Dynamic Molecular Collision (DMC) Model for General
Diatomic Rarefied Gas Flows
Takashi Tokumasu∗ , Yoichiro Matsumoto† and Kenjiro Kamijo∗∗
†
∗
Institute of Fluid Science, Tohoku University
Department of Mechanical Engineering, The University of Tokyo
∗∗
Institute of Fluid Science, Tohoku University
Abstract.
The Dynamic Molecular Collision (DMC) model can accurately estimate energy transfer between the translational and
rotational degrees of freedom at a collision. In this model, a probability density function (PDF) of energy after collision at
each degree of freedom is modeled using an exponential function. Properties of the model are obtained by results of the
Molecular Dynamics (MD) method. A total and inelastic collision cross section are also constructed. The defect of the model
is that a large number of binary collisions of diatomic molecules have to be simulated in advance in order to construct a table
of the properties. In this paper, the dependence of initial energy on the properties of the DMC model is analyzed in detail and
some relations between these properties and the initial energy are obtained. Using these results, each property is expressed by
fundamental functions of the initial energy. In order to verify the validity of the model function, equilibrium or nonequilibrium
flows are simulated by the model and the results are compared with theoretical or experimental results.
INTRODUCTION
Simulations of highly nonequilibrium flows such as freejets or shock waves are becoming mechanically important.
Relaxation of diatomic molecules in nonequilibrium flows is very different from that of monatomic molecules due to
the internal degrees of freedom. It is important to study the effect of the internal degree of freedom upon the energy
transfer between colliding diatomic molecules. The Direct Simulation Monte Carlo (DSMC) method is the best scheme
to analyze these flows [1] [2], and based on previous research, monatomic rarefied gas flows can be simulated by the
method. However, diatomic rarefied gas flows are difficult to simulate because energy can be transferred between
translational and rotational degrees of freedom in the flows. In this case the accuracy of the method highly depends on
a model function which determines the amount of energy transfer between these degrees of freedom.
The authors have constructed the Dynamic Molecular Collision (DMC) model [3] for nonpolar diatomic molecules
to calculate the amount of energy transfer which occurs at a collision of diatomic molecules. This model is constructed
based on collision dynamics, and it is found that strong nonequilibrium flows such as shock waves can be simulated by
this model without adjustable parameters. The defect of the model, however, is that a large number of binary collisions
of diatomic molecules have to be simulated by the Molecular Dynamics (MD) method in advance in order to construct
a table of the properties of the model function. The model is very useful for obtaining the properties by fundamental
functions and moreover, the model can be easily extended to various diatomic molecules.
In this paper, the dependence of initial energy on the properties of DMC model are analyzed in detail and these
properties are expressed by fundamental functions of initial energy. Nitrogen is used as a collision molecule. In order
to verify the validity of the model functions, equilibrium or nonequilibrium flows are simulated by the model and the
results are compared with theoretical or experimental results.
DYNAMIC MOLECULAR COLLISION (DMC) MODEL
A large number of collision of diatomic molecules must be simulated by the MD method to construct the DMC model.
Nitrogen, N2 , molecules are used as the collision molecules. Both collision molecules can be assumed to be rigid
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
312
Probability Density Function [-]
rotors, and vibration and dissociation of the molecules can be neglected. Moreover, the quantum effect of rotational
energy can be ignored and the rotational energy can be assumed to be continuous. In the present study, the 2 center
Lennard–Jones (12–6) model [4] is used. The potential parameters, σa and εa , are determined as σa = 3.17 × 10−10 m
and εa = 6.52 × 10−22 J, respectively. The distance between atoms of a molecule is chosen to be l = 1.094 Å and the
mass of the nitrogen atom is set at ma = 2.32 × 10−26 kg [3].
Using the potential, a collision of two nitrogen molecules is simulated by the Molecular Dynamics method. Details
√
of the method are described in Ref. [3]. In the previous paper, the impact parameter, b, is determined by b = bmax R
so that b is distributed according to the probability proportional to b. However, this method is not appropriate because
the number of results at a smaller impact parameter which greatly influences the probability density function (PDF)
are relatively small, and therefore a larger number of MD simulations must be performed to obtain the PDF accurately.
In this paper, b is√determined by b = bmax R2 . Then the impact parameter is distributed according to the probability
proportional to 1/ b. The results are evaluated by multiplying the weighting factor of b3/2 so that the statistical results
in which b is distributed according to the probability proportional to b is obtained. The simulation mentioned above
is performed 80 000 times by changing the impact parameter, b, and initial Euler angle of molecule 1 or 2 to obtain
the total and inelastic collision cross section and the PDF of energy after collision at a combination of initial energy,
(etr , er1 , er2 ).
Using the data, the PDF is obtained in the same manner as in Ref. [3]. The shapes of the distributions are shown
by the dotted line in Fig. 1. In this figure, the initial energy is etr = 4.0εa , er1 = 8.0εa and er2 = 12.0εa , and energy
0.35
etr
0.30
0.25
0.20
er1
er2
10
15
0.15
0.10
0.05
0.00
0
5
20
Energy [-]
FIGURE 1. The PDF of energy after collision. Bold line: model function, Dotted line: MD results. Initial energy is etr = 4.0εa ,
er1 = 8.0εa and er2 = 12.0εa .
is reduced in εa . In the DMC model, the PDF of energy after collision is constructed by fitting the shape of the MD
results using the following exponential function [3]:

: left side
 Al exp{−Bl (ei − e )}
F e =
(1)
 A exp {−B (e − e )}
: right side,
r
r
i
where ei is the initial translational, rotational 1 or rotational 2 energy and e is the translational, rotational 1 or rotational
2 energy after collision. In the previous paper [3], the parameters of model function, Al , Ar , Bl and Br , are obtained
using the left and right side probabilities, Pl and Pr , and deviations, σl and σr . In the present paper, however, the left
and right side averages, Sl and Sr , are used instead of σl and σr considering the convenience of the modeling of Sl and
Sr mentioned in a later section. These properties, Pl , Pr , Sl and Sr , are obtained from MD results by
Pl =
Nl
,
Nl + Nr
Pr =
Nr
,
Nl + Nr
Sl =
Nl 1
e − ei
∑
Nl + Nr i=1
and Sr =
Nr 1
e − ei ,
∑
Nl + Nr i=1
(2)
and the parameters are obtained by
Al = −Pl2 /Sl ,
Ar = Pr2 /Sr ,
313
Bl = −Pl /Sl ,
and Br = Pr /Sr ,
(3)
where Nl is the number of molecules in which the energy after collision is less than ei , and Nr is the number of
molecules in which the energy after collision is greater than ei . The shapes of the model functions are shown by the
bold line in Fig. 1. It is found that the shape of the model function is similar to that obtained by MD data.
As mentioned above, the 6 parameters, dt , di , Pl , Pr , Sl and Sr , are determined by the result of MD data at a
combination of initial energy, (etr , er1 , er2 ). Sets of simulations are carried out for 858 combinations of initial energy
in the same manner as in Ref. [3] and tabulated. These properties are analyzed in a later section. In this section energy
is reduced in εa and length in σa .
CHARACTERISTICS OF THE PROPERTIES AND THEIR MODELING
Left and right side average of the probability density function
In the present paper, six paths of energy transfer are considered as shown in Fig. 2. In this figure, ∆eba denotes the
er1
∆etrr1
∆ertr1
etr
∆etrr 2
∆ertr2
er 2
∆err12
∆err12
FIGURE 2. Paths of energy transfer between each degree of freedom.
amount of energy transfer from a degree of freedom of a to another degree of freedom of b. The symbol, a or b,
denotes the degree of freedom of translation, rotation 1 or rotation 2. Using the amount of energy transfer, the left and
right side averages of the PDF of each degree of freedom are expressed by the following equations:
r1
r2
2Str
l = −∆etr − ∆etr ,
tr
tr
tr
2Sr = ∆er1 + ∆er2 ,
r2
Slr1 = −∆etr
r1 − ∆er1 ,
r1
r1
r1
Sr = ∆etr + ∆er2 ,
tr
Slr2 = −∆er1
r2 − ∆er2 ,
r2
r2
r2
Sr = ∆er1 + ∆etr .
(4)
In the present paper the amount of energy transfer is modeled and the average of the PDF is expressed using the model.
Considering that ∆eba is of first degree of energy and that it increases with the increase in the initial energy of degree
of freedom of a, ea , the form of the model function is assumed in the following equation:
∆eba = Fab (etr , er1 , er2 )ea ,
(5)
where Fab (etr , er1 , er2 ) denotes the efficiency of energy transfer from the degree of freedom of a to the degree of
freedom of b. In the present paper, the efficiency of energy transfer, Fab , is modeled by fundamental functions and Sl
and Sr are expressed using ∆eba .
The amount of energy transfer from rotation to rotation. First, the amount of energy transfer from rotation to
rotation is analyzed. Using Eq. (4), the equation for the amount of energy transfer from rotation to rotation mentioned
below is obtained.
r1
r1
r2
tr
∆er2
(6)
r1 + ∆er2 = Sr + Sr + 2Sl .
r1
The typical distribution of ∆er2
r1 + ∆er2 is shown by the dotted line in Fig. 3. In this figure, the initial translational energy
is etr = 4.0. Based on the distribution, the amount of energy transfer from rotation 1 to rotation 2, ∆er2
r1 , is modeled.
r1 at each e can be expressed well when the efficiency,
After repeated trial and error, the distribution of ∆er2
+
∆e
tr
r1
r2
Fr1r2 , is modeled as the sum of two planes. It is assumed that one of the planes expresses the distribution at er1 < er2
r1
and it includes the three points of (er1 , er2 , ∆er2
r1 + ∆er2 )=(0, 0, 0), (0, 40, 0) and (40, 40, 0.5B) and that the other plane
expresses the distribution at er1 > er2 and it includes the three points of (0, 0, 0), (40, 0, A) and (40, 40, 0.5B), where
314
r1
r2
+∆ e r2
∆ e r1
7
6
5
4
3
2
1
0
-1
0
5
10
15
20
e r1
25
30
35
40
0
5
10
15
20
25
30
35
40
e r2
r1
FIGURE 3. Typical distribution of the amount of energy transfer, ∆er2
r1 + ∆er2 . Dotted line: MD result, Bold line: model function.
The initial translational energy is etr = 4.0
r1
A and B are the value of ∆er2
r1 + ∆er2 at (er1 , er2 ) = (40, 0) and (40, 40), respectively, at each etr . The efficiency,
r2
Fr1 (etr , er1 , er2 ), is expressed by the following equation:
r2
(etr , er1 , er2 )
Fr1
=
 B


 80 ,
(er1 < er2 ),
(7)


 A (1 − x) + B x,
40
80
x = er2 /er1
(er1 ≥ er2 ),
r1
In general, A and B are functions of etr . However, it is confirmed that the value of ∆er2
r1 + ∆er2 at er1 = er2 hardly
changes although etr changes, and therefore B can be chosen to be constant. The value, A(etr ), is chosen so that
r1
∆er2
r1 + ∆er2 obtained by the model are consistent with the MD data by least square fitting. The values, A(etr ) and B,
are expressed by
b
A = aetr
,
(a = 1.142, b = 0.4341),
B = 6.196.
(8)
r1
The value, ∆er2
r1 + ∆er2 , obtained by the model is shown by the bold line in Fig. 3. As shown in this figure, this model
can express the distribution well.
The amount of energy transfer from rotation to translation. Using ∆er2
r1 modeled above, the amount of energy
r1
r2
transfer from rotation to translation, ∆etr
,
is
modeled.
From
Eq.
(4),
the
equation
∆etr
r1
r1 = −Sl − ∆er1 is obtained.
tr
The dotted line in Fig. 4 shows the typical distribution of ∆er1 obtained by MD simulations. In this figure, the left
tr
tr
∆ e r1
∆ e r1
4
3.5
3
2.5
2
1.5
1
0.5
0
-0.5
0
6
5
4
3
2
1
0
5 10
15 20
25 30
35 40
e r1
FIGURE 4.
0
5
40
35
30
25
20
15
10
e r2
0
5 10
15 20
25 30
35 40
e r1
0
5
40
35
30
25
20
15
10
e r2
Amount of energy transfer, ∆etr
r1 . Dotted line: MD results, Bold line: model function. Left: etr = 1.0, Right: etr = 12.0.
side shows the distribution at etr = 1.0 and the right side shows that at etr = 12.0. As shown in this figure, the shape
of the distribution is almost a plane at higher translational energy but changes complicatedly at lower translational
tr (e , e , e ) is modeled by dividing the efficiency into two contributions. The one is the
energy. In this paper, Fr1
tr r1 r2
tr ) . The
contribution of energy transfer from rotation 1 to translation directly, the efficiency being denoted by (Fr1
1
tr
other is the contribution via rotation 2, the efficiency being denoted by (Fr1 )2 . Moreover, the efficiencies decrease as
tr ) is relatively small when e is large because
x = log(er1,2 /etr ) increases. It is considered that the contribution of (Fr1
2
r2
315
tr ) , is therefore expressed from the
x = log(er2 /etr ) becomes large at lower translational energy. The efficiency, (Fr1
1
tr
at er2 = 40.0 at each
MD data at which er2 = 40.0. The left side of Fig. 5 shows the distribution of efficiency of Fr1
etr against x = log(er1 /etr ). The distributions are normalized using the maximum value at each translational energy,
0.40
1.0
0.35
etr=0.5
0.30
0.8
Fr1tr/h
Fr1tr
0.25
0.20
0.15
0.10
0.4
0.2
0.05
etr=20.0
0.00
-0.05
-5
0.6
-4
-3
-2
-1
0
0.0
1
2
3
4
5
-0.2
-5
log(er1/etr)
FIGURE 5.
-4
-3
-2
-1
0
1
2
3
4
5
log(er1/etr)-xo
Efficiency of energy transfer at er2 = 40.0 at each etr .
tr /h). The normalized values of these
h(etr ), and the x value at which the maximum value is obtained, xo , as (x − xo , Fr1
distributions are shown in the right side of Fig. 5. This distribution is modeled by

2

(x < 0) (σl = 1.761),

 0.65 exp − (x/σl ) + 0.35,
y(x) =
(9)


 exp − (x/σr )2 ,
(x ≥ 0) (σr = 1.350).
This model function is shown by the black line in the right side of Fig. 5. As shown in this figure, this model can
express the normalized value of the distribution well. Using the equation, the efficiency from rotation 1 to translation
directly is expressed by
tr
)1 (etr , er1 , er2 ) = h(etr )y {x − xo (etr )} ,
(Fr1
where
−b
+ c,
h(etr ) = aetr
−b
xo (etr ) = aetr
+ c,
x = log(er1 /etr ),
(a = 0.111, b = 1.022, c = 0.165),
(a = 0.830, b = 0.622, c = 0.2).
(10)
(11)
tr ) , is obtained by F tr − (F tr ) . In order to express the shape of the distribution, the following model
The efficiency, (Fr1
2
r1
r1 1
function is used:

2

e
1
1

r2

,
(er1 ≥ er2 ),
−
−
 A(etr )
er1 2
4
tr
(Fr1
)2 (etr , er1 , er2 ) =
(12)




0,
(er1 < er2 ),
where
A(etr ) = −0.1exp (−aetr + b),
(a = 0.1856, b = 1.3760).
(13)
The distribution obtained by the model function is shown by the bold line in Fig. 4. As shown in this figure, this
model can express the tendency that the shape of the distribution is almost a plane at higher translational energy
but changes complicatedly at lower translational energy. Moreover, this model can also express the tendency of the
variation even lower translational energy. However, this model cannot express the distribution quantitatively. The effect
of the difference is discussed in a later section.
The amount of energy transfer from translation to rotation. Finally, the amount of energy transfer from translation
r1 , is modeled. From Eq. (4), the equation ∆er1 = Sr1 − ∆er1 is obtained. The distribution of ∆er1 is
to rotation, ∆etr
tr
r
tr
r2
plotted and analyzed. It is confirmed that the distribution can be expressed by the sum of the two planes in a region
316
of 2etr < er2 and 2etr ≥ er2 , respectively, as with the distribution of ∆er2
r1 . In this paper, the efficiency of energy from
translation to rotation is expressed by

B


(2etr < er2 )

 20 ,
r1
(14)
∆Ftr =

A − B er2
A


 −
(2etr ≥ er2 )
+ ,
40 etr 20
r1 )=(0, 0, 0), (0, 40, 0) and
considering that the model function at 2etr < er1 includes the three points of (etr , er1 , ∆etr
(20, 40, B) and that at 2etr > er1 includes the three points of (0, 0, 0), (20, 0, A) and (20, 40, B), where A and B are the
values of ∆er2
r1 at each er1 . The values, A and B, are the functions of er1 and are expressed by
A(er1 ) = exp(aer1 + b) + c,
B(er1 ) = exp(aer1 + b) + c,
(a = −0.123, b = 1.278, c = 2.0),
(a = −0.197, b = 1.236, c = 1.0).
(15)
It is confirmed that this model can express the distribution of MD data well.
Left and right side probability of the probability density function
In this section, the left and right side probability of the PDF, Pl and Pr , are modeled. First the left side probability of
PDF of translational energy, Pltr , is modeled. The right side probability, Prtr , is obtained by Prtr = 1 − Pltr . The typical
distribution of Pltr is shown by the dotted line in the left side of Fig. 6. In this figure the initial translational energy is
pl r1
pltr
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
5 10
15 20
25 30
35 40
e r1
0
5
40
35
30
25
20
15
10
r2
e
0
5 10
15 20
25 30
35 40
e r1
0
5
40
35
30
25
20
15
10
e r2
FIGURE 6. The left side probability of the PDF. Dotted line: MD results, Bold line: model function. Left: probability of the PDF
of translational energy, Pltr , Right: probability of the PDF of rotational energy, Plr1 . The initial energy is etr = 8.0
etr = 8.0 As shown in this figure, the probability decreases rapidly and converges as the rotational energy increases at
lower translational energy. It is confirmed that the tendency becomes gentle as the translational energy increases. In
order to fit the shape of the distribution, the distribution mentioned above is expressed by the following equation:
Pltr (etr , er1 , er2 ) =
1−A
{exp(−Bx1 ) + exp(−Bx2 )} + A,
2
x1 = er1 /etr ,
x2 = er2 /etr ,
(16)
where A(etr ) is the minimum value of the distribution at each translational energy and B determines the degree of
decrease of the probability. These coefficients are determined by least square fitting and are obtained by
A(etr ) = − exp(aetr + b) + c,
(a = −0.382, b = −2.154, c = 0.33),
B = 1.5788.
(17)
The distribution of Pltr obtained by the model function is shown by the bold line in the left side of Fig. 6. As shown in
this figure, this model function can express the distribution obtained by the MD data well.
The typical distribution of left side probability of the PDF of rotational energy, Plr1 , is also plotted in the right side
of Fig. 6. As shown in this figure, the dependence of er1 on the probability is relatively small. For this reason, the
approximation function mentioned below is used to express the distribution.
Plr1 (etr , er1 , er2 ) = A(1 − exp(−Ber1 )).
317
(18)
The coefficient, A, is the converged value at er1 → ∞ and is obtained from the distribution of Plr1 at er1 = 40. The
coefficient, B, is obtained if A is obtained. Using A, B is expressed by exp(−Ber1 ) = 1 − Plr1 /A. The coefficient B is
obtained by least square fitting using the relation mentioned above. In the present paper, the coefficients, A and B, are
expressed by
(a = −2.258 × 10−3 , b = 0.6755),
A = aer2 + b,
(19)
B = exp(aetr + b) + c,
(a = −0.1228, b = −0.7750, c = 0.1).
The distribution of Plr1 obtained by model function is shown by the bold line in the right side of Fig. 6. As shown in
this figure, this model function can express the distribution obtained by MD data well.
Total and inelastic collision cross section
First the radius of inelastic collision cross section, di , is modeled. It is confirmed that the distribution of di at
higher rotational energy is a plane but it changes complicatedly by changing er1 and er2 at lower rotational energy. In
this paper the distribution of di is expressed by neglecting the change at the lower rotational energy for purposes of
simplicity. The model function of the distribution is obtained by
di (etr , er1 , er2 ) =
T −B
40A − T + B
er,max +
er,min + B,
40
40
er,max = max(er1 , er2 ),
er,min = min(er1 , er2 ), (20)
considering that the function is symmetrical with regard to the exchange of er1 and er2 and that it includes the three
points of (etr , er1 , er2 )=(0, 0, B), (40, 40, 40A + B) and (40, 0, T ). The coefficients, A, B and T are obtained by
A = aetr + b,
−b
B = aetr
+ c,
−b
T = aetr + c,
(a = 4.898 × 10−4 , b = −8.824 × 10−3 ),
(a = 0.941, b = 0.4318 × 10−3, c = 0.9),
(a = 0.757, b = 0.4993, c = 1.0).
It is confirmed that the model function can express the distribution of di except at lower rotational energy.
The total collision cross section is obtained by
∆etr 2 1 1 2
3QMD
3 2 1 N
(1)
(2)
2
− sin χ
σT =
= π bmax ∑ sin χ +
= σT + σT ,
(2etr )2
2
N i=1
etr
3 2
where
(1)
σT
1 N
3
= π b2max ∑ sin2 χ ,
2
N i=1
(2)
σT
3 2 1 N ∆etr 2 1 1 2
− sin χ .
= π bmax ∑
2
N i=1 etr
3 2
(21)
(22)
(23)
(1)
It is confirmed that the distribution of σT decreases exponentially as the translational energy increases and does not
(1)
greatly depend on the rotational energy. Therefore σT is expressed as only the function of etr by
(1)
−B
σT = Aetr
,
(A = 11.55, B = 0.4176).
(24)
(2)
The value σT changes greatly according to er1 and er2 at lower translational energy (etr < 0.5). However, the value
(2)
is relatively small compared with Q(1) at higher translational energy. In the present paper, therefore, the value σT is
neglected for the purposes of simplicity.
VERIFICATION OF THE MODEL
In order to verify the validity of the model, the translational and rotational energy distributions at equilibrium condition
are simulated by the DSMC method using our model and the results are compared with the theoretical results. The
details of the simulation method are described in Ref. [3]. The results are shown in the left side of Fig. 7. The
temperature is T = 300 K and pressure is P = 1.013 × 105 Pa. The number of molecules is N = 7338. As shown in
this figure, the rotational energy distribution at lower energy departs a little from the theoretical result. It is considered
318
2.0
1.8
1.8
1.6
1.6
: etr , DSMC
: erot , DSMC
: MB distribution
1.4
1.2
1.0
1.4
0.8
0.6
1.0
0.2
0.2
0.0
0.0
1.0
1.5
2.0
2.5
: Trot
(exp)
ρ
-0.2
-4
Trot
-3
-2
-1
0
1
2
3
4
Distance [-]
Energy [-]
FIGURE 7.
shock wave.
(exp)
Ttr , yz
0.6
0.4
0.5
: ρ
0.8
0.4
-0.2
0.0
Ttr , x
1.2
ρ, T
Probability Density Function [-]
that this is because the properties of the DMC model change complicatedly due to the change in the initial energy at
lower energy and because in this region this model function cannot express the properties of the DMC model well.
However, the whole distributions are consistent with theoretical results and therefore this model can be considered
to simulate equilibrium state at moderate temperature. Moreover, a one–dimensional normal shock wave is simulated
and the results are compared with experimental results. In this simulation, the upstream Mach number is Min = 7.0, the
upstream temperature is Tin = 28.78 K and the upstream pressure is Pin = 0.3704 Pa. Details of the simulation method
are also described in Ref. [3]. The results are shown in the right side of Fig. 7. As shown in this figure, the results are
consistent with experimental results, and therefore it can be said that this model can calculate normal shock wave well.
Verification of the model. Left: Equilibrium state at T = 300 K. Right: Shock profiles of a one–dimensional normal
CONCLUDING REMARKS
The properties of the DMC model were expressed by fundamental functions of initial energy in order to extend the
DMC model for various kinds of molecules. The impact parameter was distributed by R = bmax R2 and the weighting
factor was multiplied to obtain the PDF accurately. Moreover, the average of the PDF was used instead of the deviation
for convenience of modeling. The average of the probability was modeled using the amount of energy transfer between
each degree of freedom. The amount of energy transfer between each degree of freedom was modeled assuming that
the shape of the model was consistent with that obtained by MD data. The probability of the PDF, the total and inelastic
collision cross section were also modeled so that the model function was similar to the distribution obtained by MD
data. In order to verify the validity of the model, the equilibrium state and a one–dimensional normal shock wave were
simulated using this model. The results were in good agreement with the theoretical or experimental data and therefore
it was verified that the properties of the DMC model can be expressed by the fitting function. The dependence of the
species of molecules on the coefficient of model function will be analyzed in a future study.
ACKNOWLEDGMENTS
All simulations were performed on Origin2000 at the Institute of Fluid Science, Tohoku University.
REFERENCES
1.
2.
3.
4.
5.
Bird, G. A., Molecular Gas Dynamics and the Direct Simulation of Gas Flows: Clarendon, Oxford, (1994).
Nanbu, K., J. Phys. Soc. Jpn. 49, 2042, (1980)
Tokumasu, T. and Matsumoto, Y., Phys. Fluids, 11, 1907, (1999).
Singer, K. and Taylor, A., Mol. Phys. , 33, 1757, (1977).
Allen, M. P. and Tildesley, D. J., Computer Simulation of Liquids: Clarendon, Oxford, (1986).
319