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Algebraic self-modeling solutions of two-dimensional transonic gas flow
ALGEBRAIC SELF-MODELINGSOLUTIONSOF TWO-DIMENSIONAL TRANSONIC GAS FLOW (~GE~RAICHESKIE
AVTO~OD~‘NYE
RESHENDA
URAVNENII
OKOLOZWUKOVOGO~LOSKO~TECHEN~AGAZA) PMM Vol. 30, No. 5, 1966, pp. 848665 S.V. FAL’KOVICH and I.A. CBERNOV (Saretov) (Received
March 12, 1966)
Approximate equations of the flow of gas in the traneonic velocity range possess an important class of selknodeling solutions. Marty of the transonic flow properties such as, for example, the character of flow at some distauce from a body, the flow in Lava1 nozzles, etc., were analysed with such solutions as the main tool[l to 31. An analysis is made below of cases in which the self-modeling solutions are represented by algebraic functions. By resorting to parametric representation of the unknown magnitudes, it is poesible to indicate in all cases a form of solution convenient for gas dynamical computations. Certain exact solutions of the Tricomi equation have been obtained in this manner, solutions which may be need iu the analysis of new properties of transonic flows such as flow in a Lava1 nozzle with linked supersonic zones, flow in a nozzle with breaks in its wall, flow in the neighborhood of the intersection point of the sonic line with the sonic stream boundary, etc. 1. We shall consider a plane irrotational flow of a perfect compressible flnid iu which the velocities are nearly sonic throughout. Such a flow is approximated in the hodograph plane by the system of equations of the form [4] (1.1) Here $ is the stream function, cp is the velocity potential, 11is a velocity function which becomes zero at the critical velocity, and 0 is the angle of inclination of the velocity vector. System (1.1) is equivalent to the single Tricomi equation of the stream function (1.2) Let us consider the self-modeling solutions of system (1.1) of the form 133
Putting I) from the first equation of (1.3) into Equation (1.21, we obtain the following 1004
Self modeling solutions of tronsonic gas flow
1005
hypergeomet~c equation which defines the function f
E (1 - Qf” + w3 - ‘f&If
+ %k W2k + ‘lalf = 0
(1.4)
c =
(1.5)
The parameters of this eqnation are: a=
b =
--Yak,
II&
+
l/e,
“Is
It is convenient to present the solution of the hypergeometric equation in terms of the Riemann P-function which indicates the position of singnlar points of this equation, and the indices of singularities of this solution at such points
i”
O”
0
--Yak
l/a
“la k +
*
0 “I3
1
(1.6)
4 j
‘I2
In the neighborhood of singular points function f can be found by means of series. Thus, in the neighborhood of the singular point [= 1, two linearly independent eohio~ of Equation (1.4) can be presented in the form
fl = F C-‘l$, Y& + ‘la; ‘lo 1 - 43 fa = vm P (‘la - Vzk, “Is + ‘I&; ‘1s; 1 - U
f1.7)
The general solution of the hypergeometric equation contains two constant
f = Clfl + Gfa* Since 1 - F,= 02 / p-2, it follows from (1.3) and (1.7) that the stream function $ which corresponds to solution fi will be even in 8, while that pertaining to solution A, by virtue of the factor d/p will be odd in 8. It is easy to see that solution f, defines flows which are symmetric about the n-axis in the physical plane. The flow in a Lava1 nozzle, and flows at some distance from a body belong to the latter category. In order to find function g we substitute ti and Cpfrom (1.3) into Eqaations (1.11, and obtain two ordinary differential equations of first order with respect to f and g, from which we find g=
-
3&J
5”” (1 -
Q’V
(E)
(1.8)
Substituting for fits expression from (l.S), and using the formula for differ~~ating hypegeometric function, we obtain for g the expression (0 go-_-----
03
- %k-%
I5
3k + 1
1
a
‘I
o
(1.9)
‘12
2. We shall find for which values of k from (1.31, algebraic solutions of Eqnation (1.4) exist. We shall make use of the results obtained by Schwarz who had solved this problem for hypergeometric equations of the general type [6]. Firstly we shall find those valuee of k for which particular algebraic integrals of Equation (1.4) exist, when the second of the two linearly independent solutions may not be an aIgebraic function. This arises if, and only if one of the numbms -Y2kc ‘12k + 1/6, 1/2k + This condition determines the following values k = p,
131.
k = -
‘is + P
(P=*o,
2/3, and -‘/,k
1, 2,.
. .)
+
‘/,
is an integer.
(2.1)
Corresponding solutions in the form of polynomials can be found in Guderley’s book
S.V.
1006
We shall
now determine
hypergeometric differences
equation
Fai’kovich
and I.A.
values
of k for which
those
(1.4)
is algebraic.
h=1-c, approx~ated equation
to their (1.5)
occur
~=a-&
integral
considered
parts,
here
of the if the exponent
to the known
the following
(2.2)
Schwarz
five
table
[6f.
For the
possibilities
p =
t/z,
v
=
“12
for
X =
l/s,
I_L-
t/s,
v
=
‘62
for
‘k =lJe
h =
1/S,
E” =
t14,
v =
‘I2
for
k =I,$:!
h =
r/s,
j& =
I,!&,
y =
1/Z
for
k ==l/zo + p,
lc x -
11,& + p
(2.6)
h =
‘,$,
1-1=
2jr,,
“v =
‘,$
for
k =7,&
k =: -
‘:/3O -, I>
(2.7)
proved
defining
by Frank1
the flow
[2f,
the value
in a Lava1
nozzle
k
=I/3
symmetric
about
the x-axis
+
P
a curved
in the physical
(2.3) k = -
+ p, ‘- p,
k
+ p,
k = y from (2.31,
with
noted [4] that in this case the solution is in algebraic integral. Frank1 [s] gave later the whole family (2.3) flows
integral
if, and only
Y=c--_a--_b
conform
we obtain
the general
cases
h =. ‘In,
As was tions
Such
Chemov
l/x $ p
~---5/,~
leads
(2.4)
j-
(2.5)
p
to sel~odeling
transition
line.
solu-
Falkovich
had
form and found the relevant general and the form of integrals,describing With k = - S/J
plane.
from the same
family we obtain a solution for the fundamental singularity of a flow at some distance we have a particular integral which defines the from a body [l]. When k = 4/3, flow
around
derived also
a corner.
gave
particular
and k =
(2.51,
and (2.51,
however,
3. We shall metric
function
stituting
into
use
their
numerical
dealt
manner,
with [IO]
and had
particular for the
We reduce
indicated
we
the families
integrals,
computation
(1.4)
where
in (2.41, k = ~[!a in the same particular
algebraic
actual
equation
soIution
form by as in [8],
of k = l/6 [9] h ave derived
and Ryzhov
method
in (1.6).
the relevant
for the cases
in a different
work
the Schwarz
appearing
exist
Lifschitz
and k = I/IZ
that
in an analytical
obtained
which
in (2.6). l/6
incidentally,
[7}, was
integrals
i/30
for k =
integrals (2.4)
We may note,
by Yaglio-Laurin
of the hypergeo-
to the normal
form by sub-
it 12 (E)
For the determination
=
VJ (1 --
of h we obtain
Q1’4f G)
(3.1)
Equation
-$+Ih=O Let
fr and ft be two linearly
necessarily
congruent
of Equation
(3.2).
ratio
(3.2)
with
We introduce
of two particular
independent
solutions
(1.7).
into
our analysis
now
integrals correspond
the Schwarz
of equation
(1.4).
two integrals function
defined
not
h, and h, as the
integrals s =
We shall
particular To these
derive
the equation
fI : f2 = which
h, : ii,
is satisfied
(3.3) by function
s. As each
of the
integrals jr and fi contain two arbitrary constants, function s, which is determined with an accuracy of the order of the multiplication constant, contains three constants. The equation for s will, therefore, be of the third order. Denoting differentiation with respect to t by primes, we can write II,” Replacing second
equation
-,
h, in the first of (3.41,
iit1
=-- 0,
of above
we obtain
equations
1&.1f1 -;m lit, with
sh,.
and
- 0 taking
(3.4) into
account
the
Self modeling
=_2$
S”
7 Differentiating together
this equation
with Equation
(3.5)
of transonic
solutions
with respect
flow
1007
(3.9
t
to c,
for the elimination
gas
andusing
the second
of h, and its derivatives,
equation
of (3.4)
we obtain for s
the expression II, S --S_
If a particular obtain solution
solution
the solution
la% =
of this equation
Taking
into account
of the hypergeometric
3
of Equation
(3.6)
a=21
(3.2),
and then utilising
s (s')-g. The general
=
h 5
(s’)-‘lx
relationships
(3.1)
equation
(3.61
!
is found, then by integrating
*A of Equation
(s’)h,
SW
z-z-C
(C,
solution
s +
(3.51, we first
(3.31,
of (3.4)
the second
is then of the form (3.71
C,)
we can write down the generalised
integral
(1.4) (3.8)
4. Having the physical
found the solution
plane.
Reverting
in the hodograph
to variables
plane,
C$ and $,
we must transpose
we obtain instead
it back onto
of (1.1) the follow-
ing system (4.1) We shall limit our considerations plane parallel rforr$,yfor$,of the sonic
to the analysis
of flows
not much different
from a
flow along the z-axis of the physical plane. We cau then substitute in (4.11 u for ‘I, and v for 6, where n and w are the dimensionless components stream perturbation
velocity
along the axes
of the orthogonal
system x, y. With this condition, system (4.1) becomes approximate eqnations derived by Karman for transonic In the physical
plane the self-modeling
u =
ys(n-1) u
Here n is related
solutions
?,I=
(5),
equivalent flows [ll]. are expressed
y3wt
v(g),
coordinate
to the system
of
by 112 and 133 (4.2)
r;=: z3-n
to k of (1.31 by
3k+ f n=--7gFunctions
& $+
[I and V satisfy
the following
z--2@-4)U=0, d5
We substitute
System (4.4) dz 7
-
for U and V variables
is now equivalent
as follows
UC-=,
equations
1)V + .+o
[I41 (4.4)
[15]
z =
(4.5)
vsa
to
2(n-I)P+3tz-3~ 2t~-2nt-3(n-i)z
equations
of differential
u -f$-3(n-
t =
These
system
(4.3)
have the following
’ singular
df T= points
(ns -
t) dt
2t2-2nt-3(n-~)z at finite
distances
(4.6) from the origin
1003
S. V. Fal’kovick
A (t = 0, z = O),
IC (t =
and I.A. Chemov
r&z, z =
2/gzp,
D (t =
1, z =
--a/a)
(4.7)
Point A corresponds to the z-axis of the physical plane, while the singular point C represents the limit characteristic. Infinity in the t %plane, which we shall denote by point B, corresponds to the y-axis. Second equation of the system (4.6) shows that, when the integral curve t = t (7) of the first equation of (4.6) intersects line t = nr at a point other than the singular point C, then d[= 0. This indicates the appearance of a limit line in the corresponding flow. The r, -plane called the phase plane, is convenient for analysing flows in the presence of shock waves. Denoting parameters related to the two sides of a shock wave by indices 1 and 2, we write conditions at discontinnities as followa [16]
t, = 2n2 -
z, = “1 + znt, -
t,,
2113,
51 = 5s
(4.8)
To determine the streamline we must integrate the approximate equation
df
for
x=”
(4.9)
y = const
Is integrate the second equation of (4.1) over the area bounded by line c= const, andlinesy=O,x=-eo and y = ye. Using Green’s formula; we obtain
$(-$dy+
vds)=
(4.10)
0
Al urg y = ye the first term of (4.10) is equal to zero, and the second one, to y,, . The integral can be easily computed along line jr= const, if for u and v we substitute their expressions from (4.3) (4.11) The analysis carried out in section 2 established all possible csses in which the nonlinear system (4.4) can be solved in terms of algebraic functions. Corresponding valnes of n can be easily found by resorting to the transformation formula (4.3), and to results obtained in (2.1) and (2.3) to (2.7). 5. We shall now find the general integral for one single value of k from each of the infinite families (2.4) to (2.7). Other hypergeometric functions of the same family can be found by consecutive differentiation of the function thus obtained. Family (2.3) is assumed known [S], and will not be considered. We take k = l/6 from family (2.4). For this value of k there exists Scliwarz equation [6] which is Hr (s) = c + 2 J&s
F=[$$je*
Fr(s)=S4-2
J&2-
-
a solution of the
1 1
(5.1)
Polynomials H1 and F, are such that the following identity hdds H$ -
Frs = fZyX[T,
(s)P,
z+l(s) = s (s4 + 1)
Using Expression (5.1) for eand the identity (5.2), we find I-_E=-12
Differentiation of (5.1) yields
y’sT&
(5.2)
Self modeling solutions of transonic
dE --ds -
1009
gas flow
(S.4)
24 1/3T1H12~1-a
We shall now find function f by using Formula (3.8) j = Eqnations metric
function
(5.1)
and (5.5)
f = f (0
derive the expression
C,)F,+
(C,s +
together
define
(5.5)
parametrically
in terms of s the hypergeo-
w h’tc h is the solution of Eqnation (1.4) for k = 116. We shall now of g = g ([) using Formula (1.8). Differentiating (5.5) with respect
to s, we obtain
df ,,=--[M~(s)lFi
Y*>
into (1.8)
for c,
1 -
g = _ In this case it is possible
[and
3%
Substituting Equation
(1.4).
1/$+1 2v2:
into (5.5) We shall
expression
(5.1)
_
(5.41, (5.3)
and
in terms of s, we
f = f (5‘1, and
for
can be solved
with respect
8-l J/--
8 Jf/1+eS'/d-1
ni 6=exp1T=
(5.6),
expressions
(5.6)
M 11 F -%
Equation
8 1/l _&'
Jf/-i
+ Cl
and Equations
df/d[their
to derive the explicit
quently for Function g = g (cl. In fact, Function s in the radical form [6]
s=-
1/39 sa + I/a
df / dE = (df / ds) (ds / df)
Using the relationship (5.1) , and sabstituting obtain
Ml (s) = c$$ -
1/
V/3--1 +21/2i,
consto
=I*
1-g-1p
')
6.8)
2ni e=expT
the expression
find the explicit
of s from (5.8), expressions
=
--
we obtain the general
for the particular
integral
solutions
of
fi and fi
given by
fl=F(-&,+;9;
1-
5) =
&v
3 1/-
+8-l
fm
(5.9)
In this work, however, preference is given to the parametric presentation of all unknown functions, as this permits a complete solution of the problem, in a form convenient for gasdynamic computations in al1 of the cases (2.4) to (2.7). We shall derive the solution of system (4.4) for k = 116. The variable Expression (4.2) is fonnd by using (1.3). (5.5) and (5.7)
1+
Jf/3Es-
~%-j-
EsS (
(E + 8)” Taking
into account
substitution
(4.5)
E= 2
= const,
we can rewrite Equations
cdefined
G = const) (4.2)
by
(5.11)
as follows (5.12)
If we substitute the previously found expressions ofq and I/J into the right-hand sides of those equalities, we obtain the general solution of the first of Equations (1.6) in a parametric form
S. V. Fal’kovich
and I.A. Chemov
(5.13) Functions
U and V are determined
from Formula (4.5)
by using the expression
for 5
given in (5.11) U =--3 Equations and V = V (0
1/$G2
nozzle
2 Jf@-
1
fE+Q“
s($+i)
determine
the parametric
of the problem of an asymptotic
of the center of a Lava1 nozzle,
corresponding
solutions
type of plane parallel
Lifschitz
to the value n = 3. It is possible
final form, In a hodograph
(5.14)
V = 36 Gs (E + r)*
’
(5.14) together with (5.11) of system (4.4).
In their analysis neighborhood
St-
and Ryzhov
U = U (<), flow in the
[14] had considered
to write down this solution
plane this is given by the function
fi which was defined
a
in its in (5.lOf.
We shall derive the corresponding solution in the physical plane. A flow which is symmetric about the x-axis is defined in the hodograph plane by the following expansion of Equation (4.6)
in the neighborhood
2 (n -
T= In the case
of the singular 1)
n
i?a +y
point A (4.7)
2 (1 -
of n = 3, this expansion
n) (12/&Z- 25th -{- 12) f3 + l&a
is obtained
from (5.13),
(5.15)
...
on the assumption
that
E=O t=
---
3 Jf/5($--2 (I-
The corresponding
(s4 + 1)
J&z-1)s” T =
’
1/Q)
36
(1 _
s4
curve is shown on fig. I. Point A in the t Fplane, in the physical parameter
plane,
(5.16)
r/t&‘)3
and the x-axis
correspond
s = 0. By assigning
to the value of
to parameter
s in-
creasing real values we move in the .t~-plane along curve I in the direction of t > 0. When s (2-_3’/‘)“z , the singular point C is reached. A further increase of s to s = 3-s yields curve CR, which stretches to infinity, and then reappears from the direction 7 < 0 (point B,). At s = 3% the curve intersects line t = 9 at the point L which indicates
of
a limit
line. When s -+ 00, the integral curve t = t (7) becomes infinite at the point B,, with its asymptotic behavior defined by T= 4t. It can be easily shown that in the case modeling,
of an arbitrary index of sclf-
if the integral
ally in the t %plane
z-2(n--l)t
FIG. 1 then the corresponding
analytical
flow in the physica
curve behaves
asymptotic-
as
as
t+
00
plane will be symmetric
(5.17) about the
y-axis. It appears that when n = 3, a fiow symmetric about the x-axis is also symmetric about the y-axis. Therefore, an analytic continuation beyond the y-axis results in the above curve in the t% plane being retraced in the reverse order, namely B,LB,B,Gld. From a physical point of view the corresponding flow is of little interest, because of the presence of a triple coverage of the physical plane which cannot be eliminated by the introduction of discontinilities.
Self modeling soiutions
Equations to parameter s = is,
(5.16)
show that it is possible
s not only real,
and substitute
The behavior increasing
this
of this
s the curve
but also
into Formulas
curve
increase
we follow
about the y-axis. curve
With an analytic
in the opposite Let
us construct
Formulas
(5.11)
formulas
the notation
1. With s1 = 0 we have point A. With
velocity point
of t < 0, and then intersects
is reached
C, when
sr =
when
sr =
2
Ij3’WX.r .T
‘I/ 2 $
i
ur
e
indicates
continuation
we move along the plotted
namely
beyond
the y-axis,
a flow symmetric
B,CZA. flow. We specify
E = 0, s = is,
and G = iG, in
and obtain v
--__:16c13~
(5.18)
'
define
of velocity
functions
components
U = U (0
the straight
which characterize line y = const
half-plane we have to use expressions are shown on fig. 2, where the branch
FIG, in the right-hand
and V = V (0
u and u along
half-plane << 0. In the ~ght-hand sume Gr = - G1. These functions
be taken
by assigning
When s1 + m , then T .X 48. This
the corresponding
s1
t = t (7)
We introduce
Jf&ss,z
s--
These magnitude
direction,
and (5.141,
1+
c=-Gt
CR,.
curve
values.
point in the direction
curve 2, and reach
1011
gas jlow
(5.16)
that sonic
of si we plot curve
to plot a real imaginary
is shown on fig.
runs from this
the line t = 0, which indicates From here,
purely
0 j fransonic
half-plane.
the
in the left-hand
(5.18) and asindexed 2 should
2
The plotting
of streamlines
in accordance
with
formula (4.11) shows that we have found a flow in a Lava1 nozzle, possessing two planes of symmetry, x and y. At the entry the flow is subsonic, then sonic velocity is reached, the stream is acceleratedrthen it stagnates in the critical section, then accelerates again, and finally smoothly changes to subsonic. This is the limiting flow in a Lava1 nozzle with local
supersonic
contour nozzle
zones
linked
together
at the z-axis
and lines center,
of the nozzle.
u = const, with CT being the limiting and C”t; downstream of it. It is important
Fig.
3 shows
the wall
characteristic upstream of the to note that this flow is analytical
throughout, except at the coordinate origin, where there is a singularity the convergence point of supersonic zones. We find that the distribution component of the velocity LI along the axis of the nozzle is it = - const
which indicates of the longitudinal x4j3 from which
we can see that for x = 0 the second derivative dzuidx2 becomes infinite. We may note that similar flows considered in the work of Tomotiki and Tamada [17], and in that of Ryzhov
[18]
are not analytical,
neither
along
the C,
characteristic
upstream
of the nozzle
1012
S. V. Fol’kovich
center,
and I.A.
Chemov
nor along the characteristic
c d ownstream of it, and can only be obtained with form, while the fIow shown on fig. 3 will obtain apparently in any nozzle of its center.
nozzles of special in the neighborhood
FIG. 3 Besides
the flow shown
continuities
on fig. 3 a wide
along the CP,
n = 3, aa was done by Ryzhov ASCB,B,CXA C$
characteristic
discontinuity
with weak dis-
can be analysed
a weak
with that analysed
discontinuity,
and is symmetric
about
beyond
the x-axis.
of components
above, which
cc,
Along
while
along
With the aid of the derived coincides
of gas
Wo shall
consider
symmetric
velocity
with increasing
Away from these zones, symmetry.
E Lava1
At eabsonic
the flow remains generally
case,
symmetric
As another the one which
require
important occurs
which
waves
zones
while in the zonea
which will upset the
link together,
as is shown on
but in the next moment, as shown on
should
gradually
at the nozzle
of equations
flow defined
center,
disappear,
5 s h ows a stream
line.
and a body placed
in it. The flow upstream
solution
of the intersection
point
with the critical
of the body andup
corresponds
accelerated
and becomes
supersonic
downstream
of the body,
Ob the line
of horizontal
inclination
of velocity,
Oc the second
A strict
for n = 3, we shall stream
note bound-
at its boundary,
to it is subsonic, with 00 being sonic
to the
proof of
nomtationary
of a sonic
velocity
of
and the flow
and which
of a two-dimensional,
by a self-modeling
ary with the sonic
which
section.
flow pattern with an abmpt disturbance
zones
in the neighborhood
is also
zones
of the critical
that the flow shown on fig. 4 is unstable.
the solution
Fig.
which coincides
supersonic
aboat the y-axis, shock
be preserved,
to a Lava1 nozzle
value n = 2. It can be assumed flow.
about the y-axis of gas, local
speaking,
should revert to one, which is analytical would
the chaugc
the Oy-axis,
the field of flow in this nozzle
when the supersonic
After that, linked supersonic
supersonic
about
rate of output appear on the two sides
fig. 4, we have a transition
the above
symmetric
symmetric
velocities
3, symmetry of the flow may still
symmetry.
1 is to be
arc shown on
problem we shaI1 analyse
nozzle
During the acceleration
there may appear, In the limiting
u = const
and Ryzhov [I4]. of Tricomi’s
the flow in a nozzle
section.
and lines
a
u and v exists.
section.
about the y-axis.
themselves
solutions
fIow through
with the critical
with the critical
by Lifschitz
walls
the
the C$ characteristic,
of perturbation
For m of nozzle
fig. 4. This flow was analysed of various modes
for
the flow is supersonic
U and V of this new flow are shown in fig. 2, where the branch indexed
Function
fig.
curve
of third derivatives
taken in the zone of [>
increase
flows
waves,
We shall consider one of such flows. Let in the t%plane. This means that at the entry, up
the flow coincides
there exists
right up to the axis,
and shock
for n = 2 in [18].
be the representative
to the C-f; characteristic,
nozzle
range of Laval
and C$ characteristics
line,
then
it is
the sonic
line,
Od the limit
Self modeling
FIG. 6
FIG. 5 characteristic
C-t;, Oc the line of horizontal
characteristic
C”, .
We shall consider lines corresponding
1013
solutions of transonic gas flow
the neighborhood
inclination
of velocity,
and O/the
of point 0 in the hodograph plane (fig. 6), where
to those of fig. 5 are denoted by the same letters.
feature of the flow considered line on which the condition
limit
here is, that the streamline
7 = 0 is fulfilled.
A characteristic
00 is at the same time the sonic
In the hodograph plane this condition is equation of r~ = 0 the hypergeomet~c
formulated thus: $ = 0 on line 00. In the neighborhood (1.4) has a particular
solution
-f8 Solution f, defines
defined by the series
-f+LL+;;;;$$) (8,2;)” F( the stream function I;t which fnlfils
for ?I = 0. We now continue f, (5.19) analytically acteristic
Od, as shown in fig. 6, and stipulate
there. The analytical
continuation
(5.191 the boundary condition
into the neighborhood the regularity
of fs into the neighborhood
I) = 0
of the limit char-
of the stream fnnction tf, of line Ob is given by
Formula [ 191
(5.20) r P/3) r (W Q1 = I’ (I,&k + 1) r (- ‘I2 k + Sfsf ’ We shall then make use of relationships the neighborhood
of Oc
r (w r (- 114 Qz = r (-
%
k +
‘Id
required for the analytical
r
f%
k +
‘M
continuation
into
S. V. Fal’kovich and I.A. Chetnov
1014
The substitution of functions (5.20) yields for the neighborhood
(5.21) and (5.22) into the right-hand of iine Oc
side of Equation (5.23)
fs =
fl
El
(312 p)"/"
It remains now to carry out the ~ontinn~tion into the neighborhood acteristic Od. This is done by resorting to Formulas [I91
of the &nit char-
We substitute functions derived in (5.24) and (5.25) into Equation (5.23), and write down the expression of the stream function $ (1.3) corresponding to solution fS fC
Self modeling
Ry substituting known r-function expressed
solutions
for Q, D and R their expressions formula, r(z)
I” (z + 1) = v/sin
4
If in Expression
=
from (5.20)
(5.26)
1‘(‘/3)
the second
the following
In particular
and using the
Nr and Nz can be
nf point A in the t%plane
l/L?)
values
(5.27)
I/;? h_ +
l/3)
term coefficient
4 3 (P =
*o,
+
yields
according
This
solution
belongs
obtained
by Barantsev
to n = 3. in the neighborhood
“)*’ Pa2 -
that particular
t -
(a
solution
-
“!_ZP! 1)
t3
(1 +
. . .
which is defied
s, we obtain the following
9)2 =
+
1/3t) (t -
(3.6)
by Formulas
relationship (5.30)
9)2
Hsa -
[20].
I/a Cc” +
Xj&Z~ -t_ ‘irZG3
(5.31)
from the family (2.5). Follower
solutions
for this value of k [6].
computations 2233FZ4 =
and (6.2)
we shall
use the identity
[T, (s)]r
(T, (s) =
1 -
33s4 -
33.P +
$2)
(6.2)
we find Expression f --
Differentiating
1’ =
lirz G2,
now the value k = t/12
exists
In the following
E=
-
2-23-3
T,F,-’
(6.3)
(6.1) we obtain the derfvative /IF -) = rln for 5, 1 - & and d&ds for f
3-a li’Z~T~l,‘,-”
(6.4)
their expressions
from (6.1).
(6.3)
and (6.4)
res-
we obtain
f z= (C,s -i- C2)F2-“‘ The corresponding fi =
(5.29)
to the class
Cc +
6. We shall consider
Substituting
is detenined
of solutions indicated in [8]. A similar solution Th e relevant curve is shown on fig. 1 by a dotted are found with the aid of U =
Using (6.1)
by the equality
(5.28)
to Formula (4.3)
f3 given in (5.19)
El iminating
i/s;
was also
of equation
then the stream
defined
1, 2, * . -)
3) t” + (::‘L -
(z -
Velocities
l/3)
by the expansion
For n = 3 this expansion v2
-t-
of k
t + +;Y$(=;
1; =
n (k
N, is equal to zero,
5h3
when
ME
of the limit characteristic
which corresponds
k = t/6
i-
“/o)
I’ (-
It is easy to prove that the solution
,,: L; J&
k-
k +5/4
~~---_-P 2
pectively,
to (5.25),
71%. the coefficients
I‘(k
r P/a) I? (l/3
$ is regular in the neighborhood
p = 0. We obtain
line.
1015
/cl’ (t/r Ic) r (l/q k + 1/2) ccc3 ‘lc
b
2 3 f (-
Nz =
(5.13)
gas flow
by N1
function
of transonic
__ 5,.
solution
,-‘5, 3-l!’ F,-?”
for g is derived it& (s)
f6.5)
from Formula (1.8)
Ma (9) =
c,
-
5ClS
-
5ces4 -f
c,s5
(6.6)
1016
S. V. Fal’kovich
Equations andg=g([)
(6.5)
and (6.6) together
whiohdeffneth
We may note the hypergeometric the form
Using Goursat’s
2r (- '12) 1' p/3) ri_,,24~r~4,24~
The right-hand
derived
function.
with (6.1) yield
e solution
that the solution
tained from the solution
and I.A. Chemov
considered
section
the particular
table of qnadratic
functions
could
also have been ob-
by a quadratic
soIutions
transformations
f= f (5)
plane.
in this section
in the preceding
In fact,
the parametric
in the hodograph
[21],
of (1.7)
transformation
for k = t/12
of
are of
we obtain
(6.8) ?-4F(&;:$;M)=
sides
of Equations
(6.8) contain
function
F (--‘/rr,
s/ts, a/a, w), but
udng
i”
M
-
1
'/I2 5/12
t U)
0 l/3
0
} =
w1/‘2P
I
0
O”
-
IfI
! 113
v4
3
()
w-_l
-
w i
v2
we return to function fas defined by (1.6) for k = t/6, but with a changed argument. This, together with the concInsion reached in the preceding section as to the possibility of deriving the explicit form of function be found for the case k = 1112.
f= f(t)
for k = 1/6 shows,
We shall now obtain the solution of system using the results obtained in (6.4) and (6.6)
C=G We write the general
tt
52
solution
’
(E-~s---~Es~+s~)~ and find the solution
of system
(4.4)
1;14s4+sS
u = GG)”
(E$_
@3
,
= I/IS.
We find variable
E-55s-5Efi+s5
z = 2 3
(4.7)
v=&)"
in a parametric
53 (i -
from Equation
gby
(6.9)
W + ST
of Equation
(1 -I- i4.e + $1(E + s12
(4.) fork
that such a form can also
form
33.~49 - 33~~ + ~9 (E + 8)s (E-5s-5Es4+s5)8
(6.9)
together *
-
s3.d -
(6 1oJ
-
with Equations
33.93+ $2
(6.11)
(E + +*
We shall now find the value of E in Equation (6.10) which for n = 5 would yieId a solution coinciding with expansion (5.15), and which defines a flow symmetric about the r-axis in the physical plane. It is easily seen that we must select for E a value equal in magnitude to one of the roots of the equation II’, (sf = 0, but of the opposite sign. Function Z’* (s) defined by (6.2) appears in the numerator of the expression of T in (6.10). Let us find the real roots of equation T, (a) = 0
Self modeling solutions
S 1,s
E = -
Let and plot
(I$
I/a.
the corresponding
schematically
on fig.
With
s =
fy
point
s = 0.840,
the
along
With s = 0.655
curve
becomes
infinite
the curve
point
A further
decrease
of the parameter
direction,
namely
the
of t < 0. However,
only
of the real
values
functions
6,
imaginary
values
with
real
assigned
of parameter
question
s in order
In order
to answer
function
[6]
(3.3),
yields
h, p and u are defined since
in the case
expressed
by [,
complex
s-plane.
to move,
in this
circumference The
In order case,
use
of complex
to a linear
using
the property
solution along
solution
curve
parameters
of the
t = t (7)
equation
beyond
would
function, [lo].
of Equation
the new
the
H3 (s) = s7 F3
Instead
of the identity
(6.2)
we shall F34
-
64
(s)
=
1 +
vz
into also
again
5 1/5 s3 -
the
It is preferable
circle
following
along
1/x
a real yields
to
axis, a
by s, we obtain solution
‘147 Jf/2,.+-
._4 +
in the
A it will be necessary
transformation function
for k = t/12,
,
here, are
triangle
for computations.
its linear
D enoting (3.6)
considered
of radius
the above
the inside
parameters
s =
s = 1, and
transform that
ton
of this
of
pro-
by Equation
to X~T, g 7~ and UT
dynamic the point
on the functions
following
of 5 are
the area
from the value
at the point
to the defined
the
and purely
the x-axis
of the
equal
values
all the gas
real
about
of variable
angles
real
is not convenient which
Schwarz
(6.1)
half-plane
only
plane
turn
into
not
for which
both
values
s = s (t),
inner
v/2,
in the
is to be followed
we shall
to go beyond
its center
transformation
of the Schwarz with
to extend with
its
solutions
unnecessary
on the parameter
of a circle
resort
with
1 -
extended
values
symmetric
0
0, and
by t ~ 8t.
be also
section,
path
function
of 7< 7=
the consideration
of the real
question
Because
of self-modeling
traversed
flows
of the lower arcs,
by (2.2).
it is clearly
all
s2 =
being
obviously
of the complex
which
line
s =
necessitate
those
: the Schwarz
by circular
then
defined
curve can
s, when
this
L,),
(4.6).
M, (s),
on the side
behavior
In the preceding arises,
representation
delineated
would
to obtain
consideration.
a conformal
curve
move
of equations
of the polynomial
t = 25 (point
in the same
to parameter
The
of the Schwarz
and
to (6.11).
is shown
parameter,
first
reappears
asymptotic
This
values.
perty
where
(6.9)
curve
the decreasing
With s approaching
its
s , but also
of parameter
were
time.
B, with
under
of a triangle
Equations of this
by the
the root
line
an extension
U, I/, t and 7 have
plane
intersects
results
such
with
B1, and then
B,CL,B,B,CA.
n = 3 were considered.
complex
behavior
defined
through
along
along
opposite area
(16.2)
of E into
The
curve
C for the second
to infinity
stretches
value
A, and then,
point
the integral
C at s = 1. When s passes curve
at s = 0, we reach the
this
on the t-plane.
we have
of t > 0 along
We reach B,.
1017
=--lfJf/Z
S3,4
We substitute
curve
gas flow
7.
1 +
in the direction
0,
1f
=
of transonic
s
(6.13)
1
have J’5H33=[7’3(s)]2
(6.14)
T,(~)=-~1~+22~%~+22 We find f and g from Formulas
j = (C,s + M3 (s) =
-
(3.8)
and (1.8)
c2)F3-*‘*, V/2cass
l/%3+1
+
by repeating
g = 5+3
the above
2’/35-‘F3-5’”
M3 (s)
-
JfFc,
5c,s2
-
procedure
(6.15)
by
1018
Self modeling
Variable
[is
determined 5= G -
The integral
curves
solutions
gas flaw
of transonic
with the aid of relationships If2 $5 + T,/
5s” -
_.-_ (/is .+
in the t%plane
(6.15)
+fz B
are easily
Cl -Ly
Ir: =
>
1)”
(6.16)
found
(6.17)
These
equations
with Equation
have to be considered
(6.16).
In order to separate
general
solution
defines
the fIow symmetric
necessary
(6.17)
that particular
together from the
solution
about the x-axis,
to find the real roots
which it is
of Equation
I’, (s) = 0. There are two such roots
FIG, 7 We assume
in Formulas
The corresponding
when s = 0.518, the integral
to transition
curve stretches
t = 25 is intersected 8t. Decreasing
into infinity
still
Equations
velocity.
fi-
.r/i-.
(6.I77) yield tha
the line t = 0. which in the physical The singular
parameter passes
point C is reached
through the values
along B,, and then reappears
For s = 0 we have t = 0 and 7=
the curve stretches the parameter
E to be equal to s =
a =&+\/s,
of the parameter we move along the curve iu the
through sonic
when s = 0.0673.
r/‘
direction,
The solution
the constant
th e curve intersects
and when the decreasing
a --_) se 5-: 1;z_ the opposite
to (6.18)
With the decrease
of t < 0. When a = f2,
plane corresponds
7~
(6.16)
curve is shown on fig. 7. For
point A on the t7-plane. direction
FIG. 8
to infinity
further results
s = 0.286,
along Br. Line -458.8,
and for
along Bb with an asymptotic
behavior
in the same curve being followed
namely &&&B&J~.
derived
in this section
may be used for the analysis
of certain
special
in
Self modeling
types
of flow in Lava1
and velocities
exist
nozzles.
We may consider t%plane
as follows:
part of a Laval
nozzle)
along the characteristics
One of such fIows is represented
perturbation
line t = 0 (the sonic
; we intersect
CB, (along
velocity
the characteristic
components
of t > 0 (the supersonic
we return along the cmrve B,C part of a Lava1
about the y-axis
We shall consider of n = 5. Let us assume
nozzle).
another
possible
the presence
inlet part and in its supersonic
; we move into
application
of this flow is the symmetry
of the derived
solution
for the case
from this point. If the break is
to be small,
the rarefied
In this case
the Prandtl
can be approximated continuity. connection sonic
zone [22].
It is interesting
of a Laval
nozzle,
supersonic
multiplication 70.5IG nuity.
constant
in Expressions The position
dimensionless 7. We shall
Points
G is retained
’
by c/C
in the dir
so, as to satisfy
sI=cl and (6.18),
= 13.34.
the first
can be satisfied
if the
and made equal to of the disconti-
The behavior
of the
U and V is shown on fig. 9. for the case
obtained 114(s)=
[ff4(s)13
c = 43.33 (Fp (s)]S
(6.16)
smooth nozzles.
from the subsonic
A &Z,B,BICA,
which define the flow downstream
is determined
components
solution
The third condition
in Expressions
and (6.11),
derive the solution
We use the relevant
Polynomials
(4.8).
of discontinuity
velocity
that it is possible
in nonanalytic
by curve
that is an
a sufficiently
Z1 and 2, have been selected
conditions (6.9)
in thi t r-plane
at the need
which shows
plane of the fIow under consideration
zone is represented
two of the discontinuity
stream
to note
characteristic
one. This means
to obtain in practice
in
super-
meeting
at its axis,
a reflection,
the flow along the C$ analytic
such
was first made by Frankl’
with the problem of a local
not produce
FIG. 9
dis-
to consider
that two such discontinuities center
wave must be - Meyer flow
by a rarefaction
The suggestion
discontinuities
by arrows.
in its
flow or the Prandtl-Meyer
spreads
assumed
indicated
nozzle
which widens the stream (point Z on fig. 8)
zone,
narrow.
ection
line
that point from the direction
of a small break in the wall of a Lava1
of the physical
continua-
to the point C (characteristic
A rarefied
The traversing
con-
fourth derivatives
the two characteristics.
wave,
to the supersonic
line) and
; then by an analytical
A peculiarity
in the area between
Cl,,
will be discontinuous)
in the real plane)
; from C we move to the point A along the curve reaching
of the stream
in the
; then, instead of moving along the analytical
along the branch B, (the y-axis
tion beyond the y-axis, C$)
center.
line C<)
CB, , we follow the curve stream
to flows in which subsonic
with weak discoatinuities
of the nozzle
point C (characteristic
infinity
1019
from point A of the curve shown on fig. 7 we move in the direction
of t > 0 (the subsonic
of the sonic
gas flow
We shall limit our analysis
a flow pattern
and downstream
tinuation
of transonic
simultaneously.
upstream
reach
solutions
of k = I/SO belonging
to the family (2.6).
by Schwarz [6] 1 .+ 228s~
i; 4!%s'O
-
22%‘5 +
1”‘4(s)=s-i11s6-s
11
H, (s) and F, (s) are such that the following
equality
520 (7.1)
is fulfilled
1020
S. V. Fol’kovich
fr43 (2“ (4 = 1 The solution of Equation
48.39F4b = [T, (s)]z
522.75 (1.4)
and I.A. Chemov
10005 s”’ for k =
(7.2)
10005P’ + 522 $5 + $0) is found from (7.1)
l/30
in conjunction
with
Eqnation f = The corresponding g -
solution
C,) F*-‘/rt
(7.3)
ia in the form of
Fd-“l# M, (s) ((Ma (s) =
The parametric
(Cl S +
c, -
form of solution
11&s
SSCas” +
-
of system
(4.4)
66C,s6 -
in the case
i1c,s’o
(7.4)
i_ ~1)
of n = 11 considered
here
is 1;=
G(E-tIls--
66Es5 + 668 -
U = (11G)2 (1 + 228~5 + 494P V = r/a (11G)* (IThe corresponding separate
solution
from the general
metric about the x-axis 7’. (a) = 0. There
10005~1~ -
(7.5)
in the physical
+
(E + s)-l1
~20) (E + s)-20
10005s20 + 522~25 +
in the t-plane
solution
228P
830)
(E
is found from Formulas
the particnlar plane,
solution
(7.5)
+
s)-30
(4.5).
which defines
In order to the fIow sym-
we have to find the real roots of equation
are four of such roots
The corresponding
solution
in (5. lS), if in Formolas schematically
522~6 -
liEsJo + P)
-
(7.5)
in the tpplane
we put E = -
will coincide
a1. This solution
with the solution gives
indicated
part of the curve shown
on fig. 10. When we decrease
the parameter
to s = s1 the curve in the t%plane point A in the direction results
the opposite
of t > 0 along the path
. A farther decrease
ACB,B,LQ&R&,B, parameter
s from s = sr,
runa from the of this
in this path being followed
in
direction.
We shall now obtain an analytical
continu-
ation of the plotted curve beyond the point A in the direction
of t < 0.
In this case resorting E = -
it is possible
to do so without
to a new form of solutions.
aJ in Formulas
(7.5).
We assume
We decrease
para-
meter s from s = a, to s = s, and plot in the t % plane the corresponding
curve which mns
from the point A in the direction along the path ACB~~,L3C~~ of this parameter versed FIG.
It will be easily
10
tion viewed because
of the presence
results
in the opposite
of three limit lines
L, , L1
of t < 0 (fig.
in this path being tra-
direction. seen that the obtained
as a whole has no physical and
10)
A further decrease
L, . Lifschitz
and Ryshov
solu-
meaning, have
Self modeling
suggested
in [14]
following
part
CB,B,
along
the introduction
of the plotted , a jump
of t > 0. The varies
of the nozzle A shock ation
a Lava1
analysis
that
nozzle
center.
to the law u = const flow
is analytic,
is propagated
beyond
which
of the given
through
through
the x-axis
is again
about
from Z, to Z, , or from Z, to 2, onto
A from the direction
becomes
again
The
solution form,
the
stream
of system
problems.
linear
transformation
behind
s’ +
(4.4)
which
ff5
consider
solution
a
:
wave
the continue the stream
in two ways,
continues
namely:
approaches
to be decelerated,
solution
of the Schwarz
(s) =
-
3 Jf/5szo -
~~s~~
-
and
225 ~%a 570285
of Equation
(1.4)
‘;
of system
(4.4)
G=G(E+s)-*‘MS(s), MS(s) = -
-
-
-
$,
v/5$5
V51
prime
2280~1~ + 1482 -
19osx -
33 Jf5.+
+
is then
l/30
(C,s +
(7.7)
is omitted)
-
Jf5sr2
+
(7.8)
3 v/s
22s2 +
Jf/s
(41”
WMOS~
-
+ 225
is then
1/%’
-
-
200100~‘~
104400s’
33
33 Jf5Es3 that
(7.9)
+
I&
given
by Equation
(7.8)
and (7.10)
bv
(E+s)-*“H5,
above
v/5sa1
90045
C2)Fj-*,+2
expressed
1ls”J + 55Es9 -
obtained
IT5
+ 2OOlOOsi’ + 90045
for k =
66 Jf.%~ f
from the results
out the
(s)]5
v/5@ + 29008
U =-(1iG)z 3 J’%sll-
F55 =
1/5:n’
f = solution
form (the
228Os6 + 57 Jf/sfi
15921
+ 15921 solution
of
We carry
the identity
+
+ 570285
J&17
problem.
(s)la
[ffs
27/s IF6
33 JGSS + 44se -
2900927 _
-
in another
for computations
1/5+1/50---o
in the
+ 57 Jf5sle
-
we have H5a
1/10-2
equation
i90sls J&8
22s’O -
(7.2)
convenient
of the Schwarz
I/s+
a=-
of identity
It follows
out that
of k = 1130 can also be expressed (7.1)
[-2-2
I’
f5 (s) =
The
00.
a.
+ 494Os’o + 4482
The
upstream
downstream
CA which
of curve
may be more
4= -
-
in the
point
a
d2u / dz2 =
as to maintain
be obtained
branch
C:
where
center
a way,
can
that
in the case
of fnnction
a new
T5 (s) = -
itself,
flow
of such
the characteristic
We may
the shock
from (7.5),
We shall
s’a-
Instead
wave,
asymptotic
the axis
of t < 0.
case
different
certain
and obtain
the shock
along
of the
C, then
A from the direction
a certain
from the nozzle in such
use
subsonic.
parametric
SC--
Along
Z,,and
Z,
of t < 0 to point
to point
at the center
accelerated.
symmetry
In the second
x zo’11.
wave
by a jump point
velocity
except
the shock
beyond
wave
defines
flow
the stream
the stream
flow
solution
The
according the
of a shock
A in the direction
this
1021
gas flow
Z, , with a return
Z, to point
center
wave
direction,
the
of transonic
from point
of [ 141 consider
in the neighborhood nozzle
into
curve:
from point
authors
solutions
V=&(ilC)9(E+s)-a”TS
I’-5~8 + 66 1/5Es7 + 66sa 55s” 7
(7.11) 66E.+ -
11 Es + 3 Jf5
the solutions
of system
(4.4)
for
1022
S. V. Fal’kovich
k = I/C,
and
l/12,
of parameter
when expressed
f/30,
s, in other words,
form is, in these j = (;,lli
cases,
(S)
(B
where E, G, k,,
in a parametric
s is the homogenizing
.P)-“,
L’
and
Mil fli,
the solution
for k =
in [6f that in this case
the solution
of Equation
ence.
of that equation
We denote
function
for k =
the Schwarz
for k =
t/30,
as
function
before,
again
(3.6)
for the case
($.7 +
of f(c)
y= d~lds
we shall
of k =
had shown
algebraically
by
by
7/30
s1 ,
and the Schwarz
formula is then valid (8.1)
use Formula
dj/ds
(3.8)
where we substitute
but in terms of a, and not of
representation,
where
dsidsl,
can be expressed
’:I
polynomials.
Schwarz
7x2) (75” _;_ I)-“
use the parametric
find dgldq
and Ti the corresponding
in the family (2.7).
y/30
by s. The following
In order to find the solution st . Weshall
solution
Klein f23] had found the form of this depend-
l/30.
“t ::
s1 for s, We shall
functions
The generalised
V = Jc”G”Ti(s) (1; _+$sn
/~,G’lli (S) (I:’ _t S)-Z’L+2,
=
derive
the solution
form, are single-valued
variable.
as follows
and k, are constants,
8. We shall
and I.A. Chemov
is defined
by (‘7.1),
and ds;ttsl
by
(8.1) de -=tls Function
f ((1
is defined
by the formula
f = together
with Equation
Having (4.4)
7
(C,Y’ +
plane
$7 -
we determine
17Es’r
7c,s2 $ C,) F-?/o
in the usual
+ 118s“J + 187Es’e
(s’ -j- 7Bs6-
I! = G?H* (s) (s’ + 7.&S The same solution (7.8)
-
manner the solution
of system
for n = 17/r
C=,C
Equation
7c,s5
(7.1).
found the solution,
in the hodograph
5.4-“.3-“1&“T~F1-e
7s’ + E)
for the case
is used instead
-yy
+ 187s’-
119Es5+
7~2 + E) ‘% 2
, V = 3 GV’,, (s) (~7
of k = ~/SO can be obtained
+
17.~2+ E
76~5 -
7~2 + I?)-~/?
in another
form, if
of (7. If. BIBLIOGRAPHY
1.
2.
Frankl’ F.I., Issiedivanie po teorii kryla beskonechnogo razmakha, dvizhushchegosia so skorost’iu zvuka. (Analysis of the theory of a wing of infinite span at sonic velocity). Dokl. Akad. Nauk SSSR, Vol. 57, No. 7, 1947. Frankl’
F.I.,
Nauk SSSR,
3. 4.
Gnderley
K teorii ser.
K., Theory
Fal’kovich
S.V.,
sopla
matem.,
Lavalia.
of Transonic
K teorii
(On the theory
of the Lava1 nozzle).
Izv. Akad.
Vol. 9, No. 5, 1945.
sopla
Flows Lavalia.
(Russian
transI.1
(On the theory
M., Izd. inostr.
of Lava1 nozzle).
lit., Pr&f,
1960. Vol.
No. 4, 1946.
5.
Frankl’
F.I.,
Ob odnom semeistve
chastnykh
reshenii
uravneniia
Darbu-Tricomi.
(On
10,
Self modeling
one
6.
family
of particular Vol.
SSSR,
uber
die jenigen F&lle,
Vaglio Vol.
8.
9.
292-335,
9, No,
S.V.
(The
sonic
284,
1964.
K&mahTh.,
of gas
and Ryshov, techenii
flows).
Il.
die Gauss’sche
rotational
Chemov
flow
Yu.R.
Forsyth,
equations).
Dokl.
Akad.
hypergeometrische
darstellt.
J. reine
Reiche u. angew.
eiue
Math.,
1873.
and
transzvukovykh
10.
in welchen
R., Transonic
gaza.
gas
of the Darboux-Tricomi
1023
flow
over
a convex
comer.
J. Fluid
llech.,
1, 1960.
Fal’kovich
Lifshits,
gas flow
56, No. 7, 1947.
Laurin
-
of transonic
Function ihres vierten Elements
Bd. 75, pp. 7.
solutions
Nauk
algebrsische
solutions
a body
OS.,
0 nekotorykh
i matem.
vrashcheniia
phyz.,
Vol.
flow.
potokom
Vol. 28, No. 2, pp. 280-
resheniiakh
solutions
uravnenii
of equations
of transonic
4, No. 5, pp. 954-957,
Equations.
law of transonic
zvukovym
Ptftf,
tochnykh
exact
on Differential
similarity
tela
of rotation)
(On certain
matem.
A Treatise
The
Obtekauie
over
gaza.
J. vychisl. A-R.,
I.A.,
London, J. Wath.
1964.
1921.
and Phys.,
Vol.
26, No..~,
1947. 12.
13.
14.
C,uderley K, and Yoshihara Ilath., Vol. 8, Uo. 4, 1951, ?yzhov
$.O.,
(fin
the
flow
t’o.
1, I’fi8. Yu.?.
techeniia
v okrestnosti
Vol.
flow
154,
Germain
and
I\pplication
17.
18.
20.
14S,
$. and fluids.
Ryzhov waves
19.
hodies.
Tomotika pressible
O.S.,
nozzles).
E.T. and
Analysis).
2-nd
Barantsev
R.G.,
Dynamics).
of a Lava1
thborzmes
tipe
nozzle
center).
Watson
ed.,
Lavalia.
PM&#, Vol.
type
Dokl.
22,
of plane-
Akad.
SSSR,
Paris, forms
aux dcoulements
Vol. 2S2,
No.
of ahock
waves
plans
17, pp. 2511-2513, in flows
over
1961. sym-
20, No. 7, 1953.
udamykh
voln
v soplakh
Lavalia.
Vol.
27, No. 2, pp. 309-337,
1963.
D.N.,
Kurs
sovremennogo
(Course
Univ.,
of com-
of shock
of Modem
1963.
po transzvukovoi
Leningr.
analiza.
flows
(Fo~ation
PMM,
pt. 2. Fizmatgiz,
Lektsii Izd.
Appl. 1953.)
ploskoparallelnogo
(On the asymptotic
de conservation
rend.,
K.. Asymptotic
Vol.
Obrazovanie
in Lava1
Lavalia.
v sopiakh surface).
I’amada K., Studies on two-dimensional transonic Part I. Quart. Appl. Math., Vol. 7, No. 4, 1950.
Wittaker
Gas
sopla
Compt.
1).‘1‘. ,Infi (iuderley
metrical
transition
Quart. No. 2,
1964.
des
hornog&es.
perekhoda
nozzle
Ob asimptoticheskom
O.S.,
tsentra
Yo. 2, pp. 290-293, P.,
:3arish
Ryzhov,
transonic flow pattern. in Sb. per. Mekhanika,
poverkhnosti
of Lava1
in the neighborhood
transsoniques 16.
v okrestnosti
in the ne~ghborhooii
I,ifshits, parallel
15,
‘) techenii
II., An axisymmetric (Russian translation
1965.
gazodinamike.
(Lectures
on Transonic
S. V. Fal’kooich
1024
21.
Bateman
H, and Erddlyi
Functions). 22.
Frankl’
Vol. F.I.,
skorostei, vnutri limited
23.
Klein
Primer
okolozvukovogo vniz
(An example
downstream
F.,
transtsendentnye
funktsii.
(Higher
Transcendental
1965.
ogranichennoi
techeniia.
PMM, Vol.
A, Vysshie
1, M., ‘Nauka’,
and I. A. Chemov
techeniia
gaza
po techeniiu
skachkom.
of transonic
gas
by a compression
discontinuity
s oblastiu
sverkhzvukovykh
uplotneniia
okanchivaiushchimsia
flow ‘with a supersonic terminating
within
velocity
zone
the flow).
19, No. 4, 1955. Vorlesungen
riber
die hypergeometrische
Function.
Berlin,
1933.
Translated
by J.J.D.
AVTO~OD~‘NYE
RESHENDA
URAVNENII
OKOLOZWUKOVOGO~LOSKO~TECHEN~AGAZA) PMM Vol. 30, No. 5, 1966, pp. 848665 S.V. FAL’KOVICH and I.A. CBERNOV (Saretov) (Received
March 12, 1966)
Approximate equations of the flow of gas in the traneonic velocity range possess an important class of selknodeling solutions. Marty of the transonic flow properties such as, for example, the character of flow at some distauce from a body, the flow in Lava1 nozzles, etc., were analysed with such solutions as the main tool[l to 31. An analysis is made below of cases in which the self-modeling solutions are represented by algebraic functions. By resorting to parametric representation of the unknown magnitudes, it is poesible to indicate in all cases a form of solution convenient for gas dynamical computations. Certain exact solutions of the Tricomi equation have been obtained in this manner, solutions which may be need iu the analysis of new properties of transonic flows such as flow in a Lava1 nozzle with linked supersonic zones, flow in a nozzle with breaks in its wall, flow in the neighborhood of the intersection point of the sonic line with the sonic stream boundary, etc. 1. We shall consider a plane irrotational flow of a perfect compressible flnid iu which the velocities are nearly sonic throughout. Such a flow is approximated in the hodograph plane by the system of equations of the form [4] (1.1) Here $ is the stream function, cp is the velocity potential, 11is a velocity function which becomes zero at the critical velocity, and 0 is the angle of inclination of the velocity vector. System (1.1) is equivalent to the single Tricomi equation of the stream function (1.2) Let us consider the self-modeling solutions of system (1.1) of the form 133
Putting I) from the first equation of (1.3) into Equation (1.21, we obtain the following 1004
Self modeling solutions of tronsonic gas flow
1005
hypergeomet~c equation which defines the function f
E (1 - Qf” + w3 - ‘f&If
+ %k W2k + ‘lalf = 0
(1.4)
c =
(1.5)
The parameters of this eqnation are: a=
b =
--Yak,
II&
+
l/e,
“Is
It is convenient to present the solution of the hypergeometric equation in terms of the Riemann P-function which indicates the position of singnlar points of this equation, and the indices of singularities of this solution at such points
i”
O”
0
--Yak
l/a
“la k +
*
0 “I3
1
(1.6)
4 j
‘I2
In the neighborhood of singular points function f can be found by means of series. Thus, in the neighborhood of the singular point [= 1, two linearly independent eohio~ of Equation (1.4) can be presented in the form
fl = F C-‘l$, Y& + ‘la; ‘lo 1 - 43 fa = vm P (‘la - Vzk, “Is + ‘I&; ‘1s; 1 - U
f1.7)
The general solution of the hypergeometric equation contains two constant
f = Clfl + Gfa* Since 1 - F,= 02 / p-2, it follows from (1.3) and (1.7) that the stream function $ which corresponds to solution fi will be even in 8, while that pertaining to solution A, by virtue of the factor d/p will be odd in 8. It is easy to see that solution f, defines flows which are symmetric about the n-axis in the physical plane. The flow in a Lava1 nozzle, and flows at some distance from a body belong to the latter category. In order to find function g we substitute ti and Cpfrom (1.3) into Eqaations (1.11, and obtain two ordinary differential equations of first order with respect to f and g, from which we find g=
-
3&J
5”” (1 -
Q’V
(E)
(1.8)
Substituting for fits expression from (l.S), and using the formula for differ~~ating hypegeometric function, we obtain for g the expression (0 go-_-----
03
- %k-%
I5
3k + 1
1
a
‘I
o
(1.9)
‘12
2. We shall find for which values of k from (1.31, algebraic solutions of Eqnation (1.4) exist. We shall make use of the results obtained by Schwarz who had solved this problem for hypergeometric equations of the general type [6]. Firstly we shall find those valuee of k for which particular algebraic integrals of Equation (1.4) exist, when the second of the two linearly independent solutions may not be an aIgebraic function. This arises if, and only if one of the numbms -Y2kc ‘12k + 1/6, 1/2k + This condition determines the following values k = p,
131.
k = -
‘is + P
(P=*o,
2/3, and -‘/,k
1, 2,.
. .)
+
‘/,
is an integer.
(2.1)
Corresponding solutions in the form of polynomials can be found in Guderley’s book
S.V.
1006
We shall
now determine
hypergeometric differences
equation
Fai’kovich
and I.A.
values
of k for which
those
(1.4)
is algebraic.
h=1-c, approx~ated equation
to their (1.5)
occur
~=a-&
integral
considered
parts,
here
of the if the exponent
to the known
the following
(2.2)
Schwarz
five
table
[6f.
For the
possibilities
p =
t/z,
v
=
“12
for
X =
l/s,
I_L-
t/s,
v
=
‘62
for
‘k =lJe
h =
1/S,
E” =
t14,
v =
‘I2
for
k =I,$:!
h =
r/s,
j& =
I,!&,
y =
1/Z
for
k ==l/zo + p,
lc x -
11,& + p
(2.6)
h =
‘,$,
1-1=
2jr,,
“v =
‘,$
for
k =7,&
k =: -
‘:/3O -, I>
(2.7)
proved
defining
by Frank1
the flow
[2f,
the value
in a Lava1
nozzle
k
=I/3
symmetric
about
the x-axis
+
P
a curved
in the physical
(2.3) k = -
+ p, ‘- p,
k
+ p,
k = y from (2.31,
with
noted [4] that in this case the solution is in algebraic integral. Frank1 [s] gave later the whole family (2.3) flows
integral
if, and only
Y=c--_a--_b
conform
we obtain
the general
cases
h =. ‘In,
As was tions
Such
Chemov
l/x $ p
~---5/,~
leads
(2.4)
j-
(2.5)
p
to sel~odeling
transition
line.
solu-
Falkovich
had
form and found the relevant general and the form of integrals,describing With k = - S/J
plane.
from the same
family we obtain a solution for the fundamental singularity of a flow at some distance we have a particular integral which defines the from a body [l]. When k = 4/3, flow
around
derived also
a corner.
gave
particular
and k =
(2.51,
and (2.51,
however,
3. We shall metric
function
stituting
into
use
their
numerical
dealt
manner,
with [IO]
and had
particular for the
We reduce
indicated
we
the families
integrals,
computation
(1.4)
where
in (2.41, k = ~[!a in the same particular
algebraic
actual
equation
soIution
form by as in [8],
of k = l/6 [9] h ave derived
and Ryzhov
method
in (1.6).
the relevant
for the cases
in a different
work
the Schwarz
appearing
exist
Lifschitz
and k = I/IZ
that
in an analytical
obtained
which
in (2.6). l/6
incidentally,
[7}, was
integrals
i/30
for k =
integrals (2.4)
We may note,
by Yaglio-Laurin
of the hypergeo-
to the normal
form by sub-
it 12 (E)
For the determination
=
VJ (1 --
of h we obtain
Q1’4f G)
(3.1)
Equation
-$+Ih=O Let
fr and ft be two linearly
necessarily
congruent
of Equation
(3.2).
ratio
(3.2)
with
We introduce
of two particular
independent
solutions
(1.7).
into
our analysis
now
integrals correspond
the Schwarz
of equation
(1.4).
two integrals function
defined
not
h, and h, as the
integrals s =
We shall
particular To these
derive
the equation
fI : f2 = which
h, : ii,
is satisfied
(3.3) by function
s. As each
of the
integrals jr and fi contain two arbitrary constants, function s, which is determined with an accuracy of the order of the multiplication constant, contains three constants. The equation for s will, therefore, be of the third order. Denoting differentiation with respect to t by primes, we can write II,” Replacing second
equation
-,
h, in the first of (3.41,
iit1
=-- 0,
of above
we obtain
equations
1&.1f1 -;m lit, with
sh,.
and
- 0 taking
(3.4) into
account
the
Self modeling
=_2$
S”
7 Differentiating together
this equation
with Equation
(3.5)
of transonic
solutions
with respect
flow
1007
(3.9
t
to c,
for the elimination
gas
andusing
the second
of h, and its derivatives,
equation
of (3.4)
we obtain for s
the expression II, S --S_
If a particular obtain solution
solution
the solution
la% =
of this equation
Taking
into account
of the hypergeometric
3
of Equation
(3.6)
a=21
(3.2),
and then utilising
s (s')-g. The general
=
h 5
(s’)-‘lx
relationships
(3.1)
equation
(3.61
!
is found, then by integrating
*A of Equation
(s’)h,
SW
z-z-C
(C,
solution
s +
(3.51, we first
(3.31,
of (3.4)
the second
is then of the form (3.71
C,)
we can write down the generalised
integral
(1.4) (3.8)
4. Having the physical
found the solution
plane.
Reverting
in the hodograph
to variables
plane,
C$ and $,
we must transpose
we obtain instead
it back onto
of (1.1) the follow-
ing system (4.1) We shall limit our considerations plane parallel rforr$,yfor$,of the sonic
to the analysis
of flows
not much different
from a
flow along the z-axis of the physical plane. We cau then substitute in (4.11 u for ‘I, and v for 6, where n and w are the dimensionless components stream perturbation
velocity
along the axes
of the orthogonal
system x, y. With this condition, system (4.1) becomes approximate eqnations derived by Karman for transonic In the physical
plane the self-modeling
u =
ys(n-1) u
Here n is related
solutions
?,I=
(5),
equivalent flows [ll]. are expressed
y3wt
v(g),
coordinate
to the system
of
by 112 and 133 (4.2)
r;=: z3-n
to k of (1.31 by
3k+ f n=--7gFunctions
& $+
[I and V satisfy
the following
z--2@-4)U=0, d5
We substitute
System (4.4) dz 7
-
for U and V variables
is now equivalent
as follows
UC-=,
equations
1)V + .+o
[I41 (4.4)
[15]
z =
(4.5)
vsa
to
2(n-I)P+3tz-3~ 2t~-2nt-3(n-i)z
equations
of differential
u -f$-3(n-
t =
These
system
(4.3)
have the following
’ singular
df T= points
(ns -
t) dt
2t2-2nt-3(n-~)z at finite
distances
(4.6) from the origin
1003
S. V. Fal’kovick
A (t = 0, z = O),
IC (t =
and I.A. Chemov
r&z, z =
2/gzp,
D (t =
1, z =
--a/a)
(4.7)
Point A corresponds to the z-axis of the physical plane, while the singular point C represents the limit characteristic. Infinity in the t %plane, which we shall denote by point B, corresponds to the y-axis. Second equation of the system (4.6) shows that, when the integral curve t = t (7) of the first equation of (4.6) intersects line t = nr at a point other than the singular point C, then d[= 0. This indicates the appearance of a limit line in the corresponding flow. The r, -plane called the phase plane, is convenient for analysing flows in the presence of shock waves. Denoting parameters related to the two sides of a shock wave by indices 1 and 2, we write conditions at discontinnities as followa [16]
t, = 2n2 -
z, = “1 + znt, -
t,,
2113,
51 = 5s
(4.8)
To determine the streamline we must integrate the approximate equation
df
for
x=”
(4.9)
y = const
Is integrate the second equation of (4.1) over the area bounded by line c= const, andlinesy=O,x=-eo and y = ye. Using Green’s formula; we obtain
$(-$dy+
vds)=
(4.10)
0
Al urg y = ye the first term of (4.10) is equal to zero, and the second one, to y,, . The integral can be easily computed along line jr= const, if for u and v we substitute their expressions from (4.3) (4.11) The analysis carried out in section 2 established all possible csses in which the nonlinear system (4.4) can be solved in terms of algebraic functions. Corresponding valnes of n can be easily found by resorting to the transformation formula (4.3), and to results obtained in (2.1) and (2.3) to (2.7). 5. We shall now find the general integral for one single value of k from each of the infinite families (2.4) to (2.7). Other hypergeometric functions of the same family can be found by consecutive differentiation of the function thus obtained. Family (2.3) is assumed known [S], and will not be considered. We take k = l/6 from family (2.4). For this value of k there exists Scliwarz equation [6] which is Hr (s) = c + 2 J&s
F=[$$je*
Fr(s)=S4-2
J&2-
-
a solution of the
1 1
(5.1)
Polynomials H1 and F, are such that the following identity hdds H$ -
Frs = fZyX[T,
(s)P,
z+l(s) = s (s4 + 1)
Using Expression (5.1) for eand the identity (5.2), we find I-_E=-12
Differentiation of (5.1) yields
y’sT&
(5.2)
Self modeling solutions of transonic
dE --ds -
1009
gas flow
(S.4)
24 1/3T1H12~1-a
We shall now find function f by using Formula (3.8) j = Eqnations metric
function
(5.1)
and (5.5)
f = f (0
derive the expression
C,)F,+
(C,s +
together
define
(5.5)
parametrically
in terms of s the hypergeo-
w h’tc h is the solution of Eqnation (1.4) for k = 116. We shall now of g = g ([) using Formula (1.8). Differentiating (5.5) with respect
to s, we obtain
df ,,=--[M~(s)lFi
Y*>
into (1.8)
for c,
1 -
g = _ In this case it is possible
[and
3%
Substituting Equation
(1.4).
1/$+1 2v2:
into (5.5) We shall
expression
(5.1)
_
(5.41, (5.3)
and
in terms of s, we
f = f (5‘1, and
for
can be solved
with respect
8-l J/--
8 Jf/1+eS'/d-1
ni 6=exp1T=
(5.6),
expressions
(5.6)
M 11 F -%
Equation
8 1/l _&'
Jf/-i
+ Cl
and Equations
df/d[their
to derive the explicit
quently for Function g = g (cl. In fact, Function s in the radical form [6]
s=-
1/39 sa + I/a
df / dE = (df / ds) (ds / df)
Using the relationship (5.1) , and sabstituting obtain
Ml (s) = c$$ -
1/
V/3--1 +21/2i,
consto
=I*
1-g-1p
')
6.8)
2ni e=expT
the expression
find the explicit
of s from (5.8), expressions
=
--
we obtain the general
for the particular
integral
solutions
of
fi and fi
given by
fl=F(-&,+;9;
1-
5) =
&v
3 1/-
+8-l
fm
(5.9)
In this work, however, preference is given to the parametric presentation of all unknown functions, as this permits a complete solution of the problem, in a form convenient for gasdynamic computations in al1 of the cases (2.4) to (2.7). We shall derive the solution of system (4.4) for k = 116. The variable Expression (4.2) is fonnd by using (1.3). (5.5) and (5.7)
1+
Jf/3Es-
~%-j-
EsS (
(E + 8)” Taking
into account
substitution
(4.5)
E= 2
= const,
we can rewrite Equations
cdefined
G = const) (4.2)
by
(5.11)
as follows (5.12)
If we substitute the previously found expressions ofq and I/J into the right-hand sides of those equalities, we obtain the general solution of the first of Equations (1.6) in a parametric form
S. V. Fal’kovich
and I.A. Chemov
(5.13) Functions
U and V are determined
from Formula (4.5)
by using the expression
for 5
given in (5.11) U =--3 Equations and V = V (0
1/$G2
nozzle
2 Jf@-
1
fE+Q“
s($+i)
determine
the parametric
of the problem of an asymptotic
of the center of a Lava1 nozzle,
corresponding
solutions
type of plane parallel
Lifschitz
to the value n = 3. It is possible
final form, In a hodograph
(5.14)
V = 36 Gs (E + r)*
’
(5.14) together with (5.11) of system (4.4).
In their analysis neighborhood
St-
and Ryzhov
U = U (<), flow in the
[14] had considered
to write down this solution
plane this is given by the function
fi which was defined
a
in its in (5.lOf.
We shall derive the corresponding solution in the physical plane. A flow which is symmetric about the x-axis is defined in the hodograph plane by the following expansion of Equation (4.6)
in the neighborhood
2 (n -
T= In the case
of the singular 1)
n
i?a +y
point A (4.7)
2 (1 -
of n = 3, this expansion
n) (12/&Z- 25th -{- 12) f3 + l&a
is obtained
from (5.13),
(5.15)
...
on the assumption
that
E=O t=
---
3 Jf/5($--2 (I-
The corresponding
(s4 + 1)
J&z-1)s” T =
’
1/Q)
36
(1 _
s4
curve is shown on fig. I. Point A in the t Fplane, in the physical parameter
plane,
(5.16)
r/t&‘)3
and the x-axis
correspond
s = 0. By assigning
to the value of
to parameter
s in-
creasing real values we move in the .t~-plane along curve I in the direction of t > 0. When s (2-_3’/‘)“z , the singular point C is reached. A further increase of s to s = 3-s yields curve CR, which stretches to infinity, and then reappears from the direction 7 < 0 (point B,). At s = 3% the curve intersects line t = 9 at the point L which indicates
of
a limit
line. When s -+ 00, the integral curve t = t (7) becomes infinite at the point B,, with its asymptotic behavior defined by T= 4t. It can be easily shown that in the case modeling,
of an arbitrary index of sclf-
if the integral
ally in the t %plane
z-2(n--l)t
FIG. 1 then the corresponding
analytical
flow in the physica
curve behaves
asymptotic-
as
as
t+
00
plane will be symmetric
(5.17) about the
y-axis. It appears that when n = 3, a fiow symmetric about the x-axis is also symmetric about the y-axis. Therefore, an analytic continuation beyond the y-axis results in the above curve in the t% plane being retraced in the reverse order, namely B,LB,B,Gld. From a physical point of view the corresponding flow is of little interest, because of the presence of a triple coverage of the physical plane which cannot be eliminated by the introduction of discontinilities.
Self modeling soiutions
Equations to parameter s = is,
(5.16)
show that it is possible
s not only real,
and substitute
The behavior increasing
this
of this
s the curve
but also
into Formulas
curve
increase
we follow
about the y-axis. curve
With an analytic
in the opposite Let
us construct
Formulas
(5.11)
formulas
the notation
1. With s1 = 0 we have point A. With
velocity point
of t < 0, and then intersects
is reached
C, when
sr =
when
sr =
2
Ij3’WX.r .T
‘I/ 2 $
i
ur
e
indicates
continuation
we move along the plotted
namely
beyond
the y-axis,
a flow symmetric
B,CZA. flow. We specify
E = 0, s = is,
and G = iG, in
and obtain v
--__:16c13~
(5.18)
'
define
of velocity
functions
components
U = U (0
the straight
which characterize line y = const
half-plane we have to use expressions are shown on fig. 2, where the branch
FIG, in the right-hand
and V = V (0
u and u along
half-plane << 0. In the ~ght-hand sume Gr = - G1. These functions
be taken
by assigning
When s1 + m , then T .X 48. This
the corresponding
s1
t = t (7)
We introduce
Jf&ss,z
s--
These magnitude
direction,
and (5.141,
1+
c=-Gt
CR,.
curve
values.
point in the direction
curve 2, and reach
1011
gas jlow
(5.16)
that sonic
of si we plot curve
to plot a real imaginary
is shown on fig.
runs from this
the line t = 0, which indicates From here,
purely
0 j fransonic
half-plane.
the
in the left-hand
(5.18) and asindexed 2 should
2
The plotting
of streamlines
in accordance
with
formula (4.11) shows that we have found a flow in a Lava1 nozzle, possessing two planes of symmetry, x and y. At the entry the flow is subsonic, then sonic velocity is reached, the stream is acceleratedrthen it stagnates in the critical section, then accelerates again, and finally smoothly changes to subsonic. This is the limiting flow in a Lava1 nozzle with local
supersonic
contour nozzle
zones
linked
together
at the z-axis
and lines center,
of the nozzle.
u = const, with CT being the limiting and C”t; downstream of it. It is important
Fig.
3 shows
the wall
characteristic upstream of the to note that this flow is analytical
throughout, except at the coordinate origin, where there is a singularity the convergence point of supersonic zones. We find that the distribution component of the velocity LI along the axis of the nozzle is it = - const
which indicates of the longitudinal x4j3 from which
we can see that for x = 0 the second derivative dzuidx2 becomes infinite. We may note that similar flows considered in the work of Tomotiki and Tamada [17], and in that of Ryzhov
[18]
are not analytical,
neither
along
the C,
characteristic
upstream
of the nozzle
1012
S. V. Fol’kovich
center,
and I.A.
Chemov
nor along the characteristic
c d ownstream of it, and can only be obtained with form, while the fIow shown on fig. 3 will obtain apparently in any nozzle of its center.
nozzles of special in the neighborhood
FIG. 3 Besides
the flow shown
continuities
on fig. 3 a wide
along the CP,
n = 3, aa was done by Ryzhov ASCB,B,CXA C$
characteristic
discontinuity
with weak dis-
can be analysed
a weak
with that analysed
discontinuity,
and is symmetric
about
beyond
the x-axis.
of components
above, which
cc,
Along
while
along
With the aid of the derived coincides
of gas
Wo shall
consider
symmetric
velocity
with increasing
Away from these zones, symmetry.
E Lava1
At eabsonic
the flow remains generally
case,
symmetric
As another the one which
require
important occurs
which
waves
zones
while in the zonea
which will upset the
link together,
as is shown on
but in the next moment, as shown on
should
gradually
at the nozzle
of equations
flow defined
center,
disappear,
5 s h ows a stream
line.
and a body placed
in it. The flow upstream
solution
of the intersection
point
with the critical
of the body andup
corresponds
accelerated
and becomes
supersonic
downstream
of the body,
Ob the line
of horizontal
inclination
of velocity,
Oc the second
A strict
for n = 3, we shall stream
note bound-
at its boundary,
to it is subsonic, with 00 being sonic
to the
proof of
nomtationary
of a sonic
velocity
of
and the flow
and which
of a two-dimensional,
by a self-modeling
ary with the sonic
which
section.
flow pattern with an abmpt disturbance
zones
in the neighborhood
is also
zones
of the critical
that the flow shown on fig. 4 is unstable.
the solution
Fig.
which coincides
supersonic
aboat the y-axis, shock
be preserved,
to a Lava1 nozzle
value n = 2. It can be assumed flow.
about the y-axis of gas, local
speaking,
should revert to one, which is analytical would
the chaugc
the Oy-axis,
the field of flow in this nozzle
when the supersonic
After that, linked supersonic
supersonic
about
rate of output appear on the two sides
fig. 4, we have a transition
the above
symmetric
symmetric
velocities
3, symmetry of the flow may still
symmetry.
1 is to be
arc shown on
problem we shaI1 analyse
nozzle
During the acceleration
there may appear, In the limiting
u = const
and Ryzhov [I4]. of Tricomi’s
the flow in a nozzle
section.
and lines
a
u and v exists.
section.
about the y-axis.
themselves
solutions
fIow through
with the critical
with the critical
by Lifschitz
walls
the
the C$ characteristic,
of perturbation
For m of nozzle
fig. 4. This flow was analysed of various modes
for
the flow is supersonic
U and V of this new flow are shown in fig. 2, where the branch indexed
Function
fig.
curve
of third derivatives
taken in the zone of [>
increase
flows
waves,
We shall consider one of such flows. Let in the t%plane. This means that at the entry, up
the flow coincides
there exists
right up to the axis,
and shock
for n = 2 in [18].
be the representative
to the C-f; characteristic,
nozzle
range of Laval
and C$ characteristics
line,
then
it is
the sonic
line,
Od the limit
Self modeling
FIG. 6
FIG. 5 characteristic
C-t;, Oc the line of horizontal
characteristic
C”, .
We shall consider lines corresponding
1013
solutions of transonic gas flow
the neighborhood
inclination
of velocity,
and O/the
of point 0 in the hodograph plane (fig. 6), where
to those of fig. 5 are denoted by the same letters.
feature of the flow considered line on which the condition
limit
here is, that the streamline
7 = 0 is fulfilled.
A characteristic
00 is at the same time the sonic
In the hodograph plane this condition is equation of r~ = 0 the hypergeomet~c
formulated thus: $ = 0 on line 00. In the neighborhood (1.4) has a particular
solution
-f8 Solution f, defines
defined by the series
-f+LL+;;;;$$) (8,2;)” F( the stream function I;t which fnlfils
for ?I = 0. We now continue f, (5.19) analytically acteristic
Od, as shown in fig. 6, and stipulate
there. The analytical
continuation
(5.191 the boundary condition
into the neighborhood the regularity
of fs into the neighborhood
I) = 0
of the limit char-
of the stream fnnction tf, of line Ob is given by
Formula [ 191
(5.20) r P/3) r (W Q1 = I’ (I,&k + 1) r (- ‘I2 k + Sfsf ’ We shall then make use of relationships the neighborhood
of Oc
r (w r (- 114 Qz = r (-
%
k +
‘Id
required for the analytical
r
f%
k +
‘M
continuation
into
S. V. Fal’kovich and I.A. Chetnov
1014
The substitution of functions (5.20) yields for the neighborhood
(5.21) and (5.22) into the right-hand of iine Oc
side of Equation (5.23)
fs =
fl
El
(312 p)"/"
It remains now to carry out the ~ontinn~tion into the neighborhood acteristic Od. This is done by resorting to Formulas [I91
of the &nit char-
We substitute functions derived in (5.24) and (5.25) into Equation (5.23), and write down the expression of the stream function $ (1.3) corresponding to solution fS fC
Self modeling
Ry substituting known r-function expressed
solutions
for Q, D and R their expressions formula, r(z)
I” (z + 1) = v/sin
4
If in Expression
=
from (5.20)
(5.26)
1‘(‘/3)
the second
the following
In particular
and using the
Nr and Nz can be
nf point A in the t%plane
l/L?)
values
(5.27)
I/;? h_ +
l/3)
term coefficient
4 3 (P =
*o,
+
yields
according
This
solution
belongs
obtained
by Barantsev
to n = 3. in the neighborhood
“)*’ Pa2 -
that particular
t -
(a
solution
-
“!_ZP! 1)
t3
(1 +
. . .
which is defied
s, we obtain the following
9)2 =
+
1/3t) (t -
(3.6)
by Formulas
relationship (5.30)
9)2
Hsa -
[20].
I/a Cc” +
Xj&Z~ -t_ ‘irZG3
(5.31)
from the family (2.5). Follower
solutions
for this value of k [6].
computations 2233FZ4 =
and (6.2)
we shall
use the identity
[T, (s)]r
(T, (s) =
1 -
33s4 -
33.P +
$2)
(6.2)
we find Expression f --
Differentiating
1’ =
lirz G2,
now the value k = t/12
exists
In the following
E=
-
2-23-3
T,F,-’
(6.3)
(6.1) we obtain the derfvative /IF -) = rln for 5, 1 - & and d&ds for f
3-a li’Z~T~l,‘,-”
(6.4)
their expressions
from (6.1).
(6.3)
and (6.4)
res-
we obtain
f z= (C,s -i- C2)F2-“‘ The corresponding fi =
(5.29)
to the class
Cc +
6. We shall consider
Substituting
is detenined
of solutions indicated in [8]. A similar solution Th e relevant curve is shown on fig. 1 by a dotted are found with the aid of U =
Using (6.1)
by the equality
(5.28)
to Formula (4.3)
f3 given in (5.19)
El iminating
i/s;
was also
of equation
then the stream
defined
1, 2, * . -)
3) t” + (::‘L -
(z -
Velocities
l/3)
by the expansion
For n = 3 this expansion v2
-t-
of k
t + +;Y$(=;
1; =
n (k
N, is equal to zero,
5h3
when
ME
of the limit characteristic
which corresponds
k = t/6
i-
“/o)
I’ (-
It is easy to prove that the solution
,,: L; J&
k-
k +5/4
~~---_-P 2
pectively,
to (5.25),
71%. the coefficients
I‘(k
r P/a) I? (l/3
$ is regular in the neighborhood
p = 0. We obtain
line.
1015
/cl’ (t/r Ic) r (l/q k + 1/2) ccc3 ‘lc
b
2 3 f (-
Nz =
(5.13)
gas flow
by N1
function
of transonic
__ 5,.
solution
,-‘5, 3-l!’ F,-?”
for g is derived it& (s)
f6.5)
from Formula (1.8)
Ma (9) =
c,
-
5ClS
-
5ces4 -f
c,s5
(6.6)
1016
S. V. Fal’kovich
Equations andg=g([)
(6.5)
and (6.6) together
whiohdeffneth
We may note the hypergeometric the form
Using Goursat’s
2r (- '12) 1' p/3) ri_,,24~r~4,24~
The right-hand
derived
function.
with (6.1) yield
e solution
that the solution
tained from the solution
and I.A. Chemov
considered
section
the particular
table of qnadratic
functions
could
also have been ob-
by a quadratic
soIutions
transformations
f= f (5)
plane.
in this section
in the preceding
In fact,
the parametric
in the hodograph
[21],
of (1.7)
transformation
for k = t/12
of
are of
we obtain
(6.8) ?-4F(&;:$;M)=
sides
of Equations
(6.8) contain
function
F (--‘/rr,
s/ts, a/a, w), but
udng
i”
M
-
1
'/I2 5/12
t U)
0 l/3
0
} =
w1/‘2P
I
0
O”
-
IfI
! 113
v4
3
()
w-_l
-
w i
v2
we return to function fas defined by (1.6) for k = t/6, but with a changed argument. This, together with the concInsion reached in the preceding section as to the possibility of deriving the explicit form of function be found for the case k = 1112.
f= f(t)
for k = 1/6 shows,
We shall now obtain the solution of system using the results obtained in (6.4) and (6.6)
C=G We write the general
tt
52
solution
’
(E-~s---~Es~+s~)~ and find the solution
of system
(4.4)
1;14s4+sS
u = GG)”
(E$_
@3
,
= I/IS.
We find variable
E-55s-5Efi+s5
z = 2 3
(4.7)
v=&)"
in a parametric
53 (i -
from Equation
gby
(6.9)
W + ST
of Equation
(1 -I- i4.e + $1(E + s12
(4.) fork
that such a form can also
form
33.~49 - 33~~ + ~9 (E + 8)s (E-5s-5Es4+s5)8
(6.9)
together *
-
s3.d -
(6 1oJ
-
with Equations
33.93+ $2
(6.11)
(E + +*
We shall now find the value of E in Equation (6.10) which for n = 5 would yieId a solution coinciding with expansion (5.15), and which defines a flow symmetric about the r-axis in the physical plane. It is easily seen that we must select for E a value equal in magnitude to one of the roots of the equation II’, (sf = 0, but of the opposite sign. Function Z’* (s) defined by (6.2) appears in the numerator of the expression of T in (6.10). Let us find the real roots of equation T, (a) = 0
Self modeling solutions
S 1,s
E = -
Let and plot
(I$
I/a.
the corresponding
schematically
on fig.
With
s =
fy
point
s = 0.840,
the
along
With s = 0.655
curve
becomes
infinite
the curve
point
A further
decrease
of the parameter
direction,
namely
the
of t < 0. However,
only
of the real
values
functions
6,
imaginary
values
with
real
assigned
of parameter
question
s in order
In order
to answer
function
[6]
(3.3),
yields
h, p and u are defined since
in the case
expressed
by [,
complex
s-plane.
to move,
in this
circumference The
In order case,
use
of complex
to a linear
using
the property
solution along
solution
curve
parameters
of the
t = t (7)
equation
beyond
would
function, [lo].
of Equation
the new
the
H3 (s) = s7 F3
Instead
of the identity
(6.2)
we shall F34
-
64
(s)
=
1 +
vz
into also
again
5 1/5 s3 -
the
It is preferable
circle
following
along
1/x
a real yields
to
axis, a
by s, we obtain solution
‘147 Jf/2,.+-
._4 +
in the
A it will be necessary
transformation function
for k = t/12,
,
here, are
triangle
for computations.
its linear
D enoting (3.6)
considered
of radius
the above
the inside
parameters
s =
s = 1, and
transform that
ton
of this
of
pro-
by Equation
to X~T, g 7~ and UT
dynamic the point
on the functions
following
of 5 are
the area
from the value
at the point
to the defined
the
and purely
the x-axis
of the
equal
values
all the gas
real
about
of variable
angles
real
is not convenient which
Schwarz
(6.1)
half-plane
only
plane
turn
into
not
for which
both
values
s = s (t),
inner
v/2,
in the
is to be followed
we shall
to go beyond
its center
transformation
of the Schwarz with
to extend with
its
solutions
unnecessary
on the parameter
of a circle
resort
with
1 -
extended
values
symmetric
0
0, and
by t ~ 8t.
be also
section,
path
function
of 7< 7=
the consideration
of the real
question
Because
of self-modeling
traversed
flows
of the lower arcs,
by (2.2).
it is clearly
all
s2 =
being
obviously
of the complex
which
line
s =
necessitate
those
: the Schwarz
by circular
then
defined
curve can
s, when
this
L,),
(4.6).
M, (s),
on the side
behavior
In the preceding arises,
representation
delineated
would
to obtain
consideration.
a conformal
curve
move
of equations
of the polynomial
t = 25 (point
in the same
to parameter
The
of the Schwarz
and
to (6.11).
is shown
parameter,
first
reappears
asymptotic
This
values.
perty
where
(6.9)
curve
the decreasing
With s approaching
its
s , but also
of parameter
were
time.
B, with
under
of a triangle
Equations of this
by the
the root
line
an extension
U, I/, t and 7 have
plane
intersects
results
such
with
B1, and then
B,CL,B,B,CA.
n = 3 were considered.
complex
behavior
defined
through
along
along
opposite area
(16.2)
of E into
The
curve
C for the second
to infinity
stretches
value
A, and then,
point
the integral
C at s = 1. When s passes curve
at s = 0, we reach the
this
on the t-plane.
we have
of t > 0 along
We reach B,.
1017
=--lfJf/Z
S3,4
We substitute
curve
gas flow
7.
1 +
in the direction
0,
1f
=
of transonic
s
(6.13)
1
have J’5H33=[7’3(s)]2
(6.14)
T,(~)=-~1~+22~%~+22 We find f and g from Formulas
j = (C,s + M3 (s) =
-
(3.8)
and (1.8)
c2)F3-*‘*, V/2cass
l/%3+1
+
by repeating
g = 5+3
the above
2’/35-‘F3-5’”
M3 (s)
-
JfFc,
5c,s2
-
procedure
(6.15)
by
1018
Self modeling
Variable
[is
determined 5= G -
The integral
curves
solutions
gas flaw
of transonic
with the aid of relationships If2 $5 + T,/
5s” -
_.-_ (/is .+
in the t%plane
(6.15)
+fz B
are easily
Cl -Ly
Ir: =
>
1)”
(6.16)
found
(6.17)
These
equations
with Equation
have to be considered
(6.16).
In order to separate
general
solution
defines
the fIow symmetric
necessary
(6.17)
that particular
together from the
solution
about the x-axis,
to find the real roots
which it is
of Equation
I’, (s) = 0. There are two such roots
FIG, 7 We assume
in Formulas
The corresponding
when s = 0.518, the integral
to transition
curve stretches
t = 25 is intersected 8t. Decreasing
into infinity
still
Equations
velocity.
fi-
.r/i-.
(6.I77) yield tha
the line t = 0. which in the physical The singular
parameter passes
point C is reached
through the values
along B,, and then reappears
For s = 0 we have t = 0 and 7=
the curve stretches the parameter
E to be equal to s =
a =&+\/s,
of the parameter we move along the curve iu the
through sonic
when s = 0.0673.
r/‘
direction,
The solution
the constant
th e curve intersects
and when the decreasing
a --_) se 5-: 1;z_ the opposite
to (6.18)
With the decrease
of t < 0. When a = f2,
plane corresponds
7~
(6.16)
curve is shown on fig. 7. For
point A on the t7-plane. direction
FIG. 8
to infinity
further results
s = 0.286,
along Br. Line -458.8,
and for
along Bb with an asymptotic
behavior
in the same curve being followed
namely &&&B&J~.
derived
in this section
may be used for the analysis
of certain
special
in
Self modeling
types
of flow in Lava1
and velocities
exist
nozzles.
We may consider t%plane
as follows:
part of a Laval
nozzle)
along the characteristics
One of such fIows is represented
perturbation
line t = 0 (the sonic
; we intersect
CB, (along
velocity
the characteristic
components
of t > 0 (the supersonic
we return along the cmrve B,C part of a Lava1
about the y-axis
We shall consider of n = 5. Let us assume
nozzle).
another
possible
the presence
inlet part and in its supersonic
; we move into
application
of this flow is the symmetry
of the derived
solution
for the case
from this point. If the break is
to be small,
the rarefied
In this case
the Prandtl
can be approximated continuity. connection sonic
zone [22].
It is interesting
of a Laval
nozzle,
supersonic
multiplication 70.5IG nuity.
constant
in Expressions The position
dimensionless 7. We shall
Points
G is retained
’
by c/C
in the dir
so, as to satisfy
sI=cl and (6.18),
= 13.34.
the first
can be satisfied
if the
and made equal to of the disconti-
The behavior
of the
U and V is shown on fig. 9. for the case
obtained 114(s)=
[ff4(s)13
c = 43.33 (Fp (s)]S
(6.16)
smooth nozzles.
from the subsonic
A &Z,B,BICA,
which define the flow downstream
is determined
components
solution
The third condition
in Expressions
and (6.11),
derive the solution
We use the relevant
Polynomials
(4.8).
of discontinuity
velocity
that it is possible
in nonanalytic
by curve
that is an
a sufficiently
Z1 and 2, have been selected
conditions (6.9)
in thi t r-plane
at the need
which shows
plane of the fIow under consideration
zone is represented
two of the discontinuity
stream
to note
characteristic
one. This means
to obtain in practice
in
super-
meeting
at its axis,
a reflection,
the flow along the C$ analytic
such
was first made by Frankl’
with the problem of a local
not produce
FIG. 9
dis-
to consider
that two such discontinuities center
wave must be - Meyer flow
by a rarefaction
The suggestion
discontinuities
by arrows.
in its
flow or the Prandtl-Meyer
spreads
assumed
indicated
nozzle
which widens the stream (point Z on fig. 8)
zone,
narrow.
ection
line
that point from the direction
of a small break in the wall of a Lava1
of the physical
continua-
to the point C (characteristic
A rarefied
The traversing
con-
fourth derivatives
the two characteristics.
wave,
to the supersonic
line) and
; then by an analytical
A peculiarity
in the area between
Cl,,
will be discontinuous)
in the real plane)
; from C we move to the point A along the curve reaching
of the stream
in the
; then, instead of moving along the analytical
along the branch B, (the y-axis
tion beyond the y-axis, C$)
center.
line C<)
CB, , we follow the curve stream
to flows in which subsonic
with weak discoatinuities
of the nozzle
point C (characteristic
infinity
1019
from point A of the curve shown on fig. 7 we move in the direction
of t > 0 (the subsonic
of the sonic
gas flow
We shall limit our analysis
a flow pattern
and downstream
tinuation
of transonic
simultaneously.
upstream
reach
solutions
of k = I/SO belonging
to the family (2.6).
by Schwarz [6] 1 .+ 228s~
i; 4!%s'O
-
22%‘5 +
1”‘4(s)=s-i11s6-s
11
H, (s) and F, (s) are such that the following
equality
520 (7.1)
is fulfilled
1020
S. V. Fol’kovich
fr43 (2“ (4 = 1 The solution of Equation
48.39F4b = [T, (s)]z
522.75 (1.4)
and I.A. Chemov
10005 s”’ for k =
(7.2)
10005P’ + 522 $5 + $0) is found from (7.1)
l/30
in conjunction
with
Eqnation f = The corresponding g -
solution
C,) F*-‘/rt
(7.3)
ia in the form of
Fd-“l# M, (s) ((Ma (s) =
The parametric
(Cl S +
c, -
form of solution
11&s
SSCas” +
-
of system
(4.4)
66C,s6 -
in the case
i1c,s’o
(7.4)
i_ ~1)
of n = 11 considered
here
is 1;=
G(E-tIls--
66Es5 + 668 -
U = (11G)2 (1 + 228~5 + 494P V = r/a (11G)* (IThe corresponding separate
solution
from the general
metric about the x-axis 7’. (a) = 0. There
10005~1~ -
(7.5)
in the physical
+
(E + s)-l1
~20) (E + s)-20
10005s20 + 522~25 +
in the t-plane
solution
228P
830)
(E
is found from Formulas
the particnlar plane,
solution
(7.5)
+
s)-30
(4.5).
which defines
In order to the fIow sym-
we have to find the real roots of equation
are four of such roots
The corresponding
solution
in (5. lS), if in Formolas schematically
522~6 -
liEsJo + P)
-
(7.5)
in the tpplane
we put E = -
will coincide
a1. This solution
with the solution gives
indicated
part of the curve shown
on fig. 10. When we decrease
the parameter
to s = s1 the curve in the t%plane point A in the direction results
the opposite
of t > 0 along the path
. A farther decrease
ACB,B,LQ&R&,B, parameter
s from s = sr,
runa from the of this
in this path being followed
in
direction.
We shall now obtain an analytical
continu-
ation of the plotted curve beyond the point A in the direction
of t < 0.
In this case resorting E = -
it is possible
to do so without
to a new form of solutions.
aJ in Formulas
(7.5).
We assume
We decrease
para-
meter s from s = a, to s = s, and plot in the t % plane the corresponding
curve which mns
from the point A in the direction along the path ACB~~,L3C~~ of this parameter versed FIG.
It will be easily
10
tion viewed because
of the presence
results
in the opposite
of three limit lines
L, , L1
of t < 0 (fig.
in this path being tra-
direction. seen that the obtained
as a whole has no physical and
10)
A further decrease
L, . Lifschitz
and Ryshov
solu-
meaning, have
Self modeling
suggested
in [14]
following
part
CB,B,
along
the introduction
of the plotted , a jump
of t > 0. The varies
of the nozzle A shock ation
a Lava1
analysis
that
nozzle
center.
to the law u = const flow
is analytic,
is propagated
beyond
which
of the given
through
through
the x-axis
is again
about
from Z, to Z, , or from Z, to 2, onto
A from the direction
becomes
again
The
solution form,
the
stream
of system
problems.
linear
transformation
behind
s’ +
(4.4)
which
ff5
consider
solution
a
:
wave
the continue the stream
in two ways,
continues
namely:
approaches
to be decelerated,
solution
of the Schwarz
(s) =
-
3 Jf/5szo -
~~s~~
-
and
225 ~%a 570285
of Equation
(1.4)
‘;
of system
(4.4)
G=G(E+s)-*‘MS(s), MS(s) = -
-
-
-
$,
v/5$5
V51
prime
2280~1~ + 1482 -
19osx -
33 Jf5.+
+
is then
l/30
(C,s +
(7.7)
is omitted)
-
Jf5sr2
+
(7.8)
3 v/s
22s2 +
Jf/s
(41”
WMOS~
-
+ 225
is then
1/%’
-
-
200100~‘~
104400s’
33
33 Jf5Es3 that
(7.9)
+
I&
given
by Equation
(7.8)
and (7.10)
bv
(E+s)-*“H5,
above
v/5sa1
90045
C2)Fj-*,+2
expressed
1ls”J + 55Es9 -
obtained
IT5
+ 2OOlOOsi’ + 90045
for k =
66 Jf.%~ f
from the results
out the
(s)]5
v/5@ + 29008
U =-(1iG)z 3 J’%sll-
F55 =
1/5:n’
f = solution
form (the
228Os6 + 57 Jf/sfi
15921
+ 15921 solution
of
We carry
the identity
+
+ 570285
J&17
problem.
(s)la
[ffs
27/s IF6
33 JGSS + 44se -
2900927 _
-
in another
for computations
1/5+1/50---o
in the
+ 57 Jf5sle
-
we have H5a
1/10-2
equation
i90sls J&8
22s’O -
(7.2)
convenient
of the Schwarz
I/s+
a=-
of identity
It follows
out that
of k = 1130 can also be expressed (7.1)
[-2-2
I’
f5 (s) =
The
00.
a.
+ 494Os’o + 4482
The
upstream
downstream
CA which
of curve
may be more
4= -
-
in the
point
a
d2u / dz2 =
as to maintain
be obtained
branch
C:
where
center
a way,
can
that
in the case
of fnnction
a new
T5 (s) = -
itself,
flow
of such
the characteristic
We may
the shock
from (7.5),
We shall
s’a-
Instead
wave,
asymptotic
the axis
of t < 0.
case
different
certain
and obtain
the shock
along
of the
C, then
A from the direction
a certain
from the nozzle in such
use
subsonic.
parametric
SC--
Along
Z,,and
Z,
of t < 0 to point
to point
at the center
accelerated.
symmetry
In the second
x zo’11.
wave
by a jump point
velocity
except
the shock
beyond
wave
defines
flow
the stream
the stream
flow
solution
The
according the
of a shock
A in the direction
this
1021
gas flow
Z, , with a return
Z, to point
center
wave
direction,
the
of transonic
from point
of [ 141 consider
in the neighborhood nozzle
into
curve:
from point
authors
solutions
V=&(ilC)9(E+s)-a”TS
I’-5~8 + 66 1/5Es7 + 66sa 55s” 7
(7.11) 66E.+ -
11 Es + 3 Jf5
the solutions
of system
(4.4)
for
1022
S. V. Fal’kovich
k = I/C,
and
l/12,
of parameter
when expressed
f/30,
s, in other words,
form is, in these j = (;,lli
cases,
(S)
(B
where E, G, k,,
in a parametric
s is the homogenizing
.P)-“,
L’
and
Mil fli,
the solution
for k =
in [6f that in this case
the solution
of Equation
ence.
of that equation
We denote
function
for k =
the Schwarz
for k =
t/30,
as
function
before,
again
(3.6)
for the case
($.7 +
of f(c)
y= d~lds
we shall
of k =
had shown
algebraically
by
by
7/30
s1 ,
and the Schwarz
formula is then valid (8.1)
use Formula
dj/ds
(3.8)
where we substitute
but in terms of a, and not of
representation,
where
dsidsl,
can be expressed
’:I
polynomials.
Schwarz
7x2) (75” _;_ I)-“
use the parametric
find dgldq
and Ti the corresponding
in the family (2.7).
y/30
by s. The following
In order to find the solution st . Weshall
solution
Klein f23] had found the form of this depend-
l/30.
“t ::
s1 for s, We shall
functions
The generalised
V = Jc”G”Ti(s) (1; _+$sn
/~,G’lli (S) (I:’ _t S)-Z’L+2,
=
derive
the solution
form, are single-valued
variable.
as follows
and k, are constants,
8. We shall
and I.A. Chemov
is defined
by (‘7.1),
and ds;ttsl
by
(8.1) de -=tls Function
f ((1
is defined
by the formula
f = together
with Equation
Having (4.4)
7
(C,Y’ +
plane
$7 -
we determine
17Es’r
7c,s2 $ C,) F-?/o
in the usual
+ 118s“J + 187Es’e
(s’ -j- 7Bs6-
I! = G?H* (s) (s’ + 7.&S The same solution (7.8)
-
manner the solution
of system
for n = 17/r
C=,C
Equation
7c,s5
(7.1).
found the solution,
in the hodograph
5.4-“.3-“1&“T~F1-e
7s’ + E)
for the case
is used instead
-yy
+ 187s’-
119Es5+
7~2 + E) ‘% 2
, V = 3 GV’,, (s) (~7
of k = ~/SO can be obtained
+
17.~2+ E
76~5 -
7~2 + I?)-~/?
in another
form, if
of (7. If. BIBLIOGRAPHY
1.
2.
Frankl’ F.I., Issiedivanie po teorii kryla beskonechnogo razmakha, dvizhushchegosia so skorost’iu zvuka. (Analysis of the theory of a wing of infinite span at sonic velocity). Dokl. Akad. Nauk SSSR, Vol. 57, No. 7, 1947. Frankl’
F.I.,
Nauk SSSR,
3. 4.
Gnderley
K teorii ser.
K., Theory
Fal’kovich
S.V.,
sopla
matem.,
Lavalia.
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K teorii
(On the theory
of the Lava1 nozzle).
Izv. Akad.
Vol. 9, No. 5, 1945.
sopla
Flows Lavalia.
(Russian
transI.1
(On the theory
M., Izd. inostr.
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lit., Pr&f,
1960. Vol.
No. 4, 1946.
5.
Frankl’
F.I.,
Ob odnom semeistve
chastnykh
reshenii
uravneniia
Darbu-Tricomi.
(On
10,
Self modeling
one
6.
family
of particular Vol.
SSSR,
uber
die jenigen F&lle,
Vaglio Vol.
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9.
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K&mahTh.,
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Il.
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Yu.R.
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Lifshits,
gas flow
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Laurin
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Function ihres vierten Elements
Bd. 75, pp. 7.
solutions
Nauk
algebrsische
solutions
a body
OS.,
0 nekotorykh
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vrashcheniia
phyz.,
Vol.
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potokom
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uravnenii
of equations
of transonic
4, No. 5, pp. 954-957,
Equations.
law of transonic
zvukovym
Ptftf,
tochnykh
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tela
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A Treatise
The
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C,uderley K, and Yoshihara Ilath., Vol. 8, Uo. 4, 1951, ?yzhov
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t’o.
1, I’fi8. Yu.?.
techeniia
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flow
154,
Germain
and
I\pplication
17.
18.
20.
14S,
$. and fluids.
Ryzhov waves
19.
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Tomotika pressible
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2-nd
Barantsev
R.G.,
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thborzmes
tipe
nozzle
center).
Watson
ed.,
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PM&#, Vol.
type
Dokl.
22,
of plane-
Akad.
SSSR,
Paris, forms
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Vol. 2S2,
No.
of ahock
waves
plans
17, pp. 2511-2513, in flows
over
1961. sym-
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udamykh
voln
v soplakh
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Vol.
27, No. 2, pp. 309-337,
1963.
D.N.,
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sovremennogo
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Univ.,
of com-
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analiza.
flows
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PMM,
pt. 2. Fizmatgiz,
Lektsii Izd.
Appl. 1953.)
ploskoparallelnogo
(On the asymptotic
de conservation
rend.,
K.. Asymptotic
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Obrazovanie
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Lavalia.
v sopiakh surface).
I’amada K., Studies on two-dimensional transonic Part I. Quart. Appl. Math., Vol. 7, No. 4, 1950.
Wittaker
Gas
sopla
Compt.
1).‘1‘. ,Infi (iuderley
metrical
transition
Quart. No. 2,
1964.
des
hornog&es.
perekhoda
nozzle
Ob asimptoticheskom
O.S.,
tsentra
Yo. 2, pp. 290-293, P.,
:3arish
Ryzhov,
transonic flow pattern. in Sb. per. Mekhanika,
poverkhnosti
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v okrestnosti
in the ne~ghborhooii
I,ifshits, parallel
15,
‘) techenii
II., An axisymmetric (Russian translation
1965.
gazodinamike.
(Lectures
on Transonic
S. V. Fal’kooich
1024
21.
Bateman
H, and Erddlyi
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skorostei, vnutri limited
23.
Klein
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s oblastiu
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the flow).
19, No. 4, 1955. Vorlesungen
riber
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Function.
Berlin,
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Translated
by J.J.D.