(SANITIZED)UNCLASSIFIED SOVIET PAPER ENTITLED, "CONTRIBUTION TO THE THEORY OF STRONG INTERACTION OF A BOUNDARY LAYER WITH AN INVISCID HYPERSONIC FLOW"(SANITIZED)

• Document Type: CREST
• Collection: General CIA Records
• Document Number (FOIA) /ESDN (CREST): CIA-RDP80-00247A003000050001-1
• Release Decision: RIPPUB
• Original Classification: C
• Document Page Count: 30
• Document Creation Date: December 27, 2016
• Document Release Date: February 27, 2014
• Sequence Number: 1
• Publication Date: November 24, 1964
• Content Type: REPORT

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CONTRIBUTION TO TEIE THEORY OF STRONG INTERACTION OF
A BOUNDARY LAYM WITH AN INVISCID HYPERSONIC FLOW
Moscow, 1964
V. V. SYCREV
STAT
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SUIVARY
The paper discusses the higher-order approximations in the theory of strong
interaction between a boundary layer and an external inviscid flow. Known results
concerning the problem of the unsteady motion of a gas past an infinite plate, and
the problem of the steady flow past a semi-infinite plate are refined. The pmlysis
leads to asymptotic expressions for the transverse displacements of a plate, or its
shape, that correspond to the pressure-distribution law of the first approximation.
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INTRODUCTION
The effect of the viscosity and heat conductivity of a gas upon the flaw
field near abody:moving at hypersonic speed is known to lend itself to an
approximate analysis on the basis of the theory of boundary-layer interaction with
the external viscous region oethe flow /1/. If the body is sufficiently slender,
Cumber
and the Nadi and Reynolds numberfoffthec.ppoblpm are such that the ratio
663
Ovt,0 /4 / g5,404 ...=?? 1 , then a-strong interaction takes place, in which
the pressure field in the pertUrbed region of the flow is primarily influenced
by the displacing effect of the boundary layer, and to a much lesser degree by
the shape of the surface of the body that is situated in the flow.The-most-tYpical
examples of a plane flow of this type, namely, the unsteady flaw past an infinite
plane that has been abruptly set in motion at constant speed, and the steady flow
past a semi-infinite plate, have been discussed in /2, 3/.
The solutions obtained in these papers are based on combining the exact
(self-similar) solutions of the boundary-layer equations with the solutions of
the equations in the smPll-perturbation theory of hypersonic flow; the procedure
of combining these solutions having been developed only in the first approximation.
The result of this, is the peculiar behavior of the solutions in the intermediate
region (at the external boundary of the boundary layer), which manifests itself In
that the enthalpy of the gas inthis region tends to zero, while the density
undergoes an infinite increase.
The papers /2, 3/ include also accuracy estimates of the first-apprffrimation
theory.
The object of the present work is .to develop the higher-order approximations
to these problems, or more precisely, to.; problems associated with the asymptotic
behavior of thefloilfield of a viscous heat-conducting gas behind shock waves
the propagation of which is. controlled, by the same law ( J1/ and
in the limiting case where Mcr,'CIV.
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I. NONSTEADY MOTION
1. Let us examine the one-dimensional nonsteady motion of a viscous heat-
conducting gasp caused,by an infinite plate that has been set in motion at a
velocity that has a constant longitudinal component lico. We assume a linPar
dependence of the viscosity coefficient of the gas upon the specific enthalpy:
-
J.4 .
(1.1)
In this ease, the Navier-Stokes equations may be written in the form
'at,()=2_ 01,1- )
} v
?
2it
_Zt. ( In. )
'DV
211')
I
where the velocity-vector components tit., e refer to the longitudinal velocity of
the plate, the pressure p) refers to the quantity foolf; , the density 20 refers
to the density of the unperturbed flow J200 , the specific enthalpy 4? refers to the
114
quantity ; the dimensionless independent variables 17, ip refer to the
oa
quanti#eS ? - and respectively; E7)and a th
Z.e respectively e
Tc0 ,x4 _Yea
Prandtl number and the ratio of specific heats. of the gas.
Introducing,fon,theJlasisnofl.the cOntinuity equation,Lthe. ftmotion , defined
by the relations
TIP
El"
-S))
( I .
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wewrite the system (1.2) in terms of the independent variables t. ,
As a result we get
'OP 4 '3 (
+ -TY T "b7i Z'Y
(811,0
'bt t6 J
(6Ne
r),6111.:r .112.11,
3
k(b1). \2
(1.4)
As already stated, our object is to obtain an asymptotic solution of these equations
that corresionds to the one-dimensional motion of a gas propagating according to the
law
that satisfies the condition of attachment
tt =
and the condition of the absence of a heat flux
rah,
= C)
Z141 c
on the plate surface 0, 0, which is thus postulated as a heat insulated surface.
2. Fbrothe external portion of the flow field, adjoining to the surface of
the shock wave, such a solution is known to have the form
A
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. .
where the independent variable
3
= r\rt, 7* (2.2)
1.113Stituting the expression (2.1) into the initial system Of eqiitiOns (1.4), and
retaining the principal terms in these equations, we obtain a system of ordinary
differential equations for the well-known progressive motion of an inviscid gas:
6
s
ko(4-v)-11.-i-H0)=--327-vPol+Po.
R.?Yc: =j
Y:? + vo =o
(2.3)
We should note that taking a gas in-the external region of the flaw to be
inviscid and nonconducting involves a relative error on the order of t , since
this is the order of smallness of the relation of the terms neglected in equations
(1.4) to the principal terms.
The solution of the system of equations (2.3) must satisfy the set 'of
boundary conditions at the surface of the shock wave, the propagation of which
follows the equation (1.5). In the limiting ease of a.flow at 144*-- 0410" , these
?boundary conditions take the form:
Y o c) = c
V? (c) 234C44)
()29, C
c
(c)
?Ho (C) = r
? 8 -z
(2.4)
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_ .7
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where the constant C. has to be determined.
For the following, it will be essential to have the expressions for the sought
functions of the external flow at))-10.0. To obtain these equations it is necessary
to note that the second of the equations (2.3) is integrable with the aid of the
last ecivation), giving:
a-5)
where the constant of integration A,, is determined fromhe boundary conditions !
(2.4):
9 cl )f
A
?
( 2 .6)
Making use of (2.5) and the remaining equations of the system (2.3), it is now
easy to obtain the following expressions, valid for
(
y 2 = y +yo) + oo)
0.
72.7s )
2.
V =
?
0V -v
0. +.. o
PO = Poo+ 0(\))
\
= o
= H00 1,)- 21'+ ())
The coefficients in these formulas are related bythe following relation:.
-
= A7 P3f
3.
00 VO
00
V-2(312) POO ?r
1-1.- A-17 p
0 40 00
= A0. .
. 8)
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?3. To investigate the internal region of the flow field we, as usual,
introduce the independent variable
_
i\r=eY?
(3.1)
_ . .
To determine the form of the asymptotic expansion in this region, we express
the external flow function through the independent variable of the external
expansion
X't
and consider the limit t?io-Mat fixed values of _Ar.
Using the expressions (2.7), we get:
(3.2)
uz
[voo?voi 4r1Y-i- 0 (?;"r19]
Poo+ ? ().]
2 IL
(_i_)
?ir ? - 2, -
= kooK* + 0
k= H00
(3.3)
These expressions predict the form in which to seek the asymptotic solution for
the internal region of the flow, namely:
3 r 4 a
? ? ?
? ?U=t (W)..+?.-24.57 1)1 ?
? O.= t-41.0o(g)+. 2.? 31(1).(1\11 +.. ? (3.10
?
?
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Indeed, the combination of the internal and external expansions can be now
-performed if, in conformity with the simplest form of the combination principle,
the following boundary conditions for the external expansion functions are
satisfied:
in the first approximation, and
at
N cs,0
(3.5)
at N c=