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On the theory op hypersonic flow of a viscous gas past a blunt body
ON TJ3ETHEORY OF HYPERSONICFLOW OF A VISCOUS GAS TAST A BLUNT BODY (K TEORII OBTEKANIII'. TUPONOSOQO TPLA QIPERZVUKOVYM POTOKOM VIAYKOQO QAZA) PMM ~01.28, NO 6, 1964, pp. 1008-1014 B.M. BULAKH (Saratov) (Received April 25, 1964)
We consider the plane and axisymmetric problems of flow of a uniform hypersonic stream of viscous perfect gas past a blunt-nosed body in the case when the ordinary boundary-layer theory is inadequate, and higher approximations in the solution of the Navier-Stokes equations are required. Ey means of the well-known method of Inner and outer expansions we obtain the conditions on the bow shock wave for the second aooroxlmation to the solution outside the boundary layer (the first approximation being the inviscid flow). We consider the boundary-value problem arising in the determination of the secondapproximatlon. In flight at high altitude with very great speed the continuum theory becomes invalid, and it is necessary to use the kinetic theory of gases. However, so long as the Knudsen number remains less than 0.15, most aerodynamic problems can be treated by means of the Navler-Stokes equations taking Into account slip at the surface of the body (see the works of Street, Sherman and Talbot, and others in Under these conditions the boundary-layer Cl]). theory is inadequate, and the next approximation to the solution of the Navier-Stokes equations must be considered. A systematic review of second-order effects is given In the work of Van Dyke [2], who used the well-known method of Inner and outer solutions. In particular, he found the boundary condlFig. 1 tlons on the body for the determination of. thesecond . . approximation outsl.ie the boundary layer. In the present work we ODtaln tne boundary conditions on the bow shock wave for the second approximation by constructing the asymptotic representations of the solutions of the NavierStokes equations outside and Inside the region of the bow wave and their matching. 1. We consider the plane or axlsymmertic problem of hypersonic flow of a uniform stream of viscous perfect gas past a contour (Fig. 1). Here AOA’ is the contour of the body. Region 4 Is the boundary layer, and region 2 the shock wave, which Is regarded as a region of large gradients consists of of gas properties. We assume that the smooth contour AM’ 1219
1220
B.M.
analytic
the arc of the contour
arcs,
to the limiting Its T
equation
characteristic
of state
the absolute
at constant 1s
Is
(see
temperature,
pressure
and
We denote region sonic
2 collapses
i?to
(lr/%, >
curvature
cUrve from point
the gas is perfect,
6); is
the pressure,
the internal
of viscosity
9
Is constant;
the gas flow
and
regime
P
h
0
that heats energy
are functions
only
is described
by
Is lamlnar.
stream by the subscript
Mach number, and
is,
the density
the specific
the gas constant; oV are constant;
of the undisturbed
V, its
speed.
m ,
For w,X - 0
At hyperCBC', and region 4 disappears. 2 and 4 are always distinct [ 21, and the
the surface
1) regions
temperature
in regions 3 and 4 are _oC,l and li” c of the bod;l and the radius of
and enthalpy
the radius
respectively;
an analytic
p
and the flow
the parameters
characteristic II,’
o
M, Is the free-stream
speeds
R
cp and volume
the coefficients
e=o,T;
being Section
where
p = RpT
the Prandtl number of T; the Navler-Stokes equations;
$0 that
Wllakh
of
curvature
of the shock wave at the points
B
We introduce nates
s,
ured along
and
0
a curvilinear
(Fig.2).
n
are of
the arc
system of
Here BC
the same order.
s
and
and along
n
coordlare meas-
the normal to
Then if lengths are referred to the quantity it. a , the gas velocity to U, , the pressure to Pm, the temperature to PJL= I the density to v,ac-’P ’ the entropy and enthalpy of the gas to and the coefficients of u,” respectively,
cp and
viscosity
to the value
then the equations energy, Fig.
2
of
and tie equations
sen coordinate
of
p
at
continuity,
T = Umaapel, momntum and
of state, assume in the cho-
system the following
form [2] :
Iirpersonlc
flow
Ii
-k r x
[ &ii@+
X (? +
n cos
(L V~SOOUS
+vvn--
&-2P ,q-& +
of
&
(
)v,-2p
e)8 +
$3
past
a blunt
1221
body
1 pn3=2 w&t)* + +
g$ j], +2J$$i-l-
[ P @n-l-
i cos 0 + n cos @ n cos
k 1 + kn
&as
9)s +
[iI G
21 cm
fhvnSr
---
k
us + kv
1 + kn 1+ e] +
kn
-
2ip cob e (p + n C~S w
X
(1.3)
u’s -ku fjL %a -I- (1 + kn) (r + n cos 0) ( I-+kn + i”,,,
fj (&
(r+n
coS e)S+v
1 x
cos O)],
Here u and v are the componentsof velocity in the directionsof increasing s and s respectively; r = r(s) and 9 = 8(s) are the distance from the angle of inclinationof the tangent with the x-axis (Fig.2) for a point on the arc CBC', and k = k(a) Is its curvature,referred to a-'. The subscripts s and n on a functionalsymbol indicate differentiation. For the plane case j = 0 and for the sxlsymmetriccase j = 1 . For a gas with p = const the small parameter e = R; 3 , where .R_ Is the Reynolds number of the undisturbed stream. 2, To find the solution of the problem In the boundary layer (region 4 In Flg.l), Van Eyke [2] used the method of Inner and outer expansionsIn the parameter c. The first terms of the inner expansion In region 4 give the usual boundary-layertheory, and the next ones account for second-order effects (2n particular, slip and temperaturejump at the body surface).For the flow dn region 3 Van Qke obtalned asymptotic expansionsof the form (f= P9 P,u, *,T) f = p, (s+n) + EF, (a,n) + . . . (2.4) To flnd the coefficients F1 in (2.1), knowledge of which is required to improve the results of boundary-layertheory, It Is necessary to give the
component tween in
of
the
[2]
velocltg
normal
E’, on the
to
bow shock
the
surface
of
CEC’
(Fig ,l)
. . The first
on the
shock
wave
and a derlvatlon
of
the
For
of
this
conditions
the
body,
and
relations is
wave
be-
avallable
follows
here-
after.
3.
the
expansions.
solution
In
expansions
of
(region
2 of
regions
the
form
Fig.1)
tlon):
1 and
problem 3 (Fig.1)
(2.1)
(the
we use
outer
expanslons
we use
the
we use
for
method U,
the
following
type
slonal the
of
case,
and
functions
then
for
/ =
by the
Is
I&o
n
are
we obtain
(es) + p’,,
to
U$Ml (s)
equation
must
by
Thus
(2.1).
and
I + E l&J
(the
let
of us
wave
Inner
solu-
(3.1)
the
assume
one-dimenthat
for
by power
n-+ 0 series;
(3.2) (s) t F,,
regrouping
f
T
shock
(3.1)
b)l
represent
and outer
and
the
asymptotically
* * ’ 1 -t- E I&,
fl
p
sort
consideration
representable
from
(s) R +
transforming / =
suggested
expanslons
Fo, F, , . . .
small
After
This
expanslon
inner
i\i zzzTIE-2
I = frJ(s, ,I-) -t ef1 (s, 11:)+ . . . )
This
of P,
InsIde
solution). of
u,
terms
+ 22 IF,, for
large
(s) n +
in
the
. . .] +
series
(s) + I?,, (sp-I N
and
small
. , .
(3.2)
+
we have
(3.3)
. . .
n (the
matching
conditions).
follows from (3.3)
It
that fUIC“
We
note
that
Navier-Stokes consists
of
and
E
the
1’01~ the ti1c
to (s
terms,
one
having
Ls taken
l;iil
in
f,,
-
Is
shown
by
represented
(s)
are the
the
Ijo (X, ?J, 2) +
different equations to
+ r,,
v(u,,uy
equations
of
fact that
the
with subscript 2 :_‘n
with
more
[3]
the
Carteslan
the of
(3.1)
and
solution
special to
accurately
for
4, (s ,m)
of
series
large
N
and
so
on.
viscous
m .)
coefficients
F,
coordinates
X,
7J, 2) +
. . .,
in y,
7’ = I’, (x, y, z) +
region H
.
1 (Fig.1).
The
expansion
velocity
momentum terms
do not
ET1 (x,y,z)
+ . . .
1’ = A-0 (.T, ?/, 2) + &VI (z, y, 2) + . . . Substituting
vector. and affect
energy,
and
the
terms
(4.1)
taking in
zero are constant, corresponding to the
, we obtain
(The
(4-Q
, u, ) is the
continuity,
In
1 + exponen-
form ‘JQ(&
in
There-
s-l.
_ 7, o (s
terms,
the
examples,
by a Power respect
+ exponential
N + f
1’ = po (cc, 3, 2) + t’l’l (d., YJ, L) + . . ., Here
fluid
terms,
(s)l’j
form
(3.4)
asymptotically
for for
use
the
consideration
character
may be written
_ r”,O (s)
has
N--+&-cc
0,
exponential
(3.4)
expansion
an Incompressible
as
?arts,
It is convenient
&re
for
Inner
,371 - Fo, (9 ) + exponential
We derive
4,
as
“YZ(s J)
Yunctlons
J’ =
two other
condltlons
for111
tial
just
equatlons,
the
into E
into account
, while
uniform
the
stream
the the terms at
Hypersolllc
flow
We require that.
of
a viscous
gas
PM Pl --) 0
Vl,
past
as
a
blunt
1223
body
x-3-00
(4.3) Section 7).
J-nany direction different from the characteristic one (see From (4.2) and (4.3)
we obtain Pl PO
P1=--1x,
th
%,
Pl
y-=---,
PO
that Is, In the plane and axlsynrmetric cases where
av
-_+=__2_~=o i3X dY
av
3X
curl
(4.4)
az
V,=
0
and
grad cp ,
V,=
cp satisfies Equation
-m2ag+;g+&o, We now show that the condition (4.3) finite portion of the arc the plane case.
Here
a/al = 0
given on
(4.5)
Imposes no llmltatlons on
CRC' (Flg.1).
q = g,(x + my) + B,(x -my)
1)“’
m = (Mm2 -
v,
on a
This is most easily obtained for
and the general solution of (4.5) Is
, where
B, and
Q2 are determined If
v,
Is
CBC' (Fig.1).
The condition (4.3)
Is satisfied If gI' (5 oo) =gs'(fm)=
Indicates differentiation.
0.
The prime
The same result Is obtained also for the axlsym-
metric case If the solution of (4.5)
Is represented by the well-known formula
of Volterra. Consequently the condition of decay of disturbances In region 1 as y--m (Flg.1) leaves u1 (a,n) and u,(s,n) arbitrary on the finite segment CRC', whereas p, and p,are related by the condition (4.4), which we write In the form
Pl
5.
1PO =
7%
1 Pot
-
Pl =
(UI Cos 0 +
ul
sin 0)
(4.6)
We consider the flow Inside the shock wave (region 2 of Flg.1).
The
expansions for the solution are taken In the form (3.1)
P =
PO
(s, N) +
&PI
6,
N)
+
. . . . ,
T
P = PO(s, N) + Ep1 (s, N) + - . . , v = zr, (*v,N) + ev, (s, N) + . . . , Transforming to
N
and
s
=
T, (s, N) + eT, (s, N) + . . .
‘U=U&N)+eul(s, N = ns+
N)+... (5.1)
In Equations (1.1) to (1.4) we obtain the sys-
tem (P@,
=
0 (89
Pvv, +
puTN -
vpN = -
PN =
PVU, [(A +
(pUN)N +
2p) 'NlN +
0-l @TN)N + (A f p = r(r -
The
=
0 (e2b
(5.2)
o (8")
2~) ZX,~+ puN2 -I- 0 (e”)
1) / 71 pr
curvature of the shock wave affects only i;ermsof order
ga In (5.2),
B.X. ailSkn
1224
so that the equations for the coefficientswith subscripts 0 and 1 are obtained just as in the one-dimensiomalcase, if n, = 0 . We give only the equations from which it follows that nOH= m!,,=0 . From the relations pcz2c = a = const,
ag*jy = (~~U~~~~,
cl0 = PLO(TLJ
it follows that Since 0 # pee m , then ncN+ 0 as N - f m only In the case uO,,2 0 . Bar I.+,,, using mOn = 0 , we obtain aulN= (ecu,,),,; hence under the condition that u,N* 0 as N 4 f 0~ it follows that uIN = 0 We introduce for conveniencethe symbol $13 = (I),,_,,,(J')~_,_, . for the one-dimensionalproblem there exist the well-known relations (PfJ) = 0,
(PU2 +
$9 = 0,
_LX_+;j=o 1r---1 P
Then
(5.3)
which follow simply from the conditions VN'
PN
PN'
TN --, 0, N-t-&cc
We may therefore expect that the relationS ConneCting PO 3 Pi ) PO) Pl 9 will be obtained from (5.3) upon substituas Ndfm U, > ‘41, ub and ‘JI ting (5.1) and equating coefficientsof like powers of e . They have the form LB+ $} = 0 {P&} = 0 (5 4) (%) = 0, @olJo2+ PO> = 0, PO i7-l {UJ = 0,
{2%P&
+ T
V,2Pl + Pl) = 0,
{PO”1 +
Plqd
= 0
~-~)+0+0
i r---1 t PO
*
(5.5)
The author has also obtained (5.4) and (5.5) directly from the equations for the quantitieswith subscripts 0 and 1 and the conditionsthat the derivativesof those quantitiestend to zero as N-i m The relations (5.4) are the usual condi(see (3.4)). tions on a shock wave for an inviscid stream. In the relations (5.5) the quantities n1 and u1 are arbitrary but pi and p1 are determined in terms of its N---m, them by means of (4.6). If we eliminate ur and v1 from 15.5) as N - - - , we obtain two relations connecting Ul' ul, pi and pi as N - + m fn -)0), that is, the desired relations on the shock wave. pig. 3 n1 MS 0 +
Omitting the rather easy computations,we give the final result VOVl +
IT ! (7 -
$)I jpIPo-l
-
PoPo-ZPJ = 0
(5.6)
Et1 COSQ- p. sin Bv, + (voz - v. sin 8) p1 + p1 = 0 All quantitiesare taken on approaching CBC' from the side of region (Fig. 1).
3
Hypersonic
The equations
6.
for
flow
viscous
a
01
g.m
the coefficients
past a
with
blunt
1225
body
subscript
I
in Expansions
(2.1) in region 3 (Flg,l) are actually obtained from the equations cld flow (the viscous terms being of order ca ) and have the form [(r +
n cos 8)’ (PO% + +
Uo%
uld
(1 +
(~1,
+
WJ,)
(1 +
P141, + [(I + W (r + n cos Qi (povl + pIvo)],= 0
JW-l
+
+
q,uln
(1 + k4-l W-l
+
+
vlu,,, +
(Po-1P,8 -
q,~n
+
+
~1’16s) (1 +
Pl
The type of Equations
=
is
region
is
k (1 +
PlPo-2Po,
v,i;,
+
+
PJO)
=
kn)-l -
=
+ u,v,,) +
0
+
kn)-1-2uou,
0
vlTon1 +
p1 [noTo, (1 +
vopln -
kn)-l
determined
+
V,Pon = 0
(6.1)
(7 -
the characteristics
of the lnviscid
The transonlc
(POT1
(6.1)
In particular,
acterlstics be found
+
-
voTo,] - (uoPls + zwo,) (1 +
+
problem.
W-l
+vkn)-l(u,vl
k (1
PIPO-"PO,)
v,v,
+ PlnPO-l PO [(u,T,,
of lnvls-
by the solution of
(6.1)
of
coincide
the lnviscld with
the char-
problem.
(Flg.3)
in which
bounded by the llmltlng
u,,
ul,
characteristic
pi,
pi and T, must first EK , the surface of the
body OK , the axis of symmetry BO , and the segment BE of the shock wave (DK being the sonic line). It is necessary to find the solution of (6.1) with the following
boundary
conditions:
the normal component of
v,
Is given
the condition of symmetry on BO ; the two conditions (5.6) on on OK [2l; is known from the solution of the lnvlscld probthe arc EB , whose location lem;
and on
ferentials Equations
EK , whose location
Is also
of the unknown quantities (6.1)
(boundedness
of
tiown,
a relation
between
appearing
In the characteristics
the derivatives
on the characteristic
the dlfof
EK ).
The question of existence and uniqueness for such. a problem requires furIn the supersonic region C.KA the answer to that ther consideration. question can be obtained by means of an Investigation of the possibility of constructing the flow by the numerical method of characteristics. Assuming the flow In the that uI, ul, p1 pI and T, are known on KK , we calculate step In the calculation Is the determination region CEKA . The essential K (Flg.3) from the data at points E of PI f PI 9 T, 1~~ and v at point - t_hf chara_cterlstic EI of Equations (6.I), and C . Along a streamilne Replacing the differential by we may find &S, , where S, = a finite Increment, we find t Ekp~a~u~“~!! S* at point I from S at point I an& G we find S, at pikit d; by K , and from values of S, at points From the relations between dp linear extrapolation. dP, > au another’conalong the characteristic flG of Equations (6.1) we obtain still (5.6) ditlon on U, , v,, p, and p, at point H . Then adding the condltlons we obtain a system of four Independent linear algebraic equations for deterH. mining u,, p,, vl and p, at point The correctness of the flow calculation In the supersonic region allows us to hope that the boundary-value problem considered above for the transonlc region Is also correct.
B.M.
1226
&l&h
In the lnvlscld problem there springs from a point on the contour 7. where the curvature Is discontinuous a characteristic L along which the derlvatlces of u, v, p, p and T ,wlth respect to n s;ffer dlscontlnulties (n now being measured normal to L and s along L ). If viscosity Is considered, there exists, just as In the case of a shock wave, a transltlon region outside of which expansions of the form (2.1) are assumed to be valid. The terms of order E In these expansions may evidently suffer dlsFrom the point of Intersection of L with the shock contlnultles on wave CBC' (Fig.14' two characteristicsemerge Into region 1, for which d&r = i m-l, with m = (M,~ - I)lia,and near which there exists transition region outside of which the solution Is represented by expansions of the form (2.1), and Inside by expansions of the form f = f. + sflh, s)+ . . .,
f. = const,
q = n&-l
Let us use the symbols ff)= (f)+,+,- (f)+,-,; the expansion (7.1) there are the relations
{PO)=
(PO)= (a01= (uo)= 0,
(PI,= ro"(P,),
(7.1)
for the coefficients of
(sl PO 4: (P3 u0 = 0,
(a3 = 0
Hence It follows that
that Is, It Is found that the Formulas (4.4) are valid in the presence of dlscontlnultles In the terms of order c in region 1 (Fig.1). As x+--m the transition reglons near the characteristics determined by Equatldns dy/dx= f m-l broaden, so that the characteristic directions must be excluded In conditions (4.3). 8. In conclusion we note that the method of Inner and outer expansions was applied by Germaln [4] to obtain conditions on the shock wave, but he expanded the Inner solution In powers of e2 and so missed the terms of order E that are of Interest In the present problem. The author thanks S.V. Fal'kovlch and S.A. Kaganov problems considered here.
for discussion of the
BIHLIOGRAPHY 1.
2.
3.
4.
Rarefied Gas Dynamics. Foreign Literature Publishing House, Moscow,I963. Van Dyke, M., Second-order compressible boundary layer theory with appllcation to blunt bodies In hypersonic flow. Hypersonic Flow Research, Academic Press, N.Y. - London, pp.37-76, 1962. Hulakh, B.M., 0 vysshlkh priblizhenllakh v teorll pogranichnogo sIoia (On higher approximations In the boundary-layer theory). PMM~01.28, Ng 3, 1964. Germaln, Paul and Gulraud, Jean-Pierre, Conditions de chcc et structure des ondes de choc dans un ecoulement statlonnalre de flulde disslpatlf. OMERA Publ. Ng 105, 1962.
Translated by M.D.V.D.
We consider the plane and axisymmetric problems of flow of a uniform hypersonic stream of viscous perfect gas past a blunt-nosed body in the case when the ordinary boundary-layer theory is inadequate, and higher approximations in the solution of the Navier-Stokes equations are required. Ey means of the well-known method of Inner and outer expansions we obtain the conditions on the bow shock wave for the second aooroxlmation to the solution outside the boundary layer (the first approximation being the inviscid flow). We consider the boundary-value problem arising in the determination of the secondapproximatlon. In flight at high altitude with very great speed the continuum theory becomes invalid, and it is necessary to use the kinetic theory of gases. However, so long as the Knudsen number remains less than 0.15, most aerodynamic problems can be treated by means of the Navler-Stokes equations taking Into account slip at the surface of the body (see the works of Street, Sherman and Talbot, and others in Under these conditions the boundary-layer Cl]). theory is inadequate, and the next approximation to the solution of the Navier-Stokes equations must be considered. A systematic review of second-order effects is given In the work of Van Dyke [2], who used the well-known method of Inner and outer solutions. In particular, he found the boundary condlFig. 1 tlons on the body for the determination of. thesecond . . approximation outsl.ie the boundary layer. In the present work we ODtaln tne boundary conditions on the bow shock wave for the second approximation by constructing the asymptotic representations of the solutions of the NavierStokes equations outside and Inside the region of the bow wave and their matching. 1. We consider the plane or axlsymmertic problem of hypersonic flow of a uniform stream of viscous perfect gas past a contour (Fig. 1). Here AOA’ is the contour of the body. Region 4 Is the boundary layer, and region 2 the shock wave, which Is regarded as a region of large gradients consists of of gas properties. We assume that the smooth contour AM’ 1219
1220
B.M.
analytic
the arc of the contour
arcs,
to the limiting Its T
equation
characteristic
of state
the absolute
at constant 1s
Is
(see
temperature,
pressure
and
We denote region sonic
2 collapses
i?to
(lr/%, >
curvature
cUrve from point
the gas is perfect,
6); is
the pressure,
the internal
of viscosity
9
Is constant;
the gas flow
and
regime
P
h
0
that heats energy
are functions
only
is described
by
Is lamlnar.
stream by the subscript
Mach number, and
is,
the density
the specific
the gas constant; oV are constant;
of the undisturbed
V, its
speed.
m ,
For w,X - 0
At hyperCBC', and region 4 disappears. 2 and 4 are always distinct [ 21, and the
the surface
1) regions
temperature
in regions 3 and 4 are _oC,l and li” c of the bod;l and the radius of
and enthalpy
the radius
respectively;
an analytic
p
and the flow
the parameters
characteristic II,’
o
M, Is the free-stream
speeds
R
cp and volume
the coefficients
e=o,T;
being Section
where
p = RpT
the Prandtl number of T; the Navler-Stokes equations;
$0 that
Wllakh
of
curvature
of the shock wave at the points
B
We introduce nates
s,
ured along
and
0
a curvilinear
(Fig.2).
n
are of
the arc
system of
Here BC
the same order.
s
and
and along
n
coordlare meas-
the normal to
Then if lengths are referred to the quantity it. a , the gas velocity to U, , the pressure to Pm, the temperature to PJL= I the density to v,ac-’P ’ the entropy and enthalpy of the gas to and the coefficients of u,” respectively,
cp and
viscosity
to the value
then the equations energy, Fig.
2
of
and tie equations
sen coordinate
of
p
at
continuity,
T = Umaapel, momntum and
of state, assume in the cho-
system the following
form [2] :
Iirpersonlc
flow
Ii
-k r x
[ &ii@+
X (? +
n cos
(L V~SOOUS
+vvn--
&-2P ,q-& +
of
&
(
)v,-2p
e)8 +
$3
past
a blunt
1221
body
1 pn3=2 w&t)* + +
g$ j], +2J$$i-l-
[ P @n-l-
i cos 0 + n cos @ n cos
k 1 + kn
&as
9)s +
[iI G
21 cm
fhvnSr
---
k
us + kv
1 + kn 1+ e] +
kn
-
2ip cob e (p + n C~S w
X
(1.3)
u’s -ku fjL %a -I- (1 + kn) (r + n cos 0) ( I-+kn + i”,,,
fj (&
(r+n
coS e)S+v
1 x
cos O)],
Here u and v are the componentsof velocity in the directionsof increasing s and s respectively; r = r(s) and 9 = 8(s) are the distance from the angle of inclinationof the tangent with the x-axis (Fig.2) for a point on the arc CBC', and k = k(a) Is its curvature,referred to a-'. The subscripts s and n on a functionalsymbol indicate differentiation. For the plane case j = 0 and for the sxlsymmetriccase j = 1 . For a gas with p = const the small parameter e = R; 3 , where .R_ Is the Reynolds number of the undisturbed stream. 2, To find the solution of the problem In the boundary layer (region 4 In Flg.l), Van Eyke [2] used the method of Inner and outer expansionsIn the parameter c. The first terms of the inner expansion In region 4 give the usual boundary-layertheory, and the next ones account for second-order effects (2n particular, slip and temperaturejump at the body surface).For the flow dn region 3 Van Qke obtalned asymptotic expansionsof the form (f= P9 P,u, *,T) f = p, (s+n) + EF, (a,n) + . . . (2.4) To flnd the coefficients F1 in (2.1), knowledge of which is required to improve the results of boundary-layertheory, It Is necessary to give the
component tween in
of
the
[2]
velocltg
normal
E’, on the
to
bow shock
the
surface
of
CEC’
(Fig ,l)
. . The first
on the
shock
wave
and a derlvatlon
of
the
For
of
this
conditions
the
body,
and
relations is
wave
be-
avallable
follows
here-
after.
3.
the
expansions.
solution
In
expansions
of
(region
2 of
regions
the
form
Fig.1)
tlon):
1 and
problem 3 (Fig.1)
(2.1)
(the
we use
outer
expanslons
we use
the
we use
for
method U,
the
following
type
slonal the
of
case,
and
functions
then
for
/ =
by the
Is
I&o
n
are
we obtain
(es) + p’,,
to
U$Ml (s)
equation
must
by
Thus
(2.1).
and
I + E l&J
(the
let
of us
wave
Inner
solu-
(3.1)
the
assume
one-dimenthat
for
by power
n-+ 0 series;
(3.2) (s) t F,,
regrouping
f
T
shock
(3.1)
b)l
represent
and outer
and
the
asymptotically
* * ’ 1 -t- E I&,
fl
p
sort
consideration
representable
from
(s) R +
transforming / =
suggested
expanslons
Fo, F, , . . .
small
After
This
expanslon
inner
i\i zzzTIE-2
I = frJ(s, ,I-) -t ef1 (s, 11:)+ . . . )
This
of P,
InsIde
solution). of
u,
terms
+ 22 IF,, for
large
(s) n +
in
the
. . .] +
series
(s) + I?,, (sp-I N
and
small
. , .
(3.2)
+
we have
(3.3)
. . .
n (the
matching
conditions).
follows from (3.3)
It
that fUIC“
We
note
that
Navier-Stokes consists
of
and
E
the
1’01~ the ti1c
to (s
terms,
one
having
Ls taken
l;iil
in
f,,
-
Is
shown
by
represented
(s)
are the
the
Ijo (X, ?J, 2) +
different equations to
+ r,,
v(u,,uy
equations
of
fact that
the
with subscript 2 :_‘n
with
more
[3]
the
Carteslan
the of
(3.1)
and
solution
special to
accurately
for
4, (s ,m)
of
series
large
N
and
so
on.
viscous
m .)
coefficients
F,
coordinates
X,
7J, 2) +
. . .,
in y,
7’ = I’, (x, y, z) +
region H
.
1 (Fig.1).
The
expansion
velocity
momentum terms
do not
ET1 (x,y,z)
+ . . .
1’ = A-0 (.T, ?/, 2) + &VI (z, y, 2) + . . . Substituting
vector. and affect
energy,
and
the
terms
(4.1)
taking in
zero are constant, corresponding to the
, we obtain
(The
(4-Q
, u, ) is the
continuity,
In
1 + exponen-
form ‘JQ(&
in
There-
s-l.
_ 7, o (s
terms,
the
examples,
by a Power respect
+ exponential
N + f
1’ = po (cc, 3, 2) + t’l’l (d., YJ, L) + . . ., Here
fluid
terms,
(s)l’j
form
(3.4)
asymptotically
for for
use
the
consideration
character
may be written
_ r”,O (s)
has
N--+&-cc
0,
exponential
(3.4)
expansion
an Incompressible
as
?arts,
It is convenient
&re
for
Inner
,371 - Fo, (9 ) + exponential
We derive
4,
as
“YZ(s J)
Yunctlons
J’ =
two other
condltlons
for111
tial
just
equatlons,
the
into E
into account
, while
uniform
the
stream
the the terms at
Hypersolllc
flow
We require that.
of
a viscous
gas
PM Pl --) 0
Vl,
past
as
a
blunt
1223
body
x-3-00
(4.3) Section 7).
J-nany direction different from the characteristic one (see From (4.2) and (4.3)
we obtain Pl PO
P1=--1x,
th
%,
Pl
y-=---,
PO
that Is, In the plane and axlsynrmetric cases where
av
-_+=__2_~=o i3X dY
av
3X
curl
(4.4)
az
V,=
0
and
grad cp ,
V,=
cp satisfies Equation
-m2ag+;g+&o, We now show that the condition (4.3) finite portion of the arc the plane case.
Here
a/al = 0
given on
(4.5)
Imposes no llmltatlons on
CRC' (Flg.1).
q = g,(x + my) + B,(x -my)
1)“’
m = (Mm2 -
v,
on a
This is most easily obtained for
and the general solution of (4.5) Is
, where
B, and
Q2 are determined If
v,
Is
CBC' (Fig.1).
The condition (4.3)
Is satisfied If gI' (5 oo) =gs'(fm)=
Indicates differentiation.
0.
The prime
The same result Is obtained also for the axlsym-
metric case If the solution of (4.5)
Is represented by the well-known formula
of Volterra. Consequently the condition of decay of disturbances In region 1 as y--m (Flg.1) leaves u1 (a,n) and u,(s,n) arbitrary on the finite segment CRC', whereas p, and p,are related by the condition (4.4), which we write In the form
Pl
5.
1PO =
7%
1 Pot
-
Pl =
(UI Cos 0 +
ul
sin 0)
(4.6)
We consider the flow Inside the shock wave (region 2 of Flg.1).
The
expansions for the solution are taken In the form (3.1)
P =
PO
(s, N) +
&PI
6,
N)
+
. . . . ,
T
P = PO(s, N) + Ep1 (s, N) + - . . , v = zr, (*v,N) + ev, (s, N) + . . . , Transforming to
N
and
s
=
T, (s, N) + eT, (s, N) + . . .
‘U=U&N)+eul(s, N = ns+
N)+... (5.1)
In Equations (1.1) to (1.4) we obtain the sys-
tem (P@,
=
0 (89
Pvv, +
puTN -
vpN = -
PN =
PVU, [(A +
(pUN)N +
2p) 'NlN +
0-l @TN)N + (A f p = r(r -
The
=
0 (e2b
(5.2)
o (8")
2~) ZX,~+ puN2 -I- 0 (e”)
1) / 71 pr
curvature of the shock wave affects only i;ermsof order
ga In (5.2),
B.X. ailSkn
1224
so that the equations for the coefficientswith subscripts 0 and 1 are obtained just as in the one-dimensiomalcase, if n, = 0 . We give only the equations from which it follows that nOH= m!,,=0 . From the relations pcz2c = a = const,
ag*jy = (~~U~~~~,
cl0 = PLO(TLJ
it follows that Since 0 # pee m , then ncN+ 0 as N - f m only In the case uO,,2 0 . Bar I.+,,, using mOn = 0 , we obtain aulN= (ecu,,),,; hence under the condition that u,N* 0 as N 4 f 0~ it follows that uIN = 0 We introduce for conveniencethe symbol $13 = (I),,_,,,(J')~_,_, . for the one-dimensionalproblem there exist the well-known relations (PfJ) = 0,
(PU2 +
$9 = 0,
_LX_+;j=o 1r---1 P
Then
(5.3)
which follow simply from the conditions VN'
PN
PN'
TN --, 0, N-t-&cc
We may therefore expect that the relationS ConneCting PO 3 Pi ) PO) Pl 9 will be obtained from (5.3) upon substituas Ndfm U, > ‘41, ub and ‘JI ting (5.1) and equating coefficientsof like powers of e . They have the form LB+ $} = 0 {P&} = 0 (5 4) (%) = 0, @olJo2+ PO> = 0, PO i7-l {UJ = 0,
{2%P&
+ T
V,2Pl + Pl) = 0,
{PO”1 +
Plqd
= 0
~-~)+0+0
i r---1 t PO
*
(5.5)
The author has also obtained (5.4) and (5.5) directly from the equations for the quantitieswith subscripts 0 and 1 and the conditionsthat the derivativesof those quantitiestend to zero as N-i m The relations (5.4) are the usual condi(see (3.4)). tions on a shock wave for an inviscid stream. In the relations (5.5) the quantities n1 and u1 are arbitrary but pi and p1 are determined in terms of its N---m, them by means of (4.6). If we eliminate ur and v1 from 15.5) as N - - - , we obtain two relations connecting Ul' ul, pi and pi as N - + m fn -)0), that is, the desired relations on the shock wave. pig. 3 n1 MS 0 +
Omitting the rather easy computations,we give the final result VOVl +
IT ! (7 -
$)I jpIPo-l
-
PoPo-ZPJ = 0
(5.6)
Et1 COSQ- p. sin Bv, + (voz - v. sin 8) p1 + p1 = 0 All quantitiesare taken on approaching CBC' from the side of region (Fig. 1).
3
Hypersonic
The equations
6.
for
flow
viscous
a
01
g.m
the coefficients
past a
with
blunt
1225
body
subscript
I
in Expansions
(2.1) in region 3 (Flg,l) are actually obtained from the equations cld flow (the viscous terms being of order ca ) and have the form [(r +
n cos 8)’ (PO% + +
Uo%
uld
(1 +
(~1,
+
WJ,)
(1 +
P141, + [(I + W (r + n cos Qi (povl + pIvo)],= 0
JW-l
+
+
q,uln
(1 + k4-l W-l
+
+
vlu,,, +
(Po-1P,8 -
q,~n
+
+
~1’16s) (1 +
Pl
The type of Equations
=
is
region
is
k (1 +
PlPo-2Po,
v,i;,
+
+
PJO)
=
kn)-l -
=
+ u,v,,) +
0
+
kn)-1-2uou,
0
vlTon1 +
p1 [noTo, (1 +
vopln -
kn)-l
determined
+
V,Pon = 0
(6.1)
(7 -
the characteristics
of the lnviscid
The transonlc
(POT1
(6.1)
In particular,
acterlstics be found
+
-
voTo,] - (uoPls + zwo,) (1 +
+
problem.
W-l
+vkn)-l(u,vl
k (1
PIPO-"PO,)
v,v,
+ PlnPO-l PO [(u,T,,
of lnvls-
by the solution of
(6.1)
of
coincide
the lnviscld with
the char-
problem.
(Flg.3)
in which
bounded by the llmltlng
u,,
ul,
characteristic
pi,
pi and T, must first EK , the surface of the
body OK , the axis of symmetry BO , and the segment BE of the shock wave (DK being the sonic line). It is necessary to find the solution of (6.1) with the following
boundary
conditions:
the normal component of
v,
Is given
the condition of symmetry on BO ; the two conditions (5.6) on on OK [2l; is known from the solution of the lnvlscld probthe arc EB , whose location lem;
and on
ferentials Equations
EK , whose location
Is also
of the unknown quantities (6.1)
(boundedness
of
tiown,
a relation
between
appearing
In the characteristics
the derivatives
on the characteristic
the dlfof
EK ).
The question of existence and uniqueness for such. a problem requires furIn the supersonic region C.KA the answer to that ther consideration. question can be obtained by means of an Investigation of the possibility of constructing the flow by the numerical method of characteristics. Assuming the flow In the that uI, ul, p1 pI and T, are known on KK , we calculate step In the calculation Is the determination region CEKA . The essential K (Flg.3) from the data at points E of PI f PI 9 T, 1~~ and v at point - t_hf chara_cterlstic EI of Equations (6.I), and C . Along a streamilne Replacing the differential by we may find &S, , where S, = a finite Increment, we find t Ekp~a~u~“~!! S* at point I from S at point I an& G we find S, at pikit d; by K , and from values of S, at points From the relations between dp linear extrapolation. dP, > au another’conalong the characteristic flG of Equations (6.1) we obtain still (5.6) ditlon on U, , v,, p, and p, at point H . Then adding the condltlons we obtain a system of four Independent linear algebraic equations for deterH. mining u,, p,, vl and p, at point The correctness of the flow calculation In the supersonic region allows us to hope that the boundary-value problem considered above for the transonlc region Is also correct.
B.M.
1226
&l&h
In the lnvlscld problem there springs from a point on the contour 7. where the curvature Is discontinuous a characteristic L along which the derlvatlces of u, v, p, p and T ,wlth respect to n s;ffer dlscontlnulties (n now being measured normal to L and s along L ). If viscosity Is considered, there exists, just as In the case of a shock wave, a transltlon region outside of which expansions of the form (2.1) are assumed to be valid. The terms of order E In these expansions may evidently suffer dlsFrom the point of Intersection of L with the shock contlnultles on wave CBC' (Fig.14' two characteristicsemerge Into region 1, for which d&r = i m-l, with m = (M,~ - I)lia,and near which there exists transition region outside of which the solution Is represented by expansions of the form (2.1), and Inside by expansions of the form f = f. + sflh, s)+ . . .,
f. = const,
q = n&-l
Let us use the symbols ff)= (f)+,+,- (f)+,-,; the expansion (7.1) there are the relations
{PO)=
(PO)= (a01= (uo)= 0,
(PI,= ro"(P,),
(7.1)
for the coefficients of
(sl PO 4: (P3 u0 = 0,
(a3 = 0
Hence It follows that
that Is, It Is found that the Formulas (4.4) are valid in the presence of dlscontlnultles In the terms of order c in region 1 (Fig.1). As x+--m the transition reglons near the characteristics determined by Equatldns dy/dx= f m-l broaden, so that the characteristic directions must be excluded In conditions (4.3). 8. In conclusion we note that the method of Inner and outer expansions was applied by Germaln [4] to obtain conditions on the shock wave, but he expanded the Inner solution In powers of e2 and so missed the terms of order E that are of Interest In the present problem. The author thanks S.V. Fal'kovlch and S.A. Kaganov problems considered here.
for discussion of the
BIHLIOGRAPHY 1.
2.
3.
4.
Rarefied Gas Dynamics. Foreign Literature Publishing House, Moscow,I963. Van Dyke, M., Second-order compressible boundary layer theory with appllcation to blunt bodies In hypersonic flow. Hypersonic Flow Research, Academic Press, N.Y. - London, pp.37-76, 1962. Hulakh, B.M., 0 vysshlkh priblizhenllakh v teorll pogranichnogo sIoia (On higher approximations In the boundary-layer theory). PMM~01.28, Ng 3, 1964. Germaln, Paul and Gulraud, Jean-Pierre, Conditions de chcc et structure des ondes de choc dans un ecoulement statlonnalre de flulde disslpatlf. OMERA Publ. Ng 105, 1962.
Translated by M.D.V.D.