Investigation of Thermal Conductivity in High-Temperature
Nitrogen Using the DSMC Method
Nuno Alves and John Stark
Department of Engineering, Queen Mary – University of London,
Mile End, London E1 4NS, United Kingdom
Abstract. The Direct Simulation Monte Carlo (DSMC) method is investigated with regard to its ability to represent
transport processes in high temperature simple flows and in particular in an idealized Couette-type flow. Nitrogen is
the test gas, with heat transport coefficient to a temperature of 10,000 K being evaluated, assuming that the species
may be represented in terms of either the variable hard sphere (VHS) or the variable soft sphere (VSS) molecular
models. These DSMC coefficients are then compared to available analytical studies. It is found that for energy fluxes
the accuracy derived from the current standard formulation of DSMC is limited to 3,000 K. This limit corresponds to
the start of high chemical activity in the flow. Investigation of recombination processes within the framework of
Bird’s DSMC formulation reveals that detailed balance is not maintained, and that this may be the cause of poor heat
flux representation at high temperatures. It appears that a revised methodology is required to represent chemical
activity wherein more than two colliding species interact.
INTRODUCTION
The direct simulation Monte Carlo (DSMC) introduced by Bird [1] in the 1960’s has developed as a practical
methodology for simulating engineering flows under rarefied gas conditions. The principal feature in this method is
the molecular model adopted to reflect the fundamental aspect of the Boltzmann equation - the collision integral.
The vast majority of simulations have been performed however with highly idealized scattering potentials, which
reflect spherical symmetry in the potential field. An in-depth study has been made by the authors [2] on the effects
of scattering laws on the fluid properties, as calculated by DSMC. That study showed that, compared to the
theoretically superior variable hard sphere (VHS) and variable soft sphere (VSS) models, statistically similar
properties are obtained even with non-physical scattering laws. It can be interpreted from this apparent insensitivity
that more sophisticated scattering models do not to predict more representatively the flow physics when used in
conjunction with the current DSMC methodology. Because of its universal use, this result demonstrates the need to
examine the range of applicability of DSMC as well as the accuracy of some of the DSMC procedures and
algorithms.
DSMC procedures may be broadly classified as being theoretically based or phenomenological in nature. Since
DSMC is a stochastic numerical method, it makes use of statistical algorithms that tend to simulate physical
phenomena on an ensemble basis, rather than on a single event basis. Our previous investigation of mainly
phenomenological procedures has identified that by trying to avoid the complexity of a fully theoretical approach,
numerical/physical artefacts may be introduced into the physical problem at hand. In this paper we look at one
particular case of high temperature heat flux where, relative to experimental data, a distortion in heat transport
process arises and we trace the source of this within the DSMC procedure. In this investigation we first look at the
general relationship between thermal conductivity and temperature on which the VHS (and VSS) molecular model
is based, over the temperature range: 4,000 K to 10,000 K. This range coincides with significant chemical activity,
with changing gas composition. Transport properties, as predicted by DSMC in Couette type flows are analysed and
compared to theoretically calculated values. Analysis of the procedure handling chemical reactions, specifically
recombination is presented, and this appears to be the source of the noted deviation from experimental data in heat
flux. Improvements to the existing methodology are proposed.
TRANSPORT PROPERTIES
In order to make a proper assessment on the reliability of the temperature variation of transport properties it is
important to ensure consistency in the definitions for all methods to be examined. Briefly we note how the viscosity
coefficient is defined for the VHS molecular model, before addressing the issue of thermal conductivity.
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
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Theoretical Properties
The difficulties of recreating the non-equilibrium low-density conditions in a laboratory environment have
recently been well documented in a review paper by Harvey & Gallis [3]. In our study we focus upon Nitrogen as
the test gas. The temperature range is from 273 K to 10,000 K. This maximum temperature is selected since below
10,000 K ionization processes need not be included in the simulation. To the knowledge of the authors, detailed
experimental information on viscosity coefficients at low-density conditions covering the complete temperature
range, for a heavy neutral diatomic molecule like N2, is not available. However, experiments have been performed
by Vargaftik [4] and by many others [5], at atmospheric pressure for temperatures up to 1,000 K. These experiments
are generally performed under static flow conditions.
Various approaches to calculate transport coefficients have been made based on kinetic theory. Some of these
have been determined in the framework of the Chapman-Enskog solution [6,7,8] to the BE with the collision
integrals, Ω(l,s) (l and s determine the order of the integral) given by
Ω
(l , s )
=
2πkT
m
∞∞
∫∫e
0 0
−ξ 2




ξ 2 s +3 1 − cos l χ bdbdξ ,
(1)
where m is the particle mass and the reduced relative collision speed is ξ = cr(m/2kT)1/2. These integrals appear in the
theoretical expressions for the derived transport properties. Capitteli and his co-workers [6,9,10] have computed
detailed transport properties based upon the Chapman-Enskog approach for Nitrogen. Others [5,11] have based their
work on equivalent expressions given by the classical kinetic theory of Taxman [11], wherein internal energy is
included in rotational modes. Herein we have focused upon the Capitteli [6,9] approach for comparison with the
DSMC method. Their method considered the contribution to the transport coefficients from each possible
permutation of collision partners independently, and this included a linking procedure recognizing the different
nature of high and low temperature gas collisions. Their results agree well with experiment.
Calculation of transport coefficients ‘mixing’ rules may be used to account for the different contributions from
each gas component, or from the different combinations of collision partners. For example the following expression
for the first approximation to the viscosity coefficient pertaining to the mixing rule, [µMIX ]1, is applied [12]
µi
v
[µ MIX ]1 = ∑
i
x
µi
1 + 1.385∑ k
k =1 x i ρ i [Dik ]1
v
,
(2)
k ≠i
where v is the total number of species considered, with the subscripts i and k denoting each specie; x and ρ
correspond to the fraction and the density of each specie, respectively. Dik is the diffusion coefficient.
In our simulation gas composition is taken from thermodynamic properties [13]. The composition of N2 is
shown in figure 1(a). Molecular Nitrogen is the dominant specie until a temperature of around 4,000 K. Thereafter,
increasing chemical activity results so that by 8,000 K, the atomic form is dominant. As may be inferred from figure
1(b), even at the highest temperature in our simulation the fraction of N+ form less than 1% of the gas. As a result
ionization influences upon transport coefficients are omitted from our study.
The calculation of the energy transport in dissociating N2 is treated in detail by Yun et al [7]. They identify
contributions from the reactive part of the heat conductivity coefficient, κR, and the non-reactive, or frozen one, κF.
The frozen part may be further separated into two components, one that relates to translational exchange, the other
to internal energy exchange. These are accounted for by the coefficients κtr and κint, respectively, such that κF =
κtr+κint. The values of κint are less certain than those of κtr, [7] since they rely on the accurate consideration of the
internal degrees of freedom. On the other hand, κR has been determined accurately and is a function of both the
temperature and the exact composition. Unfortunately, available gas-mixture laws for the heat conductivity
coefficient are not as accurate as those for viscosity, defined in equation (2). Our study results using DSMC will
therefore only be compared directly with the theoretical calculations of Yun et al., which were validated by
comparison with the experimental results of Burhorn [14], obtained also for a Nitrogen gas. Both Yun’s calculation
and Burhorn’s experiments provide evidence for a large transfer of energy, due to dissociation, in the temperature
region around 5,000 K.
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1.0
1.0
0.8
0.8
0.6
0.6
N2
N2
N
0.4
0.2
0.0
4000
N
0.4
0.2
6000
8000
0.0
4000
10000
6000
8000
T, K
T, K
(a)
(b)
10000
Figure 1. Plot of equilibrium composition of a Nitrogen gas: (a) Nitrogen molecule and atom and (b) logarithmic plot of the
composition of minor species.
Simulation Calculation
The description of the DSMC procedures is thoroughly covered by Bird [1], and our own code adheres to
Bird’s methodology. Our focus in this paper is the evaluation of transport properties, via the molecular model. Two
models have been widely accepted as ‘standard’ for comparative simulations: the VHS and VSS models. Morokoff
and Kersch [19] showed that the main justification for the VSS model is that it should reproduce the correct Schmidt
number, Sc = µρ/D, linking together the viscosity coefficient, µ, the density, ρ, and the diffusion coefficient, D. The
theoretical value of the viscosity coefficient for the VHS model, according to the first approximation of the
Chapman-Enskog expansion method, was found by Bird [1] to be given by

5
Γ − ω 
 15  2
  2
µ0 =   
 8  Γ 9 − ω   Tref


2





ω−
1
2
1
1  mk  2 ω
 T .
2 
d ref
 π 
(3)
In DSMC the equilibrium viscosity coefficient dependence on temperature is implicitly implemented through
the expression in equation (3), more precisely in terms of dref and ω specified for the molecular model, and by
linking the collision speed with the actual temperature. The scattering parameters adopted for our simulations are
ω=0.75, α=1.36 and, Sc-1=1.34 with a number density of 2.68×1025 m-3. The assumed value for the VHS model
atomic parameters are such that the ratio between the molecular to the atomic reference diameter is 1.3, and the
corresponding mass ratio is 2. [1]. In our simulations of Couette flow 200 physical cells are identified spanwise,
physically representing a 1m separation between walls that bound the flow; each of the cells is of the same size.
Simulation is performed with 100,000 molecules; iteration is performed until steady flow is reached. All molecular
properties are initially assigned randomly from a Maxwellian equilibrium distribution according to an initial flow
temperature. A single velocity gradient, given by an upper-wall speed of 1,000 m s-1, is applied in all simulations
86
with the flow and wall temperatures being equal. The imposed temperature gradient of near zero is so that when
steady state is reached, the curvature in the equilibrium temperature profile is small. The reason for this is that with
this condition we achieve a static temperature, which is the underlying assumption in most of the available
calculated [13,14] and experimental data [8].
Discussion of Results
Simulation by DSMC has been undertaken for the Couette flow of the thermal conductivity in Nitrogen, κ, as a
function of temperature. In figure 2, this coefficient is plotted together with thermal conductivity calculated by Yun
et al., for both the total conductivity, κ, and for that specified for a chemically-frozen mixture, κF. Significant
differences are apparent between these data. First consider the comparison between κ, theoretically determined by
Yun et al., and the DSMC prediction. The data of Yun et al., shows a steep rise in κ, starting at 4,500 K. This
corresponds to the start of high chemical activity. A maximum value occurring at ~7,000 K is observed, followed by
a sharp reduction. In contrast, the trend given by DSMC shows a regular monotonically increasing conductivity with
temperature. It can be inferred from this result that the DSMC method does not fully include the energy exchange
when chemical reactions are initiated. This observation is confirmed when considering the alternative calculation of
Yun et al. for a Nitrogen equilibrium gas mixture, also plotted in figure 2. The frozen thermal conductivity
coefficient, κF, results seem to be qualitatively closer to the DSMC result. At low temperatures the DSMC results
correspond closely to Yun et al.’s κF. However as the temperature is increased, the DSMC evaluation seems to
underestimate the value of this thermal conductivity, with an increasingly divergent trend as the temperature
increases.
Total, calc. by
Yun
4.0
Frozen, calc. by
Yun
3.0
κ , J K-1m -1s-1
DSMC-VHS
DSMC-VSS
2.0
1.0
0.0
0
2500
5000
T /K
7500
10000
Figure 2. Plot of the sampled DSMC and calculated heat conductivity coefficients as a function of static
temperature for Nitrogen gas.
There are two issues that arise from these observations, which need discussion. One inference is that, as the
DSMC simulated value of κ is governed by only repulsive-type molecular interaction models, wherein transport
coefficients have a simple power-law dependence on temperature, the DSMC method fails to account for the change
in behaviour of the interaction potential. The ideal approach would therefore comprise an interaction model that
shows better consistency in terms of the potential over a wider temperature range, but desirably with the same
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computational efficiency as the VHS model. One such model already exists and is the VSS model. The VSS model
was therefore also employed to analyse the effect on static temperature of heat conduction, and the results obtained
are also included in figure 2. It is encouraging to identify that there is some increase in κ using this model,
particularly at the temperature range of high chemical activity. However, this increase is marginal and the value
predicted still remains well below the values predicted in Yun et al.’s calculations.
The second issue to note is that since the discrepancies between simulated and theoretical values for total
thermal conductivity coincides with the start of reactivity, it may be argued that the inability of DSMC, to match the
frozen κ, lies in the procedures which handle reactive collisions. These procedures combine a pseudo-equilibrium
transfer of energy between the vibrational and translational modes using Borgnakke-Larsen (BL) models for
dissociation, with an ‘equilibrium collision theory’ [1] approach to recombination. The main criteria for the
dissociation procedure is the ‘maximum vibrational level’ theory, which is performed on an individual collision
basis. In contrast, for recombination a cell-based cumulative probability function is employed. This function was
derived in such a way that it matches expected equilibrium reaction rates. In the recombination procedure the
contributions from all inter-atomic collisions, occurring in the cell, are added together until the value of the function
exceeds unity. The atoms involved in the last contributing collision are then marked for recombination. While this
last procedure is justified by the fact that it allows DSMC to reproduce expected equilibrium rates, its use is less
satisfactory for particle simulations relative to that employed for dissociation, since the exact state of the atoms is
not considered apriori. Finally, in terms of energy-transfer modelling in the DSMC method, a different approach to
BL-type models already exists. Recently, Tokumasu and Matsumoto[20] devised the dynamic molecular collision
(DMC) model, which is based upon the Molecular Dynamics (MD) method. However, in verification of model
transport properties, Tokumasu and Matsumoto [21] obtained results with similar characteristics to those presented
here, with inconsistency found in the heat conduction simulated results. Again the discrepancies reported in that
study were seen to increase with temperature. It is noteworthy that Tokumasu & Matsumoto observed that the
viscosity was well modelled, in agreement with our own earlier findings [22].
This result identifies the need for an alternative approach to the VHS, and VSS physical models, to represent
energy transport properties at the high temperature conditions wherein chemical activity predominates.
DSMC RECOMBINATION
After implying in the above discussion that the DSMC algorithm handling the recombination part of reactive
collisions may not be consistent with the microscopic approach inherent in the method it is pertinent that we now
look at the details of Bird’s algorithm, [1] to analyse its theoretical accuracy (or consistency) in the original
framework of the DSMC method.
As aforementioned, the strategic theoretical approximation for DSMC is that it should only be applied for
dilute gases, pre-supposing that the majority of collisions will be binary. The mechanics and energetics of binary
elastic collision partners, before and after the collision, can be easily calculated by applying the basic physical
principles of momentum and energy conservation. This is thoroughly covered by Bird in his manuscript. The main
result is that the translational mode can be described in a centre-of-mass frame of reference by the relative speed, cr,
and the reduced mass, mr, of the two partners, yielding:
1
1
1
1
m A c A2 + m B c B2 = (m A + m B )c m2 + mr c r2 ,
2
2
2
2
(4)
where A and B denote the individual collision partners; cm is the centre-of-mass speed. The magnitude of cr is an
invariant of the collision and the whole problem of finding the post-collision velocities is reduced to the calculation
of the azimuth angle of the post-collision relative velocity, cr. Moreover, DSMC processes all collision quantities on
an averaged basis, meaning that at each collision step it introduces a factor in order to maintain the temperature
trend of equation (3). This factor is calculated over all possible collisions in a VHS or VSS gas, and under the
previous assumption it is made directly proportional to cr, such that
c r = ∫ ∫ c r f A f B dc A dc B .
88
(5)
Although this treatment is strictly only valid for binary elastic collisions we witness, however, that throughout
the simulation the same principle is employed for binary collision events even though some collisions are clearly
inelastic, i.e. there is energy interchange between the translational and internal modes. This apparent limitation is
solved by the use of phenomenological BL models which govern the proportional re-distribution of the total
collision energy, ET, according to the available energy modes; thus (5/2-ω) of ET , goes into the translation and ζ/2
into the internal modes. Once this re-distribution takes place the collision can proceed as if it were elastic. This
circumvents the complexities that would be associated with an exact deterministic approach, whilst maintaining
detailed balance. This approach also ensures that the DSMC methodology retains its required high computational
efficiency.
We have now seen procedures devised for binary ‘elastic’ collisions requiring additional methods so that their
use in binary ‘inelastic’ collisions is justified, or more precisely, the mechanics remain the same with extra
modelling introduced for the energetics. The imperative question is: What is the justification for the use of this
methodology in reactive collisions?
Reactive collisions involve more than two molecules and the amount of energy transferred between the internal
and translational modes is of the order of kT, where k is the Boltzmann constant and T is the temperature; this
amount of energy transfer is significantly more than that seen in inelastic collisions [15]. In this context it becomes
unclear whether the BL model is suitable to represent the energetics when such a large amount of energy is
interchanged. Perhaps more importantly binary collisions are replaced by tertiary collisions. This implies that the
whole structure of the collision mechanics devised and described by equation (5), no longer strictly applies.
Reactive Procedure
Bird [1] recognized the need for an alternative approach for reactive collisions, whilst at the same time
retaining the original approach for the collision mechanics. In short he introduced a new ‘reaction’ cross section.
This cross-section is then linked to the original elastic cross-section through the formulation of available rate
coefficients, which have been largely obtained under near-equilibrium conditions. We mentioned previously the
difficulties in acquiring reliable laboratory data for non-equilibrium high-temperature flows. The adoption of the
near equilibrium data may provide a potential source of inconsistency between DSMC and experimental
observations, such as Burhorn’s data [14], which is well modelled by Yun’s full theoretical treatment. Our own
simulations however have also been performed in such a manner as to retain near-equilibrium conditions, and thus
Bird’s approximation should therefore be of less import for the Couette flow simulation
The major concern then must be about the suitability of binary-type collision mechanics in a reactive collision,
its impact upon the efficiency of energy interchange and, most importantly, the issue of detailed balance. We have
therefore investigated the energetics of the recombination procedure of Bird [1]. The essence of our approach is to
track the energy of all partners contributing to both 2-body and 3-body collisions, inclusive of translational and
internal energy modes, in addition to the binding energy associated with the difference between atomic and
molecular states. These energies were tracked in a simple, single cell problem. In the model gas; initially equal
numbers of molecular and atomic species were presumed.. The number density assumed was 2.68×1025 m-3, with a
total energy content equivalent to there being a final equilibrium temperature of 1,000 K in the cell. Clearly at this
low temperature the molecular species is completely dominant (see Fig. 1), and thus propagation of the collision
algorithm led to recombination of atomic species. The collisions between species were simulated in the context of
the standard DSMC paradigm, and those atoms identified for potential recombination were marked so that their
initial and final total energies could be tracked. Our analysis enabled both elastic and inelastic collisions to be
evaluated. In our preliminary study, reported here, only 246 recombination events were identified. These results
were statistically analysed.
It should be noted that our original DSMC simulation code was modified to prevent molecules already marked
for recombination not only being re-marked but also from their being selected as possible third body molecules. In
our analysis the post collision energy of the recombining particles, as calculated in the standard DSMC method, is
subtracted from the corresponding value of total energy, prior to the particle interaction. This has the effect that a
positive value plotted in our results means that energy has been lost from the system. In our results we have
evaluated this energy difference for all 246 events and normalized the net total energy difference by the Boltzmann
constant. We can then relate the net energy difference directly in terms of the equivalent dissociation temperature,
which for Nitrogen is 118,200 K. Our results in the form of a probability distribution function are shown in figure 3.
Although the size of the sample is limited, we note that despite the fact that binding energy is added to
individual recombination events, on average energy is lost from the system. To our surprise these losses are in some
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cases even greater than the binding energy. The sign of the energy difference depends on various factors, of which
the relative energy of the colliding atoms seems to be the most important, and this explains why in fewer cases
energy is gained. On the basis of this result we conclude that this procedure does not satisfy detailed balance, as has
been formerly claimed. Furthermore, and taking into account the results plotted in figure 2, this procedure may
contribute to discrepancies observed earlier in heat transport processes. The method as conventionally employed
underestimates the energy of the system.
1.0
dissociation
barrier
probability function
0.8
0.6
0.4
0.2
0.0
-60,000
-30,000
0
30,000
60,000
90,000
120,000
150,000
Energy/BOLTZ, (K)
Figure 3. Plot of the sampled net lost (positive) and gained (negative) energy over the DSMC recombination
procedure, shown as a ratio to the Boltzmann constant.
It is important to note that our analysis of both elastic collisions and inelastic dissociation reactions does not
demonstrate this lack of detailed balance: at each collision tracked total energy was clearly seen to be conserved in
the simulation process.
The new question posed is then how does this failure arise in the DSMC methodology? Our brief investigation
shows that it is only the recombination procedure that fails. In reality, as well as in the simulation model, a third
body particle is required to accommodate the binding energy being added into the collision system. Because of this,
expressions (4) and (5) become invalid. The problem appears to be in the use of two separate relative energies, and
two centre of mass systems, in the calculation of ET: one between the pre-collision of the atoms and the other
between the third body and the newly combined molecule, whose speed is set to that of the pre-collision centre-ofmass speed of the atoms[1]. One can trace these steps in the following manner.
First consider the original translational energies of the 3 collision partners as modified in the following
intermediate step derived from equation 4:
1
1
1
1
1
1
m A c A2 + m B c B2 + m3 c32 → (m A + m B )c m2 AB + mrAB c r2AB + m3 c32 ,
2
2
2
2
2
2
(6)
where
c m AB =
m A c A + mB c B
.
m A + mB
90
(7)
If one now sets m2=mA+mB, and c2=cmAB, and then we re-write this expression, one identifies two ‘molecules’ which
when combined together in the same way as in (4) give
1
1
1
1
1
1
mrAB c r2AB + m2 c 22 + m3 c32 → mrAB c r2AB + (m2 + m3 )c m2 23 + mr23 c r223 .
2
2
2
2
2
2
(8)
On the right-hand side of (8) we clearly note the two relative speeds employed in the recombination procedure,
however the centre-of-mass expression is only related to the second relative energy. Therefore it is inappropriate to
treat (8) in the same way as in (4) because the first relative energy is unrelated to the new centre-of-mass frame of
reference and use of two relative energies violates the simplification of the law governing binary elastic collisions in
DSMC as explained at the beginning of this section. It is our belief that this treatment of the translation energy leads
to further imbalance of the energy modes and it is inevitable that detailed balance is therefore lost.
What methods can be used to overcome this problem? Clearly it is not possible to use the computationally
efficient process for binary collisions. A starting point would be to reframe the original DSMC concept and extend
those techniques for collision quantities such as those inherent in expressions (4) and (5) to tertiary type events. This
would require significant computational effort due to the fact that the centre-of-mass frame of reference changes
during recombination. Whilst matching reaction rates is credible, however, alone this approach does not suffice to
simulate microscopic processes. Here a more robust approach is needed. This approach would have to consider the
energetics of the three collision partners simultaneously, in such a way that ensures detailed balance as well as in a
manner totally independent of the other binary-elastic-collision-type procedures.
Though an exact solution to this problem is not proposed here, the authors feel that in pointing out this failing
in Bird’s DSMC recombination methodology users should become more aware of the limitations with regard to the
method’s ability to simulate heat flux processes. As it is the case with all simulation methods it is important to grasp
these limitations before full assessment of results can be made.
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