Kinetic-Fluid Coupling in the Field of the
Atomic Vapor Laser Isotopic Separation:
Numerical Results in the Case
of a Monospecies Perfect Gas
DELLACHERIE Stéphane
Commissariat à l’Énergie Atomique
91191 Gif sur Yvette, France
E-mail: [email protected]
Abstract. To describe the uranium gas expansion in the field of the Atomic Vapor Laser Isotopic
Separation (AVLIS; SILVA in french) with a reasonable CPU time, we have to couple the resolution
of the Boltzmann equation with the resolution of the Euler system. The resolution of the Euler
system uses a kinetic scheme and the boundary condition at the kinetic - fluid interface – which
defines the boundary between the Boltzmann area and the Euler area – is defined with the positive
and negative half fluxes of the kinetic scheme. Moreover, in order to take into account the effect
of the Knudsen layer through the resolution of the Euler system, we propose to use a Marshak
condition to asymptoticaly match the Euler area with the uranium source. Numerical results show
an excellent agreement between the results obtained with and without kinetic - fluid coupling.
July 2002
INTRODUCTION
The aim of the Atomic Vapor Laser Isotopic Separation (AVLIS) is to separate uranium235 from uranium-238 to obtain the fuel for nuclear plants (cf. [1]). From this point
of view, the AVLIS process vaporizes uranium by using an intense electronic beam
which heats an uranium liquid source up to 3000 Kelvin (the uranium output is of
some kilogrammes per hour). Then, the uranium vapor is irradiated by a laser beam
which ionizes the uranium-235 (and, ideally, not the uranium-238) further collected as
a liquid on collectors which are negative electrods. To simulate the uranium expansion,
we have to evaluate the distribution function f x v of the uranium expansion which is
the stationary solution of the classical mono-species Boltzmann equation
1
Q f f
(1)
ε
because the gas is almost rarefied (in (1), ε defines the order of the mean free path). Let
us note that some papers have already focuss on the simulation of this Boltzmann equation for AVLIS applications: see [2], [3], [4] and [5]. Nevertheless, near the source of
uranium, it exists a tiny area where the vapor is very dense and near the thermodynamic
∂t f
v ∇x f
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
947
equilibrium (then ε 1) which makes the CPU time of the Monte-Carlo technics used
to discretize the Boltzmann equation dramaticaly increase. To diminish this CPU time,
we discretize the fluid limit of the Boltzmann equation in the dense area where the gas
is at the thermodynamic equilibrium – i.e. in the fluid area –, limit which is the Euler
system
∂t ∇x 0
(2)
closed with the equation of perfect gas (with γ 5 3), knowing that ρ ρ u ρ E and that ρ u ρ u u P ρ E P u by using the classical notations. Of
course, in the remaining part of the physical domain – i.e. in the rarefied or kinetic
area –, we solve the Boltzmann equation. In other words, we have to solve a domain decomposition problem which is named here kinetic - fluid coupling. Moreover, between
the uranium source and the fluid area, the gas is not at the thermodynamic equilibrium
although it is very dense: this very tiny area is called Knudsen layer (see the figure 1).
To optimize the gain in CPU time, we would like to asymptoticaly match the fluid area
where the Euler system is solved with the uranium source. The technics chosen to obtain a boundary condition between the kinetic and fluid domains was previously used
in [6] for classical aerodynamics problems (see also [7] and the references herein). This
technics uses a kinetic scheme (cf. [8]) in the fluid domain to discretize the Euler system since it allows to define a natural boundary condition at the kinetic - fluid interface
with no overlaping between the kinetic and the fluid domains. To define the boundary
condition for the asymptotic matching of the fluid domain with the uranium source, we
use a Marshak condition which, when it is coupled with the previous kinetic scheme,
defines a condition similar to the previous one applied to the kinetic - fluid interface.
This Marshak condition is well adapted since, for example, it does not suppose the value
of the Mach number at the exit of the Knudsen layer as it should be done if we applied a
Dirichlet condition deduced from the study of the half space problem (cf. [12], [13] and
[14]).
The plan of this paper is the following: in the following section, we recall the basic
properties of the kinetic schemes. Then, we describe the kinetic - fluid coupling algorithm, the boundary condition at the kinetic - fluid interface and the Marshak condition
designed for the asymptotic matching of the fluid area with the uranium source. In the
last section, we present numerical results.
Let us note that to simplify the notations, the algorithms are written in monodimensional
cartesian geometry although the numerical results are obtained for a bidimensional axisymmetrical geometry (see also [11]).
KINETIC SCHEME
Let us define the monodimensional conservative scheme
i
n 1
ρin ∆t
∆x
n
i 1 2 n
i 1 2 (3)
for the Euler system (2) (i and n are respectively the space and time subscripts). The
kinetic schemes are constructed from the following lemma due to B. Perthame (cf. [8]):
948
Lemma (B. Perthame) Let us define the initial conditions ρ 0 x , u 0 x and E 0 x of the Euler system (2) which are supposed to be regular and let us define the function
χ vx 0 such that
IR
1 v2x χ vx dvx
1 1 and χ
vx χ vx . Let h t x vx be
solution of the monodimensional transport equation
where M ρ u P vx ∂t h vx ∂x h 0 h t 0 x vx M ρ u P vx ρ 0 x
P 0 x ρ 0 x χ
vx u 0 x P 0 x ρ 0 x Then ρ t x , u t x and E t x defined by
!!
!!
ρ t x $%
"
!!
!!#
IR
(4)
u2 3 P
.
2 2 ρ
with P such that E h t x vx dvx ρ t x u t x &'
ρ t x E t x (
IR
h t x vx vx dvx IR
h t x vx *) v2x 2 P t x + ρ t x -, dvx
is an approximation in ∆t 2 of the solution of (2) when t
∆t (in 1D cartesian geometry).
.
Then, by using an upwind scheme to solve (4) and by taking χ vx / 0
we obtain a first order numerical scheme defined by the numerical flux
i 1 2 '
i 1 2 1
1
2π
exp
i 1 2
v2x 2 ,
(5)
where the positive and negative half fluxes are defined by
i 1 2
ρi ui Ei vx 2 0
1
vx
v2x 2 Pi ρi
vx
34
34
1
vx
v2x 2 Pi 56
M ρi ui Pi vx dvx
(6)
and by
i 1 2
ρi 1
ui 1
Ei 1 $7
vx 8 0
vx
56
1
ρi M ρi 1
M ρ u P vx ρ
exp
2π P ρ
: 1
Pi 1
vx dvx 1
(7)
M ρ u P vx being the classical monodimensional maxwellian
9
ui vx u P ρ
2
;=<
(8)
An important property of this scheme is that it is possible to prove that it is positive and
entropic under a classical CFL criteria (cf. [9]).
949
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KINETIC - FLUID COUPLING
When ε ˆ 0 in (1), we can formally prove that f t x v converges to the maxwellian
distribution function (v ‰ vx vy vz [Š IR3 )
‹
ρ u P v Œ
ρ
2π P ρ 3
exp
2
 vx u 2
v2y v2z
P ρ
Ž
(9)
where ‰ ρ ρ u ρ E is solution of (2), except in some areas as in the Knudsen layer.
We now use the following notations (see the figure 1): Sq 1 2 is the uranium source,
Xq 1 is the mesh having a frontier on the interface Sq 1 2 , Im 1 2 is the interface which
separates the kinetic mesh Xm and the fluid mesh Xm 1 , and n in the normal entering in
the fluid domain. The algorithm of the kinetic - fluid coupling is the following:
- We initialize the kinetic and fluid domains by solving everywhere the transport equation ∂t f v ∇x f 0;
- First stage: knowing the boundary condition Γ f luid  kinetic on the interface Im 1 2 , we
solve the Boltzmann equation (1) in the kinetic area;
- Second stage: knowing the boundary condition Γkinetic  f luid on the interface Im 1 2 ,
we solve the Euler system (2) with the previous kinetic scheme in the fluid area;
- If the gloval level of convergence is not enough, we come back to the first stage.
We now define the boundary conditions Γ f luid 
kinetic
and Γkinetic 
f luid .
Boundary conditions at the kinetic - fluid interface
We define these conditions on the interface Im nx  0 (see the figure 1).
950
1 2
where the normal n is such that
Boundary condition Γ f luid  kinetic : We now suppose that ρ u P x ‘ fluid domain is
known. Since the gas is supposed to be at the thermodynamic equilibrium in the fluid
mesh Xm 1 , we impose that
f x Š Im where
‹
1 2 v $‰
‹
x Xm 1
v
if
is the maxwellian defined by (9) with ρ u P &
vx
.
0
ρ u P
(10)
x ’ Xm “
1
.
Boundary condition Γkinetic  f luid : We now suppose that f x Š kinetic domain v is
known. We use the kinetic decomposition of the macroscopic flux mn 1 2 used in
the numerical scheme (3) and deduced from the previous kinetic scheme built with
the particular choice χ vx ” 0 12π exp v2x 2 . Then, this condition is defined with
& n
n
where
m
1 2 % m 1 2 •
m 1 2
and where & n
m 1 2
m 1 2 %–7
vx 2 0
vx
34
1
vx
v2 2
56
f x Xm v dv
(11)
is defined by the formula (7).
It is easy to prove that the boundary conditions (10) and (11) make conservative the
kinetic - fluid coupling algorithm.
Asymptotic matching: the Marshak condition
The boundary condition at the uranium source Sq
f x Š Sq 1 2 v @‰ Φ v 1 2
is defined by
if vx

0
(12)
where Φ v is a physical data coming from the modelization of the interaction of the
electronic beam with the uranium source. This boundary condition induces that the gas
is not at the thermodynamic equilibrium between the evaporation source Sq 1 2 and the
fluid domain: this defines the Knudsen layer (cf. figure 1). Then, to asymptoticaly match
the fluid domain on the source Sq 1 2 , we have to define a good boundary condition
for the macroscopic flux qn 1 2 . A priori, this can be done by studying the half space
problem and by supposing that the exit of the Knudsen layer is sonic, and, then, by
using the results of [12], [13] and [14]: this boundary condition defines a (Dirichlet)
sonic condition. But, in AVLIS applications, there are situations where the exit of the
Knudsen layer is not sonic: these situations appear when the temperature of the source
is not uniform. Nevertheless, an other reason is that the results of [12], [13] and [14]
can only be applied for the classical mono-species Boltzmann equation (1) and we
would like to extend the results to the semi-classical multispecies Boltzmann equations
951
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Figure 2: Physical domain and mesh
(cf. [10]). An other idea is to apply a boundary condition deduced from the Marshak
condition proposed for the Navier-Stokes system (see [15]: in the case of the NavierStokes system, the Marshak condition defines a Robin condition). Let us note that the
Marshak condition was firstly used for radiative transfer and neutron transport problems
around the year 1940 in Los Alamos. It is possible to show (cf. [10]) that this Marshak
condition when it is coupled with the previous kinetic scheme is simply defined by the
macroscopic flux
n
q 1 2 %
q 1 2 1
& n
q 1 2
where q 1 2 is defined by (11) by replacing f with Φ and where the formula (7).
( n
q 1 2
is defined by
NUMERICAL RESULTS FOR A SUBSONIC KNUDSEN LAYER
The physical domain, the mesh and the kinetic - fluid interface are defined on the figure
2 whose geometry is bidimensional and axisymmetrical. The evaporation condition
defined by Φ v on the source S is a centered maxwellian being at the non uniform
temperature defined on the figure 2 (3400, 3200 and 3000 Kelvin: the first temperature
is due to the impact of the electronic beam). The density of the maxwellian is given by
the saturation density of the uranium. The figures 3 and 4 show the uranium density with
and without kinetic - fluid coupling, and with the sonic condition (figure 3) or with the
Marshak condition (figure 4) for the asymptotic matching of the fluid domain on the
source S. We can see that the kinetic - fluid coupling algorithm is correct only with the
Marshak condition: indeed, the exit of the Knudsen layer is not sonic everywhere (and
not monodimensional) because of the discontinuity 3400 / 3200 Kelvin at R R1 8 5
<
10 3 m (cf. figure 5). Nevertheless, near the X axis, the sonic condition gives good
results since the exit of the Knudsen layer is almost sonic at this place (cf. figure 5).
952
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Figure 3:
Sonic condition
for the asymptotic matching
56 5$5$;
56 57
56 57 <
56 57 =
56 57 9
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5
4 56 585$9
Figure 4:
Marshak condition
for the asymptotic matching
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9 10 3
m)
CONCLUSION
To diminish the CPU time used to simulate the uranium gas expansion in the AVLIS
process, we have proposed to couple the resolution of the Boltzmann equation with the
resolution of the Euler system by using the technics initialy used in [6] for classical
aerodynamics problems. Moreover, we have shown that the Marshak condition coupled
with a kinetic scheme is well adapted to asymptoticaly match the Euler area with the
uranium source in order to take into account the effect of the Knudsen layer in the
resolution of the Euler system. Indeed, it gives a boundary condition without supposing
the value of the Mach number at the exit of the Knudsen layer and without solving any
non linear system.
Numerical results show that the results obtained with and without kinetic - fluid coupling
are quasi similar and that the Marshak condition takes into account without difficulties
953
4 56 57
-/. 021
a Knudsen layer having a subsonic exit which is not the case with a sonic boundary
condition deduced from the study of the half space problem (cf. [12], [13] and [14]).
Finally, let us note that the technics proposed in that paper are extended in [10] to the
semi-classical case and to the multispecies case where we take into account quantified
energy transfers between the electronic metastable energy levels of atoms of different
species, and that these technics can be applied to other vaporization problems as, for
example, for the description of the gas expansion in the tail of a comet produced by sun
radiations (cf. [16]).
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