A Multigroup M1 Model for Radiation Hydrodynamics
and Applications
R. Turpault
CEA-CESTA, Bp2 33114 Le Barp, France
and MAB, UMR CNRS 5466, LRC-CEA M03,
Université Bordeaux I, 33400 Talence, France
Abstract. Several applications of radiative transfer need to take into account quantities which are frequency dependant.
However, when coupling radiation with other physical phenomena, it is crucial to use radiation models which are less
expensive than kinetic methods. We introduce a moments model which aims at extending the possibilities of the M1 model
proposed by B.Dubroca and J.L.Feugeas [3] in order to solve such applications and keeps the main mathematical and physical
properties.
INTRODUCTION OF THE PHYSICAL PROBLEM
A lot of applications can be found involving several different domains of physics that introduce a form of coupling
between hydrodynamics and radiative transfer. These applications include combustion, hypersonic processes -for
example probe atmospheric (re)entry-, Inertial Confinement Fusion, hot plasmas and high temperature industrial
processings such as material pyrolysis or glass manufacture.
In order to get relevant results, it is very important to use convenient radiation models when dealing with this kind of
problems. These models have to be both accurate enough to give a good approximation of the physical phenomena
and cheap enough to avoid unreasonable calculation times, espacially for multi-dimensionnal coupling.
Among the admissible models, the M1-model (for the grey formulation see[3]) is a good compromise. Very cheap, as a
moments model, it keeps a lot of good physical properties such as the possibility to take into account strong directionnal
non-equilibria (high anisotropy factors). However, as a grey model, it is still approximative when considering problems
where the quantities are frequency dependant. Our goal is to extend it into a multigroup model in order to get the rid
of this deficiency. This multigroup model is to be implemented inside a fully coupled hydro-rad code.
MULTIGROUP M1 MODEL WITH ENTROPIC CLOSURE
The radiative transfer equation describes the evolution of the radiative intensity I, which is linked the the photons’
distribution function. Since it does not conserves the energy, we have to couple it with an equation that describes the
evolution of the material temperature. For the purpose of this article, we used a very simple one. More general energy
equations, such as hydrodynamics equations for example, can be treated the same way. The system is then given by:
1
∂ I Ω Ω ∇x Iν Ω σ a Bν T Iν Ω c t ν
σd
σ Iν Ω 4π
d
ρ Cv ∂t T 4π
pν Ω Ω Iν Ω dΩ
0
σ a ν Bν T Iν d ν d µ
CP663, Rarefied Gas Dynamics: 23rd International Symposium, edited by A. D. Ketsdever and E. P. Muntz
© 2003 American Institute of Physics 0-7354-0124-1/03/$20.00
T is the material temperature, σ x t ν the opacities (σ a is the absorption opacity and σ d the scattering opacity), Ω
the direction of propagation of the photons, ν the frequency, p ν Ω Ω the scattering angular redistribution function
and Bν T Planck’s function which realizes the minimum of the radiative entropy.
This kinetic system can be solved thanks to Monte-Carlo or deterministic (see [9]) methods. However, these
methods are very expensive, especially for multidimensionnal problems and since our goal is to couple the radiation
with other physics, we will look for cheaper macroscopic models. Hence we have to integrate our equations. The first
step is to integrate the radiative transfer equation over all the directions of propagation to get the intermediate system:
∂t E ν ∇x F ν cσ a ν 4π B T ν E ν (1)
∂t F ν c2 ∇x P ν c σ a ν σ d 1 ν F ν (2)
This intermediate system has now to be integrated over several groups of frequency, thus we consider
ν 1 m 1 M 1 such as ν 1 0 and ν 1 ∞ and we integrate the previous system from ν 1 to ν 1 to
m 2 get 1 the system:
M 2
2
∂t m
m 2
∇x m m
ρ Cv ∂t T c ∑
m 2
(3)
m m
aθm4 T Em (4)
m
cσma
F
0
Em
cσma aθm4 T , m 2 m , m and m .
Fm c Pm 0
c σma 1 m σmd 0
the mth group.
Em , Fm and Pm are the radiative energy, the radiative flux vector and the radiative pressure tensor inside
where: m θm has the dimension of a temperature and is defined by caθm4 S2
ν
m
ν
m
is given by:
1
2
1
2
Bν T d ν dΩ. The radiative entropy density
2kν 2
n lnnI nI 1 ln nI 1 c3 I
c2
where nI is the occupation number defined by: nI I
2hν 3 ν
The system 3 is not closed yet. We choose to close it by using the minimum entropy principle.
Iν (5)
2hν 3
hν
1
exp
m α m 1!
;
ν
2
1
c
k
m
m 1
m
2
2
realizes the minimum of the radiative entropy under the condition to adequately reconstruct the moments of the
considered model.
Furthermore, if Em Fm are physically relevant, then α m exist and are unique and independent from one another.
Theorem 1. If we note α the lagrange multiplier, the distribution ν
Ω ∑ 1 ν
With this closure, we are able to know Pm as a function of Em and Fm so that 3-4 is a closed nonlinear system.
Moreover, it is possible to write it in an Eddington form Pm ∑m Dm Em where Dm is the Eddington tensor defined by:
Dm 1 χm
3 χm 1
Id m#
"
2
2
"
m
(6)
Fm
$ and χm is the eigenvalue of Dm associated with m .
"
"
Fm
Unfortunatly, as opposed to the grey case, it cannot be explicited. We also have:
where
m
$
Theorem 2. The multigroup M1 model has the following properties:
1. The system is hyperbolic.
1 We chose here to consider the opacities σ to be constants σ inside each group. The way to choose a correct mean is most of the times very
m
difficult and is not developped in this article.
2. The eigenvalues of the system are smaller than c in module.
3. The system has a total entropy which is locally dissipated.
4. The total energy of the system (radiative+matter) is conserved.
Fm
5. The model has a natural limitation of the flux ie
1.
cEm
6. The closure distribution allows to consider any physically relevant anisotropy. Furthermore, all the anisotropy
lies in the direction colinear to the radiative flux.
These properties are very useful to implement the model. The first one allows us to work with classical schemes
used to solve hyperbolic problems, even at the boundaries. The third and fourth show that the system has a realistic
physical behaviour. The fifth point is very important since it tells us that the model is relevant even far away from
the equilibrium. As opposed to some other models, the M1-model still has a realistic behaviour even for nearly
unidirectionnal flows. In particular, the photons will never travel faster than c, which can be greatly violated by P1
and other models.
The last point, linked with (6), shows that even for multidimensionnal cases, only a scalar (χ m ) is needed to
determine the radiative pressure. This means that the cost of 2D and 3D computations is nothing more than the ’usual’
cost. Another very important point is that this model is relevant for free-streaming as well as very diffusive regimes
(see [1]) and in-between. Since these regimes can be found very close from one another in practice (due to a huge
spatial variation of the opacities at the transition from a very opaque to a transparent zone such as smoke/air for
example), all the calculation can be made without shifting models.
NUMERICAL APPROXIMATION
For numerical approximation, we use an explicit finite volumes scheme of the following form:
n 1
i
m
n
i
m
∆t
Ci
∑
n
i j
m
j
∆t n
i
m n
i
m n
i
m (7)
1
n x dx where C are all the cells of the mesh. n
i j m are the numerical fluxes normal to the
i
Ci Ci m
interfaces. We used two different numerical fluxes: one associated with the HLLE approximate Riemann solver and a
kinetic one based on a kinetic interpretation of 3 .
with:
n
i
m
The numerical HLLE flux has the following form:
n
ij
b
b
n
i
b
b
n
j
b b b b U jn Uin (8)
The coefficients b and b are written to include the system’s eigenvalues. We can hence choose b b c,
however, this choice gives some numerical diffusion. To reduce this numerical diffusion, it is possible to choose values
which are closer to the system’s eigenvalues. The numerical scheme we get is positive, decreases the entropy and natu∆t
rally limits the flux ( Fm ni c Em ni m i n) under a very restrictive CFL condition: c c∆t σma 1 m σmd 1.
∆x
This condition is most of the times too restrictive when coupling the radiative transfer with other phenomena, such as
hydrodynamics, which means that in these cases, implicit schemes are to be used.
The kinetic numerical flux is computed as the sum of two different half-fluxes: Finj ν
Fi j m
m 1
2
S2 νm 1
2
ν
Ω i d ν dΩ and Fi j m
m 1
2
S2 νm 1
2
Ω
j d ν dΩ,
where S2 Ω S2 Ω n 0 et S2 Fi j n Fi j n with:
Ω S2 Ω n 0 ,
n is the incoming normal vector.
This method is slighly more expensive than an HLLE-type flux, but allows to use classic upwind schemes and does
not require to evaluate the eigenvalues of the system.
The main difficulty introduced by using a multigroup method instead of a grey one is that the value of the radiative pressure Pm is no longer an explicit function of the radiative energy and the radiative flux. In fact, it depends on
ν η3
d η , which is only analytical if ν 0 or ν ∞. Since we obviously have to compute
the function Ξ ν η
0 e 1
this function a lot of times during a time step, a lot of CPU time is wasted in the process.
A way to overcome this drawback consists in tabulating the radiative pressure (and the half-fluxes in the cas of our
kinetic half-fluxes scheme). It is nearly the same way that the pressure is tabulated for the resolution of the real-gas
Euler equations. Since the variation of the Eddington factor χ (which is the only scalar we need to know in order to
compute the pressure in our model) is smooth, the tabulation is very efficient: we do not need a huge amount of points
and a linear interpolation is good enough to have a very accurate approximation.
A boundary condition frequently used in radiative transfer consists in imposing the incoming radiative flux. For
instance, in one dimension, the left boundary condition is done by giving Fi j m . Such a condition is easily implementable with the kinetic scheme since it is the natural condition one may expect for it. However, the HLLE scheme
does not take half fluxes into account, hence this condition is not so obvious to implement for it. In fact, the whole
flux at the boundary for HLLE is replaced by the kinetic one: the total flux at the boundary is given by the sum of the
incoming half flux, which is given, and the outgoing half flux, which is computed.
NUMERICAL RESULTS
Comparison between the M1 model and flux-limited diffusion
Here we are going to compare the results given by the M1 model with results given by flux-limited diffusion models.
We consider a simple 1D application: radiation comes from the left boundary inside an initially cold medium. We use
the following set: σ a 100, σ d 0, ρ Cv 10 3, Tm0 300K et Tmg 1000K. We look at the evolution of the M1
model, a flux-limited diffusion model using Kershaw limiter [5] and a kinetic model which solves the full radiative
transfer equation [9] for 3 different times: t0 1 33ns t1 10t0 and t2 100t0 .
1000
M1
flux−limited diffusion/Kershaw
kinetic model
900
800
700
600
500
400
300
0
0.02
0.04
0.06
Figure 2 (a)-Radiative temperatures
0.08
0.1
1000
900
800
700
600
500
400
300
0
0.02
0.04
0.06
0.08
0.1
Figure 2 (b)-Material temperatures
1000
900
P1
M1
kinetec RTE
flux limited diffusion/Kershaw
flux limited diffusion/Levermore
800
700
600
500
400
300
0
0.01
0.02
0.03
0.04
Figure 2 (c)-Material temperatures at t1
As expected, for t t2 the three models give similar results since we are in a diffusion regime. However for smaller
times, the flux-limited diffusion overestimates the temperature and the error can be as great as 100 %. On the other
hand, the results of the M1 model are still satisfying.
Figure 2 c also gives the results for the P1 model and a flux-limited diffusion using Levermore’s limiter [5]. The
P1 model is another moment model which is good near equilibrium. Since there are high anisotropy factors in the
precursor, this model yield some errors in this region. The other flux-limited diffusion method is a little bit better than
the previous one, but the difference is still important compared to the solution.
Multigroup results
The case which result is given in Figure 1 shows why it can be crucial to consider some frequency non-equilibrium
to solve several applications: a hot wave penetrates inside an initially cold medium. Opacities are quite important inside
IR (σ a 4) and UV(σ a 3 2) regions but very small inside visible region(σ a 10 3). In this 1D case, the scattering
is neglected. The results showed here are given by 3 models: grey-M1 (with a mean opacity), multigroup-M1 and a
kinetic model which solves the full radiative transfer equation. All the results are second-order and given for a time of
approximately 1 05ns.
Trad/grey−M1
Tmat/grey M1
Trad/kinetic
Tmat/kinetic
Trad/mg−M1
Tmat/m1−M1
5000
4000
3000
2000
1000
0
0.1
0.2
0.3
0.4
0.5
Figure 1-Material and radiative temperature of the 3 models.
The difference between the 2 M1 models is obvious: in the grey model the thermal wave is too fast since it
overestimates the effects of the visible regions and underestimates the effects of the IR and UV regions. The M1
model still conserves the energy, the result is that, as the radiation is too fast, the material heating is too slow. Hence
it is not very relevant for unstationnary results. On the other hand, the multigroup model gives a good approximation
compared to the kinetic model which is more expensive.
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