The Rosseland approximation for radiative transfer problems in heterogeneous media

J. Quanr. Spectrosc. Radial. Tramfer Vol. 58, No. I. pp. 33S-43. 1997

Pergamon PII: S00224073(97)00041-1

!: 1997 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0022-4073/97 $17.00 + 0.00

THE ROSSELAND APPROXIMATION FOR RADIATIVE TRANSFER PROBLEMS IN HETEROGENEOUS MEDIA J.-F. CLOUET Commissariat

g I’Energie

Atomique,

Centre d’Etudes de Limeil-Valenton, 94195 Villeneuve. St Georges. France

(Received 5 December 1996)

Abstract-We address the problem of the validity of the Rosseland approximation in a binary mixture when the opacities of the two materials exhibit a high contrast ratio. We derive a new formulation for the set of radiative equations. Some numerical simulations on effective opacities are presented. 0 1997 Elsevier Science Ltd

1. INTRODUCTION

The problem of solving the radiative transfer equations is of crucial interest in various fields of nuclear physics and has been studied for a long time (see Pomraning,’ Mihalas and Mihalas’ and references therein). In particular, it is well known that the behaviour of the particles (i.e., the photons) is very different according to the value of the opacity of the surrounding medium. Indeed, in an optically thick medium, the particles diffuse and the transport equation of transfer has to be replaced by a diffusion equation: this is the so-called Rosseland approximation. On the other hand, in many physical situations, the medium in which the particles evolve is not homogeneous and can rather be described by a binary mixture. In this case, it is usual to look for an effective homogeneous medium, equivalent to the heterogeneous medium. When the two components of the mixture have the same optical thickness (i.e., their opacities are approximately of the same order of magnitude), the effective behaviour is well understood and we can use an homogenized diffusion equation or an homogenized transport equati0n.j We shall restrict ourselves to the case where the two opacities exhibit a high-contrast ratio: one of the component of the mixture is diffusive whereas the other one is not. We shall show, under some geometric hypotheses on the mixture and on the various scales involved, that a modified version of the Rosseland approximation is valid which means that the mixture is globally diffusive. However, the thermal equilibrium is not achieved in the effective medium. For our exposition, we shall use a gray model (opacities do not depend explicitly on the frequency): the frequency dependancy could be included in a multi-group formalism. We consider also the medium to be unbounded in all directions because we do not want to worry about boundary layers (see Refs. l-4 for boundary conditions). With these simplifications, the radiative transfer equations reduce to

~a,z+i2.v~z+tJz-b~T4=0

(1)

ka,e(T) + mcT4 -

o

ZdsZ= 0. ss

Here, I = Z(x,n,t) is the radiative intensity (x is in IV, R in the two dimensional sphere S and t is the time in an observation interval [0, t,,,]), T E T(x,t) is the temperature, e(T) is the specific 33

34

J.-F. Clouet

internal energy, a and c denote respectively the Stefan constant and the speed of light. We suppose that we are given two smooth isotropic initial conditions I(x,Q,O) = &(X) i T(x,O) = T,(x).

(2)

The opacity of the background material is denoted by r~and should, in general, depend on x and t if the geometry of the mixture is time-dependent. If the characteristic evolution time of the geometry is large compared to the observation time t,,, or if its statistical properties are time invariant and independent of the radiation field, we can suppose that the opacity is only a function of the position CT= a(x). We introduce now the different lengths of the problem. Let L denote the scale at which the evolution of (1) is observed (this will be the size of the domain when we include boundary conditions). The opacities cr, and g2 of the two components of the mixture define two mean-free-path

Lastly, we suppose that the mixture can be characterized by a scale I which measures the typical size of the inhomogeneities. Of course, I < L is a natural assumption for an heterogeneous medium (otherwise there is no need for an effective medium as we can solve (1) separately in each component of the mixture). The component 1 shall be the diffusive medium and the component 2 non diffusive which means that 1, < L and 1, - L. We have now to compare 2, and 1 and we have basically three possibilities This is certainly the more difficult case and there exists only partial results. The main point is that Eq. (1) is no longer adequate to describe the radiation field. We should do first the diffusion approximation in medium 1, derive appropriate boundary conditions and then try to homogenize the medium. Some authors have suggested to adopt a billiard model with appropriate reflexion conditions (see Golse 5 and references therein). We shall not consider this case in detail. 2. l<
where c,, IS~are two dimensionless constants of order 1 and r,, & are the indicator functions of medium 1 and 2, which vary rapidly on the scale 1. We will use dimensionless variables x’=

-Y L

t’=

g.

Rosseland approximation

for RT problems in heterogeneous media

35

We drop the primes and (1) becomes

(3) 8,e(T) + (

FI L(,( F

) +

F

2 L&(

ZdQ) = 0.

))(acT4 -

ss

If we introduce the small parameter E-G--5--,

1

4

L

L

Eq. (3) reduces to

1

a,r + Q.V,1”

I

+ ayl”

-

2

(yy)

= 0

(4)

d,e( T”) + o’(ac( T)4 - u

Z”dQ) = 0, ss

with

where &, and ~5~denote the scaled opacities (&, = a,,$ and 6z = a&). In the sequel, we will however keep the original notations C, and c2. We suppose that the initial conditions (2) are independent from the small scale factor c and we are interested in the behaviour of (I’, T’) as 6 goes to zero. The remainder of the paper is organized as follows. In the next section, we present the asymptotic formulation that arise when letting c going to zero in Eq. (4). In Sec. 3, we give the formal derivation of this formulation and lastly, in Sec. 4, we give some results, both theoretical and numerical on the effective opacity of the mixture. 2. THE 2.1.

ROSSELAND

APPROXIMATION

Setting of the problem

2.1.1. Description of the mixture. So far, we have not specified anything about the nature of the mixture (except the typical size of the inhomogeneities). Our analysis of Eq. (4) is based on asymptotic expansions, following the ideas of Bensoussan et ak6 Papanicolaou and Varadhan’ and Kozlov,* and is well adapted to periodic mixtures and to stochastic mixtures when the randomness can be expressed with the help of ergodic transformations on an abstract probability space. For the sake of simplicity, we shall use only the periodic framework but the results remain true for more general (stochastic) mixtures. Our main result states that the particles diffuse in the medium. Of course this could not be true in general for any mixture (even periodic) if particles can escape to infinity without crossing the diffusive medium. We have to add an assumption on the geometry of the mixture (remind that the medium fills the whole space [w3): Cl: The non-diffusive medium 2 contains no half-line.

In fact, this qualitative assumption has to be fulfilled only for almost every half-line but it is still rigorously not strong enough to prove the diffusion result and a technical hypothesis has to be made which gives a quantitative expression to Gl. Here, as we do not pretend to full mathematical rigor, we will only use Gl and we refer to Ref. 9 for more details. To simplify also the exposition, we shall restrict ourselves to the isotropic case: the medium is periodic with period I and we suppose that the geometry is invariant under rotation (see Remark 2.2 for the anisotropic case).

36

J.-F. Clouet

2.1.2. Linearization. The problem [Eq. (4)], as it stands, is too complex to handle because it is non-linear. The strategy consists in linearizing Eq. (4), making the asymptotic analysis on the linearized version and then coming back to a non-linear model. Of course, this is not entirely satisfying and the validity of the model can be discussed when high temperatures gradients occur in the system. We shall linearize Eq. (4) with respect to the variable c#qx,t) =

2

(zyx,t))“,

by introducing y =

,

WT

w

in medium i.

These functions should depend on x and x/s which leads to the following linearized version of Eq. (4):

a,r + 52.V.J” +

o”(x)(l” - 4”) = 0 y”(x)a,qY + d(x)(t#f - P) = 0,

(6)

with

and with smooth initial conditions I’(x,R,O) = Z,(x) and &(x,0) = E,(x). 2.2. Asymptotic formulation of the linearized equation When studying the Rosseland approximation in an homogeneous mediums using asymptotic analysis, the usual procedure consists in performing a time resealing to observe the evolution of (6) over a large time-period (see Larsen, Pomraning, Badham4). With our notations this would lead to change of time t + t/c. However, this scaling would impose a thermal equilibrium in the whole medium and in particular in the transparent medium 2. So, if we want to model the energy exchanges in medium 2, we cannot perform this time resealing. The price to pay for this is that the limiting diffusion coefficient will be of order 6. Let rreflbe the effective opacity of the mixture (we postpone to Sec. 4 its exact definition) and let D be the diffusion coefficient

Let d’ be the differential operator defined by

0 (

&+

(1 + y,(xlfi)cw

+ dXl - 4xN(U - II/)rdx)W + Mf+ - WY

,

where f; and f2are the volume fraction and a(x)

=

Yl(x)2~2fi(l+ YdXlf, + r*(xI!a r*(-~)ol(l+ r,(x~f;)’ .

This defines the limiting operator, which depends on E because there is no scaling in time.

(7)

Rosseland approximation

37

for RT problems in heterogeneous media

Our main result is the following Theorem 2.1

Let (U”$“) be the solution of:

(8) subject to initial conditions U”(x,O) = I,(x) @“(x,0) = E,(x). Then (U”,$“) is an approximation

of order two for (I’,@) in the following sense

s s zyX

+

y,~,t)

g

dy - U’(x,t) = o(e’)

(9)

rxs

MYW(X + v,Wy -fi$“(x,~) = W2).

r

Equation (8) is a linearized version of the Rosseland approximation, as will become clear in Sec. 2.3, which means that the effective medium is diffusive. There is however a major difference with the classical Rosseland approximation which lies in the energy balance equation: the thermal equilibrium is not achieved in the non-diffusive medium. The specific internal energy e’ for the limiting system [Eq. (8)] is defined as cc = (y,(xzf, + eB(x))UYx,t) + (y2(x)fi - aB(x))P(x,t)

(10)

,

with

B(x) =

w(x)2~fi odl + YdXlf,).

Dropping the index t for U, Y and e and adding the two equations of Eq. (8), we see that Eq. (8) can be rewritten as d,e +

a,u =

EDA u

4e - (~dxlfi +

amu + f&

(11) (e - df;yd-~) +.MxW)

= 0.

It is now easy to check that the total energy is conserved

a, (U

+ e)dx = 0.

s Remark 2.2

In the anisotropic case, the diffusion is no longer isotropic: the diffusion operator DA is replaced by C,d,a, where (do) is a non-negative matrix (see Sec. 4 for its definition). Moreover, in some assymetric cases, a small drift term enters into the energy balance equation but it is negligible in a first approximation.” Of course, taking anisotropic effects into account requires a more deeper knowledge on the geometrical structure of the mixture which can be very difficult to achieve. 2.3. Asymptotic

ormulation

of the non-linear

equation

We now recover a non-linear equation which will constitute the Rosseland approximation in the mixture. In this paragraph, we will write all the equations in terms of the physical variables, i.e., we will drop the parameter E in the definition of operator 4’(7).

38

J.-F. Clouet

We know the limit of &(x/~)&(x,t); &,r by

this enables us to define an intermediate

temperature

(12) that is T”,,, = limF&(x/c),

so that 47c a+?“,,,,) Y*(X)- - ac aT:‘,,,, .

Remark that this temperature is, in some sense, not physical: in particular it is not really the temperature of the matter in medium 2 because we now consider the transfer equations in an homogeneous medium. We introduce also the limiting radiative temperature

With these definitions (8) can be rewritten as:

a,u+fia,e(L)

+ cfz(U -

z Em,)= DA lJ

a&C,,,) + cr*(acC,,,, - 47rU) = 0,

(13)

and the limiting specific internal energy is: e =J; e(Z’& +fie(T,,,,,).

(14)

This defines a matter temperature T,,,,,,,, such that e(Tmatter) = e. Note that we have neglected the &a(x) and s/I(x) terms which are small in front of 1 in practical situations. We now consider two limiting cases for Eq. (13): l

Iffy = 0, there is only one diffusive material and we find the classical Rosseland approximation:

a,u + a&r) = DAU, with D = $. l

I

IfJ; = 0, Eq. (13) is not valid in general (the diffusion result is false in a non-diffusive material) and we find:

I a,u+ 1

az(~ - ~T*)=EDAU

a,e(T) + a,( z

T4 - U) = 0,

with D = l/30, . This is the so-called grey diffusion model (see Ref. 9). 3. DERIVATION

OF THE LIMITING

EQUATION

We present only a formal derivation of Eq. (8). It uses the same techniques as in Refs. 4-6, the main new technical point being the resolution of Eq. (20). The idea is to look for (I”,$“) in the form of the asymptotic expansion

I

= W-w) + ~U,(x,Q,t,y) + W) eGta> = h(x,~,_Y) -I- 4I(XJ,Y) + Ok%

I”wu,Y)

where y denotes the rapidly oscillating space variable X/C. We shall adopt the following notation:

(15)

Rosseland

approximation

for RT problems

in heterogeneous

39

media

First, we can solve explicitly the linearized energy balance equation

(16)

We put Eq. (15) into Eq. (16)

(

@~(s,t,p) = C,(y) U&,t)

+

1

&O,(S,f,?‘) - y

a,u,(.u,t) + 0(&I)

52(Y)(vk?(x,t) + E$,(-w,y) + 0(&Q),

+

where $,, and $, are defined by

i

y,cx)i?$o+ az($o- mx,t)) = 0 ,1&W, + @?($I- O,(.wt)) = 0 ~,(x,O) = 0. h(x,O) = J%(x) i

(17)

Adding the two equations of Eq. (6) we get a,p + L?.V,Z”+ y”a,@ + frE(li - P) = 0 .

(18)

We remark that

LvJ(x,%) =

(

Q.V, +

+v,>f (.Y,y),r~r’tr

so that Eqs. (15),( 18) give f

PV,I/, + 0,5,(Y)(Uo-

aI)1+ bdX)5,(Y)+ Y2(X)
+ WA + Q~V,~llf ~*~,(Y)(~o - ml + PV,U + ~,S,(Y)(U,- G)l + 4YdX)Sd.Y)+ ydxMY)l44, + &w, + Q.V,U, + aYt*(y)(U,- ml = W).

(19)

As U, = U,,(x,r) the l/t term vanishes (actually, this is a necessary condition). For the order 1 term to be independent from the direction R, we should have s2.v,u, + [Q.VJ,

+ o,<,(y)(U, - &>I = 0 .

(20)

Under assumption Gl, this equation admits a unique solution, up to a constant which can be taken to be zero (see Ref. 9 for the details) U,(x,y,t,S2) = - Z(Q,y)~v,u~(X,t).

(21)

We put Eq. (21) into Eqs. (16) and (17) to get MGYJ)

= 5,0?)u&J)

+ L(u)V%l(xJ)

&(x,y,r) = 5,(_Y)(- %V?~0(xJ)

- +)

(22)

&U,(x,f))

- 52(~)&~)~V,($o(~,0 - e- zlMSx)). Note that s(y) = l,E(Q,y)dS2/4 rt is non-zero even in the isotropic case when solving Eq. (20) in a non-homogeneous medium. From Eq. (22) we obtain (Y,(X)5,(Y)+ Y*(X)S*ti))%#%=

rl(x)~,(YHUO

-

r2(YM1(1”

-

&I)

(23)

40

J.-F. Clouet (YI(X)SIGy) + YdXMY))%h

=

-

+ wxv)%).(w0

51(Y) % -

a:&

- Y,(x)5,(Y)~(Y).c7,a,uo

ul) -

We put now Eq. (21) and Eq. (23) into Eq. (19) to obtain ~,w,wa,u~

- 52w~,wo

- 6)

+ a,6

- 4~lw

9

+ ~[~~5~(y)~(~).(~.~(~~ - UJ - &qt2,y).v,a,u,

- ~~~v,8(i2,~p~u,

a:u, + y,(x)~,(y)~~).~,a,U,l

&

e- 2

WdxNl

+ to2t2~)(u,

- 0,) = o(2)

.

(24)

We shall now take the average, denoted with < . > , of this equation with respect to the periodic variable y and the angle variable R. We have (S(s2,y))

= 0 (&))

(&(Y)> =fi

= 0

(MY)) =fi*

The limiting diffusion operator is defined by: (25) which, in the isotropic case, reduces to DA. Lastly, in the symmetric case, we have (@)3(&Y))

= (MJV(%Y))

= 0,

(in the assymetric case, this vector is not, in general, equal to zero: this gives a drift term of order 6 in Eq. (26)). Taking the mean in Eq. (24) we obtain

+ &$Q a;(u,) + o(2)

4(w + waa,(uo) + Y2mha,(lcIo)) = EDA

(26)

Y2wa,(h) f ~2((tio)- (uo)) = m, which can be rewritten as

wh) + Yddma,(uo) +_&a(1- wm(Uo) - (II/o)WA(Uo) Y2b3a,(tio) + 62((h) - (uo)) = o(E*),

(27)

by differentiating Eq. (26) with respect to t. Up to the O(E*) term, this is just Eq. (8), satisfied by the limit (UE,t,bE). As the initial conditions are smooth, the maximum principle shows that: (I”(x,Qt))

- UE(X,f) = (uO) - UE(X,f) f O(EZ)= 0(&l),

which is exactly the statement of theorem (2.1). The approximation result for t2@ is obtained from the same manner. It is possible to state a result without taking the mean with respect to the periodic space variable y provided that the convergence is understood in a weak sense: For all test function F, smooth and with compact support, we have

F)) = O(e*), ’

(28)

Rosseland approximation

for RT problems in heterogeneous media

41

where < < . > > denotes integration with respect to the space variable x. Note that we can not conclude that Cr’(x,t) is close to

because there is no maximum principle associated with this weak formulation. The expression of the specific energy Eq. (10) and Eq. (11) follows now from a straight-forward computation. 4. EFFECTIVE

OPACITY

OF THE MIXTURE

4.1. Theoretical results In the isotropic case, the effective opacity is simply obtained from the diffusion coefficient D by 0 eR= l/30 but in the anisotropic case we should rather define a diffusion matrix (di,) which can be computed by [see Eq. (25)] 4, =

(Q’~(sZ,y)‘).

(29)

where 3(R,y) is solution of a stationary transport equation on the cell I Q.V,.E + a,l$,(y)(Z - E’) = - !G?,

(30)

and < > denotes the mean with respect to the periodic variable 4’ and the direction R. We can use the probabilistic representation of Eq. (30) to solve this equation. It is then easy to see that the diffusion coefficients are simply given by the Einstein-Kubo formula d, =

k2$2:dt, s0

(31)

where (Y, R,) is a Markov random process on the periodic cell (I- x S) defined by: l l

0

Y, is a random variable with a uniform distribution on I. !& is random variable with a uniform distribution on the sphere S.

a,I:=n,

R, is a pure jump process. The jump points have a uniform distribution on the sphere S and the time between two jumps has an exponential distribution with parameter a,t,(Y). l

There is no explicit formulation for Eq. (31) (even when the geometry of the mixture is simple), but we can exhibit a bound (32) (see Ref. 9 for the proof). The previous definitions extend easily to the random case by replacing the integration with respect to the periodic variable y by the expectation with respect to the randomness of the medium. An example of random medium (introduced in Ref. 11) is a randomly stratified medium: in this case, the distribution along each line is known and the bound Eq. (32) can be explicitly computed (see Refs. 10-12 for the computation) and depends on the Laplace transform of the length distribution. This model is unfortunately not very realistic. The coefficients di are nevertheless much easier to compute numerically than d,. Remark that the opacity of the transparent medium do not enter in the definition of the effective opacity. This is not surprising since our result rests on the hypothesis that cz is negligible in front of 0, but it may be not accurate in practical situations. We can solve simply this problem by replacing 0,(,(y) by o,{,(y) + a,&(v) in Eqs. (30),(32) and in the definition of the random process (Y, Q,). This will be done in the numerical simulations of the next paragraph.

42

J.-F. Clouet Arithmetic_mean 0 Imfp + 5mfp 0 IO mfp x

0

0.2

0.4

0.6

0.8

1

Fig. I. Effective opacity as a function of the volume fraction for a distance between spheres equal to 1, 5 and 10 mean-free-path.

4.2. Numerical results 4.2.1.

The cuse of dimension one. In dimension 1, numerical simulations show that the effective

opacity is the arithmetic mean

Although it is not obvious that (31) gives this simple result, it is not surprising because, in the one dimensional case, the homogenization and diffusion approximation procedures commute (this fact had already been observed in Ref. 5), the effective opacity being always the arithmetic mean. 4.2.2. Three-dimensional simulations. In this paragraph, we just want to point out that the effective opacity is not, in general, the arithmetic mean of the opacities of the constituent. So we concentrate on a single example: a more complete study of effective opacities is a work in progress. We have considered an heterogeneous medium where periodical inclusions of spheres of opacity 0, = 100 are imbedded in a background medium of opacity o2 = 0. The geometry is entirely described by the distance 1 between the centers of the spheres and the volume fraction fi of the spheres. The medium is isotropic and the diffusion coefficient is

so be/f=

1 *im ‘11 ~112 ’ I-z 2t

The following pictures? (Fig. 1) represent the effective opacity as a function of the volume fraction tThese simulations were done by F. Thiout from C.E.A. Limeil.

Rosseland approximation

for RT problems in heterogeneous media

43

for a distance 1between the centers of the spheres equal to 1, 5 and 10 mean free path in the spheres. We have also represented the arithmetic mean of the opacity. The effective opacity is always lower than the arithmetic mean and the difference can be large (more than 50%) in particular for small fraction of diffusive material. We observe that the effective opacity decreases when a,l increases and get closer to the arithmetic mean as a,1 decreases (this is coherent with the results of Golse’). So the typical length scale is an important parameter for the computation of the effective opacity. 5. CONCLUSION

Using radiation different. medium

asymptotic expansions, we have derived a model which describe the time evolution of the field in a binary mixture when the optical thickness of the two components are very This model is intermediate between the classical Rosseland approximation in a diffusive and the grey diffusion model. In an isotropic medium, it writes as:

The matter temperature

r,,,,,, is given by

The effective opacity creRis defined with the help of the Einstein-Kubo formula. Numerical simulations show that it can be much lower than the arithmetic mean of opacities in particular for small volume fraction of diffusive material. As there is no explicit formulation for the effective opacity, there is now a need for analytic approximations, valid over a large range of volume fractions and opacities. REFERENCES G. C., The Equations of Radiation Hydrodynamics. Pergamon Press, Oxford, 1973. Mihalas, B. W. and Mihalas, D., Radiation Hydrodynamics. Pergamon Press, Oxford. 1984. Golse, F. and Sentis, R., unpublished. Larsen. E. W., Pomraning, G. C. and Badham, V. C., JQSRT, 1983, 29, 4. Golse, F., in Mathematical Aspect of Fluid and Plasma Dynamics, eds G. Toscani, V. Boffi and S. Roinero. Lect. Notes in Maths, Springer, Berlin, 1991. Bensoussan, A., Lions, J. L. and Papanicolaou, G. C.. Publications qf the Research Institute for Mathematical Sciences, Kyoto University, 1979, Vo. 15, p. 1. Papanicolaou, G. C. and Varadhan, S. R. S.. Colloquia Mathematics Societas Janos Bolyai. Budapest,

1. Pomraning,

2. 3. 4. 5. 6. I. 8. 9. IO. II. 12.

1978. Kozlov,

S. M., Dodlaky Akad.Nauk,

1978, 241, 5.

Clouet, J. F., SIAM J. Appl. Math., 1996. Submitted. Clouet, J. F., Internal report, CEA, 1996. Levermore, C. D., Pomraning, G. C., Sanzo, D. L. and Wong, J., J. Math. Phys.. 1986. 27, 2526. Vanderhaegen. D., JQSRT, 1988, 39, 333.