




Cu"
3'C

cv



3 u z
E= E
Cu0L
Va t
.^cv

1vtm-
'-cI



ass

enacv


0 -
i Ed E : ,-





0 N -
U:&En 3 3) u -






o oo U








E -



UZ m 0 0 vM
SzOGC 0 2 0. a.i H>


oC
5 a C, E q


in 0 C3
3~0



aen rni. r -
en .U i


rO- .0 m*0

C c


Cu V

3 3 o !! l-"
-n en- a
o ",= -00
6'3a"-5





0 = g 5 4
582En g
Cuvr vi
cu
- EU "
0Um N S -'.
'Sm
~srnap,
~~m 1







eq -
'-CV Min




a k. 'a -CT 1c.26

aa r6
r 0 o <;- S






x. b E U ZU-
pf oi n U20


0r R
a: C :z gEu
Li L) 01
.< .0 *eq| -a






) 0 ur.





in >,Z In ;T
-O cu ca. C M
b U ,3






o o o
ci. [M C) O -c







in < 0 aj tn ^
eq E-
> -90 -LC4



cu n o.ku Cu 4,
oj0 CL a in F










U2 C.;u
0 : 11 M L








w 0'
C*3 ~ ~ auI iB- J ; C




a)NJ rz 5 u M




eq Q
L.,.
(jCu nO g ia nI







m Li



01 C 0 Jq CuO
a. U U O LM f-a












U~ 0 ;
3 -- Qe^ ^ r
5 U S.U 5E-."u









cci .c u
0j it Li 0 J w o






N o u




o& z
oQ Z 9= H












9-U~ 01gWo
o a 0 a 2 j 0

S00 .,4 M ^S 0














0
0* ;: wg 1 ..% 1 -













. o 0 o 5 -M
M 0 U -
> i N := 4g
-J fu *^ 3

z < a *SO a. L 1% A


<

I-
A4

w


o i.


t o w .u
-C.- u01

50 0 0 >
en~ 0
a Li U. Cu



0" :3 6 Q '0

Sm o uS


I. u Lo
-a r
Ln L 3. CU V
0 0o
>c;n >.Mu

M w3 E! um
to 0 3 Ca
C'5*
oS
L*f Un u tu5
04a Cu 11%a
CJOW~ifu
S- l| ~.
Cu~~o.. 0

Cu2" oA g
W S-CuSuU


Li 091'

-3 LLGJ.J..

in 0} t C0
it E -S p MO

5L, ,n po"
L 0 L.
0 23 E '0

ini

w VACuL
in Cuu
' 01 Cu -~ S%




Mc i V"T
- n & C


.0 U A 4,.0
0 gi C0
0 tS t-.=a
.2 ^
Cu' 5 v <


E in
Li i -& -

- a 1- --ra M
o c u


.2 c :-
L s. am .
L F- !z o

o0 a
I.. CM ) IS0.E
- eq g i


0





I-Ed

bi
L in 1









r Zz
-0~
01x
!2

Zu:
Q.S

"-0
S-U

1o
u z u
;.,o
0> -





A Z U.<


. 01
V


w. .0
Uo u 'A
M 2n






CCC

2 r
A a,."
M a) c c5
C 0 V t

Li c u



to ^u0 n "
.5 3~ o 5J I

o5tru5 -'

^1..5^ .


.-. 3ruS E2~ C







UNIVERSITY OF FLORIDA
3 1262 0 103 370




Full Text
xml version 1.0 encoding UTF-8
REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd
INGEST IEID EIQ6NCU0G_CNMA25 INGEST_TIME 2012-03-02T22:28:14Z PACKAGE AA00009231_00001
AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
FILES