From Boltzmann Kinetics to the Navier-Stokes Equations without a Chapman-Enskog Expansion

Transcription

1 From Boltzmann Kinetics to the Navier-Stokes Equations without a Chapman-Enskog Expansion Benoit Cushman-Roisin and Brenden P. Epps Thayer School of Engineering; Dartmouth College; Hanover, New Hampshire 3755, USA (Dated: March 4, 28 Abstract It is well known that the Navier-Stokes uations can be derived from the Boltzmann Equation, which governs the kinetic theory of gases, upon (i assuming the Bhatnagar-Gross-Krook collision formulation (a simple relaxation toward an uilibrium distribution, (ii assuming the Maxwell-Boltzmann form of this uilibrium distribution, and (iii performing the so-called Chapman-Enskog perturbation expansion under the assumption of a short relaxation time. Herein, we demonstrate that there is an alternate path from Boltzmann to Navier-Stokes (in lieu of the Chapman-Enskog expansion and that the particular form of the uilibrium distribution is inconsuential, as long as it meets some basic properties such as isotropy in velocity space and integrability of several moments. The essential ingredients are the relaxation formulation of the collision term and the assumption of a short relaxation time. This analysis provides new insights into the connections between kinetic theory and continuum mechanics, and it suggests new possibilities for fluid flow modeling using kinetic theory. Benoit.Cushman.Roisin@dartmouth.edu Brenden.Epps@dartmouth.edu

2 6 I. INTRODUCTION Boltzmann kinetic theory describes the microscopic evolution of a large number of colliding particles, such as in a monoatomic ideal gas. This theory has become accepted as a valid rep- resentation of fluid mechanics since it was shown that the microscopic description of Boltzmann kinetics produces the Navier-Stokes uations at the macroscopic level []. This strong connection between Boltzmann and Navier-Stokes has subsuently ustified numerical simulation of contin- uum fluid flows based on the Boltzmann Equation (which governs kinetic theory; in particular, the lattice-boltzmann method (LBM has enoyed great success due to the ease of coding, ease of accommodating complex moving boundaries, and computational efficiency [2 4]. Today, the LBM is used across a very wide array of applications from improving vehicle aerodynamics [5, 6] to flow through porous media [7]. Historically, the link between microscopic Boltzmann kinetics and macroscopic fluid mechanics was made through the so-called Chapman-Enskog perturbation expansion [, 8 ]. The expansion is based on the assumption that the mean free time τ between successive collisions is very short compared to the time scale of evolution of the fluid flow L/V, where L and V are characteristic length and velocity scales of the macroscopic flow. In other words, the fluid flow is assumed to evolve only slightly over a great many particle collisions. This short-time assumption is consistent with assuming a very small Knudsen number ɛ λ/l = Uτ/L, where λ Uτ is the mean free path, U is the thermal speed, and it is assumed that V O(U. Based on the smallness of the Knudsen number, the particle mass probability distribution function f (i.e. the probability of finding mass moving at a certain velocity at a certain location; SI units: kg/[(m/s 3 m 3 ] can be formally expanded: f = f ( + ɛ f ( + ɛ 2 f ( ( with time likewise split into multiple time variables, each one evolving more slowly than the previous one: t = t + ɛ t + ɛ 2 t (2 Upon inserting ( and (2 into the Boltzmann Equation, the Navier-Stokes uations can be derived. The procedure is systematic and straightforward, although the algebra is tedious. Although the Chapman-Enskog procedure is beyond debate, it is worth exploring other con- nections between the Boltzmann and Navier-Stokes uations, because these connections could provide a fresh perspective on turbulence modeling [ 4]. In this paper we show that there is an alternate path from the Boltzmann Equation to the Navier-Stokes uations that does not involve the Chapman-Enskog expansion. Instead, a formal solution for the distribution f is first established, and then using this solution the constitutive relations for the stress and energy flux are obtained. The assumption of a short relaxation time (uivalent to a small Knudsen number simplifies the expressions by means of Taylor series expansions, and the Navier-Stokes uations emerge. This theoretically-minded paper provides a crucial step towards establishing new perspec- tives on turbulence modeling via Boltzmann kinetics, although such turbulence modeling efforts 2

3 42 are beyond the scope of this contribution. What follows instead is a detailed derivation of the continuum mass, momentum, and energy conservation uations from the Boltzmann Equation in a novel way that does not involve the Chapman-Enskog expansion. 45 II. FROM BOLTZMANN KINETICS TO CONTINUUM MECHANICS 46 A. Boltzmann Kinetics Our premise is Boltzmann kinetic theory, which governs the evolution of a mass probability distribution f (t,x,u. The quantity f (t,x,u dx du represents the mass of fluid particles that, at time t, are located within volume dx = dx dx 2 dx 3 surrounding position x = [x, x 2, x 3 ] and endowed with velocity within the range du = du du 2 du 3 surrounding u = [u, u 2, u 3 ]. Like the particle positions x, the particle velocities u are independent variables, and the ensemble-averaged hydrodynamic variables are derived from f(t, x, u via integrals over all possible velocities [3, 5]: density: ρ(t,x f (t,x,u du (3 velocity: ū i (t,x ρ ui f (t,x,u du (4 internal energy: 3 2 U 2 (t,x ρ 2 u ū(t,x 2 f (t,x,u du (5 total energy: ě(t,x ρ 2 u 2 f (t,x,u du = 2 ū(t,x U 2 (t,x (6 stress: σ i (t,x (u i ū i (t,x(u ū (t,x f (t,x,u du (7 energy flux: q i (t,x 2 u ū(t,x 2 (u i ū i (t,x f (t,x,u du, (8 in which (... du is shorthand for the definite integral (... du du 2 du 3. For convenience, we will hereafter use index notation for vectors (and tensors and adopt the convention of implicit summation over repeated indices. The velocity U of the internal energy (5 can be understood as the thermal agitation velocity; that is, the average velocity of Brownian motions of particles between collisions. The right hand side of uation (6 follows from (4 and (5 and the fact that u ū 2 = u 2 2u ū + ū 2. The evolution of the mass probability distribution is governed by the Boltzmann Equation [3] (written here with gravity as the sole external force: t + u i g = C(f, (9 x i u 3 in which g is the downward gravitational acceleration (with x 3 directed upward. This evolution uation states that mass distribution function f is advected in physical space by the particle velocities, altered in velocity space by the gravitational acceleration, and redistributed by collisions. The collision term C(f may be expressed in a variety of ways depending on assumptions made about individual collisions [3, 6], but for the purpose of the recovery of the Navier-Stokes Equa- tions, it turns out that a fairly simple form is sufficient: It is assumed that the accumulated 2 The factor 3 2 in front of U 2 is reflective of the three dimensions of the physical space. This factor becomes 2 2 in two dimensions and 2 in one dimension. 3

4 6 effect of collisions is merely to relax the distribution f toward an uilibrium distribution f via C(f = τ (f f. This is the so-called BGK formulation [7], with a relaxation time τ that can be interpreted as the averaged time between successive collisions. This formulation leads to rewriting Equation (9 as the Boltzmann-BGK uation: t + u i The form of the uilibrium distribution f is discussed next. x i = τ (f f + g u 3. ( 66 B. Form of the Equilibrium Distribution Since Ludwig Boltzmann [8] considered gas particles animated by Brownian motions, it has been traditional to adopt for f the Maxwell-Boltzmann distribution: f ρ(t,x u ū(t,x 2 (t,x,u = (2π 3/2 U 3 exp ( (t,x 2U 2, ( (t,x which is a Gaussian distribution of velocities u i centered on their averages ū i and with standard deviation U. Alternative distributions are worth considering, especially Lévy α-stable distributions [4]. So, we will proceed using a generic distribution of the form f (t,x,u = ρ(t,x U 3 (t,x F ( (t,x,u with (t,x,u = u ū(t,x 2 U 2 (t,x. (2 To ensure that the collision term (f f/τ on the right hand side of ( conserves mass, momentum, and energy, f is ruired to obey uations (3, (4, and (5 [3]. Clearly, this is the case for the Maxwell-Boltzmann distribution (. For the generic distribution (2, condition (4 is automatically satisfied by symmetry, whereas satisfaction of (3 and (5 imposes respectively: I I 3 2 F ( d = 2π, (3 3 2 F ( d = 3 2π. (4 Equations (3 and (4 are obtained by inserting (2 into (3 and (5, passing to spherical coordinates, and evaluating the angular integrals. Equations (3 and (4 impose two constraints on the moments of F (. Not only must they be finite, but but they must take on these specific values in order to conserve mass and energy during collisions. If F ( corresponds to Maxwell- Boltzmann, these constraints are satisfied naturally. If F ( has a heavy tail (as in a Lévy α-stable distribution with α < 2, then a truncation and renormalization are ruired. 83 C. Continuum Governing Equations as Moments The governing uations of continuum mechanics can be derived by taking moments of the Boltzmann-BGK Equation (. In particular, the zeroth, first and second moments of ( result 4

5 86 in the continuum uations for conservation of mass, momentum, and energy. This development is well known, but it is reviewed here both for completeness and to contextualize the novel solution presented in III. 89. Mass Mass conservation is obtained by the zeroth moment of Equation (, which is simply its integration over the velocity space: { t + u i x i } { du = τ (f f + g } du. (5 u 3 The temporal and spatial derivatives commute with the velocity integrations because they are independent variables. The gravity term vanishes because f as u 3 ±. The f f term vanishes because collisions conserve mass (i.e. definitions (3 and (4, Equation (5 reduces to the continuity uation: both distributions satisfy (3. Then, with ρ t + (ρū i =. (6 x i Momentum 97 The momentum uations are obtained by the first-order moments of (: u { t + u i x i } du = { u τ (f f + g } du. (7 u Upon expansion of u i u = (u i ū i (u ū + u i ū + u ū i ū i ū, the advective term in (7 evaluates to x i ui u f du = x (ui i ū i (u ū f du }{{} σ i via (7 + x (ui i ū + u ū i ū i ū f du (8 }{{} =ρū i ū The gravity term is evaluated by making use of u 3 u 3 = (u 3f u 3 f and the fact that u 3 f as u 3 ±. The f f term vanishes because collisions conserve momentum (i.e. both distributions satisfy (4. Thus, (7 with (8 simplifies to the Cauchy momentum uation: t (ρū + (ρū i ū = σ i ρgδ 3. (9 x i x i At this point, we note that if the stress (7 can be reduced to the Newtonian constitutive uation, then continuity uation (6 and momentum uation (9 would together become the Navier-Stokes uations. 5

6 6 3. Energy 7 The total energy uation is obtained as (half of the second moment of ( { } { 2 u iu i t + u du = x 2 u iu i τ (f f + g } du. (2 u The unsteady term readily evaluates to t ( 3 2 ρu ρū iū i by definition (6. The collision term integrates to zero because collisions conserve energy (i.e. both distributions satisfy (6. The gravitational term integrates by parts to ρgū 3, assuming u 2 3 f as u 3 ±. The advective term can be partitioned via u i = (u i ū i + ū i to obtain: 2 u iu i u f du = 2 (u i ū i (u i ū i (u ū f du }{{} =q via (8 + ū 2 (u i ū i (u i ū i f du }{{} = 3 2 ρu 2 via (5 = q ū i σ i + ( 3 2 ρu ρū iū i ū. + ū i (ui ū i (u ū f du } {{ } = σ i via (7 + ( 2ūiū i ū f du }{{} =ρ via (3 (2 2 With this expression, the total energy budget (2 becomes: ( 3 t 2 ρu ρū [( iū i + 3 x 2 ρu ρū ] iū i ū = (ū i σ i ρgū 3 q. (22 Equation (22 can be simplified by making use of the earlier momentum uation (9 to eliminate the mean kinetic energy. Upon switching the i and indices in (9, multiplying by ū i, and then using (6 we obtain the kinetic energy evolution uation: t ( 2 ρū iū i + ( x 2 ρū σ i iū i ū = ū i ρgū 3. (23 Then subtracting (23 from (22 yields the uation for the evolution of only the internal energy: ( 3 t 2 ρu 2 + ( 3 x 2 ρu 2 ū = σi ū i q. (24 Equations (6, (9 and (24 provide a set of three uations for the three hydrodynamic variables ρ, ū i, and U 2. The key is to be able to express the stress tensor σ i and energy flux vector q in terms of only the same hydrodynamic variables in order to have a closed set of governing uations. 2 III. CONSTITUTIVE EQUATIONS FOR THE STRESS AND ENERGY FLUX With the development presented in the preceding section, the Chapman-Enskog expansion could then be used to derive the constitutive relations for the stress and energy flux. Herein, we present a novel route towards deriving these constitutive relations. The end result of the present derivation 6

7 25 is the same: Assuming the uilibrium distribution f is the Maxwell-Boltzmann distribution, the derivation leads to the constitutive relations consistent with a compressible Newtonian fluid that behaves as an ideal gas. The present derivation, however, lays a framework for considering other uilibrium distributions (such as the Lévy α-stable distributions [4], although such considerations are beyond the current scope. 3 A. General Solution of the Boltzmann-BGK Equation Our derivation begins with the realization that the Boltzmann-BGK uation ( is linear in its unknown f. Therefore, the closed form solution is immediate: f(t, x i, u i = τ t f ( t, x i u i (t t δ i3 2 g (t t 2, u i + δ i3 g (t t e t t τ dt, (25 where δ i stands for the Kronecker delta (the identity tensor. Note that the uilibrium distribu- tion within the integrand is shifted in time, space, and velocity to an earlier time t, corresponding upstream position, and corresponding vertical velocity u 3 + g (t t. This earlier time t varies from long ago to the present time t. The exponential decay indicates that contributions from more remote times are attenuated compared to more recent times and is attributable to memory loss caused by collisions. 3 A more compact expression of (25 can be obtained by switching from the earlier time t to the dimensionless elapsed time s (t t /τ, which runs from to +. We further introduce the tilde notation to indicate a variable with space-time shift so that (25 becomes f f ( t sτ, x i u i sτ δ i3 2 gs2 τ 2, u i + δ i3 gsτ, (27 f = f e s ds. (28 The spatial domain is taken as infinite so as to avoid the complications of boundary conditions. 44 B. General Form of the Stress and Energy Flux The stress tensor σ i and the energy flux vector q were defined in uations (7 and (8, respectively. Inserting the general solution (28 into (7 and (8, we can write σ i = i ū i (u ū f e s ds du (29 q = 2 u ū 2 (u ū f e s ds du. (3 3 If one is uneasy about going back in time forever (t starting at, one can write the solution in terms of an initial condition (with time t now starting from : t f(t, x i, u i = f ( t, x i u i (t t δ i3 τ g (t 2 t 2, u i + δ i3g (t t e t t τ dt ( + f xi u it δ i3 2 gt2, u i + δ (26 i3gt e τ t, in which f (x i, u i is the initial distribution. Note how its contribution is fading with time. 7

8 47 These integrals (over ds and du would be straightforward to calculate if it were not for the space time shift in f. Indeed, without the space-time shift, f would be replaced by f, the integrals would decouple, and e s ds =. Denoting the resulting stress and energy flux as σ ( i and q (, we have σ ( i = (u i ū i (u ū f du = ρu 2 δ i (3 q ( = 2 u ū 2 (u ū f du =, (32 Note that the diagonal elements of the stress are ual to one another because f is isotropic in velocity space, and the off-diagonal elements vanish because f is symmetric in (u i ū i. We recognize here the pressure component of the stress tensor, with pressure defined as p = ρu 2. The stress tensor can thus be decomposed as follows: σ i = p δ i + τ i, ( with the deviatoric component τ i corresponding to the space-time shift in f. The conclusion at this point is that, if the space-time shift is ignored, the non-dissipative Euler uations are recovered. Put another way, the space-time shift is what introduces viscosity in the momentum uation and diffusion in the energy uation. The integral expressions of the deviatoric stress tensor and energy flux vector are: τ i = e s ds (ui ū i (u ū ( f f du (34 q = e s ds 2 u ū 2 (u ū ( f f du, (35 in which we recall that the tilde indicates a time shift by sτ, space shift by u i sτ, and velocity shift by δ i3 gsτ, so the integrals are coupled. The problem is to find a way to express the integrals in (34 and (35 in terms of only ρ, ū i and U. We shall proceed by considering the deviatoric stress (34 in III C and the energy flux (35 in III D. 64 C. Deviatoric stress In this subsection, we evaluate the deviatoric stress (34. To start, we invoke the assumption of a short relaxation time. In the spirit of Chapman-Enskog, we assume τ is so short compared to the time scale over which the overall system evolves that only short time shifts t t = sτ need to be considered, because the time exponential in (34 vanishes rapidly. This means that the space-time shift in f is relatively small, and a Taylor expansion of f may be performed with respect to sτ. Recalling that the space-time shift occurs inside the hydrodynamic variables ρ, ū i and U on which f depends, we apply the chain rule to f = f (ρ(t,x i, ū i (t,x i, U 2 (t,x i, u i : f = f + ρ ( ρ ρ + ( ū m ū m + ū m (U 2 (Ũ 2 U 2 + gsτ + O(s 2 τ 2 (36 u 3 8

9 72 and then expand the shifted hydrodynamic variables (e.g. ρ = ρ(t sτ,x i u i sτ δ i3 2 gs2 τ 2 : ( ρ ρ = ρ + t + u ρ n ( sτ + O(s 2 τ 2 (37 ( ūm ū m ū m = ū m + + u n ( sτ + O(s 2 τ 2 (38 t ( (U Ũ 2 = U 2 2 (U u n ( sτ + O(s 2 τ 2. (39 t 73 Using (36 (39, the deviatoric stress (34 becomes τ i = +τ s e s ds (u i ū i (u ū }{{} = [ ( ρ ρ t + u ρ n + ( ūm ū m + u n + ū m t ( (U 2 (U 2 (U 2 + u n g t ] u 3 du. ( The u n terms within the square bracket are evaluated with the aid of u n = (u n ū n + ū n and the following arguments: Since f is ruired to be isotropic in the velocity space according to (2, it must be an even function of (u m ū m, and its derivatives with respect to ū m and u 3 are odd. However, the derivatives /ρ and /(U 2 remain even functions of their argument. These symmetries and anti-symmetries lead to the cancellation of numerous integrals in (4, including that with gravity. The integral over s, which now stands alone, can be readily evaluated (=. After this series of simplifications, (4 becomes ( ρ τ i = τ t + ū ρ n (u i ū i (u ū ρ du + τ ū m (u i ū i (u ū (u n ū n du ū ( m (U 2 (U 2 + τ + ū n (u i ū i (u ū t (U 2 du. ( For the generic uilibrium distribution f given in (2, its derivatives with respect to ρ, ū m and U 2 are: ρ = F ( U 3 (42 = 2ρ ū m U 5 (u m ū m F ( (43 (U 2 = 3ρ 2U 5 F ( ρ U 5 F (, (44 9

10 83 and the expression (4 for τ i becomes: τ i = τ ( ρ U 3 t + ū ρ n (u i ū i (u ū F ( du 2ρτ ū m U 5 (u i ū i (u ū (u m ū m (u n ū n F ( du 3ρτ ( (U 2 (U 2 2U 5 + ū n (u i ū i (u ū F ( du t ρτ U 5 ( (U 2 t + ū n (U 2 (u i ū i (u ū F ( du. Using spherical coordinates in velocity space (r, θ, φ with (u ū, u 2 ū 2, u 3 ū 3 = (r sin φ cos θ, r sin φ sin θ, r cos φ, such that d(u ū d(u 2 ū 2 d(u 3 ū 3 = r 2 sin φ dφ dθ dr, and ( φ π, θ 2π, = r 2 /U 2, the preceding integrals (some after integration by parts can be expressed in more compact forms and then evaluated using (4: (u i ū i (u ū F ( du = + 2π ( 3 U F ( d (u i ū i (u ū F ( du = 5π ( 3 U F ( d (45 δ i = U 5 δ i (46 δ i = 5 2 U 5 δ i (47 (u i ū i (u ū (u m ū m (u n ū n F ( du ( πu F ( d = 3 2 U 7 if all indices are ual ( = π 3 U F ( d = 2 U 7 if indices make two pairs if at least one index is unique = 2 U 7 (δ i δ mn + δ im δ n + δ in δ m. ( in which the combination (δ i δ mn + δ im δ n + δ in δ m uals 3 if i = = n = m, uals if the set (i,, m, n consists of two pairs, and uals otherwise. Upon inserting (46 (48 into (45, the deviatoric stress tensor is found to be: [ ] τ i = τ t (ρu 2 + ū n (ρu 2 δ i + ρτu 2 ū m (δ i δ mn + δ im δ n + δ in δ m, (49 This expression can be further simplified by making use of the continuity uation (6 and internal energy uation (24. Taking the energy uation (24 at leading order in sτ (i.e. with σ i taken as σ ( i = ρu 2 δ i and q taken as q ( =, which is sufficient at this order of sτ, we arrive at the final expression for the deviatoric stress tensor: [( τ i = ρτu 2 ūi + ū 2 ( ūm x i 3 x m ] δ i. (5 Equation (5 represents the constitutive uation for a compressible Newtonian fluid with viscosity µ = ρτu 2 = ρuλ. (5

11 97 It is important to note that this value is independent of the particular form of the uilibrium distribution, so long as it is isotropic and satisfies the constraints (3 and (4. Upon inserting (5 and (5 into the momentum uation (9, we recover the compressible flow Navier-Stokes uation. Thus we have derived the Navier-Stokes momentum uation from the Boltzmann Equation without a Chapman-Enskog expansion. 22 D. Energy flux We now proceed likewise to obtain the energy flux vector so as to complete the closure of the hydrodynamic uations. convenience: q = Our starting point is expression (35, which is repeated here for e s ds 2 u ū 2 (u ū ( f f du. (52 Note that because (52 contains all scales of u, the quantity q represents both the flux of both heat and kinetic energy due to velocity departures from the mean. Thus, we expect q to be related to the heat flux (as in Fourier s law and may possibly contain an additional term to account for a kinetic energy flux. We again make use of the Taylor expansion of f performed in (36 (39. With the time integral now decoupled, it evaluates readily: s e s ds =. Using u ū 2 = U 2 by virtue of (2, Equation (52 turns into: q = τu2 2 (u ū [ ρ + ( ρ t + u ρ n + ( (U 2 (U 2 Using again the set (42 (44, to which we add t ( ūm ū m t g + u n (U 2 ū m + u n ] du. u 3 (53 u 3 = 2ρ U 5 (u 3 ū 3 F (, (54 24 replacing u n by (u n ū n + ū n, and observing that many integrals vanish by symmetry, we obtain: q = τ ρ (u ū (u n ū n F ( du 2U + ρτ ( ūm ū m U 3 + ū n (u ū (u m ū m F ( du t + 3ρτ (U 2 4U 3 (u ū (u n ū n F ( du + ρτ (U 2 2U 3 (u ū (u n ū n 2 F ( du + ρgτ U 3 (u ū (u 3 ū 3 F ( du. (55

12 Clearly, the remaining integrals vanish unless uals one of m, n, or 3. The non-zero integrals can be evaluated by passing to spherical coordinates in velocity space: (u ū (u n ū n F ( du = 2π 3 U 5 I 5 δ n (56 (u ū (u m ū m F ( du = 5π 3 U 5 I 3 δ m (57 (u ū (u n ū n 2 F ( du = 7π 3 U 5 I 5 δ n (58 in which I 3 was defined in (4 and I 5 is defined as: 4 I F ( d. (59 Upon inserting (56 (58 into (55, we obtain for the energy flux: q = π 3 I 5 τu 4 ρ 5π ( 3 I 3 ρτu 2 ū t + ū ū n + gδ 3 2π 3 I 5 ρτu 2 (U 2. (6 This expression ( can be simplified by making use of the momentum uation (9 taken at leading order in τ which reads ū ū t + ū n + gδ 3 = ρ = (ρu 2 ρ : σ i x i q = π 3 I 5 ρτu 2 (U 2 π 3 (I 5 5I 3 τu 2 (ρu 2. (6 The first term in Equation (6 constitutes the heat flux in the classical sense. In the context of classical Boltzmann kinetic theory, the uilibrium distribution is restricted to be the Maxwell- Boltzmann distribution, for which I 5 = 5 2π = 5I 3. This leads to the cancellation of the second term in (6, leaving q = 5 2 ρτu2 (U 2. The classical theory relates the thermal speed U to the absolute temperature by U 2 = k B m T, where k B = J/K is the Boltzmann constant and m is the mass of a particle. Then (6 is uivalent to q = k T, with k = 5 2 nk BU 2 τ, (62 where k is the thermal conductivity (W/m K and n = ρ/m is the number of particles per volume. Thus, with f taken as the Maxwell-Boltzmann distribution and the preceding relation between thermal speed and absolute temperature, we have recovered Fourier s Law of heat conduction without performing a Chapman-Enskog expansion. But when the uilibrium distribution is other than Maxwell-Boltzmann, the difference I 5 5I 3 does not necessarily vanish, and the second energy flux term of (6 may remain. The presence of this term can be rationalized with the following analogy: The Reynolds-average uation 5 for the turbulent kinetic energy tke ρ u i u i is: (tke + (tke ū = ρ u ū i i t x u ( x 2 ρ u i u u + u p µ u i s i 2µ S i S i. (63 4 Using the Schwartz inuality, it can be shown that I I 5 I 2 3, and thus I 5 9/2π for all distributions. 5 The form given here is for incompressible flow in the absence of gravity (Pope, 2, page 25 in order to make the argument as brief as possible. Primed quantities denote turbulent fluctuations, in accord with the classical Reynolds decomposition. 2

13 where S i u i + u x i is the fluctuating strain rate. The analogy is not perfect, and it suffices to ignore the viscous terms in (63. Mapping the non-viscous terms in (63 onto the energy budget (24 of Boltzmann kinetic theory, the analogy yields: σ i 3 2 ρu 2 = 2 ρ u i u i tke (64 ū i = ρ u i u ū i (65 q = 2 ρ u i u i u + u p, (66 The uivalence expressed in (64 is obvious from the form of (5, but now interpreting U as the rms velocity fluctuation rather than the thermal speed. Likewise, the mapping (65 is clear from the definition of the stress tensor (7. This leaves Equation (66 to be matched with (6. With p = ρu 2 according to (33, the natural partitioning is: π 3 I 5 ρτu 2 (U 2 = 2 ρ u i u i u (67 π 3 (I 5 5I 3 τu 2 p = u p (68 with correlations expressed as down-gradient fluxes, as it could be expected. From this consider- ation, the second term of (6 represents redistribution of kinetic energy by pressure fluctuations (modeled as a gradient of the mean pressure. What is rather unexpected is that this term vanishes when the uilibrium distribution happens to be Gaussian. 244 IV. CONCLUSIONS The preceding analysis leads to three important conclusions: First, the Chapman-Enskog expansion is not the only path to obtain the Navier-Stokes uations from the Boltzmann Equation. Second, the alternative path followed here shows that the particular form of the uilibrium distribution is rather inconsuential as long as it obeys a few basic properties, chiefly isotropy in velocity space and integrability. The two essential ingredients in the Boltzmann Equation are the BGK formulation for the collision term and the smallness of its relaxation time. Third, there is a new term in the expression for energy flux when the uilibrium distribution f is not Gaussian, which is found to be similar to the pressure redistribution flux of the Reynolds-average uation for the turbulent kinetic energy. 254 ACKNOWLEDGMENTS The authors are grateful to Dr. Hudong Chen for valuable insight and discussions that helped shape this article [] S. Chapman and T. G. Cowling, The Mathematical Theory of Non-uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion of Gases, 3rd ed. (Cambridge 3

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