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Propagation of weak shock waves in non-uniform, radiative magnetogasdynamics
ARTICLE IN PRESS Acta Astronautica 67 (2010) 296–300
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Acta Astronautica journal homepage: www.elsevier.com/locate/actaastro
Propagation of weak shock waves in non-uniform, radiative magnetogasdynamics L.P. Singh, S.D. Ram , D.B. Singh Department of Applied Mathematics, Institute of Technology, Bananas Hindu University, Varanasi 221005, India
a r t i c l e i n f o
abstract
Article history: Received 5 October 2009 Received in revised form 15 December 2009 Accepted 2 January 2010 Available online 6 February 2010
A problem of propagation of weak planer shock wave in radiative magnetogasdynamics is theoretically investigated. The gas is taken to be sufficiently hot for the effects of thermal radiation to be significant, which are of course treated by the optically thin approximation to the radiative transfer equation. The density of the medium ahead of the shock wave is assumed to be exponentially varying. A systematic perturbation scheme is used to obtain the analytical solution of the flow field. Also, an approximate solution near the front shock is determined. The effect of radiation and magnetic field strength on the shock wave propagation is examined. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Weak shock wave Radiating gas Magnetogasdynamics Perturbation scheme
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 great interest both from mathematical and physical point of view. In classical gasdynamics, the transfer of energy through radiation is usually neglected. However, on the account of the high temperature that prevail in many phenomena it is of interest to consider the effects of thermal radiation in gasdynamics. Since at high temperatures a gas is likely to be partially ionized, electromagnetic effects may also be significant. A complete analysis of such a problem should, therefore consist of the study of the gasdynamic flow and of the electromagnetic and radiation fields simultaneously. A number of problems relating to wave propagation with radiative effects and some considerations of the non-linear effects have been studied analytically in the
Corresponding author. Tel.: + 91 9451895174.
E-mail address: [email protected] (S.D. Ram). 0094-5765/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.actaastro.2010.01.002
past see, for example [1–9]. Pai and Hsieh [10] used the method of strained coordinates to study the problem of supersonic flows over weak compression and expansion corner and over biconvex airfoil. Problems of propagation of weak planar and non-planar shock waves in uniform and non uniform gases have been investigated using the perturbation scheme [11–13]. The propagation of weak shock waves near the nose of a body of revolution in real gas flows have been analyzed in [14–17], using the perturbation technique. Xue [18] investigated the problem of non-planar dust-ion acoustic shock waves with transverse perturbation and deduced the Kadomtsev–Petviashvili Berger equation that describes the dust-ion acoustic shock waves. Krasnobaev et al. [19] calculated the distribution of flow parameters in the region between shocks. Also, the possible influence on the model of galactic hydrogen neutral atoms penetrating into interplanetary medium is estimated. Krasnobaev [20] investigated the problem of propagation of the linear and non-linear wave in HII region by taking into account the relaxation processes to the temperature and ionization fraction. In the present paper we use the perturbation scheme to study theoretically the problem of propagation of weak shock waves in a non-uniform radiating and electrically conducting gas permeated by a transverse magnetic field.
ARTICLE IN PRESS L.P. Singh et al. / Acta Astronautica 67 (2010) 296–300
The shock wave is generated by the motion of a piston. The piston path may be prescribed by a given function eFðtÞ where t is time and e may be regarded as measure of the magnitude of the disturbance generated by the piston. The analytical solution of the flow field has been presented up to the first order of e. Also an approximate solution near the front shock is determined. The effect of thermal radiation and the magnetic field on the flow parameters is assessed. 2. Basic equations Assuming the electrical conductivity to be infinite and direction of the magnetic field to be orthogonal to the trajectories of gas particles, the basic equations for a one dimensional unsteady planar flow in radiative magnetogasdynamics, where the effects of thermal radiation are treated by the optically thin approximation to the radiative transfer equation, can be written down in familiar form [12] ð2:1Þ
~ @u~ @u~ 1 @ðp~ þ hÞ þ u~ þ ¼ 0; ~ r~ @x~ @x~ @t
ð2:2Þ
ð2:4Þ
~ 2 =2; the magnetic where r~ is the gas density, h~ ¼ uH ~ the magnetic field strength and u the pressure with H magnetic permeability, p~ the pressure, u~ the velocity, Q~ the rate of energy loss by the gas per unit volume through radiation and is given by 4 4 Q~ ¼ 4ksðT~ T~ 0 Þ;
ð2:5Þ
where k is the Planck mean absorption coefficient depending on the density r~ and temperature T~ of the gas, s is Stefan–Boltzmann constant and T~ 0 is uniform temperature of the body along which the flow is 4 envisaged. The additional energy gains 4ksT~ ; have 0
4 been added to the energy loss 4ksT~ due to the radiating gas so that infinite optically thin gas has no radiating boundary. The boundary wall with constant temperature T~ 0 has no loss or gain of energy from the radiating gas.
The initial and boundary conditions associated to the system are
@u @u 1 @ðp þhÞ þu þ ¼ 0; @t @x r @x
ð2:7Þ
@p @p gp @r @r þu þ ðg1ÞQ ¼ 0; þu @x @t @x r @t
ð2:8Þ
@h @h 2h @r @r þu ¼ 0: þu @x @t @x r @t
ð2:9Þ
dFðtÞ ; dt
at
x ¼ eFðtÞ;
ð2:10Þ
and unperturbed flow field is specified by
r0 ¼ ex ;
u0 ¼ 0;
h0 ¼ dð0 o d o 1Þ:
ð2:11Þ
3. Characteristic transformation The hyperbolic system of Eqs. (2.6)–(2.9) possesses four families of characteristics dx=dt ¼ u; u; the trajectory of fluid particle, and dx=dt ¼ u 7c; the outgoing and incoming wavelets. Here, c ¼ ða2 þb2 Þ1=2 ; is the magnetosonic speed
with
a ¼ ðgp=rÞ1=2 ;
r~ 0 ¼ r elx~ ;
u~ 0 ¼ 0;
h~ 0 ¼ h :
the
speed
of
sound
and
1=2
b ¼ ð2h=rÞ ; the Alfve´n speed. To study the nonlinear effects on the wave pattern, we introduce a new coordinate system ða; bÞ where a is constant along an outgoing wave if it originates from the piston path at a time t ¼ t ; the trajectory will be labeled as a ¼ t ; b is simply x. The transformation relation between ðx; tÞ; and ða; bÞ can be deduced immediately from the relation dt ¼ ta da þ tb db where subscripts a and b denote partial differentiation with respect to a and b respectively. Also, @a=@t ¼ 1=ta ; @a=@x ¼ ðtb =ta Þ In terms of new coordinates a and b, the equations (2.6)–(2.9) become ðu þ cÞtb ¼ 1;
c2 tb2 ¼ u2 tb2 2utb þ1;
ð3:1Þ
ra þurb ta ura tb þ rub ta rua tb ¼ 0;
ð3:2Þ
rua þ ruub ta ruua tb þ ðpb þ hb Þta ðpa þ ha Þtb ¼ 0;
ð3:3Þ
~ x~ ¼ eF~ ðtÞ:
The conditions of the undisturbed region ahead of the propagating shock wave generated by the piston motion are taken as p~ 0 ¼ p ;
ð2:6Þ
@h~ @h~ 2h~ @r~ @r~ ¼ 0; þ u~ þ u~ r~ @t~ @x~ @x~ @t~
at
@r @r @u þu þr ¼ 0; @t @x @x
p0 ¼ 1; ð2:3Þ
~ dF~ ðtÞ ; dt~
Consequently, Eqs. (2.1)–(2.4) are written in the following dimensionless form
u¼e
@p~ @p~ gp~ @r~ @r~ þ ðg1ÞQ~ ¼ 0; þ u~ þ u~ r~ @t~ @x~ @x~ @t~
u~ ¼ e
Introducing the flow variables in terms of their corresponding undisturbed quantities as 1=2 h~ r~ p p~ ~ ; p ¼ ; r ¼ ; h ¼ ; x ¼ lx; t ¼ lt~ r p r p 1=2 1=2 r r u ¼ u~ ; Q ¼ Q~ =lp : p p
Also, the initial condition at the piston becomes
@r~ @r~ @u~ þ r~ ¼ 0; þ u~ ~ ~ @ x @x~ @t
297
pa þ upb ta upa tb
gp ðr þ urb ta ura tb Þ þ ðg1Þta Q ¼ 0; r a ð3:4Þ
ha þuhb ta uha tb
2h
r
ðra þ urb ta ura tb Þ ¼ 0:
ð3:5Þ
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L.P. Singh et al. / Acta Astronautica 67 (2010) 296–300
Similarly, Eq. (2.10) becomes 0
u ¼ eF ðaÞ;
at
b ¼ eFðaÞ;
ð3:6Þ
where the superscript prime denotes the total differentiation with respect to its argument. Consequently, the labeling of ða; bÞ coordinates gives t ¼ a;
at
b ¼ eFðaÞ:
ð3:7Þ
In addition to the above boundary conditions, the usual Rankine–Hugoniot conditions must be satisfied at each point of the front shock. If U be the shock velocity, these conditions are
Similarly making use of Taylor’s expansion, (3.7) yields the following set of boundary conditions
e0 : t0 ¼ a at b ¼ 0;
ð4:4Þ
e1 : t1 ¼ FðaÞt0b at b ¼ 0;
ð4:5aÞ
u1 ¼ F 0 ðaÞ at b ¼ 0:
ð4:5bÞ
5. Jump conditions at the shock wave
rðUuÞ ¼ r0 U;
ð3:8aÞ
Substituting the expansion (4.1) into the jump conditions (3.8a–d) and collecting the terms of like power of e, we get
rðUuÞ2 þ ðp þ hÞ ¼ r0 U 2 þ ðp0 þ h0 Þ;
ð3:8bÞ
e1 : r1 U0 r0 u1 ¼ 0;
ð5:1aÞ
hðUuÞ2 ¼ h0 U 2 ;
ð3:8cÞ
ðp1 þh1 Þr0 U0 u1 ¼ 0;
ð5:1bÞ
h1 U02 2h0 U0 u1 ¼ 0;
ð5:1cÞ
ðgp1 a20 r1 Þ þ ðg1Þð2h1 b20 r1 Þðg1Þr0 U0 u1 ¼ 0;
ð5:1dÞ
e2 : r2 U0 r0 u2 ¼ r1 u1 r1 U1 ;
ð5:2aÞ
ðp2 þh2 Þr0 U0 u2 ¼ r0 u1 U1 ;
ð5:2bÞ
h2 U02 2h0 U0 u2 ¼ 3h0 u21 2h0 u1 U1 ;
ð5:2cÞ
" # b20 ðg1Þ b2 ðg1Þ 2 2 2 ; a2 1 þ ðUuÞ þ ð g 1Þ 1 þ U þð g 1Þ ¼ a 0 2a2 a2 2a20 a20
ð3:8dÞ and Q ¼ Q0 . The position of the front shock is, of course, not known a priori. However, it must assume a form such that dx=dt ¼ U; at every point of the shock, which in the (a,b) plane takes the following form ta
da 1 þtb ¼ ; db U
ð3:9Þ
subject to the conditions, a ¼ 0 at b ¼ 0. 4. Characteristic perturbation Here, it is assumed that the fluid is weakly disturbed by the front shock generated by the forward piston motion. It is also assumed that the maximum fluctuation of the flow properties depend on the order of e. Solution of the system (2.1)–(2.4) can be constructed in the power series of e in (a,b) plane as
jða; bÞ ¼ j0 ða; bÞ þ ej1 ða; bÞ þ e2 j2 ða; bÞ þ ::::::::::::
ð4:1Þ
where j can be any dependent variable and subscript ‘‘0’’ denote free stream conditions. Using Eq. (4.1) in Eq. (3.1)–(3.5) and collecting terms of like powers of e, we have
e0 : c02 t02b ¼ 1;
ð4:2Þ
ðgp2 a20 r2 Þ þ ðg1Þð2h2 b20 r2 Þðg1Þr0 U0 u2 ðg1Þ ¼ r0 u21 þ ðg1Þr0 u1 U1 : 2
It may be noted here that the homogeneous parts of the systems (5.1) and (5.2) are identical. Therefore, the following condition for the system (5.2) to be consistent, we have U1 ¼
ðg þ1Þ ½6h0 þ g þ g2 u1 ; 4m
e : c1 t0b þ c0 t1b þu1 t0b ¼ 0;
ð4:3aÞ
r1a þu1 r0b t0a þ r0 u1b t0a r0 u1a t0b ¼ 0;
ð4:3bÞ
r0 u1a þðp1b þh1b Þt0a ðp1a þh1a Þt0b ¼ 0;
ð4:3cÞ
6. Solution of the problem 6.1. Zeroth order solution The zeroth order flow parameters are identical to the ones in the unperturbed region except t0 from Eq. (2.11) we have
r0
ð4:3dÞ h1a þ2h0 ðu1b t0a u1a t0b Þ ¼ 0:
ð4:3eÞ
h0 ¼ d;
r0 ¼ eb ;
u0 ¼ 0;
ð6:1:1Þ
the variable t0 may be obtained by integrating Eq. (4.2) and using Eq. (4.4) as t0 ¼
r ðp1a a20 r1a Þ þ gu1 t0a þ 16ðg1Þsk0 T04 t0a p1 1 ¼ 0;
ð5:3Þ
where, m ¼ ð2h0 þ gÞ.
p0 ¼ 1; 1
ð5:2dÞ
2
g1=2
ð1eb=2 Þ þ a:
ð6:1:2Þ
6.2. First order solution The solution of the first order quantities can be obtained from the Eqs. (4.3b)–(4.3e) by the method of Laplace transform as follows. Denoting the Laplace
ARTICLE IN PRESS L.P. Singh et al. / Acta Astronautica 67 (2010) 296–300
transform of a function fða; bÞ by FðzbÞ; i.e.
f ðz; bÞ ¼
Z
1
The equation of the front shock wave in the physical ðx; tÞplane can be obtained as follows: 2 F 0 ðaÞ U0 RðbÞ ; 1 ð7:2Þ t ¼ a þ 1=2 ð1eb=2 Þe 2 c0 m
fða; bÞeaz da;
0
and using the boundary conditions we have zr 1 þ r0b t0a u 1 þ r0 u1b t0a zr0 u 1 t0b ¼ 0;
ð6:2aÞ
zr0 u 1 þ ðp 1b þ h 1b Þt0a zðp 1 þh 1 Þt0b ¼ 0;
ð6:2bÞ
r zðp 1 a20 r 1 Þ þ 16ðg1Þk0 sT04 p 1 1 t0a þ gt0a u 1 ¼ 0; ð6:2cÞ ð6:2dÞ
Solving the above equations, we get the first order quantities in the transformed plane as p 1 ¼ G e2le
b=2
K0 ð2Oeb=2 Þ; 9 > > > > h 1 ¼ 2G h0 g e K0 ð2Oeb=2 Þ; > = 2leb=2 b=2 p1 ¼ G e K0 ð2Oe Þ; > > b=2 > u 1 ¼ G m1=2 g1 eðb=22le Þ K0 ð2Oeb=2 Þ; > > ; 1 2leb=2 b=2 h 1 ¼ 2G h0 g e K0 ð2Oe Þ; u 1 ¼ G m1=2 g1 e
1 2leb=2
r 1 ¼ G g1 eðb þ 2le
b=2 Þ
ð6:3Þ
K0 ð2Oeb=2 Þ;
and t 1b ¼
F 0 ðaÞ b 2 b=2 þ l ½6h þ g þ g exp ðe 1Þ : 0 d 4 2m1=2
ð6:4Þ
G ¼
ð6h0 þ g þ g2 Þ
Z
2
m
b 0
Z exp þld ðeZ=2 1Þ dZ: 4
Eliminating a from the Eqs. (7.2) and (7.3) and replacing b by x, we obtain the equation of the front shock in the physical (x,t)-plane.
The effects of the thermal radiation and the magnetic field strength on the first order shock wave solutions are clearly indicated in the system of Eqs. (7.1). It may be noted here that the radiation and magnetic field effects enter into the first order solutions through the radiative flux parameter L0 and the magnetic pressure term h0 . The case, L0 ¼ 0 and h0 ¼ 0 corresponds to non-radiative and non-magnetic case. The typical values of physical quantities involved in computation are taken as
g ¼ 1:67;
s ¼ 5:67 105 erg cm2 s1 K4 ;
k0 ¼ 1:38 1016 erg K1 ; and the temperature range taken is 1:0 105 Kr
where
O¼
RðbÞ ¼
8. Result discussion
K0 ð2Oeb=2 Þ; ðb=22leb=2 Þ
and using (7.2) in (3.9) we get the following equation of the shock in the ðx; tÞ-plane Z a e 02 e F ðaÞRðbÞ ¼ F 0 ðxÞ 1 1=2 eb=2 F 0 ðxÞ dx; ð7:3Þ 2 m 0 where
r0
zh 1 þ 2h0 ðu 1b t0a þ zu 1 t0b Þ ¼ 0:
299
1=2
z L0 þ U0 1 L0 =ðg1ÞU0 þ z 2m1=2
;
l ¼ z=m1=2 ;
ge2l F 0 ðaÞ; m1=2 K0 ð2OÞ
is the arbitrary constant, K0 is the modified Bessel function of second kind of order zero, ld ¼ L0 þ U0 =m1=2 , is the radiative decay length, B0 ¼ r0 a20 =ðg1ÞsT04 , is the Boltzmann number and L0 ¼ 16ðg1Þk0 a0 =B0 , is the radiative flux parameter.
T0 o 1:5 105 K. Figs. 1–3 show the profiles of the pressure, magnetic pressure and shock velocity distribution in the perturbed region. The effect of an increase in radiating flux parameter results in falling-off in gas pressure, magnetic pressure and shock velocity which is evident from Figs. 1–3. Further, an increase in magnetic field strength results in falling-off in the gas pressure and shock velocity. However, there is an increase in magnetic pressure with an added magnetic field. This is, of course, what is expected physically. It may also be noted that to attain the same value of shock speed as in h0 = 0 h0 = 0.2 h0 = 0.4
1.2
7. Approximate solution near the front shock
1.0 0.8 p˜ p*
Now taking the inverse Laplace transformation of the Eqs. (6.3) we have 9 > b > > p1 ¼ gm1=2 F 0 ðaÞexp þld ðeb=2 1Þ ; > > > 4 > > > > > 3b > 0 b=2 > u1 ¼ F ðaÞexp þ ld ðe 1Þ ; > = 4 ð7:1Þ b > > h1 ¼ 2h0 m1=2 F 0 ðaÞexp þ ld ðeb=2 1Þ ; > > > 4 > > > > > 3b 1=2 0 b=2 > r1 ¼ m F ðaÞexp þ ld ðe 1Þ : > > > ; 4
Λ0 = 0 Λ0 = 4 Λ0 = 7
0.6 0.4 0.2 0.0 0.0
0.5
1.0
1.5 x˜
2.0
Fig. 1. Variation of pressure versus distance.
2.5
3.0
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0.5 0.4
through the Figs. 1–3. Also, an approximate analytical solution near the front shock is obtained.
Λ0 = 0 Λ0 = 4 Λ0 = 7
h0 = 0.4 h0 = 0.2
Acknowledgments
h˜ p*
0.3
The second author acknowledges the financial support from the UGC, New Delhi, India, and the third author acknowledges the financial support from the CSIR, India, under the JRF scheme.
0.2 0.1 0.0 0.0
0.5
1.0
1.5 x˜
2.0
2.5
3.0
Fig. 2. Variation of magnetic pressure versus distance.
Λ0 = 0 Λ0 = 4 Λ0 = 7
h0 = 0 h0 = 0.2 h0 = 0.4
4
U
3 2 1 0 0.0
0.5
1.0
1.5 x˜
2.0
2.5
3.0
Fig. 3. Variation of shock speed versus distance.
non-magnetic case, the general level of radiation must be greater. In other words we can say that the effect of magnetic field interaction is to diminish the effect of radiative transfer.
9. Conclusion A systematic perturbation scheme is used to study the problem of the propagation of weak planar shock waves in radiative magnetogasdynamics. The effect of coupling between the radiative transfer and magnetogasdynamics phenomena on the flow field is analyzed. The analytical solution of the flow field has been obtained up to the first order e. The effect of radiative transfer and the magnetic field strength in the flow field variables is illustrated
References [1] G.B. Whitham, Linear and Non-Linear Waves, Wiley, New York, 1974. [2] V.D. Sharma, Development of jump discontinuities in radiative magnetogasdynamics, International Journal of Engineering Science 24 (1986) 813–818. [3] W.J. Lick, The propagation of small disturbances in radiating gas, Journal of Fluid Mechanics 18 (1964) 274–288. [4] H.R. Long, W.G. Vincenti, Radiation driven acoustic waves in a confined gas, Physics of Fluids 10 (1967) 1365–1376. [5] H. Schmitt, Fortpflanzung Schwacher Unstetigkeiten beinichtli nearen Wellenausbreitungsvorganzen, ZAMM 48 (1986) 241–244. [6] V.D. Sharma, R. Shyam, L.P. Singh, Shock formation distance in a two dimensional steady supersonic flow over a concave corner in radiative magnetogasdynamics, ZAMM 67 (1987) 87–92. [7] L.P. Singh, R. Shyam, Flow behind an attached shock wave in a radiating gas, Astrophysics and Space Science 106 (1984) 81–92. [8] R. Shyam, L.P. Singh, V.D. Sharma, Steepening of wave in radiative magnetogasdynamics, Acta Astronautica 13 (1986) 95–100. [9] S.I. Pai, Magnetogasdynamics and Plasma Dynamics, Springer, New York, 1962. [10] S.I. Pai, T. Hsieh, A perturbation theory of an isentropic flow with radiative heat transfer, Z. Flug. 18 (1970) 44–50. [11] V.D. Sharma, R. Shyam, V.V. Menon, Behavior of finite amplitude waves in a radiating gas, ZAMM 61 (1981) 443–448. [12] D.C. Chou, S.Y. Maa, Propagation of weak shock waves in a vibrational nonequilibrium, nonuniform gasTransaction of the ASME: Journal of Applied Mechanics (1975) 564–568. [13] B.D. Pandey, D.C. Chou, Uniform solutions for characteristic and weak shock waves in reactive medium, Progress in Astronautics and Aeronautics (AIAA publication) 95 (1985) 610–628. [14] D.C. Chou, Weak shock wave propagation in a relaxing gas, Acta Astronautica 17 (1972) 625–632. [15] D.C. Chou, B.T. Chu, On the decay of weak shock waves in axisymmetric non-equilibrium flow, Journal of Fluid Mechanics 50 (1971) 355–367. [16] L.P. Singh, V.D. Sharma, R. Ram, Radiating flow past a slender body with an attached shock wave, ZAMM 69 (1988) 299–303. [17] H. Takigami, Z. Hasimato, Weak non-linear shock waves in steady two dimensional flow for a non-equilibrium gas along a curve wall, Journal of the Physical Society of Japan 53 (1984) 123–133. [18] J.-K. Xue, Nonplaner dust-ion acoustic shock waves with transverse perturbation, Physics of Plasma 12 (2005) 0114-1-5. [19] V.B. Baranov, K.V. Krasnobaev, M.S. Ruderman, On the model of the solar wind-interstellar medium interaction with two shocks, Astrophysics and Space Science 41 (1976) 481–490. [20] K.V. Krasnobaev, Nonisothemal wave motion in the HII region, Astrophysics and Space Science 274 (2000) 307–314.
Contents lists available at ScienceDirect
Acta Astronautica journal homepage: www.elsevier.com/locate/actaastro
Propagation of weak shock waves in non-uniform, radiative magnetogasdynamics L.P. Singh, S.D. Ram , D.B. Singh Department of Applied Mathematics, Institute of Technology, Bananas Hindu University, Varanasi 221005, India
a r t i c l e i n f o
abstract
Article history: Received 5 October 2009 Received in revised form 15 December 2009 Accepted 2 January 2010 Available online 6 February 2010
A problem of propagation of weak planer shock wave in radiative magnetogasdynamics is theoretically investigated. The gas is taken to be sufficiently hot for the effects of thermal radiation to be significant, which are of course treated by the optically thin approximation to the radiative transfer equation. The density of the medium ahead of the shock wave is assumed to be exponentially varying. A systematic perturbation scheme is used to obtain the analytical solution of the flow field. Also, an approximate solution near the front shock is determined. The effect of radiation and magnetic field strength on the shock wave propagation is examined. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Weak shock wave Radiating gas Magnetogasdynamics Perturbation scheme
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 great interest both from mathematical and physical point of view. In classical gasdynamics, the transfer of energy through radiation is usually neglected. However, on the account of the high temperature that prevail in many phenomena it is of interest to consider the effects of thermal radiation in gasdynamics. Since at high temperatures a gas is likely to be partially ionized, electromagnetic effects may also be significant. A complete analysis of such a problem should, therefore consist of the study of the gasdynamic flow and of the electromagnetic and radiation fields simultaneously. A number of problems relating to wave propagation with radiative effects and some considerations of the non-linear effects have been studied analytically in the
Corresponding author. Tel.: + 91 9451895174.
E-mail address: [email protected] (S.D. Ram). 0094-5765/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.actaastro.2010.01.002
past see, for example [1–9]. Pai and Hsieh [10] used the method of strained coordinates to study the problem of supersonic flows over weak compression and expansion corner and over biconvex airfoil. Problems of propagation of weak planar and non-planar shock waves in uniform and non uniform gases have been investigated using the perturbation scheme [11–13]. The propagation of weak shock waves near the nose of a body of revolution in real gas flows have been analyzed in [14–17], using the perturbation technique. Xue [18] investigated the problem of non-planar dust-ion acoustic shock waves with transverse perturbation and deduced the Kadomtsev–Petviashvili Berger equation that describes the dust-ion acoustic shock waves. Krasnobaev et al. [19] calculated the distribution of flow parameters in the region between shocks. Also, the possible influence on the model of galactic hydrogen neutral atoms penetrating into interplanetary medium is estimated. Krasnobaev [20] investigated the problem of propagation of the linear and non-linear wave in HII region by taking into account the relaxation processes to the temperature and ionization fraction. In the present paper we use the perturbation scheme to study theoretically the problem of propagation of weak shock waves in a non-uniform radiating and electrically conducting gas permeated by a transverse magnetic field.
ARTICLE IN PRESS L.P. Singh et al. / Acta Astronautica 67 (2010) 296–300
The shock wave is generated by the motion of a piston. The piston path may be prescribed by a given function eFðtÞ where t is time and e may be regarded as measure of the magnitude of the disturbance generated by the piston. The analytical solution of the flow field has been presented up to the first order of e. Also an approximate solution near the front shock is determined. The effect of thermal radiation and the magnetic field on the flow parameters is assessed. 2. Basic equations Assuming the electrical conductivity to be infinite and direction of the magnetic field to be orthogonal to the trajectories of gas particles, the basic equations for a one dimensional unsteady planar flow in radiative magnetogasdynamics, where the effects of thermal radiation are treated by the optically thin approximation to the radiative transfer equation, can be written down in familiar form [12] ð2:1Þ
~ @u~ @u~ 1 @ðp~ þ hÞ þ u~ þ ¼ 0; ~ r~ @x~ @x~ @t
ð2:2Þ
ð2:4Þ
~ 2 =2; the magnetic where r~ is the gas density, h~ ¼ uH ~ the magnetic field strength and u the pressure with H magnetic permeability, p~ the pressure, u~ the velocity, Q~ the rate of energy loss by the gas per unit volume through radiation and is given by 4 4 Q~ ¼ 4ksðT~ T~ 0 Þ;
ð2:5Þ
where k is the Planck mean absorption coefficient depending on the density r~ and temperature T~ of the gas, s is Stefan–Boltzmann constant and T~ 0 is uniform temperature of the body along which the flow is 4 envisaged. The additional energy gains 4ksT~ ; have 0
4 been added to the energy loss 4ksT~ due to the radiating gas so that infinite optically thin gas has no radiating boundary. The boundary wall with constant temperature T~ 0 has no loss or gain of energy from the radiating gas.
The initial and boundary conditions associated to the system are
@u @u 1 @ðp þhÞ þu þ ¼ 0; @t @x r @x
ð2:7Þ
@p @p gp @r @r þu þ ðg1ÞQ ¼ 0; þu @x @t @x r @t
ð2:8Þ
@h @h 2h @r @r þu ¼ 0: þu @x @t @x r @t
ð2:9Þ
dFðtÞ ; dt
at
x ¼ eFðtÞ;
ð2:10Þ
and unperturbed flow field is specified by
r0 ¼ ex ;
u0 ¼ 0;
h0 ¼ dð0 o d o 1Þ:
ð2:11Þ
3. Characteristic transformation The hyperbolic system of Eqs. (2.6)–(2.9) possesses four families of characteristics dx=dt ¼ u; u; the trajectory of fluid particle, and dx=dt ¼ u 7c; the outgoing and incoming wavelets. Here, c ¼ ða2 þb2 Þ1=2 ; is the magnetosonic speed
with
a ¼ ðgp=rÞ1=2 ;
r~ 0 ¼ r elx~ ;
u~ 0 ¼ 0;
h~ 0 ¼ h :
the
speed
of
sound
and
1=2
b ¼ ð2h=rÞ ; the Alfve´n speed. To study the nonlinear effects on the wave pattern, we introduce a new coordinate system ða; bÞ where a is constant along an outgoing wave if it originates from the piston path at a time t ¼ t ; the trajectory will be labeled as a ¼ t ; b is simply x. The transformation relation between ðx; tÞ; and ða; bÞ can be deduced immediately from the relation dt ¼ ta da þ tb db where subscripts a and b denote partial differentiation with respect to a and b respectively. Also, @a=@t ¼ 1=ta ; @a=@x ¼ ðtb =ta Þ In terms of new coordinates a and b, the equations (2.6)–(2.9) become ðu þ cÞtb ¼ 1;
c2 tb2 ¼ u2 tb2 2utb þ1;
ð3:1Þ
ra þurb ta ura tb þ rub ta rua tb ¼ 0;
ð3:2Þ
rua þ ruub ta ruua tb þ ðpb þ hb Þta ðpa þ ha Þtb ¼ 0;
ð3:3Þ
~ x~ ¼ eF~ ðtÞ:
The conditions of the undisturbed region ahead of the propagating shock wave generated by the piston motion are taken as p~ 0 ¼ p ;
ð2:6Þ
@h~ @h~ 2h~ @r~ @r~ ¼ 0; þ u~ þ u~ r~ @t~ @x~ @x~ @t~
at
@r @r @u þu þr ¼ 0; @t @x @x
p0 ¼ 1; ð2:3Þ
~ dF~ ðtÞ ; dt~
Consequently, Eqs. (2.1)–(2.4) are written in the following dimensionless form
u¼e
@p~ @p~ gp~ @r~ @r~ þ ðg1ÞQ~ ¼ 0; þ u~ þ u~ r~ @t~ @x~ @x~ @t~
u~ ¼ e
Introducing the flow variables in terms of their corresponding undisturbed quantities as 1=2 h~ r~ p p~ ~ ; p ¼ ; r ¼ ; h ¼ ; x ¼ lx; t ¼ lt~ r p r p 1=2 1=2 r r u ¼ u~ ; Q ¼ Q~ =lp : p p
Also, the initial condition at the piston becomes
@r~ @r~ @u~ þ r~ ¼ 0; þ u~ ~ ~ @ x @x~ @t
297
pa þ upb ta upa tb
gp ðr þ urb ta ura tb Þ þ ðg1Þta Q ¼ 0; r a ð3:4Þ
ha þuhb ta uha tb
2h
r
ðra þ urb ta ura tb Þ ¼ 0:
ð3:5Þ
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Similarly, Eq. (2.10) becomes 0
u ¼ eF ðaÞ;
at
b ¼ eFðaÞ;
ð3:6Þ
where the superscript prime denotes the total differentiation with respect to its argument. Consequently, the labeling of ða; bÞ coordinates gives t ¼ a;
at
b ¼ eFðaÞ:
ð3:7Þ
In addition to the above boundary conditions, the usual Rankine–Hugoniot conditions must be satisfied at each point of the front shock. If U be the shock velocity, these conditions are
Similarly making use of Taylor’s expansion, (3.7) yields the following set of boundary conditions
e0 : t0 ¼ a at b ¼ 0;
ð4:4Þ
e1 : t1 ¼ FðaÞt0b at b ¼ 0;
ð4:5aÞ
u1 ¼ F 0 ðaÞ at b ¼ 0:
ð4:5bÞ
5. Jump conditions at the shock wave
rðUuÞ ¼ r0 U;
ð3:8aÞ
Substituting the expansion (4.1) into the jump conditions (3.8a–d) and collecting the terms of like power of e, we get
rðUuÞ2 þ ðp þ hÞ ¼ r0 U 2 þ ðp0 þ h0 Þ;
ð3:8bÞ
e1 : r1 U0 r0 u1 ¼ 0;
ð5:1aÞ
hðUuÞ2 ¼ h0 U 2 ;
ð3:8cÞ
ðp1 þh1 Þr0 U0 u1 ¼ 0;
ð5:1bÞ
h1 U02 2h0 U0 u1 ¼ 0;
ð5:1cÞ
ðgp1 a20 r1 Þ þ ðg1Þð2h1 b20 r1 Þðg1Þr0 U0 u1 ¼ 0;
ð5:1dÞ
e2 : r2 U0 r0 u2 ¼ r1 u1 r1 U1 ;
ð5:2aÞ
ðp2 þh2 Þr0 U0 u2 ¼ r0 u1 U1 ;
ð5:2bÞ
h2 U02 2h0 U0 u2 ¼ 3h0 u21 2h0 u1 U1 ;
ð5:2cÞ
" # b20 ðg1Þ b2 ðg1Þ 2 2 2 ; a2 1 þ ðUuÞ þ ð g 1Þ 1 þ U þð g 1Þ ¼ a 0 2a2 a2 2a20 a20
ð3:8dÞ and Q ¼ Q0 . The position of the front shock is, of course, not known a priori. However, it must assume a form such that dx=dt ¼ U; at every point of the shock, which in the (a,b) plane takes the following form ta
da 1 þtb ¼ ; db U
ð3:9Þ
subject to the conditions, a ¼ 0 at b ¼ 0. 4. Characteristic perturbation Here, it is assumed that the fluid is weakly disturbed by the front shock generated by the forward piston motion. It is also assumed that the maximum fluctuation of the flow properties depend on the order of e. Solution of the system (2.1)–(2.4) can be constructed in the power series of e in (a,b) plane as
jða; bÞ ¼ j0 ða; bÞ þ ej1 ða; bÞ þ e2 j2 ða; bÞ þ ::::::::::::
ð4:1Þ
where j can be any dependent variable and subscript ‘‘0’’ denote free stream conditions. Using Eq. (4.1) in Eq. (3.1)–(3.5) and collecting terms of like powers of e, we have
e0 : c02 t02b ¼ 1;
ð4:2Þ
ðgp2 a20 r2 Þ þ ðg1Þð2h2 b20 r2 Þðg1Þr0 U0 u2 ðg1Þ ¼ r0 u21 þ ðg1Þr0 u1 U1 : 2
It may be noted here that the homogeneous parts of the systems (5.1) and (5.2) are identical. Therefore, the following condition for the system (5.2) to be consistent, we have U1 ¼
ðg þ1Þ ½6h0 þ g þ g2 u1 ; 4m
e : c1 t0b þ c0 t1b þu1 t0b ¼ 0;
ð4:3aÞ
r1a þu1 r0b t0a þ r0 u1b t0a r0 u1a t0b ¼ 0;
ð4:3bÞ
r0 u1a þðp1b þh1b Þt0a ðp1a þh1a Þt0b ¼ 0;
ð4:3cÞ
6. Solution of the problem 6.1. Zeroth order solution The zeroth order flow parameters are identical to the ones in the unperturbed region except t0 from Eq. (2.11) we have
r0
ð4:3dÞ h1a þ2h0 ðu1b t0a u1a t0b Þ ¼ 0:
ð4:3eÞ
h0 ¼ d;
r0 ¼ eb ;
u0 ¼ 0;
ð6:1:1Þ
the variable t0 may be obtained by integrating Eq. (4.2) and using Eq. (4.4) as t0 ¼
r ðp1a a20 r1a Þ þ gu1 t0a þ 16ðg1Þsk0 T04 t0a p1 1 ¼ 0;
ð5:3Þ
where, m ¼ ð2h0 þ gÞ.
p0 ¼ 1; 1
ð5:2dÞ
2
g1=2
ð1eb=2 Þ þ a:
ð6:1:2Þ
6.2. First order solution The solution of the first order quantities can be obtained from the Eqs. (4.3b)–(4.3e) by the method of Laplace transform as follows. Denoting the Laplace
ARTICLE IN PRESS L.P. Singh et al. / Acta Astronautica 67 (2010) 296–300
transform of a function fða; bÞ by FðzbÞ; i.e.
f ðz; bÞ ¼
Z
1
The equation of the front shock wave in the physical ðx; tÞplane can be obtained as follows: 2 F 0 ðaÞ U0 RðbÞ ; 1 ð7:2Þ t ¼ a þ 1=2 ð1eb=2 Þe 2 c0 m
fða; bÞeaz da;
0
and using the boundary conditions we have zr 1 þ r0b t0a u 1 þ r0 u1b t0a zr0 u 1 t0b ¼ 0;
ð6:2aÞ
zr0 u 1 þ ðp 1b þ h 1b Þt0a zðp 1 þh 1 Þt0b ¼ 0;
ð6:2bÞ
r zðp 1 a20 r 1 Þ þ 16ðg1Þk0 sT04 p 1 1 t0a þ gt0a u 1 ¼ 0; ð6:2cÞ ð6:2dÞ
Solving the above equations, we get the first order quantities in the transformed plane as p 1 ¼ G e2le
b=2
K0 ð2Oeb=2 Þ; 9 > > > > h 1 ¼ 2G h0 g e K0 ð2Oeb=2 Þ; > = 2leb=2 b=2 p1 ¼ G e K0 ð2Oe Þ; > > b=2 > u 1 ¼ G m1=2 g1 eðb=22le Þ K0 ð2Oeb=2 Þ; > > ; 1 2leb=2 b=2 h 1 ¼ 2G h0 g e K0 ð2Oe Þ; u 1 ¼ G m1=2 g1 e
1 2leb=2
r 1 ¼ G g1 eðb þ 2le
b=2 Þ
ð6:3Þ
K0 ð2Oeb=2 Þ;
and t 1b ¼
F 0 ðaÞ b 2 b=2 þ l ½6h þ g þ g exp ðe 1Þ : 0 d 4 2m1=2
ð6:4Þ
G ¼
ð6h0 þ g þ g2 Þ
Z
2
m
b 0
Z exp þld ðeZ=2 1Þ dZ: 4
Eliminating a from the Eqs. (7.2) and (7.3) and replacing b by x, we obtain the equation of the front shock in the physical (x,t)-plane.
The effects of the thermal radiation and the magnetic field strength on the first order shock wave solutions are clearly indicated in the system of Eqs. (7.1). It may be noted here that the radiation and magnetic field effects enter into the first order solutions through the radiative flux parameter L0 and the magnetic pressure term h0 . The case, L0 ¼ 0 and h0 ¼ 0 corresponds to non-radiative and non-magnetic case. The typical values of physical quantities involved in computation are taken as
g ¼ 1:67;
s ¼ 5:67 105 erg cm2 s1 K4 ;
k0 ¼ 1:38 1016 erg K1 ; and the temperature range taken is 1:0 105 Kr
where
O¼
RðbÞ ¼
8. Result discussion
K0 ð2Oeb=2 Þ; ðb=22leb=2 Þ
and using (7.2) in (3.9) we get the following equation of the shock in the ðx; tÞ-plane Z a e 02 e F ðaÞRðbÞ ¼ F 0 ðxÞ 1 1=2 eb=2 F 0 ðxÞ dx; ð7:3Þ 2 m 0 where
r0
zh 1 þ 2h0 ðu 1b t0a þ zu 1 t0b Þ ¼ 0:
299
1=2
z L0 þ U0 1 L0 =ðg1ÞU0 þ z 2m1=2
;
l ¼ z=m1=2 ;
ge2l F 0 ðaÞ; m1=2 K0 ð2OÞ
is the arbitrary constant, K0 is the modified Bessel function of second kind of order zero, ld ¼ L0 þ U0 =m1=2 , is the radiative decay length, B0 ¼ r0 a20 =ðg1ÞsT04 , is the Boltzmann number and L0 ¼ 16ðg1Þk0 a0 =B0 , is the radiative flux parameter.
T0 o 1:5 105 K. Figs. 1–3 show the profiles of the pressure, magnetic pressure and shock velocity distribution in the perturbed region. The effect of an increase in radiating flux parameter results in falling-off in gas pressure, magnetic pressure and shock velocity which is evident from Figs. 1–3. Further, an increase in magnetic field strength results in falling-off in the gas pressure and shock velocity. However, there is an increase in magnetic pressure with an added magnetic field. This is, of course, what is expected physically. It may also be noted that to attain the same value of shock speed as in h0 = 0 h0 = 0.2 h0 = 0.4
1.2
7. Approximate solution near the front shock
1.0 0.8 p˜ p*
Now taking the inverse Laplace transformation of the Eqs. (6.3) we have 9 > b > > p1 ¼ gm1=2 F 0 ðaÞexp þld ðeb=2 1Þ ; > > > 4 > > > > > 3b > 0 b=2 > u1 ¼ F ðaÞexp þ ld ðe 1Þ ; > = 4 ð7:1Þ b > > h1 ¼ 2h0 m1=2 F 0 ðaÞexp þ ld ðeb=2 1Þ ; > > > 4 > > > > > 3b 1=2 0 b=2 > r1 ¼ m F ðaÞexp þ ld ðe 1Þ : > > > ; 4
Λ0 = 0 Λ0 = 4 Λ0 = 7
0.6 0.4 0.2 0.0 0.0
0.5
1.0
1.5 x˜
2.0
Fig. 1. Variation of pressure versus distance.
2.5
3.0
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0.5 0.4
through the Figs. 1–3. Also, an approximate analytical solution near the front shock is obtained.
Λ0 = 0 Λ0 = 4 Λ0 = 7
h0 = 0.4 h0 = 0.2
Acknowledgments
h˜ p*
0.3
The second author acknowledges the financial support from the UGC, New Delhi, India, and the third author acknowledges the financial support from the CSIR, India, under the JRF scheme.
0.2 0.1 0.0 0.0
0.5
1.0
1.5 x˜
2.0
2.5
3.0
Fig. 2. Variation of magnetic pressure versus distance.
Λ0 = 0 Λ0 = 4 Λ0 = 7
h0 = 0 h0 = 0.2 h0 = 0.4
4
U
3 2 1 0 0.0
0.5
1.0
1.5 x˜
2.0
2.5
3.0
Fig. 3. Variation of shock speed versus distance.
non-magnetic case, the general level of radiation must be greater. In other words we can say that the effect of magnetic field interaction is to diminish the effect of radiative transfer.
9. Conclusion A systematic perturbation scheme is used to study the problem of the propagation of weak planar shock waves in radiative magnetogasdynamics. The effect of coupling between the radiative transfer and magnetogasdynamics phenomena on the flow field is analyzed. The analytical solution of the flow field has been obtained up to the first order e. The effect of radiative transfer and the magnetic field strength in the flow field variables is illustrated
References [1] G.B. Whitham, Linear and Non-Linear Waves, Wiley, New York, 1974. [2] V.D. Sharma, Development of jump discontinuities in radiative magnetogasdynamics, International Journal of Engineering Science 24 (1986) 813–818. [3] W.J. Lick, The propagation of small disturbances in radiating gas, Journal of Fluid Mechanics 18 (1964) 274–288. [4] H.R. Long, W.G. Vincenti, Radiation driven acoustic waves in a confined gas, Physics of Fluids 10 (1967) 1365–1376. [5] H. Schmitt, Fortpflanzung Schwacher Unstetigkeiten beinichtli nearen Wellenausbreitungsvorganzen, ZAMM 48 (1986) 241–244. [6] V.D. Sharma, R. Shyam, L.P. Singh, Shock formation distance in a two dimensional steady supersonic flow over a concave corner in radiative magnetogasdynamics, ZAMM 67 (1987) 87–92. [7] L.P. Singh, R. Shyam, Flow behind an attached shock wave in a radiating gas, Astrophysics and Space Science 106 (1984) 81–92. [8] R. Shyam, L.P. Singh, V.D. Sharma, Steepening of wave in radiative magnetogasdynamics, Acta Astronautica 13 (1986) 95–100. [9] S.I. Pai, Magnetogasdynamics and Plasma Dynamics, Springer, New York, 1962. [10] S.I. Pai, T. Hsieh, A perturbation theory of an isentropic flow with radiative heat transfer, Z. Flug. 18 (1970) 44–50. [11] V.D. Sharma, R. Shyam, V.V. Menon, Behavior of finite amplitude waves in a radiating gas, ZAMM 61 (1981) 443–448. [12] D.C. Chou, S.Y. Maa, Propagation of weak shock waves in a vibrational nonequilibrium, nonuniform gasTransaction of the ASME: Journal of Applied Mechanics (1975) 564–568. [13] B.D. Pandey, D.C. Chou, Uniform solutions for characteristic and weak shock waves in reactive medium, Progress in Astronautics and Aeronautics (AIAA publication) 95 (1985) 610–628. [14] D.C. Chou, Weak shock wave propagation in a relaxing gas, Acta Astronautica 17 (1972) 625–632. [15] D.C. Chou, B.T. Chu, On the decay of weak shock waves in axisymmetric non-equilibrium flow, Journal of Fluid Mechanics 50 (1971) 355–367. [16] L.P. Singh, V.D. Sharma, R. Ram, Radiating flow past a slender body with an attached shock wave, ZAMM 69 (1988) 299–303. [17] H. Takigami, Z. Hasimato, Weak non-linear shock waves in steady two dimensional flow for a non-equilibrium gas along a curve wall, Journal of the Physical Society of Japan 53 (1984) 123–133. [18] J.-K. Xue, Nonplaner dust-ion acoustic shock waves with transverse perturbation, Physics of Plasma 12 (2005) 0114-1-5. [19] V.B. Baranov, K.V. Krasnobaev, M.S. Ruderman, On the model of the solar wind-interstellar medium interaction with two shocks, Astrophysics and Space Science 41 (1976) 481–490. [20] K.V. Krasnobaev, Nonisothemal wave motion in the HII region, Astrophysics and Space Science 274 (2000) 307–314.