Energy Exchange Modelling of Internally Excited Gas
Surface Interactions
A A Agbormbai
Department of Computing
City of London College, Whitechapel Road, London E1 1DU, UK.
ABSTRACT
Gas surface interactions are an important class of molecular interactions in rarefied gas dynamics because they determine the heat transfer and forces that act on a body submerged in a gas. In this study the problem of vibrationally
excited single-body gas surface interactions is considered in the absence of chemical reactions, and reciprocity theory is applied to calculate the energy exchange process. The introduction of gas internal energy creates opportunities
for the gas translational energy to be diverted away from being exchanged with the surface, in preference for an exchange with the internal mode. The application of reciprocity theory follows a standard approach described in earlier
works. However, this presentation does not start from first principles. Instead, simple rules of thumb are used to
write down the symmetrised contracted reciprocity equation for internally excited interactions. It is assumed that the
scattering proceeds as in monatomic interactions. Two classes of models are constructed: a number of fully correlated models and a partially correlated model. Numerical simulations are performed which illustrate the properties
of the models. The models are formulated for use in Direct Simulation Monte Carlo computations of rarefied gas
phenomena.
INTRODUCTION
The reciprocity formalism has long been recognised as the proper direction for modelling complex interactions
in rarefied gas dynamics. Proponents of the approach include Fowler (Ref. 1), Cercignani (Refs. 2,3), Pullin
(Ref. 4), Agbormbai (Refs. 5,6) and Lord (Refs. 7,8). These workers have applied reciprocity ideas to modelling
gas surface interactions and molecular collisions in rarefied gas dynamics. Previously, the author applied the
formalism to monatomic single-body gas surface interactions (Ref. 9). In this paper we consider the impact of
internal degrees of freedom. The analysis is limited to single-body gas surface interactions in which chemical
reactions do not occur.
When internally excited gas molecules strike a surface not only is the translational energy of the gas
exchanged with the surface; it is also exchanged with the internal modes, which may also exchange energy directly with the surface. In this paper a number of statistical models are explored for calculating internally excited gas surface interactions. The models are formulated using reciprocity concepts. However, instead of starting from first principles, the procedure for constructing a reciprocity equation is summarised and then the contracted reciprocity equation is written down directly based on simple rules of thumb. The reciprocity formalism
generates models for use in Direct Simulation Monte Carlo (DSMC) computations, which is the established
standard for computing rarefied gas phenomena.
RECIPROCITY CONCEPTS IN GAS SURFACE INTERACTIONS
Fowler (Ref. 1) was the first person to recognise that gas surface models should be derived from a reciprocity
principle. He wrote down an expression for the elementary number flux of incident particles to the surface, for a
gas in equilibrium at the surface temperature. This represents the number of molecules that would be destroyed
CP663, Rarefied Gas Dynamics: 23rd International Symposium, edited by A. D. Ketsdever and E. P. Muntz
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in unit time by unit area of surface. For equilibrium to be preserved Fowler argued that an equal number of these
molecules, moving in the opposite direction, must be thrown off by the wall in unit time. This is the simplest
formulation of a reciprocity principle for gas surface interactions. In searching for models that satisfy his reciprocity expression Fowler listed specular reflection, perfect backscattering (i.e. reflection by direct reversal of
path), diffuse reflection and Maxwell reflection. For diffuse reflection with incomplete accommodation
Fowler’s expression for the elementary number flux describes steady states, i.e. equilibrium states with temperatures intermediate to the incident gas and surface temperatures. This means that the temperature appearing in the
Fowler expression is intermediate to the incident gas and surface temperatures, its magnitude being determined
by the degree of accommodation to the surface (i.e. by the thermal accommodation coefficient). For diffuse reflection with complete accommodation Fowler’s expression describes equilibrium with the surface, and the
temperature appearing in the expression is now the surface temperature. In concluding Fowler noted that a more
correct analysis would fuse both the specular and perfectly diffuse reflections into a single law, with a continuously varying correlation between the incident and reflected states.
Cercignani (Refs. 2,3) was the first person to write down a rigorous mathematical equation for the reciprocity principle in gas surface interactions. He used the concept of a transition probability or interaction kernel to characterise the interaction, and then he formulated the elementary number flux terms using this kernel. It
is possible to recover Fowler’s reciprocity equation by performing a suitable integration of the Cercignani reciprocity. Cercignani and Lampis (Ref. 10) later constructed a simple model that satisfies their reciprocity equation.
They also demonstrated that their approach satisfies the H theorem. The trouble with the foregoing reciprocity
equations is that they ignore the effects of the solid atoms. Therefore they are best used to model the scattering
process. Explicit modelling of the energy exchange between the gas and surface requires the inclusion of solid
terms in the reciprocity equation. This paper shows how to do this.
The Cercignani-Lampis (C-L) model was originally limited because its implementation in DSMC
computations appeared impossible. However, Lord (Ref. 11) resolved this problem and also extended the model
to include classical internal degrees of freedom. Later, Lord developed quantised versions of the model for calculating harmonic (Ref. 12) and anharmonic (Ref. 7) vibrational energy exchange. The anharmonic version uses
the Morse potential for the vibrational potential energy. Lord has also pointed out a number of limitations of the
C-L model. These are that:
1) The participating degrees of freedom behave independently through the interaction. This means that no explicit energy exchange is modelled.
2) A relationship exists between the accommodation coefficient for tangential momentum and that for the kinetic energy associated with the tangential velocity component. This relationship leads to the result that the
former is always less than the latter whereas experimental evidence indicates the converse.
3) The C-L model does not cover diffuse reflection with complete or incomplete accommodation. In fact, the
model does not provide a continuous variation between specular and perfectly diffuse reflection.
To resolve these problems he extended the C-L model to include both fully and partially diffuse reflection with
partial energy accommodation (Refs. 7, 12). He used the Borgnakke-Larsen (Ref. 13) redistribution scheme to
partition the energy totals into components, and also suggested the alternative use of Pullin’s partial exchange
scheme (Ref. 4) to achieve partial equilibration. For the scattering distributions he suggested the use of his own
elliptical model (Ref. 14). Finally Lord (Ref. 15) proposed a scheme that combined the translational and rotational degrees of freedom together so that they do not behave independently in the interaction.
In spite of these modifications the Cercignani-Lampis-Lord (CLL) models derive essentially as
mathematical conveniences rather than from physical plausibility. They do not reflect naturally the basic experimental finding that most gas surface interactions are intermediate to specular and perfectly diffuse reflections, i.e. they do not provide a continuous variation from specular to perfectly diffuse reflection. The modifications incorporated by Lord to emulate the fully and partially diffuse reflection laws are merely ad hoc adjustments that constitute a patchwork of remedies to compensate for what are serious shortcomings in the philosophical outlook of the model. What we want is a phenomenological approach that reflects some of the basic physics of gas surface interactions, emulates the partial exchange approach of Larsen and Borgnakke (Ref. 16) and of
Pullin (Ref. 4) for molecular collisions, is mathematically rigorous, and provides a smooth variation between the
extremes of specular and perfectly diffuse reflection. Before constructing this approach let us first review the
path to partial exchange modelling.
Bird (Ref. 17) and Borgnakke and Larsen (Ref. 13) introduced the phenomenological approach in 1970
and 1975 respectively. Bird proposed the energy sink model which compares the rotational energy for a given
collision against the equilibrium value and then transfers a fraction of the rotational energy to the translational
energy such that the final state is closer to equilibrium. This guarantees the attainment of macroscopic equilibrium but because the model employs no concept of inverse collisions it cannot lead to microscopic equilibrium
satisfying reciprocity. In 1975 Borgnakke and Larsen proposed the two-class phenomenological exchange
scheme which has been popular and influential. It has been popular because it is the simplest phenomenological
model that satisfies most of the requirements for a DSMC interaction model (it has optimal efficiency for
DSMC computations while giving reasonable results), but more because Bird adopted it and applied it extensively to many problems. Hence everyone who begins a career in DSMC, and who therefore uses Bird’s books,
adopts the model. It is not the best model to use as it represents only crudely the underlying physics of the process. Moreover, with the rapid advances in computer technology, the constraints imposed by efficiency requirements are diminishing rapidly. This means that accuracy (and design safety) takes on even more importance,
especially in practical engineering applications that involve very complex geometry and flow fields – hence the
need for more realistic phenomenological models. The Borgnakke-Larsen (B-L) approach is influential because
it has led to the enactment of highly sophisticated phenomenological approaches that satisfy all the requirements
for a practical DSMC interaction model (e.g. Refs. 4, 5).
The B-L two-class scheme treats a collisional exchange as either elastic or perfectly inelastic. A single
model parameter determines the fraction of collisions that are perfectly inelastic and it controls the relaxation
rate by limiting, over a large number of collisions, the gross amount of energy that is exchanged in collisions.
This parameter is used to match relaxation data. In elastic collisions no energy exchange occurs between the
translational and internal modes, whereas in perfectly inelastic collisions energy exchange occurs involving all
the energy and degrees of freedom of each mode. In fact, the post-collision energies are sampled from a local
equilibrium distribution for the collision.
In 1974 Borgnakke and Larsen attempted to refine this model by restricting the amount of energy, and
hence the number of degrees of freedom, that participates in the exchange to only a certain fraction of the total
collision energy. In this case all collisions are inelastic but to different degrees. The single parameter now defined the fraction of energy or of degrees of freedom that participates in the exchange. The parameter varied
between elastic collisions at the lower end and perfectly inelastic collisions at the upper end. In 1978 Pullin
showed that this model does not satisfy reciprocity. B-L’s new approach was called restricted exchange, but is
more aptly called partial exchange – a term suggested by Pullin (Ref. 4). The term ‘restricted exchange’ is better fitted to models wherein the number of energy components that participate in the exchange is restricted by
combining a number of the components into lump sums (a process called lumping) which are then used in the
exchange (Ref. 5). In fact, restricted exchange models are a type of partial exchange models. The concept of
partial exchange has been fruitful in the construction of highly refined phenomenological models that satisfy
reciprocity while also having all the desirable properties of a DSMC model. For instance, Pullin exploited the
partial exchange concept to construct a phenomenological model that satisfies reciprocity at equilibrium. The
partial exchange concept strikes the best compromise among efficiency, accuracy and physical realism. In view
of the effectiveness of the approach the author has adopted it as the basis for his own works. In particular, it is
this approach that is applied below for gas surface interactions. However, mathematical details are not given.
THE PROBLEM
The gas surface system consists of a gas molecule incident on a solid block of atoms as shown in Figure 1a. In
the course of the interaction the translational and internal energies of the gas molecule are exchanged with the
vibrational energy of the solid block. However, to formulate the gas surface problem we transform the description into the system shown in Figure 1b. In this system the gas molecule has been structurally enhanced so that
it now has translational, internal and solid vibrational energy. The gas surface energy exchange has become an
exchange between the translational, internal and solid vibrational energy of the enhanced gas molecule. This
exchange occurs only when the enhanced gas molecule strikes the surface. The solid block has become simply
an idealised plane with no structure. Its role is simply to scatter the incident atoms and to trigger the translational-internal-solid exchange in the enhanced molecules.
THE INTERNALLY EXCITED GAS SURFACE RECIPROCITY EQUATION
Invoke a number of simple rules of thumb to write down directly the contracted reciprocity equation for the energy exchange. Each energy mode that appears in the contracted reciprocity equation has the Gamma distribution term:
(
)
νj
εj
Ga (x µ ) =
1
x µ −1e − x , ( µ > 0, 0 ≤ x ≤ ∞)
Γ( µ )
is the Gamma distribution for x with parameter µ , Γ( µ ) is the gamma function. In these expressions
Ga ξ j γ j
where γ j =
2
, ξj =
kT
and
ν is the number of degrees of freedom of the mode (4 for translational, 6 N s for the solid, 2 or 3 for rotational
and 2 or more for vibrational), k is Boltzmann' s constant, and T is the equilibrium temperature.
(1)
Also appearing in the symmetrised reciprocity equation are the correlation densities G(s), which allow formulation of the statistical models of the exchange.
Using these rules we can write for the symmetrised, contracted, reciprocity equation for energy:
Ga (ξ t′ γ t )Ga (ξ s′ γ s ) Ge (s′e )ds′e dξ t′dξ s′
= Ga (ξ t′′γ t )Ga (ξ s′′ γ s
∏ Ga (ξ ′ γ )dξ ′
) G (s′′ )ds′′dξ ′′dξ ′′ ∏ Ga (ξ ′′ γ )dξ ′′
e
e
e
t
ik
s
ik
ik
ik
ik
( 2)
ik
where subscript k denotes a specific internal mode. To formulate statistical models we assume that only two
internal modes are excited: rotational and vibrational. Therefore, we simplify the reciprocity equation into:
Ga (ξ t′ γ t )Ga (ξ s′ γ s )Ga (ξ r′ γ r )Ga (ξ v′ γ v ) Ge (s′e )ds′e dξ t′dξ s′dξ r′dξ v′
= Ga (ξ t′′γ t )Ga (ξ s′′ γ s )Ga (ξ r′′ γ r )Ga (ξ v′′ γ v ) Ge (s′e′ )ds′e′dξ t′′dξ s′′dξ r′′dξ v′′
(3)
THE STOCHASTIC MODELS
To formulate the statistical transformation models we split each energy mode into active and inactive components. Both the total active and total inactive energies are conserved in the interaction. Thus, we split the incident energies into active and inactive components and then recombine the active components into the total active incident energy. This total active incident energy is split (redistributed) into reflected active components,
which are finally recombined with the incident inactive components in order to get the final reflected energies.
The whole point is that only the active components are interchanged in the interaction. Inactive components are
not. This means that the reflected inactive components remain equal to their incident inactive counterparts. For
restricted exchange schemes we first lump or combine the energy components for each mode into a total energy
for each mode before proceeding to decompose the total energies into active and inactive parts (molecular collisions). In unary gas surface interactions, the loosely restricted exchange scheme is implemented by lumping all
internal modes into a total internal energy before decomposing this energy into active and inactive parts. Severely restricted exchange schemes for unary gas surface interactions lump the internal and translational modes
into a total gas energy before exchanging with the surface. We consider rotational excitations separately from
vibrational excitations.
ROTATIONAL EXCITATIONS
Two types of models can be formulated: free exchange and restricted exchange.
Free Exchange
The model is:
ξ t′′ = (1 − st′ )ξ t′ + (1 − sa′ )ξ a , ξ r′′ = (1 − sr′ )ξ r′ + s′I sa′ ξ a , where ξa = st′ξ t′ + sr′ξ r′ + s′sξ s′ is the total active energy, ( 4a)
with correlation densities:
Ge (se ) = β (st′ α tγ t , (1 − α t )γ t )β (sr′ α rγ r , (1 − α r )γ r )β (ss′ α sγ s , (1 − α s )γ s )β (s′I α rγ r ,α sγ s )β (sa′ α rγ r + α sγ s ,α tγ t ) ( 4b)
To determine the total active energy, which is an integral part of the overall transformation, we sample the solid
energy from:
(
f s (ξ s′ ) = Ga ξ s′ γ s
)
( 4c)
Equation (4c) applies to all models. Note that we have omitted the equations for the post-collision solid energy
and for the post-collision correlation variates, as they are not required in numerical simulations.
Restricted Exchange
The model is:
ξ t′′ = (1 − st′ )ξ t′ + (1 − sa′ )ξ a , ξ r′′ = s1′ξ I′′, where ξa = st′ξ t′ + s′I ξ I′ , ξ I′′ = sa′ ξ a + (1 − s′I )ξ I′ ,
ξ I′ = ξ r′ + ξ s′
(5a)
with correlation densities:
Ge (se ) = β (st α tγ t , (1 − α t )γ t )β (sI α I γ I , (1 − α I )γ I )β (s1 γ r , γ s )β (sa α I γ I ,α tγ t )
(5b)
VIBRATIONAL EXCITATIONS
Three types of models can be formulated: free exchange, loosely restricted exchange and severely restricted
exchange. However, the severely restricted exchange scheme is not formulated. This is left to the reader as an
exercise.
Free Exchange
The model is:
ξ t′′ = (1 − st′ )ξ t′ + (1 − s′g ) sa′ ξ a ,
where
ξ r′′ = (1 − sr′ )ξ r′ + s′I s′g sa′ ξ a , ξ v′′ = (1 − sv′ )ξ v′ + (1 − s′I ) s′g sa′ ξ a ,
(6a)
ξ a = st′ξ t′ + sr′ξ r′ + sv′ξ v′ + ss′ξ s′ = st′′ξ t′′ + sr′′ξ r′′ + sv′′ξ v′′ + ss′′ξ s′′ is the total active energy.
with associated correlation densities:
Ge (s′e ) = β (st′ α tγ t , (1 − α t )γ t )β (sr′ α rγ r , (1 − α r )γ r )β (sv′ α vγ v , (1 − α v )γ v )β (ss′ α sγ s , (1 − α s )γ s )
(
)
×β (s′I α rγ r ,α vγ v )β s′g α rγ r + α vγ v ,α tγ t β (sa′ α tγ t + α rγ r + α vγ v ,α sγ s )
(6b)
Only the transformation equations for the gas energies have been written down. Considering the entire
set of overall transformation equations we can use algebraic manipulation to invert the transformation to demonstrate that it satisfies the symmetry property, i.e. that the forward and inverse transformations have the same
form. We can also verify that the overall transformation satisfies energy conservation as well as reciprocity. For
these reciprocity proofs see Ref. 20.
Loosely Restricted Exchange
Restricted exchange schemes are based on lumping a certain group of energy modes together before formulating
the exchange. This lumping is not done in free exchange schemes. For instance, the rotational and vibrational
energies can be lumped together as internal energy before exchanging with the other modes. This gives rise to a
separate model. This sort of restriction is not severe.
We can also formulate a severely restricted exchange model wherein the internal and translational
modes are lumped together into the gas energy, before being exchanged with the solid energy. Another formulation for severe restriction involves lumping the internal and solid energies together before exchanging with the
translational energy. All these restricted exchange models represent different physics of the interaction and suit
different gas dynamic situations, again illustrating the versatility of the reciprocity approach.
Only the loosely restricted model is constructed here. This lumps the gas rotational and vibrational energies into the internal energy before exchanging with the translational and solid modes. The overall model
transformation is:
ξ t′′ = (1 − st′ )ξ t′ + (1 − s′g ) sa′ ξ a ,
ξ r′′ = s1′ξ I′′,
where ξ a = st′ξ t′ + s′I ξ I′ + ss′ξ s′ ,
ξ I′′ = s′g sa′ ξ a + (1 − s′I )ξ I′ ,
with associated correlation densities:
ξ v′′ = (1 − s1′ )ξ I′′,
ξ I′ = ξ r′ + ξ v′
(8a)
Ge (s′e ) = β (st′ α tγ t , (1 − α t )γ t )β (s′I α I γ I , (1 − α I )γ I )β (ss′ α sγ s , (1 − α s )γ s )
(
)
×β (s1′ γ r , γ v )β s′g α I γ I ,α tγ t β (sa′ α tγ t + α I γ I ,α sγ s )
where
(8b)
γ I = γ r + γ v.
We can verify that the overall transformation equations satisfy energy conservation as well as reciprocity.
We can obtain a simplified form of this model, called the simplified loosely restricted exchange model,
by averaging over the starting variates of the exchange process – these variates are st , ss and sI. This gives:
ξ t′′ = (1 − α t )ξ t′ + (1 − s′g ) sa′ ξ a ,
ξ r′′ = s1′ξ I′′,
ξ v′′ = (1 − s1′ )ξ I′′,
where ξ a = α tξt′ + α I ξ I′ + α sξ s′ ,
ξ I′′ = s′g sa′ ξ a + (1 − α I )ξ I′ , ξ I′ = ξ r′ + ξ v′
(9a)
with correlation densities:
(
)
Ge (s′e ) = β (s1′ γ r , γ v )β s′g α I γ I ,α tγ t β (sa′ α tγ t + α I γ I ,α sγ s ) ,
where
γ I = γ r + γ v.
(9b)
This model satisfies energy conservation but does not satisfy reciprocity. However, it has only three Beta distributions to sample from and is thus faster.
PARTIAL CORRELATION
The preceding models are fully correlated models because they exhibit both longitudinal correlation and transverse (or cross) correlation. Longitudinal correlation is correlation between the pre- and post-interaction states
of each mode, i.e. for each mode the pre-interaction state is correlated to the post-interaction state. Cross correlation is correlation between one energy mode and the other modes. This is enforced through energy conservation. We can formulate a partially correlated model that bears longitudinal correlation but no transverse correlation. The model does not explicitly satisfy energy conservation.
We arrive at the model by assuming that the surface block grows so large, and the solid vibrational
energy so large, that surface interactions do not really affect the solid energy. Under these conditions we can
leave out the solid energy from the reciprocity equation, and from the energy conservation law. In fact, it becomes meaningless to express an energy conservation law for the interaction. With these assumptions we get the
following partially correlated model:
ξ t′′ = st′ξ t′ + ξ td′ ,
′
′ , ξ v′′ = sv′ξ v′ + ξ vd
ξ r′′ = sr′ξ r′ + ξ rd
(10a)
with associated densities:
′ α rγ r )
Ge (s′e ) f (ξ d′ ) = β (st′ (1 − α t )γ t ,α tγ t )Ga (ξ td′ α tγ t )β (sr′ (1 − α r )γ r ,α rγ r )Ga (ξ rd
′ α vγ v )
× β (sv′ (1 − α v )γ v ,α vγ v )Ga (ξ vd
(10b)
SIGNIFICANCE OF THE MODEL PARAMETERS
For each energy splitting Beta distribution of the form:
(
β s′j α jγ j , (1 − α j )γ j
)
(11a)
the mean and variance of the correlation variate are:
sj = α j,
σ 2j =
α j (1 − α j )
1+γ j
(11b)
Thus each parameter is the average of an energy-splitting correlation variate. These correlation variates also
appear in the expression for the total active energy. When we average the total active energy over each variate
we find that each model parameter represents the average fraction of each energy mode that participates in the
partial exchange – the remaining fraction being inactive. Note that the model parameters also appear in the parameters of the energy splitting Beta distributions. The first parameter of these Beta distributions represents the
active degrees of freedom whereas the second parameter represents the inactive degrees of freedom. Therefore
the model parameters also represent the fraction of degrees of freedom, for each energy mode, that participates
in the exchange. This is tantamount to saying that each model parameter reflects the average degree of excitation of each mode in the collisions. A zero model parameter value signifies zero excitation of the corresponding
mode in the collisions. Therefore no amount of this mode participates in the exchange, and none of its degrees
of freedom is involved. A unit value for a model parameter signifies full excitation of the corresponding mode in
the collisions. All the energy in this mode participates in the exchange, and all the degrees of freedom are involved. The energy exchange is maximal. A related interpretation is that each model parameter reflects the efficiency of the corresponding mode in redistributing its energy in the interactions, the efficiency rising from zero
at zero parameter values to maximal at unit parameter values. This efficiency affects the rate of accommodation
of the given mode in the collisions, and in fact the model parameters are determined by matching accommodation coefficient data.
PERFECTLY INELASTIC REFLECTION
Let us examine the behaviour of the models at the upper limit of the parameters. At this limit the free exchange
model takes the form:
ξ t′′ = (1 − s′g ) sa′ ξ a , ξ r′′ = s′I s′g sa′ ξ a ,
ξ v′′ = (1 − s′I ) s′g sa′ ξ a ,
where
ξ a = ξ t′ + ξ r′ + ξ v′ + ξ s′ .
(12a)
with correlation densities:
(
)
Ge (s′e ) = β (s′I γ r , γ v )β s′g γ r + γ v , γ t β (sa′ γ t + γ r + γ v , γ s )
(12b)
The restricted exchange model also degenerates to this form, at the upper parameter limit, except that s1 is used
instead of sI. We find that the fully correlated models do not quite make diffuse exchange with complete accommodation, at the upper limit of the parameters. Call this exchange perfectly inelastic. The reason for not
being able to attain complete accommodation is down to the need to satisfy energy conservation, which transversely correlates the gas modes and forces them to exchange energy with themselves rather than exchange all
the energy with the surface. In fact, we see in the preceding equation that a reflected energy mode is not directly
correlated to its incident energy mode – i.e. the longitudinal correlation is zero. However, each reflected mode is
correlated to the total system energy and hence to all the incident modes (or all the reflected modes) taken together. This is what cross correlation is about. Such latent correlation means that the models cannot attain complete lack of correlation between the incident and reflected states, and this is what diffuse exchange with complete accommodation requires. Therefore, the fully correlated models cannot attain diffuse exchange with complete accommodation. They can only attain perfectly inelastic exchange which, for gas surface interactions, is a
type of diffuse reflection with incomplete accommodation. These results agree with the analysis of accommodation coefficients given below in the model parameter determination theory.
When cross correlation is absent, as in the partially correlated model, we find that we can attain diffuse
reflection with complete accommodation. For the partially correlated model the upper parameter limit gives:
ξ t′′ = ξt′2 ,
ξ r′′ = ξ r′2
ξ v′′ = ξ v′ 2
(13a)
with associated densities:
f (ξ ) = Ga (ξ t′2 γ t )Ga (ξ r′2 γ r )Ga (ξ v′ 2 γ v )
(13b)
We thus get equilibrium distributions for the normalised energies. These equilibrium distributions define diffuse
exchange with complete accommodation. Note that complete accommodation is only attained if the surface temperature is used as the equilibrium temperature. If the freestream temperature is used as the equilibrium temperature the partially correlated model only gives incomplete accommodation at its upper parameter limit.
NUMERICAL SIMULATIONS
The simulations are performed by sampling the correlation variates from their probability densities and using
these to compute the interactions. Thus, to perform a calculation for a given incident molecule we sample the
required correlation variates from their correlation densities. We then substitute these into the model equations,
along with the incident state. Evaluating the right hand side of the equations gives the reflected energy. By repeating these calculations for a number of incident molecules we can classify the reflected states into elementary
ranges. This way we can determine the reflected energy distributions for any given incident pattern. Combining
an energy exchange model with a scattering model allows us to obtain reflected velocity distributions. To sample from the Gamma and Beta distributions we use the schemes proposed by Ahrens and Dieter (Refs. 21 and
22) and also described in Ref. 23.
For the studies, collimated-thermal beams were used to provide the incident distribution. A collimated
thermal beam streams particles at a fixed angle, but the energy is distributed among the particles following a
Boltzmann distribution. Velocity and energy distributions were obtained by centring the statistics on a given
value y″c, such that:
yc′′ + ( 2n − 1)
δy ′′
2
≤ y ′′ ≤
δy ′′
2
( 2n + 1) + yc′′
where
n = − N , ..., − 1, 0, 1, ..., N .
(47)
For the purpose of the computations we reduced the model parameters to a single parameter, called the coincident parameter, by writing:
αt = αr = αv = α I = α E
The simulation conditions were as follows:
Number of Particles = 100 000
Translational d.o.f. = 4,
Solid d.o.f. = 10000
Rotational d.o.f. = 2,
Solid parameter = 1
Vibrational d.o.f. = 2
Incident Translational Temperature to Solid Temp. = 5
Incident Rotational Temperature to Solid Temp. = 7
Incident Vibrational Temperature to Solid Temp. = 9
Class width:
Energy = 0.3
Incident angle = 135 degrees,
Velocity = 0.3
Incident plane = 90 degrees
Velocity calculations were based on the Zero-Correlated Scattering Model (ZCSM) described in Ref. 5. The
parameters for this model were set equal to the energy parameters. The velocity components are:
u = c sin θ cosφ , v = c sin θ sin φ ,
w = c cosθ
where θ is the polar angle and φ is the azimuthal angle in the spherical co - ordinate system.
Energies and velocities were normalised with respect to:
Energy :
kTs ,
Velocity :
2kTs m
where m is the molecular mass.
Note that gas surface interactions depend on temperature only as a ratio of the gas temperature to the solid temperature. Therefore the actual gas and surface temperatures are immaterial.
Sample results from the simulations are shown in Figures 2 – 5 which depict a continuous variation
from specular to diffuse reflection with complete accommodation. The large number of solid atoms (3333.33)
used for these simulations means that the perfectly inelastic reflection becomes coincident with diffuse reflection with complete accommodation. This is seen by the fact that the accommodation coefficients for each model
are unity at the upper parameter extreme. Another pointer is the fact that the accommodation behaviour of the
fully correlated models becomes identical to the accommodation behaviour of the partially correlated model.
This contrasts with the results of Ref. 5 wherein only 5 solid atoms were used and it was observed that the accommodation coefficients for the fully correlated models were generally less than the values obtained for the
partially correlated model. Hence the perfectly inelastic reflection was not coincident with perfectly diffuse reflection.
Figures 2a to 2c compare the translational energy distributions (i.e. probability densities) for monatomic and diatomic gases. The general shapes are similar but there are important differences. First of all at lower
parameter values the peaks are flatter for monatomic gases than for diatomic gases. Secondly for monatomic
gases the specular peak is more widely separated from the diffuse peak. For diatomic gases the peaks are concentrated within a narrow region. Thirdly, for monatomic gases, specular to diffuse reflection spans a wider
range than for diatomic gases. All these results show that the presence of internal energy affects significantly the
behaviour of the translational mode. The translational behaviour for the free exchange model is similar to the
behaviour for the restricted exchange model. However, the restricted exchange peaks are generally taller.
Figures 2d to 3b compare the rotational distributions for the partially and fully correlated models for a
diatomic gas. It should be pointed out that a classical treatment of internal energy misrepresents the internal distributions. Nevertheless, the accommodation behaviour is properly represented because for any macroscopic
average over a quantised distribution at a given model parameter value one can find an identical average over a
classical distribution simply by changing the model parameter value. This argument means that all macroscopic
properties would be correctly represented by classical reciprocity theory. However, a correct treatment of internal energy requires quantum reciprocity theory. Figures 2d to 3b show that there are important differences between the rotational distributions and that the degrees of correlation and restriction are important. First of all the
distribution peaks for partial correlation span a range of 0 – 4 whereas for free exchange they span 2 – 5.7 and
for restricted exchange they span 2 to 6.5. The shape of the distributions for the partially correlated model is also
somewhat different from that for the fully correlated models. It should also be noted that the pattern of restriction used for the vibrational model is different from that used for the rotational model. In the rotational model
the rotational energy is combined with the solid energy before interchanging with the translational mode. This
leads to rotational patterns and rotational accommodation coefficients that always stay close to the perfectly
diffuse extreme (Ref. 5). In the vibrational model discussed here the rotational and vibrational energies are
combined before interchanging with the translational and solid modes. This allows the rotational patterns to
span a wider range. Still another manner of restriction is to combine the rotational, vibrational and translational
energies into a total gas energy before exchanging with the surface.
The vibrational patterns are compared in Figures 3c to 4b. Again the pattern spans a wider range for the
partially correlated model than for the fully correlated models. The pattern is also wider for the restricted exchange model than for the free exchange model. The shape for the partially correlated model is also somewhat
different from that for the fully correlated models. Note that the vibrational patterns generally span a wider
range than the rotational patterns which also span a wider range than the rotational patterns. This is because the
incident vibrational temperature is higher than the incident rotational temperature which is also higher than the
incident translational temperature.
Figure 5c shows the accommodation patterns for the restricted exchange model. The patterns for the
other models are not shown because the accommodation coefficients are all equal to the model parameter values, this being caused by the large number of solid atoms accounted for in the solid block. For restricted exchange this is also the case for the translational and gas accommodation coefficients. The vibrational accommodation coefficient is always larger than the model parameter whereas the rotational accommodation coefficient
is always lower. This unrealistic result is caused by the trivial choice of parameter values, which makes the parameters coincident. In reality the translational accommodation coefficient would be greater than the rotational
accommodation coefficient which would also be greater than the vibrational accommodation coefficient.
ESTABLISHING THE NUMBER OF SOLID ATOMS
The choice of solid block size depends on the efficiency of the Gamma-Beta sampling algorithms. Efficient algorithms (such as those described in Refs. 22 – 24) get faster the larger the solid block while inefficient algorithms (such as those based on convenient sampling methods – see Ref. 24) get slower, and grind to a halt, the
larger the solid block. Therefore, with the commonly used inefficient schemes the solid block should be 13 – 15
atoms large, which is similar to how dynamical theories of gas surface scattering set the solid block size. With
efficient schemes the number of solid atoms could be as large as 5000 or more! The accommodation coefficient
for perfectly inelastic reflection becomes 1, as illustrated in the simulations described here. Therefore, with efficient sampling schemes reciprocity models get faster the larger the solid block whereas dynamical theories get
faster the smaller the solid block!
CONCLUSION
This paper has discussed how to model energy exchange processes for vibrationally excited single-body gas
surface interactions for which no chemical reactions occur. Two general types of models were formulated: fully
correlated and partially correlated. The fully correlated models exhibit both longitudinal and transverse correlation whereas the partially correlated model exhibits only longitudinal correlation. A theory was also given for
obtaining the model parameters from thermal accommodation coefficients. This theory allowed terminal accommodation coefficient values to be calculated, from which it was found that for a small solid block the fully
correlated models do not establish complete accommodation at the upper limit of the model parameter whereas
the partially correlated model does. The fully correlated models provide a continuous variation between elastic
and perfectly inelastic reflection whereas the partially correlated model permits a continuous variation between
elastic and diffuse reflection with complete accommodation. However, complete accommodation is only obtained if the surface temperature, rather than the freestream temperature, is used as the equilibrium temperature.
Perfectly inelastic reflection is a type of diffuse reflection with incomplete accommodation. It approaches perfectly diffuse reflection as the solid block size increases.
ACKNOWLEDGEMENTS
This work was sponsored by the Ministry of Defence of Great Britain under contract numbers AT 2037 /250 and
331 in the period 1984 to 1988 at Imperial College, Department of Aeronautics. There was also financial support from the Cameroon Government.
REFERENCES
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Cercignani C (1975), “Theory and Application of the Boltzmann Equation’’, Scottish Academic Press, London.
4
Pullin D I (1978), Phys. Fluids, 21:2, 209.
5
Agbormbai A A (1988), PhD Thesis, Imp. Coll., Univ. London.
6
Agbormbai A A (1997), “The General Principles of Reciprocity Theory”, Minerva Press, London.
7
Lord R G (1995a), Rarefied Gas Dynamics, 19th symp., p563.
8
Lord R G (1998), Phys. Fluids, 10:3, 742.
9
Agbormbai A A, “Energy Exchange Modelling of Monatomic Gas Surface Interactions”, submitted to AIAA 41st Aerospace
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10
Cercignani C and Lampis M (1971), Transp. Theory Stat. Phys., 1, 101.
11
Lord R G (1991a), Rarefied Gas Dynamics, 17th symp., p1427.
12
Lord R G (1991b), Phys. Fluids, A:3, 706.
13
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14
Lord R G (1995), Phys. Fluids, 7, 1159.
15
Lord R G (1999), Rarefied Gas Dynamics, 21st symp., p.
16
Larsen P S and Borgnakke C. (1974), Rarefied Gas Dynamics, 9th symp., paper A7.
17
Bird G A (1970), AIAA Journ., 8, 1988.
18
Agbormbai A A, “Reciprocity Equation for Gas Surface Interactions in Rarefied Gas Dynamics”, submitted to AIAA 41st
Aerospace Sciences Meeting and Exhibit.
19
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20
Agbormbai A A, “Reciprocity Proofs for Gas Surface Interactions in Rarefied Gas Flows”, submitted to AIAA 41st Aerospace Sciences Meeting and Exhibit.
21
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22
Ahrens J. H. and Dieter U. (1982), Comm. ACM, 25, p47.
23
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2
3
(a)
Incident atom
Reflected atom
Solid block of
vibrating atoms
(b)
Enhanced incident
atom
(contains
solid d. o. f.)
Enhanced reflected
atom (contains solid
d. o. f.)
Idealised surface that only scatters and triggers energy exchange (no solid atoms)
Figure 1.
(a) Gas surface system showing solid block of atoms that are affected by an interaction.
(b) Transformed gas surface system showing idealised surface and enhanced gas atom. The solid
degrees of freedom have been transferred to the gas atom.
(a)
Diatomic Gas, Free Exchange Model, Coincident Parameters
= 0.05 - 1.0
2.5
2
0.05
0.2
0.4
0.6
0.8
1
1.5
1
0.5
0
0
2
4
6
8
10
Norm alised Translational Energy
12
14
16
(b)
M o n a to m ic G a s , F u lly C o rre la te d M o d e l, T ra n s la tio n a l
P a ra m e te r = 0 .0 5 - 1 .0
4
3 .5
3
0 .0 5
0 .2
0 .4
0 .6
0 .8
1
2 .5
2
1 .5
1
0 .5
0
0
2
4
6
8
10
12
14
16
N o r m a lis e d T r a n s la tio n a l E n e r g y
(c)
D iato m ic G as, R estricted E xch an g e M o d el, C o in cid en t
Param eters = 0.05 - 1.0
3
2 .5
0 .0 5
0 .2
0 .4
0 .6
0 .8
1
2
1 .5
1
0 .5
0
0
2
4
6
8
10
12
14
16
N o r m a lis e d T ra n s la tio n a l E n e r g y
(d)
Diatom ic G as, Partially C orrelated M odel, C oincident
Param eter = 0.05 - 1.0
4.5
4
3.5
0.05
0.2
0.4
0.6
0.8
1
3
2.5
2
1.5
1
0.5
0
0
2
4
6
8
10
12
14
16
N o rm alised R o tatio n al En erg y
Figure 2. Distributions for (a) diatomic gas translational energy with free exchange (b) monatomic gas
translational energy with full correlation (c) diatomic gas translational energy with restricted exchange, and
(d) diatomic gas rotational energy with partial correlation.
(a)
Diatom ic G as, Free Exchange M odel, Coincident Param eters
= 0.05 - 1.0
6
5
0.05
0.2
0.4
0.6
0.8
1
4
3
2
1
0
0
2
4
6
8
10
12
14
16
N o rm alised R o tatio n al En erg y
Diatomic G as, Restricted Exchange M odel, Coincident
Parameters = 0.05 - 1.0
(b)
7
6
5
0.05
0.2
0.4
0.6
0.8
1
4
3
2
1
0
0
2
4
6
8
10
12
14
16
N orm alised R otational Energy
(c)
Diatomic G as, Partially Correlated M odel, Coincident
Parameter = 0.05 - 1.0
8
7
6
0.05
0.2
0.4
0.6
0.8
1
5
4
3
2
1
0
0
2
4
6
8
10
12
14
16
N orm alised Vibrational Energy
Figure 3. Distributions for (a) diatomic gas rotational energy with free exchange (b) diatomic gas rotational energy with restricted exchange, and (c) diatomic gas vibrational energy with partial correlation.
(a)
Diatom ic G as, Free Exchange M odel, C oincident Param eters
= 0.05 - 1.0
5
4.5
4
0.05
0.2
0.4
0.6
0.8
1
3.5
3
2.5
2
1.5
1
0.5
0
0
2
4
6
8
10
12
14
16
N o rm alised Vib ratio n al En ergy
Diatom ic G as, R estricted Exchange M odel, C oincident
Param eters = 0.05 - 1.0
(b)
6
5
0.05
0.2
0.4
0.6
0.8
1
4
3
2
1
0
0
2
4
6
8
10
12
14
16
N o rm alised Vib ratio n al En erg y
D iatom ic G as, R estricted E xchange M o del, Therm al
Accom m odation C o efficients
(c)
1.2
1
0.8
T ranslational
R otational
V ibrational
G as
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
1.2
-0.2
C o in cid en t M o d el P aram eter
Figure 4. Distributions for (a) diatomic gas vibrational energy with free exchange (b) diatomic gas vibrational energy with restricted exchange, and (c) thermal accommodation coefficients for a diatomic gas
with restricted exchange.