Acta Astronautica Vol. 13. No. 3, pp. 95-100. 1986 0094-5765/86 $3.00 + .DO Printed in Great Britain. © 1986 Pergamon Press Ltd.

STEEPENING OF WAVES IN RADIATIVE MAGNETOHYDRODYNAMICS

RADHE SHYAM, L. E SINGH and V. D. SHARMA Applied Mathematics Section, Institute of Technology, Banaras Hindu University, Varanasi 221005,

India

(Received 3 January 1985)

Abstract--Singular surface theory is used to determine the modes of wave propagation and development of discontinuities at planar and cylindrical wave fronts. It is investigated as to how the effects of thermal radiation and magnetic field strength influence the steepening and flattening of wave fronts. Although the effect of magnetic field, whether axial or azimuthal, is to increase the shock formation distance, it, unlike the effect of thermal radiation, cannot offset the tendency of a compression wave-head, carrying a jump discontinuity (no matter how weak initially), to grow into a shock wave after a finite running length. In an optically thick gas with negligible radiation pressure and energy, an increase in the radiative flux reduces the shock formation distance; this is in contrast to the corresponding result for an optically thin gas where an increase in the radiative flux causes shock formation distance to increase. However, numerical calculations show that in a temperature range 9 × 104 < T < 2 × 105, the combined effect of radiation pressure, energy and flux is to delay the formation of a shock wave, It is found that the decaying of expansion waves is enhanced (slowed down) due to the presence of thermal radiation (magnetic field).

1. INTRODUCTION

It is well known that weak discontinuities of a quasi- linear hyperbolic system of equations propagate along characteristics. When the magnitudes of the jumps in the derivatives of the flow variable gradients across the wave front become unbounded, the weak discontinuity be- comes a strong discontinuity and the propagation then ceases to be along the leading characteristic. The occur- rence of such an unbounded jump discontinuity is iden- tified with the physical process of steepening or breaking of a wave[l]. A general method for the study of the development of jump discontinuities in quasi-linear hy- perbolic systems has been applied by Whitham[ 1 ] to study the breaking of finite amplitude shallow water waves, flood waves and tidal waves with planar geometry. The growth and decay of weak discontinuities headed by a wave front of arbitrary shape in a radiative gas has been studied in [2]. The breaking of planar as well as non- planar waves in magnetohydrodynamics[3] has been in- vestigated and remarkable differences have been noted with regard to the influence of magnetic field on the process of steepening or flattening of wave fronts.

In the present paper, following the singular surface theory[4], we shall study the development of a jump discontinuity in plane and cylindrically symmetric flow of a plasma with thermal radiation. The plasma is as- sumed to be an ideal gas with infinite electrical conduc- tivity and to be permeated by a magnetic field orthogonal to the trajectories of gas particles. In the cylindrical case, the magnetic lines of force can be straight lines parallel to the axis of symmetry or concentric circles with centres on the axis of symmetry. The effects of radiation are treated by optically thick approximation to the radiative transfer equation. The transport equations for the jump

discontinuities at the wave-heads which propagate with magnetosonic speed, satisfy a Bernoulli type equation. A fairly complete analysis of such an equation modifying and generalizing several known results, quoted in 15], has been performed in [3,6-8].

2. BASIC EQUATIONS

The equations which describe the one-dimensional mo- tion in radiative magnetogasdynamics, where the effects of thermal radiation are treated by the optically thick approximation to the radiative transfer equation, and the applied magnetic field, being either axial or azimuthal in a cylindrical symmetry, is orthogonal to the trajectories of gas particles, can be written down in the familiar form[9,10]:

d p + p + = 0 , dt

( ])

P d~ + a~ p + p~ + + x = O, (2)

dH (du m(I - n)u) - - + H + = 0 , dt \0x x

d ( ER P + PR) d p ~ e + --9 + P - ~ ( P + PR)

(3)

Ox x ax (4)

p = p . ~ T , (5)

95

96

where p, p, u, T, e and ,<' are, respectively, the gas pressure, density, velocity, temperature, internal energy and the universal gas constant. PR and ER are, respec- tively, the radiation pressure and radiation energy den- sity, which are connected by the relation

ER = 3p,~ = ( 4 o ' / c ) T ~,

with o- as the Stefan Boltzman constant and c the velocity of light. K is the radiative thermal conductivity given by:

K = 16~T3/(3e0,

R. SHYAM el al.

and A and A are the quantities defined on 5Z. The square brackets [ ] stand for the value of the quantity enclosed immediately behind the wave Y., minus its value just ahead of the wave 2~; we shall use the suffix o to denote a quantity in the region ahead of E.

The gas, in the medium ahead of the wave front, is assumed to be at rest, to have a constant density p,,, a uniform pressure p<, and to be permeated by a transverse magnetic field, which when axial is of constant magni- tude H<,, but when azimuthal is of magnitude l,,/2"rrx, where I,, is the constant axial current.

The law of conservation of energy across Y_, yields:

with a as the Rosseland mean absorption coefficient de- pending on 9 and T. H is the transverse magnetic field, which in a cylindrically symmetric (m = 1) motion is either axial (n = 0) or azimuthal (n = 1); for a plane (m = 0) motion, we have n = 0.

d l d t = (OlOt) + u(O/Ox)

denotes material time derivative with t as time and x as distance, which is either axial or radial as the case may be.

Equation (4), in view of (1) and (5), can be conven- iently transformed into

IOT/Ox] = 0. (9)

Evaluating eqns (1), (2), (3) and (6) on the inner boundary of ]~ and using (7), (8) and (9), we get:

G~ = p,,X, (10)

4p<,R.~ P<,GX - (1 + 4R,,)~ + - IxH,,'q = 0, (11)

Po

G-q = H<,R, (12)

(1 + 1 2 ( - / - I )Rp) ( -G{ + p,,Fh)

= (y - 1)0, (13)

d t ~xx + 1 + 1 2 ( ` / - I)R,,

× K Ox~_ + ~ + - - , (6)

where R I, = (PR/P) is the radiation pressure number and F is the modified heat exponent given by

F = y + 16(`/ - I)R~, I + 1 2 ( ` / - 1)R,,"

where

= [O2TlOx:],

= [OplOx],

= [~plOx],

X = [Ou/Ox] and ~ = [OH/Ox].

If we differentiate eqn (5) with respect to x and evaluate the resulting equation on the inner boundary of ~,. we obtain, on using (7) and (9), that

3. VELOCITY OF WAVE PROPAGATION

Let x = X(t) be the equation of wave front 2~ across which the flow variables u, p, p and H etc. are essentially continuous but discontinuities in their derivatives are per- mitted. In this case, the geometrical and kinematical com- patibility conditions of first and second order for a sin- gular surface derived by Thomas[4] reduce to

= a ~ , (14)

where a<, = (p,,/p,,)'"- is the isothermal speed of sound. Equation (11), on using (10), (12) and (14), yields G

= -+c<, where c,, = (a~, + b?,) ~2 is the magnetosonic speed with b,, = (IxH~/p,,) ~'-" as the Alfv6n speed. For an advancing wave, we have G to be positive, and thus we have

[aZlOx] = A ,

la:Z/ax-'l = A,

[OZlat] = - G A

[a'-Ziaxat] = G ( ~ - - f~)

G = c,. (15) (7)

Equations (10), ( I l L (12) and (14), on using (15), (8) yield

where Z may represent any of the flow variables u, p, p and H etc.; G = d X / d t is the speed of propagation of E

X = ' " - ' ~ = '"' ~ = " " (16)

Steepening of waves

4. T R A N S P O R T E Q U A T I O N

If we differentiate eqns (1), (2) and (3) with respect to x, evaluate behind the wave E, and then make use of (7), (8), (9), (15) and (16), we get

d/; _ c,,~ + p,/~ + 2X{ + mp,,?~ C. ~ X

dR

= 0, (17)

1 6 o ' T ~ - + ~ + - - 0 + IXH,,~q + Ix'q: + 21X~l

3c

= 0 ,

d'q c , ~ - c,,~l + H,,h + 2Mq + h

( m(1 - n)H,,) × 2H,~, +

X

97

5. R E S U L T S A N D D I S C U S S I O N

Let S,, be the value of S at x = x,,, then the above transport equation can be written in the fol lowing di-

mensionless form:

dII + ~(1 + 4Rp)'-

d r I. 2e_Q

m ( n ( C - - 1 ) ) } n ~2

(3e-' - l)to [I 2 = 0,

2E 4 (23)

(18) where r = x/x~, is the distance parameter , H = S/So is the parameter for the jump discontinuity S, R r, = PR/P is the radiation pressure number defined earlier, ~ = c,,/ao = (1 + (by,/a~,)) "2 is the Alfv6n number , to = S,,r,,/a~, is a d imensionless measure for the strength of

(19) initial j ump discontinuity S,,, and

where the quantit ies ~., ~, ~ and fi are, respectively, the values of the second order derivatives of u, p, p and H with respect to x evaluated just at the rear of the wave

Similarly, eqn (5) yields the fol lowing relation

= - a ; O / P . . . . . (20)

If we denote [au/Ot] by S, then since

Q = (aoKo/.~p,,x,,)=- 16"//{3("/ - 1)[36.}

is a measure of the importance of radiative flux with 6~ = ax,, as the dimensionless Bouger number and [3 = a,,p,,h,,/ (6/',4,) as the Bol tzman number representing the ratio of convect ive energy flux to the black body heat flux; here ho = "yp,,/p,,('Y - 1) is the specific enthalpy of the gas.

When the applied magnet ic field is axial (n = 0), like other parameters e is also a constant , and eqn (23), on integration, yields

[Ou/Ot] + c.[Ou/Oxl = O,

we have

S = - c , , M (21)

where h is same as defined earlier i.e. h = [Ou/Ox]. Eliminat ing h, ~. /; and ~ from (17) - (20) and using (16) and (21), we obtain

dS + S {p, , .~ '( l + 4Rp) 2

d-S t

+ Z-~ 1 -

---; --Z = O, - 1 . 5 - 2c; , c ;

(22)

where use has been made of eqn (2), which, in fact, implies that

n~H~, ~H Jl,,, -

X

Equation (22) is the required transport equation for the jump discontinuity S = [Ou/Otl which we have been seeking.

r ,,,2 exp ( - A ( r - 1)/e) l l = (24)

1 - xtol(m, e, A, r)

where

A = (1 + 4R,,)2/(2Q), X = (3~-' - 1)/(2~ 4)

and

fl r l(m, e, A, r) = z ='''2 exp ( - A ( z - 1)/e)dz.

Since A and X are positive and l(m, ~, A, r) is a monotonic increasing function of r, it is evident from (24) that the behaviour of H with r will depend on the sign of S,, and hence that of to. Therefore, when S,, is negative (i.e. an expansion wave front with to < 0), [1 will increase mon- otonically to zero with distance r. If S,, > 0 (i.e. a compress ion wave front with to > 0) and has magni tude greater than to, given by to, = (xl(m, ~)) i, then the jump discontinuity [1 at the wave-head will steepen up into a shock wave after a finite running length r~ given by I(m, r~) = (Xto) '. However , i f to = to,, the wave nei ther grows into a shock nor it decays completely; in fact H approaches a constant value independent of r as r ~ ~ . Finally if to > 0 and has a magni tude less than to,, then H at the wave-head decreases monotonical ly to zero. We also notice that (Or,/OA) > 0 and (OrflOe) >

98

0, which mean that an increase (decrease) in A or e causes shock formation distance to increase (decrease).

When the temperature is not very high, radiation pres- sure and radiation energy can be neglected]l 1,12] and thus A = I /2Q, which shows that an increase in Q causes A to decrease. Thus, in an optically thick gas, with negligible radiation pressure and energy, an increase in the radiative flux causes the shock formation distance to decrease as one might expect: it is to be noted that this result is in contrast to the corresponding result for an optically thin gas l l3 l , where an increase in the ra- diative flux causes shock formation distance to increase. At high temperatures, the radiation pressure and energy become significant, but owing to a complex dependence of the absorption coefficient ¢x on density and tempera- ture, it is rather difficult to analyse analytically the effect

R. SHYAM Cl (ll.

of A on the shock formation distance. However, the true character can be computed using an approximate rela- tionship of the form c~ = Kp'-T ~, which has a reasonable validity for a specific range of temperature] 14]; the value of K depends on corresponding range, Following[ 12,14], one finds that for densities of the order of the ambient value at mean sea level, viz. p = 1.29 × 10 ~ g / cm ' , K - 7 x 10 t ' f o r 9 x 10 a < T < 2 x 10 ~.Numerical integration of eqn (23) has been carried out for planar (m = 0) and cylindrical (m = 1 } motion with axial mag- netic field (n - 0) for the above mentioned values of o~, K and p in the given temperature range using the Runge -

Kutta method. The calculations are carried out for dif- ferent values of Alfv4n numbers with y = 5/3, and m = 0.5; the value of Stcfan's constant (r, the gas constant ,,4', and the velocity of light c are taken to be 5.735 x

8"0

T • v

6"0 W

5"0 Z 0 U1

m 4.0 IAI n , I3-

~E o 3"0 rY 0

" 2 '0

1"0

!I I) . . . . Shock formation ,l il Cylindrical waves i I . . . . Plane w a v e s

iI 11

/ / /7'

/ /

~ ' ~ ~ - ~ ' - - - ~ C 1:1 : 1"8x105 A= 0'005, E :1"0

~ C z :T =9'5x104, A=0"045, E~:I'0

~ C 3 : T = 1"0x105 A=0"005, E=2"0

C/~ :T = 9"5x10~ A= 0"045, E=2"0

C5:T = 5"0x10{ ^= 0"430, C==3"3 --._~.__.. . . . . . : . _ . ~ / . ~ . C.6:T 5"0x 104 ̂ = 0"430, 6=2"0

0 , 0 l ~ , ~ i r ,

11-0 5 " ~ ,s'___9_o ~20--0

> " . - ~ ' . ~ - ~ - ~ - _-

o , L I I I I i;~,"/7"~'~<~-~'<"---~ ~ ~ ~ : t : ~o ~,o ~, ̂ --o-~3o,~:2o

, , , - o , lo; ^ : o

I III1~!/I// ~---- E6:T = 1.8 x 105A= 0.005, E=2.0 ,°!F Fig. I. Effects of thermal radiation (A) and magnetic field strength (e) tm the steepening and flattening of plane

and cylindrical waves for y = 5/3 and to = 0.5.

Steepening of waves

10 ~ergs /cm- 'sec ' K ', 2.882 × 1 0 O e r g s / g d e g K a n d

3.0 x 10 ~'' cm/sec , respectively. Figure 1 exhibits some of the solution curves o f e q n (23); the parameter A, which is a measure of the importance of thermal radiation is found to be an increasing function of T in the above range. The flattening of expansion waves is illustrated by curves E,-E,,, which show that the decaying of ex- pansion waves is enhanced by an increase in A whereas it is s lowed down by an increase in the Alfvdn number e. The steepening and flattening of compression waves are illustrated by solution curves C,-C,,. Curves C,-C~ correspond to the situation when to > co, and show that in each case after a certain running length, a vertical tangent is reached indicating the occurrence of a shock wave. Notice that the effects o f magnetic field strength and the thermal radiation are to increase the shock for-

99

mation distance; curves C~ and C,, are representative for the case co < to .

When the applied magnetic field is azimuthal (n = 1, m = 1), ~ is a function of distance r and is given by

= (I + (6-'/r-')Y -', where 6-' = p, Ij,/(4w2a~,X~, po) is a dimensionless measure of the strength of constant axial current L,. In this case also the behaviour is similar to the earlier case, i.e. all expansion waves decay whereas not all compression waves grow into shock waves; in fact there exists a critical value of the initial jump dis- continuity such that compression waves with initial dis- continuity greater than this critical value steepen into a shock after a finite running length whereas waves with initial discontinuity less than this critical value flatten out with distance r. Figure 2 exhibits some of the solution curves o f e q n (23) for m = I and n = I; the calculations

8"0

l 7"0

LU > 6"0

Z o 5"0 m

t./) i l l

n- 4 '0 O .

I[ o

3-0 r e

O

I= 2.0

1'0

. . . . Shock formation

CI:T = l'8x105, A=0"005,1~:0"5

C2:T : g'5x 10,4A: 0"045,8:0"5

c~:r: 1.8 xlO,S^= o.0o5, S:3-0

C4:T : 9"5x104, A:0"045,6:3"0

C5:T : 5"0x104, A: 0"430, ~:3"0

Ce: T : 5"0x104, A : 0"430, ~:1"0

o . o l I I I'0 5"0 r 100 15"0

-0.I

>m -0'2

- 0 ' 3 g

-0"4

- o.5 ~ : r : 5.oxlo ~, .A: 0 -430 , 5 :1 .0

/ \ \ \ "E2 : r : o 3o. :3.o

o: \ \ \ ,,: 0.0,5, :0.5 " -o., s:o.5

\E 5 :T : 95x104,A : @045, 6:3'0 -0"8 I ~E 6 :T : 1"8x105A:0"005, 6:3"0

-0"9

-1"01

Fig, 2. Effects of thermal radiation (A) and the axial current strength (8) on the steepening and flattening of cylindrical waves for 3~ = 5/3 and ~0 = 0.5.

100 R. SI-IVAM et al.

are carried out for different values of the current strength 4. T. Y. Thomas, The general theory of compatibility condi- 8 and for a similar range of other parameters a , K, 9, T

etc. as before. In this case also the effect of therrnal

radiation is to enhance (slow down) the decaying (steep-

ening) o f expans ion (compression) waves. However , both

the decaying of expans ion waves and the s teepening of

compress ion waves are s lowed down by an increase in

the axial current strength 8. This feature is somewhat

different f rom other effects such as geometrical conver-

gence (divergence) and dissipation which increase one

rate but d iminish the other.

Acknowledgement--One of us (R. S.) is grateful to the Uni- versity Grants Commission tbr the award of a Research Asso- ciateship.

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