A BGK model for a mixture of chemically reacting gases
Maria Groppi∗ and Giampiero Spiga∗
∗
Dipartimento di Matematica, Università di Parma, Via D’Azeglio 85, 43100 Parma, Italy
Abstract.
A four species reacting mixture is studied by generalizing a recently proposed consistent BGK–approach for inert gases.
The model guarantees exact macroscopic balance equations, and exact collision equilibrium, satisfying the mass action law
of chemistry. Some numerical illustrative results for the space homogeneous relaxation problem are presented.
INTRODUCTION
The so called BGK equations [1, 2] constitute a well known model of the nonlinear Boltzmann equation of gas
dynamics. Their extension to gas mixtures has always represented a delicate problem [3, 4]. In the last of the quoted
papers an interesting consistent approach has been proposed, based on the simple idea of introducing only one BGK
collision operator for each species s, taking into account all interactions with whatever species r. This operator drifts the
distribution function f s towards a suitable local Maxwellian Ms , with parameters that guarantee “exact” transfer rates
for momentum and kinetic energy, and differ from the moments of f s . Such an approach allows to fulfil fundamental
principles like indifferentiability and positivity of density and temperature fields. On the other hand, one of the most
significant physical situations involving a multispecies gas is that of a chemically reacting mixture [5]. The collisional
part of the kinetic equations becomes much heavier, and a BGK–type approach seems to be even more motivated. The
model should now adequately describe also transfer of mass and of energy of chemical link. Preliminary steps have
been moved already in this direction [6, 7] for the reversible chemical reaction
A1 + A2 ­ A3 + A4 .
(1)
In particular, Ref. [7] presents an extension of the BGK strategy of [4] to include chemical interactions along with
mechanical (elastic) collisions. Such an extension is made possible by the knowledge of the exact rates of exchange
of mass, momentum, and total energy for the case corresponding to the classical (inert) Maxwellian molecules, as
derived in [8] in the frame of a Grad’s 13–moment closure of the kinetic equations. This requires a large number of
unknown parameters to be determined, since not even number densities are preserved by collisions. In this paper we
shall follow such a BGK strategy for a four species gas in which reaction (1) takes place. The analytical details and
an ample discussion of the scheme may be found in [7]. The main facts are however summarized in the next Section
for the readers’ convenience. We only recall here that our BGK equations guarantee the exact macroscopic laws and
the exact collision equilibria, including mass action law of chemistry. The kinetic H–functional has been numerically
verified to decrease along solutions. The final Section is instead devoted to some illustrative numerical calculations
relevant to the space–homogeneous relaxation problem for varying control parameters. A comparison with a simple
exact solution is also shown. Future work will include a proof of an H–theorem, consideration of space–dependent
problems and boundary conditions, and comparison of the present scheme to other BGK or kinetic algorithms.
REACTIVE BGK EQUATIONS
In order to deal with the reactive mixture described in the Introduction one should solve extended non–linear kinetic
Boltzmann equations of the type
∂ fs
∂ fs
+v·
= Qs
∂t
∂x
s = 1, . . . , 4,
Qs = I s + J s ,
Is =
4
∑ I sr .
r=1
(2)
In (2), I sr [ f s , f r ] is the elastic scattering collision operator [9] for the binary (s, r) interaction, involving the differential
cross section σ sr , depending on the relative speed g = |v − w| and on the deflection angle χ . The chemical collision
terms J s takes into account mass transfer (with m1 + m2 = m3 + m4 = M) and energy of chemical bond E s , and is
affected by an energy threshold for the endothermic reaction ∆E = − ∑4s=1 λ s E s > 0 (with λ 1 = λ 2 = −λ 3 = −λ 4 = 1).
It can be cast in terms of the reduced masses µ sr and, thanks to microreversibility, of only one differential cross section
34 [5]. We report only the microscopic collision frequencies that will be used later,
σ12
νksr (g) = νkrs (g) = 2π g
Z π
0
σ sr (g, χ )(1 − cos χ )k sin χ d χ
with ν1sr ≤ 2ν0sr , and
34
ν12
(g) = 2π g
Z π
0
k = 0, 1 ,
(3)
34
σ12
(g, χ ) sin χ d χ .
(4)
For the global collision operator {Qs } there exist only seven independent collision invariants, and they may be
chosen as three combinations of number densities ns + nr , for instance (r, s) = (1, 3), (1, 4), (2, 4), the global mass
velocity u, and the internal (kinetic plus chemical) energy U = 3/2 nKT + ∑4s=1 E s ns , where n and T denote global
number density and global temperature respectively. All moments of the distribution functions, including viscous stress
tensor p and heat flux vector q, both global and for single species, are defined in the standard way and may be found
in [7]. Correspondingly, we get seven exact, non-closed, macroscopic conservation equations, and a seven-parameter
family of Maxwellians
¶3
·
¸
µ
2
ms
ms
exp −
(v − u)2
s = 1, . . . , 4
(5)
M s (v) = ns
2π KT
2KT
as collision equilibria, with equilibrium densities bound together by the well known mass action law of chemistry
µ
µ 12 ¶ 23
¶
n1 n2
µ
∆E
exp
=
.
(6)
n3 n4
µ 34
KT
A strict entropy inequality for relaxation to equilibrium has also been established in terms of the H functional [5]
4
H=
∑
Z
3
s=1 R
£
¤
f s log f s /(ms )3 d 3 v.
(7)
The collision rates for mass, momentum and kinetic energy of each species can be cast in closed analytical form for
Maxwellian molecules, a restriction which is consistent with the BGK approach, and which will also be adopted here.
For the mechanical part {I s } we have [8, 4]
Z
R3
Z
R3
Z
R3
Is d3v
ms v I s d 3 v
ms 2 s 3
v I d v
2
= 0
4
∑ ν1sr µ sr ns nr (ur −us )
=
r =1
4
=
∑
r =1
ν1sr
(8)
·
¸
2µ sr s r 3
1 r r
r
s
s s
r
s
nn
K(T − T ) + (m u + m u ) · (u − u ) ,
ms + mr
2
2
where of course the νksr are now constant. For the chemical contributions from {J s }, again in a pseudo-Maxwellian
34 independent of g, calculations are much more complicated [8], and, for slow chemical reaction,
assumption of ν12
yield
Z
R3
Z
R3
Z
R3
Js d3v
ms v J s d 3 v
ms 2 s 3
v J d v
2
= λ sS
= λ s S ms u

(9)
µ
¶3
∆E 2 − ∆E
e KT
1
1−λ s M−ms
M−ms
KT
 s 2 3
s
¶
= λ S  m u + (KT ) −
∆E +
KT µ
3 ∆E
2
2
M
M
2
Γ
,
2 KT







µ
¶
µ 1 2 ¶ 32
∆E
2
3
∆E
m
m
34
n3 n4
√ Γ
S = ν12
,
e KT − n1 n2 
2 KT
m3 m4
π
where
(10)
and Γ denotes incomplete gamma function [10].
Now, the BGK approximation to equations (2) in the spirit of [4] consists in the modified equations
¡
¢
∂ fs
∂ fs
es ≡ νs Ms − f s
+v·
=Q
∂t
∂x
s = 1, . . . , 4,
(11)
where Ms is a local Maxwellian with 5 disposable parameters ns , us , Ts ,
µ
Ms (v) = ns
ms
2π KTs
¶3
2
·
¸
ms
exp −
(v − us )2
2KTs
s = 1, . . . , 4.
(12)
The factor νs is a (macroscopic) collision frequency, i.e. inverse collision time, independent of v. Here, contrary to [4],
the single ns are not conserved quantities, and therefore also the auxiliary number densities ns are unknown fields to
be determined. The exchange rates (8), (9) of the Boltzmann level become now simply
Z
R3
Z
Z
R3
R3
es d 3 v
Q
es d 3 v
ms v Q
ms 2 es 3
v Q d v
2
= νs (ns − ns )
= νs ms (ns us − nsus )
·
¸
3
3
1
1
= νs ns KTs − ns KT s + ns ms u2s − ns ms (us )2 .
2
2
2
2
(13)
Though this approximation is quite crude, a good description of the evolution is expected if we use the 20 overall
disposable parameters in such a way that the same exact exchange rates of the kinetic level are prescribed for the
main macroscopic observables of each species. It suffices to equate the right hand sides of (13) with the sum of the
corresponding right hand sides of (8) and (9), which provides, for given νs > 0, a set of 20 uniquely solvable linear
algebraic equations for the 20 free parameters (functions of x and t) previously introduced, ns , us , Ts . Such a choice
implies in particular that BGK equations (11) fulfil exactly the seven correct conservation laws. In addition, our BGK
equations (11) share with kinetic equations (2) the correct collision equilibrium (5), (6). The interested reader may
find in [7] also a discussion on the proper choice of the collision frequencies νs , which measure the strength at which
model equations (11) push distributions towards equilibrium. We quote here only the conclusions, based on a suitable
counting of the actual average number of collisions taking place for each species, that suggest the choice
ν1
ν2
ν3
ν4
µ
¶
2
3 ∆E
1r r
34 2
√
ν
n
+
ν12
n
Γ
,
∑ 0
2
KT
π
r=1
µ
¶
4
2
3 ∆E
34 1
= ∑ ν02r nr + √ Γ
,
ν12
n
2
KT
π
r=1
µ
¶ µ 12 ¶ 32
4
∆E 34 4
2
3 ∆E
µ
3r r
= ∑ ν0 n + √ Γ
e KT ν12
n
,
34
2 KT
µ
π
r=1
¶ µ 12 ¶ 32
µ
4
∆E 34 3
2
3 ∆E
µ
4r r
√
,
= ∑ ν0 n +
Γ
n .
e KT ν12
34
2
KT
µ
π
r=1
4
=
(14)
NUMERICAL EXAMPLES
The proposed BGK approximation (11) has been applied here in the case of a spatially homogeneous mixture and of
isotropic distribution functions, namely f s (v,t) = f s (|v|,t), s = 1, . . . , 4, that yields us = 0, and then u = 0 and us = 0.
The numerical values used in our test, chosen only for illustrative purposes, have to be considered as dimensionless,
2
2
final
f1
final
f1
BKW
BKW
1.5
t=0.23
1.5
BGK
BGK
t=1.16
1
1
t=0.08
t=0.46
0.5
0.5
t=0
t=0
0
0
1
2
3
4
0
0
1
3
4
|v|
|v|
FIGURE 1.
2
Comparison of BGK and BKW solutions: f 1 versus |v| for different time instants.
and corresponding to arbitrary scales. Moments of the distribution functions f s needed in the expression of Ms have
been numerically evaluated in spherical coordinates by means of the composite trapezoidal quadrature rule. The set of
physical parameters characterizing elastic scattering between pairs of components are given in Table 1.
An important test to be performed is comparison with exact analytical solutions. For this reason we start by switching
off the chemical reaction and considering first a binary mixture with masses m1 = 8 and m2 = 7.3 and with ν1sr = ν0sr
(isotropic differential cross section). Microscopic collision frequencies have been chosen in such a way that the relevant
Boltzmann equations admit a BKW–mode [11] as exact solution [12]. We refer to the latter paper for the detailed
analytical expression of the two distribution functions versus time. Initial conditions have been selected by taking
n1 (0) = 10,
n2 (0) = 13,
T (0) = 4,
(15)
with the constraint that one of the distributions (in our case, f 1 ) just crosses the positivity threshold at t = 0, so that
f 1 (v, 0) vanishes at v = 0. With such an option there results T 1 (0) = 4.6415 and T 2 (0) = 3.5065. In this case of course
both ns are conserved quantities and remain constant versus t, and the same occurs to the global temperature T . The
solution then describes relaxation towards the pair of Maxwellians determined by such macroscopic parameters. The
two temperatures T s (t) (not shown here for brevity) evolve instead in time and approach exponentially the equilibrium
value. The two initial BKW–modes have then been given as initial data for our BGK scheme, and the relevant results
have been compared to the exact solution. The agreement is very good after an initial transient taking place in a layer
of the order of the macroscopic collision time. As a specific feature of this method, evolution of the temperatures
is completely correct, and in fact BGK results for the T s overlap with the exact results to computer accuracy, and
equilibrium is practically achieved in a time of the order of 5 or 6 dimensionless units. Discrepancies occur, as
expected, in the initial layer, where it seems that the BGK collision model pushes too strongly towards a Maxwellian
shape, and thus artificially enhances approach to equilibrium. This effect is however quite limited, as shown by Figure
1, which corresponds to one of the cases with the largest observed discrepancies, and exhibits the initial BKW–mode
for species 1, the relevant equilibrium Maxwellian, and the velocity distribution functions (BGK and exact) for four
selected small values of time. Indeed, the BGK and exact solutions become undistinguishable already after about 3
time units, even before temperature equalization. Discrepancies for species 2, which does not present initially a particle
depletion at small speeds, are quite lower. An interesting point that deserves further investigation is the optimal choice
of the control parameters νs , in order to establish a proper level of collisionality in the BGK mechanism even in
the initial layer. As discussed in [7], such parameters do not affect at all, instead, the evolution of the macroscopic
quantities. This point will be subject of future investigations.
A second test that we have considered concerns effects of anisotropy in the scattering cross sections on the BGK
solution for an actually reactive mixture. We start from initial conditions given by Maxwellian distributions with the
TABLE 1. Elastic collision frequencies relevant to
the four species reacting mixture
ν0sr
1
2
3
4
1
2
3
4
0.4
0.1
0.2
0.7
0.1
0.69
0.5
0.8
0.2
0.5
0.4
0.3
0.7
0.8
0.3
0.6
5
5
Ts
T
4.5
4.5
1
2
3
4
4
3.5
s
0
5
10
15
20
1
2
3
4
4
25
3.5
0
1
2
4
5
t
t
FIGURE 2.
3
Temperature T s versus time for ν1sr = 0.1ν0sr (left) and ν1sr = 1.5ν0sr (right).
following macroscopic parameters
n1 (0) = 10
n2 (0) = 12
n3 (0) = 14
n4 (0) = 13
T 1 (0) = 4
T 2 (0) = 4.3
T 3 (0) = 3.7
T 4 (0) = 3.5.
(16)
34 = 0.01, namely one order of
The collision frequency affecting the chemical reaction has been chosen as ν12
magnitude smaller than the typical mechanical collision frequencies, with energy difference ∆E = 10 between products
and reactants. Finally, masses take the values m1 = 11.7, m2 = 3.6, m3 = 8, m4 = 7.3 . We report in Figure 2 only the
most interesting macroscopic parameters for this problem, the species temperatures T s . The left part of this figure is
relevant to the option ν1sr = 0.1ν0sr , namely to a rather forwardly peaked angular distribution of the mechanical cross
sections. The overall number of elastic scattering in thus much higher than the number of chemical reactions, but,
due to the almost grazing mechanical encounters, transfer of momentum and kinetic energy is scarcely effective, and
goes on a scale which is comparable to the chemical collision times. The right part of the same figure is relevant to the
option ν1sr = 1.5ν0sr , corresponding to rather backwardly peaked elastic cross sections (head–on mechanical collisions).
In this way transfer of momentum is considerably enhanced, and the mechanical scale is expected to be much faster
than the chemical one. All of that is clearly reproduced by our numerical results, where the global temperature is not
a conserved quantity, because of the chemical reaction, and varies with time from the initial value T (0) = 3.8551 to
the equilibrium value T = 4.675. The single species temperatures T s also vary with time, and get equalized due to
scattering. And in fact, it can be seen from the right plot that equalization is a fast process followed by a slow relaxation
to equilibrium when the mechanical transfer mechanism is stronger than the chemical one, whereas in the left plot,
where the transfer scales are comparable, equalization and equilibrium are practically reached in about the same time.
Notice that equations for densities and temperatures are self–consistent in our hypothesis, so that these results are
universal, namely independent from the actual initial distributions, provided the initial values of these macroscopic
parameters are kept the same.
Finally, in order to show the actual velocity distributions obtained from our BGK computations, we consider initial
distributions of disparate type for the four species, namely "bump–on–tail", linear compactly supported functions
3
1
2
3
4
fs
2
2.5
f3 2
1.5
1
1
0.5
0
4
0
0
1
2
3
4
2
t
0
2.5
0
5
|v|
|v|
FIGURE 3.
Initial and final distribution functions versus |v| (left) and distribution function f 3 versus |v| and t (right).
or Maxwellians (see Figure 3, left). The mechanical momentum and energy transfer are determined by the option
ν1sr = ν0sr . Equilibrium is given by Maxwellians at a common temperature and with number densities related by mass
action law. They are shown in the same left part of the figure. An example of the overall evolution is provided by
Figure 3 (right), a three–dimensional plot of f3 versus v and t. The initial shape is quite rapidly smoothed towards a
Gaussian one by elastic scattering, and then it adjusts to the slower evolution of macroscopic parameters controlled by
the chemical reaction.
ACKNOWLEDGMENTS
This work was performed in the frame of the activities sponsored by MIUR (Project “Mathematical Problems of
Kinetic Theories”), by CNR, by GNFM, and by the University of Parma (Italy), and by the European TMR Network
“Hyperbolic and Kinetic Equations: Asymptotics, Numerics, Analysis”.
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