Freepaper.me 10.1007 sdf Aerodynamical Applications of the Boltzmann Equation

RMSTA DEL NUOVO CIMENTO VOL. 18, N. 7 1995

Aerodynamical Applications of the Boltzmann Equation.

C. CERCIGNANI

Dipartimento di Matematica, Politecnico di Milano - Milano, Italy

1 1. Introduction. 6 2. Continuum vs. molecular models of a gas. 7 3. Rarefaction regimes. 8 4. Gas-surface interaction.

18 5. Polyatomic gases and mixtures. 24 6. Chemical reactions and thermal radiation. 26 7. Solving the Boltzmann equation. Analytical techniques. 31 8. Solving the Boltzmann equation. Numerical techniques. 33 9. Vortices and turbulence in a rarefied gas. 34 10. Concluding remarks.

1 . - I n t r o d u c t i o n .

This paper deals with an important application of the Boltzmann equation, i.e. the study of upper-atmosphere flight, which occurs, e.g., in connection with the re-entry of a space shuttle.

This is one of the fields of modern technology, where the concepts and tools introduced by Boltzmann are essential. This would have pleased Boltzmann, who was very much interested in technological advances (he also wrote a paper, in which he correctly predicted the superiority of airplanes over dirigible airships) and is the author of the following sentence written in 1902 [1], which nowadays may sound a bit trivial:

However much science prides itself on the ideal character of its goa~ looking down somewhat contemptuously on technology and practice, it cannot be denied that it took its rise from a striving for satisfaction of purely practical needs. Besides, the victorious campaign of contemporary natural science would never have been so incomparably brillian~ had not science found in technologists such capable pioneers.

To this we may add another sentence, written in 1905 [1]:

That is why I do not regard technological achievements as unimportant by-products of natural science but as logical proofs. Had we not attained these practical achievements, we should not know how to infer. Only those inferences are correct that lead to practical success.

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Before entering the main subject of the paper, we want to deal very briefly with the objections which are frequently still voiced against the Boltzmann equation. This will also serve the purpose of fLxing some notation.

We start by recalling this famous equation [2-6]:

(1.1) af + a / + x . a f 3t ~x ~

- I I ( ' f f , - f f , ) B ( n . V , IVI)d~, (ln; R 3 ~+

f = f(x, ~, t) is the distribution function of the molecules, which can be normalized in various ways (as a number density, as a probability density, as a mass density). t denotes time, x the position of a molecule, ~ the velocity of the same molecule, X a body force acting on each molecule (such as, e.g., gravity). The right-hand side contains a quadratic expression Q(f, f) , given by

(1.2) Q(f' f )= I I (f'f * -ff,)B(n.V, IVI)d ,d . R 3 S 2

Here B(n. V, IV I ) is a kernel containing the details of the molecular interaction, f ' , f ' , , f . are the same as f, except for the fact that the argument ~ is replaced by ~', ~ . , ~ , , respectively, ~. being an integration variable (having the meaning of the velocity of a molecule colliding with the molecule of velocity ~, whose path we are following). ~' and ~. are the velocities of two molecules entering a collision that will bring them to have velocities ~ and ~., whereas n is a unit vector giving the direction of the straight line joining the centres of the molecules at the moment of impact and describes a unit sphere S 2 . The relations between ~', ~ . , on the one hand, and ~ and ~ . , on the other hand, read as follows:

(1.3) ~ ' = ~ - n [ ( ~ - ~ , ) - n ] , ~ , = ~ , + n [ ( ~ - ~ , ) . n ] .

We shall denote by G(f, f ) and fLf, respectively, the gain and loss parts of Q(f, f) . Here we have restricted ourselves to monatomic molecules; polyatomic gases will

be considered in sect. 5. One important property of eq. (1.1) is that the collision integral vanishes when f is

a Maxwellian distribution M:

(1.4) M = Q(2zRT)-~/2 exp [ - I ~ - v l 2/(2RT)],

where the parameters Q, T, and v have the meaning of density, temperature and bulk velocity of the gas.

If we multiply both sides of eq. (1.1) by log f and integrate with respect to ~, we obtain

a:~v a (1.5) - - + - - . J = ~f,

at ax

AERODYNAMICAL APPLICATIONS OF THE BOLTZMANN EQUATION 3

where

(1.6) f

= / f log f d ~ , ha

(1.7) J = f / i f log f d~, ha

(1.8) J" = ~ f log fQ(f, f) d~. ha

We know, however, that a famous inequality due to Boltzmann applies:

(1.9) ~f ~< 0 and ~f = 0 iff f is a Maxwellian.

Because of this inequality, eq. (1.5) plays an important role in the theory of the Boltzmann equation. We illustrate the role of eq. (1.9) in the case of space- homogeneous solutions. In this case the various quantities do not depend on x and eq. (1.5) reduces to

(1.10) 8~V _ J" ~< O. 8t

This implies the so-called H-theorem (for the space-homogeneous case): ~ is a decreasing quantity, unless f is a Maxwellian (in which case the time derivative of ~V is zero). Since in this case the density Q, the momentum density Q$ and the internal energy e are constant in time, we can build a Maxwellian M which has, at any time, the same ~), ~ and e as any solution f corresponding to given initial data. Since decreases unless f is a Maxwellian (i.e. f= M), it is tempting to conclude that f tends to M when t -~ oo. The temptation is strengthened when we realize that ~ is bounded from below by ~M, the value taken by the functional .~V when f = M. In fact :~V is decreasing, its derivative is non-positive unless it takes the value ~VM; one feels that ~ tends to :~VM! This conclusion is, however, unwarranted, without a more detailed consideration of the source term ~f in eq. (1.10). This is shown in detail in ref. [4].

If the state of the gas is not space-homogeneous, the situation becomes more complicated. In this case it is convenient to introduce the quantity

(1.11) H = f ~ dx, ~2

where ~2 is the space domain occupied by the gas (assumed here to be time independent). Then eq. (1.5) implies

(1.12) dH f J-, ,do, dt J

where n is the inward normal and da the measure on 8~2. Clearly, several situations may arise. Among the most typical ones, we quote:

1) ~2 is a box with periodicity boundary conditions (flat toms). Then there is no

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boundary, dH/dt


negative sign in front of the collision term, and hence describing only motions with increasing H. We must remark that, in order to derive the Boltzmann equation, one takes special (although highly probable) initial data; thus certain special data are excluded. As the discussion in ref. [3, 4] shows, these excluded data correspond to a state in which the molecular velocities of the molecules that are about to collide show an unusual correlation. This situation can be simulated by studying the dynamics of many interacting particles on a computer and leads to an evolution in which there is an increasing H, as expected, while ,,randomly- chosen initial data invariably lead to an evolution with decreasing H [3, 4]. In other words, the fact that H decreases is not an intrinsic property of the dynamical system but a property of the level of description.

It is not the place here to discuss the relation of the H-theorem with the notions of past and future [2,9].

It is not, however, perhaps out of place to comment on a statement which is frequently made, to the effect that no kind of irreversibility can follow by correct mathematics from the analytical dynamics of a conservative system and hence some assumption of kinetic theory must contradict analytical dynamics. It should be clear that it is not a new assumption that is introduced, but the fact that we study asymptotic properties of a conservative system in the so-called Boltzmann-Grad limit (see below), under the assumption that the initial probability distributions are restricted to a special (but highly probable) set [3, 4].

There is another objection which can be raised against the H-theorem when presented as a rigorous consequence of the laws of dynamics. The starting point is a theorem of Poincar~ [10] (the so-called ,,recurrence theorem,), which says that any conservative system, whose possible states form a compact set in phase space, will return arbitrarily close to its initial state, for almost any choice of the latter, provided we wait long enough. This applies to a gas of hard-sphere molecules, enclosed in a specularly reflecting box, because the set of the possible states S with a given energy is compact and has a finite measure/x(S) (induced by the Lebesgue measure).

This theorem implies that our molecules can have, after a ,,recurrence time,, positions and velocities so close to the initial ones that the one-particle distribution function f would be practically the same; therefore, H should also be practically the same, and if it decreased initially, then it must have increased at some later time. This paradox goes under the name of Zermelo, who stated it in 1896 [11], but was actually mentioned before in a short paper by Poincar~ [12]. The traditional answer to Zermelo's paradox was given by Boltzmann himself[13]: the recurrence time is so large that, practically speaking, one would never observe a significant portion of the recurrence cycle. In fact, according to an estimate made by Boltzmann [13], the recurrence time for a typical amount of gas is a huge number even if the estimated age of the Universe is taken as time unit.

Nowadays we know that we can claim validity for the Boltzmann equation in the Boltzmann-Grad limit only (when the number of molecules goes to infinity and the sphere diameter a goes to zero in such a way that N a 2 is finite), we do not have to worry about the recurrence paradox; in fact, the set S is no longer compact when N --~ oo and the recurrence time is expected to go to infinity with N (at a much faster rate).

The Boltzmann equation was written by Boltzmann in 1872 and has become a practical tool for upper-atmosphere aerodynamics in the last twenty years. In order to be useful for applications it must, of course, be matched with appropriate initial and

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boundary conditions. While the initial data are specified by assigning the initial distribution, the boundary conditions are more complicated [2-4]. A discussion of this aspect of rarefied-gas dynamics will be given, as indicated above, in sect. 4.

2. - C o n t i n u u m v s . m o l e c u l a r m o d e l s o f a gas .

In this section we shall explain why, when dealing with upper-atmosphere flight, one may easily be led to questioning the validity of the Euler and Navier-Stokes equations as an accurate description of the fluid dynamics of the atmosphere. In fact, as we saw in the previous section, according to the kinetic theory of gases, the basic description of the state of a gas is in terms of the distribution function, a function of position x, velocity ~ and time t, which gives the probability density of finding a molecule at position x with velocity ~ at time t. In terms of this non-negative function, f = f ( x , ~, t), one can define the typical quantities used in macroscopic fluid dynamics, such as density Q, bulk velocity v, stress components p~j, pressure p, temperature T, heat flow q, in the following way:

(2.1)

Q(x, t ) : If(.,:, t)d , Qv(x, t ) : f(x, t)d+,

p~/(x, t) = I ci cj f (x , ~, t) d~

p(x, t) = QRT = (1/3) I [c[2f(x, ~, t )d~ ,

q(x, t) = (1/2) f c lc l2 f (x , ~, t )d~

(c~ = ~i - v i ) ,

(c= ~ - v ) .

(here the subscripts range from 1 to 3 and R is the gas constant, equal to k/m, k being the Boltzmann constant and m the molecular mass).

In order to see that the Navier-Stokes equations become invalid on a sufficiently small scale, it is sufficient to remark that the definition of the components of the stress tensor in eq. (2.1) implies (because of the elementary inequality 21abl


so that eq. (2.4) becomes

(2.6) [Su/3y I ~< 3(2RT)1/2/,~..

In order to appreciate this simple result, we remark that, in the Earth's atmosphere, the mean free path )~ is about 1 metre at about 130 km. More typical rarefactions are met in the study of the flight of aircrafts transporting men or robots for the planned exploration of other planets, such as Mars or Venus. In any case, eq. (6) indicates that for rarefied gases and/or high speeds, one cannot rely on the usual Navier-Stokes equations for a compressible fluid and must resort to kinetic theory.

It is to be remarked that rarefied gases appear in many other areas of technology: vacuum technology, micromachines, dynamics of aerosols, etc. All these problems have in common the fact that the mean free path is not negligible with respect to some other characteristic length.

3. - Rarefact ion regimes.

In order to orient ourselves in a new field and chart the phenomena that can be expected, it is very useful to consider the typical non-dimensional parameters associated with the equations which rule these phenomena. Rarefied-gas dynamics is no exception: it makes use of two basic non-dimensional numbers, the Knudsen number

(3 .1) i n = ;/L,

where L is a characteristic length, such as the diameter of a pipe, the thickness of the viscous boundary layer, the curvature radius of the nose of an airfoil, or t h e wavelength of a sound wave, and the molecular speed ratio

(32) S = u/C,

where C is the thermal speed V2R-T. Kn and S are related to the Mach and Reynolds numbers, Ma and Re, in the

following way:

(3.3) Ma = V2V2V2V~ S; Kn = ~ f ~ Ma/Re . ,

This indicates that one might use the same parameters as in the usual continuum theory, based on the Navier-Stokes equations. For rarefied gases, however, the Knudsen number is more handy than the Reynolds number. Of course, one can consider several Knudsen numbers, based on different characteristic lengths, exactly as one does for the Reynolds number. Thus, in the flow past a body, there are two important macroscopic lengths: the local radius of curvature and the thickness of the boundary layer 5, and one can consider Knudsen numbers based on either length. Usually the second one, (Kna = k/a), gives the most severe restriction to the use of Navier-Stokes equations; when Kn~ > (say) 0.01, the presence of a thin layer near the wall, of thickness - )~ (Knudsen layer), influences the viscous layer in a significant way. This regime is called the slip regime because the gas slips upon the boundary with a velocity us and, at the boundary, has a temperature different from the

8 C. CERCIGNANI

temperature of the boundary itself. The velocity slip is given by

(3.4) us = ~(~u/Sn)w,

where ~ is the slip coefficient. A similar formula holds for the temperature jump. Maxwell computed ~ to be approximately equal to (V~/2)s (for complete diffusion of the molecules at the wall see below). Actually ~ is about 15% larger [2,3].

This and other regimes to be described below are met in high-altitude flight; in particular, they are all met by a shuttle when returning to the Earth.

When the mean free path increases, one witnesses a thickening of the bow shock wave which forms in front of a vehicle travelling at a supersonic speed in a gas. This thickness is of the order of 6)[ and eventually the shock merges with the viscous layer; that is why this regime is sometimes called the merged-layer regime. Another frequently used name is transition regime.

When Kn is large (few collisions), phenomena related to gas-surface interaction play an important role. One distinguishes between free-molecule and nearly free- molecule regimes. In the first case the collisions between molecules are completely neglegible, while in the second they can be treated as a perturbation.

4. - G a s - s u r f a c e i n t e r a c t i o n .

The problem of gas-surface interaction is basic in rarefied-gas dynamics, since it is to this interaction that one can trace back the origin of the drag and lift exerted by the gas on the body and the heat transfer between the gas and a solid boundary. This has its mathematical counterpart in the fact that, if we want to describe a physical situation where a gas flows past a solid body or is contained in a region bounded by one or more solid bodies, the Boltzmann equation must be accompanied by boundary conditions, which describe the above-mentioned interaction of the gas molecules with the solid walls. Hence, in order to write down the correct boundary conditions for the Boltzmann equation, we must possess information which stems from a discipline which may be regarded as a bridge between the kinetic theory of gases and solid-state physics.

The difficulties of a theoretical investigation are due, mainly, to our lack of knowledge of the structure of surface layers of solid bodies and hence of the effective interaction potential of the gas molecules with the wall. When a molecule impinges upon a surface, it is adsorbed and may form chemical bonds, dissociate, become ionized or displace surface molecules. Its interaction with the solid surface depends on the surface finish, the cleanliness of the surface, its temperature, etc. It may also vary with time because of outgassing from the surface. Preliminary heating of a surface also promotes purification of the surface through emission of adsorbed molecules. In general, adsorbed layers may be present; in this case, the interaction of a given molecule with the surface may also depend on the distribution of molecules impinging on a surface element.

The first observations of the interaction of gases with solid surfaces (apart from early studies which arrived at the conclusion that the gas does not slip on the wall under standard conditions) are due to Kundt and Warburg [14]. They noted that the flow rates through tubes at very low pressures are appreciably higher than predicted by the familiar Poiseuille formula and attributed that effect to slip at the boundary. Maxwell [15] suggested that slip would be a consequence of kinetic theory and


computed the amount of slip for a particular model of the gas-surface interaction, still very popular and known under his name. Smoluchowski [16] described temperature jumps in a similar way. A more systematic effort started with Knudsen[17]. Full-scale research, however, only began about thirty years ago, under the impetus of space flight and thanks to the developments in high-vacuum technology.

We shall try here to give an idea of the physical models and computations which aim at simulating the complex phenomena to which we have briefly alluded, following the surveys of Ku~Ser [18] and Cercignani [19].

The possible events, that a gas molecule hitting a solid (or liquid) surface may experience, have been briefly described above. A more systematic classification may read as follows:

1) Scattering. specular reflection;

Elastic scattering (zero-phonon scattering) ~diffraction into discrete peaks; / [diffuse elastic scattering.

Ione -phonon scattering; Inelastic scattering [multi-phonon scattering. Sputtering and other high-energy effects.

2) Adsorption.

Ite mporary; Adsorption at specific sites [permanent. Mobile adsorption (surface diffusion).

Diffusion into the condensed phase.

3) Condensation (in case of identical composition of gas and condensed phase).

4) Reactive interaction.

Chemical reactions.

Ionization.

Elastic scattering is strong only if the surface behaves as it were almost rigid; this may occur if both the energy of the gas molecule and the thermal energy of the wall are comparable to or below kB TD (where kB is the Boltzmann constant and TD the Debye temperature). In this case specular reflection can occur only if the equi- potential surfaces are approximately flat on the molecular scale. Thermal motions in the solid reduce elastic scattering, till, if the molecule kinetic energy is several times in excess of kB TD, incident molecules become capable of ejecting atoms or whole chunks of the solid (,~sputtering,).

Gas molecules with strong chemical affinity to the solid become adsorbed, i.e. attached to the surface. Even in the absence of such affinity, attachment can occur at sufficiently low temperatures, because of van der Waals forces. Although the difference is mainly in the magnitude of the binding energy, one usually distinguishes the two cases by means of the names chemisorption and physisorption. Any adsorption is, in principle, temporary [20], but for strong binding and low surface

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temperatures the sitting time may well exceed the duration of any experiment or even the age of the Universe; in this case one talks about permanent adsorption. Heating strongly accelerates the desorption process. Strongly adsorbed molecules are usually localized at particular adsorption sites, but even a face of an ideal crystal offers several kinds of such sites. Polycrystals and amorphous solids are expected to be far richer in this respect. The behaviour of the adsorbed substance very much depends on the fraction of adsorption sites actually occupied. A complete layer looks like a part of the crystal and has a similar if not the same symmetry. Above it a second layer can sometimes be adsorbed.

The exchange of matter in condensation and evaporation is analogous to adsorption and desorption, except for the identical composition of both phases. This has the important consequence that the surface remains unchanged.

Chemical reactions may occur at the wall; e.g., the adsorbate can dissociate during adsorption, or, if the gaseous phase is a mixture, the wall can catalyze reactions that would not occur in the gas. Analogously one may have ionization of the adsorbate.

In general, a molecule striking a surface with a velocity ~' re-emerges from it with a velocity ~ which is strictly determined only if the path of the molecule within the wall can be computed exactly. This computation is very hard, because it depends upon a great number of details, such as the locations and velocities of all the molecules of the wall and an accurate knowledge of the interaction potential (see below). In fact a sufficiently accurate calculation should be able to predict all the phenomena that we have briefly described above.

Hence it is more convenient to think in terms of a probability density R(~'--~ $; x, t; r) that a molecule striking the surface with velocity between ~' and ~' + d~' at the point x and time t will re-emerge at practically the same point with velocity between $ and ~ + d$ after a time interval v (adsorption or sitting time). If R is known, then we can easily write down the boundary condition for the distribution function f(x, ~, t). To simplify the discussion, the gas will be presently assumed to be monatomic and the surface at rest.

The mass (or number, depending on normalization) of molecules emerging with velocity between ~ and ~ + d~ from a surface element dA about x in the time interval between t and t + dt is

(4.1) d * ~ =f(x , ~, t ) l ~ . n l d t d A d ~ (x E ~Y2, ~ 'n > 0),

where n is the unit vector normal to the surface ~t2 at x and directed from the wall into the gas. Analogously, the probability that a molecule impinges upon the element dA with velocity between ~' and ~ '+ d~' in the time interval between t - r and t - r + d t ( r > 0 ) is

(4.2) d * ~ ' =f(x , ~', t - r ) l~ ' . n ]d tdAd ~' (x 9 ~Y2, ~ ' .n < 0).

If we multiply d 'A// ' by the probability of a scattering event from velocity ~' to a velocity between ~ and ~ + d~ with an adsorption time between r and r + dr (i.e. R(~'--~ ~; x, t; r )d~ dr) and integrate over all the possible values of ~' and r, we must obtain d * ~ (here, for simplicity sake, we assume that each molecule re-emerges from the surface element into which it entered, which is not so realistic when r is large):

c c

(4.3) d * A ' / = d ~ I d r I R(~'- - -~;x , t ;v)d*. ' r ( x 9 0 ~ < 0


Equating the expressions in eqs. (4.1) and (4.3) and cancelling the common factor dA d$ dt, we obtain

(4.4) f(x, ,t)l 'nl = i f o ~ 0).

If ~ is an average adsorption time, ~ the average normal velocity with which the gas molecules impinge upon the surface and n the number density of the gas, n~ dA molecules will impinge, per unit time, on a surface element of area dA and stay there an average time ~ ; ff a0 is an effective range of the gas-surface interaction, each molecule will occupy an area of the order of a~ and the total area occupied by adsorbed molecules will be n~-~a~ dA, i.e. a fraction n ~ a ~ of the surface will be occupied. This is, of course, just a rough order-of-magnitude argument, since the molecules may penetrate somewhat into the solid and not necessarily remain at its surface.

If na~ ~ is not close to zero, the nature of the interaction of each incident molecule depends on the total number and energy of the incident molecules. Under conditions of extremely low density (as, for example, in the case of an orbiting satellite) na~V~

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(4.7)

and, as a consequence,

(4.8)

1) Non-negativeness, i.e. R cannot take negative values:

R(~'--+ ~; x, t; ~) >>- O

R(~'--~ ~; x, t) >~ O.

2) Normalization, if permanent adsorption is excluded; i.e. R, as a probability density for the totality of events, must integrate to unity:

~c

(4.9) f d r f R ( ~ ' - - + ~ ; x , t ; r ) d ~ = l 0 ~,, > 0


(4.10) f R(~'--+~;x,t)d~= l .

3) Reciprocity; this is a subtler property related to the fact that the microscopic dynamics is time-reversible and the wall is assumed to be in a local equilibrium state, not significantly disturbed by the impinging molecule. It reads as follows:

(4.11) I ~ ' . n j M w ( ~ ' ) R ( ~ ' - - ~ ; x , t ; r ) = l ~ ' n I M w ( ~ ) R ( - ~ - * - ~ ' ; x , t ; ~ )


(4.12) I ~ ' . n l M w ( ~ ' ) R ( ~ ' - - ~ ; x , t ) = I ~ ' n ] M w ( ~ ) R ( - ~ - ~ - ~ ' ; x , t ) .

Here Mw is a (non-drifting) Maxwellian distribution having the temperature of the wall, which is uniquely identified apart from a factor.

We remark that the reciprocity normalization relations imply another property:

3') Preservation of equilibrium, i.e. the Maxwellian M~. must satisfy the boundary condition (4.4):

OC

(4 .13) Mw(~) l~ 'n l=I d~ I R(~'--*~;x ' t ;r)Mw(~') l~"nld~' 0 ~ < o

equivalent to

(4.14) Mw(~)lv'nl = I R(~'o~;x't)Mw(~')l~"nld~" ~ < 0

In order to obtain eq. (4.13) it is sufficient to integrate eq. (4.11) with respect to ~' and v, taking into account eq. (4.9) (with - ~ and - ~ ' in place of ~' and ~, respectively). We remark that frequently, one assumes eq. (4.13) (or (4.14)), without mentioning eq. (4.11) (or (4.12)); although this is enough for many purposes, reciprocity is very important when constructing mathematical models, because it places a strong restriction on the possible choices. A detailed discussion of the physical conditions under which reciprocity holds has been given by B~irwinkel and Schippers [27].

The scattering kernel is a fundamental concept in gas-surface interaction, by means of which other quantities should be defined. Frequently its use is avoided by


using the so-called accommodation coefficients (a.c.'s), with the consequence of lack of clarity, misinterpretation of experiments, bad definitions of terms and misunder- standing of concepts. The basic information on gas-surface interaction, which should in principle be obtained from a detailed calculation based on a physical model as discussed in the next section, is summarized in a scattering kernel. The further reduction to a small set of a.c.'s can be advocated for practical aims, provided this use is firmly related to the scattering kernel, as indicated, e.g., in two previously quoted books [3, 4].

The concepts of scattering kernel and a.c. are very useful schemes, where to fit the information arising from a study of the complicated physical phenomena associated with the interaction of the molecules of a gas with the atoms of a wall. This study is by necessity difficult, since we are dealing with a many-body problem. Any theoretical model leading to more than purely formal results turns out to be either a crude one or a rough approximation, with the exception of special situations. The support that may come from experimental data is sometimes deceptive, because these data are scanty.

A systematic review of existing models and approximations prior to 1976 may be found in the book by Goodman and Wachman [28]. Here we only sketch a few attempts and indicate a few recent developments.

An early attempt is due to Baule [29], who considered a model of the solid in the form of an array of smooth hard spheres, exercising independent thermal motions; later Logan et al. [30] replaced the spheres with hard cubes. Few examples are simple enough to yield to a closed-form solution even in the case of a single-scattering event. The scattering kernel for this model has been evaluated by B~irwinkel and Schmidt[31]. We remark that the kernels obtained from single-scattering events must be cut off in order to discard the molecules still moving inwards after the interaction. Hence they are not normalized and violate reciprocity, because of the arbitrariness in their way of dealing with multiple-scattering events.

Some of the difficulties are avoided by the soft-cube model, proposed by Logan and Keck [32]. In this model each cube carries a rather fiat potential well, with the purpose of explaining scattering events at energies of the order of the thermal energy without having to deal with multiple scattering, which is now insignificant. The typical lobular patterns in the angular distributions of scattered molecules, which are seen in experiments, are well reproduced [28]. The model also explains adsorption [32], but does not account for tangential momentum exchange.

A realistic picture can only be obtained by taking an entire piece of solid wall and simulating the dynamics on a computer[33,28], through the so-called trajectory analysis. Recent examples of this kind of calculations will be briefly discussed.

At higher energies the impinging molecule begins to feel the surface waviness, which results into a modified scattering pattern (,,structure scattering,,). A conspicuous pattern shows two peaks near certain limiting angles (the so-called ,,rainbow effect,, [28]).

Models capable of dealing with multiple scattering were proposed by this author [25, 3] but have received little if any attention. They are based on the idea of using a transport equation for the molecule inside the solid, which is regarded as a half-space. Then one has to solve the following equation:

3P 3P = L P , (4.15) ~ . - - ~ + X(x ) . ~

14 C. CERCIGNANI

where P is the probability density of finding the molecule in the half-space which simulates the wall (say xl < 0) at position x with velocity ~. X is the many-body force exerted on the molecule by the solid atoms and L a linear operator (of the Boltzmann and/or Fokker-Planck type) which describes the shorter-range interactions. This type of equation is not so easy to deal with, except in artificial but interesting cases; in fact by simulating the many-body force by a rigid surface and using collision frequencies (in the Boltzmann-like term) or diffusion coefficients (in the Fokker-Planck term) proportional to vl, it is possible to recover the Maxwell model and the Cercignani- Lampis model (to be discussed in the next section) for R(~'--~ ~). In fact, once eq. (4.15) is solved with the boundary condition

(4.16) P = ~(~ - ~') , xl = 0, ~1 = ~.n < 0,

then R(~'--~ ~) is immediately recovered by

I 'nl (4.17) R(~'-~ ~) - - - [ P ( x , ~)]xl ~0.

I "nl

The idea of using a statistical description of the motion of a molecule inside the wall appears in a current approach known as the three-dimensional generalized Langevin model [34, 35]. Details of the gas-surface interaction are computed as in a trajectory simulation, but the surface atoms are coupled to the lattice ones through two additional terms in the equation of motion: one of them accounts for the dissipation in the lattice and the second, a fluctuating force, accounts for the effects of lattice vibrations on surface atoms. The lattice atoms, in turn, are harmonically coupled to each other.

Quantum-mechanical models have also been developed. Some interesting results concerning elastic scattering have been obtained in the framework of the semi- classical theory by Doll [36,37], Masel et al. [38, 39], Dion and Doll [40]. The results appear reliable in the case of an incident molecule of high energy and a direction of approach almost normal to the wall. Fully quantum-mechanical calculations have also been performed starting with Tsuchida [41] and Cabrera et al. [42, 43]. Details on this topic are given in the book by Goodman and Wachman [28] as well as in the paper by Armand and Lapujoulade [44].

In view of the difficulty of computing the kernel R(~'---~ ~) from a physical model of the wall, we shall presently discuss a different procedure, which is less physical in nature. The idea is to construct a mathematical model in the form of a kernel R(~ ' -~ ~) which satisfies the basic physical requirements expressed by eqs. (4.8), (4.10), (4.12) and is not otherwise restricted except by the condition of not being too complicated.

A possible approach to the construction of such models is through the eigenvalue equation

(4.18) I R(~'--~ ~)Mw(~')~(~')l~"nl d~ ' = ,~Mw(~)l~'nlPt~f, ~ > 0

where Pt is the operator which reflects the tangential components of $. Let the eigenvalues ;tk's form a discrete set and let ~ be the corresponding eigenfunctions.


Then one can show [8] that

(4.19) R($'--->$)=Mw(~)l~'nl ~ 2k~bk(--$')~k($), k=O

where

(4.20) ~k($) = P t [ ~ ( $ ) ] .

In order to construct a model, it is necessary to choose the eigenfunctions ~ and the eigenvalues )~k. Of course, it is convenient to make the choice in such a way that the series in eq. (4.19) can be evaluated in finite terms. One possibility would be to take only a finite number of eigenvalues different from zero [2,3], but this has the disadvantage that, generally speaking, the positivity requirement, eq. (4.8), cannot be met. The simplest choice, then, is to take the first eigenvalue to be unity (with a constant eigenfunction, as required by eq. (4.14)) and the others all equal to the same value 1 - a(0 ~< a ~< 1). The kernel then turns out to be

(4.21) R(~'--~ ~) = aMw(~) l$-n I + (1 - a)5($ - $ ' + 2n(~ ' .n) ) .

This is the kernel corresponding to Maxwell model, according to which a fraction (1 - a) of molecules undergoes a specular reflection, while the remaining fraction a is diffused with the Maxwellian distribution of the wall Mw. This is the only model for the scattering kernel that appeared in the literature before the late 1960s. Since this model was felt to be somehow inadequate to represent the gas-surface interaction, Nocilla [45] proposed to assume that the molecules are re-emitted according to a drifting Maxwellian with a temperature, which is, in general, different from the temperature of the wall. While this model is useful as a tool to represent experimental data and has been used in actual calculations, expecially in free molecular flow [46], when interpreted at the light of later developments, it does not appear to be tenable, unless its flexibility is severely reduced [5, 47]. While the idea of a model like Nocilla's can be traced back to Knudsen [10], the full development of these ideas led to the so-called Cercignani-Lampis (CL) model [23]. From the point of view taken here, this model can be easily obtained by taking the eigenfunctions ~k to be products of Hermite polynomials in the tangential components of $ times Laguerre polynomials in the square of the normal component v, = v- n of the same vector. The reason of the different treatment of the components will be clear if one thinks that the range of v~ is [ 0, r162 ) rather than ( - ~ , ~ ) and the weight factor in the natural scalar product is not Mw(~) but rather ~Mw($) . The eigenvalues 2k are taken to be subsequent powers of two parameters. Then the series in eq. (4.19) can be summed up by means of the so-called Miller-Lebedev formula [23,3] to yield

(4.22) R($' --> $) = [ a na t ( 2 -- a t ) ]

p2 9 fl~w~nexp -f lw v ~ + ( 1 - a n ) v n -flw

~Z n

I~ t - - (1 - - (2 t )~ 12 ] ( 2(1- an )~/2 Vn v" ) at(2- a t ) Io ~w an

16 r CERCIGNANI

where Io denotes the modified Bessel function of first kind and zeroth order defined by

(4.23) 2~

Io(Y) = (2z) -1 I exp [ycosq~] dq~. 0

The two parameters at and an have a simple meaning; the frrst one is the a.c. for the tangential components of momentum, the second one the a.c. for v 2 , hence for the part of kinetic energy associated with the normal motion. This model became rather popular because it was found by other methods as well: through an analogy with Brownian motion by KugSer et al. [47], under a special mathematical assumption by Cowling [48], through an analogy with the scattering of electromagnetic waves from a surface by Williams[49]. Finally, it followed from the solution of the steady Fokker-Planck equation describing a (somewhat artificial) physical model of the wall [25], as was already mentioned. Also, the comparison with the data from beam scattering experiments were quite encouraging [23, 50, 3]. In spite of this, it must be clearly stated that there are no physical reasons why this model should be considered better than others. In particular, we remark that any linear combination of scattering kernels with positive coefficients adding to unity is again a kernel that satisfies all the basic properties. Thus, from a kernel with two parameters, such as the CL model we can construct a general model, containing an arbitrary function of those parameters.

We want to mention another method to produce scattering kernels in a simple way. This method was described about 15 years ago [51, 3] but does not appear to have been ever used. The starting point is any positive function K(~, ~'), defined for ~n >I 0 and ~', >I 0, symmetric in its arguments and such that

(4.24) H(~') = I K(~, ~')Mw(~)~n d~ ~ . > 0

is not larger than unity for any ~' .n i> 0. Then we form the kernel

(1 - H ( - ~ ' ) ) ( 1 - / - / ( ~ ) ) (4.25) R(~'--~ ~) = ~nMw(~)K(~, - ~ ' ) + ~nMw(~) J "(1

H ( w ) ) w n M w ( w ) d w

In fact in this way both reciprocity and normalization are clearly satisfied and positivity follows from the assumption on H(~).

In order to illustrate this procedure, one might think of costructing a mode] chosen to be as close as possible to Nocilla's model and still satisfy the requirements expressed by eqs. (4.8), (4.10) and (4.12). This was attempted by the author[19], starting from the assumption (to be corrected later, according to the method just described) that the emerging distribution is a drifting Maxwellian, when the impinging distribution is a delta-function. Then the candidate kernel is

(4.26) R 0 ( ~ ' ~ ~ ) = A ' I~" I-l~n e x p [ - f i ' I ~ - u' [el,

where A ' = A ( - ~ ' ) , f i ' = f i ( - ~') and u ' = u ( - ~') can be chosen to be dependent on ~'. We satisfy reciprocity first; this requires

~n exp [-fiw [ ~'[2]A' exp [ - f i ' l $ - u'l z ] = I~" I exp [-flw 1~12)]Aexp[-fl l~ ' + u12],


where A = A(~), fl = fl(~) and u = u($). From this equation we conclude that the only resonable choice is u(~) = a~R($R = ~ -- 2n(v "v)) and fl(~) = b (a and b constants). Then

(4.27) flw [~' 12 + bl~ + a~'R 12 = flw I~12 + bl~' + a~R 12 and flw + ba2 = b or b = f l w ( 1 - a2) -1. Thus A ' = [~'1 and

(4.28) Ro(~'-~ ~) = c~, exp [ - f iw(1 - a2) -1 I~ - a ~ 12],

where c is a constant. The symmetric kernel K(~, $') associated with this choice is

(4.29) K ( ~ , ~ ' ) = c e x p [ - f l w ( 1 - a 2 ) - l l ~ + a ~ ' R ] 2 + f l w l ~ ] 2 ] .

If we now calculate H(~) as defined in eq. (4.21), we obtain a rather lengthy expression[19]. Unfortunately, this function does not turn out to satisfy the requirement to be not larger than unity [52], due to the behaviour at large speeds. A damping factor may be introduced [52] and thus one obtains a somewhat clumsier kernel, which satisfies all the requirements. Some results have been compared [52] with the experimental data by Legge [53].

All these models contain pure diffusion according to a non-drifting Maxwellian as a limiting case. The use of this model is justified for low-velocity flows over technical surfaces, but is inaccurate for flows with orbital velocity. In fact, the elaboration of the measurements of lift-to-drag ratio for the Shuttle Orbiter in the free molecular regime [54] implies a significant departure from diffuse, fully equilibrated re-emission of molecules at the wall. The experimental data by Legge [53] seem to exclude most of the known models, which indicates that more work is required in this area.

The traditional methods of testing a model for gas-surface interactions is to compare with molecular-beam experiments and drag and heat transfer measurements in free molecular flow. The first type of test is, in principle, the best one can think of, but requires a lot of systematic work, which does not appear to have been done. Also the information from molecular beam experiments can be overwhelming for certain applications, such as drag, lift and heat transfer calculations, and one must think carefully about how the amount of material arising from experiments should be represented and used. Good recent surveys of experimental work and its comparison with existing theories are due to Hurlbut [55, 56].

An early example of a partial success of trajectory calculations is also due to Hurlbut [33]. Recently, there have been several calculations of this kind that shed light on gas-surface interaction. In particular, Hulpke and Mann [57] and Tenner et al. [58-60] conducted studies on the scattering of potassium ions from clean tungsten surfaces and analysis of ion trajectories and found reasonable agreement with experiment. Among the important result arising from these studies, one can quote the following ones: a) trajectory analysis can yield simulations representing details of the physical scattering process very accurately; b) three-dimensional analyses are required, because two-dimensional analyses miss the lower-energy tail in the energy distribution of scattered particles; c) the exact value of the parameters in the interaction potential can be fixed only after a comparison with experiments is made.

Another example of trajectory modelling accompanying experimental data is provided by the paper of Kolodney et al. [61], who considered mercury scattering from magnesium oxide.

18 c. CERCIGNANI

With a few exceptions, scattering studies have used single-crystal surfaces as targets. Hurlbut [55] made the important remark that, while the use of these well-defined surfaces is essential for progress in surface science, one should, in order to understand the interaction of the upper atmosphere with the surfaces of orbiting shuttles, study scattering events and the attendant physics and chemistry on surfaces with the appropriate composition.

An experimental validation of the CL model was attempted by Bellomo, Dankert, Legge and Monaco [62]. According to these authors, the CL model does not reproduce the experimental drag and the heat flow on a metal plate placed in a supersonic nearly free molecular flow. The interpretation of the experimental results is, however, based on a misinterpretation of the CL model; in fact, the above-mentioned authors try to fit the experimental data for the drag and heat transfer coefficients vs. the plate temperature by means of expressions obtained from the CL model under the additional tacit assumption that the parameters in the model are temperature independent. The only fLrm conclusion that one can draw from the experimental data is that the recovery temperature does not depend on the angle of attack 0 (at least in the range of the experiments, 45 ~ ~< 0


at least at room conditions, only a few changes in the equations occur, the most remarkable being the change of the ratio of specific heats y. Incidentally, we remark that the value of ~, caused some problems from the very beginning of kinetic theory. In fact, the value for a monatomic gas is easily found to be 5/3 (the first calculation goes back to Clausius). Now, at that time, no monatomic gas was known, except mercury vapour. Indeed the data for this gas existed thanks to the experiments of Kundt and Warburg, but they were old and were soon forgotten. Maxwell assumed the molecules to be solid bodies different from perfectly smooth spheres and found the value 4/3. Neither value agreed with the ratio for the most common gases, like nitrogen, oxygen, hydrogen, etc. and this caused a difficulty, till it was pointed out that a diatomic molecule, modelled as two mass points at a fixed distance, and hence having all the mass placed on one axis, had only five degrees of freedom, which led to the value y = 7/5, in agreement with the experimental data. Later Rayleigh and Ramsay discovered the first rare gases and the value 5/3 was found to apply to the ratio of their specific heats. More problems arise when vibrational degrees of freedom are considered, which certainly is the case at high temperatures.

We shall first deal with the case of mixtures of monatomic gases, then with the case of a single monatomic gas; the case of a mixture of polyatomic gases easily follows by combining these two cases. In the case of a mixture of monatomic gases, the differences between the various species will be in the values of the masses and in the law of interaction between molecules of different species; in the simplest case, when the molecules are pictured as hard spheres, the second difference is represented by unequal values of the molecular diameters. In the mathematical treatment, a first diffrence will be in the fact that we shall need n distribution functions j~ (i = = 1, 2, ..., n) if there are n species. The notation becomes complicated, but there is no new idea, except, of course, for the fact that we must derive a system of n coupled Boltzmann equations for the n distribution functions. The result is

(5.1) f f O'--"t O"--'X ----~ ~--'kZl= ( f ' f ; , - f ~ A , ) B ~ ( n ' ~ , I V l ) d r (in, R 3 ffJ+

where Bik is computed from the interaction law between the i-th and k-th species, while in the k-th term in the left-hand side, V = $ - $ . is the relative velocity of the particle of the i-th species (whose evolution we are following) with respect to a particle of the k-th species (against which the former is colliding). The arguments $' and ~. are computed, as before, from the laws of conservation of mass and energy in an impact with the following result:

(5.2) $ ' = ~ - 2/~ik n [ ( $ - ~,).n], ~ , = $ , + 2 /~ikn[($ - $ , ) - n ] , mi mk

where I~ik = mimkl(mi + ink) is the reduced mass [3]. The description of polyatomic gases may also be reduced to the case of a mixture of

gases, with the following modification. We remark that eq. (5.1) can be rewritten as

2 0 C. C E R C I G N A N I

follows [3]:

(5.3) 5J~ + ~. ~fi + X,.. ~3~ _ ~t ~x ~

f k = l R 3 R 3 R 3

(~' f ; , - j~ j~ , ) W~k(~, ~, I~', ~ , ) d ~ , d~' d ~ , ,

where now ~', ~,, ~, ~. are independent variables (i.e. they are not related by the conservation laws) and

(5.4) W, ik(~, ~, I ~ ' ~ ) = S ~ k ( n ' L 1~1)5(m~$, + m k ~ , -mi~'-mk~,)"

.,~(m~ 1~, I e + . ~ I~, I ~ - ~n~ I~' I ~ - m~ I ~ 1~),

w h e r e n = ( ~ - ~ ' ) / 1 ~ - ~ ' 1 a n d

(5.5) S~k (n- ~, I ~1 ) = B~k(n'~, IVI)

2n. V ( m~ + m k )3 m i m k 9

Conservation of momentum and energy is now taken care of by the delta-functions appearing in eq. (5.4)[3].

With a slight modification, eq. (5.3) can be extended to the case of a mixture in which a collision can transform the two colliding molecules of species j, 1 into two molecules of different species k, i (a very particular kind of chemical reaction). In this case the relations between the velocities before and after the encounter are different from the ones used so far, but we may still write a set of equations for the n species:

(5.6) + ~ - -~' + x , . - Dt ~x ~

= E ~ (~'~' k,l,j=~ - J~J2* - )~fk , )W~(~ , ~, [~', ~ , ) d ~ , d~' d~, R 3 R 3 R 3

where W[ j gives the probability density that a transition from velocities ~', ~, to velocities ~, ~, takes place in a collision which transforms two molecules of species l, j, respectively, into two molecules of species i, k, respectively. It is clear how the previous model is included into the new one when the species do not occur.

A possible picture of a molecule of a polyatomic gas, suggested by quantum mechanics, is as follows [70]. The molecule is a mechanical system, which differs from a mass point by having a sequence of internal states, which can be identified by a label, assuming integral values. In the simplest cases these states differ from each other because the molecule has, besides kinetic energy, an internal energy taking different values E~ in each of the different states. A collision between two molecules, besides changing the velocities, can also change the internal states of the molecules and, as a consequence, the internal energy enters in the energy balance. From the viewpoint of writing evolution equations for the statistical behaviour of the system, it is convenient to think of a single polyatomic gas as a mixture of different monatomic gases. Each of these gases is formed by the molecules corresponding to a given internal energy, and a collision changing the internal state of at least one molecule is


considered as a reactive collision of the kind considered above, w[J(~, ~. I~' , ~ ) giving the probability density of a collision transforming two molecules with internal states l, j, respectively, and velocities ~', ~ , respectively, into molecules with internal states i, k, respectively, and velocities $, ~ . , respectively.

This model is amply sufficient to discuss aerodynamic applications. We want to mention, however, that it requires non-degenerate levels of internal energy, if there are, e.g., strong magnetic fields which can act on the internal variables such as (typically) the spin of the molecules. In that case, if the molecule has spin s, the distribution function f becomes a square matrix of order 2s 1 and the kinetic equation reflects the fact that matrices in general do not commute and, as remarked by Waldmann [71,72] and Snider [73], the collision term contains not simply the cross-section but the scattering amplitude itself, which may not commute with f.

It is appropriate now to enquire why we started talking about quantum rather than classical, mechanics. The main reason is not related to practice, but rather to history. Classical models of polyatomic molecules are regarded with suspicion since 1887 when Lorentz found a mistake in the proof of the H theorem of Boltzmann [74] for general polyatomic molecules. The question arises from the fact that when one proves the H theorem for monatomic gas one usually does not explicitly underline (because it is irrelevant in that case) that the velocities $' and ~. are not the velocities into which a collision transforms the velocities ~ and $ . , but the velocities which are transformed by a collision in the latter ones; this is conceptually very important, but the lack of a detailed discussion does not lead to any inconvenience because the expressions for ~. and ~' are invariant with respect to a change of sign of the unit vector n, which permits an equivalence between velocity pairs that are carried into the pair ~ . , ~ and those which originate from the latter pair, as a consequence of a collision. The remarkable circumstance which we have just recalled is related to the particular symmetry of a collision described by a central force, which allows to associate to a collision [~. , ~]--) [~ . , ~ '] another collision, the so-called ,,inverse collisiom, [ ~ , ~ ' ] - -~[~ . , ~], which differs from the former just because of the transformation of the unit vector n into - n . When polyatomic molecules are dealt with, the states before and after a collision require more than just the velocities of the mass centres to be described (the angular velocity, e.g., if the molecule is pictured as a solid body). Let us symbolically denote by [A, B] the state of the pair of molecules. Then there is no guarantee that one can correlate with the collision [A, B] ---> [A', B ' ] an ,,inverse collisiom~ [A', B ' ] --, [A, B], differing from the previous one just because of the change of n into - n. Now in the original proof of the H theorem for polyatomic molecules proposed by Boltzmann [74], the assumption was implicitly made that there is always such collision. Lorentz remarked[75-77] that this is not true in general. Boltzmann recognized his blunder and proposed another proof based on the so-called ,,closed cycles of collisions~, [75-78]; the initial state [A, B] is reached not through a single collision but through a sequence of collisions. This proof, although called unobjectionable by Lorentz [75] and Boltzmann [77], never statisfied anybody [79, 72].

For a while the matter was forgotten till a quantum-mechanical proof showed that the required property followed from the unitarity of the S matrix [72]. A satisfactory proof of the inequality required to prove the H theorem for a purely classical, but completely general, model was given only about 15 years ago [80].

For aerodynamic applications all these aspects are not so relevant and, in fact, the main problem is to find a sufficiently handy model for practical calculations.

22 c. CERCIGNANI

Lordi and Mates [81] studied the ,~two centres of repulsion- model and found that a rather complicated numerical solution was required for a given set of impact parameters. The lack of closed-form expressions makes the model impractical for applications, where the numerical solution describing the impact should be repeated many millions of times. Curtiss and Muckenfuss[82] developed the collision mechanics of the so-called sphero-cylinder model, consisting of a smooth elastic cylinder with two hemispherical ends. Whether or not two molecules collide depends on more than one parameter; in addition, there are several impacts in a single-collision event. The loaded-sphere model had been already developed by Jeans [83] in 1904 and was subsequently developed by Dahler and Sather [84] and Sandler and Dahler [85]. Although it is spherical in geometry, the molecules rotate about the centre of mass, which does not coincide with the centre of the sphere, with the consequence that it has essentially the same disadvantages as the sphero-cylinder model.

The only exact model which is amenable to explicit calculations is the perfectly rough-sphere model, first suggested by Bryan [86] in 1894. The name is due to the fact that the relative velocity at the point of contact of the two molecules is reversed by the impact. This model has some obvious disadvantages. First, a glancing collision may result in a large deflection; second, all collisions can produce a large interchange of rotational and translational energy, with the consequence that the relaxation time for rotational energy is unrealistically short; third, the number of internal degrees of freedom is three, rather than two, which makes the model inappropriate for a description of the main components of air, which are diatomic gases. One can disregard the first disadvantage and put a remedy to the second by assuming that a fraction of collisions follow the smooth-sphere rather that rough-sphere dynamics; but there is obviously no escape from the third difficulty.

In practical calculations, one has learned, since long time, that one must compromise between the faithful adherence to a microscopic model and the computational time required to solve a concrete problem. This was true in the early days of rarefied-gas dynamics (and may still be true nowadays when one tries to f'md approximate closed-form solutions or spare computer time) even for monatomic gases, as we shall discuss in sect. 7 (in connection with the so-called BGK model) and 8 (in connection with Monte Carlo simulation). As a matter of fact, when trying to solve the Boltzmann equation one of the major shortcomings is the complicated structure of the collision term; if to this problem, present even in the simplest case, one adds the complication of the presence of internal degrees of freedom, any practical problem becomes intractable, unless one is ready to accept the above-mentioned compromise. Fortunately, when one is not interested in fine details, it is possible to obtain reasonable results by replacing the collision integral by a phenomenological collision model, e.g., a simpler expression which retains only the qualitative and average properties of the true collision term. As computers become more and more powerful, the amount of phenomenological simplification diminishes and the calculations may more closely mimic the microscopic models.

For polyatomic gases, the basic new fact with respect to the monatomic ones is that the total energy is redistributed between translational and internal degrees of freedom at each collision. Those collisions for which this redistribution is neglegible are called elastic, while the others are called inelastic. The simplest approach would be to calculate the effect of collisions as a linear combination of totally


elastic and completely inelastic collisions, the second contribution being described by a model analogous to the BGK model shortly described in sect. 7.

Let us now consider in more detail the case of a continuous internal-energy variable. In this case, it is convenient to take the unit vector n in the centre-of-mass system and use the internal energy E i and Ei, of the colliding molecules. As usual, the values before a collision will be denoted by a prime. Equation (1.2) is replaced by

E

(5.7) Q(f, f ) : f dg, f E[:-2)/2dSi, f f(E,')(~ dSi'. o sr

E - Ei'

I (Ei ' , ) ( ' - e)/2 d E i ' , ( f ' f ' , - f f , ) B ( E ; n . n ' ; El ' , El',---~ El, E i , ) . 0

Here E = ml g 12/4 + Ei + Ei, is the total energy in the center-of-mass system which is conserved in a collision. The kernel B satisfies the reciprocity relation

(5~) [ V [ B ( E ; n ' n ' ; E ~ ' , E i ' , - ~ E i , E~ , )= [ V ' [ B ( E ; n . n ' ; E ~ , E i , - ~ E i ' , E i ' , ) .

Here we follow a paper by Kug~er [87] and look for a one-parameter family of models, assuming that the scattering is isotropic in the centre-of-mass system. The second crucial assumption will be that the redistribution of energy among the various degrees of freedom only depends upon the ratios of the various energies to the total energy E, e i = E i / E , etc. This assumption is valid for collisions of rigid elastic bodies and can be considered as a good approximation for steep repulsive potentials. It is then possible to write B in the following form:

(5.9) ]V[B(E; n . n ' ; Ei' , Ei',----> Ei, Ei, -- IV' [2 gtot(E ) [V[ 4 z E n

- - - - O(E[, E ; , ---> s ~i* ] T)"

The denominator on the right takes care of normalization. Then the function 0 satisfies the following relations:

1 l - e

(5.10) I e ( ' - 2>/2 de I e:-e>/2 de . 0(el, e ; , ~ el, el. ; v ) = 1, 0 0

(5 .11) ( 1 - 8i ~ - E l , ) 0 ( c i ' , E l , - - ) Ei , E i , ; T) -~ (1 - E i -- E i , ) 0 ( E i , E i , - - > E l , E l , ; "g).

The dependence of ato t on E makes it possible to adjust the model to the correct dependence of the viscosity on temperature. The parameter v will be chosen in such a way as to represent the degree of inelasticity of the collisions, r = 0 will correspond to elastic collisions:

(5.12) 0(e[, E[,-- '->Ei, E i , , O) e - ( n - 2 ) / 2 E . (n-2) /2 6 ( E i 8 [ ) 6 ( e ; -- e i , ) ;

r = oo will correspond to maximally inelastic collisions:

(5.13) 0(el, e' ---~ ei, ei , ; oo) = e - ( , - 2)/2 e , ( ' - 2)/2 F(n + 2) ( r ( n ) ) 2

( 1 - e i - s

24 C. CERCIGNANI

A mixture of the two extreme cases will give the model first proposed by Borgnakke and Larsen [88] in 1975:

P ~ (5.14) 0(ei', e j , ---) e i , e~,, r) =

= exp[ -v ]0 (e ( , e~.--~ ei, el. ; 0) + (1 - exp[ - r ] )0 (e ( , e~. ~ c i , el.; ~ ) .

Kug~er [87] notices an analogy between this model and Maxwell model for gas-surface interaction, as discussed in ref. [3], and introduces another model, called the theta- model, which would correspond to the CL model in this analogy.

The Larsen-Borgnakke model has become a customary tool in numerical simulations of polyatomic gases. It can be also applied to the vibrational modes through either a classical procedure that assigns a continuously distributed vibrational energy to each molecule, or through a quantum approach that assigns a discrete vibrational level to each molecule. It would be out of place here to discuss this point in more detail, for which we refer to the book of Bird [89]. We also refrain from discussing the interesting recent developments [90,91] of an old idea of Boltzmann [92] to interpret, in the frame of classical statistical mechanics, the circumstance that at low temperatures the internal degrees of freedom appear to be frozen, as due to the extremely long relaxation times for the energy transfer process.

6. - C h e m i c a l r e a c t i o n s a n d t h e r m a l r a d i a t i o n .

Chemical reactions are important in high-altitude flights because of the high temperatures which develop near a vehicle flying at hypersonic speed. Up to 2000 K, the composition of air can be considered the same as at standard conditions. Beyond this temperature, Ne and 02 begin to react and form NO. At 2500 K dyatomic oxygen begins to dissociate and form atomic oxygen O, till 02 completely disappears at about 4000 K. Nitrogen begins to dissociate at a slightly higher temperature (about 4250 K). Thus NO disappears at about 5000 K. Ionization phenomena start at about 8500 K.

As we have implicitly remarked when we have written eq. (5.6), the kinetic theory of gases is an ideal tool to deal with chemical reactions of a particular kind, i . e . bimolecular reactions, which can be written schematically as

(6.1) A + B ,-* C + D,

where A, B, C and D represent different molecular species. We already used the term -molecule-, as usual in kinetic theory, to mean also atom (a monotomic molecule); in this section we shall further enlarge the meaning of this term to include ions, electrons and photons as well, when we have to deal with ionization reactions and interaction with radiation.

As long as the reaction takes place in a single step with the presence of no other species than the reactants, it is a well-known circumstance that the change of concentration of a given species (A, say) in a space-homogeneous mixture can be written as follows:

(6.2) ~dn---~ = k b (T)n c n D - - kf(T)nAnB 9 dt


We remark that, in chemistry, one uses the molar density in place of the number density used here, the two being obviously related through Avogadro's number.

The rate coefficients kf and kb for the forward and backward (or reverse) reactions, respectively, are functions of temperature and are usually written by a semi-empirical argument, which generalizes the Arrhenius formula, in the form

(6.3) k (T ) = A T " exp - - ~ ,

where A and y (= 0 in the Arrhenius equation) are constants, and Ea is the so-called act iva t ion energy of the reaction. It is clear that these equations, though having a flavour of kinetic theory, are essentially macroscopic and can be assumed to hold when the distribution function is essentially Maxwellian.

In fact, the above reaction theory can be obtained by assuming that the distribution functions are Maxwellian, the role of internal degrees of freedom may be ignored and the react ion cross-section vanishes if the translational energy Et in the centre-of-mass system is less than E a and equals a constant aR if the energy is larger than E a . A more accurate theory is obtained [93, 94, 89] by assuming that the ratio of the reaction cross-section to the total cross-section is zero when the to ta l collision energy Ee (equal to the sum of Et and the total internal energy of the two colliding molecules Ei) is less than Ea and proportional to the product of a power of Ee - E a and a power of Ec. The exponents and the proportionality factor are essentialy dictated by the number of internal degrees of freedom, the exponent of the temperature in the diffusion coefficient of species A in species B and the empirical exponent ~] appearing in eq. (6.3). This theory provides a microscopic reaction model that can reproduce the conventional rate equations (6.2), (6.3) in the continuum limit. The model is, however, as in the case of gas-surface interaction and models for polyatomic gases, largely based on phenomenological considerations and mathematical tractability. The ideal microscopic model would consist of complete tabulations of the differential cross- sections as functions of the energy states and n. Some microscopic data, coming from extensive quantum-mechanical computations, supported by experiments, are available, but, unfortunately, not very much is known for reactions of engineering interest. When comparisons can be made, the reaction cross-sections provided by the phenomenological models is of the correct order of magnitude. This provides some reasons of optimism about the validity of the results obtained with these models for the highly non-equilibrium rarefied-gas flows.

Termolecular reactions provide some difficulty to kinetic theory, because the Boltzmann equation essentially describes the effect of binary collisions. They are, however, of essential importance in high-temperature air, where the reverse (or backward) reaction of a dissociation one is a recombination reaction, which is necessarily termolecular, as we shall presently explain. A typical dissociation- recombination reaction can be represented as

(6.4) AB + T ~ A + B + T ,

where AB, A, B and T represent the dissociating molecule, the two molecules produced by the dissociation and a third molecule (of any species), respectively. The latter molecule, in the forward reaction, collides with AB and causes its dissociation. This process is described by a binary collision and is an endothermic reaction, requiring a certain amount of energy, the dissociation energy Eo.

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The recombination process is an exothermic reaction and it might seem that one could dispense with the ,,third body, T and consider it as a bimolecular reaction

(6.5) AB ~-- A + B.

However, one can easily see that the momentum and energy balance for a binary collision cannot be simultaneously satisfied in the presence of energy release. The molecule T is thus required to describe the recombination process.

In order to keep the binary-collision analysis, appropriate for a rarefied gas, we must think of the recombination process as a sequence of two binary collisions. The first of these forms an (unstable) orbiting pair, that is stabilized by the second collision of this pair with T, as long as this collision occurs within a sufficiently small elapsed time. One can then extend the previous theory, assuming that the activation energy is zero and the cross-section acquires a factor proportional to the number density of the species T.

Ionization reactions involve the electronic states and it is unlikely that a purely classical theory will be successful in describing them, because of the selection rules. Yet, one can use the phenomenological approach to provide at least an upper bound for the reaction rates.

As mentioned above, one can, in principle, think of interaction with a radiation as if it were a reaction involving photons as ,,molecules-. Here spontaneous emission should also be taken into account. It becomes harder to develop phenomenological models, because one should consider as many species as three excited levels for each molecule.

7. - Solving the Boltzmann equation. Analytical techniques.

In the early sections of this paper we defined (in a qualitative fashion, to be sure) how to attack the problems of rarefied-gas dynamics and what phenomena should be looked for in their solutions. But time has come to say something about the way these problems are solved: how does one handle the already complicated Boltzmann equation with similarly complicated boundary conditions?

The history of approximate solutions goes back, after the first attempts of Maxwell and Boltzmann, to Hilbert [95], Chapman [96] and Enskog [97]. As is well known, they obtained solutions valid in the continuum limit, very useful to compute the transport coefficients; there are standard monographs [98,99], which deal with this part of the theory.

In 1949 Grad [100] devised a systematic method to deal with the solutions of the Boltzmann equation, i.e. his famous 13-moment method. Although there is some rat ionale in his approach and his equations give better results than the Navier-Stokes equations for certain problems, it appears fair to say that they turned out to be rather useless for the progress of rarefied-gas dynamics. The most notable failure is related to the problem of shock structure, for which the 13-moment equations fail to give a solution for a Mach number larger than 1.65.

An important early approximate solution of the shock wave problem was Mott-Smith's solution [101], remarkable for its simplicity and resistance to any simple improvement; we shall discuss it later.

All these methods paid little, if any, attention to the problem of boundary


conditions, which, as we have already remarked, is vital in any application to upper-atmosphere flight.

The lesson taught by Mott-Smith's solution is that it is not so useful to look for general methods with the aim of obtaining continuum-like equations, but one should rather devise approximate methods for dealing with particular problems. This is the rationale behind the use of models and the linearization techniques [2,3].

The linearized Boltzmann equation is obtained by assuming that the solution is a small perturbation of a basic Maxwellian M0, and is useful for low Mach number flows and, as such, is used in the applications of rarefied-gas dynamics to environmental problems. It is also useful in order to deal with the kinetic layers and, as such, retains its validity to compute the boundary conditions in the slip regime and hence can still play an important role in the computations of re-entry problems. An example of this kind of work is the evaluation of the slip coefficient and the temperature jump for arbitrary models of gas-surface interaction, based on a variational principle for the integro-differential form of the linearized Boltzmann equation [102,2,3].

In the area of environmental problems, understanding and control of the formation, motion, reactions and evolution of particles of varying composition and shapes, ranging from a diameter of the order of 0.001 ~m to 50 ~tm, as well as their space-time distribution under gradients of concentration, pressure, temperature and the action of radiation have grown in importance, because of the increasing awareness of the local and global problems related to the emission of particles from electric- power plants, chemical plants, vehicles as well as of the role played by small particles in the formation of fog and clouds, in the release of radioactivity from nuclear-reactor accidents, and in the problems arising from the exhaust streams of aerosol reactors, such as those used to produce optical fibers, catalysts, ceramics, silicon and carbon whiskers.

When the mean free path is appreciable but still small with respect to size of the particles, the study of the Knudsen layers to obtain the correct slip boundary conditions is useful. When the Knudsen number becomes larger, the transfer processes between an aerosol particle can be studied with the linearized Boltzmann equation and hence variational techniques of the type hinted at above prove useful. In fact already twenty years ago variational results [50] for the drag on a spherical particle were obtained for the entire range of values of the Knudsen number based on the radius of the particle. Similarly, if one considers the diffusion of a trace species (water vapour in air) that may condense on a particle, the variational method can be successfully applied, as indicated by Loyalka [103]. The heat transfer from a particle was similarly studied by Cercignani and Pagani [104].

These problems have also been solved with numerical methods that produce results in a very good agreement with the variational calculations. Other problems, such as those related to the forces experienced by a particle in a thermal or concentration gradients (thermophoresis and diffusiophoresis) have been treated by numerical methods only [105].

Attempts to use the linearized Boltzmann equation in rarefied-gas dynamics began in the late 1950s and early 1960s. One of the first fields to be explored was that of the ,,simple flows,,, such as Couette and Poiseuille flows in tubes and between plates; here it turned out that the equation to be solved is still formidable and various approximation methods were proposed. Some of these were perturbation methods: for large or small Knudsen numbers or about an equilibrium solution (Maxwellian). The first two approaches gave useful results in the limiting regimes, while the third

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method led to studying the so-called linearized Boltzmann equation, which produced predictions which are in a spectacular agreement with experiment and have shed considerable light on the basic structure of transition flows, whenever non-linear effects can be neglected [2, 3]. This gave confidence to further use of the Boltzmann equation for practical flows. Other problems which were treated with the linearized equation were the half-space problems which are basic in order to understand the structure of Knudsen layers and to evaluate the slip and temperature jump coefficients.

This explains the importance of the linearized equation for the hypersonic flows met in the aerodynamics of space vehicles; in fact there is a large portion of upper- atmosphere aerodynamics (vital to the dynamics of a re-entering body) for which the Navier-Stokes equations may still be considered to be valid (0.01 < 1 /L < 0.1), except in a thin layer near the body, having a thickness of the order of a mean free path (Knudsen layer). This layer can be described by means of linearized equations and this circumstance has led to a complete understanding of the phenomena occurring in this situation. Actual calculations can be performed in an approximate analytical form by means of variational techniques and also by numerical methods. This has led, in particular, to confirming the existence of a minimum in the flow rate in the Poiseuille flow of a rarefied gas as a function of the Knudsen number, discovered by Knudsen about eighty years ago [2, 3], and to computing the drag upon a sphere at low subsonic speeds with results in a very good agreement with the experimental data that Millikan obtained as a pre-requisite to his celebrated oil-drop experiment to measure the electron charge [106, 2, 3].

A particularly simple and useful technique to obtain appproximated but accurate results from the linearized Boltzmann equation is provided by the variational technique [102].

As is well known, a variational principle, which does not reduce to a triviality, i.e. to something analogous to a least-square method, is based on a property of symmetry of the operator appearing in the equation. For linear equations this property reduces to the symmetry of a linear operator with respect to a suitable scalar product. In the steady case, the linearized Boltzmann equation reads as follows:

~h (7.1) ~. - - - L h = g o ,

~x

where h is the perturbation of the basic Maxwellian M0 in the distribution function f = M0 (1 +h), go a source (which might arise from some inhomogeneity in M o ) and L the linearized collision operator, related to the quadratic operator defined in eq. (1.2) by

(7.2) 2 M o L h = Q(M0(1 + h), M0(1 + h)) - Q(M0(1 - h), Mo(1 - h)).

For simplicity, we have omitted the body force term in eq. (7.1). This equation has to be accompanied by the linearized version of the boundary conditions discussed in sect. 4.

Now, it is easy to see [102, 3] that L is self-adjoint with respect to a scalar product weighted with Mo, while the differential operator in eq. (7.1) (which we shall henceforth abbreviate to D) is (with suitable boundary conditions) antisymmetric with respect to the same scalar product. Thus the variational formulation of eq. (7.1) is not immediate. There is, however, a property [102,3] that opens the way to such


formulation. Let P denote the operator which changes $ into - ~ (i.e. the parity operator in velocity space). Then P commutes with L (at least for monatomic gases and polyatomic gases with molecular interaction possessing spherical symmetry) and P D is self-adjoint with respect to the above-mentined scalar product and we can obtain a variational principle,/~e, we can fred a functional J(f~) whose first variation vanishes if and only if f~ coincides with the solution h of eq. (7.1) satisfying the appropriate boundary conditions. It is sufficient to replace the equation D h - L H = = go by the equivalent equation P D h - P L H = Pgo. More work is, of course, needed [102,3] to write down the principle in an appropriate way, when we want to vary the functions on the boundary as well.

To have a variational principle at our disposal would be a matter of idle curiosity, unless we knew that there is an important property associated with it. This is provided by the circumstance that if we take a subclass of functions {f~} and try to fred the one that best approximates the solution in the sense of the variational principle, the above-mentioned functional J, evaluated at the approximate stationarity point, differs from the value at the exact statioparity point by ~ if ~ is the order of magnitude of the difference between the optimal h in the restricted class and the solution of the problem. This paves the way to very accurate estimations of the functional. The last step that remains to be done is to relate the value of J to a physically important quantity, which will be thus computed with analogous accuracy. This has been done for many problems including all those mentioned above.

A particularly interesting problem, which cannot be treated with the linearized Boltzmann equation, has been already mentioned and is related to the structure of a shock wave. The latter is not a discontinuity surface as in the theory of compressible Euler equations, but a thin layer (having, usually, a thickness of the order of a few mean free paths). In the case of a normal shock wave, one can imagine that it occurs in the entire space without boundaries; finding the shock wave structure means solving the Boltzmann equation when the solution (which depends on one space coordinate, say x, and the three velocity components $~ (i = 1, 2, 3), but not on time and the other two coordinates) tends to two different Maxwellians when x tends to + ~ and - ~ . The two Maxwellians have the following shape:

(7.3) fo 2 = 0 ( 2 z r R T )-8/2 e x p [ - ] ~ - u 2 i[2 / ( 2 R T )],

where i is the unit vector of the x-axis and the superscripts _+ refer to down- and up-stream states, respectively.

The shock wave problem for high values of the upstream Mach number provides an example of a flow where very large local Knudsen numbers can occur at any density, due to the sharp changes of the physical quantities in a very thin layer.

An early approach that was moderately successful in dealing with the shock wave problem was the Mott-Smith method [101]. This method postulates that there is a bimodal distribution, i.e. a linear combination of the two Maxwellians defined in eq. (7.3):

(7.4) f = vfi + + (1 - v)fi- .

Here v = v ( x ) is a function that goes from 0 to 1 through the shock. Equation (7.4) is easily shown to be compatible with the balance of mass, momentum and energy, provided the constant values Q 2, u 2, T 2 satisfy a set of compatibility conditions, which are nothing else than the Rankine-Hugoniot relations, familiar from the ideal

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fluid theory of shock waves. In order to determine v(x), several procedures have been presented, none of which is very satisfactory, since they are essentially arbitrary. Grad suggested [107, 3] that the problem of shock wave structure had a limit for the upstream Mach number tending to infinity in the form of a delta-function plus a relatively tame function. This matter has been revived by several authors.

The results obtained by this method for several physical quantities, including the thickness of the shock, are considerably more accurate than the Navier-Stokes values for other-than-low Mach numbers. In addition to the unsatisfactory status from a mathematical point of view, the Mott-Smith approach suffers, however, the further drawback of being restricted to the shock structure problem. One should also remark that, while the values for the macroscopic quantities (density, bulk velocity, pressure) are in a reasonable agreement with the most accurate numerical solutions of the Boltzmann equation, this is not the case for the distribution function itself (see sect. 8).

When trying to solve the Boltzmann equation for practical problems, one of the major shortcomings is the complicated structure of the collision term, eq. (2). When one is not interested in fine details, it is possible to obtain reasonable results by replacing the collision integral by a so-called collision model, a simpler expression J(f) which retains only the qualitative and average properties of the collision term Q(f, f). The equation for the distribution function is then called a kinetic model or a model equation.

The most well-known analytical (or semi-analytical) solutions in kinetic theory are obtained through the simplest collision model, usually called the Bhatnagar, Gross and Krook (BGK) model, although Welander proposed it independently at about the same time as the above-mentioned authors [108, 109]. It reads as follows:

(7.5) J(f) = u[q)($) - f ( ~ ) ] ,

where the collision frequency v is independent of g (but depends on the density Q and the temperature T) and q~ denotes the local Maxwellian, i.e. the (unique) Maxwellian having the same density, bulk velocity and temperature as f :

(7.6) = O(2zRT) -3/2 exp [ - ] ~ - v]2/(2RT)].

Here 0, v, T are chosen is such a way that we have

(16) f ~fa (~) r d~ = I ~f. (~) f(~) d ~ (a = O, 1, 2, 3, 4),

where F0 = 1, Fi = ~i (i = 1, 2, 3) and ~4 = [g[2. It should be remarked that the non-linearity of the BGK collision model, eq. (14), is

much worse than the non-linearity in Q(fi f ) ; in fact the latter is simply quadratic in f, while the former contains f in both the numerator and denominator of an exponential, because v and T are functionals of f, defined by eq. (2.1).

The main advantage in the use of the BGK model is that for any given problem one can deduce integral equations for Q, ~, T, which can be solved with moderate effort on a computer. Another advantage of the BGK model is offered by its linearized form.


The BGK model has the same basic properties as the Boltzmann collision integral, but has some shortcomings. Some of them can be avoided by suitable modifications, at the expense, however, of the simplicity of the model [2, 3].

8. - Solving the Boltzmann equation. Numerical techniques.

The kinetic models have been very useful in obtaining approximate solutions and forming qualitative ideas on the solutions of practical problems, but, in general, do not provide us with the detailed and precise answers to the sort of question that is posed by the space engineer. Various numerical procedures exist which either attempt to solve for f by conventional techniques of numerical analysis or efficiently bypass the formalism of the integro-differential equation and simulate the physical situation that the equation describes (Monte Carlo methods). Only recently proofs have been given that these partly deterministic, partly stochastic games provide solutions that converge (in a suitable sense) to solutions of the Boltzmann equation. Numerical solutions of the Boltzmann equation based on finite-difference methods meet with severe computational requirements due to the large number of independent variables. In practice the only method that has been used for inhomogeneous problems in more than one space dimension is the technique of Hicks-Yen-Nordsiek [110,111], which is based on a Monte Carlo quadrature method to evaluate the collision integral. This method was further developed by Aristov and Tcheremissine[ll2,113] and has been recently applied to a few two-dimensional flows [114,115].

An additional difficulty for these methods is the fact that chemically reacting and thermally radiating flows (and even simpler flows of polyatomic gases) are hard to describe with theoretical models having the same degree of accurateness as the Boltzmann equation for monatomic non-reacting and non-radiating gases. These considerations paved the way to the development of simulation methods, which started with the work of Bird on the so-called direct-simulation Monte Carlo (DSMC) method [116] and have become a powerful tool for practical calculations. There appear to be very few limitations to the complexity of the flow fields that this approach can deal with. Chemically reacting and ionized flows can be and have been analysed by these methods.

In the DSMC method, the intermolecular collisions are considered on a probabilistic rather than a deterministic basis. Furthermore, the real gas is modelled by some thousands of simulated molecules on a computer. For each of them the space coordinates and velocity components (as well as the variables describing the internal state, if we deal with a polyatomic molecule) are stored in the memory and are modified with time as the molecules are simultaneously followed through representative collisions a

Freepaper.me 10.1007 sdf Aerodynamical Applications of the Boltzmann Equation

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