Reconsideration of DSMC Models for Internal Energy
Transfer and Chemical Reactions
N.E. Gimelshein∗ , S.F. Gimelshein∗ , D.A. Levin∗ , M.S. Ivanov† and I.J. Wysong∗∗
†
Penn State University, University Park, PA 16802
Institute of Theoretical and Applied Mechanics, Novosibirsk 630090, Russia
∗∗
EOARD, London NW1 5TH, UK
∗
Abstract. A general model to calculate thermally and chemically inelastic collisions in the direct simulation Monte Carlo
method is discussed. The advantages of using discrete internal energy distributions are summarized. The principal aspects
of the model, such as the selection methodology for inelastic collision events, and matching internal energy transfer and
chemical reaction rates to the experimental data, are examined. The correction procedure to maintain accurate rates for the
vibration-translation energy transfer and chemical reactions with discrete internal energies are given.
INTRODUCTION
Extensive application of the direct simulation Monte Carlo (DSMC) method[1] to different rarefied gas dynamics
problems over the last decade has stimulated the development of different models of energy transfer and chemical
reactions suitable for this method. Among the many models developed for the vibration-translation (VT) and rotationtranslation (RT) energy transfer we mention only those based on the Larsen-Borgnakke technique that include the
discrete nature of the internal energy modes,[2]−[4] and some others that use level-by-level transition probabilities.[5,
6]
These and other sophisticated models are based mostly on quasiclassical or quantum mechanics results and imply
the utilization of different parameters (such as the anisotropy parameters in Ref. [6]), that have to be determined using
available experimental data. The determination of these parameters is problematic for cases where gases different from
an N2 /O2 mixture need to be considered, and experimental data do not exist. The models therefore can not be used in
many modern applications where the physics of polyatomic molecules is important.
Relaxation rates based on Arrhenius fits and Jeans and Landau-Teller relaxation equations are used in continuum
computations. Direct use of these models in the DSMC method is not feasible, but it is possible to use them to construct
phenomenological DSMC models for energy transfer between internal and translational energy modes and chemical
reactions.[7]
For polyatomic molecules, only the Larsen-Borgnakke (LB) model is available now for modeling internal energy
transfer, and collision theory based models such as [8] are available for chemical reactions (both total collision energy
and with vibration-dissociation coupling). It is important that the relaxation rates obtained using these models may be
related to those obtained from continuum methods.
The main goal of the paper is to discuss a general DSMC model for the energy transfer and chemical reactions
that is both computationally effective, generally applicable to any gas mixtures, and uses energy transfer and chemical
cross sections that match Jeans, Landau-Teller, and Arrhenius rate equations at equilibrium. The description of the
methodology for selection of the reaction or energy transfer event out of all possible inelastic collision processes is
given in the next section. Then, the advantages of using discrete internal energy levels instead of the conventional
continuum description are explained. The correction factor for the translation-vibration energy transfer rate to be used
in the DSMC modeling is then presented. Finally, an approach that enables one to match Arrhenius rates for chemical
reactions at equilibrium using discrete internal energy distributions is suggested, and examples that examine the CO 2
dissociation process in a shock front both in the Earth and Martian atmosphere are presented.
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
349
SELECTION METHODOLOGY
Modeling of real gas effects in the DSMC method usually implies simulation of inelastic collision processes, that
is those with the energy transfer between the translational and internal energy modes of molecules and chemical
reactions. A selection methodology may be introduced as the algorithm that determines the rules of how and in what
sequence inelastic processes are modeled.
Let us first consider the internal energy transfer and chemical reactions as the two major groups of inelastic
processes. The first group includes the energy transfer between rotational, vibrational, and translational modes. The
second group includes various chemical transformations that may occur upon a collision of two molecules, such as
dissociation, exchange, ionization, radiation, etc.
The principal step in modeling that allows one to significantly simplify a collision algorithm is to assume that the
rates (or the energy dependent cross sections) of the internal energy transfer and chemical processes are independent
of each other. There are several possible ways to model these two groups of processes. The most general is to calculate
the total probability of an inelastic (both thermal and chemical) event, accept the collision as inelastic with this
probability, and then choose the specific process to occur according to the ratio of its probability to the total inelastic
collision probability. This approach, however, requires one to compute all possible inelastic event probabilities for
each collision.
We have used here another approach, where first we evaluate whether a reaction occurs. In fact, when two molecules
collide, there is a possibility that one of several chemical processes will occur (these may be, for example, dissociation
or exchange reactions). Without the detailed knowledge of reaction paths depending on reactant energies, it is
convenient to assume the independence of each reaction process, and calculate the total reaction probability Pr as
the sum of all possible chemical probabilities, Pr = ∑i Pri . If Rnd < Pr , where Rnd is the random number uniformly
distributed between 0 and 1, then a reaction occurs. The specific reaction path i will be selected with the probability
Pri /Pr .
In case Rnd > Pr , a reaction does not occur and a check for internal energy transfer will be performed. To accurately
model the process of energy transfer between translational, rotational, and vibrational modes one needs to know
the cross sections for the transition (r1 , r2 , v1 , v2 ) → (r10 , r20 , v01 , v02 ), where 1 and 2 are molecules, and r and v are
rotational and vibrational states, respectively. If the energy-dependent transition probabilities are not known, the
conventional Larsen-Borgnakke model is used. This model is general and may be used for virtually any type of
molecular interaction.
As was discussed in [9], the internal energy transfer according to the LB model may be applied to each particle
individually, regardless of the outcome of its partner (particle selection), or to each particle pair collectively (pair
selection). Pair selection methodology was used in the DSMC method prior to [9]. In the most general case of a
collision between two molecules the energy is transferred between all modes of the molecules.
It was shown [9] that the pair selection methodology does not allow one to match the relaxation rates dictated by
Jeans and Landau-Teller equations. The relaxation rates can be independent only if multiple relaxation events (that is,
events where more than one internal state is changed) are prohibited in one collision. A particle selection methodology
was proposed in [9] that allows one to maintain independent vibrational and rotational relaxation rates for different
species in the mixture.
The selection methodology of Ref. [9], however, leads to an unnecessarily complicated functional dependence
between the probability of relaxation and the collision number. To avoid the complexity, a different selection methodology, illustrated in Fig.1 is suggested. The first step in this straightforward approach is to select a random number
Rnd uniformly distributed between 0 and 1. Then, using the value of Rnd, the possible relaxation events are tested,
so that either rotational or vibrational relaxation may occur for particle A or particle B. Multiple relaxation events are
prohibited. In Fig. 1, P denotes the probability of particle to undergo rotational or vibrational relaxation, and subscripts
r and v refer to rotational and vibrational modes, respectively.
Even though the LB model with particle selection methodology may generally be extended to model rotationrotation and vibration-vibration energy transfer, it is usually applied only to translation-rotation and translationvibration energy transfer as the most important internal energy transfer processes.
Let us now consider the issues of the applicability of the considered selection methodology. As was mentioned
above, the probability of reaction is applied first, before the internal energy transfer consideration. The probabilities
of selected dissociation reactions calculated according to the TCE reaction model (here and below, all internal energy
modes are assumed to completely contribute to the total collision energy) are shown in Fig. 2 (left). It is seen that for
large energies the probabilities increase sharply and may even be larger than one. This may be due to an inadequacy
of the TCE probability form or the VHS total collision cross sections at high temperatures. The increase itself also
350
Rnd < P 1
Rotational relaxation of A
P 1 < Rnd < P 2
Rnd
Rotational relaxation of B
P 2 < Rnd < P 3
Vibrational relaxation of A
P 3 < Rnd < P 4
Vibrational relaxation of B
Rnd > P 4
FIGURE 1.
10
Elastic collision
P 2 = Pr,A+Pr,B
P 3 = Pr,A+Pr,B+Pv,A
P 4 = Pr,A+Pr,B+Pv,A+Pv,B
Suggested selection methodology for inelastic and elastic collisions.
0
x
N2+N2->N+N+N2
H 2O+N2->OH+H+N2
CO2+N2->CO+O+N2
x
x
x
x
x
x
10
x
-2
10
-3
10-2
Probability
10
-1
x
10-1
Probability
P 1 = Pr,A
x
10-3
0
50000
FIGURE 2.
100000
E c/k, K
150000
10-4
200000
P rot(N2)
P vib(N2)
P rot(H 2O)
P vib(H 2O)
5000
10000
T, K
15000
20000
Reaction probabilities (left) and internal energy transfer probabilities (right) in DSMC
may cause some interference with the rotational and vibrational relaxation rates if there are sufficient pairs with total
energies larger than the reaction threshold.
The probabilities of energy transfer between the translational and internal modes of molecular nitrogen and water
molecules colliding with N2 are given in Fig. 2 (right). Details regarding the calculations are given in later sections.
Note that the total probability of the energy transfer (sum of Prot and Pvib ) is significantly lower than 1, which shows
the applicability of the selection methodology to model these types of inelastic processes.
DISCRETE VERSUS CONTINUUM INTERNAL ENERGIES
Both continuous and discrete representations of the internal modes of molecules are currently used in the DSMC
method. Even when a form of the LB approach is used for these two cases, there are several key issues in which they
differ.
1. Relaxation rates for continuous model. The use in the LB method of the acceptance-rejection technique to
distribute energy between two modes requires the determination of the maximum of the joint distribution function.
For the energy exchange between translational and vibrational modes of diatomic molecules this maximum occurs at
zero energy and is infinite, since the number of vibrational degrees of freedom is less than two. Therefore, the LB
scheme must be modified this case [1]. Usually, separate TR and TRV energy exchange events are considered for the
continuous model. In the latter case, the energy is redistributed first between translational and internal modes using the
LB technique, and then between rotational and vibrational modes using the energy splitting proportional to the number
of degrees of freedom. The use of TRV energy exchange does not permit one to match the relaxation rates governed
by Jeans and Landau-Teller equations [9].
2. Relaxation rates for discrete model. The discrete model requires a separate consideration of all rotational and
351
1
0.5
discrete
continuous
0.9
0.8
0.4
0.7
0.35
0.3
0.5
0.25
f
r
fr
0.6
0.4
0.2
0.3
0.15
0.2
0.1
0.1
0.05
0
0
discrete
continuous
0.45
1
2
3
Er /kTr
4
5
6
0
0
1
2
3
4
E /kT
r
r
5
6
7
8
FIGURE 3. The discrete and continuous rotational distribution functions for diatomic and linear polyatomic molecules (left) and
spherical top polyatomic molecules (right). Tr /θr = 1000.
non-degenerate vibrational modes, and the application of a particle selection technique prohibiting double relaxation,
such as [9] or Fig. 1, enables one to accurately capture Jeans and Landau-Teller relaxation rates.
3. Chemical reactions. With the assumption of the particular form of reaction probability dependence on the
reactants energies, the continuous model allows one an accurate analytic determination of unknown coefficients
in reaction probabilities. Accurate here means that being integrated over all possible energies using equilibrium
distribution functions, these probabilities give reaction rates that strictly follow experimental curves written in an
Arrhenius form. The discrete model requires a special correction procedure to be used for the reaction rates to follow
the Arrhenius dependence (see the next Section).
4. Energy of dissociation. The rotational and vibrational energies in the continuum model may be greater than the
dissociation energy. This is definitely an unphysical behavior, since such molecules should dissociate. In the discrete
model, the energy of internal modes can not be greater that the dissociation energy.
5. Assumptions used for the energy transfer in collisions. Two problems arise when the LB scheme is used for energy
exchange between translational and vibrational modes. First, there is an assumption about the temperature used in the
continuous energy model. Second, a special cutt-off has to be used in the LB scheme when the number of vibrational
degrees of freedom is less than two. Typically one uses proportional splitting instead but this causes deviations of the
internal energy distribution functions from the Boltzmann form at equilibrium.
6. Distribution functions. An important issue is the difference between the continuous and discrete models that
pertains to the shape of the equilibrium energy distribution of internal modes. The shape of the distribution function
may be critical for accurate prediction of chemical reactions whose cross sections directly depend on rotational
or vibrational states. Also, the internal energy distribution strongly affects the calculated flow radiation. Radiative
phenomena are determined by the internal molecular states, and the internal distributions need to be determined as
accurately as possible.
Consider now the rotational energy of diatomic and linear polyatomic molecules. The two distributions are compared
in Fig. 3 (left) for Tr /θr = 1000, where Tr is the rotational temperature, and θr is the rotational characteristic
temperature. It is seen that even though the maximum of the discrete distribution function is located at J > 0 whereas
for the continuous distribution function it occurs at E r = 0, the profiles are very close for most energy values. This
statement is also valid for spherical top polyatomic molecules which are compared in Fig. 3 (right). The conclusion
with respect to the comparison of discrete versus continuous rotational energies is therefore that the continuous energy
model approximates well the discrete spectrum up to the dissociation energy and may be recommended for most cases
(the concern expressed in previous item 4 is still applicable, though).
The difference between the discrete and continuous representations of vibrational energy is much larger than for the
rotational energy. For diatomic molecules, the equilibrium continuous distribution is proportional to
Ev
ξ /2−1
,
exp −
f (Ev ) ∝ Ev v
kT
352
0
0
10
10
energy conservation
rounding to nearest
SHO
−1
10
energy conservation
rounding to nearest
SHO
−2
10
−3
fraction
fraction
10
−1
10
−4
10
−5
10
−6
10
−2
10
−7
10
0
1
2
3
4
vibrational level
5
6
7
0
1
2
3
4
vibrational level
5
6
7
FIGURE 4. Comparison of SHO and discretized continuum vibrational energy distributions at equilibrium. Tv /θv = 0.5 (left)
and Tv /θv = 2.0 (right).
and the discrete distribution for SHO is
θv v
f (v) ∝ exp −
,
T
where Ev is the vibrational energy and v is vibrational level. Obviously, the shape of the continuous distribution
is different from exponential. Convolution of a continuous vibrational distribution to discrete distribution based on
quantum mechanical expressions is subject to a much larger quantization error than for the corresponding rotational
distribution transformation.
One can attempt to obtain a discrete distribution from a continuous one, and compare it with the actual discrete
distribution at equilibrium. Since continuous vibrational energy distribution is constructed in such a way that the
energy of a continuous vibrational mode is equal to that of discrete SHO vibrational mode, it makes sense to compare
these two cases. To this end, it is necessary to assign discrete vibrational levels to all energies of the continuous energy
spectrum. There are several ways to discretize the spectrum.
The obvious one is to simply round the energies towards the nearest vibrational level energy. In this case, molecules
with the energies less than 0.5kθv will be assigned to the ground vibrational level, molecules with the energies between
0.5kθv and 1.5kθv will be assigned to the first vibrational level, etc. The comparison of discrete distributions obtained
using this rounding-to-the-nearest-level approach and actual SHO discrete distributions is shown in Fig. 4 for two
Tv /θv ratios. It can be seen that the discrete vibrational distribution (open circles) is poorly approximated, with the
approximation (triangles) significantly underpredicting higher levels populations. The population of the ground and
first levels is almost equal at higher temperature (Tv /θv = 2). Moreover, this way of discretization does not even
conserve the energy originally contained in the continuous vibrational mode.
One can also discretize a continuous distribution conserving the energy originally contained in the continuous
vibrational mode [10]. The discrete distributions obtained using the energy conservation principle are also shown
in Fig. 4. This distribution is close to that obtained using the rounding-to-the-nearest-level approach. The accurate
discrete distribution can therefore not be obtained from the continuous, and the discrete energy model has to be used
when the level-by-level vibrational energy distribution is important.
The main conclusion of this section is that the use of discrete vibrational energy levels has several important
advantages compared to the continuous internal energy distribution, and is therefore recommended.
ROTATIONAL AND VIBRATIONAL RELAXATION RATES
As was mentioned above, the LB model is still considered the most general model used in the DSMC method to
calculate the energy transfer in collisions. In this model, a collision number Z is used, which is defined as Z ≡ 1/Fi ,
where Fi is the fraction of total collisions that lead to thermal relaxation of i-the energy mode.
An important issue in modeling the energy transfer between the translational and internal modes is the ability of the
DSMC method to reproduce the continuum and experimental relaxation rates (these rates are usually determined with
353
1
0.95
1
exact ZDSMC
v
approximate ZDSMC
0.95
v
ZDSMC/Zcont
0.85
v
0.85
0.8
0.8
v
cont
/Zv
DSMC
Zv
v
0.9
0.9
0.75
0.75
0.7
0.7
0.65
0
exact ZDSMC
v
approximate ZDSMC
0.65
0.2
0.4
0.6
Tv/θv
0.8
1
1.2
1.4
0.6
0
0.5
1
1.5
Tv/θv
2
2.5
3
3.5
FIGURE 5. Comparison of accurate (Eq. (2)) and approximate (Eq. (4)) Z vDSMC normalized by Zvcont for different total energies
E: E = kθ (left), E = 3kθ (right).
the Jeans and Landau-Teller relaxation equations). The selection methodology in this case needs to prohibit multiple
mode relaxation (see above).
In order to match the Jeans relaxation equation with rotational relaxation number Z r used in the continuum approach,
a correction factor [11] has to be applied,
ξT
ZrDSMC = Zrcont
,
(1)
ξT + ξ r
where ξT = 4 − 2α , α is the exponent in the VHS/VSS model, and ξ r is the number of rotational degrees of freedom
of the relaxing rotational mode.
The main difference between rotation-translation and vibration-translation relaxation is that while the rotational
mode can be described as having a constant number of degrees of freedom at all temperatures except cryogenic,
the number of vibrational degrees of freedom varies with the temperature. The correction factor that connects the
continuum and DSMC vibrational relaxation numbers was recently derived in [12], and is written as
ξT (Tt − T 0 )
,
ξv (Tt )Tt − ξv (Tv )Tv
(2)
ξv (Tv )Tv + ξT Tt = ξv (T 0 )T 0 + ξT T 0
(3)
ZvDSMC = Zvcont
where the temperature T 0 may be found numerically from the energy conservation equation,
/T
Here, Tt and Tv are translational and vibrational temperatures, respectively, ξ v (T ) = exp(2θθvv/T
)−1 , and θv is the characteristic vibrational temperature. Note that the temperatures here are local quantities, i.e. those in a cell where the
collision occurs. Instead of solving Eqs.2-3, a simple approximate expression may be used,
ZvDSMC = Zvcont
ξT
ξT + A
(4)
ξ 2 (T ) exp(θ /T )
where A = v
. It was found that if the translational temperature is used as T , Eq. (4) reproduces well the
2
correct expression in Eq. (2) for all cases where the vibrational temperature is lower than the translational one.
The values of ZvDSMC calculated using Eq. (2) and Eq. (4) are presented in Fig. 5. The two cases are given for two
different values of the total energy E = ET + Ev , with E being constant for each case. It is seen that for Tv < Tt the
difference between the correct and approximate Z vDSMC is about 1% for the lowest considered E of kθ , and decreases
further for larger energies.
CHEMICAL RATES FOR DISCRETE INTERNAL ENERGY MODEL
The existing chemical reaction models derived for continuous internal energies need special modifications to be applied
for discrete energies. Let us show how existing DSMC models for continuous energy can be adapted to the discrete
354
case. The total collision energy (TCE) model [8] is used here, but the approach can be employed for other models as
well. Note also that it can also be easily adapted to the continuous rotational and discrete vibrational energies.
The TCE model is constructed in such a manner that it reproduces the rate constant in the Arrhenius equation
(k(T ) = AT B exp(−Ea /kT )) for an equilibrium continuous distribution of reactant energies. When the continuous
distribution is replaced with the discrete one, the reaction rate produced by this model can differ from the Arrhenius
rate (a factor of 1.2 to 3 for most reactions). However, the input parameters of the model (namely, the parameters A
and B of the Arrhenius equation) can be modified so that when these new parameters are used to calculate reaction
probability according to the TCE model, the resulting rate will match the original Arrhenius rate.
To calculate new Arrhenius parameters one first needs to calculate the reaction rate constants k d (T ) using the TCE
model probabilites for discrete internal energies at several temperatures, and using the original Arrhenius parameters
A and B. Then one needs to determine parameters A 0 and B0 in the Arrhenius expression that fit the calculated rates, so
0
that kd (T ) ≈ A0 T B exp(−Ea /kT ). Then the new parameters to be used in the DSMC modeling may be calculated as
Anew = A2 /A0 , Bnew = 2B − B0
(5)
When these parameters are used to determine the TCE probability, the original rate will be reproduced.
The calculation of the discrete reaction rate requires the evaluation of the following expression
kd (T ) = ∑ ∑ ∑ ∑ f (v1 ) f (v2 ) f (r1 ) f (r2 )
v1 v2 r 1 r 2
Z∞
p(Ec )Ω(Et ) f (Et )dEt ,
(6)
0
where v1 ,v2 ,r1 ,r2 denote the summation over all vibrational and rotational modes of the first and the second molecules,
Ω(Et ) is the total collision cross section multiplied by the relative collision velocity, E t is the relative translational
energy of pair, Ec is the total collisional energy, and p(Ec ) is the TCE probability of the reaction. Eq. 6 may be
evaluated using a Monte Carlo approach.
To find A0 and B0 , a linear least squares fit to the logarithm of the calculated rate constant, k d (T ), is used
log kd (T ) + Ea /kT ≈ Ã0 + B0 log T,
where Ã0 = log A0 . In this case, the following expression is minimized,
kd
∑(log kd − log k f it )2 = 2 ∑ log( k f it ),
i
i
0
and where k f it = A0 T B exp(−Ea /kT ).
As a result, equal weight is assigned to the calculated values at all temperatures. Note that if the calculated constants
were fitted directly, the values at the lower temperatures would be considered with a lower weight. This is because the
calculated values vary over many orders of magnitude, and even a large relative error in k d at low temperatures would
not change significantly the absolute error and, therefore, would not affect parameters A 0 and B0 .
After A0 and B0 are determined, the new parameters are calculated to be used in the TCE model according to Eq. (5).
These parameters enable one to reproduce the correct rate at equilibrium using discrete internal energy levels.
The adaption of other continuous DSMC chemistry models to the discrete distribution of internal energy can be
performed similarly, but the probability of reaction as a function of the reactants energies in Eq. (6) has to be specified
according to those models.
Let us now give some examples of the application of the above procedure to correct reaction rates. Two CO 2
dissociation reactions are considered, CO2 +CO2 →CO2 +CO+O, typical for the Martian atmosphere entries, and
CO2 +N2 →CO+O+N2 , which is the important mechanism for CO2 dissociation in Earth atmosphere. The values
of the original and corrected Arrhenius parameters are given in Table 1. Comparison of the temperature dependence
of corrected and uncorrected reaction rate coefficients for the discrete internal energy model with the Arrhenius rate
coeffiecient is given in Fig. 6. The results depend primarily on the reaction parameters, namely, the constants in the
Arrhenius equation. It is seen that the uncorrected reaction rates are much larger for the first reaction, but differ from
the Arrhenius values not more than 25 percent for the second reaction. The above empirical correction procedure
enables one to reproduce the Arrhenius curves with the accuracy of better 3 percent.
In order to examine the sensitivity of a hypersonic flow over a spacecraft to the corrected chemical rates that are
used in the discrete internal energy model, the computations were performed of a reacting flow over an axisymmetric
355
TABLE 1.
Uncorrected and corrected constants in Arrhenius fit
CO2 +CO2 →CO2 +CO+O
CO2 +N2 →CO+O+N2
B
A0 , m3 /s
B0
0.2 · 10−18
1.146 · 10−8
0.5
-1.5
0.2 · 10−21
2.917 · 10−10
1.166
-1.11
-17
10-18
10
-16
k, m3/molecule s
10
k, m3/molecule s
10
A, m3 /s
-19
(1)
(2)
(3)
5000
10000
15000
T, K
20000
25000
(1)
(2)
(3)
10-17
30000
10000
15000
T, K
20000
FIGURE 6. Temperature dependence of CO2 + CO2 → CO2 +CO+O (left) and CO2 +N2 →CO+O+N2 (right) reaction rates. (1)
Original Arrhenius equation, (2) Monte Carlo integration result for discrete energies with corrected parameters, (3) results obtained
using standard TCE reaction probabilities with discrete internal energies.
blunted body in the Martian and Earth atmosphere. For the first case, the free stream velocity and temperature are
4.7 km/s and 143 K, and the Knudsen number based on the body radius is 1.2 × 10 −3 . The free stream composition
corresponds to that of 65 km above the surface of Mars with a CO 2 mole fraction 95 percent. For the second case, the
free stream velocity and temperature are 5 km/s and 267 K, the Knudsen number is 3.3×10 −4 , the and gas composition
at 50 km assumes a CO2 mole fraction of 3.4 × 10−4 ) [13].
The process of interest here was CO2 dissociation and the formation of CO (note that there is no CO in the free
stream). The profiles of CO mole fraction, a variable most sensitive to the reaction model, are given in Fig. 7 along the
stagnation streamline. The models used are the conventional continuum LB model with the TCE model for chemical
reactions, and the discrete LB model with TCE reaction probabilities obtained from corrected and uncorrected reaction
rates. It is seen that the corrected discrete model gives reaction rates about 10 percent higher than the continuous model,
which is attributed to the difference in reaction probability as a function of collision energy that is produced in the
corrected discrete model due to the difference between the value of B’ and B. The difference in the internal energies
and their distribution functions for these models may also play a role. The correction of the reaction rates for discrete
model leads to a much lower CO concentration for the Martian atmosphere, but almost does not change the results for
the Earth atmosphere, which is consistent with Fig. 6.
CONCLUSIONS
A selection methodology to treat different inelastic collision events is discussed. The impact of large probabilities of
chemical reactions on the internal energy transfer rates need to be further investigated. The advantages of the discrete
vibrational energy model as compared to the conventional discrete model are clarified. The correction algorithms for
the discrete internal energy model that allow one to maintain the Landau-Teller and Arrhenius relaxation rates for
vibration-translation energy transfer and chemical reactions, respectively, are presented. The impact of chemical rate
correction on the CO2 dissociation process in hypersonic flow is estimated.
356
0.30
Continuous
Discrete, corrected
Discrete, uncorrected
0.25
Continuous
Discrete, corrected
Discrete, uncorrected
3.5E-04
3.0E-04
0.20
Mole fraction
Mole Fraction
2.5E-04
2.0E-04
0.15
1.5E-04
0.10
1.0E-04
0.05
0.00
5.0E-05
-0.06
-0.04
X, m
-0.02
2.4E-08
0
-0.004
X, m
-0.002
0
FIGURE 7. CO mole fraction profiles along the stagnation streamline (surface is at X=0) for the Martian (left) and Earth (right)
atmosphere.
ACKNOWLEDGMENT
Research performed at the Pennsylvania State University was carried out under AF Grant F49620-02-1-0104. Support
was provided for the Institute of Theoretical and Applied Mechanics under INTAS Grant 99-0701.
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