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Journal of Quantitative Spectroscopy &Radiative Transfer

Journal of Quantitative Spectroscopy & Radiative Transfer 110 (2009) 293–299

0022-40

doi:10.1

� Cor

E-m

journal homepage: www.elsevier.com/locate/jqsrt

Angular anisotropy of group averaged absorption coefficient and itseffect on the behavior of diffusion approach in radiative transfer

Jiwei Li �, Jinghong Li

Institute of Applied Physics and Computational Mathematics, P.O. Box 8009-14, Beijing 100088, China

a r t i c l e i n f o

Article history:

Received 17 April 2008

Received in revised form

15 October 2008

Accepted 5 November 2008

Keywords:

Radiative transfer

Multi-group method

Diffusion approximation

Anisotropy of group averaged absorption

coefficient

73/$ - see front matter Crown Copyright & 2

016/j.jqsrt.2008.11.001

responding author.

ail address: li_jiwei@iapcm.ac.cn (J. Li).

a b s t r a c t

Multi-group method, a generally accepted procedure for handling the radiative transfer

equation, is accompanied by the group averaged absorption coefficient, and actually the

coefficient is angularly anisotropic. In the paper, we present a brief discussion how the

anisotropy of the coefficient makes the material absorb photons at different rate in each

direction, also study its effect on the diffusion approach by comparing the results

calculated using multi-group diffusion and multi-group discrete ordinate SN for the

isotropic and anisotropic group averaged absorption coefficients respectively, and find

that the anisotropy deteriorates the behavior of diffusion approach.

Crown Copyright & 2008 Published by Elsevier Ltd. All rights reserved.

1. Introduction

Time-dependent non-equilibrium radiative transfer can be important in astrophysics, inertial confinement fusion, andhigh energy density physics applications. It is always of great cost to solve the equation of radiation transport accuratelydue to its complexity. To deal with the frequency-dependent variables in the equation, multi-group method is introduced[1,2], which really amounts to a discretization of the frequency variable. That is to say, the photons within one frequencygroup are treated the same, assigning them the same frequency-independent average properties, such as emission andabsorption coefficients. The method is least accurate for coarsen frequency groups and it gives increasingly better resultsas more and finer groups are used; of course, it also becomes very costly. To effectively reduce the computational time,multi-group diffusion approximation is preferable. The approach is a method of approximately describing the angulardistribution of radiation field in which the radiation flux in one group is assumed to be proportional to the gradient of theradiative energy density in the group and actually it solves the equations of the energy density and the flux for each groupregardless of the anisotropy of the radiation intensity in angular direction. Considerable work [3–7] has been done to studythe diffusion approximation and compare its results with the exact solutions for the gray case in which the absorptioncoefficient is presumed to be independent on the frequency. Su and Olson [8] also compared them for non-gray cases wherethey assumed the mean absorption coefficients for each group are constants, respectively. However their work does notconsider the angular anisotropy of the group averaged absorption coefficient in the multi-group radiative transfer equation.Actually the group averaged absorption coefficient strictly introduced in the multi-group method is angularly anisotropic[1] though it is independent on frequency, and the anisotropy may affect the behavior of diffusion approach. In this paper,first we will give a discussion on the anisotropy of the group averaged absorption coefficient in multi-group radiative

008 Published by Elsevier Ltd. All rights reserved.

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J. Li, J. Li / Journal of Quantitative Spectroscopy & Radiative Transfer 110 (2009) 293–299294

transfer equation, then compare the multi-group diffusion approximation to the multi-group discrete ordinate SN for boththe isotropic and anisotropic group averaged absorption coefficients and study the effect of the anisotropy in one group onthe accuracy of diffusion approximation for the group.

2. Multi-group method

Under the assumption of local thermodynamic equilibrium (LTE), the time-dependent radiative transfer equationwithout considering the source and scattering terms in a one-dimensional planar geometry can be given by

1

c

qInðx;m; tÞqt

þ mqInðx;m; tÞqx

¼ s0aðn; TmÞ½BnðTmÞ � Inðx;m; tÞ�, (1)

where x, m and t are the space, angular direction cosine, and time variables, respectively; Iv is the spectral radiationintensity; BnðTmÞ ¼ ð2hn3=c2Þð1=ehn=kTm � 1Þ the spectral intensity for equilibrium radiation; c the speed of the light; Tm thematerial temperature and sa

0 the effective absorption coefficient.Taking integration of both sides of Eq. (1) over the photon frequency interval of each group (frequency interval Dvg), we

obtain the multi-group radiative transfer equation

1

c

qIgðx;m; tÞqt

þ m qIgðx;m; tÞqx

¼ sbgBgðTmÞ �

ZDng

s0aInðx;m; tÞdn, (2)

where Ig ¼RDng

In dn, Bg ¼RDng

Bn dn and sbg ¼

RDng

s0aBnðTmÞdn=Bg is the Planck’s mean value in each group for sa0. In order

to deal with the absorption term, we expand the radiative intensity in Legendre polynomials according to

Inðx;m; tÞ ¼X1n¼0

2nþ 1

4pInðx; n; tÞPnðmÞ. (3)

where due to the orthogonality of the PnðmÞ, the expansion coefficients are given by

Inðx; n; tÞ ¼ 2pZ 1

�1PnðmÞIðx; n;m; tÞdm.

Inserting Eq. (3) into the frequency-integrated absorption term in Eq. (2), the multi-group radiative transfer equation iswritten as

1

c

qIgðx;m; tÞqt

þ m qIgðx;m; tÞqx

¼ sbgBgðTmÞ �

X1n¼0

2nþ 1

4p PnðmÞsngIg

n, (4)

where sng ¼

RDng

s0aInðx; n; tÞdn=Ign, the nth order group averaged absorption coefficient and Ig

n ¼RDng

Inðx; n; tÞdn. For n ¼ 0,I0ðnÞ ¼ 2p

R 1�1 InðmÞdm ¼ cEn, here Ev is the spectral radiative energy, then we have s0

g ¼RDng

s0aEn dn=Eg and name it theisotropic group averaged absorption coefficient, where Eg ¼

RDng

En dn. Because Ev here is unknown, the zeroth order meancoefficient is always used with an approximation s0

g ¼ spg ¼

RDng

s0aBnðTrÞdn=BgðTrÞ, the Planck mean for each group, wherethe Planck function having a local radiation temperature Tr is to get a better spectrum [9–11] where the radiative field is outof equilibrium with the material.

Then Eq. (4) can be written as

1

c

qIg

qtþ m

qIg

qx¼ sb

gBg � spgIg þ

X1n¼1

2nþ 1

4p PnðmÞðspg � s

ng ÞI

gn. (5)

2.1. Group averaged absorption coefficient is angularly isotropic

If we assume that the effective absorption coefficient sa0 happens to be constant in each group, the high order terms

(nX1) in Eq. (5) will get disappeared. Then Eq. (5) degenerates into be

1

c

qIg

qtþ m qIg

qx¼ sb

gBg � spgIg . (6)

clearly the group averaged absorption coefficient spg is angularly isotropic.

Now we will derive the multi-group P1 diffusion equation by virtue of the angular moments of Eq. (6). Actually here ourprocedure is to take integration of radiative transfer Eq. (1) first over frequency then over angle rather than the traditionalone to take integration first over angle then over frequency [12]. Multiplying Eq. (6) successively by 1 and m, and integratingover angle, we can obtain the equations of spectral radiative energy and radiative flux in each group

qEg

qtþqFg

qx¼ 4psb

gBgðTmÞ � cspgEg , (7)

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J. Li, J. Li / Journal of Quantitative Spectroscopy & Radiative Transfer 110 (2009) 293–299 295

1

c

qFg

qtþ

Z 1

�1mm qIg

qxdm ¼ �sp

gFg , (8)

here Fg ¼RDng

Fn dn> and Fn ¼ 2pRDng

mIn dm, the spectral radiative flux. To obtain the P1 diffusion approximation, we makethe further assumption that the time derivative of the radiation flux is negligible and substitute Ig ¼ ðc=4pÞEg þ ð3=4pÞmFg

into Eq. (8), then the radiative diffusion flux in one group is given by [6]

Fg ¼ �c

3spg

qEg

qx. (9)

Eqs. (7) and (9) are the corresponding multi-group diffusion equations of Eq. (6). Since we neglect the angle-dependentabsorption terms in Eq. (5), the group diffusion coefficient in Eq. (9) is determined by the Planck mean free path rather thanthe traditional Rosseland mean free path.

For this case that the group averaged absorption coefficient is isotropic, Su and Olson [8] compared the results of two-group diffusion and two-group radiative transfer and concluded that the accuracy and validity of two-group diffusionapproximation will be deteriorated as the material’s absorption becomes weaker, which also means that the diffusionapproach behaves poorly for optically thin materials.

We can also know that it seems invalid to only keep the zeroth order term in Eq. (5) unless the effective absorptioncoefficient sa

0 is constant in each group because the corresponding diffusion coefficient is determined by the Planck meanfree path rather than the Rosseland mean free path. However it does not matter because our motivation here is to study theaccuracy and validity of diffusion approach.

2.2. Group averaged absorption coefficient is angularly anisotropic

Because the absorption coefficients encountered in practice are generally complex and widely varying functions offrequency, the higher order terms in Eq. (5) have to be kept. Considering them, we can introduce an effective groupaveraged absorption coefficient sg

eff which satisfies

seffg Ig ¼ sp

gIg �X1n¼1

2nþ 1

4pPnðmÞðsp

g � sng ÞI

gn. (10)

It is obvious that the sgeff is angularly anisotropic because sp

gasng for nX1. As we know, the validity of diffusion

approximation is dependent on the angular distribution of the radiative intensity, so the anisotropy may affect the accuracyof diffusion approximation for it will redistribute the radiative intensity in angular direction.

From the angular moments of Eq. (5), we could obtain its corresponding multi-group diffusion equations

qEg

qtþqFg

qx¼ 4psb

gBgðTeÞ � cspgEg , (11)

Fg ¼ �c

3s1g

qEg

qx. (12)

Now we can see that the diffusion coefficient here is determined by the first order group averaged absorption coefficient(it can be approximated by Rosseland mean free path as discussed below), which agrees with the traditional multi-groupdiffusion approximation. The two different procedures to derive the multi-group diffusion reach the same result. Physicallyspeaking, the multi-group Eq. (5) is much more reasonable and accurate than Eq. (6).

Allowing for the convenience to achieve the numerical simulation, we truncate the high order terms at n ¼ 1, thenEq. (5) becomes

1

c

qIg

qtþ mqIg

qx¼ sb

gBg � spgIg þ

3

4pmðspg � s

1g ÞFg . (13)

For n ¼ 1, I1ðnÞ ¼ Fn, the radiation flux and s1g Fg ¼

RDng

s0aFn dn. Actually here we handle the group absorption term in

Eq. (5) with P1 approximation, so we can assume s0aFn ¼ �ðc=3ÞqEn=qx, then s1g

RDngð1=s0aÞqEn=qx dn ¼

RDngðqEn=qxÞdn.

Further we have s1g ¼ sR

g ¼RDng

qBnðTrÞ=qTr dn=RDngð1=s0aÞqBnðTrÞ=qTr dn, the Rosseland mean for each group, then Eq. (13)

becomes the new multi-group method [1] and its effective absorption coefficient for each group

seffg ðmÞ ¼ sp

g �3

4pmðsp

g � sRg ÞFg

Ig. (14)

From this equation, we can see that the absorption rate of forward moving photons becomes lower than that of backwardmoving photons, and the anisotropy plays a role where the Rosseland and Planck means vary widely.

Comparing Eq. (11) to Eq. (7), we would find that the energies transported from radiation to materials have the samedependence on the radiation energy density, though the group averaged absorption coefficient is isotropic in Eq. (6) andanisotropic in Eq. (5). That is to say, though the anisotropy makes the material become absorbing the photons within onegroup at different rate for each direction, the total absorption integrated over all the direction and frequency remains

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literally sameP

g

R4p s

pgIg dO ¼

Pg

R4p s

effg Ig dO ¼ c

Pgs

pgEg . In the following numerical simulation, we will compare the

multi-group diffusion to the multi-group discrete ordinate SN for the two cases respectively to study the effect of theanisotropy of the group averaged absorption coefficient on the accuracy and validity of diffusion approximation.

3. Numerical simulation

The simulation is achieved by virtue of the code RDMG [13] (1D radiation diffusion multi-group) with an average atomopacity model. In this simulation, the plastic (CH) planar slab is exposed to a constant radiation field with T ¼ 149.3 eV onone side and the other side is the vacuum boundary, and the radiation field is assumed to be Planckian in frequency andisotropic in angle. The hydrodynamics is treated by solving the standard mass, momentum and energy conservationequations. For the material CH in the simulation, the 20-group opacities and tabulated equation of state are used [10].Results are compared between the calculations using multi-group discrete ordinate S8 and multi-group diffusion for theisotropic and anisotropic group averaged absorption coefficient, respectively. To better compare the 20-group diffusionapproach and 20-group S8, first we get the time-dependent net radiation flux of each group at the exposed boundary fromS8 results, then set those net flux of each group as the drive source (net diffusive flux) of the multi-group diffusion. Bydoing that, we can guarantee that the drive spectrum of the diffusion approach is closely matched to that of full transfer[14]. Compared to the isotropic case, the anisotropic group averaged absorption coefficient will result in the net radiationflux at the exposed side increase as time goes by though it contributes nothing at the initial stage, correspondingly for somegroups, such as group No. 11 (348.1�425.7 eV), their net radiation flux at the exposed side also increase, as shown in Fig. 1,and the heat wave also propagates faster [15]. To clearly illustrate the effect of the anisotropy on the diffusion approach, wenormalized the space by the position of heat wave front xF and the radiation flux in one group by its net radiation flux at theexposed side Fnet

g_b, respectively when comparing the multi-group SN and multi-group diffusion for the two cases.Fig. 2 is the spatial distribution of the radiation temperature and Fig. 3 is the spectrum of the radiation flux at the

position x=xF ¼ 0:8 at time t ¼ 1.5 ns for two cases, respectively. For the isotropic case, the diffusive heat wave propagatesonly a little ahead of the transport theory and the corresponding spectrum also agrees well with that of the S8 results. Bythe comparison between the anisotropic case and isotropic one, it is clear that the anisotropy of the group averagedabsorption coefficient makes the diffusion approach predict the worse frequency spectrum compared to the S8 results,though the worse of radiation temperature is not too significant. The radiation flux of those photons whose energy are300 eV or so is overestimated by the diffusion approach, and that will also be shown in Fig. 6.

The angular anisotropy caused by the high order term (n ¼ 1) in Eq. (13) retards the decrease of the radiative intensityfor those photons moving forward and the increase of that for those photons moving backward, therefore it willredistribute the radiative field in angular direction. Fig. 4 is the angular distribution of the group’s mean absorptioncoefficient and radiative intensity of group No.11 at time t ¼ 1.5 ns at the position x=xF ¼ 0:6 where the Planck andRosseland means for the group vary widely. Near the position, the distribution of the absorption coefficient becomesanisotropic seriously, correspondingly the radiative intensity also becomes much more anisotropic. That is to say, the roleof the anisotropy of group mean absorption coefficient is to prevent the radiative field from tending to being isotropic. Thespectral radiation flux is determined by the angular distribution of radiative intensity, so it is sensitive to the anisotropy ofgroup averaged absorption coefficient and easy to be affected by it, as shown in Fig. 1. As the radiative field becomesangularly anisotropic, the behavior of the diffusion approach will become deteriorated, such as the diffusion flux. From thecomparison of the radiation flux for group No.11 between the diffusion approach and discrete ordinate S8 shown in Fig. 5,

Fig. 1. The net radiation flux for group No. 11 at the exposed side for the isotropic and anisotropic group averaged absorption coefficients respectively.

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Fig. 2. Comparison of the radiation temperature between 20-group S8 and 20-group diffusion at time t ¼ 1.5 ns for the isotropic and anisotropic group

averaged absorption coefficients respectively.

Fig. 3. Comparison of the spectral radiation flux between 20-group S8 and 20-group diffusion at the position x=xF ¼ 0:8 at time t ¼ 1.5 ns for the isotropic

and anisotropic group averaged absorption coefficients, respectively.

Fig. 4. The angular distribution of the mean absorption coefficient (normalized by sgp) and radiative intensity (normalized by I0max the maximum for all

the direction each case) for group No.11 at the position x=xF ¼ 0:6 and time t ¼ 1.5 ns.

J. Li, J. Li / Journal of Quantitative Spectroscopy & Radiative Transfer 110 (2009) 293–299 297

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Fig. 5. Comparison of the radiation flux in group No.11 at time t ¼ 1.5 ns between 20-group S8 and 20-group diffusion for the isotropic and anisotropic

mean absorption coefficients respectively.

Fig. 6. Comparison of the radiation flux in group No.10 at time t ¼ 1.5 ns between 20-group S8 and 20-group diffusion for the isotropic and anisotropic

mean absorption coefficients respectively.

J. Li, J. Li / Journal of Quantitative Spectroscopy & Radiative Transfer 110 (2009) 293–299298

we can see that the diffusion description becomes less accurate behind the heat wave for the case that the group averagedabsorption coefficient is anisotropic. For the radiation flux of group No.10 (284.7�348.1 eV) shown in Fig. 6, it is much moreevident that the diffusive flux is impracticably overestimated near the heat wave front where the anisotropy disables thediffusion approach.

4. Conclusion and discussion

In the paper, we analyzed the angular anisotropy of the group averaged absorption coefficient and compared the multi-group diffusion and multi-group discrete ordinate SN for the coefficients are isotropic and anisotropic, respectively. Theanisotropy plays a significant role where the Rosseland and Planck means for each group vary widely—for some groupsnear the heat wave front, while others behind the heat wave, and it deteriorates the results predicted by diffusionapproximation, especially for the spectrum and the radiation flux for each group. For the fine enough groups, theanisotropy will become less important because two means are closed to each other.

Here we only consider the anisotropy caused by the high order term with n ¼ 1. If other higher order terms are included,the anisotropy will become more complicated and may result in a more significant effect on the accuracy of diffusionapproximation.

The scattering is not considered in all the work here, otherwise, the group averaged absorption coefficient should beredefined by sn

g ¼R

nngðs0a þ ssÞInðx; n; tÞdn=Ig

n, where the ss is the scattering coefficient.

Acknowledgements

We are grateful to the radiation transport crew at IAPCM and Prof. Wenbing Pei for fruitful and valuable discussion. Wewould like also to thank Prof. Xujun Meng for his Atomic Parameter Databases.

This work was jointly supported by the National Natural Science Foundation of China under Grant No. 10775019 and bythe Laboratory of Computational Physics, China.

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