transonic flows of a radiating gas in channels
TRANSCRIPT
130 G. V. Voskresenskii and A. I. Plis
in the problems discussed above, our method of solution makes it possible to analyse numerically the solution in the range ka G 20, where a is the characteristic linear parameter. It may be expected that the use of bilinear substitutions will enable a solution to be obtained in some other interest~g boundary value problems of the mathematical theory of diffraction.
Danslated by D. E. Brown
1. NOBLE, B., Methods bused on the Wiener-Hopf technbue, Pergamon, 1959.
2. VAINSHTEIN, L. A., Diffraction theory and the method of factorization (Teoriya difraktsii i metod faktorizatsii), Sov. Radio, Moscow, 1966.
3. AVDEEV, E. V., and VOSKRESENSKII, G. V., Radiation of a charged thread moving ~~orrnly close to a comb structure; generat solution, Rad~tekhn. Elek~on~ka, 12, No. 3,469-478, 1967.
4. MITTRA, R., and LEE, S. W., Analytic methods of waveguide theory, Mir, Moscow, 1974.
5. AVDEEV, E. V., and VOSKRESENSKII, G. V., Electromagnetic excitation of a periodic lattice structure of conducting tapes, Plane wave diffraction, Radiotekhn. i Elektronika, 14, No. 5, 839-850, 1969.
6. BEREZHNOI, V. A., and PLIS, A. I., Electroma~etic excitation of a periodic structure, in: Brief communications on physics (Kratkie soobshcheniya po fizike), No. 5,82-87, Fiz. in-t Akad. NaukSSSR.
7. VEREZHNOI, V. A., VOSKRESENSKII, G. V., and PLIS, A. I., Electromagnetic excitation of a periodic structure with a channel, Izv. Vuzov. Radiofizika, 16, No. 3,449-460, 1973.
8. BOLOTOVSKII, B. M., VOSKRESENSKII, G. V., and PLIS, A. I., Electromagnetic wave scattering in waveguides with an i~omogeneity of finite length, Preprint FIAN, No. 154, 1971.
TRANSONIC FLOWS OF A RADIATING GAS IN CHANNELS*
V. N. KOTEROV
Moscow
(Received 26 July 1974)
ASYMPTOTIC equations are obtained which describe the transonic flows of a radiating gas in channels of finite optical width. Examples are given of the c~cula~on of the converse problem of finding the channel wall temperature for a given gas flow. In the case where the characteristic length of the channel wall temperature variation is much greater than the mean free path of the radiation, a simple equation connecting the gas temperature with the wall temperature is derived and studied.
Stationary plane and axisymmetric flows of a radiating gas k channels with weakly varying cross-sectional area are considered in this paper. It is assumed that the Rows do not differ too strongly from a uniform transonic flow.
Linear problems simulating two-dimensional subsonic and supersonic motions of a radiating gas were studied in [l-4] , In [S] non-linear equations for the transonic flows of a viscous heat- conducting radiating gas were constructed, which were obviously valid at great optical distances from the radiating surfaces, when the radiative transfer is one-dimensional to a first approximation.
The purpose of this paper is to construct non-linear equations describing the transonic flows
of a radiating gas at finite optical distances from the radiating surfaces, where the radiative transfer
is essentially not one~mension~. In section I we consider some questions of the non-univ~ate
*Zh. vychisl. Mat. mat. Fiz., 15,3,682-694,1975.
Transonic flows of a radiating gas in channels 131
transfer of selective radiation in media with weakly varying properties. Section 2 is devoted to the
derivation of asymptotic equations valid for the transonic flows of a gas in channels of finite optical
width. In section 3 we study the converse problem of finding the channel wall temperature distribution for a given gas flow. In section 4 we consider separately the case where the characteristic length of the wall temperature variation is much greater than the mean free path of the radiation.
1. Consider a region G occupied by a radiating gas. Suppose there is no scattering of the radiation and the gas is in a state of local thermodynamic equilibrium. The surface S bounding the
region G radiates like an absolutely black body. We neglect the contribution of the radiation to the internal energy and pressure of the medium. The radiative intensity I, satisfies the transfer equation
[61
Q gradI,@, 52)=x,(r) {B,[T(r) I-Zv(r, Q)}, Iv(rs, Q>=B,[T,(r.J I, rsS, &z-CO, B, (T) =2hv3c-’ ( ehvikT- 1) --I.
EG,
(1.1)
Here v is the frequency of the radiation, r and rs are the radius-vectors of points in the region G and on the surface S, R is the unit vector in the direction of the light ray, n is the outward normal
to the surface S, xv is the volume absorption coefficient, T is the temperature of the gas in the region G, T, is the’temperature of the surface, S, B, is Planck’s function, h is Planck’s constant, k
is Boltzmann’s constant, and c is the velocity of light.
The spectral density of the radiative energy wy, of the spectral heat flow qr, and of the total heat flow 4 of the radiation are given by the formulas
wv=c-’ I, dQ, J qv= JZVR cm, q= J qv dv. (1.2)
The first two integrals in (1.2) are evaluated over the unit sphere 1 Q 1 =i_
We will suppose that the gas temperature in the region G and the temperature of the surface
S differ little from the constant temperature To :
T=T,(I+ETTI), T,=To (1+aJ,,), ET=O(l). (1.3)
We denote the unperturbed state of the gas by the subscript 0 and use the expansions
z,=B,,+ETT,H,i,+ . . . , Wv=43V+ (&,+E,T,H,W,,) + , . . ,
q,=4neTT,H,q,,+ . . . , q = 16wT,4q,+ . . . ,
H,= (X?, / aT,>, o=2n5k4 / ( 15h3C2).
From Eqs. (1 .l)-( 1.4) to a first approximation we obtain
(1.4)
8 grad iv@, Q) =x,,[T,(r) -L(r, 8) I, r=G,
iv (rs, Q) =T,, (rJ , r,d, Qn-KO,
1 w,,= -
43-c J i, c&J,
1 qv,= 4; J &s-2 as&
(1.5)
132 V. N. Koterov
Integrating the equation in (1.5) over the unit sphere, we obtain
div ql= j x,H,[T, -uL,]dv (j,,) -: 0 0
(1.6)
Introducing the averaged absorption coefficient x0 and the perturbation of the mean radiative energy density w1 by the formulas
we write relation (1.6) in the form
div Q~=~~(T,--wJ. (1.8)
We can integrate equation (1.5) and find the relation between the perturbation of the spectral
radiative energy density wV, and the temperature perturbation 2’1 (see, for example, [7]). If Eqs.
(1.7) are then used, we have
1 {JJ cD(xolr-r81) WI= -
4n Tw, (r.,)
a(r) Ir-rs13 I (r-r,) n I dr,
--x0 JJJ Ti b-i) ~‘(3101r-ril) dr
lr-ri12 1 3
G(r) 1 (1.9)
@t(R)= jx,H~~~p(-x~~,x,)dv( x.jH,dv) -1
W(R) =od@ (R) / dR, S(r) cs, ‘G(r)cG.
One of the simplified transfer models is the grey radiation model in which it is assumed that
the absorption coefficient of the gas is independent of the frequency Y. If this coefficient is denoted
by xl, then in the case considered
div qi=~.(Ti--wi),
1
4n (SJ Tw, bs) exp(--)G.lr-rsl) wi= -
I r-r8 I 3 I (r-r,)nldr,
S(r)
SX. JJJ Ti b-d exp(--x.lr-ril) dr i .
G(T) Jr-ri12 }
(1.10)
Comparison of Eqs. (1.8), (1.9) with Eq. (1.10) shows that in media with weakly varying properties the transfer of selective radiation is to a first approximation equivalent to the transfer of grey radiation with e-R replaced by CD (R) and averaging of the absorption coefficient by formula (1.7). A similar fact is known for plane-parallel transfer problems (see, for example, [5 1). Apparently, its validity for multidimensional problems has not been mentioned previously.
It is obvious from the definition of the function 0 (R) in Eqs. (1.9) that it can be approximated to any degree of accuracy by the sum of exponents:
(D(R)= &jexp(-aJ1). j=i
(1.11)
Transonic jlows of a radkting gas in channels 133
Because of the linearity of the transfer problem, the appro~mation (1.11) transforms Eq.
(1.5) into the system Qgrad ij(r, Q)=~*j[Ti(r)--i,(r, Q)], r=G,
ij(r,, 9) =T,, G-J, r&C 52n-=co, (1.12) N
1 X*j=X(JEjy j=l,Z,...,N, wi=
IS PjW*j, W$j= z J i j dSZ+
1-i Ix For every j Eq. (1.12) is the same as the equation of the grey radiation model in which the
absorption coefficient equals X*j. Therefore, the approximation (1 .l 1) of the kernel iI, (I?) in the
expression (1.9) is in some sense equivalent to the multi~oup approach (see, for example, [7])_ We emphasize, however, that the assumption of the smallness of the perturbations enables us to formulate the criteria of (1 .l 1) for the best determination of both the frequency groups, and also of the effective values of the absorption coefficient..
The problem of the best choice of the constants &i and @, connected with the actual relation between the absorption coefficient x y and the frequency, is not investigated here. For plane- parallel transfer problems a similar question was considered, for example, in [8] . We merely mention that for the best approximation of the functios@ (R) in the neighbourhood of R = 0 it is desirable
V lj=1. to choose the constants /3j in such a way that +I This requirement is necessary for the
correct description of radiative transfer in reg&ns of total thermodynamic equilibrium. It can also
be shown that at the limit of large optical thicknesses of the region G the exponential approximation procedure gives an exact solution of the transfer problem, if the constants CI~ and /3i are so chosen that they satisfy the relation
From the form of the function 0 (R) we may conclude that the apprtixrmation (1 .I 1) should give excellent results even for small values of IV. In particular, satisfactory results should be obtained even for N = 1. Therefore, in what follows we use the simplest one-term approx~ation CD (R) “eeaR, leading to the equation
Q grad i(r, Q)=x*[T,(r) -i(r, Q) I, r=G,
i(r,, Sz) =T,, (rS>, rsd, QKO, (1.13)
1
wi= 4n; 17 J i dS-2, X.=X&L
An efficient method of solving Eqs. (1 .13) is the method of spherical harmonics, in which the radiative intensity i is sought as a series of spherical harmonics (see, for example, [7] )_ The lowest is known as the PI -approximation
i=zl:,--)c,-‘Q grad r~,, Aw,=3x.‘(w,-T,). (1.14)
The heat flow of the radiation is found by Eq. (1 .S>.
The boundary condition for the perturbation of the radiation energy density can be obtained from the balance of the heat flows on the surface S [9, lo] :
134 V. N. Koterov
Because of Eqs. (1.14) it has the form
w + 2 dWi --- ’ 3x. dn
-Tto,. (1.15)
2. We consider stationary flows of an inviscid non-heat-conducting radiating gas in channels with plane or axial symmetry. Let x be the distance along the channel, y the distance from the plane or axis of symmetry, u and v the longitudinal and transverse velocity components, p the density, p the pressure, s the specific entropy, T the gas temperature, and q the radiative heat flow
vector. We will assume that the flow possesses plane (m = 0) or axial (m = 1) symmetry. Then the equations of motion of the gas take the form [6]
aPu as+ 1 aY"Pv_O au 1 QJ y”‘ay ’
ua$+v-+--0, aY P ax
(2.1)
The radiative heat flow q is determined from Eqs. (1.1) and (1.2). The system (2.1) must be
supplemented by two relations connecting the thermodynamic functions p, p, s, T, and the
absorption coefficient xv must be defined as a function of two of them, for example
P’P(PYS), T=T(p, s>, xv=% (p, s> . (2.2)
Let the generatrix of the channel wall and its temperature be defined by the relations
y =L,+GLf WL) , T,(x) =T,[li-&TT,,(~~L) I. (2.3)
Here Ly is the characteristic half-width of the channel, L, is the characteristic dimension along the x-axis, 6 and eT are small parameters, To is the characteristic wall temperature, andfand T,, are dimensionless functions of the order of unity.
On the wall the non-penetration condition
vlu=6f’(x/L,), f’(E) =df (8 /a. (2.4)
is satisfied.
On the plane or axis of symmetry the transverse velocity component and the derivatives with respect to the coordinate y of the remaining functions vanish.
In what follows it is convenient to use a corollary of Eqs. (2.1) [S] :
(2.5) 1 aP
pZT as pdivq, ( ) ap s-m - aa’= F *. ( 1
Transonic flows of a radiating gas in channels 135
We assume that the gas flow differs little from a uniform transonic flow with temperature To.
We denote the unperturbed state of the gas by the subscript 0. We define the small parameter E as
the deviation of the Mach number M of the unperturbed flow from unity:
M=uo / a,,=l+Ke. (2.6)
For K<O the unperturbed flow is subsonic, for K>O it is supersonic.
We introduce the dimensionless variables
x1=x IL,, y,=3’“x.y,
where 3~~ is the spectral mean of the absorption coefficient defined in section 1. We note that the
dimensionless coordinate y1 has the meaning of the effective optical distance measured from the
plane or axis of symmetry.
It is convenient to defme the characteristic optical thicknesses along the x and y axes by the formulas
Tx=3”%Lxr r,=3%.L,.
We introduce the small parameter eS, characterising the order of the perturbation of the entropy in the flow, and expand all the gasdynamical functions in the following series:
u=u,(l+EUl)+. . . ,
p=po(I+epi) + *. . , S’S0 (IS&&) + . . . (
u=u&,f . . . , q=16&TTo4q,+. . . ,
p=po (If&P,) f. - * 1 T=To(l+eTT,)+. . . .
(2.7)
Using the usual procedures of the method of asymptotic expansions (see, for example, [ 111) we obtain from Eqs. (2.1), (2.6) and (2.7) equations of the first approximation (the subscripts of the perturbations and the dimensionless variables are omitted here and below)
PO dP --so, poaso2 ax
and determine the ratios of the order of the small parameters
dP - =o, ds
8Y -=O dX
w3)
6=O(&Tt-‘), &~=O(&,BOT,-l)) Bo=pOaao3 / (oT,~). (2.9)
The parameter Bo is the ratio of the characteristic convective energy flow to the characteristic
radiative heat flow.
It follows from Eqs. (2.8) that s=s (y) , and to the first approximation the entropy along the streamline is preserved. In [5, 121 flows of this type are called quasi-isentropic.
We put
E~=E (a In T/d In po) S, a,=& (a In s/a In po) T.
Then to a first approximation Eqs. (2.2) and (2.7) imply
p0aso2
p=- PO
T=p-s.
(2.10)
(2.11)
136 V. N. Koterov
Here To =cPO / cvo is the ratio of the specific heats of the gas.
iteration of Eqs. (2.8) taking into account Eq. (2.11) leads to the final relations
u(x, Y)=u(~)+~(YJ. P(X) =-pa&Wr) /PO,
P(X, Y> =-LW+(“(o--1MY) /To, T(s, Y>=-W+-s(Y) /%I. (2.12)
The functions V(Y) and S(P) in Eqs. (2.12) are determined by the monitions in the flow entering the stream domain considered. If the oncoming flow is uniform and homogeneous, then V (y) =0, s (y) =0 and the density, pressure, temperature and longitudinal velocity component to a first approximation depend only on the coordinate x.
To a first approbation
a,=a,,{1+EI (m 80- 1) p+s,2 (dp / as0> T (aa, / dsCJ,s 1 (PL%> I>,
ms,= (d2p / wo2> s ! (2p03G9, V=l/p. (2.13)
Using Eqs. (IS), (2.6), (2.7), (2.12) and (2.13), we obtain from Eq. (2.5) the equation
(2.14)
C(y>= X+-JJ(Y)
mso
and determine the connections between the small parameters:
(2.15)
We note that Eqs. (2.10) and (2.15) do not contradict Eq. (2.9).
We will describe the radiative field by the PI -approximation of the spherical harmonics method. Then the radiative energy density w satisfies Eq. (1.14), which in the case considered has the form
1 PW(X, y)+ 1 d -- 2 ax2 Y” @Y [
Y" f%fJ (2, y) I =4w)--%Y).
TX 8Y
It follows from Eqs. (2.3) and (2.15) that to a first approbation the channel wall
corresponds to the coordinate y=rtv. The non-penetration condition (2.4) and the boundary
condition (1.15) have the form
y=z,: v=f’ (x), w-l- 2 aw ~-=Tdx)* Y
(2.16)
(2.17)
On the plane or axis of symmetry
y=o: v=o, dW
JG=*- (2.18)
Transonic flows of a radiating gas in channels 137
In the approbation of the radiative energy transfer by the ~~-appro~mation, Eqs. (2.14) (2.16), relations (2.12), and the boundary conditions (2.17) and (2.18) completely describe the flow, if appropriate conditions are specified at the channel input and output. The functions V@) and $0) in (2.12) and the function CYy) in (2.14) describe the non-uniformity and inhomogeneity of the oncoming flow. The parameter b in (2.14) describes the radiative energy transfer which is negiigibly small for b=w. In this case from Eqs. (2.14), (2.17) and (2.18) we obtain the formulas
The first of these equations can be obtained if to the flow considered we apply the hydraulic
approximation of [ 131, regarding the flow as transonic. The gas stream at b=m is independent of
its opticaf properties, and the quantity ru appears in these equations due to non~ension~i~ng the transverse coordinate by the length of the mean-free path of the radiation.
Relations (2.10) and (2.15) permit us to establish the region of app~cab~ty of the equations obtained. In particular, we note the relation Bo-* ==O (ET=-'1 . This implies that the equations hold
for z,=O (1) if the characteristic heat flow of the radiation is much less than the convective
energy flow. This requirement is not obligatory for z;=o (I)
3. Let the generatrix of the channel wall be such that f’( 5) -0 as z+ * 00. With respect
to perturbations of the wall temperature we suppose that T, (--m) =O, T, (+m) =0=const, For simplicity we will consider that for x=--00 the gas flow is uniform and homogeneo~: V(y) =O, s(y) =o.
Because of the assumptions made the gas is in a state of complete ~ermodyn~ic equilibrium for s=fm Therefore
T(--oo)=w(-m, y)=O, T (+-) =w (+m, y) =8. (3.1)
From Eqs. (2.12) and (2.14) we obtain the equation
1 dy”u ++T--m)= ---,
Y” %J k=L.
me (3.2)
From Eqs. (2.6), (2.7), (2.12) and (2.13) it may be concluded that the stream is subsonic in the range T>k and supersonic in the range T<k
Equations (2.16) and (3.2) with the boundary conditions (2.17), (2.18) and (3.1) completely describe the problem. They may be further simplified.
We represent w as a sum
wb, Y>=W(s)fZ(s* 3J>, W(--m) =o, W(f=) =e. (3.3)
Then Eqs. (2.26)-(2.18) imp!y: the relation
(3 -4)
the equation
138 V. N. Koterov
(3.5)
the boundary condition on the axis of symmetry
y=o: a2 / dy=O, (3 -6)
and the boundary condition on the channel wall
2 dZ y==zu: z+ -- +w=r,.
13 dy (3.7)
If we integrate Eq. (3.2) with respect toy, then taking into account Eqs. (2.18) and (3.3) we obtain
““-&.L[ fk-T)$f+v) J+$j q”.whrl)drl. (3.8) ”
The non-penetration condition in Eqs. (2.17) and (3.8) give the relation
Integrating Eq. (3.5) with respect toy between the limits 0 and ru and using Eqs. (3.6) and (3.9), we arrive at the formula
(3.10)
With the aid of these relations it is comparatively simple to sdve the converse problem: for a
given flow field in a channel with a specified shape, determine the wall temperature. Indeed, if the temperature Tis specified, the problem reduces to the integration of Eq. (3.5) in the strip O<‘y<r, with Neumann’s conditions (3.6) and (3.10) at the upper and lower limits. Then the wall temperature is found from Eq. (3.4) and the boundary conditions (3.7).
Examples of c~c~a~ons of the converse problem are shown in Figs, 1 and 2. The nurne~c~ integration of Eq. (3.5) was performed by the method of alternating directions 1141.
Figure 1 shows the wall temperature T,,, for which monotonic acceleration of the gas from a subsonic to a supersonic velocity is possible in a plane channel,m=O, f (2) =O. The curves correspond to different values of the parameter b. The gas temperature whose graph is denoted by the dashed curve in Fig. 1 was specified by the formula T(s) =-0.5 (1 +th r) . The parameters were assumed to be given by k=-0.5, z,=r,=l.
For the same values of the parameters and the same gas temperature Fig. 2 shows graphs of the wall temperature of a channel with the generatrix f(x) = --eexz, m-0.
Transonic flows of a radiating gas in channels
+ku@
7” ---ii-
b=l
0.5
~
.I 1 2 3x --_
n
139
FIG. 1 FIG. 2
4. In the above analysis no assumptions have been made about the order of the quantity rx,
We consider separately the case of optically thick perturbations of the wall temperature.
Let r=+-* ==o (1). We first consider the plane channel case, m=O, f(z) =O.
To a first approximation Eq. (2.16) assumes the form
Pw / dy”=w-T.
Integrating this equation taking into account conditions (2.17) and (2.18) we obtain
w=T(s)+c,lT,(s)--T(z)]chy, ~~==3”~ (2sh r,+3’” ch ‘c,) -*.
From Eqs. (3.2), (4.2) and (2.18) it follows that
Using the non-~netration condition in (2.17), we obtain from (4.3)
dl’ 1 I’,-1
-z==bck--T7 bo=z,b (2+3”’ cth .tU) 3-l’?.
(4.1)
14.2)
(4.3)
(4.4)
Equation (4.4) connects the gas temperature Tin the flow with the wall temperature T,. It follows from Eqs. (4.3) and (4.4) that the transverse velocity component v depends to a first appro~mation only on the temperature slip at the wall:
140 V. N. Koterov
We recall that the temperature and velocity of the gas in the flow are connected by the
relation (2.12): u=-T, and for x=f 00 the gas is in a state of complete thermodynamic equilib~um T(&oo) =T,(=tm).
If the wall temperature nowhere attains the value k, then the flow is everywhere isentropic subsonic (KO) or isentropic supersonic(DO behaviour of the integral curves of Eq. (4.4) in this case is shown in Fig. 3. ‘The dashed curve shows the variation of T, (2).
FIG. 3
FIG. 4
Tranmnic flows of a radbting gas in channels 141
If at some point r=z, the wail tem~rature T, (5,) =k, then Eq. (4.4) Implies that
T(x, ) =k. At the point xS the velocity of the gas equals the local isentropic speed of sound.
The point X=X,, T=k for Eq. (4.4) is a saddle point if (dT, f dx) .=,,<O, a node if
o< (dT, / d5) XC%< Q&f -I, and a focus if (dT, / dx) x=xs> (&I,,) -‘.
Figure 4 shows the fields of the integral curves of Eq. (4.4) on the passage of the flow through the isentropic speed of sound. A discontinuity may appear in the flow for (dT, / ds),=,,>O. A
sufficient condition for its existence is given by the inequality (dT,ldx)._> (46,) -i.
Let the ~scont~~ty occur at the point ;c=zo.We put T(z&O) =T,. Then
w(s&O, y) =w+ (y) =T,+co[Tw(~) -T,lch Y.
We notice that Hugoniot’s conditions, expressing the requirements of continuity of mass, momentum and energy at a ~scontin~~, are satisfied to within 0 (E’) for any values of T+ and T -.
The relation connecting the temperatures Tk follows from Eq. (3.2), if we note that the quantities du / dy, w and T can only have d&continuities of the first kind:
T,i-T_=2k.
This relation and Eq. (4.4) enable us to find the coordinate x0 of the discontinuity and determine the values of the temperature T+ at the discontinuity.
The radiative energy density at the ~scont~uity must be cont~uous. The solution (4.2) of Eq. (4.1) always has a ~scontinuity if the temperature T is discontinuous. Therefore close to the discontinuity Eq. (4.1) is invalid.
In the neighbourhood of the discontinuity we introduce the new variablez=t,(s-z,) ,r,-l = o (1). Then Eqs. (2.16) and (3.2) are written in the form
r,(k-T)$++(T-m)= -3, azw d”w
a9 - = w--T. dz”+ dy” (4.5)
The equations T=T, for ~=*a are conditions for matching the internal solution, valid
in the nei~bourhood of the ~scont~ui~, with the external solution, satisfying Eq. (4.4). From the first equation in (4.5) we obtain to a first approximation
(4.6)
Therefore, the temperature discontinuity is retained in the internal solution also.
In the neighbourhood of the discontinuity the radiative energy density w satisfies to a first approximation the second equation in (4.9, in which the temperature T is defined by Eq. (4.6). The bounda~ con~tions for this equation at the upper and lower limits have the form
y=(): d”=o, dY
y=T,: w+ ~~=Tw(z,).
142 PC iv. Korerov
The conditions for z=fw are determined by matching the internal sofution with the external solution (4.2): w (km, y) =w* (9).
We consider an axisymmetric channel with a curvilinear genera&ix. We present the fmal formulas, valid for G-~==o (1) :
dT 1 k--T
dx-l;;k-T’ w=T-i-c,(T,-T)Z,(y),
t, = %(T,-T) [Z*(y)-$z,(r,)]++f’, c,=3”~[2Z,(T,)+3~l’Zc(z,)]-‘, (4*7)
O,=T,-Zb,f’/ q,, b,=q,b[2+3%(z,) /Z,(z,) ] (2.3”‘)-‘.
Here I,(g), p=O, 1, are Bessel functions of imaginary argument.
The equa~on in (4.7) is obtained by the same method as Eq. (4.4). These equations are identical in form. Therefore the whole of the previous analysis is also valid for gas flows in curved
axisymmetric channels, if in place of the wall temperature T, we introduce the effective temperature 0,.
For example we consider a gas stream in a channel of variable cross-section, whose wall
temperature has the ~st~bution T,(z) =2b,,f’(s) / z#, so that 8,--O. If f/(&w) =O, then for z=fa, the gas is in a state of complete thermodynamic equilibrium: T (ztm) =O. With these
conditions Eq. (4.7) implies that T (ST) =0, and to a first approximation the longitudinal velocity component remains constant. For the transverse velocity component and the radiative energy density the following expressions are obtained:
u=!*(y)/‘(z) /11(%), w=I&)f’(x) ‘/r,(‘G,).
5. The equations obtained in this paper are valid at finite optical distances from radiating surfaces of small curvature. In the streams considered the perturbation of the transverse velocity component is much less than the perturbation of the ~on~tu~n~ com~nent. Therefore the flows
are close to one-dimensional.
The equations obtained are not sufficient for the c~culation of the transonic streams in
channels of great optical width, where ‘r, -‘=o (1)) These equations are valid in the boundary layer, but in the neighbourhood of the centre the flow must be described by the equations obtained in [S ] . The need to use a matching procedure makes the problem more complex.
The author is indebted to V. V. Aleksandrov for his continued interest and for valuable
comments.
Danslated by J. Berry
REFERENCES
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Isentropie gas jlow in a constant gravitational field 143
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SOLUTION OF THE CAUCHY PROBLEM FOR A PLANE ONE-DIMENSIONAL ISENTROPIC GAS FLOW IN A CONSTANT GRAVITATIONAL FIELD*
V.LGOLIN'KO
Cor’kii
(Received 3 September 1973)
A SOLUTION of the Cauchy problem is obtained for a quasilinear system of differential equations
of hyperbolic type, describing the one-dimensional isentropic motion of a gas in a constant gravitational field for any value of the polytropy factor.
The one-dimensional isentropic motions of a gas are one of the most studied sections of gas
dynamics. Special and general solutions of the differential equations describing these motions have been obtained [l] . General solutions have been obtained for a discrete series of values of the polytropy factor n. However, the use of these general solutions, dependent on two arbitrary
functions, gives rise to well-known difficulties. Therefore, in the solution of various gasdynamical
problems approximate methods, the chief of which is the numerical method of characteristics, are often used.
In [2] the Cauchy problem for the gasdynamical system of equations is formulated. In the case of an isentropic flow the exact solution of the Cauchy problem is obtained for n = 3. However,
in the simplest case of the plane isentropic motion of a gas the Cauchy problem can be easily solved in the range OGt<tO, where to is the instant of formation of the shock wave, for any value of the polytropy factor n.
*Zh. vFchis1. Mat. mat. Fiz., 15, 3,695-701, 1975.