Shock Waves (1997) 7:25–28

Possibility of acceleration of the threshold processes
for multi-component gas in the front of a shock wave

S.V. Kulikov

Institute of Chemical Physics in Chernogolovka RAS, Chernogolovka 142432, Moscow region, Russia

Received 4 August 1995 / Accepted 25 April 1996

Abstract. Studies of translational nonequilibrium in the
front of a shock wave propagating in a three-component gas
were performed by the Monte Carlo simulation method. Sim-
ulations were performed for mixtures of components with
molecular mass ratios m

3

/m

1

= 80, 3

≤ m

2

/m

1

≤ 60

and shock Mach number M = 4. The distribution of rel-
ative velocities g for pairs of molecules of heavy low-
concentration additives 2 and 3 substantially exceeded, in
the front, its equilibrium values behind the wave at high
values of g. The maximum value of this superequilibrium
was about 10

11

for the numerical density ratio: 1000 : 1 : 1

and m

2

/m

1

= 30. Calculations showed that high values of

the effect of superequilibrium take place up to a ratio of
densities 200 : 1 : 1.

Simulations performed for M = 4 and a mixture of He,

molecular oxygen and Xe with the numerical density ratio
200 : 1 : 1 showed also the high value of the superequilib-
rium effect at g corresponding to dissociation threshold of
oxygen. Thus, dissociation of oxygen by collisions with Xe
in the front of a wave may have a considerably higher rate
than total dissociation behind the wave.

Key words: Monte Carlo simulation, Shock wave, Transla-
tional nonequilibrium, Gas mixture

1 Introduction

Studies of translational nonequilibrium in the front of a
shock wave propagating in an especially multi-component
gas is important for understanding peculiarities of the thresh-
old physicochemical processes initiated by the shock wave
(Zel’dovich et al., 1979). The study of the shock-wave front
in a one-component gas (Kulikov et al., 1993; 1994) showed
that the distribution of relative velocities exceeded weakly
in front of a shock wave its downstream equilibrium distri-
bution.Such effect of superequilibrium is more considerable

An abridged version of this paper was presented at the 15th Int. Colloquium

on the Dynamics of Explosions and Reactive Systems at Boulder, Colorado,
from July 30 to August 4, 1995.

for the case of a two-component gas when the concentration
of impurity is small and the impurity has a much higher
molecular mass in comparison with the main gas (Genich
et al., 1991; 1992). Zel’dovich, Genich and Manelis (1979)
have considered a three-component gas containing two very
heavy impurities in a light gas and have estimated the ve-
locities of components in the front. They have shown that
the translational nonequilibrium in this case may strongly
affect the rate of the threshold physicochemical interaction
of impurities initiated in the shock wave front.

The results of continuation of this investigation at the

level of velocity distributions will be reported below.

2 Numerical procedure

The studies were performed by the use of the Monte Carlo
method of unstationary statistical simulation with constant
weighting factors (Genich et al., 1991; 1992). Molecules
were considered as hard spheres. The collision stage was
simulated using the sophisticated ballot-box scheme (Ku-
likov and Serikov 1993). The modeling of a planar stationary
shock wave was carried out in a one-dimensional coordinate
space and a three-dimensional velocity space.

3 Results and discussion

3.1 Numerical simulation of mixture of components
with molecular mass ratio 1 : 20 : 80

Results obtained for mixture of components 1, 2 and 3 with
the molecular mass ratio m

1

: m

2

: m

3

= 1 : 20 : 80 and

the numerical density ratio n

1

: n

2

: n

3

= 100 : 1 : 1 are

shown in Figs. 1 and 2. In this case the ratio of elastic cross-
sections of molecules was σ

1

: σ

2

: σ

3

= 1 : 2 : 2 and the

shock Mach number M = 4. The size of the cell ∆x was
equal to 0.3. Distance x is given everywhere in units of the
mean free path λ of molecules in the flow ahead of the wave.
The mean number N of model particles in a cell ahead of
the wave was equal to 9. The time of splitting of collision
and displacement stages was equal to ∆t = 0.2λ/u where u

26

Table 1. Dependence of the effect of superequilibrium on m

2

/m

1

m

2

/m

1

3

5

8

12

16

20

30

40

50

60

maximum overshoot of G

23

6

· 10

3

7

· 10

3

10

7

7

· 10

7

3

· 10

8

10

9

10

11

6

· 10

9

3

· 10

7

8

· 10

4

Fig. 1. Profiles of relative velocities u

0i

= (u

i

− u

b

)/(u

a

− u

b

) and relative

temperatures T

0i

= (T

i

− T

a

)/(T

b

− T

a

) (solid curves 1, 2, 3, 4, 5, 6) of

components 1, 2 and 3 with concentration ratio: n

1

: n

2

: n

3

= 100 : 1 : 1

Fig. 2. Distributions G

23

of pairs of molecules of components 2 and 3 over

relative velocities g in the wave front, concentration ratio of components:
n

1

: n

2

: n

3

= 100 : 1 : 1

is the most probable thermal velocity of particles of lightest
component in undisturbed upstream flow.

Figure 1 shows the profiles of relative velocities u

0i

=

(u

i

− u

b

)/(u

a

− u

b

) (solid curves 1, 2 and 3) and relative

temperatures T

0i

= (T

i

− T

a

)/(T

b

− T

a

) (solid curves 4, 5

and 6) of components 1, 2 and 3, respectively, (i = 1, 2, 3).
Here indices a and b refer to variables ahead of and behind
the shock wave, respectively. Analogous calculations were
carried out by Ruyev, Fomin and Shavaliev (1990) using
equations similar to Navier-Stokes equations and the profiles
of u

0i

and T

0i

obtained, shown by the broken curves in Fig. 1,

are close to those given above. This fact confirms once more
the reliability of the obtained results of the Monte Carlo
simulation. (The repetition of the Monte Carlo simulation
with different N = 18, ∆t = 0.1λ/u and ∆x/λ = 0.15 have
led to the same profiles of u

0i

and T

0i

.)

Figure 2 shows the distribution functions G

23

of pairs

of molecules of components 2 and 3 over relative velocities

g. Molecular and relative velocities are normalized to the

Fig. 3. Profiles of relative velocities u

0i

= (u

i

− u

b

)/(u

a

− u

b

) and relative

temperatures T

0i

= (T

i

− T

a

)/(T

b

− T

a

) (solid curves 1, 2, 3, 4, 5, 6) of

components 1, 2 and 3 with concentration ratio: n

1

: n

2

: n

3

= 1000 : 1 : 1

velocity of sound a in the gas mixture ahead of the wave.
Curves 1 and 5 are the equilibrium distributions ahead of and
behind the wave. Curves 2, 3 and 4 correspond to distance

x = −16, −14 and −10 in the front, respectively. One can
see that, at g > 1.5, the distributions 2, 3 and 4 substantially
exceed the equilibrium distribution 5. These overshoots are
about 15 at g = 2.4 and about 50 000 at g = 4.57 for curve
4. (The mean-square error of determining limited values of

G

23

at high relative velocities (g

≈ 4.6) was about 50%.)

The last value is the maximum observed value in this case.
The effect is more considerable than for the case of a two-
component gas. The overshoot appears at the leading edge of
the wave, is retained at the distance about 30 and has maxi-
mum at x =

−10 where the difference between the velocities

of components 2 and 3 has also a maximum. In this case,
the number of collisions of molecular pairs moving at high
relative velocities will be in the front substantially greater
than behind the wave in equilibrium. The superequilibrium
considered should appreciably affect the threshold physic-
ochemical processes initiated by the shock wave in anal-
ogous reactive mixtures. This influence should take place
in spite of the narrowness of the zone of nonequilibrium
due to the high value of the overshoot. The effect may be
particularly pronounced behind the wave in the avalanche-
type chain processes when even a relatively small degree of
physical and chemical transformations in the nonequilibrium
zone may have a significant influence on the entire flow.

It should be noted that the maximum value of the ob-

served overshoot rises with growth of the number of model
particles, because this permits to obtain information about
distribution for the region of higher relative velocities. Thus,
the two-fold increase of N leads to the ten-fold increase of
the maximum value of overshoot.

Distributions of relative velocities for pairs of molecules

of grades 1 and 3 G

13

, 2 and 2 G

22

, 3 and 3 G

33

also

exceeded in the front their equilibrium values behind the
wave. But these maximum overshoots were not so high (not
higher than 20).

The stronger effect was calculated for the analogous case

with different numerical density ratio: n

1

: n

2

: n

3

= 1000 :

27

Fig. 4. Distributions G

23

of pairs of molecules of components 2 and 3 over

relative velocities g in the wave front, concentration ratio of components:
n

1

: n

2

: n

3

= 1000 : 1 : 1

1 : 1 and ∆t = 0.15λ/u. The results obtained are shown in
Figs. 3, 4 and 5. Figures 3 and 4 show the profiles of u

0i

,

T

0i

and G

23

. The notation is the same as in the Figs. 1 and

2. One can see that, at g > 1.5, the distributions 2, 3 and
4 substantially exceed the equilibrium distribution 5. These
overshoots are about 1000 at g = 2.4 and about 1

· 10

7

at

g = 3.9 for curve 4.

As previously,maximum overshoots of G

13

, G

22

, G

33

were not so high (not higher than 20).

Figure 5 shows the distributions F

1

, F

2

, F

3

of longitudi-

nal molecular velocities c for components 1, 2 and 3 respec-
tively. Curves 1 and 8 are equilibrium distributions ahead of
and behind the wave. Curves 2, 3, 4, 5, 6 and 7 show the
distributions in the front at distance x =

−18, −16, −14,

−10, −5 and 7.3 respectively. The behaviour of these func-
tions explains the great value of the effect. It is connected
with behaviour of F

2

and F

3

. The function F

2

relaxes more

quickly to translational equilibrium distribution behind the
shock wave than F

3

(see curves 5). Moreover, F

2

and F

3

are not of bimodal form, in contrast to the distribution of
the light molecules F

1

and their width are narrow enough.

Calculations showed that such high values of the effect of

overshoot takes place up to a ratio of concentrations n

1

: n

2

:

n

3

= 200 : 1 : 1. For instance, in the latter case maximum

overshoot was about 2

· 10

7

.

3.2 Dependence of the effect of superequilibrium on m

2

/m

1

Simulation for mixture with n

1

: n

2

: n

3

= 1000 : 1 : 1

was performed in order to investigate the dependence of
the effect on m

2

/m

1

. In this case, as previously, m

3

/m

1

=

80, σ

1

: σ

2

: σ

3

= 1 : 2 : 2, ∆t = 0.15λ/u, ∆x/λ =

0.3, M = 4 and N = 9. The obtained results are shown
in Table 1. The maximum overshoot corresponds to m

2

/m

1

30. If m

2

/m

1

< 30 then the width of F

2

increases and the

effect decreases. If m

2

/m

1

> 30 then F

2

relaxes slowly to

translational equilibrium behind the wave and the effect also
decreases.

Fig. 5. Distributions F

1

, F

2

, F

3

of longitudinal molecular velocities c for

components 1, 2 and 3 with concentration ratio: n

1

: n

2

: n

3

= 1000 : 1 : 1

The two-fold increase of N from 9 to 18 has led to the

maximum observed value of overshoot 10

10

at m

2

/m

1

= 12.

3.3 Numerical simulation of mixture of components
with another molecular mass ratio

Calculations for M = 5 and mixture with m

1

: m

2

: m

3

=

1 : 5 : 10 and σ

1

= σ

2

= σ

3

were carried out for several

numerical density ratios of components 1, 2 and 3 : (1)100 :
1 : 1; (2)100 : 10 : 1; (3)100 : 1 : 10; (4)10 : 1 : 1. In these
cases N = 10, ∆x/λ = 0.15, ∆t = 0.1λ/u. Obtained results
show the absence of the high effect of superequilibrium (not
higher than 20).

28

Table 2. Dependence of the effect of superequilibrium on the shock Mach
number for a mixture of He, O

2

and Xe

M

4

4.5

10

maximum overshoot of G

23

10

4

10

4

4

· 10

5

2

· 10

6

overshoot of G

23

at g/a = 6.7

−

−

6

· 10

4

8

N

18

72

72

72

3.4 Investigation of mixture of He, O

2

and Xe

Simulations performed for a mixture of He, O

2

and Xe (m

1

:

m

2

: m

3

= 1 : 8 : 33 and σ

1

: σ

2

: σ

3

= 1 : 2.77 : 5.10) with

n

1

: n

2

: n

3

= 200 : 1 : 1 at ∆x/λ = 0.15, ∆t = 0.075λ/u,

M = 4, 4.5 and 10 showed also the high value of the effect
for all M . The observed absolute maximum overshoots of

G

23

are given in Table 2. If the temperature ahead of the

wave is equal to 293 K than the threshold of dissociation
of O

2

by collisions with Xe corresponds to g = 6.7. The

maximum overshoots of G

23

at this g are also shown in

the Table 2. In this case, calculations were carried out with
the mean number N of model particles in a cell ahead of
the wave of 18 and 72. At M = 4.5 absolute maximum
overshoot was about 4

· 10

5

and the maximum overshoot

at threshold velocity was about 6

· 10

4

. Thus, the reaction

of dissociation of O

2

by collisions with Xe in the front of

a wave may have a considerably higher rate than the total
dissociation behind the wave for the considered case.

4 Conclusion

The calculations have confirmed the conjecture made by
Zel’dovich, Genich and Manelis on the possible strong in-
fluence of translational nonequilibrium of gas mixture in the
front of a shock wave at threshold physicochemical pro-
cesses.

References

Genich AP, Kulikov SV, Manelis GB, Chereshnev SL (1991) Front Struc-

ture and Effects of the Translational nonequilibrium in Shock Waves
in a Gas Mixture. In: Beylich AE(ed) Proceedings of 17th IS on RGD,
pp 175–182, (Aachen, 1990)

Genich AP, Kulikov SV, Manelis GB, Chereshnev SL (1992) Thermo-

physics of Translational Relaxation in Shock Waves in Gases. Sov.
Tech. Rev. B Therm. Phys. part 1, pp 1–69

Kulikov SV, Serikov VV (1993) Weighting algorithms for the Monte Carlo

simulation of multicomponent reactive gas flows and their application
to the shock-wave problem. Rus. J. Comput. Mechan. 3, pp 49–69

Kulikov SV, Ternovaia ON, Chereshnev SL (1993) Peculiarity of the trans-

lational non-equilibrium in the shock-wave front in a one-component
gas. Khim. Fizika 12, pp 340–342 (in Russian)

Kulikov SV, Ternovaia ON, Chereshnev SL (1994) Peculiarity of evolution

of distributions of relative velocities for one-component gas in the front
of SW. Fiz. Gor. Vzr.4, pp 140–144 (in Russian)

Ruyev GA, Fomin VM, Shavaliev MSh (1991) Shock-Wave Structure in

Ternary Disparate-Mass Gas Mixture. In: Beylich AE(ed) Proceedings
of 17th IS on RGD, pp 183–190, (Aachen 1990)

Zel’dovich YaB., Genich AP, Manelis GB (1979) Features of Translational

Relaxation in a Shock Wave Front in Gas Mixture. Docl. Acad. Nauk
SSSR, 2, pp 349–351 (in Russian)