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Evolution of weak discontinuities in a non-ideal radiating gas
Commun Nonlinear Sci Numer Simulat 16 (2011) 690–697
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Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
Evolution of weak discontinuities in a non-ideal radiating gas L.P. Singh, Akmal Husain *, M. Singh Department of Applied Mathematics, Institute of Technology & DST-CIMS, Banaras Hindu University, Varanasi 221 005, Uttar Pradesh, India
a r t i c l e
i n f o
Article history: Received 21 February 2010 Received in revised form 23 April 2010 Accepted 24 April 2010 Available online 29 April 2010 Keywords: Weak discontinuity Wavefront analysis Van der Waals gas Thermal radiation
a b s t r a c t A method of wavefront analysis is used to study the formation of shock waves in a twodimensional steady supersonic flow of a non-ideal radiating gas past plane and axisymmetric bodies. The gas is taken to be sufficiently hot for the effect of thermal radiation to be significant, which is, of course, treated by the optically thin approximation to the radiative transfer equation. Transport equations, which lead to the determination of the shock formation distance and also to conditions which insure that no shock will ever evolve on the wavefront, is derived. The influence of the parameter of the non-idealness, upstream flow Mach number in the presence of thermal radiation on the behavior of the wavefront are examined. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction While studying the wave propagation in a medium, one often encounters certain kinds of discontinuities known as shock waves, acceleration waves and weak waves. Therefore, for nonlinear systems, the analysis of these waves has been the subject of the great interest both from mathematical and physical point of view. In recent years, the analysis of possible steepening of compression waves leading to shock waves has received considerable attention in the literature. It is well known that the solutions of nonlinear systems generally develop discontinuities even with the initial smooth data {see, Jeffrey [1]}. However, if the initial data are not regular but C1 piecewise, that is, the dependent variables are continuous across the manifold n(x, t) = 0, while their normal derivatives suffer finite jump discontinuities, then a discontinuity wave will be produced whose wavefront is represented by the manifold itself [1–3]. By following these discontinuities along wavefronts, it is possible to calculate explicitly the position and time of shock formation, which also serve as an important parameter in characterizing the relative importance of convective nonlinear steepening and dissipative flattening and setting a limit for the use of certain approximate theories. A number of problems relating to wave propagation and some considerations of the nonlinear effects have been studied by several investigators [1–4] in various gas dynamic regimes. The problem of interaction of shock waves with weak discontinuities and determination of shock formation distance for quasilinear hyperbolic system has been carried out by Chen [5], Sharma [6], Sharma et al. [7] and Radha et al. [8]. Sharma [6] calculated the shock formation distance and time for weak discontinuities over a concave wall while Radha et al. [8] studied the problem of interaction of shock waves with weak discontinuities and examined the existence and uniqueness of the reflection and transmission coefficients. The propagation of disturbances through a gas may be significantly affected by the presence of thermal radiation, when it is treated as an important mode of energy transfer within the gas. A further contribution towards the study of thermal radiation effects on the propagation of small disturbances in gas flows under steady and unsteady conditions have been discussed previously, in particular, by Pai [9], Penner and Olfe [10], Lick [11], Schmitt [12], Sharma et al. [13] and Singh
* Corresponding author. Tel.: +91 9452060150. E-mail address: [email protected] (A. Husain). 1007-5704/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2010.04.037
L.P. Singh et al. / Commun Nonlinear Sci Numer Simulat 16 (2011) 690–697
691
et al. [14] among others. However, if the temperature of the gas is very high and density is too low the assumption that the gas is ideal is no longer valid. The popular alternative to the ideal gas is a simplified van der Waals model. In recent years, several studies have been performed concerning the problem of strong shock waves in a van der Waals gas (see, Wu and Roberts [15], Pandey and Sharma [16,17], Singh et al. [18]). Wu and Roberts [15] used this model to study the problem of structure and stability of a spherical shock wave in a sonoluminescence bubble. Also, Pandey and Sharma [17] used the method of Lie group transformation analysis to study the problem of interaction of weak discontinuity in a van der Waals gas. In the present study, we employ the method of wavefront analysis to investigate the propagation of weak shock waves in a two-dimensional steady supersonic flow of a non-ideal gas past plane and axisymmetric bodies with thermal radiation. The medium is taken to be sufficiently hot for the effects of thermal radiation to be significant, which are, of course, treated by optically thin approximation to the radiative transfer equation. Transport equations are derived which lead to the determination of the shock formation distance and also to conditions which insure that no shock will ever evolve on the wavefront. The influence of the parameter of the non-idealness and upstream flow Mach number in the presence of thermal radiation on the shock formation distance is assessed. 2. Governing equations The basic equations for a two-dimensional inviscid steady axisymmetric flow of a non-ideal radiating gas with where the effects of thermal radiation are treated by the optically thin limit can be written in the form (see, [9,10])
uqx þ v qy þ qðux þ v y þ mv =yÞ ¼ 0;
ð1Þ
quux þ qv uy þ px ¼ 0; quv x þ qvv y þ py ¼ 0; cp ðuq þ v qy Þ þ ðc 1ÞQ ¼ 0; upx þ v py qð1 bqÞ x
ð3Þ
ð2Þ
ð4Þ
where q is the gas density, p the pressure, u and v the velocity components directed along the x and y axes, respectively; x is the distance along the axis of the symmetry from the tip of the body in the direction of oncoming flow and y is the radial distance from the x-axis; m is a constant which takes values 0 and 1 for plane and axisymmetric flows, respectively; is the rate of energy loss by the gas per unit volume through radiation and is given by
Q ¼ 4kr T 4 T 4b ;
ð5Þ
where k is the Planck mean absorption coefficient depending on the density q and temperature T of the gas, r is the Stefan – Boltzmann constant and Tb is the uniform temperature of the body along which the flow is envisaged. The additional energy gains 4krT 4b have been added to the energy loss 4krT4 due to the radiating gas so that infinite optically thin gas has no radiating boundary. The boundary wall with constant temperature Tb has no loss or gain of the energy from the radiating gas. The system of Eqs. (1)–(4) is supplemented with a van der Waals equation of state of the form (see, [15,18])
pð1 bqÞ ¼ qRT;
ð6Þ
where b is the van der Waals excluded volume which is known in terms of the molecular interaction potential in high temperature gases and R is the gas constant. It may be noted that the case b = 0 corresponds to an ideal gas (ideal in the sense that the particle interactions are absent). cp Here the equilibrium speed of sound for a non-ideal gas is c2 ¼ qð1b qÞ, where c is the specific heat ratio of the gas; therefore the relation between the equilibrium speed of sound of a non-ideal gas and a perfect gas may be defined as, c2 = {a2/ (1 bq)}, where the entity a2 is the equilibrium speed of sound for a perfect gas. 3. Characteristic formulation Using the summation convention on the repeated indices; the governing Eqs. (1)–(4) can be written as
@U i @U i þ Aij þ F i ¼ 0 i; j ¼ 1; 2; 3; 4; @x @y
ð7Þ
where the column vector (Ui)41 and (Fi)41are defined as
2
2 3
q
6u 7 6 7 U ¼ 6 7; 4v 5 p
F¼
6 6 1 6 6 2 2 2 ½ðu a Þ bqu 6 4
mquv ð1bqÞ y
þ ðc1ÞQuð1bqÞ
mvya ðc1ÞQqð1bqÞ 2
0 mquv a2 y
þ uðc 1ÞQ ð1 bqÞ
3 7 7 7 7; 7 5
ð8Þ
692
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and the non-zero components of the matrix (Aij)44, are as follows
A11 ¼ A33 ¼ A14 ¼
u
A12 ¼
;
qv ð1 bqÞ ; fðu2 a2 Þ bqu2 g
v ð1 bqÞ ; ufðu2 a2 Þ bqu2 g a2
A23 ¼ A34 ¼
v
fðu2
1
;
qu
a2 Þ
bqu2 g
A42 ¼
A22 ¼ A44 ¼ A24 ¼
;
A13 ¼
quð1 bqÞ ; fðu2 a2 Þ bqu2 g
uv ð1 bqÞ ; fðu2 a2 Þ bqu2 g
v ð1 bqÞ ; qfðu2 a2 Þ bqu2 g qua2 43
qv a2 ; fðu2 a2 Þ bqu2 g
A
¼
fðu2 a2 Þ bqu2 g
ð9Þ :
The matrix Aij has the eigenvalues
kð1;2Þ ¼
½uv ð1 bqÞ a2 fM 2 ð1 bqÞ 1g1=2 ; ½ðu2 a2 Þ bqu2
kð3;4Þ ¼
v u
;
ð10Þ
with the corresponding left eigenvectors
Lð1Þ ¼ ½0; 1; u=v ; fM 2 ð1 bqÞ 1g1=2 =qv ; Lð2Þ ¼ ½0; 1; u=v ; fM 2 ð1 bqÞ 1g1=2 =qv ; Lð3Þ ¼ ½1; 0; 0; ð1 bqÞ=a2 ;
Lð4Þ ¼ ½0; 1; v =u; ð1=quÞ:
9 > = > ;
ð11Þ
Here, M = q/a, with q = (u2 + v2)1/2, is the upstream flow Mach number. It is obvious from Eq. (11) that the system (7) possesses, except along stream line on which k = v/u, two families of characteristics along which dy/dx = k(1, 2); these characteristics represent the propagation of waves in opposite directions with characteristic speeds k(1, 2). It may be remarked that these characteristic velocities are real if, and only if, the flow is supersonic, i.e., M > 1. 4. Evolution of transport equations for jump discontinuities Let us assume that k(1) describes the initial wavefront trace n(x, y) = 0, which passes through the point (x0, y0). The medium ahead of the initial wavefront is assumed to have a uniform temperature T0 = Tb and a uniform velocity u0 in the x direction with v0 = 0. In the foregoing analysis, we shall use the suffix 0 to denote a quantity in the region ahead of the wavefront n = 0. Now, we derive the transport equations for the jump discontinuities in U as they move along the wavefront n = 0. We introduce new curvilinear co-ordinates n, y0 defined as (see Jeffrey [2])
)
nx þ kð1Þ ny ¼ 0
ð12Þ
nðx; y0 Þ ¼ x x0 ;
and y = y0 . Then n has the required co-ordinate property that n is positive (negative) behind (ahead of) the leading characteristic on which n = 0. In view of these new co-ordinates, Eq. (7) on pre-multiplying by L(i) reduces to the form
LðiÞ U n þ
kð1Þ kðiÞ ð1Þ
k
ðiÞ
k
xn LðiÞ U y0 þ
kð1Þ k
ð1Þ
kðiÞ
xn LðiÞ B ¼ 0;
ð13Þ
where xn = 1/nx is the Jacobian of the transformation. Across the wavefront n = 0, U and U y0 are continuous and have their subscripts-0 values whilst Un and xn are discontinuous. On using Eq. (11) and the flow conditions ahead, the Eq. (13) at the rear side of n = 0 for i = 2, 3 and 4 becomes:
ð1 bq0 Þ pn ; a20 1 p; un ¼ q0 u0 n
qn ¼
vn ¼
½M20 ð1
bq0 Þ 1 q0 u0
ð14Þ ð15Þ 1=2
pn :
ð16Þ
We set i = 1 in Eq. (13), differentiate the resulting equation with respect to n, and then evaluate it at the rear of n = 0, we obtain
h i1=2 mq0 u0 u M 20 ð1 bq0 Þ 1 pny0 þ q0 u0 v ny0 þ v n þ 20 ð1 bq0 Þðc 1ÞQ n ¼ 0: y0 a0
ð17Þ
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693
After substituting T from Eq. (6) and differentiating Eq. (5) with respect to n at the rear of n = 0 yields;
16ðc 1ÞkrT 4b pn : q0 a20
Qn ¼
ð18Þ
Plugging in Eqs. (16) and (18), Eq. (17) can be reduces of the form
pny0 þ
m X p ¼ 0; þ ð1 bÞ n 2y0
ð19Þ
h i1=2 ¼ bq is the parameter of non-idealness of the gas and X ¼ 8ðc1ÞkM0 M 2 1 where b > 0 is a measure of importance 0 0 d ð1bq Þ 0
q a3
0 0 , the Boltzmann number, representing the rate of convective energy flux. of thermal radiation with d ¼ ðc1Þ rT 4 b
Also, along n = constant we obtain
xy0 ¼
u2 a2 bqu2 uv ð1 bqÞ þ a2 ½M2 ð1 bqÞ 11=2
ð20Þ
:
Differentiating Eq. (20) with respect to n and evaluated at the rear of wavefront n = 0, yields
xny0 ¼
1=2 ðc þ 1ÞM 20 ð1 bÞ 2 2q0 a0
( M 20
)1=2 m=2 y0 0 expfXð1 bÞðy 0 y Þgpn0 : 0 y ð1 bÞ 1
ð21Þ
Eqs. (19) and (21) are the required transport equation for the discontinuities pn and xn, which shall be used to study the behavior of waves in the disturbed region. 5. Nonlinear distortion of waves Integrating Eq. (19), with respect to y0 , we obtained
pn ¼
m=2 y0 0 expfXð1 bÞðy 0 y Þgpn0 ; y0
ð22Þ
where pn0 ¼ lim pn , taken along n = 0. 0 y !y0
On substituting Eq. (22) into Eq. (21) and then integrating we get
xn ¼ 1
1=2 ðc þ 1ÞM20 ð1 bÞ 2 2q0 a0
( M20
1 ð1 bÞ
)1=2
ðy0 Þm=2 eXð1bÞy0
Z
y
wm=2 expfXð1 bÞwgdw;
ð23Þ
y0
where we have used the fact that xn0 = xnjn=0 = xnjn=0+ = 1; this follows from the Eq. (12). Let y = Y(x) be the equation of the body contour with tangent, being parallel to the velocity of the stream line, at the leading body edge. We, thus, have dy/ dx = v/u, which on differentiating with respect to n and evaluating it at the rear of n = 0 yields
v n0 ¼ u0 Y 000 ;
ð24Þ
where Y 000 is the body curvature at the tip. With the help of the Eqs. (16) and (24), Eq. (23) may be cast in the following form
ðc þ 1ÞM40 xn ¼ 1 2
( M 20
1
)1
ð1 bÞ
ðy0 Þm=2 eXð1bÞy0 Y 000
Z
y
wm=2 expfXð1 bÞwgdw:
ð25Þ
y0
The left hand side of Eq. (25) is the Jacobian of co-ordinate transformation in the region immediately behind n = 0, therefore for some y = yw this Jacobian vanishes, the neighboring characteristics of the family n = constant must intersect on the wavefront n = 0 and a strong discontinuity known as shock wave then occurs in the solution vector U. This will be the case if Un is finite at y = yw as xn = 0, for then just behind the wavefront n = 0, Ux = Un/xn becomes infinite; this describes the phenomenon of the steepening of the wavefront. A detailed analysis of result (25) for plane (m = 0) and axisymmetric (m = 1) flow configuration is presented in the following section: 6. Results and discussion In this section we shall deal with a supersonic flow past a plane beak (m = 0) and a sharp edged ring (m = 1). The described phenomenon is sketched in Fig. 1.
694
L.P. Singh et al. / Commun Nonlinear Sci Numer Simulat 16 (2011) 690–697
y
Wavefront ξ =0 ξ <0 ξ >0
x0 M0 >1 y0
0
x
Fig. 1. Flow field and convergence of characteristic for a plane and axisymmetric supersonic flow.
6.1. Plane beak case (m = 0) In case of a plane beak with body contour y ¼ Y 00b ðxÞ, the initial disturbance is released by a sharp edge of the contour with a vanishing small initial tangent (beak) and therefore Eq. (25) reduces to
xn ¼ 1
Y 00b ð0Þ ½1 expfXð1 bÞðy y0 Þg; D
ð26Þ
h i ð1bÞ 2 1 > 0, and Y00b (0) is the radius of curvature of the body shape at the tip where the body contour where D ¼ ð2cXþ1ÞM 4 M 0 ð1bÞ 0
begins to bend. As discussed earlier, the formation of shock is characterized by the vanishing of the Jacobian xn, i.e., when the characteristic begins to coalesce. Since X > 0, from Eq. (26), it is obvious that the Jacobian will vanish on the leading wavefront for y > y0 only when Y 00b ð0Þ > 0 (which corresponds to the situation when the body shape has a compressive corner at x = 0) with Y 00b ð0Þ > D. For Y 00b ð0Þ 6 D, the Jacobian remains positive for finite y > y0 and consequently a shock will not form on the leading wavefront. Thus, the parameter D represents a critical level such that when this level is exceeded by the radius of curvature Y 00b ð0Þ at the body tip, a shock will form at a finite distance away from the body. This is in contrast with the corresponding case of an ideal gas in the absence of the thermal radiation, where one always finds a shock after a finite length of run, no matter how small the initial body curvature may be. It may be recalled that at the wavehead n = 0, vx and vn are related according to vx = vn/xn; it is therefore immaterial whether we seek an expression for vx or vn at the wavehead. Since vx has a slightly more direct physical interpretation we shall opt to work in terms of that quantity and note from Eqs. (14), (19) and (26) that for the plane beak (m = 0) case
vx ¼
a0 M0 Y 00b ð0Þ½expfXð1 bÞðy y0 Þg : 1 ðY 00b ð0Þ=DÞ½1 expfXð1 bÞðy y0 Þg
ð27Þ
The condition for the growth and decay behavior of the wave can be derived from Eq. (27). It is evident from Eq. (27) that for Y 00b ð0Þ is positive and Y 00b ð0Þ > D, a shock will form and the corresponding shock formation distance, y = yw, is given by
yw ¼ y0 þ
Y 00b ð0Þ ; log Y 00b ð0Þ D Xð1 bÞ 1
ð28Þ
and the denominator of (27) becomes zero whereas numerator remains finite, i.e., the velocity gradient at the wavehead becomes unbounded at y = yw, thus signifying the steepening of the wave into a shock wave; the coincidence of this behavior with the vanishing of the Jacobian xn is clear from (28). For the case Y 00b ð0Þ 6 D, the wave is still compressive but the steepening of the velocity gradient does not occur. On the contrary vx either diminishes out along the wavehead or it remains stationary according as Y 00b ð0Þ < D or Y 00b ð0Þ ¼ D, respectively, and no shock will ever evolve on the leading wavehead n = 0. the parameter of non-idealness, c The shock formation distance specified by Eq. (28) is computed for various values of b, the specific heat ratio, X the measure of importance of thermal radiation and M0 upstream flow Mach number. For computational purpose y0 is taken as 1. The typical values of parameters involved in computation are taken as [16,17]
L.P. Singh et al. / Commun Nonlinear Sci Numer Simulat 16 (2011) 690–697
695
1.20
Shock - formation distance (yw)
b =0
1.15
b = 0.1 b = 0.2 b = 0.3 b = 0.4
1.10
b = 0.5
1.05
1.00 1.5
2.0
2.5
3.0
3.5
4.0
Mach number (M0) Fig. 2. The effect of upstream flow Mach number on shock formation distance for c = 1.4 and X = 0.5.
Shock - formation distance (yw)
1.35
b =0
1.30
b = 0.1 1.25
b = 0.2 b = 0.3
1.20
b = 0.4 1.15
b = 0.5
1.10 1.05 1.5
2.0
2.5
3.0
3.5
4.0
Mach number (M0) Fig. 3. The effect of upstream flow Mach number on shock formation distance for c = 1.67 and X = 0.8.
¼ 0; 0:1; 0:2; 0:3; 0:4; 0:5; b
c ¼ 1:4; 1:67; Y 00b ð0Þ ¼ 0:7:
causes a shock It is clear from Figs. 2 and 3, that the dissipative mechanism due to presence of non-idealness of the gas b wave to attenuate faster as compared to what it would be in an ideal gas case (b = 0). Also, it may be noted here that the causes to decrease the shock formation distance. effect of increasing value of b Also from Eq. (27) and Figs. 2 and 3, it is remarked that for M0 1, the shock formation distance yw, is given by
yw y0
2b
00 ð0Þ ; ðc þ 1Þð1 bÞY b
whereas for M0 1
yw
2b ðc þ 1ÞM40 Y 00b ð0Þ
( M20
1 ð1 bÞ
) ;
this is a decreasing function of M0, which exhibits that an increase in the value of M0 causes an early shock formation; the corresponding situation is illustrated by the curve in Figs. 2 and 3. From Figs. 4 and 5, it is evident that an increase in the value of parameter of thermal radiation in the presence of nonidealness enhances the shock formation distance. However, for the ideal gas case b = 0, the analysis presented in [14] is recovered. Also, an increase in the value of c causes to increase the shock formation distance in a non-ideal gas. Here it is interesting to note that, the presence of non-idealness of the gas reduces the shock formation distance as compared to the ideal gas (b = 0). 00 00 Also, when Y b ð0Þ < 0, which corresponds to the situation when the body shape has an expansive corner at x = 0, then for Y ð0Þ D, Eq. (27) given that b
L.P. Singh et al. / Commun Nonlinear Sci Numer Simulat 16 (2011) 690–697
Shock - formation distance (yw)
696
1.35 b =0 b = 0.1
1.30
b = 0.2 b = 0.3
1.25
1.20 0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Thermal radiation (Ω) Fig. 4. The effect of thermal radiation on shock formation distance for M 20 = 3; solid line: c = 1.4 and dashed line: c = 1.67.
Shock - formation distance (yw)
1.30 1.28 1.26
b =0
b = 0.2
b = 0.1
1.24
b = 0.3
1.22 1.20 1.18 0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Thermal radiation (Ω) Fig. 5. The effect of thermal radiation on shock formation distance for M 20 = 4; solid line: c = 1.4 and dashed line: c = 1.67.
vx ¼
00 ð0Þ 2a0 M 0 Xð1 bÞY b ðc þ 1ÞM 40
" M20
1
#
expfXð1 bÞðy y0 Þg
½1 expfXð1 bÞðy ð1 bÞ y0 Þg
;
which is an expression for the velocity gradient at the head of the Prandtl–Meyer expansion flow. 6.2. Axisymmetric case (m = 1) Here, we consider a ring shaped body y = Yw(x) with a sharp edged inlet releasing the initial disturbance which runs both inwards and outwards along characteristic lines. Now, putting m = 1, Eq. (25) can be written as
" xn ¼ 1 /Y 00w ð0Þ 1
# 1=2 erfcfXð1 bÞyg ; g1=2 erfcfXð1 bÞy
ð29Þ
0
where
" #1 ðc þ 1ÞM40 ðpy0 Þ1=2 efXð1bÞy0 g 1 2 g1=2 : /¼ M0 erfcfXð1 bÞy 0 1=2 ð1 bÞ 2½Xð1 bÞ
ð30Þ
From Eq. (29), it is evident that for y > y0 the entity within the square bracket is always positive and less than unity. Therefore the left hand side of (29) will vanish, leading to the formation of shock, provided Y 00w ð0Þ is positive and exceeds the critical value /1; when Y 00w ð0Þ 6 /1 the Jacobian xn is always positive and hence no shock wave will ever form on the leading wavehead. We, thus deduce that the shock formation will take place only when radius of curvature at the body tip exceeds the critical level 1// and consequently for the ordinate y = yw of the beginning of shock, we obtained
g1=2 ¼ ½1 f/Y 00 ð0Þg1 erfcfXð1 bÞy g1=2 : erfcfXð1 bÞy w 0 w
ð31Þ
L.P. Singh et al. / Commun Nonlinear Sci Numer Simulat 16 (2011) 690–697
697
From Eqs. (28) and (31), it is obvious that the shock formation distance yw depends upon the upstream flow Mach number 1 1 and the parameter X, M0, the initial body curvature g being either Y 00b ð0Þ or Y 00w ð0Þ , the parameter of non-idealness b which represents the effect of thermal radiation. is negative, which means that an increase in body curvature or decrease in Since @yw/@ g, @yw/@ X are positive and @yw =@ b Boltzmann number enhances the shock formation distance while an increase in the value of van der Waals excluded volume causes to decrease the shock formation distance. It is evident from Eqs. (28) and (31) that in non-ideal gas there is an early shock formation in axisymmetric case as compared to plane flow under the same initial condition. 7. Concluding remarks A problem of propagation of weak shock wave in a two-dimensional steady supersonic flow of a non-ideal gas for past plane and axisymmetric bodies in the presence of thermal radiation is theoretically investigated. The effect of thermal radiation is treated by the optically thin approximation. Using the method of wavefront analysis, Transport equations are derived which lead to the determination of the shock formation distance and also to conditions which insure that no shock will ever evolve on the wavefront. The influence of the parameter of the non-idealness, upstream flow Mach number and thermal radiation on the shock formation distance is illustrated through the Figs. 2–5. The effect of an increase in the upstream flow Mach number and the thermal radiation is to enhance the shock formation distance which, however decreases with an increase in the parameter of non-idealness. Acknowledgements The second author acknowledges the financial support from DST-CIMS, Banaras Hindu University, Varanasi, India under the JRF scheme. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
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Evolution of weak discontinuities in a non-ideal radiating gas L.P. Singh, Akmal Husain *, M. Singh Department of Applied Mathematics, Institute of Technology & DST-CIMS, Banaras Hindu University, Varanasi 221 005, Uttar Pradesh, India
a r t i c l e
i n f o
Article history: Received 21 February 2010 Received in revised form 23 April 2010 Accepted 24 April 2010 Available online 29 April 2010 Keywords: Weak discontinuity Wavefront analysis Van der Waals gas Thermal radiation
a b s t r a c t A method of wavefront analysis is used to study the formation of shock waves in a twodimensional steady supersonic flow of a non-ideal radiating gas past plane and axisymmetric bodies. The gas is taken to be sufficiently hot for the effect of thermal radiation to be significant, which is, of course, treated by the optically thin approximation to the radiative transfer equation. Transport equations, which lead to the determination of the shock formation distance and also to conditions which insure that no shock will ever evolve on the wavefront, is derived. The influence of the parameter of the non-idealness, upstream flow Mach number in the presence of thermal radiation on the behavior of the wavefront are examined. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction While studying the wave propagation in a medium, one often encounters certain kinds of discontinuities known as shock waves, acceleration waves and weak waves. Therefore, for nonlinear systems, the analysis of these waves has been the subject of the great interest both from mathematical and physical point of view. In recent years, the analysis of possible steepening of compression waves leading to shock waves has received considerable attention in the literature. It is well known that the solutions of nonlinear systems generally develop discontinuities even with the initial smooth data {see, Jeffrey [1]}. However, if the initial data are not regular but C1 piecewise, that is, the dependent variables are continuous across the manifold n(x, t) = 0, while their normal derivatives suffer finite jump discontinuities, then a discontinuity wave will be produced whose wavefront is represented by the manifold itself [1–3]. By following these discontinuities along wavefronts, it is possible to calculate explicitly the position and time of shock formation, which also serve as an important parameter in characterizing the relative importance of convective nonlinear steepening and dissipative flattening and setting a limit for the use of certain approximate theories. A number of problems relating to wave propagation and some considerations of the nonlinear effects have been studied by several investigators [1–4] in various gas dynamic regimes. The problem of interaction of shock waves with weak discontinuities and determination of shock formation distance for quasilinear hyperbolic system has been carried out by Chen [5], Sharma [6], Sharma et al. [7] and Radha et al. [8]. Sharma [6] calculated the shock formation distance and time for weak discontinuities over a concave wall while Radha et al. [8] studied the problem of interaction of shock waves with weak discontinuities and examined the existence and uniqueness of the reflection and transmission coefficients. The propagation of disturbances through a gas may be significantly affected by the presence of thermal radiation, when it is treated as an important mode of energy transfer within the gas. A further contribution towards the study of thermal radiation effects on the propagation of small disturbances in gas flows under steady and unsteady conditions have been discussed previously, in particular, by Pai [9], Penner and Olfe [10], Lick [11], Schmitt [12], Sharma et al. [13] and Singh
* Corresponding author. Tel.: +91 9452060150. E-mail address: [email protected] (A. Husain). 1007-5704/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2010.04.037
L.P. Singh et al. / Commun Nonlinear Sci Numer Simulat 16 (2011) 690–697
691
et al. [14] among others. However, if the temperature of the gas is very high and density is too low the assumption that the gas is ideal is no longer valid. The popular alternative to the ideal gas is a simplified van der Waals model. In recent years, several studies have been performed concerning the problem of strong shock waves in a van der Waals gas (see, Wu and Roberts [15], Pandey and Sharma [16,17], Singh et al. [18]). Wu and Roberts [15] used this model to study the problem of structure and stability of a spherical shock wave in a sonoluminescence bubble. Also, Pandey and Sharma [17] used the method of Lie group transformation analysis to study the problem of interaction of weak discontinuity in a van der Waals gas. In the present study, we employ the method of wavefront analysis to investigate the propagation of weak shock waves in a two-dimensional steady supersonic flow of a non-ideal gas past plane and axisymmetric bodies with thermal radiation. The medium is taken to be sufficiently hot for the effects of thermal radiation to be significant, which are, of course, treated by optically thin approximation to the radiative transfer equation. Transport equations are derived which lead to the determination of the shock formation distance and also to conditions which insure that no shock will ever evolve on the wavefront. The influence of the parameter of the non-idealness and upstream flow Mach number in the presence of thermal radiation on the shock formation distance is assessed. 2. Governing equations The basic equations for a two-dimensional inviscid steady axisymmetric flow of a non-ideal radiating gas with where the effects of thermal radiation are treated by the optically thin limit can be written in the form (see, [9,10])
uqx þ v qy þ qðux þ v y þ mv =yÞ ¼ 0;
ð1Þ
quux þ qv uy þ px ¼ 0; quv x þ qvv y þ py ¼ 0; cp ðuq þ v qy Þ þ ðc 1ÞQ ¼ 0; upx þ v py qð1 bqÞ x
ð3Þ
ð2Þ
ð4Þ
where q is the gas density, p the pressure, u and v the velocity components directed along the x and y axes, respectively; x is the distance along the axis of the symmetry from the tip of the body in the direction of oncoming flow and y is the radial distance from the x-axis; m is a constant which takes values 0 and 1 for plane and axisymmetric flows, respectively; is the rate of energy loss by the gas per unit volume through radiation and is given by
Q ¼ 4kr T 4 T 4b ;
ð5Þ
where k is the Planck mean absorption coefficient depending on the density q and temperature T of the gas, r is the Stefan – Boltzmann constant and Tb is the uniform temperature of the body along which the flow is envisaged. The additional energy gains 4krT 4b have been added to the energy loss 4krT4 due to the radiating gas so that infinite optically thin gas has no radiating boundary. The boundary wall with constant temperature Tb has no loss or gain of the energy from the radiating gas. The system of Eqs. (1)–(4) is supplemented with a van der Waals equation of state of the form (see, [15,18])
pð1 bqÞ ¼ qRT;
ð6Þ
where b is the van der Waals excluded volume which is known in terms of the molecular interaction potential in high temperature gases and R is the gas constant. It may be noted that the case b = 0 corresponds to an ideal gas (ideal in the sense that the particle interactions are absent). cp Here the equilibrium speed of sound for a non-ideal gas is c2 ¼ qð1b qÞ, where c is the specific heat ratio of the gas; therefore the relation between the equilibrium speed of sound of a non-ideal gas and a perfect gas may be defined as, c2 = {a2/ (1 bq)}, where the entity a2 is the equilibrium speed of sound for a perfect gas. 3. Characteristic formulation Using the summation convention on the repeated indices; the governing Eqs. (1)–(4) can be written as
@U i @U i þ Aij þ F i ¼ 0 i; j ¼ 1; 2; 3; 4; @x @y
ð7Þ
where the column vector (Ui)41 and (Fi)41are defined as
2
2 3
q
6u 7 6 7 U ¼ 6 7; 4v 5 p
F¼
6 6 1 6 6 2 2 2 ½ðu a Þ bqu 6 4
mquv ð1bqÞ y
þ ðc1ÞQuð1bqÞ
mvya ðc1ÞQqð1bqÞ 2
0 mquv a2 y
þ uðc 1ÞQ ð1 bqÞ
3 7 7 7 7; 7 5
ð8Þ
692
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and the non-zero components of the matrix (Aij)44, are as follows
A11 ¼ A33 ¼ A14 ¼
u
A12 ¼
;
qv ð1 bqÞ ; fðu2 a2 Þ bqu2 g
v ð1 bqÞ ; ufðu2 a2 Þ bqu2 g a2
A23 ¼ A34 ¼
v
fðu2
1
;
qu
a2 Þ
bqu2 g
A42 ¼
A22 ¼ A44 ¼ A24 ¼
;
A13 ¼
quð1 bqÞ ; fðu2 a2 Þ bqu2 g
uv ð1 bqÞ ; fðu2 a2 Þ bqu2 g
v ð1 bqÞ ; qfðu2 a2 Þ bqu2 g qua2 43
qv a2 ; fðu2 a2 Þ bqu2 g
A
¼
fðu2 a2 Þ bqu2 g
ð9Þ :
The matrix Aij has the eigenvalues
kð1;2Þ ¼
½uv ð1 bqÞ a2 fM 2 ð1 bqÞ 1g1=2 ; ½ðu2 a2 Þ bqu2
kð3;4Þ ¼
v u
;
ð10Þ
with the corresponding left eigenvectors
Lð1Þ ¼ ½0; 1; u=v ; fM 2 ð1 bqÞ 1g1=2 =qv ; Lð2Þ ¼ ½0; 1; u=v ; fM 2 ð1 bqÞ 1g1=2 =qv ; Lð3Þ ¼ ½1; 0; 0; ð1 bqÞ=a2 ;
Lð4Þ ¼ ½0; 1; v =u; ð1=quÞ:
9 > = > ;
ð11Þ
Here, M = q/a, with q = (u2 + v2)1/2, is the upstream flow Mach number. It is obvious from Eq. (11) that the system (7) possesses, except along stream line on which k = v/u, two families of characteristics along which dy/dx = k(1, 2); these characteristics represent the propagation of waves in opposite directions with characteristic speeds k(1, 2). It may be remarked that these characteristic velocities are real if, and only if, the flow is supersonic, i.e., M > 1. 4. Evolution of transport equations for jump discontinuities Let us assume that k(1) describes the initial wavefront trace n(x, y) = 0, which passes through the point (x0, y0). The medium ahead of the initial wavefront is assumed to have a uniform temperature T0 = Tb and a uniform velocity u0 in the x direction with v0 = 0. In the foregoing analysis, we shall use the suffix 0 to denote a quantity in the region ahead of the wavefront n = 0. Now, we derive the transport equations for the jump discontinuities in U as they move along the wavefront n = 0. We introduce new curvilinear co-ordinates n, y0 defined as (see Jeffrey [2])
)
nx þ kð1Þ ny ¼ 0
ð12Þ
nðx; y0 Þ ¼ x x0 ;
and y = y0 . Then n has the required co-ordinate property that n is positive (negative) behind (ahead of) the leading characteristic on which n = 0. In view of these new co-ordinates, Eq. (7) on pre-multiplying by L(i) reduces to the form
LðiÞ U n þ
kð1Þ kðiÞ ð1Þ
k
ðiÞ
k
xn LðiÞ U y0 þ
kð1Þ k
ð1Þ
kðiÞ
xn LðiÞ B ¼ 0;
ð13Þ
where xn = 1/nx is the Jacobian of the transformation. Across the wavefront n = 0, U and U y0 are continuous and have their subscripts-0 values whilst Un and xn are discontinuous. On using Eq. (11) and the flow conditions ahead, the Eq. (13) at the rear side of n = 0 for i = 2, 3 and 4 becomes:
ð1 bq0 Þ pn ; a20 1 p; un ¼ q0 u0 n
qn ¼
vn ¼
½M20 ð1
bq0 Þ 1 q0 u0
ð14Þ ð15Þ 1=2
pn :
ð16Þ
We set i = 1 in Eq. (13), differentiate the resulting equation with respect to n, and then evaluate it at the rear of n = 0, we obtain
h i1=2 mq0 u0 u M 20 ð1 bq0 Þ 1 pny0 þ q0 u0 v ny0 þ v n þ 20 ð1 bq0 Þðc 1ÞQ n ¼ 0: y0 a0
ð17Þ
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693
After substituting T from Eq. (6) and differentiating Eq. (5) with respect to n at the rear of n = 0 yields;
16ðc 1ÞkrT 4b pn : q0 a20
Qn ¼
ð18Þ
Plugging in Eqs. (16) and (18), Eq. (17) can be reduces of the form
pny0 þ
m X p ¼ 0; þ ð1 bÞ n 2y0
ð19Þ
h i1=2 ¼ bq is the parameter of non-idealness of the gas and X ¼ 8ðc1ÞkM0 M 2 1 where b > 0 is a measure of importance 0 0 d ð1bq Þ 0
q a3
0 0 , the Boltzmann number, representing the rate of convective energy flux. of thermal radiation with d ¼ ðc1Þ rT 4 b
Also, along n = constant we obtain
xy0 ¼
u2 a2 bqu2 uv ð1 bqÞ þ a2 ½M2 ð1 bqÞ 11=2
ð20Þ
:
Differentiating Eq. (20) with respect to n and evaluated at the rear of wavefront n = 0, yields
xny0 ¼
1=2 ðc þ 1ÞM 20 ð1 bÞ 2 2q0 a0
( M 20
)1=2 m=2 y0 0 expfXð1 bÞðy 0 y Þgpn0 : 0 y ð1 bÞ 1
ð21Þ
Eqs. (19) and (21) are the required transport equation for the discontinuities pn and xn, which shall be used to study the behavior of waves in the disturbed region. 5. Nonlinear distortion of waves Integrating Eq. (19), with respect to y0 , we obtained
pn ¼
m=2 y0 0 expfXð1 bÞðy 0 y Þgpn0 ; y0
ð22Þ
where pn0 ¼ lim pn , taken along n = 0. 0 y !y0
On substituting Eq. (22) into Eq. (21) and then integrating we get
xn ¼ 1
1=2 ðc þ 1ÞM20 ð1 bÞ 2 2q0 a0
( M20
1 ð1 bÞ
)1=2
ðy0 Þm=2 eXð1bÞy0
Z
y
wm=2 expfXð1 bÞwgdw;
ð23Þ
y0
where we have used the fact that xn0 = xnjn=0 = xnjn=0+ = 1; this follows from the Eq. (12). Let y = Y(x) be the equation of the body contour with tangent, being parallel to the velocity of the stream line, at the leading body edge. We, thus, have dy/ dx = v/u, which on differentiating with respect to n and evaluating it at the rear of n = 0 yields
v n0 ¼ u0 Y 000 ;
ð24Þ
where Y 000 is the body curvature at the tip. With the help of the Eqs. (16) and (24), Eq. (23) may be cast in the following form
ðc þ 1ÞM40 xn ¼ 1 2
( M 20
1
)1
ð1 bÞ
ðy0 Þm=2 eXð1bÞy0 Y 000
Z
y
wm=2 expfXð1 bÞwgdw:
ð25Þ
y0
The left hand side of Eq. (25) is the Jacobian of co-ordinate transformation in the region immediately behind n = 0, therefore for some y = yw this Jacobian vanishes, the neighboring characteristics of the family n = constant must intersect on the wavefront n = 0 and a strong discontinuity known as shock wave then occurs in the solution vector U. This will be the case if Un is finite at y = yw as xn = 0, for then just behind the wavefront n = 0, Ux = Un/xn becomes infinite; this describes the phenomenon of the steepening of the wavefront. A detailed analysis of result (25) for plane (m = 0) and axisymmetric (m = 1) flow configuration is presented in the following section: 6. Results and discussion In this section we shall deal with a supersonic flow past a plane beak (m = 0) and a sharp edged ring (m = 1). The described phenomenon is sketched in Fig. 1.
694
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y
Wavefront ξ =0 ξ <0 ξ >0
x0 M0 >1 y0
0
x
Fig. 1. Flow field and convergence of characteristic for a plane and axisymmetric supersonic flow.
6.1. Plane beak case (m = 0) In case of a plane beak with body contour y ¼ Y 00b ðxÞ, the initial disturbance is released by a sharp edge of the contour with a vanishing small initial tangent (beak) and therefore Eq. (25) reduces to
xn ¼ 1
Y 00b ð0Þ ½1 expfXð1 bÞðy y0 Þg; D
ð26Þ
h i ð1bÞ 2 1 > 0, and Y00b (0) is the radius of curvature of the body shape at the tip where the body contour where D ¼ ð2cXþ1ÞM 4 M 0 ð1bÞ 0
begins to bend. As discussed earlier, the formation of shock is characterized by the vanishing of the Jacobian xn, i.e., when the characteristic begins to coalesce. Since X > 0, from Eq. (26), it is obvious that the Jacobian will vanish on the leading wavefront for y > y0 only when Y 00b ð0Þ > 0 (which corresponds to the situation when the body shape has a compressive corner at x = 0) with Y 00b ð0Þ > D. For Y 00b ð0Þ 6 D, the Jacobian remains positive for finite y > y0 and consequently a shock will not form on the leading wavefront. Thus, the parameter D represents a critical level such that when this level is exceeded by the radius of curvature Y 00b ð0Þ at the body tip, a shock will form at a finite distance away from the body. This is in contrast with the corresponding case of an ideal gas in the absence of the thermal radiation, where one always finds a shock after a finite length of run, no matter how small the initial body curvature may be. It may be recalled that at the wavehead n = 0, vx and vn are related according to vx = vn/xn; it is therefore immaterial whether we seek an expression for vx or vn at the wavehead. Since vx has a slightly more direct physical interpretation we shall opt to work in terms of that quantity and note from Eqs. (14), (19) and (26) that for the plane beak (m = 0) case
vx ¼
a0 M0 Y 00b ð0Þ½expfXð1 bÞðy y0 Þg : 1 ðY 00b ð0Þ=DÞ½1 expfXð1 bÞðy y0 Þg
ð27Þ
The condition for the growth and decay behavior of the wave can be derived from Eq. (27). It is evident from Eq. (27) that for Y 00b ð0Þ is positive and Y 00b ð0Þ > D, a shock will form and the corresponding shock formation distance, y = yw, is given by
yw ¼ y0 þ
Y 00b ð0Þ ; log Y 00b ð0Þ D Xð1 bÞ 1
ð28Þ
and the denominator of (27) becomes zero whereas numerator remains finite, i.e., the velocity gradient at the wavehead becomes unbounded at y = yw, thus signifying the steepening of the wave into a shock wave; the coincidence of this behavior with the vanishing of the Jacobian xn is clear from (28). For the case Y 00b ð0Þ 6 D, the wave is still compressive but the steepening of the velocity gradient does not occur. On the contrary vx either diminishes out along the wavehead or it remains stationary according as Y 00b ð0Þ < D or Y 00b ð0Þ ¼ D, respectively, and no shock will ever evolve on the leading wavehead n = 0. the parameter of non-idealness, c The shock formation distance specified by Eq. (28) is computed for various values of b, the specific heat ratio, X the measure of importance of thermal radiation and M0 upstream flow Mach number. For computational purpose y0 is taken as 1. The typical values of parameters involved in computation are taken as [16,17]
L.P. Singh et al. / Commun Nonlinear Sci Numer Simulat 16 (2011) 690–697
695
1.20
Shock - formation distance (yw)
b =0
1.15
b = 0.1 b = 0.2 b = 0.3 b = 0.4
1.10
b = 0.5
1.05
1.00 1.5
2.0
2.5
3.0
3.5
4.0
Mach number (M0) Fig. 2. The effect of upstream flow Mach number on shock formation distance for c = 1.4 and X = 0.5.
Shock - formation distance (yw)
1.35
b =0
1.30
b = 0.1 1.25
b = 0.2 b = 0.3
1.20
b = 0.4 1.15
b = 0.5
1.10 1.05 1.5
2.0
2.5
3.0
3.5
4.0
Mach number (M0) Fig. 3. The effect of upstream flow Mach number on shock formation distance for c = 1.67 and X = 0.8.
¼ 0; 0:1; 0:2; 0:3; 0:4; 0:5; b
c ¼ 1:4; 1:67; Y 00b ð0Þ ¼ 0:7:
causes a shock It is clear from Figs. 2 and 3, that the dissipative mechanism due to presence of non-idealness of the gas b wave to attenuate faster as compared to what it would be in an ideal gas case (b = 0). Also, it may be noted here that the causes to decrease the shock formation distance. effect of increasing value of b Also from Eq. (27) and Figs. 2 and 3, it is remarked that for M0 1, the shock formation distance yw, is given by
yw y0
2b
00 ð0Þ ; ðc þ 1Þð1 bÞY b
whereas for M0 1
yw
2b ðc þ 1ÞM40 Y 00b ð0Þ
( M20
1 ð1 bÞ
) ;
this is a decreasing function of M0, which exhibits that an increase in the value of M0 causes an early shock formation; the corresponding situation is illustrated by the curve in Figs. 2 and 3. From Figs. 4 and 5, it is evident that an increase in the value of parameter of thermal radiation in the presence of nonidealness enhances the shock formation distance. However, for the ideal gas case b = 0, the analysis presented in [14] is recovered. Also, an increase in the value of c causes to increase the shock formation distance in a non-ideal gas. Here it is interesting to note that, the presence of non-idealness of the gas reduces the shock formation distance as compared to the ideal gas (b = 0). 00 00 Also, when Y b ð0Þ < 0, which corresponds to the situation when the body shape has an expansive corner at x = 0, then for Y ð0Þ D, Eq. (27) given that b
L.P. Singh et al. / Commun Nonlinear Sci Numer Simulat 16 (2011) 690–697
Shock - formation distance (yw)
696
1.35 b =0 b = 0.1
1.30
b = 0.2 b = 0.3
1.25
1.20 0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Thermal radiation (Ω) Fig. 4. The effect of thermal radiation on shock formation distance for M 20 = 3; solid line: c = 1.4 and dashed line: c = 1.67.
Shock - formation distance (yw)
1.30 1.28 1.26
b =0
b = 0.2
b = 0.1
1.24
b = 0.3
1.22 1.20 1.18 0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Thermal radiation (Ω) Fig. 5. The effect of thermal radiation on shock formation distance for M 20 = 4; solid line: c = 1.4 and dashed line: c = 1.67.
vx ¼
00 ð0Þ 2a0 M 0 Xð1 bÞY b ðc þ 1ÞM 40
" M20
1
#
expfXð1 bÞðy y0 Þg
½1 expfXð1 bÞðy ð1 bÞ y0 Þg
;
which is an expression for the velocity gradient at the head of the Prandtl–Meyer expansion flow. 6.2. Axisymmetric case (m = 1) Here, we consider a ring shaped body y = Yw(x) with a sharp edged inlet releasing the initial disturbance which runs both inwards and outwards along characteristic lines. Now, putting m = 1, Eq. (25) can be written as
" xn ¼ 1 /Y 00w ð0Þ 1
# 1=2 erfcfXð1 bÞyg ; g1=2 erfcfXð1 bÞy
ð29Þ
0
where
" #1 ðc þ 1ÞM40 ðpy0 Þ1=2 efXð1bÞy0 g 1 2 g1=2 : /¼ M0 erfcfXð1 bÞy 0 1=2 ð1 bÞ 2½Xð1 bÞ
ð30Þ
From Eq. (29), it is evident that for y > y0 the entity within the square bracket is always positive and less than unity. Therefore the left hand side of (29) will vanish, leading to the formation of shock, provided Y 00w ð0Þ is positive and exceeds the critical value /1; when Y 00w ð0Þ 6 /1 the Jacobian xn is always positive and hence no shock wave will ever form on the leading wavehead. We, thus deduce that the shock formation will take place only when radius of curvature at the body tip exceeds the critical level 1// and consequently for the ordinate y = yw of the beginning of shock, we obtained
g1=2 ¼ ½1 f/Y 00 ð0Þg1 erfcfXð1 bÞy g1=2 : erfcfXð1 bÞy w 0 w
ð31Þ
L.P. Singh et al. / Commun Nonlinear Sci Numer Simulat 16 (2011) 690–697
697
From Eqs. (28) and (31), it is obvious that the shock formation distance yw depends upon the upstream flow Mach number 1 1 and the parameter X, M0, the initial body curvature g being either Y 00b ð0Þ or Y 00w ð0Þ , the parameter of non-idealness b which represents the effect of thermal radiation. is negative, which means that an increase in body curvature or decrease in Since @yw/@ g, @yw/@ X are positive and @yw =@ b Boltzmann number enhances the shock formation distance while an increase in the value of van der Waals excluded volume causes to decrease the shock formation distance. It is evident from Eqs. (28) and (31) that in non-ideal gas there is an early shock formation in axisymmetric case as compared to plane flow under the same initial condition. 7. Concluding remarks A problem of propagation of weak shock wave in a two-dimensional steady supersonic flow of a non-ideal gas for past plane and axisymmetric bodies in the presence of thermal radiation is theoretically investigated. The effect of thermal radiation is treated by the optically thin approximation. Using the method of wavefront analysis, Transport equations are derived which lead to the determination of the shock formation distance and also to conditions which insure that no shock will ever evolve on the wavefront. The influence of the parameter of the non-idealness, upstream flow Mach number and thermal radiation on the shock formation distance is illustrated through the Figs. 2–5. The effect of an increase in the upstream flow Mach number and the thermal radiation is to enhance the shock formation distance which, however decreases with an increase in the parameter of non-idealness. Acknowledgements The second author acknowledges the financial support from DST-CIMS, Banaras Hindu University, Varanasi, India under the JRF scheme. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
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