On the problem of gas flow over an infinite cascade using Chaplygin's approximation


Material Information

On the problem of gas flow over an infinite cascade using Chaplygin's approximation
Series Title:
Physical Description:
16 p. : ill ; 27 cm.
Bugaenko, G. A
United States -- National Advisory Committee for Aeronautics
Place of Publication:
Washington, D.C
Publication Date:


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


The steady flow of a compressible fluid past two-dimensional infinite cascades is obtained by using the Chaplygin's simplified pressure-density relation.
Includes bibliographic references (p. 14).
Sponsored by National Advisory Committee for Aeronautics
Statement of Responsibility:
by G.A. Bugaenko.
General Note:
"Report date May 1951."
General Note:
"Translation of "K voprosu o struinom obtekanii beskonechnoi reshetki gazon v priblizhennoi postanovke S.A. Chaplygina." Prikladnaya Matematika i Mekhanika, T. XIII, No. 4, 1949."

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University of Florida
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All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003874033
oclc - 156912677
sobekcm - AA00006205_00001
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Full Text
lc- fr \ I^ 1





By G. A. Bugaenko

1. Some well-known results of Chaplygin's method (reference 1)
are first presented. For the adiabatic law of the state of the gas
when p = kP, the following relations hold:

P = PC(1 a
> (1.1)

p = 00 2

where p is the gas pressure, p is the gas density, V is the
modulus of velocity, p0 and P0 are the values of p and p at
the critical point of the flow-at which the velocity becomes zero,
k is the coefficient of proportionality, ? is the ratio of specific
heats, and a and g are constants.

If the angle 6 between the velocity and the x-axis and the
magnitude T equal to V2/2a are considered, then, as was shown by
Chaplygin, the equations of gas motion assume the form

5" 2"T 0e
Sw (l-T) 3P I,

bT = 2-(1-T)+l e'
04p 1-(20+1)T OV

where 9p is the velocity potential and W is the stream function.

*"K Voprosu o Struinom Obtekanii Beskonechnol Reshetki Gazom v
Priblizhennoi Postanovke S. A. Chaplygina." Prikladnaya Matematika i
Mekhanika, T. X7II, No. 4, 1949, pp. 449 456.

. -1

NACA TM 1298

Chaplygin reduced these equations to the very simple form

60 6e

a2 =_- 1 6e
cp K 6V

by introducing the new variable 0 and the constant K defined by
the formulas

2= f I dT
2 (1.4)

K = 1-(2P+1) |

Chaplygin showed that for velocities far removed from the velocity of
sound, the magnitude K is approximately unity and equations (1.3)
can be integrated by assuming K equal to 1. For K = 1, these
equations go over into the conditions of Cauchy-Riemann; hence,
w = 0 + iT will be an analytical function of the complex variable
f = c + i*.

The equation for the elementary vector along a streamline in
the approximate treatment has, as is known (reference 3), the form

dz = (aei + bei)d (a = b = (
2A 2 2 2 A 2 2 )

The complex pressure is given by the equation (reference 3)

Y +i = 02(e-i e ei)d) + p0(l T2)B+ d1 (1.6)

2. The steady potential flow of a gas through an infinite
cascade according to the well-known scheme of Kirchhotf with
separations of the jet is next considered. The vanes of the cas-
cade will be assumed to be plane (fig. 1).

Velocity of the gas in the flow at infinity is denoted by
V,1 and its angle with the x-axis by Ql; velocity of the gas in
the jet at infinity is denoted by V62 and its angle with the

NACA TM 1298

x-axis by Q,2. The angle e between the velocity vectors and the
x-axis lies within the range -21
The velocity field of the flow repeats itself for each dis-
placement by the pitch of the cascade, that is, by the vector he-ik

The condition of constancy of the mass flow for steady flow of
the gas gives

Q = pl V,1 h sin (X+ -l) = Pg2 =V2 n (2.1)

where pl is the density of the gas at infinity in the flow, p,2
is the density of the gas at infinity in the jet, and n is the
width of the gas jet at infinity.

.y making use cf expressions (1.1) for p, equation (2.1) can be
represented in the form
( 2 P
V h (1 2- sit (x+ Qe) = v2 n 1 2- 2 (2.2)

The behavior of the function f = O + iy in the z-plane of the
gas flow is now considered. For simplicity, the function f is
assumed equal to zero at the critical point 0 of the flow (fig. i).
From the relation

d9 = ds = Vs ds

it follows that on moving along the streamline W = 0, the function
9 varies monotonically from -c at the point E (infinity in the
flow) to +c at the point C (infinity in the jet) and passes
through the zero value at the critical point 0 where the stream-
line branches. The value of the potential f = + iV at the
critical point 0' displaced by the period he- relative to
the point 0 is found and (fig. 1)

(0') V ds d = V, ds + Vs ds + Vs ds

where NI4' is a.cut parallel to the axis of the cascade.

NACA TM 1298

The first and last integrals mutually cancel and therefore
as MM' approaches infinity in the flow, the following equation is

V (0') = Vl h cos (X + e,,)


Furthermore, from the relation

where dQ is the quantity of gas flowing in
infinitely near streamlines, it follows that
by the pitch of the cascade the function n'
equal to Q/PO. Hence,

(0o) = = (1

- 12 l

unit time between
on being displaced
receives an increment

hV sin (X + e6l)

In this manner, the f-plane with double-sided cuts along the
half-straight lines parallel to the axis of reals (fig. 2) corresponds
to the region of the gas flow (fig. 1). All the cuts, because of the
rule by which the cascade was constructed, are obtained from the
initial one (the positive c@-axis) by simultaneous displacement along
verticals and horizontals at distances that are multiples of Q/PO
and V i h cos (X + 8 i), respectively.

* Because w = e + io is an analytic function of
the problem may be solved by relating these functions
of a parameter that varies in the upper semicircle of
as in the Levi-Civita method.

f = p + i*,
with the aid
unit radius,

By considering the rectilinearity of the cuts in the f-plane,
the analytic function f(t) is found, which brings about the con-
formal transformation of the f-plane into the semicircle t. In the
t-plane, the flow of an ideal fluid about the boundary of the semi-
circle is constructed. For this purpose, sources and vortices of
strengths and intensities are located, as shown in figure 3, at the
points t, and t.-l that are symmetrical with respect to the circle
and at the mirror reflection of these points in the diameter, that
is, at the points ,s and t,-1. At the origin of coordinates we
place a sink of strength 2q (and a similar sink at infinity).


IACA TM 1298

In the constructed flow in the t-plane, the upper semicircle of
unit radius and the diameter of the semicircle are, of course, stream-
lines so that the stream function W maintains a constant value at
the boundary of the upper semicircle t; in the f-plane, this
bcundary will correspond to straight cuts. The complex potential
of the constructed flow will have the form

f(t) = 1('Y + iq) log (t t) + (-y + iq) log (t +

(- + iq) log (t + ( + iq) log(t- o) 2iq log t + constant


f(t) = 2- ( + iq) log t + 2- 2M -

(' iq) log (t + 2 + constant (2.5)

where Y is the intensity of the vortex and q is the strength of
the sources, and

M = t+ t,

The arbitrary constant in equation (2.5) is chosen so that
f(t) becomes zero at a certain point t = eci, the position of
which will be subsequently determined.

Because the logarithm has multiple values, the upper semicircle
of the t-plane will correspond to an f-plane with an infinite number
of straight cuts v = kq(k = 0, 1, 2,...), where q changes from
k? to +-, as easily follows from equation (2.5) by substituting
t = eie (the arc of the semicircle) and t = t1 where tl is the
real amount of the interval (-1, +l), the diameter of the semicircle.

In this manner, the function (2.5) establishes a conformal
mapping of the upper semicircle of the t-plane on the f-plane with
the double-sided cuts represented in figure 2.

The function (2.5) is used in the work of N. I. Akhiezer
(reference 2) where it is obtained by successive conformal mappings:
the f-plane on the half plane, the half plane on the unit circle,
and finally the circle on the upper semicircle.

NACA TM 1298

The point t = 0 is carried by the transformation (2.5) Into
f = +m so that the radii AC and BC go over into the infinite
segments (figs. 2 and 3). The conformal property of the transfor-
mation breaks down at the points t = -1, t = +1, and t = eic
(the point eic corresponds to the origin of the double-sided cut
in the f-olane). The condition df/dt = 0 for t = eic gives

7 + iq 7 i(2.6)
=- (2.6)
cos C-M cos c-M

In order to obtain the elements of the motion, an expression for
the derivative df/dt is required that is represented in the form

df q (t-eit)(t-e-if)(t-t-1)
dt (t- (t--)(t-)(t- 1)(2.7)

The quantities 7 and q are next determined. When a point in
the z-plane of the gas flow is displaced by the pitch of the
cascade he-1 the point t, corresponding to the point in the z-plane,
goes over from one sheet of the Riemann surface to the next, passing
once around the point E (t = t<), as a result of which the function
f(t) receives an increment 7 + iq, as follows from equation (2.5).
Because the corresponding increments of the functions P and J
are equal to Va1 h cos (h + Q~1) and Q/P0, respectively, the
following equations are obtained:

7 = 1l1 h cos (h + %,)

( T r (2.8)
q = 1 2 V1 h sin (X + 8,)

From the expression for 7, it is evident, among other things,
that 7 = 0 corresponds to the case where the approaching flow has a
velocity at infinity perpendicular to the axis of the cascade.

3. The function w(t) is next determined. The function
w = 9 + ic is regular within the semicircle t and has the following

1. At the point 0(t = e1c), the real part of the function u
has a discontinuity, equal to t, because of the branching of the
streamline. On the arc AO the angle e is equal to -it and on
the arc OB it is equal to zero.

NACA TM 1298

2. On the real diameter of the semicircle, the function
u(t) = e + iT is real because its imaginary part J = 0 on the
free jets where T =T2

3. At the origin of coordinates t = 0, the function W(t)
is equal to A62 because at infinity in the jet 0 = 0 and
C = Co2

From the preceding discussion, it follows that the function
u(t) admits of analytical continuation in the lower semicircle and
may be obtained by the Schwarz formula. Thus,

i eip + t 1 eW +t t ei _- t
W(t) a diP d = i log (3.1)
2) f ei T- t 2 ei t 1 -tei
It = 1 -E

that branch of the logarithm being chosen that is equal to ic for
t = 0. If the third property is used from equation (3.1) for t = 0,
it is found that e = -9e2. The value of to is obtained from
equation (3.1) by making use of the value of the velocity at infinity
in the stream

1 to exp ioe2
w(t,) = ~l + iCl = i log exp i 1 -
exp i'662 to

whence (reference 2)

cho ,1 cos (0l 62)
t cho cos ('e1 + e2)

shcOil sin 9 2
arg to = arc tg c cos e -cos
ch(col cos 0-2 cos 9 1

The pressure of the gas on a blade of the cascade is then com-
puted. On-the forward side of the plate, which is a streamline, the
pressure is obtained by equation (1.6) where Vg2 is the velocity
in the stream.

The back side of the plate is in the gas at rest where the
pressure is constant and equal to

P = p0 (1 T0 +1

NACA TM 1298

If the fact that along a streamline dtp = df is considered, a
formula is obtained for the complex pressure in the form
Po V02 -
Y + iX = 2 f (e-i e" -) df

SiX = 2 I(e i e') df dt
-2 f dt


where the integration is taken over the upper semicircle of the
t-plane in the clockwise direction. By considering that after
analytical continuation in the lower semicircle the function W(t)
assumes conjugate values at conjugate points, the following relation
is obtained:

Y ix = -

Po V2

S 1
Iti = 1

ei((t) df dt

where the integration is taken over the entire arc of the unit
circle in the counterclockwise direction.

Substituting the value of df/dt from equation (2.7) in
equation (3.4) gives

Y iX = 2
Itl = 1

The function under the integral sign in equation (3.5) has
three poles, t = to t = t., and t = 0, all of which lie within the
unit circle. The residues of the function at these points are,

(t, ei )(t, e-i)

( c Zo)(tGco e -)1

(Eo- ell)(E.- e-rC)



exp iWl1

exp iZ.l

-exp ie 2

eiu(t) (t-eic)(t-e-i) (t-t-l)dt
(t-tw (t-to-l) (t-t< (t-% -I)

(I toc)(E, tj 1)

NACA TM 129~

From equation (2.6), however, it follows that

(t,- eic)(to,- e-ic)
7 + iq
(to- Z) ( -1) 2iq

Hence, the residues may be represented in the form

7 + i exp(ie,, -O l)

7 exp(ieO1 + ~c1)

exp iG 2

From equation (3.5), applying the theorem on residues gives

Y X = 2 [- 21q exp 1ie2 +

(7 + iq) exp(i61 -0Ga) ( iq) exp(ie01 +o ml)

If the real and imaginary parts are separated,

Y= [2q sin %2 + (7 cos e1 q sin Qcl) exp(- O1) -

(7 cos 1 + q sin Ql) expO 1] (3.6)

S= -2 2q cos ^2 + (7 sin 06l + q cos 6,l) exp(-o.l)-

(7 sin Oi q cos e8,) expOC,] (3.7)

The velocity on the contour of the blade is assumed to remain
finite; the total-pressure force on the blade will then be perpen-
dicular to the velocity and therefore X = 0, that is,

NACA TM 1298

(7 sin e,, + q cos 6l1) exp(- a, ) -

(7 sin Ql q cos ql) exp Ogl = 2q cos (2 (3.8)

Thus the pressure force of the gas on a blade of the cascade is
by equations (3.6) and (2.8) equal to

POV iV V i V2\P,
Y = =2 h 2(l sin( + ,,i) sin eQ2 +
/ 2a)
[cos( X+ 01) cos Gl + 1 I sin (0 + 10l) sin ol ]s exp(- ,) -

[cos(X+ e1) cos 6 1 2! ) sin (x+ 0e1) sin 0a1 exp Gol


If in equation (3.9) 3 is set equal to 0 and the
magnitude 0,,l, determined by equations (1.4), is correspondingly
replaced by
dTr i og 109. = log
log log -
1 2 'og2 -2

the formula for the pressure is obtained for the case of an ideal
fluid (reference 2).

In order to determine the angle X, entering equation (3.9),
between the axis of the cascade and the x-axis, the ratio (2.6)
and the values of Y' and q from equations (2.8) are used. Thus

ce i o 2 cos (M + M) (3.0)
ctg(\ + eml) = i 1 M -.10)
2c 2 M /

In order to compute the length of a cascade blade, equation (1.5)
is used. Replacing dcp by df gives

dz = a tei b t ei+ (t eiC)(t e-i)(t t-^)dt
= eic- t teiC i/ (t t.)(t %-) (t to- )t .-1)

NACA TM 1298

This expression may be put in the form

dz = a gl(t)dt + 2b g2(t)dt





(t-t) (t-tw-1) (t-Z,) (t-t-z1)

= e-i(t-ei)2 (t-t-1)
(t-t) (t-t (t-t)(t- 1

expansions of gl(t)

are of the form

g,U(t) -= + -
t-t= I

and g2(t) into the sum of simple


+-- 1

+ t

(V = 1,2)

Al = C2 = Y exp(iEl1 al)

B1 = D2 = isC- exp(iew1 + o0,l)

E1 = -e-i

C = A2 = 7 exp(aol iol)

D1 = B2 = 1- exp(- 161, -Oml)

E2 = -ei

These expressions for the coefficients are obtained if the
following relations are used:





NACA TM 1298

exp iul =

(t-eif t-e-L i +1
(to-U) (L- -1I) 2iq

exp ie-i = l

(t-e i( e- ) i-

(E-t (b-t 1-) 21q

If equation
the t-plane in a
z(-1)-z(l) = Z
a cascade blade

1 =

(3.11) is integrated over the upper semicircle in
counterclockwise direction and if relation
is used, the following expression for the length of
is obtained:

A --t -- -l-t1
[S Allg -- + Bllg + C0lg +
S1-t 1-1 l-t-

Dllg +

qb A21

D21g -

E1lg (-1) +

+ B2g +


E2lg (-1)

NACA TM 1298

t +1
(A1 + Cl)lg o- +

(B1 + D)ig -1
0to -1

E91l(+lB] + ELA2 2g -_ -1

(B2 + D2)lg

= q (A, + B1
Ti I

to -1

+ (A2 + B2

+ Cl + D1)lg

+ E2)lg (-1)]

R, +

ia (Al + Cl B1 D1) + (A, + B1 + E1)

i I +

(A, + B1 + C1 + Dl)lg R +

ia (A1 + C, B1 D1) + (Cl + D1 + E2)


The magnitudes R1, R2, and a that enter this equation are
shcwn in figure 4. Substituting the values of the coefficients and
making use of equation (3.8) g-ies the following expression for
the length of a blade:

+ = + 2 (a + b) cos G-2 Ig R +
It R2

a (a + b)[(7 cos 8 q sin cxI) exp(- Ol) +

(y cos 9,1 + q sin e.1) expol]+

a-b [(7 cos 0oS q sin ei) exp(- a01)-

(7 cos 8,,o + q sin e~1) expO,1]+ qa sin eg2

= qa

14 NACA TM 1298

The formula for the length of a blade in the case of an ideal fluid
(reference 2) is obtained from the preceding equation for P = 0

a =-

b =0

q = -Vl h sin (X + e0)

7 = V,1 h cos (A + 0l)

icol = Ig V


1. Chaplygin, S. A.:

Gas Jets. NACA TM 1063. 1944.

2. Akhiezer, N. I.: On the Plane-Parallel Flow through an
Infinite Cascade. No. 2, Nauchnie Zapi 3: Kharkovskogo
aviainstituta, 1934.

3. Slezkin, N. A.: On the Problem of the Plane Motion of a
Gas. No. 7, Uchenie Zapiski Moskovskogo Uni-ersiteta, 1957.

Translated by S. Reiss
National Advisory Committee
for Aeronautics

NACA TM 1298

Figure 1.

- I

- ---0 ~1


Figure 2.


NACA TM 1298



Figure 3.

Figure 4.

NACA-Langley 5-11-51 1000



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