ACTA
Acts Mechanics 37, 159--164 (1980)
MECHANICA
9 by Springer-Verlag 1980
P r o p a g a t i o n of a P r e s s u r e S h o c k in an Optically T h i c k M e d i u m By R. S. Singh and R i s h i Ram, Varanasi, India WiSh 4 Figures
(Received August 23, 1978; revised February 27, 1979) 1. I n t r o d u c t i o n
In recent years several authors have taken keen interest in studying the growth and decay problem of non-linear waves. Ram and Srinivasan [5] studied the growth and decay of sonic waves in an optically thick medium at very high temperature. Ram, Gaur and Tiwari [6] also investigated the non-equilibrium effects on the propagation of weak discontinuities. Thomas [8] defined the pressure shock model in ideal gases and studied its propagation without accounting for the high temperature effects of the concerned phenomenon. McCarthy [2], [3] studied the problem of growth of thermal waves and Balaban [ 1] studied the growth and decay problem of acceleration waves in elastic-plastic materials. But in dealing with certain problems like that of blast waves or pressure shocks the sudden and abrupt jumps in pressure and temperature are of primary concern. Since the gas is viscous and thermally conducting, it is found mathematically consistent and physically realistic to assume a discrete discontinuity in pressure alone, while the discontinuities in the gas velocity and density are negligible. I t is also a well known fact that when the temperature of the medium shoots up to the order of 104 ~ the he~t flux of radiation is of the same order as that of the heat transfer by thermal conduction or convection. Near the temperature range from 104 ~ to l0 s ~ the radiation stresses and the radiation energy are no longer negligible. In violent explosions the temperature of m a n y thousand degrees of Kelvin can easily be attained, as the kinetic energy will be dissipated through viscous heating and shock compression. The object of the present paper is to study the thermal radiation effects on the propagation of a pressure shoek in a radiative medium of a viscous and thermally conducting gas. 2. V e l o c i t y of P r o p a g a t i o n
The shock relations under optically thick gray gas approximation are [4]
[e(u, - G)i = o,
(2.1)
[%] nj + el(G - u , , ) [u~] = o,
(2.2)
9 '
f
1 ~
'1
(2.3)
0001-5970/80/0037/0159/$01.20
160 where
1~. S. Singh and gishi Ram: 2
a~j = - - ( p 4- PR) (~i § #(u~,j 4- uj, i) -- --~ ~Uk,kd~i 1 aRT ~ PR = -~
1 ER, -~
K ~ f = lc § 4DRaRT 3
and [/] ----] - - / 1 denotes the jump in the quantity enclosed across a shock wave, where [ and/1 are the values of ] evaluated just on the downstream and upstream side of the shock respectively. G is the speed of propagation and n~ are the components of the unit normal to the shock surface, u~, p, ~, T, ~, k, aa and DR represent the gas velocity components, the gas pressure, the density, the ~emperature, the heat exponent, the coefficient of thermal conduction, the StefanBoltznlann constant and the gosseland diffusion coefficient respectively. A comma followed by an index denotes partial differentiation. The boundary conditions for a pressure shock model of Thomas and Edstrom [8] are a) [ ~ ] = 0 ,
b) [ u , ] = g ~ = 0 ,
c) ~ = 0 ,
d) ~ = 0 .
" (2.4)
The bar appearing in (2.4) denotes evaluation at the rear or flow side of the shock surface. The velocity condition represents that the gas on the upstream side of the shock is at rest. The pressure condition states that the normal pressure gradient vanishes on the flow side. The temperature condition implies that the temperature of a material particle has a stationary value at the instant of its contact with th~ rear of the wave surface. The g~ometrical and kinematical compatibility conditions for a singular surface in continuum mechanics are [7]
[/,~] = [I.~1 ~jn~ + g~[l].~xi~
(2.5a)
--~ = --G[J,j] nj § --~ [/]
(2.5b)
where ] stands for a n y of the flow parameters, g:~ and x~ are the components of the metric tensor and projective tensor respectively of the wave surface. 5 - 7 represents the time derivative as apparent to an observer moving with the shock front. Using the boundary conditions (2.4) and the compatibility conditions (2.5) we h a v e 2
[P § PR] n~ § -~ #).jn~n~ -- #2~ -- #~n~n i : O,
(2.6)
p~G [ K e m T ~] n~ § ~ {~ § 3(~ -- 1)/~1~]/(~?)} ~ 0,
(2.7)
where
The dimensionless parameter ~] is the measure of the strength of the pressure shock. In consequence of (2.6) we can assmne ,~ = ~n~. Thus we obtain
[p § pR] = 4#/~ -~,
~. :
3pl~J 4~ {1 §
Rlp/(~)},
where Rip = PR___~is the radiation pressure number of state 1. Pl
(2.S)
Propagation of a Pressure Shock in an Optically Thick Medium
161
The Eq. (2.7) provides us a relation
p~Gv where R 1 / - -
[1 +
3(~, - - 1)
Ri~/(~])}/{1 + R~/(1 -[- ~7)3)
(2.9)
4DftaRTI~ is the radiation heat flux nmnber of state 1. Differentiating
k the equation of state p = ~ R T and using the compatibility conditions and the boundary conditions (2.4) we obtain [m,~] n, --
3PlT1V(1 + ~]) {1 @ Rlp/(~]) }. 4~G
(2.10)
Equating (2.9) and (2.10) we get a relation ~2~
((1 -~ ~ ) { 1 ,-~ RI/(1 ~ ~])a} {1 -~- Rlp/(r])[) 1 ~- 3(~ - - ]) Rlp/(~])
(2.11)
where P , = ~Cv is the Prandtl number and d = G | / 4 P r is the dimensionless ]c
c1
{
3
p a r a m e t e r for the measure of the velocity of propagation of the wave. Here c I = (rp/~) m represents the local speed of sound. The variation of d verses 7 has been shown in Fig. 1, 2, and 3 for the case ~ = - - . The numerical results 5 show that the velocity of propagation of the pressure shock increases or decreases according as the increase or decrease of its strength during propagation. The radiation pressure will cause to decrease the velocity of propagation, while the radiative heat flux will cause to increase it. Thus the radiative heat flux effects are to accelerate the rate of decrease of the velocity of propagation during decaying process of the wave. Next we shall study the variation of the strength during the course of propagation.
40O[ I
0
/
2
3
$
5
8
Y
8
8
/0
ATi~ez$ioni~d Yo~iG~l~ ~
Fig. 1. Variation of the velocity of propagation of the wave during propagation for the case R p = 0 11 Acta Mech. 37/1--2
102
~, S. Singh and ~ishi Ram:
"I
/
l,=t
i, o!
,//
oO~ ~I - - -f - - -3 ~ , ,'7
5
~ ( ~.q
Y
#
.
,3
,
I0
I]ll/i~ilJlOnl~3~Yerljble 7 2 - - ~
Fig. 2. Variation of the velocity of propagation of the wave during propagation for the case Rp := 10
~00
350
/ /
.
I
Z
3
O
~)nension/e55
.
.
5
.
.
~"
II~rlable zZ
.
.
2"
.
6
.
#
/
I0
Fig. 3. Variation of the velocity of propagation of the wave during propagation for the case R / = 10
3. Growth and Decay of the Pressure Shock Using t h e first o r d e r k i n e m a t i c c o m p a t i b i l i t y conditions (2.5) and t h e Eq. (2.10) a n d (2.4) we get ~_Z7: 6t
_ .3P1~ (1 -~ 77) (1 Jr Rlp](~7) }. 4re
(3.1)
L e t X(0) r e p r e s e n t t h e p o s i t i o n of t h e w a v e surface Z(t) a t t h e initial t i m e t ~ 0. If a r e p r e s e n t s t h e d i s t a n c e t r a v e r s e d b y t h e w a v e in t i m e t as m e a s u r e d
Propagation of a Pressure Shock in an Optically Thick Medium
163
from Z(0) along the direction of propagation, then d_~ =_ G. Using this transdt formation and the relation (2.11) in (3.1) we obtain
where F = ]3~_~ p~a is the dimensionless p a r a m e t e r for the measure of distance cltt
during propagation. The differential Eq. (3.2) governs the growth and decay of a pressure shock in an optically thick medium of viscous and heat conducting gases. The variation of the pressure shock strength ~ during propagation has been shown in Fig. 4. The solution curves of (3.2) show that the pressure shock strength ~ decreases continuously with time and tends to zero ultimately. The radiative heat flux has a stabilizing effect in the sense that it slows down the decaying process. The effects of thermal radiation on the velocity of propagation have been shown in Fig. 1, 2 and 3. /U 0
i f
8
%,
8
3
/
I
Z
3
~,
5
6~
O/mens/~n/essO/:s/~zceJt/ / ~
7
~
,9
I0
4~)
>,'-O//A/
Fig. 4. Variation of the pressure shock strength during propagation for the case RT
=
0
4. Case of Vanishing Radiation Effects Putting Rip = 0 and/~11 = 0 in (3.2) and (2.11) we get d~] :
- - • ]/1 @ ~],
dY'
G 2 = 361--'-~( l ~- ~]),
4Pr
Cl 2 ~-- Y~D'-"'~9I
Q
(4.1)
Eliminating ~] we get d_GG - - -
dF 11'
3~1 ~ - -
4c,1
4PrG 2
(4.2)
164
P r o p a g a t i o n of a Pressure Shock in a n Optically Thick Medium
Solving (4.1) and (4.2) we obtain ,~ + 1 =
O
:~ -~- Go
(~.:~)
- - (o, - - Go) e-l"
-c~ - = c~ + Go d (c~ - - Go) e r '
~/ 3
(4.4)
c* = e 1 ~/ 4 P r
where % and Go are the values of ~q and G respectively at the initial time t = 0. The Eq. (4.4) is in full agreement with the result obtained by Thomas and Edstrom [8]. The Eqs. (4.3) and (4.4) determine r) and G at any time t during propagation. As F - + ~ ,
3 ~-->0 and G-+c~ ~ / 4P~
m
This shows that the pressure shock will
ultimately degenerate and the resulting disturbance will propagate with a constant speed q ]/3~-P~. References [1] B a l a b a n , M. M.: Acceleration waves in elastic-plastic materials. I n t . J. Engg. Sci. 9 (1970). [2] McCarthy, H. F.- The growth of t h e r m a l waves. I n t . J. Engg. Sci. 9 (1971). [3] McCarthy, M . F . : T h e r m o d y n a m i c influences on the propagation of electroelastic materials. I n t . J. Engg. Sci. l l (i973). [4] Pal, S. I." Inviseid flow of r a d i a t i o n gasdynamics. J. Math. Phys. Sei. 8, 370 (1969). [5] R a m , tgishi, Srinivasan, S." The growth a n d decay of sonic waves in a radiating gas a t high temperature. ZAMP 26, 307 (1975). [6] R a m , Rishi, Gaur, M., Tiwari, T. N." Anisotropic propagation of weak discontinuities in flows of t h e r m a l l y conducting a n d dissociating gases. Proc. Ind. Acad. Sci. B 3, 65 (1976). [7] Thomas, T. u E x t e n d e d compatibility conditions for the s t u d y of surfaces of discontinuity in c o n t i n u u m mechanics. J. Math. Mech. 6, 311--322 (1957). [8] Thomas, T. Y., E d s t r o m , C. I~.: Pressure shocks in viscous a n d h e a t conducting gases. Proc. Nat. Acad. Sci. U.S.A. 47, 3 i 9 - - 3 2 5 (1961). R. S. Singh Department o/ Mech. Engg. Institute o/ Technology B.H.U., Varanasi, I n d i a
Rishi R a m Applied Math. Section Institute of Technology B.H.U., Varanasi, I n d i a
Eigentiimer, Herausgeber uncl Verleger: Springer-u M61kerbastei 5, A-101i Wien. -- Ftir den Textteil verantwortlich: Prof. Dr. H. Parkus, Technische Uni~ersits Karlsplatz 13, A-1040Wien. -- Ffir den Anzeigenteil Verantwortlich: ]~ag. Bruno Schweder, 5[61kerbastei 5, A-1011 Wien. nruck: u Druckhaus ,,~axim Gorki", DDP~-7400 Altenburg. 5000/20[80. - Printed in the German Democratic Republic.