PHYSICS OF FLUIDS
VOLUME 16, NUMBER 12
DECEMBER 2004
A Bhatnagar–Gross–Krook-type approach for chemically reacting
gas mixtures
Maria Groppi and Giampiero Spigaa)
Dipartimento di Matematica, Università di Parma, 43100 Parma, Italy
(Received 18 March 2004; accepted 21 July 2004; published online 21 October 2004)
A recently proposed consistent approach for elastically scattering gas mixtures, of the type
introduced by Bhatnagar, Gross, and Krook (BGK), has been extended to deal with a four species
gas undergoing reversible bimolecular chemical reactions. The single BGK collision operator
introduced for each species must take into account also transfer of mass and of energy of chemical
bond. Suitable auxiliary fields have then to be introduced not only for temperatures and velocities,
but also for densities, in order to fulfill correctly balance equations for mass, momentum, and total
energy. The exact collision equilibrium, satisfying the mass action law of chemistry, is also
recovered, and the proper choice of collision frequencies is discussed. Preliminary numerical results
for the relaxation problem in space-homogeneous conditions are reported and briefly commented
on. © 2004 American Institute of Physics. [DOI: 10.1063/1.1808651]
I. INTRODUCTION
indifferentiability requirements, and even to prove an entropy inequality.
This paper is aimed at extending the previous BGK strategy to the much more complicated problem, significant for
the applications, in which gas components may undergo
chemical reactions. Exchange of mass and of energy of
chemical link have then to be considered. In particular, we
shall deal with a four species gas mixture and with the reversible chemical reaction
It is well known that there are several important regimes
of gas dynamics in which the proper mathematical tool of
investigation is the nonlinear Boltzmann equation. On the
other hand, such an equation is quite complex to deal with,
and various simpler models have been introduced for it, and
among them the most popular and widely used is probably
the so-called BGK model.1,2 But it is also well known that
considerable troubles are encountered if one tries to extend
the BGK philosophy, originally devised for a single species
gas, to a gas mixture.3,4 The situation is much better established in the frame of linearized kinetic theory, where a consistent modeling of gaseous mixtures has been achieved,
which provides the correct expressions of all transport coefficients. We refer the interested reader to Ref. 5 and the bibliography therein. In the nonlinear frame, to be considered if
dealing with physical conditions far from equilibrium, one
faces immediately very basic drawbacks such as loss of positivity and breakdown of the indifferentiability principle. An
accurate and detailed discussion on the subject may be found
in Ref. 6, where the authors propose a simple and very interesting consistent BGK-type model for gas mixtures which
overcomes all previous difficulties. The main idea is the introduction of only BGK collision operator for each species s,
taking into account all interactions with whatever species r,
instead of approximating each of the binary collision operators between species s and r. The sth BGK operator drifts
then the distribution function f s towards a suitable local
Maxwellian Ms, whose macroscopic parameters are chosen
in such a way that the “exact” (i.e., for Maxwellian molecules) rates for momentum and energy exchange are rigorously reproduced. A proper choice of the collision frequencies allowed them to verify the positivity and
A1 + A2 A3 + A4
according to the kinetic model proposed in Ref. 7. Molecules
As may also interact with any, different or equal, species r
= 1 , . . . , 4 by elastic scattering. Some description of the internal structure of each molecule might be included, for instance, according to Ref. 8; but, for simplicity, we will stick
here to translational degrees of freedom only, and evaluate
accordingly both mechanical and chemical collisions. Rigorous deterministic calculations relevant to the chemical reaction (1) have been performed at various levels, such as Euler
equations, Grad’s 13-moment closure, multigroup or semicontinuous kinetic algorithms. We can quote, among others,
Refs. 9–12. A first attempt of constructing a reactive BGKtype approximation has been presented in Ref. 13, where the
Boltzmann-like chemical collision operator is handled in
such a way that each distribution function f s is replaced everywhere, except for the entry factored out from its loss collision integral, by the local Maxwellian corresponding to mechanical equilibrium. Due to the detailed balance principle,
this amounts to producing a Maxwellian factor in the gain
term, Gaussian function of the molecular velocity v. The
approach is quite appealing, especially because of its simplicity, and preliminary [homogeneous in space and onedimensional in velocity] numerical results look promising.
Indeed, essentially all of the above literature is relevant to
the assumption of dominant elastic scattering, and thus of
slow chemical reaction, which justifies the preceding ap-
a)
Present address: Dipartimento di Matematica, Università di Parma,
Via D’Azeglio, 85/A - 43100 Parma, Italy. Electronic mail:
[email protected]
1070-6631/2004/16(12)/4273/12/$22.00
共1兲
4273
© 2004 American Institute of Physics
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4274
Phys. Fluids, Vol. 16, No. 12, December 2004
M. Groppi and G. Spiga
proximation. The same physical scenario applies to the
present work, which however, intends to follow a different
policy for the inclusion of chemical reactions, namely, a generalization of the consistent BGK model proposed in Ref. 6.
This requires a large number of unknown parameter to be
determined, and an explicit analytical knowledge of the rates
for mass, momentum, and energy exchange for each species
in the chemical reaction. Again, this result can be achieved,
for Maxwellian molecules, when the elastic relaxation time
is shorter than the chemical one.9
Owing to the previous facts, we propose and discuss in
the next sections a BGK scheme for modeling the evolution
of a four species gas in which reaction (1) takes place. The
scheme guarantees that mass, momentum, and energy are
exchanged at the same pace as for the Boltzmann kinetic
level. Attention is paid to the proper choice of collision frequencies, in order to adhere to the actual number of collisions in the gas mixture. Several consistency results of the
type obtained in Ref. 6 may be reproduced; for others, like
an H theorem, we have not yet been able to get an analytical
proof, because of the additional considerable difficulties introduced by chemical encounters. This problem will be matter of future work, where we intend also to compare the
numerical performance of our scheme with kinetic calculations and with other available BGK approximations. In the
final section we show here, however, preliminary numerical
results in dimensionless form and in space-homogeneous
physical conditions, in order to test adherence to physical
expectations and response to variation of control parameters.
In particular, we shall deal with the relaxation problem from
isotropic initial distributions towards Maxwellian equilibria,
which must share the same mass velocity and temperature,
and exhibit densities related by mass action law.
II. KINETIC AND BGK EQUATIONS
Extended kinetic equations (of the Boltzmann type) for
the problem described in the Introduction read as
Is =
共2兲
Isr ,
兺
r=1
where I 关f , f 兴 is the usual elastic scattering collision
operator14,15 for the binary 共s , r兲 interaction, involving the
differential cross section ␴sr, depending on the relative speed
g = 兩v − w兩 and on the deflection angle ␹. Microscopic collision frequencies to be used later are
sr
s
R3
⫻
rs
␯sr
k 共g兲 = ␯k 共g兲 = 2␲g
冕
k = 0,1,
sr
s
with ␯sr
1 艋 2␯0 . The chemical collision term J takes into ac1
2
3
4
count mass transfer (with m + m = m + m = M) and energy
of chemical bond Es, and is affected by an energy threshold
4
⌬E = −兺s=1
␭sEs ⬎ 0 (with ␭1 = ␭2 = −␭3 = −␭4 = 1) to be over-
3
册
f 3共v⬘兲f 4共w⬘兲 − f 1共v兲f 2共w兲 d3wd2⍀̂⬘ , 共4兲
冕
␲
34
␴12
共g, ␹兲sin ␹d␹
共5兲
and to recall the crucial properties of the global collision
operator 兵Qs其, which differ slightly but significantly from the
purely elastic case. There exist in fact only seven independent collision invariants, and they may be chosen as three
combinations of number densities ns + nr, for instance 共r , s兲
= 共1 , 3兲 , 共1 , 4兲 , 共2 , 4兲, the global mass velocity u, and the in4
E sn s,
ternal (kinetic plus chemical) energy U = 3 / 2nKT + 兺s=1
where n and T denote global number density and global temperature, respectively. Global macroscopic parameters (including mass density ␳, stress tensor p, and heat flux q) are
defined by
4
4
4
兺
s=1
␳ = 兺 m sn s,
n s,
s=1
4
u=
1
m sn su s ,
␳ s=1
兺
4
1
n KT +
␳s共uskusk − ukuk兲,
nKT =
3 s=1
s=1
兺
s
s
4
pij =
共6兲
4
4
qi =
兺
psij + 兺 ␳s关共usi usj − uiu j兲 − 31 ␦ij共uskusk − ukuk兲兴 ,
兺
s=1
s=1
兺
s=1
4
qsi
+
兺
s=1
4
psij共usj
5
− u j兲 +
nsKTs共usi − ui兲
2 s=1
兺
4
+
共3兲
␮12
␮34
冊
2⌬E
34
g␴12
共g, ␹兲
␮12
0
␴sr共g, ␹兲共1 − cos ␹兲ksin ␹ d␹,
0
冋冉 冊
34
␯12
共g兲 = 2␲g
r
␲
⌰ g2 −
S2
where v⬘ and w⬘ are the postcollisional velocities attained by
the products 3 and 4, and the relative speed g⬘ = 兩v⬘ − w⬘兩 is
smaller than g because of the loss of kinetic energy spent to
increase the chemical energy. The unit step function ⌰ accounts for the fact that the reaction cannot occur if such a
threshold is not overcome. All other chemical collision terms
Js may be obtained from (4) by suitable permutations of
indices, taking microreversibility into account.16 We skip
here all technical details, for which the interested reader is
referred to Refs. 7–9. It suffices here to introduce the chemical microscopic collision frequency
s = 1, . . . ,4,
4
冕冕 冉
J1 =
n=
⳵ fs
⳵ fs
+v·
= Q s,
⳵t
⳵x
Qs = Is + Js ,
come in the endothermic reaction, conventionally assumed to
be the one from the left to the right in (1). If ␮sr denotes the
reduced mass of the 共s , r兲 pair, the collision integral J1 may
34
as7
be cast in terms of a differential cross section ␴12
1
␳s共usk − uk兲共usk − uk兲共usi − ui兲,
2 s=1
兺
in terms of the traditional single component parameters
ns , us , Ts , ps, and qs ( ␦ij denotes the usual Kronecker symbol). Consequently, we get seven exact, non-closed, macroscopic conservation equations
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Phys. Fluids, Vol. 16, No. 12, December 2004
A BGK-type approach for chemically reacting gas
⳵ s r
⳵
共n + n 兲 +
· 共nsus + nrur兲 = 0,
⳵t
⳵x
共s,r兲 = 共1,3兲,共1,4兲,共2,4兲,
⳵
⳵
共␳u兲 +
· 共␳u 丢 u + P兲 = 0,
⳵t
⳵x
冉
共7兲
冊 冋冉
4
⳵ 1 2 3
⳵
·
␳u + nKT + 兺 Esns +
2
⳵t 2
⳵x
s=1
冊
4
+
兺
s=1
1 2 3
␳u + nKT
2
2
4
E sn s u + P · u + q +
Esns共us − u兲
兺
s=1
册
= 0,
where P = nKTI + p is the pressure tensor. Analogously, we
have now a seven-parameter family of Maxwellians
M s共 v 兲 = n s
冉 冊 冋
ms
2␲KT
3/2
exp −
册
ms
共v − u兲2 ,
2KT
as collision equilibria, with equilibrium densities bound together by the well known mass action law of chemistry
冉 冊 冉 冊
␮12
n 1n 2
=
n 3n 4
␮34
3/2
exp
⌬E
.
KT
lytical representation can be obtained if the distribution function f s appearing in certain integrals are replaced by the
Maxwellians M s corresponding to mechanical equilibrium.
The latter are a family of Gaussian distributions formally
identical to those given in (8), but with number densities ns
completely uncorrelated (eight-parameter family). Such an
approximation is reasonable in all numerous physical situations in which the elastic relaxation time is much shorter
than the chemical one, and is valid in the bulk time domain
after the initial layer where a fast transient pushes distribution functions towards local mechanical equilibrium. With
these restrictions, we have, following essentially the same
steps as in Ref. 9
冕
J sd 3v = ␭ sS
冕
msvJsd3v = ␭sSmsu
冕
ms 2 s 3
vJdv
2
R3
R3
共8兲
s = 1, . . . ,4
共9兲
R3
= ␭ sS
A strict entropy inequality for relaxation to equilibrium has
also been established in terms of the H functional
4
H=
兺
s=1
冕
R3
f sln关f s/共ms兲3兴d3v.
冕
Isd3v = 0,
R3
冕
m vI d v =
冕
2␮
m 2s 3
s r 3
r
s
␯sr
v I d v=
1
s
r n n 关 2 K共T − T 兲
m
+
m
2
r=1
4
s
R3
R3
s
s 3
sr s r r
s
␯sr
兺
1 ␮ n n 共u − u 兲,
r=1
4
兺
共11兲
sr
+ 21 共mrur + msus兲 · 共ur − us兲兴 ,
␯sr
k
s
are now constant. For the chemical
where of course the
contributions from 兵J 其, even in a pseudo-Maxwellian as34
independent of g, calculations are much
sumption of ␯12
more complicated, as discussed in Ref. 9. An explicit ana-
冤
1 s 2 3
1 − ␭s M − ms
m u + 共KT兲 −
⌬E
2
2
2
M
M − ms
KT
+
M
共10兲
The crucial point for the construction of a BGK model according to the idea of Ref. 6 is the availability of the correct
collision rates for mass, momentum, and kinetic energy of
each species (they are sufficient to guarantee also the correct
exchange for internal energy). Such rates can be made explicit in closed analytical form for Maxwellian molecules, a
restriction which is consistent with the BGK approach, and
which will also adopt here. For the mechanical part 兵Is其 the
final results have been already worked out in literature9,6 and
read as
4275
where
34
S = ␯12
2
冑␲
⌫
冉 冊
冉 冊
⌬E
KT
⌫
3/2
e−⌬E/KT
3 ⌬E
,
2 KT
冥
冉 冊冋 冉 冊
3 ⌬E
,
2 KT
n 3n 4
共12兲
,
m 1m 2
m 3m 4
3/2
e⌬E/KT − n1n2
册
共13兲
and ⌫ denotes incomplete gamma function.17 Other, more
realistic, g-dependent, collision frequencies, taking into account also the chemical activation energies,18 could be dealt
with in a similar, but more complicated, manner. This task,
however, will be undertaken in a separate future investigation.
With these results at hand, the BGK approximation to
Eq. (2) in the spirit of Ref. 6 consists in the modified equations
⳵ fs
⳵ fs
+v·
= Q̃s ⬅ ␯s共Ms − f s兲,
⳵t
⳵x
s = 1, . . . ,4,
共14兲
where Ms is a local Maxwellian with five disposable parameters ns , us , Ts,
Ms共v兲 = ns
冉
ms
2␲KTs
冊 冋
s = 1, . . . ,4.
3/2
exp −
册
ms
共v − us兲2 ,
2KTs
共15兲
The factor ␯s is a (macroscopic) collision frequency, i.e.,
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4276
Phys. Fluids, Vol. 16, No. 12, December 2004
M. Groppi and G. Spiga
inverse collision time, possibly dependent on the moments of
the distribution function 兵f s其, but independent of v. Here,
contrary to Ref. 6, the single ns are not conserved quantities,
and therefore also number density of the attracting sth Maxwellian is different from that of the actual distribution function, as it occurs for drift velocity and temperature. Though
highly nonlinear, Eq. (14) allow some simplification and are
easier to implement numerically. The exchange rates (11)
and (12) of the Boltzmann level become now simply
冕
R3
4
1
␮sr s r s s
+
␯sr
n n 共m u + mrur兲 · 共ur − us兲
1
s
␯s r=1 m + mr
兺
+
冤
M − ms
␭s 1 s s 3
S m n + KT +
2
␯s 2
M
s
msvQ̃sd3v = ␯sms共nsus − nsus兲,
冕
ms 2 s 3
v Q̃ d v
2
R3
兺
Q̃ d v = ␯s共ns − n 兲,
s 3
冕
R3
4
3
3
1
1
ns KTs = ns KTs − ms关nsus2 − ns共us兲2兴 +
␺srTr
2
2
2
␯s r=1
⫻KT
共16兲
冉 冊
冉 冊
⌬E
KT
⌫
冥
3/2
e−⌬E/KT
3 ⌬E
,
2 KT
−
1 − ␭s M − ms
⌬E .
2
M
Such a choice implies, in particular, that BGK equations (14)
fulfill exactly the seven correct conservation laws. We have,
in fact, from (16) and (19)
冕
= ␯s关ns 2 KTs − ns 2 KTs + ns 2 msus2 − ns 2 ms共us兲2兴 .
3
3
1
1
R3
∀s
Q̃sd3v = ␭sS,
共20兲
and then
III. DISCUSSION
Following Ref. 6, we now impose that (2) and its approximate model (14) prescribe the same exchange rates by
collision for the main macroscopic observables of each species. One must include density, along with momentum and
kinetic energy, as required by the presence of reaction (1).
We have then to equate the right-hand sides of (16) with the
sum of the corresponding right-hand sides of (11) and (12).
For given ␯s ⬎ 0, this provides a set of 20 linear algebraic
equations for the 20 free parameters (functions of x and t)
previously introduced, ns , us , Ts. Fortunately enough, such
equations decouple very easily and the unique solution is
immediately achieved in explicit form without any matrix
inversion. Upon introducing the auxiliary symmetric matrices
4
sl sl s l
sr s r
sr
␾sr = ␯sr
1 ␮ n n − ␦ 兺 ␯1 ␮ n n
共17兲
l=1
冕
R3
4
兺
s=1
4
sl
␮sr s r sr
sl ␮
s l
n
n
−
␦
3K
␯
兺
1 s
ln n ,
ms + mr
m
+
m
l=1
共18兲
␭s
ns = ns + S,
␯s
4
m sn su s = m sn su s +
1
␭s
␾srur + msuS,
␯s r=1
␯s
兺
共19兲
冕
4
s
s 3
m vQ̃ d v =
R3
4
␺
兺
s=1
4
4
兺
兺␾
s=1 r=1
4
sr
= 0,
␭m
兺
s=1
s
4
u + Su
sr r
␭ sm s = 0
兺
s=1
共22兲
4
s
= 0,
␭s = 0,
兺
s=1
4
共23兲
1 − ␭s M − ms
= − 1,
␭s
2
M
s=1
M − ms
= 0,
␭s
M
s=1
兺
兺
it is not difficult to check that, from (16) and (19)
冕
R3
ms 2 s 3
v Q̃ d v = S⌬E
2
共24兲
and then, for the total internal energy, we have easily
4
the parameters determining all Maxwellians Ms read explicitly as
共21兲
due to the fact that mass is conserved and the sum of the
entries of any given column of matrix ␾sr is equal to zero.
Finally, due to the identities,
4
␺sr = 3K␯sr
1
共s,r兲 = 共1,3兲,共1,4兲,共2,4兲.
Similarly, again from (16) and (19)
兺
s=1
and
共Q̃s + Q̃r兲d3v = 0,
兺
s=1
冕冉
R3
冊
ms 2
v + Es Q̃sd3v = 0.
2
共25兲
Equations (21), (22), and (25) enable us to recover for the
BGK model (14) the correct macroscopic conservation equations (7).
Another essential feature of our BGK equations (14) is
that, analogously to Ref. 6, they share with kinetic equations
(2) the correct collision equilibrium. If equilibrium quantities
are labeled by a bar, by definition we have from (14) f̄ s
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Phys. Fluids, Vol. 16, No. 12, December 2004
A BGK-type approach for chemically reacting gas
= M̄s, and consequently n̄s = n̄s , ūs = ūs , T̄s = T̄s. Because of the
first of (19), this yields S̄ = 0, which explicitly reads as
n̄1n̄2
3 4
n̄ n̄
=
冉 冊 冉 冊
␮12
␮34
3/2
exp
⌬E
共26兲
KT̄
(mass action law). Entering these results into the second of
(19) yields then
4
␾srūr = 0,
兺
r=1
s = 1, . . . ,4
共27兲
which, following the same arguments exploited in Ref. 19,
since the rank of the matrix ␾sr is equal to 3, implies that all
ūs must be equal to each other, and their common value ū
remains free. With these results at hand, the third of (19)
collapses to
34 1
− ␯12
f 共v兲
s = 1, . . . ,4,
共28兲
where ␺sr has the same structure of the matrix ␾sr, and
therefore19 there is again a one-parameter family of solutions, given by T̄s = T̄ , ∀ s, where T̄ is a positive parameter. In
conclusion, collision equilibria are Maxwellians with seven
free parameters, gas temperature T̄, mass velocity ū, and
three out of four densities n̄s, which must be related by (26).
Thus the correct kinetic equilibria (8) and (9) are reproduced
by our approximation.
All of the above facts are independent of the choice of
collision frequencies ␯s, which measure the strength at which
model equations (14) push distributions towards equilibrium,
and which are linked to the actual number of collisions that
each species undergoes in the mixture per unit time. It is
clear from (19) that taking ␯s large enough would guarantee
that ns , us , Ts differ as little as desirable from ns , us , Ts, and
then would guarantee, in particular, the required positivity of
the additional fields ns and Ts. On the other hand, an excess
of collisionality would artificially enhance the relaxation
problem with respect to reality. It seems appropriate then to
estimate the actual number of collisions (both mechanical
and chemical) taking place in the gas, and to evaluate the
collision frequencies accordingly. The calculation is easy for
elastic scattering. In our assumptions the actual loss term for
r s
species s when colliding with species r is −␯sr
0 n f , so that the
sr r
relevant number of collisions per unit time is ␯0 n . Summing
over all target species we get the rate of elastic scattering for
a general species
R3
⌰ g2 −
冊
2⌬E 2
f 共w兲d3w
␮12
and the corresponding frequency of interaction turns out to
depend on v. We are led thus to consider an average by
simply counting all reactions taking place at whatever value
of v and dividing then by the number n1 of molecules of
species 1 which are present. Such an expression could be
used directly as it is in our scheme, but a simpler, more
manageable form, in agreement with our assumption of slow
chemical reaction, can be obtained by replacing again the
distribution functions f 1 and f 2 in the sixfold integral by the
local Maxwellians M 1 and M 2 corresponding to mechanical
equilibrium. Then a little algebra allows us to evaluate the
integral analytically in terms of gamma functions, and to
estimate the rate of chemical reactions for species 1 as
␯chem
=
1
4
␺srT̄r = 0,
兺
r=1
冕 冉
4277
2
冑␲ ⌫
冉 冊
3 ⌬E 34 2
␯ n ,
,
2 KT 12
共30兲
where the factor 2 / 冑␲⌫共 23 , ⌬E / 共KT兲兲 ⬍ 1 clearly accounts
for the presence of the threshold. Analogous result holds for
the other species. In particular, for species 3, the same reasoning leads to an average for the reaction frequency
1
n3
冕冕冕
R3
R3
12
g␴34
共g, ␹兲f 3共v兲f 4共w兲d3vd3wd2⍀̂⬘ ,
S2
12
34
where ␴34
can be expressed in terms of ␯12
by means of the
microreversibility condition. On using the same approximation of slow chemical reaction, we get after some algebra the
estimate
␯chem
=
3
2
冑␲
⌫
冉 冊冉 冊 冉 冊
3 ⌬E
,
2 KT
␮12
␮34
3/2
exp
⌬E 34 4
␯ n .
KT 12
共31兲
The collision frequencies to be used in (14) would read then
as
TABLE I. Elastic collision frequencies relevant to the test case.
␯sr
0
1
2
3
4
1
0.3
0.4
0.1
0.4
2
3
4
0.4
0.1
0.4
0.3
0.4
0.6
0.4
0.3
0.2
0.6
0.2
0.4
␯sr
1
1
2
3
4
1
0.5
0.6
0.2
0.7
2
3
4
0.6
0.2
0.7
0.4
0.5
0.8
0.5
0.4
0.3
0.8
0.3
0.6
4
␯smech =
r
␯sr
兺
0n .
r=1
共29兲
We pass now to the number of chemical reactions per unit
time of species 1 (necessarily, with species 2 only). The loss
term from (4) can be cast as
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4278
Phys. Fluids, Vol. 16, No. 12, December 2004
M. Groppi and G. Spiga
FIG. 1. Evolution of number densities
ns (a) and temperatures Ts (b) (zoom),
both recovered from the computed distribution functions (equilibrium temperature: T = 4.675).
4
r
␯1 = 兺 ␯1r
0 n +
r=1
2
冑␲
4
⌫
冉 冊
␯1 ⬎
冉 冊
␯3 ⬎
2
冑␲
⌫
␯4 ⬎
2
⌫
3 ⌬E 34 2
,
␯ n ,
2 KT 12
3 ⌬E 34 1
r
␯2 =
␯2r
⌫ ,
0 n +
冑␲ 2 KT ␯12n ,
r=1
2
兺
4
冉 冊冉 冊
3 ⌬E
r
␯3 =
␯3r
⌫ ,
0 n +
冑␲ 2 KT
r=1
2
兺
4
兺
␮
␮34
冉 冊冉 冊
3 ⌬E
r
␯4 =
␯4r
⌫ ,
0 n +
冑
␲ 2 KT
r=1
2
共32兲
12 3/2
␮12
␮34
34 4
e⌬E/KT␯12
n ,
2
冑␲ ⌫
冑␲
冉 冊
冉 冊冉 冊
冉 冊冉 冊
3 ⌬E 34 2
,
␯ n,
2 KT 12
␯2 ⬎
3 ⌬E
,
2 KT
␮12
␮34
3/2
3 ⌬E
,
2 KT
␮12
␮34
3/2
2
冑␲ ⌫
冉 冊
3 ⌬E 34 1
,
␯ n ,
2 KT 12
34 4
e⌬E/KT␯12
n ,
共34兲
34 3
e⌬E/KT␯12
n ,
which are equivalent to the requirement
3/2
max共− ␯1n1,− ␯2n2兲 ⬍ S ⬍ min共␯3n3, ␯4n4兲,
34 3
e⌬E/KT␯12
n .
共35兲
and the latter is just the positivity condition for the density
fields ns.
It is remarkable that such a choice implies, in particular,
IV. NUMERICAL TESTS
4
1
␯s ⬎
␯srnr ,
2 r=1 1
兺
共33兲
namely, the condition adopted in Ref. 6, which allowed to
prove positivity of the temperature fields in absence of
chemical reactions. In addition, Eq. (32) also imply
In this section we report on some preliminary numerical
calculations that have been performed for a reacting spacehomogeneous gas mixture, mainly in order to check the response of the proposed BGK approximation (14) of the collision term to varying collision parameters and to varying
FIG. 2. Distribution function f 3 vs t
and 兩v兩.
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Phys. Fluids, Vol. 16, No. 12, December 2004
A BGK-type approach for chemically reacting gas
4279
FIG. 3. Zoom of the deviation f 2
− M2 vs t and 兩v兩.
initial conditions. Space-dependent problems with boundary
conditions are scheduled as future work. Our model has been
tested extensively for a wide choice of physical situations,
under the additional restriction of isotropic distribution functions, i.e., f s共v , t兲 = f s共兩v兩 , t兲 , s = 1 , . . . , 4, that yields us = 0, and
consequently u = 0 and us = 0, as follows from (6) and (19),
respectively. We present here only a sample of the many
cases that have been run. The numerical values used in our
tests have to be considered as dimensionless, and corresponding to arbitrary scales. They have been chosen for illustrative purposes, without any reference to an actual specific problem.
Moments of the distribution functions f s needed in the
expression of Ms, namely, ns and Ts, have been numerically
evaluated in spherical coordinates by means of the composite
trapezoidal quadrature rule. To this end, the system has been
discretized in speed by using N knots on a sufficiently large
bounded interval 关0 , R兴, both N and R to be chosen depending on the initial conditions and on collisional parameters.
The resulting time-dependent 4N-dimensional system of ordinary differential equations has been numerically solved by
an adaptive method based on second- and third-order
Runge–Kutta formulas, with an accuracy of 10−6.
As reference case we have chosen a gas with initial conditions given by Maxwellian distributions with the following
macroscopic parameters
n1共0兲 = 10,
n2共0兲 = 12,
n3共0兲 = 14,
n4共0兲 = 13,
T1共0兲 = 4,
T2共0兲 = 4.3,
T3共0兲 = 3.7,
共36兲
T4共0兲 = 3.5.
The set of physical parameters characterizing elastic scattering between pairs of components are given in Table I. The
FIG. 4. Differences between actual
number densities ns (solid lines) and
“fictitious” densities ns (dashed-dotted
lines) for small times.
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4280
Phys. Fluids, Vol. 16, No. 12, December 2004
M. Groppi and G. Spiga
FIG. 5. Differences between actual
temperatures Ts (solid lines), and “fictitious” temperatures Ts (dasheddotted lines) [ s = 1 , 3 (a), s = 2 , 4 (b)]
for small times.
collision frequency affecting the chemical reaction has been
chosen as
34
␯12
= 0.01,
namely, one order of magnitude smaller than the typical mechanical collision frequencies, with energy difference
⌬E = 10 between products and reactants. Finally, masses take
the following values:
m1 = 11.7,
m2 = 3.6,
m3 = 8,
m4 = 7.3.
共37兲
For this problem (say, problem A), Fig. 1 shows the
trend of the macroscopic parameters ns and Ts versus time, as
recovered by integration from our computed distribution
functions f s. They reproduce exactly, to computer accuracy,
the results obtained in Ref. 9 by Grad’s approach for the
same physical situation. Such results, being relevant to Maxwellian molecules, are indeed exact. Referring the interested
reader to that paper, it is only worth observing here that the
prevailing reaction is exothermic, and that equalization of
temperatures occurs on the short time scale of elastic scatter-
ing, whereas relaxation of densities and global temperature
to their equilibrium values takes place at the slower pace of
chemical reaction.
Similarly, the actual distribution functions f s are expected to approach a Maxwellian shape according to the
scattering characteristic time, and then to relax to chemical
equilibrium according to the trend of the hydrodynamic
quantities. This is clearly seen in the three-dimensional plot
of Fig. 2, relevant to s = 3. An even better description of the
deviation of a distribution function from a Gaussian shape is
obtained by considering the differences f s − Ms, where Ms is
the Maxwellian sharing with f s the same moments up to
temperature. This is shown, for s = 2, in Fig. 3, where the
difference is zero initially, due to the chosen initial condition,
then undergoes a fast and relatively intense transient, and
eventually resumes the value zero, very well approximated
already for quite short times, of the order of the scattering
relaxation time, and even smaller than that (about one order
of magnitude). The absolute deviation from the corresponding Maxwellian remains in addition quite small.
FIG. 6. Equilibrium distribution functions f s compared to the relevant initial shapes for problem B.
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Phys. Fluids, Vol. 16, No. 12, December 2004
A BGK-type approach for chemically reacting gas
4281
FIG. 7. Distribution function f 3 vs t
and 兩v兩 for problem B.
The strength of the BGK collision term during the evolution, and the distance from equilibrium, are measured by
the difference Ms − f s, and then one is led to compare the
auxiliary parameters ns and Ts, characterizing the present approach, to the physical quantities ns and Ts. That is what
Figs. 4 and 5 are about. Notice that the inequality ns ⬍ ns
共ns ⬎ ns兲 always implies ṅs ⬍ 0 共ṅs ⬎ 0兲, as physically expected, and as predicted by the relevant weak forms of (14).
The same argument applies to temperatures, even in the case
when the fast equalization process in the initial layer makes
Ts nonmonotonic. This is clearly shown for s = 1 , 3 in the left
part of Fig. 5, where T1 becomes monotonic, and by s = 2 , 4
in the right part, where T2 is nonmonotonic. The relative
discrepancies between “actual” and “fictitious” macroscopic
observables remain again quite modest in all cases, and tend
to vanish, of course, for t → ⬁.
A second test problem (problem B) is obtained by simply
changing the initial conditions from Gaussian distributions to
the triangular shapes (tent functions) of Fig. 6. Now all mol-
ecules are initially confined to a compact support, and low
speeds are underpopulated. Widths and heights of these linear splines are determined in such a way that all ns共0兲 and
Ts共0兲 are the same as for problem A, and their determination
is unique. Therefore, since moment equations are closed in
our setting, all ns and Ts evolve in time exactly in the same
way as for problem A, as it has been checked by integrating
the outputs f s of our computations. The same occurs then to
ns and Ts, which will not be shown either. Figure 6 shows
also the final equilibrium shape of the different distribution
functions.
The variation with respect to the reference case affects
only the kinetic level, and it can be seen in the plot of f 3 of
Fig. 7, for which the same comments of Fig. 2 are in order.
Figure 8 is instead the counterpart of Fig. 3, and the comparison shows very clearly the expected new feature: deviations from a Gaussian shape, still confined in a very initial
stage, become much larger, about three orders of magnitude,
than for problem A, due to the strong initial distortion. An-
FIG. 8. Zoom of the deviation f 2
− M2 vs t and 兩v兩 for problem B.
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4282
Phys. Fluids, Vol. 16, No. 12, December 2004
M. Groppi and G. Spiga
FIG. 9. H functional vs t (a) and its
zoom in the initial transient (b), in the
different cases of problem A (solid
line) and of problem B (dashed line),
respectively.
other indication of this fact is provided by the numerically
computed H functional (10), given in Fig. 9. The trend is
monotonically decreasing in both situations, and the
asymptotic limits coincide. In fact, initial conditions determine uniquely the final equilibrium, but only via ns共0兲 and
Ts共0兲, which turn out to be the same for the two different
cases. However, as expected, problem B shows an extremely
faster initial descent, starting from a quite higher initial
value, due to its non-Maxwellian initial data. The two curves,
however, tend to coincide very soon in the evolution process.
In order to test finally the role played by the parameters
␯s in our model, we have run Eq. (14) for problem B, but
doubling or halving such parameters (problems C1 and C2,
respectively) with respect to the “physical” option (32). Now
we can predict, by the same arguments as before, that nothing changes in ns共t兲 and Ts共t兲, which remain the same as for
problems A and B. Therefore, the differences ns − ns and Ts
− Ts must be halved 共C1兲 and doubled 共C2兲 with respect to
problem B. This is confirmed indeed by our calculations,
and, as an example, Fig. 10 exhibits density differences versus time for one reactant and one product in the three different problems.
As regards the relaxation of f s, the variations of the factors ␯s are expected to double or halve their time derivatives,
but, by construction of the model itself, M s depends on ␯s in
such a way that the first fundamental moments ns and Ts are
kept independent from those parameters, so that variations
can affect only higher-order moments, and should then be
confined to the elastically dominated initial layer. Again, this
is confirmed by our results, and is shown here by Fig. 11,
comparing the trend of the H functional for problems B, C1,
and C2. The initial point is of course the same, and differences are once more restricted to a very initial stage, much
shorter than the temperature equalization time. After this
stage the three curves overlap, and approach then the common asymptotic equilibrium value. Obviously, the larger the
␯s, the faster the descent; in any case, relative variations are,
even in the transient region, of the order of the percent, or
less. In a sense, the facts above indicate a comforting stability of the model with respect to variations of these control
parameters over a reasonably wide range. However, the situation would change dramatically if we pushed variations beyond certain limits, though our experiments seem to indicate
that these limits are quite broad. For instance, in the preceding section we anticipated a possible loss of positivity for the
temperature fields Ts when ␯s become small. Indeed, we verified numerically that this actually occurs, and results lose
any meaning in few time steps, if such collision frequencies
FIG. 10. Differences ns − ns vs t 共s
= 2 , 3兲 when the ␯s are doubled (problem C1) or halved (problem C2) with
respect to the reference case of problem B.
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Phys. Fluids, Vol. 16, No. 12, December 2004
A BGK-type approach for chemically reacting gas
4283
FIG. 11. Zoom of the H functional vs
t in the initial transient, in the case of
problem B, when parameter ␯s are
doubled and halved with respect to the
reference case.
are suitably reduced with respect to problem B. The breakdown occurs, for our set of data, for a reduction factor ranging between 19 and 20.
V. CONCLUSIONS
We have extended a recent advanced nonlinear BGK
model for gas mixtures6 to the case of a chemically reacting
mixture. The machinery has become necessarily much more
complicated, but nevertheless it has been possible to prove
theoretically most of the significant features holding in the
non reactive case, and that are essential for any model in
order to mimic properly the relevant kinetic equations. In
particular, the correct rates of exchange for mass, momentum, and energy of all species are reproduced, and the correct
equilibria, including mass action law, are recovered. In addition, indifferentiability6 is verified as soon as the chemical
reaction is switched off. Unfortunately, due to the quite cumbersome machinery, we have been not able to extend the
theoretical proof of the H theorem of Ref. 6 to the reactive
case, nor to devise a different proof. Work is in progress in
that direction, and we hope to get some results in a not-toofar future. However, we are quite confident about the dissipativity of our model equations, as shown also by the extensive numerical calculations of the pertinent H functional (10)
along the solutions. A sample of our test computations, restricted for now to isotropic space-homogeneous conditions,
is shown in the paper, and these preliminary results seem to
be promising. In particular, the scheme exhibits a remarkable
stability with respect to variations of important control parameters such as the macroscopic collision frequencies, provided they are chosen in the relevant physically meaningful
ranges. Future work will concern the obvious extension to
space-dependent situations, like the classical evaporationcondensation and Riemann problems, and comparison to
available kinetic calculations.
ACKNOWLEDGMENTS
This work was performed in the frame of the activities
sponsored by MIUR (Project “Mathematical Problems of Kinetic Theories”), by GNFM-INdAM, 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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