nonlinear breaking of waves in an electrically conducting and radiating gas

Journal of Applied Mathematics and Physics (ZAMP) 0044-2275/82/006772-11 $ 3.70/0 Vol. 33, November 1982 �9 Birkh~iuser Verlag Basel, 1982

Nonlinear breaking of waves in an electrically conducting and radiating gas

By Radhe Shyam and V. D. Sharma, School of Applied Sciences, Institute of Technology, Banaras Hindu University, Varanasi, India

1. Introduction

The behaviour of weak discontinuities in radiating gas has been discussed by Srinivasan and Ram [1] and Ram [2]. In a recent paper, Shankar and Prasad [3] have extended the paper [1] to include the unsteady motion ahead of the wave surface. In the above references [1 - 3], apart from the fact that the discussion of the growth equation is incomplete, the approximation to the radiative transfer equation is too strong and deals with only simple cases when the gas is optically thick or thin. In [1] and [3] the ratio of mean free path of radiation to a characteristic length in the flow field is assumed to be extremely small so that the energy equation does not contain the radiative flux term and thus the basic equations and the analysis are parallel to that of Thomas [4]. In [2] the gas is assumed to be transparent with constant absorption coefficient. It should be noted that the transparent approximation for the radiative transfer equations with constant absorption coefficient is rarely satisfied for high temperature gases: Furthermore the neglect of time derivative in radiative transfer equations suppresses one mode of wave propagation excited by radiation and therefore the exact behaviour of waves in radiation gasdynamics is not fully understood as is evident from references [1 - 3].

The present paper uses a more general differential approximation for the equations of radiative transfer in contrast to the transparent or Rosseland limits and the situation is considered over the entire optical depth range from the transparent limit to the optically thick limit, and the absorption coefficient is taken to be a function of density and temperature. Since at high temperatures a gas is likely to be fully or partially ionized, electromagnetic effects may also be significant. One is thus led to study the interaction of radiative and electromagnetic effects that may arise in the problems of solar photosphere, rocket re-entry and elsewhere. The present work takes into account the electromagnetic effects; the equations clearly show the existence of radiation induced waves which are followed by modified magnetogas-

Vol. 33, 1982 Nonlinear breaking of waves 773

dynamic waves. It is found that the nonlinearity in the governing differential equations contributes nothing towards radiation induced waves; these waves are ultimately damped and the formation of a discontinuous front is not possible from a continuous flow. But in the case of modified magnetogasdynamic waves the nonlinearity in the governing equations does play an important role. The influences of thermal radiation, the magnetic field intensity, the finite electrical conductivity and that of the initial wave front curvature are discussed on the nonlinear breaking of modified magnetogasdynamic waves.

2. Fundamental equations

The equations governing the three dimensional unsteady flow of an electrically conducting and radiating gas are

dQ dt + Oui, i = O, (2.1)

dui Q - ~ + (P + PR),i + 11 Hj (Hj, i - Hi, j) = O, (2.2)

dHi dt I-Ij. uij + Hi uj,: - (11 a)-I Hi, jj = O, (2.3)

d ( p +ER 1 j2 0~-'~ 0 ( v - I) ---~-] + (p +PR) ui, i+ qi, i---=a O, (2.4)

Hi, i = 0, (2.5)

J i= eij~ Hk, j= a(Ei +11eijk uy Hk), J2 = Ji Ji , (2.6)

where d/d t = ~ /~ t + ui ~ /~x i denotes material time derivative; ui, qi, Hi, J~, and E; are respectively the components of velocity, radiative flux, magnetic field, current density and electric field; p denotes pressure, PR the radiation pressure, ER the radiation energy density, 0 the density, 11 the magnetic permeability, a the electrical conductivity, v the specific heat ratio and e,.jk the well known permutation tensor. The summation convention over repeated indices is employed, and a comma followed by an index (say 0 denotes a partial derivative with respect to a space variable xi. The range of Latin indices i, j', k is taken to be 1, 2, 3.

Within the differential approximation the equations of radiative transfer may be written as the pair [5]

eER 8--7-- + qi, i = - ke (c ER - 4aR T4), (2.7)

774 Radhe Shyam and V. D. Sharma ZAMP

1 Oqi C c at ~_3ER, i = _ ~ e q i I";- (2.8)

where/7p is the absorption coefficient depending on density and temperature, aR is the Stefan's constant and c is the velocity of light. Further, within the order of this approximation the radiative pressure is one third of the radiation energy density.

The equation of state of the gas is taken to be of the form

p = Q~T, (2.9)

where ~ is the gas constant. Equation (2.4) with the help of (2.1) yields

dpR ( v - 1) j2 dp ~- Qa z Ui, i q- 3 (V -- 1) T + (V -- 1) qi, i = 0, (2.10) dt o"

where a = {(vp + 4(v - 1)pR)/O} I/2 is the effective acoustic speed.

3. Modes of wave propagation

A moving singular surface 27 across which the flow parameters p, ~, Ui,

Hi, q; and pR etc. are essentially continuous but the discontinuities in their derivatives are permitted is called a weak wave. Suppose that S is given by f (xi , t )= 0 and that we denote by ni the unit normal v e c t o r f i / l g r a d f l and by G = - ( S f / ~ t ) / ] g r a d f ] the normal speed of advance of the surface. For definiteness, we require that the description of the surface 27 be such that G is always positive. Then the normal n; always points in the direction of propagation of 27. The jump in a flow quantity Z, across 27, is denoted by [Z] = Z1 - Z 0 , where Z0 is the value of Z immediately ahead of the wave front, and Z~ is the value of Z immediately behind it. The first order geometric and kinematic compatibility conditions derived by Thomas [6] reduces to

[Z i] = Bn;, I ~ t l = - G B , (3.1), (3.2)

where Z may represent any of the flow variables p, Q, ug, H;, qi and PR etc. and the scalar quantity B = [Z,i]ni is defined over 27.

Forming jumps, across 27, in Eqs. (2.1), (2.2), (2.3), (2.7), (2.8) and (2.10) and using (3.1), (3.2), the second order compatibility conditions [6] and the condition that the state ahead of 27 is at rest, we get

G ~ = Qo 2i ni , (3.3)

OoG 2i = ~ni + Oni, (3.4)

VoI. 33, 1982 Nonlinear breaking of waves 775

0i = (u (Hi 2j nj - I-/. 2;),

3 G O = ei ni,

G ,~i = c 2 0 ? l i ,

G ~ - ~o a~ 2i ni + (v - 1) (3 G 0 - ei ni) = O,

(3.5)

(3.6)

(3.7)

(3.8)

where the subscript 0 denotes a value at the wave head and the quantities = [p,i]ni, ( = [O,i]ni, 0 = [pR, i]ni , ai = [qi,.i]nj, 2i = [ui, j ] n j , rli = [Hi, j ] n j ,

rli = [Hi, i k ] n j n ~ and Hn = H i n i are defined on N. In getting Eqs. (3.3) to (3.8) we have used the result [ H i j ] n j = 0 which follows at once from Eqs. (2.5), (2.6) and (3.1) on using the fact that the normal component of magnetic field vector and the tangential component of electric field vector are continuous across the discontinuity surface 2.

Equations (3.3), (3.4), (3.6), (3.7) and (3.8) constitute a set of nine homogeneous equations in nine unknowns 2;, el, r ( and 0. Hence the necessary condition for the existence of non-trivial solutions of (3.3), (3.4), (3.6), (3.7) and (3.8) is that the determinant of the coefficient matrix must vanish. This yields

G = O, G = +__ C / 3 1 / 2 , G = -t- ao .

Since a frame of reference can always be found with G 4= 0, the case G = 0 may be rejected as having no physical interest. For an advancing wave surface, we shall take G to be positive. We thus find that there are two waves present in the gas, one which propagates with the speed

G = c/31/2 (3.9)

has attributes which are basically due to radiation and it may be identified as radiation induced wave. The other which propagates with the speed

G = a0 (3.10)

is essentially a modified magnetogasdynamic wave.

4. Behaviour of radiation induced waves

Let us consider discontinuities in the first derivatives of flow quantities propagating along a radiation induced wave X. The flow ahead of this wave remains undisturbed and it is convenient to take it to be of uniform state; the Eqs. (3.3), (3.4), (3.6), (3.7) and (3.8), using (3.9), yield

0 = e/31/2 c, (4.1)

( = 31/2 e / c 3 {1 -- ( 3 a ~ / c 2 ) } , (4.2)


= 31/2 a~ e/c 3 { 1 - (3a~/c2)}, (4.3)

2 = e/O0 c 2 {1 - (3a~/g2)}, (4.4)

where 2 = 2~ ni and e = ei ni. If we differentiate Eq. (2.9) with respect to x~, form jumps across 27, and

multiply the resulting equation by n~, and make use of the relations (3.1), (3.2), (4.2) and (4.3), we get

Z = 31/2 (a~ - a2o) e / ~ eo c 3 {1 - (3 a2o/c2)}, (4.5)

where Z = [7"/] ni and aro = (p0/o0) 1/2 is the isothermal speed of sound. When Eq. (2.8) is differentiated partially with respect to t and use is

made of (2.7), we get

3 02qi 6kp ~qi qj,.ii+3k'2pqi+ 16/~, aR T 3 T i

c 2 ~t 2 c ~t

+ 3q~ [ ~F.e ~ OF.? c I ~e ~----~q ~T

(e Tp eEp ) + (4an T 4 - 3cpR) \ W Q,i + " - ~ Z,i = O.

If we multiply this equation by n; and form jumps across 27, we find, on using the second order compatibility conditions [6], and the relations (3.1), (3.2), (3.9), (4.2), (4.4) and (4.5), that

62 6--7 + (As -A2) 2 = 0, (4.6)

where 6/~St is the Thomas [6] time derivative along an orthogonal trajectory of 27, & = (/~? - (O/31/2))c and

{1 - (3d/c2)) [ A2= 2-~ ~00 ~ [ 16kp aR

3 2 R To(ao a~-o) - (]/-3-cqn+cERo- 4aRoT 4)

} o + ] o ( - t

s and qn are respectively the mean curvature of 27 and the normal component of the flux vector defined as 2 0 = g ~ b~a and qn = qinz with g ~ and b~ B being the first and second fundamental forms of 27; the range of Greek letter indices, which refer to the surface coordinate y~, is taken to be 1,2.

Equation (4.6) is the differential equation governing the propagation of the discontinuity 2 associated with a radiation induced wave. The behaviour


of the solution of (4.6) can, of course, be readily established. The mean curvature/2 at any point of the wave surface 27 has the representation [7]

[2 = (f2o - K o G t ) / ( 1 - 2f2oG t - KoG2t2) , (4.7)

where/20 and K0 are respectively the mean and Gaussian curvatures of 27 at t = 0, and G is the constant speed of propagation of the wave surface which for radiation induced wave is equal to c/3 1/2.

Equation (4.6), on using (4.7), can be integrated to yield

( c isC2dt), (4.8) 2 = )~ exp (-/Sp c t) exp ~ 0

where ;to is the value of 2 at t = 0. In (4.6) the term Az, which contains c 2 in the denominator, has been neglected in comparison with A1 in presenting the result (4.8).

It is evident from Eq. (4.8) that 2 ~ 0 as t ~ ~ , i.e. the radiation induced waves are damped (i.e. there occurs no breakdown) and the formation of a front, carrying discontinuities in the flow variables, is not possible from a continuous flow. From the expressions (4.1) to (4.5), it follows that the quantities 2, ~, r and Z are small compared with e and 0 as it should be in radiation induced waves.

5. Breaking or nonbreaking of modified magnetogasdynamic waves

Now consider discontinuities in the first derivatives of flow quantities propagating along a modified magnetogasdynamic wave X. In general, the flow ahead of this wave will be disturbed by radiation induced waves. But We have just seen that for radiation induced waves 2, ~, ~ and X are small compared to e and 0 and hence the main effects can be derived for the special case in which u~ = 0, and p, Q and T are constant ahead of the wave 22. For a modified magnetogasdynamic wave, Eqs. (3.3) to (3.8), on using (3.10), yield

= 00 a0 ,~, a0 ~ = 5o 4 , (5.1) , (5 .2)

fli ni = 0, e = 0 = 0. (5.3), (5.4)

If we differentiate Eqs. (2.2), (2.7), (2.8) and (2.10) partially with respect to x~, take jumps across 27, and multiply the resulting equations by n~, then using (2.9), (3.1), (3.2), (3.10), (5.1), (5.2), (5.3), (5.4) and the second order compatibility conditions [6], we obtain

62 O 0 - ~ + ~ + 0 - 50 a0 2+/12 aHZz 2 = 0, (5.5)


a~176 16F'eaRT~-(cER~ O/S']](a~ 7:3~

- eo (cERo - 4aR T 4) \--~--~ ]0J = 0, (5.6)

ao (c2 i f_ ao ~) + Cq, Oo ~--~]o + ~ \ ~ T ]oJ 2 = 0, (5.7)

3~ ao(~_QoaoZ)+Ooao(v+1)22_2ooa~f22+(v_l)(g_3ao~)=O 6t (5.8)

where 2-= [ui, jk] nin:nk, (= [P, ij] nin:, O= [pR, ij] nins and ~ = [qi>:k] nin:nk are the quantities defined on 2?, H2ot = (H 2 -H2, ) is the magnetic-field intensity and (2 is the mean curvature of 27 given by (4.7).

Eliminating 2, ~, 0 and ~ from (5.5) to (5.8) and using (5.1), we get

32 (v + 1) 22 -----0 (5.9) 6--7+ (m - ao ) - + - - - 7 - - '

where .1)[ { kt0---2~oa02 16/~p~Rl~(a~-a2o) - Oo \ i3~O]o

\ ' - '~- ) 0 (a~ (c ERo - 4 aR 2 00

In obtaining Eq. (5.9) the terms containing c 2 in the denomina tor of/z0 are neglected. Since/~e is an arbitrary function of 0 and T, the sign of/x0 may be positive or negative depending on the form of/~? [8].

Equation (5.9) is the ~ required differential equat ion governing the propagation of the discontinuity 2 associated with the modif ied magnetogasdynamic wave 27. Equation (5.9), on using (4.7), can be integrated to yield

2 = 2o exp(-ktot) (1 - 200 ao t + Ko a~ t2) -1/2 , (5.10)

{/ i 1-r - - - - ~ 2O exp( - i zo: ) (1- 2Ooaot+ Koa2 t'2) -1/2 d{

where 20 is the value of 2 at the wave front at t = 0. To make the physical aspect more accessible, we discuss the following

two cases of plane and cylindrical waves respectively.

Case (i). Plane waves

For a plane wave front s = 0 = Ko, the Eq. (5.10) yields

2o exp ( - ao t) 2 =

1 + (2o/2c) {1 - exp ( - kto t)} '

where 2~ = 2kto (v+ 1) -1.

(5.11)


Equation (5.11) shows that if 20 > 0 (i.e. an expansive wave front) and /-to > 0 then the denominator of Eq. (5.1 1) remains positive and 2--* 0 as t ~ o% the wave decays i.e. the nonlinear breaking of wave front does not occur. Also if 20 < 0 (i.e. a compressive wave front) and if it has the magnitude less than 2+ then the denominator of Eq. (5.11) remains positive finite as t ~ o% whereas the numerator tends to zero, i.e. 2 ~ 0 as t --+ oo and the compressive wave decays and damps out ultimately. Further if 20 = - 2+ then 2 = 20 and the wave propagates with the initial discontinuity without any growth or decay (i.e. no breaking is possible at the wave front). But if 20 < 0 and has a magnitude greater than 2+ then 2 increases beyond all bounds for a finite time t+ given by

( 2+t-I t+ = 1 log 1 + . (5.12)

~0 201

Thus at the instant t+ the velocity gradient at the wave front becomes infinite. This predicts the non-linear breaking of the wave front, and after this time a shock wave with discontinuities in the flow variables p, ~, ui and Hi must be introduced. From (5.12) it follows that (~tc/61ao) > 0, i.e. an increase in ~t0 causes an increase in the time t+ of the breaking on the wave front. Thus the effects of thermal radiation, the finite electrical conductivity o- and that of the magnetic-field intensity H0, are to increase the time of breaking.

When/a0 < 0, it follows from Eq. (5.11) that if 20 > 0 when 2 ~ j2+ 1 as t ~ 0% i.e. no breaking is possible at the wave front, and the wave ultimately takes a stable wave form. This interesting feature of expansive waves does not appear in the former case in which all expansive waves decay and damp out ultimately. But if 20 < 0 and ~0 < 0 then we have the criterion

1 ( + 12+1] ~ = - ~ log 1 1201]

for the nonlinear breaking of the wave front. Thus in this case we find that a discontinuity, no matter how small, associated with a compressive wave always breaks at the wave front. Further in this case (~F~/~ 1/~0 I) < 0 i.e. F~ is a decreasing function of I/~0 l- Thus we reach to an important conclusion that the effects of thermal radiation, the finite electrical conductivity and that of the magnetic field intensity with/-to < 0 are to decrease the time of breaking of the waves.

Case (ii). Cylindrical waves

In this case the breaking and non-breaking phenomenon is very much similar to that of plane waves. If the outward travelling discontinuity surface is a cylinder of radius R0 at t = to, then at any time t > to, the radius of the cylinder is given by R = R0 + a0t; in this case 120 = - 1/2R0, K0 = 0, and the

780 Radhe Shyam and V. D. Sharrna ZAMP

Eq. (5.10) reduces to

2 = 20(Ro/R) 1/2 exp { - #0(R - R0)/ao} , (5.13) { er fc( l z~176

1 + (20/2~) 1 - erfc(#oRo/ao)l/2 �9

where

2 (go ao/rcRo) 1/2 exp ( - lloRo/ao) (5.14) (v + 1) erfc(IzoRo/ao) 1/2

is a positive critical value of the initial discontinuity in the sense discussed below and

o0

erfc(x) = (2 / f -~) S e x p ( - t 2) dt x

is the complementary error function. When/10 > 0 the term inside the cur ly bracket in the denominator of (5.13) increases monotonically from 0 to 1 as R increases from R0 to oe. Hence if 20 > 0, or if 20 < 0 and 12o I < 2c then it follows from (5.13) that 2 ~ 0 as R ~ oe, i.e. no breaking is possible at the wave front, rather the wave decays and damps out ultimately. However, if 20 < 0 and 1.201 = &c then at any time t, the discontinuity I,~J is given by

[2[ = 2(#0 ao/7~R0) 1/2 exp(-IzoRo/ao) (v + 1) erfc(ItoRo/ao) 1/2 (5.15)

which shows that a compressive wave for 1201 = 2o can neither terminate into a shock nor can it ever completely damp out.

Using l'Hospital's and Leibnitz rules, it follows from (5.15) that 2 ~ -2 /10(v+ 1) -1 (critical 20 for plane wave) as R ~ oo, i.e. no breaking is possible at the wave front and the wave ultimately takes a stable wave form. But if 20 < 0 and / 20 1 > ]c then there exists a finite t ime t~ given by

such that 12 L ~ oc as t ---, 4. This signifies the appearance of a shock wave at a finite time/'~ (i. e. the non-linear breaking will occur on the wave front in a finite time ic). The value of the initial discontinuity 20 = ).c plays a crucial role in determining the breaking or nonbreaking of the waves. Accordingly we designate ).r as the critical value of the initial discontinuity for cylindrical waves. Using the inequality e r f c ( x ) < e x p ( - x E ) / ( x ] / - ~ ) ; it follows from (5.14) that 2c > 2/z0(v + 1) -1, i.e. the critical value of the initial discontinuity for a cylindrical wave is always greater than that of a plane wave. Also it follows from (5.16) that (0t~/~/t0) > 0 and (~[~/ORo) < O, which means that an increase in/.to or an increase in the initial curvature will increase the t ime of breaking of the wave front.


For p0 < 0, the breaking and nonbreaking p h e n o m e n o n is again similar to that of plane waves, i.e. if 2o > 0 t hen / t ~ ]/1c] (critical value for a plane wave) as R ---> ~ , i.e. no breaking is possible at the wave front. But if 20 < 0 then we have the criterion

-(g~176176 2 (R0 a0/[/z0 t)-1/2 exp (-/z0 R0/a0) j" exp (x 2) dx -

(v + 1) I2ol

for the breakdown of the wave front at a finite distance R =/~.

A c k n o w l e d g e m e n t

The author (R.S.) is thankful to CSIR, India, for financial support.

References

[i] S. Srinivasan and R. Ram, The growth and decay of sonic waves in a radiating gas at high temperature. Z. angew. Math. Phys. 26, 307-313 (1975).

[2] 1L Ram, Effects of radiative heat transfer on the growth and decay of acceleration waves. Appl. Sci. Res. 34, 93-104 (1978).

[3] R. Shankar and M. Prasad, On sonic discontinuities in an ideal radiating gas. Z. angew. Math. Phys. 30, 937-942 (1979).

[4] T. Y. Thomas, The growth and decay of sonic discontinuities in ideal gases. J. Math. Mech. 6, 455-469 (1957).

[5] W. G. Vincenti and C. H. Kruger, Introduction to physical gasdynamics. Wiley, New York 1965, Chap. 12.

[6] T. Y. Thomas, Extended compatibility conditions for the study o f surfaces of discontinuity in continuum mechanics. J. Math. Mech. 6, 311 -322 (1957).

[7] E. P. Lane, Metric differential geometry of curves and surfaces. Chicago Univ. Press, Chicago 1940, p. 209.

[8] R. Shyam, V. D. Sharma, and V. V. Menon, Evolution of discontinuity at a disturbance wave- head in a radiating gas. AIAA J. t9, 410-412 (1981).

Abstract

The effects of radiative transfer are treated by the use of a differential approximation which is valid over the entire optical depth range from the transparent limit to the optically thick limit. The singular surface theory is used to determine the modes of wave propagation and to evaluate the behaviour at the wave head. It is shown that there are two modes of wave propagation namely (i) the radiation induced waves which are always damped, and (ii) the modified magnetogasdynamic waves which break at the wave front if the initial discontinuity is sufficiently strong. The effects of thermal radiation, the magnetic field intensity, the finite electrical conductivity and the initial wave front curvature on the non-linear breaking of modified magnetogasdynamic waves are discussed.


Zusammenfassung

Der Einflul3 der Strahlung wird mit Hilfe von Differential-N~iherungen behandelt, die im ganzen Bereich yon optischen Tiefen Giiltigkeit haben, yon der Transparenz bis zur optisch dicken Grenze. Die Theorie singul~irer Fl~ichen wird beniitzt um die Wellenausbreitung und das Verhalten an der Wellenfront zu behandeln. Es wird gezeigt, dab es zwei Formen yon Wellen- ausbreitung gibt, n~imlich (i) Wellen erzeugt durch Strahlung, die immer ged~impft sind, und (ii) die modifizierte magnetogasdynamische Welle, die an der Front immer bricht, wenn die urspr0ngliche Diskontinuit~it stark genug ist. Es werden die Einfltisse der Wgrmestrahlung, der magnetischen Feldst~rke, der endlichen elektrischen Leitf~ihigkeit und der ursprtinglichen Wellenkrtimmung auf die nicht-lineare Frontbildung untersueht.

(Received: October 21, 1981.)

nonlinear breaking of waves in an electrically conducting and radiating gas

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