Numerical Solution Of One-Dimensional Problems In Binary
Gas Mixture On The Basis Of The Boltzmann Equation
A. A. Raines
St. Petersburg State University, Department of General Mathematics and Informatics
Abstract. One-dimensional problems for a binary gas mixture are studied on the basis of numerical solution of the complete
kinetic Boltzmann equation. The collision integral is evaluated by conservative discrete ordinate method.
DESCRIPTION OF THE METHOD AND THE BASIC EQUATIONS
Basic equations
In this paper we consider the problems for binary gas mixtures: shock wave structure and heat transfer between two
parallel plates with diffusive reflection on the wall. Both of them are important and classical problems in rarefied gas
dynamics and have been investigated experimentally [1, 2] and theoretically [3, 4, 5, 6, 7, 8, 9, 10, 11] by different
methods.
For evaluation of the collision integral we apply a generalization of the conservative discrete ordinate method (the
kernel method for a single gas was developed by Tcheremissine [12]) for binary gas mixtures and for the case of
cylindrical symmetry [13]. This discrete ordinate method ensures strict conservation of mass, momentum and energy.
The conservativeness of the method is achieved by special projecting of integrand values calculated at non-node points
to the nodes of the momentum grid closest to them. For the evaluation of collision integrals we use the algorithm
which includes the "inverse collisions".
The system of Boltzmann equations in the momentum space for two gas components consisting of hard sphere
molecules with the masses mi and the diameters di may be written as
∂ fi
~p ∂ fi
+
= Ii = −Li + Gi ,
∂t
mi ∂~x
i = 1, 2
(1)
where fi is the distribution function which depends on the vector of momentum ~p, the vector of configuration space ~x
and the time t. Here the "loss" term Li and the "gain" term Gi are defined as
2
Li =
∑
Z+∞Z2πZπ
j=1−∞
0 0
+∞
2π π
2 Z Z Z
Gi =
∑
j=1−∞
0 0
1
fi f j∗
2
0 1
fi0 f j∗
di + d j
2
2
2
di + d j
2
q ji sin θ dθ dϕd~p∗
(2)
q ji sin θ dθ dϕd~p∗ .
(3)
2
In kinetic momentum space we have the following equalities for the vectors of momentum ~p,~p∗ before the collision
and ~p 0 ,~p∗0 after the collision
2mi m j ~p 0 = ~p +
~g ji ·~n ~n
(mi + m j )
2mi m j ~p∗0 = ~p∗ −
~g ji ·~n ~n
(4)
(mi + m j )
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
67
~p∗
~p
− ,
m j mi
~g ji =
~n =~n(cos θ , sin θ cos ϕ, sin θ sin ϕ).
Here q ji = | ~g ji ·~n |, ~n is a unit vector directed along the interaction line of molecules, ~g ji is their relative velocity, θ
and ϕ are collision angles. Here fi is the distribution function which depends on the vector of momentum ~p, the vector
of configuration space ~x and the time t.
We introduce in the limited domain of the physical space a fixed grid~x and impose limits on the momentum variables
in (1) – (3) by introducing a domain Ω of the volume V . In Ω we construct a discrete grid containing N0 equidistant
points ~pβ with the step ~h.
The Boltzmann equation in a discrete form becomes
∂ fiβ
∂t
+
~pβ ∂ fiβ
mi ∂~x
= Iiβ = −Liβ + Giβ .
(5)
Equation (1) is solved by the splitting procedure. On each interval ∆t we split the process into the two stages: freemolecular flow and collisional relaxation. For the first stage we have the divergence finite-difference scheme of the
2nd order suggested by Boris [14]. In this scheme the transport of mass, momentum and energy between the cells of
the configuration space is realized in a conservative way. For the second stage we apply the Euler’s method which
provides the satisfaction of conservation laws in each x node. The steady problems were solved by the stabilization
method based on the splitting procedure.
Conservative discrete ordinate method
Let us consider the integral operator
2
Qi (Φ) =
∑
Z+∞Z+∞Z2πZπ
Φ fi f j∗
j=1−∞ −∞
0 0
1
2
di + d j
2
2
q ji sin θ dθ dϕd~p∗ d~p
Taking for Φ a three dimensional δ -function, we reduce the integrals to the following form
1
Qi δ (~p ∗ −~p) + δ (~p ∗ −~p∗ )
2
1
∗
Gi (~p ) = Qi δ (~p ∗ −~p 0 ) + δ (~p ∗ −~p∗0 )
2
Li (~p ∗ ) =
(6)
(7)
Evaluating the integrals (6), (7), we exclude those values of variables θ , ϕ which remove the vectors ~p 0 ,~p∗0 out of the
domain Ω × Ω. For this purpose we multiply the integrand by the characteristic function χ(~p 0 ,~p∗0 ) where χ = 1 if
(~p 0 ,~p∗0 ) ∈ Ω × Ω and χ = 0 if (~p 0 ,~p∗0 ) 6∈ Ω × Ω.
While transforming the integral operator to cylindrical coordinates one should take into account the relations
~p = ~p(p, ρ, γ), d~p = ρd pdρdγ. The integral operator becomes
2
Qi (Φ) =
∑
j=1
Z2πZπ
Z
Ω×Ω 0 0
1
Φ fi f j∗
2
di + d j
2
2
q ji χ sin θ dθ dϕd~p∗ d~p
In cylindrical coordinates each node of the momentum grid will have two coordinates ~pβ (pβ , ρβ ) with the step (h, h). It
is important to note that the method needs the same grid with a constant mesh in momentum space for all components
of the mixture. At the same time the steps in each momentum coordinate could be different. We construct the cubature
formula for the evaluation of the integral Iiβ satisfying the conservation laws
∑ ψβ Iiβ = 0,
ψβ = {1,~pβ ,~p2β }
β
68
and also the condition
Iiβ ( fiMβ , fiMα ) = 0
at each nod ~pβ of the momentum grid. Here fiMβ is the Maxwellian distribution function at the nod ~pβ .
Introduce the uniform integration grid pαν , ραν , pβ , ρβ , γν , γ∗ν , θν , ϕν with Nν nodes in such a way that pαν , ραν
ν
ν
and pβ , ρβ belong to the momentum grid while the angles are distributed uniformly in the corresponding intervals.
ν
ν
The multiple integral is calculated as the 8-fold sum over all the nodes while the distribution functions do not depend
on γν , γ∗ν . One obtains the following result
Nν
e (~p ∗ ) = A ∑
L
i
2
∑ Jνi j
ν=1 j=1
Nν
e (~p ∗ ) = A ∑
G
i
2
∑ Jνi j
ν=1 j=1
δ (~p ∗ −~pαν ) + δ (~p ∗ −~pβ )
(8)
δ (~p ∗ −~pα0 ν ) + δ (~p ∗ −~pβ0 ν )
(9)
ν
where
Jνi j
= fiαν f jβ
ν
1
2
diαν + d jβ
!2
ν
q jβ
ν iαν
2
sin θν ραν ρβ
ν
and A = V 2 π 2 /Nν .
The integral Liβ is calculated straightforwardly
Nν
e =B ∑
L
iβ
2
∑
0
Jνi j + Jνi j
00
(10)
ν=1 j=1
Here B = V π 2 /(Nν /N0 ), the primes and double primes mark the values Jνi j with αν = β and βν = β , respectively.
It is not possible to use the same way for obtaining the values of Giβ because the singularities in the formula (9) do
not coincide with the nods of the momentum grid. The correct way of evaluating the values of Giβ is to decompose
each term in the parentheses in (9) into δ -functions which have singularities at discrete ordinate nodes. Here ~pλ and
ν
~pµ are the nodes nearest to ~pα0 and ~p 0 respectively. We introduce the notation ~∆ = ~pα0 − ~p , ~∆µ = ~p 0 − ~pµ
ν
ν
From (4) we have
βν
λν
ν
λν
ν
βν
~∆ = −~∆µ
ν
λ
ν
(11)
ν
e
e ,∆
e
e
We denote ~∆ = ~∆λ /h = (∆1 , ∆2 ), ~∆ = ~∆µν /h = (∆
1 2 ) where |∆k | ≤ 1/2 and |∆k | ≤ 1/2 for k = 1, 2. Let
ν
us introduce the vectors of displacements along the grid ~s (δ1 , δ2 ) and ~es (δe1 , δe2 ) where δk = 0 or δk = sign(∆k ),
e ) for k = 1, 2. Then the 2 sets of the 4 nodes surrounding the points ~pα0 and ~p 0 can be
δe = 0 or δe = sign(∆
k
k
presented as ~pλ
k
ν +~s
and ~p
ν
µν +~es
βν
respectively. These nodes form two cells being their vertices. From (11) we have
~es = −~s. However, in cylindrical coordinates on the momentum grid we have only one equality for the first coordinates
e = −∆ =⇒ δe = −δ . We replace the expressions in parentheses in (9) by
∆
1
1
1
1
δ (~p ∗ −~pα0 ν ) + δ (~p ∗ −~pβ0 ν )
= ∑ r~s δ (~p ∗ −~pλ
ν +~s
~s
) + δ (~p ∗ −~p
(12)
e)
µν +~s
The coefficients r~s can be found from the conditions of conservation of density, kinetic momentum and energy in the
decomposition (12) for a pair of above mentioned cells including their vertices. For economy of computations it would
be preferable to have a decomposition with a minimum of terms. Such a solution exists and contains due to (11) only
two non-zero coefficients: the first one (1 − r~s ∗ ) corresponds to ~s = 0, ~es = 0 and the second one r~s ∗ corresponds to
some ~s =~s ∗ 6= 0, ~es = ~se∗ 6= 0 and depends on a combination of parameters in the energy equation. Thus, the expansion
69
(12) becomes
δ (~p ∗ −~pα0 ν ) + δ (~p ∗ −~pβ0 ν )
(13)
∗
∗
= (1 − r~s ∗ ) δ (~p −~pλ ) + δ (~p −~pµν )
ν
+ r~s ∗ δ (~p ∗ −~pλ +~s ∗ ) + δ (~p ∗ −~p
e∗ )
µν +~s
ν
If we use (13) in the formula (9), then we obtain
Nν
e =B ∑
G
iβ
2
∑
0
00 Jνi j + Jνi j (1 − rν )
ν=1 j=1
∗
+ Jνi j + Jνi j
∗∗ rν
(14)
The primes and double primes, asterisk and double asterisk mark the values of Jνi j with the subscripts λν = β , µν = β ,
λν +~s ∗ = β , µν +~se∗ = β , respectively and rν = r~s ∗ . The expressions (10) and (14) define the conservative discrete
ordinate method if the coefficients have been already found. The points of a cubature formula are determined with the
use of the 8-dimensional grid distributed uniformly over the unit hyper-cube as proposed by Korobov [15].
Algorithms for the evaluation of collision integrals
Let us call a "collision" a node of the integration formula. Such a node is represented in the momentum space by two
points which coincide for example with the nodes ~p1 and ~p2 of the momentum grid. The momenta after the collision
~p10 and ~p20 do not belong to the grid. According to the formula (10) this collision gives the contributions BJνi j to the
integral Liβ at the nodes ~p1 and ~p2 . It would give the same contributions to the integral Giβ at the points ~p10 and ~p20
only if they coincide with the nodes of the momentum grid. Since this condition is not satisfied, these contributions
are distributed in the ratio 0 ≤ (1 − rν )/rν ≤ 1 between the nodes ~p3 ,~p4 nearest to ~p10 and ~p20 and another pair of nodes
~p5 ,~p6 chosen from the vertices of the cells containing the points ~p10 ,~p20 . The nodes ~p5 ,~p6 and the coefficients of the
formula (14) are determined by algebraic equations and are defined by the parameters of collision. This algorithm uses
only "direct collisions". The main algorithm was optimized by introducing the "inverse collisions" for the evaluation
of the collision integrals as it was made and proved in [16] for a single gas. In this case to each collision from the
nodes (~p1 ,~p2 ) to the nodes (~p3 ,~p4 ,~p5 ,~p6 ) we added the collision from the points (~p10 ,~p20 ) not belonging to the grid
to the nodes (~p1 ,~p2 ). In short, in the main algorithm the contributions BJνi j are modified as B(Jνi j − Jνi j(inv) ) and we
choose the method of interpolation for determining Jνi j(inv) . In this work we use the linear interpolation which is more
fast for calculations than the logarithmical one. However, the linear interpolation does not give the equality to zero of
the collision integral of the Maxwell function but it provides its equality to zero only up to O(h2 ). Numerical results
presented at this paper were obtained by modified algorithm.
ONE-DIMENSIONAL PROBLEMS
The problem of the shock wave in a binary gas mixture was studied for the Mach numbers 1.5, 2, 3 for the ratio of
masses 0.1, 0.25, 0.5 and the concentrations (for the 2nd component) χ2− = 0.07, 0.1, 0.5, 0.9, 0.95, 0.999 (Fig. 1 –
Fig. 3). The macroscopic variables of the light component start to deviate from their up-stream uniform values earlier
than the corresponding variables of the heavy component. The numerical density and flow velocity (momentum) for
the light component reach their downstream uniform values earlier than for the heavy component. The temperature of
the heavy component displays the well-known phenomenon as the "overshoot of the temperature". This phenomenon
has already been shown by the computations [3, 5]. One can see it more clearly when the shock wave is not weak and
the concentration of the light component is large.
The accuracy of computations estimated by comparing macroscopic quantities for different lattice systems and
different number of integration nodes is 10−2 − 10−3 (percentage error). For Mach = 2, m2 /m1 = 0.25, d2 /d1 = 1 and
various concentrations we take the following values of the parameters: 10 658 nodes of the momentum grid (146 points
for p and 73 points for ρ) with the step h = 0.08, 100 nodes of the x-grid with the step hx = 0.2, 198 000 integration
70
nodes and ∆t = 0.01 (on the figures). For Mach = 3, m2 /m1 = 0.5, d2 /d1 = 1 and various concentrations we take
the following values of the parameters: 11 858 nodes of the momentum grid (154 points for p and 77 points for ρ)
with the step h = 0.1, 90 nodes of the x-grid with the step hx = 0.2, 198 000 integration nodes and ∆t = 0.01 (on the
figures). The period of stabilization of the solution is equal to 20τ − 27τ. With this method we can obtain the results
on rough grids (1 800 = 60 × 30 nodes of the momentum grid with the step h = 0.26, 90 nodes of the x-grid with the
step hx = 0.2, 66 000 integration nodes, ∆t = 0.01) which coincide well with more detailed results. The computing
time for one iteration step is 4.2 sec, for the whole problem — 2.7 hours.
Our results were compared with the results obtained in [5]. Figure 4 shows the profiles of macroscopic variables
for the mixture. Here we observe a good agreement. In [5] the computing time for one iteration step in a parallel
computation using ten CPU’s on Fujitsu VPP800 computer is 142 sec. for M− = 2 and 99 sec. for M− = 3. The
computer memory for M− = 2 is 1.7 GB, for M− = 3 is 1.4 GB. In this work the computations were made on a personal
computer with the processor Pentium 3 with the frequency 550 Mhz and the memory 128 Mb. The computing time
for one iteration step for M− = 2 is 36 sec., for the whole problem is 23 hours, computer memory is 8.5 Mb. The same
data for M− = 3 are 19.3 sec, 13 hours and 8.5 Mb respectively. One can see a good agreement between numerical
results obtained in the present study and experiments [1] (Fig. 5,6,7).
The problem of the heat transfer between two parallel plates is studied for m2 /m1 = 0.5, d2 /d1 = 1, TII /TI =
2, n2av /n1av = 1 (niav is the average molecular numerical density of i-component in the domain between two parallel
plates, 0 < X < D) and Kn = 0.1, 1, 10 (Fig. 8, Fig. 9).
This method for a mixture solves problems with acceptable preciseness, small time of calculations, small computer
memory and with a good agreement with theoretical and experimental data of other authors.
1.2
1
0.8
0.6
0.4
0.2
0
0
2
4
6
8 10
X/l_
12
14
16
18
FIGURE 1. Profiles of molecular number densities, flow velocities, and temperatures for M− = 3, m1 /m2 = 0.5, d2 /d1 =
1, χ2− = 0.07. Here the solid lines indicate n,U, T for the total mixture, the small dashed lines n1 ,U1 , T1 for the first component,
and the big dashed lines n2 ,U2 , T2 for the second component.
1.2
1
0.8
0.6
0.4
0.2
0
-0.2
0
2
4
6
8 10
X/L_
12
14
16
18
FIGURE 2. Profiles of molecular number densities, flow velocities, and temperatures for M− = 3, m2 /m1 = 0.5, d2 /d1 = 1, and
χ 2 − = 0.999. See the caption on the Fig.1.
71
1.2
1
0.8
0.6
0.4
0.2
0
-0.2
0
5
10
15
20
25
X/l_
FIGURE 3. Profiles of molecular number densities, flow velocities, and temperatures for M− = 2, m2 /m1 = 0.1, d2 /d1 = 1, and
χ 2 − = 0.5. See the caption on the Fig.1.
1
0.8
0.6
0.4
0.2
0
-0.2
-10 -8
-6
-4
-2
0 2
X/l_
4
6
8
10
FIGURE 4. Comparison with the results of Kosuge et. al. [5]: profiles of molecular number density, flow velocity and temperature
for the mixture: M− = 3, m2 /m1 = 0.5, d2 /d1 = 1, and χ2− = 0.9. The results obtained by the method of Kosuge et. al. are shown
by the rhombuses (n), triangles (U), and squares (T ). The results of the present method are shown by +, ×, and ∗ respectively
1.7
1.6
1.5
1.4
1.3
1.2
1.1
1
0.9
0
10
20
30
40 50
X/l_
60
70
80
90
FIGURE 5. Profiles of parallel and radial temperatures for argon for M− = 1.58, m2 /m1 = 0.1, d2 /d1 = 0.593, and χ2− = 0.9.
The results obtained by Harnett and Muntz are shown by squares (T|| ), and rhombuses (T⊥ ). The results of the present method are
shown by × and + respectively.
72
1.6
1.5
1.4
1.3
1.2
1.1
1
0.9
0
FIGURE 6.
10
20
30
40 50
X/l_
60
70
80
90
Profiles of parallel and radial temperatures for helium with the same parameters and notation as in Fig.5.
1.6
1.5
1.4
1.3
1.2
1.1
1
0.9
0
10
20
30
40 50
X/L_
60
70
80
90
FIGURE 7. Profiles of total temperature for argon and helium for M− = 1.58, m2 /m1 = 0.1, d2 /d1 = 0.593, and χ2− = 0.9. The
results obtained by Harnett and Muntz are shown by squares (Targon ), and rhombuses (Thelium ). The results of the present method
are shown by + and × respectively.
1.5
1.4
1.3
1.2
1.1
1
0.9
0.8
0.7
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
X/D
FIGURE 8. Profiles of number densities n1 and n2 for Kn = 0.1, 1, and 10; m2 /m1 = 0.5, d2 /d1 = 1, TII /TI = 2, n2av /n1av = 1.
niav is the averaged molecular number density of the i-component in the domain 0 < X < D. Here for Kn = 0.1: n1 — rhombuses,
n2 — ∗; for Kn = 1: n1 — squares, n2 — ×; for Kn = 10: n1 — triangles, n2 — +.
73
2
1.9
1.8
1.7
1.6
1.5
1.4
1.3
1.2
1.1
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
X/D
FIGURE 9.
squares.
Profiles of the total temperature for the mixture. Here for Kn = 0.1 — rhombuses, for Kn = 1 — +, for Kn = 10 —
REFERENCES
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
Harnet L.N., Muntz E.P. Experimental investigation of normal shock wave velocity distribution functions in mixtures of
Argon and Helium, Phys. Fluids 10 (1972) 565–572.
Romani F.A. Der Entmischungeffekt infolge Thermodiffusion in mässig verdünnten Gasen, Thesis-RWTH Aachen, 1979.
Bird G.A., Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Oxford Univ. Press, Oxford, 1994.
Fernandez-Feria R., Fernandez de la Mora J., Shock wave structure in gas mixtures with mass disparity, J. Fluid Mech. 179
(1987) 21–40.
Kosuge S., Aoki K., Takata S. Shock-wave structure for a binary gas mixture: finite-difference analysis of the Boltzmann
equation for hard sphere molecules, Eur. J. Mech. B - Fluids 20 (2001) 87–126.
Kosuge S., Aoki K., Takata S. Heat transfer in a gas mixture between two parallel plates: finite-difference analysis of the
Boltzmann equation, Eur. J. Mech., B/Fluids (submitted).
Inamuro T., Sturtevant B., Numerical study of discrete-velocity gases, Phys. Fluids A 2 (1990) 2196–2203.
Rogier F., Schineider J.A., Direct method for solving the Boltzmann equation, Trasport Theory and Stat.Physics, 23, No.1–3
(1994) 313–338.
Buet C. Conservative and entropy schemes for the Boltzmann collision operator of polyatomic gases, Math. Models Methods
Appl. Sci. 7 (1997) 165.
Oguchi H., Hatakeyama M., Honma H., Computational aspects of discrete ordinate-velocity description of rarefied gases, in:
Proc. of 19th Internat. Symp. on RGD, Vol. 2, Oxford, 1995, pp. 815–821.
Bobylev A.V., Palczewski A., Schneider J.A., A consistency result for a discrete velocity model of the Boltzmann equation,
Siam J. Numer. Anal 34, No.5 (1997) 1865–1883.
Tcheremissine F.G. Conservative evaluation of Boltzmann collision integral in discrete ordinates approximation, Computers
Math. Applic., 1998, v. 35, pp. 215–221.
Raines A.A. A method for solving Boltzmann equation for a gas mixture in the case of cylindrical symmetry in the velocity
space, Zh. Vychisl. Mat. Mat. Fiz. 42 (2002) 1308–1320.
Boris J.P.and Book D.L. Flux-corrected transport, J. Comp. Phys., 1973, v. 11, pp. 38–69.
Korobov N.M. Trigonometric sums and their applications, Mir, Moscow, 1989.
Cheremisin F.G., Solving the Boltzmann equation in the case of passing to the hydrodynamics flow regime, Doklady Physics
45, No. 8 (2000), 401–404.
74