Particle Simulation of Detonation Waves in Rarefied
Gases
D. Bruno*, S. Longo#
* C.N.R. - I.M.I.P. sez. Bari, c/o Dept. Chemistry, v. E. Orabona 4, 70126 - Bari (BA) Italy
# Dept. Chemistry - University of Bari, v. E. Orabona 4, 70126 - Bari (BA) Italy
Abstract. We study the dynamics of a steady overdriven detonation wave in planar geometry by means of
DSMC simulation. The exothermic reaction is modeled as a single step, irreversible, threshold reaction
with Arrhenius kinetics. Results show that the nonequilibrium character of the velocity distribution
function in the shock front modifies substantially the kinetics of the reaction and the overall profile of the
reaction zone.
INTRODUCTION
We study the dynamics of a steady overdriven detonation wave in planar geometry by means of DSMC
simulation.
In these very fast reactive flows several nonequilibrium processes determine the time required for the onset of
chemical reaction, control the energy release rates, and supply the mechanism by which the chemical energy
sustains the leading shock wave front. The chemical kinetics is coupled strongly with the gasdynamics so that
the stability of the stationary detonation wave depends substantially on the parameters that govern the chemical
kinetics. Even if we restrict to one-dimensional problems (multidimensional and curvature effects will not be
taken into account) the study of these phenomena requires consideration of one-dimensional flow of chemically
reactive viscous thermoconductive gas. Many authors have treated the problem theoretically, modeling the flow
with the reactive Euler or, sometimes, Navier-Stokes equations and consideration of more or less complete
chemical kinetics.
In this study we want to tackle also problems which cannot be handled by fluid dynamics equations. In the
reaction zone behind the shock nonequilibrium distributions of the particle velocities can play a role in the
overall kinetics, and the degree of nonequilibrium rises as the collision frequency is decreased [1].
We treat the 1D problem of an exothermic chemical reaction developing in a rarefied gas. The gas is a perfect
gas with polytropic equation of state. The model kinetic equation is the Boltzmann transport equation so that
transport effects and rarefied gas effects are taken into account. Chemical kinetics is included by means of a
single step reaction mechanism. The model equations are solved by direct numerical simulation with a Monte
Carlo particle method, DSMC in fact [2]. The particle methods are best suited for the study of rarefied flows
and for conditions of strong nonequilibrium which occur for very fast flows. In particular, they allow for the
consideration of strong nonequilibrium distribution for the particle velocities. This includes the description of
the detailed form of shock fronts. The kinetic processes (chemical reactions) can be treated on a microscopic
basis, by introducing the appropriate set of reactive collision cross sections. Therefore, chemical nonequilibrium
processes are naturally included in the simulation [3].
Results show that, in the critical conditions of this work, the chemical kinetics is strongly altered by the
nonequilibrium character of the velocity distribution function and the overall profile of the reaction zone differs
from the results of a Euler calculation.
STATEMENT OF THE PROBLEM
We consider a reactive gas which undergoes a irreversible exothermic reaction:
A -->B
(1)
Here, A is the explosive species and B the burnt gas. Under some conditions the reaction can develop a steady
detonation wave. We therefore study the dynamics of such a detonation wave in planar geometry. We model the
chemical kinetics by simple one-step, bimolecular reaction:
A+M -->B+M
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
178
(2)
where M is either A or B. The gases are assumed ideal, calorically perfect and have constant heat capacities.
The profile in the reaction zone as a function of the molar fraction of the explosive gas can be determined by
applying the conservation equations with the speed of the detonation wave as a parameter [4].
Then, the dependence upon the spatial coordinate can be obtained by integrating the rate equation:
d[ A]
= -K [A] n ; dx = udt
dt
(3)
If T0 is the temperature of the unperturbed explosive, the velocity of the self-sustained detonation, DCJ is
determined by assigning the parameters [4]:
†
Q=
g=
E
kT0
cp
(4)
cv
E being the energy released in the exothermic chemical reaction.
The spatial profile and the stability features of the overdriven detonation are fixed by the following parameters
†
[5]:
Qth =
f =
Eth
kT0
(5)
D2
2
DCJ
Eth being the activation energy for the reaction.
The selected test case is that studied in [5] whose parameters are:
†
Q = 50
g = 1.2
Qth = 40
f = 1.6
(6)
which gives a stable detonation with DCJ=6.216.
The reaction mechanism is a simple, irreversible bimolecular threshold reaction. In collision, the probability of
†
reaction is given by:
PA ÆB
Ï0
Ô
= Ì Ê Eth ˆ
ÔPr Á1- E ˜
Ó Ë
k ¯
if Ek < Eth
if Ek ≥ Eth
(7)
where Ek is the kinetic energy so that a simple Arrhenius law results in the thermal case:
†
E th
K = Pr s
8kT - kT
e
pm
(8)
here, s is the total cross section for the A, M pair, assumed constant, m is the reduced mass of the couple.
We use here scaled quantities as follows:
†
(i) the density, the temperature and the pressure are normalized to the respective quantities in the unreacted
explosive;
(ii) the velocities are expressed in Mach numbers with respect to the speed of sound in the unreacted explosive;
(iii) the spatial coordinate is expressed in units of the 'half-reaction distance', x1/2, which is defined below.
179
The 'half-reaction time', t1/2 is the time interval required, in the steady solution, for the conversion of half of
the species A in a fluid element to species B beginning at the time of its passage through the shock front. x1/2
is t1/2 times the speed of sound in the unreacted explosive.
THE DSMC METHOD
The Direct Simulation Monte Carlo method, DSMC [2], is a particle simulation method aimed at solving the
non linear Boltzmann equation. It is used for the simulation of gases in rarefied regime. But it has been
successfully extended to treat the chemical and vibrational kinetics of molecular gases [3]. The main advantages
of the method are that it allows the simulation of elementary processes at the microscopic level, i.e. the input
physical data are in the form of cross sections, and that the velocity distribution function is explicitly
evaluated. Here we apply the method to the simulation of a steady detonation wave. A previous example is [6].
The detonation wave is simulated in a reference frame where the shock is stationary. The simulation domain
extends for 150 'half-reaction' distances. The grid is such that the cell size is always less than one fifth of the
local mean free path. Since the heat capacity ratio is 1.2 the gas particles have 7 internal degrees of freedom.
These are considered classical. At each collision the available energy is redistributed according to the LarsenBorgnakke scheme which ensures equipartition at equilibrium [7]. Therefore, the relaxation of the internal
temperatures should follow the gas temperature without delay.
The hard sphere collision model is adopted. The following values have been chosen for the explosive gas:
number density, n0=5e15 cm-3
temperature, T0=30 K
mass, m=4 a.m.u.
cross section, s=5.026e-13 cm2
reaction probability, Pr=1.0
with these numbers the half-reaction distance is x1/2=7.4e-5 cm. We note here that the mean free path in the
undisturbed gas is l=2.8e-4 cm so we expect nonequilibrium effects arising from the interplay of the chemical
kinetics with the shock front dynamics.
The simulation uses a sample of 1e6 simulated particles.
RESULTS
The classical theory of detonation waves models the flow by the reactive Euler equations. The steady
detonation wave consists of a shock wave and a reaction zone. The exothermic reaction starts after the passage
of the shock front and at each point the reaction rate is determined by the local values of composition and
temperature. If the wave speed is known, the conservation equations across the shock allow to resolve the
structure of the reaction zone in terms of a progress variable, e.g. the fraction of the burnt explosive. With
respect to this model our results can differ only in the vicinity of the shock front. In the DSMC simulation, in
fact, the shock structure is resolved and the chemical reaction can start before the shock front is passed.
In figs. 1 to 4 we compare the DSMC results with the classical results obtained from the reactive Euler
equations. In figs. 1 and 2 we plot the density and pressure, respectively, along the detonation wave as a
function of the fraction of the burnt gas, a.
180
Density
10
8
6
4
2
0
0
0,2
0,4
a
0,6
0,8
1
FIGURE 1. Profile of the normalised density as a function of the fraction of burnt explosive (Full line: Euler model;
points: DSMC)
Pressure
70
60
50
40
30
20
10
0
0
0,2
0,4
a
0,6
0,8
1
FIGURE 2. Profile of the normalised pressure as a function of the fraction of burnt explosive (Full line: Euler model;
points: DSMC)
We see that the results coincide far from the shock front, but near the shock large differences are visible. In
fig. 3 we report the spatial profile of the density as obtained by the two models. We see that, as compared to
the Euler results, now the shock front is smeared, the reaction starts earlier, and the reaction zone extends for a
much wider region. This can be better appreciated from fig. 4 which reports the profile of the gas composition.
181
Density
10
8
6
4
2
0
0
10
20
x/x
30
40
50
1/2
FIGURE 3. Spatial profile of the normalised density (Full line: Euler model; points: DSMC)
a
1
0,8
0,6
0,4
0,2
0
0
20
40
x/x
60
80
100
1/2
FIGURE 4. Spatial profile of the molar fraction of burnt explosive (Full line: Euler model; points: DSMC)
We now turn to a more thorough discussion of nonequilibrium effects. We note that the density is very low,
so that the shock front extends for several mean free paths [1].
In this rarefied regime, kinematic effects play a role. The physical model of the particles assumes that energy
relaxation takes place at each collision, so that the gas temperature and the internal temperatures should be
virtually equal. Actually, at the shock front, the gas temperature rises due to the mixing of the cold and hot
gas components from both sides of the shock, but the collisions are not enough to thermalise the internal
degrees of freedom of the reactant gas, whose relaxation lags behind. This effect is shown in fig. 5.
From this graph we can also see that the product species is formed at a higher temperature very early in the
shock front. The high temperature is due to the chemical energy released in the reaction, but its presence at an
early stage suggests that nonequilibrium velocity distribution functions may be present and give rise to high,
non thermal values of the reaction rate.
In fig. 6 we show the reaction rate constant as a function of the temperature along the flow. The thermal value,
eq. (8), is reported for comparison. In fig. 7 we show the spatial profile of the same quantity compared to the
Euler solution. We can see that the chemistry and the fluid dynamics characteristic times are comparable, so
that the reaction kinetics is strongly influenced by the shock front dynamics. The answer to this strange
behaviour lies in the structure of the velocity distribution function. In fig. 8 we report the velocity distribution
function at different points along the flow (refer to figs. 3-5 for comparison). At the first position the
distribution is still the equilibrium distribution at the upstream temperature. The second curve already shows
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strong nonequilibrium distribution in the form of a high energy tail. The curve refers to a position where the
shock front has arrived, the temperature and the pressure begin to rise but the density is still at its undisturbed
value. It is this high energy tail which produces the nonequilibrium reaction rate of figs. 6-7. As the
temperature reaches its maximum the velocity distribution function has already attained its equilibrium
structure. At the position where the velocity distribution shows its nonequilibrium character we have evaluated
separately the contribution of the two species. The reactant is the most abundant since the reaction is just
started at this stage: it shows the characteristic nonequilibrium distribution typical of shock wave fronts [1].
The product gas, instead, has a thermal distribution at a high temperature dictated by the exothermicity of the
reaction. This is a consequence of the reaction model which assumes that the reaction products redistribute
the available energy in a statistical manner [7].
Temperatures
20
15
10
5
0
0
10
20
x/x
30
40
50
1/2
FIGURE 5. Spatial profile of the gas temperature (full line), internal temperature of the reactant species (dotted line),
internal temperature of the product species (dashed line). Also plotted the Euler model results (bold line).
3
Reaction rate constant, cm /s
-7
10
-8
10
-9
10
-10
10
-11
10
-12
10
0
2
4
6
8
T/T
10
12
14
16
0
FIGURE 6. Reaction rate constant as a function of the gas temperature (Full line: Euler model; points: DSMC).
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0
Velocity distribution function (global)
10
x/x =6.76
1/2
x/x =16.2
1/2
x/x =21.6
1/2
x/x =67.6
-1
10
1/2
-2
10
-3
10
-4
10
0
1 10
5
5
2 10
3 10
v, cm/s
5
4 10
5
5 10
5
FIGURE 7. Velocity distribution function of the gas mixture at different positions along the flow.
Velocity distribution function
0
10
A
B
-1
10
global
-2
10
-3
10
-4
10
0
1 10
5
5
2 10
3 10
v, cm/s
5
4 10
5
5 10
5
FIGURE 8. Velocity distribution function of the components of the gas mixture at x=16.2 'half-reaction' distances.
The different curves refer to the reactant, species A, the product, species B, and the gas as a whole.
CONCLUSIONS
The DSMC method is capable of simulating rarefied gas flows with chemical kinetics on a microscopic basis.
When the characteristic times of the fluid dynamics and of the chemical kinetics are comparable, the interplay
can produce nonequilibrium effects which can be dealt with only by kinetic methods. In this study we have
shown the effects of the nonequilibrium velocity distributions on the reaction rate constants in an overdriven
detonation in planar geometry.
ACKNOWLEDGMENTS
This work was supported by
CSPA.ATD.SC.02.03).
M.I.U.R.
(Contract
2001031223_009) and by
A.S.I.
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