Boltzmann Kinetic Equation for Filtered Fluid Turbulence

Sharath S. GirimajiAerospace Engineering Department, Texas A&M University, College Station, Texas 77843, USA

(Received 17 November 2006; published 19 July 2007)

We develop a kinetic Boltzmann equation for describing filtered fluid turbulence applicable forcontinuum and noncontinuum effects. The effect of unresolved turbulent motion on the resolveddistribution function is elucidated and closure modeling issues of kinetic Boltzmann and Navier-Stokesdescriptions are reconciled. This could pave the way for unifying turbulence modeling at kinetic andcontinuum levels and the development of numerical methods that are valid over a wide range of flowphysics.

DOI: 10.1103/PhysRevLett.99.034501 PACS numbers: 47.11.Qr, 47.27.E�, 47.45.Ab

Turbulence is a complex flow phenomenon, ubiquitousin nature and engineering, occurring in liquids, gases, andplasma. Traditionally, Navier-Stokes (NS) equations havebeen used for analysis and computation of turbulence. Thechaotic nature of turbulence necessitates a statistical ratherthan a deterministic approach. Over a century ago,Reynolds [1] introduced the mathematical framework foraveraging NS equations for a statistical turbulence descrip-tion—the so-called Reynolds-averaged Navier-Stokes(RANS) approach. More recently, a formal frameworkfor spatially filtering NS equations was formulated (e.g.,[2]) leading to the large eddy simulation (LES) approach.In principle, the kinetic Boltzmann equation [3] is bettersuited for describing turbulence over a wider range of flowphysics than NS—e.g., rarefied, hypersonic nonequilib-rium, astrophysical, and nanoscale flows. Indeed,Boltzmann-based computational schemes [4–12] havedemonstrated great potential as computational fluid dy-namics (CFD) tools offering some key advantages overNS methods. However, their success in turbulence has beenlimited. Despite notable attempts [12–15], a critical short-coming of the current kinetic Boltzmann approach is thelack of a formal physical framework for describing statis-tical or filtered turbulence. Such a kinetic Boltzmannframework is of great scientific value—for a statisticaldescription of turbulence in fluids lacking a constitutiveor state relation or flows with rarefied effects—and vitalpractical importance for development of turbulence com-putational methods valid over a wide range of flow physics.The objectives of this Letter are (i) to develop a kineticequation for statistical or filtered turbulence descriptionrigorously from the fundamental Boltzmann equations,and (ii) reconcile the manifestation of turbulence effectsin Navier-Stokes and kinetic statistical descriptions. Wewill also perform LES of decaying isotropic turbulence tovalidate the new kinetic equation.

The Navier-Stokes equation describes the evolution ofinstantaneous flow velocity (V) in terms of pressure (P)and density (�) fields. It is well known that the nonlinearadvection or inertial term in the equation imparts chaoticcharacter to the solution that leads to turbulence at high

enough Reynolds numbers. For statistical turbulence de-scription, all flow variables are decomposed into resolved(denoted by an overbar) and unresolved parts—e.g.,

V � U� u where �V � U: (1)

The overbar can represent RANS time, space, or ensembleaverage as appropriate, or any LES filter that commuteswith spatial and temporal differentiation [2]. In the RANSapproach, the effect of the unresolved field on averagedvelocity manifests via the so-called Reynolds stress—uiuj.The filtered LES equation has the same mathematical formas the RANS equation, owing to the scale-invariance prop-erty of the Navier-Stokes equation [2]. In LES, the subgridstress is the generalized second moment of velocity—Rij.Reynolds and subgrid stress represent unresolved momen-tum transfer and originate from the chaos-prone nonlinearadvection term. Thus averaged and filtered NS descriptionscan be considered under a single statistical framework.Now we proceed to develop a similar statistical kineticBoltzmann framework for turbulence.

The kinetic Boltzmann equation describes the evolutionof the distribution function—f�x; �; t�—of particle veloc-ity �:

@f��; x; t�@t

� �i@f@xi� C�f�; (2)

where C is the collision operator. It is very important torecognize that � is a phase-space variable and is indepen-dent of space and time. The Maxwellian equilibrium dis-tribution function is

f�0���; x; t� ��

�2�RT�D=2exp

���� � V�2

2RT

�;

where D is the spatial dimension; R � kb=m is the gasconstant; kb is the Boltzmann constant; and m is molecularmass. Conserved continuum quantities such as fluid den-sity and momentum are moments of f and f�0�: � �R�f�0�; f�d�, �V �

R��f�0�; f�d�. As collisions conserve

mass, momentum, and energy, the first several moments ofC are zero. It is possible to verify that the zeroth and firstorder moments of the Boltzmann equation (2) lead to the

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macroscopic mass and momentum conservation (NS)equations. Very important to note is the fact that the non-linear advection term in the Navier-Stokes equation thatgives rise to Reynolds stress resides in the linear advectionterm of the Boltzmann equation: �i

@f@xi

. It is thereforenatural to expect that the tendency to chaos and turbulencephysics will be incumbent in this advection term of theBoltzmann equation.

For statistical turbulence description, it would appearlogical to decompose the kinetic dependent variable (f)into its filtered and unresolved parts: f � �f� f0, wherethe overbar represents the same operator as in Eq. (1). Theequation for �f is easily obtained from (2):

@ �f@t� �i

@ �f@xi� �C: (3)

Unlike the Navier-Stokes case, the instantaneous (f) andfiltered ( �f) distribution function equations are identical inform and the effect of turbulence is implicit in the collisionoperator. In [12], citing similarities between molecular andturbulent fluctuation effects, a BGK model [3] for �C isproposed:

@ �f@t� �i

@ �f@xi� C�f� � �

1

��� �f� �f�0��; (4)

where the relaxation time �� is modified to account for‘‘turbulent-eddy collisions’’ and �f�0� is Maxwellian aboutthe filtered macroscopic velocity U:

�f �0���; x; t� ��

�2�RT�D=2exp

���� � U�2

2RT

�: (5)

LES computations using Eqs. (4) and (5) have enjoyedsome degree of success [6,7,12].

While the filtering procedure in Eq. (3) is mathemati-cally correct, it does not lend itself to clear interpretation offiltered turbulence physics which is important for the de-velopment of accurate closure models and efficient com-putational schemes. There are three crucial difficultiesassociated with this type of kinetic description of turbu-lence. (i) Most importantly, the effect of the unresolvedfield on the filtered distribution function ( �f) evolution isimplicit manifesting via the averaged collision operator �C.This is in complete contrast with the averaged Navier-Stokes equation in which the turbulence effects manifestas momentum flux originating in the advection term. TheRANS equation cannot be easily reconciled with this equa-tion. Therefore, decades of turbulence insight developed inthe Navier-Stokes context including important closuretheories and models cannot be easily adapted toBoltzmann description. (ii) The next difficulty involvesthe closure modeling of �C. The collision term has theonus of representing not only molecular collisions, butalso the advective effects of unresolved eddies. Unlikemolecular collisions, eddy advection can result in small-to-large scale energy transfer—the so-called inverse en-ergy cascade phenomenon. This renders the popular BGK

collision model (4) inappropriate for �C as it permits onlyirreversible energy transfer from large to small scales (perH theorem). (iii) The final difficulty involves the closuremodeling of the filtered equilibrium distribution function�f�0�. This represents an average over Maxwellian distribu-tions of different macroscopic velocities and cannot beMaxwellian about the averaged macroscopic velocity dueto the fact that

exp���� � V�2

2RT

�� exp

���� � U�2

2RT

�(6)

Thus, the closure in Eq. (5) is also questionable.We seek an alternate kinetic framework to separate the

advective turbulent-eddy effects of the unresolved velocity(u) from the dissipative molecular collision effects.Consider a coordinate transformation in the velocityphase-space: independent variable changed from � to �and dependent variable from f�x; �; t� to g�x;�; t�. Thephase-space variable � is again independent of physicalspace and time. Let the new distribution function g bedefined by the relationship

g�x;�; t� � f�x; � � �� u; t�; (7)

where u once again is the unresolved component of totalvelocity V. All of the required information about thevelocity field is incumbent in g: Z 1

�1g���d� �

Z 1�u�1�u

f���d� �Z 1�1

f���d� � �

Z 1�1�g���d� �

Z 1�1�� � u�f���d� � �U: (8)

(As u is finite, we can write:1 u � 1.) The equilibriumdistribution g�0� is a Maxwellian about the filtered velocity:

g�0���� � f�0���� u� ��

�2�RT�D=2exp

����� U�2

2RT

�:

(9)

The next challenge is to restate the Boltzmann equation interms of g���. The spatiotemporal derivatives of f and gare related according to

@g���@t

�@f���@t�@f@�i

@�i@t

���������;

@g���@xk

�@f���@xk

�@f@�i

@�i@xk

���������:(10)

From Eq. (7) we can write

@f���@�j

�@g���@�j

;@�i���@xk

����������@ui@xk

;@�i���@t

����������@ui@t:

(11)

Substituting Eq. (11) into (10) and the ensuing results intothe Boltzmann Eq. (2), we obtain

�@g@t� �i

@g@xi

�� ui

@g@xi� aj

@g@�j� C�g�; (12)

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where aj � @uj@t � �i

@uj@xi� ui

@uj@xi�. Further the following

equalities can be derived: ui@g@xi� @gui

@xi� g @ui@xi

and ai@g@�i�

@gai@�i� g @ui@xi

, since @ai@�i� @ui

@xi. Using these relationships,

Eq. (12) can be written in alternate form as

�@g@t� �i

@g@xi

��@gui@xi�@gaj@�j

� C�g�: (13)

It can be shown that the instantaneous mass and momen-tum (NS) equations can be recovered by integratingEqs. (12) or (13) over � space. The evolution equationfor �g���, where the overbar represents the same operator asin Eq. (1), is

@ �g@t� �i

@ �g@xi�@gui@xi�@gaj@�j

� C�g�: (14)

Distribution function �g contains all filtered mass and mo-mentum information:

R�gd� � ��;

R�g�d� � �U.

For the purpose of filtered turbulence description wepropose that g��� is better suited than f��� in all threecategories discussed previously:

(i) First and foremost, the proposed transformationcauses the effects of unresolved turbulence on the filtereddistribution function �g to manifest appropriately and ex-plicitly via advective terms—gui and gai. The term guirepresents transport in physical space due to unresolvedvelocity. Its main role is to spatially redistribute �g tosmoothen any spatial gradients. The term gai representsthe advection of �g in the velocity phase space due tounresolved acceleration. It has the form of a body forceterm and can function either a source (forward energycascade) or sink (inverse energy cascade) as demandedby turbulence physics. As will be shown further below,these terms can be directly related to Navier-Stokes closuremodels bridging the divide between kinetic and macro-scopic (NS) descriptions. (ii) The advective effects of theunresolved velocity implicit in C�f� is absent in C�g�.Relieved of the onus of accounting for turbulent-eddyadvection, the filtered collision operator is now requiredto represent only the averaged effects of irreversible mo-lecular collisions. This can be modeled justifiably withsimple BGK-type closure:

C�g� � �1

�� �g� �g�0�� (15)

(iii) The transformation has the effect of recentering thelocal equilibrium distribution function about the filteredmacroscopic velocity. As the local g�0� is independent ofthe unresolved field, the filtered equilibrium function issimply

�g �0� � g�0� ��

�2�RT�D=2exp

����� U�2

2RT

�: (16)

Thus, the filtered distribution function is Maxwellian aboutfiltered macroscopic velocity. Equation (16) is exact unlikethe model Eq. (5) for �f0 which cannot be easily justified.Based on these observations, we propose Eq. (14) for fil-

tered turbulence description in continuum and noncontin-uum flows. In this kinetic description, density-weighted ve-locity (gui) and acceleration (gai) need closure modeling.Any continuum turbulence effects must reside in theseterms. Noncontinuum effects can also appear through fil-tered collision operator and equilibrium distributionfunctions.

We will now relate (gui) and (gai) to terms that requireclosure in the RANS description. To accomplish this in thesimplest possible manner, without loss of generality, wewill restrict consideration to continuum incompressibleturbulence (� � ��). The zeroth moment of Eq. (14) yieldsthe averaged macroscopic mass conservation equation.The first moment gives the RANS equation and revealsthe direct connection between the kinetic and macroscopicterms:

Z�@ �g@td� �

@�U@t

;Z�C���d� � 0

Z��i

@ �g@xi

d� � r � �@�UU� �T�

Z�@gui@xi

d� � r � ��U �u� � 0

�Z�@gai@�i

d� �@� �u

@t�r � �@�U �u� �uu� � r � �uu

(17)

In the above, T is the instantaneous molecular stress tensorwhich includes pressure and viscous effects. Two crucialinferences are the following: gui does not contribute to theaveraged macroscopic momentum equation and gai yieldsthe Reynolds stress term. The simplest microscopic clo-sures that yield the desired macroscopic behavior as de-picted in Eq. (17) are the following:

gui � �g �ui � 0; gai � �g �ai � �g@uiuj@xj

: (18)

Thus, the correct macroscopic behavior is obtained whenthe kinetic distribution function g is taken to be uncorre-lated with macroscopic turbulent fluctuating velocity u andacceleration a. This can be considered the turbulenceanalog of the Chapman-Enskog ansatz [3]. For describingincompressible turbulence, the following evolution equa-tion for �g suffices:

@ �g@t��i

@ �g@xi�@uiuj@xj

@ �g@�i�C�g���

1

�� �g� �g�0��; (19)

where � is the molecular relaxation time scale. The differ-ence between the previous and present approaches can bereadily seen by comparing the forms of Eqs. (19) and (14).Irrespective of the Reynolds stress closure model used, theprevious method can only yield an enhanced eddy-viscosity effect, whereas the current approach can accom-modate a variety of physical effects. Inverse cascade ofenergy is possible if an appropriate closure model forReynolds stress is employed. Thus, advanced turbulence

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models developed in the RANS context can be used to theirmaximum physical capability. Further, invoking the scale-invariance property of Navier-Stokes physics [2], we canalso conclude that Eq. (19) is valid for incompressible LESas well, provided uiuj is replaced by subgrid stress Rij.

We will now compare the previous approach [Eqs. (4)and (5)] with the present method (19) in LES computationof decaying isotropic turbulence using the latticeBoltzmann method (LBM) [6–10]. Decaying isotropicturbulence is the most fundamental benchmark problemfor turbulence theory or model validation. Any differencebetween the two models in this flow will be amplified inmore challenging flows. In both computations we use thesimple Smagorinsky subgrid scale closure:

Rij � �Cs�2jSjSij; (20)

where � is the grid size, Sij is the filtered-field strain rate,and Cs � 0:1 [6]. The computational details are similar tothose in [6] where the implementation of general body

force in LBM formulation is also given. A comprehensivecomparative study was performed, and we present only thesalient results here. In Fig. 1, the kinetic energy decay fromrandom initial field is plotted for three cases: (i) the refer-ence 1283 DNS-LBM calculation using Eq. (2); (ii) 323

LES using Eq. (4)—labeled old; and (iii) 323 LES usingEq. (19)—labeled new. The spectra from the three casesare compared in Fig. 2. The LES model that is closer to thereference DNS results must be considered more accurate.The present LES predicts the kinetic energy and spectramore accurately. The energy content of the intermediateand large scales is preserved better by the new method.Even in this simple problem, the advantage of the presentmethod over the previous approach is clear.

In this Letter, we develop a kinetic Boltzmann equationfor filtered turbulence description valid for continuum andnoncontinuum effects. In the continuum limit, direct cor-respondence with the RANS equation is analyticallyshown. The validity of the equation and its advantagesover previous kinetic models is clearly demonstrated inthe fundamental problem of decaying isotropic turbulence.This work represents an important step in unifying kineticand continuum statistical turbulence description, closuremodeling, and computational tool development.

This work was supported by AFOSR-MURI GrantNo. FA9550-04-1-0425 (Program Manager, Dr. JohnSchmisseur).

[1] O. Reynolds, Phil. Trans. R. Soc. A 186, 123 (1895).[2] M. M. Germano, J. Fluid Mech. 238, 325 (1992).[3] S. Harris, An Introduction to the Theory of the Boltzmann

Equation (Dover Publications Inc., New York, 2004).[4] G. R. McNamara and G. Zanetti, Phys. Rev. Lett. 61, 2332

(1988).[5] H. Chen, S. Chen, and H. W. Matthaeus, Phys. Rev. A 45,

R5339 (1992).[6] H. D. Yu, S. S. Girimaji, and L. S. Luo, Phys. Rev. E 71,

016708 (2005).[7] H. D. Yu and S. S. Girimaji, Phys. Fluids 17, 125106

(2005).[8] R. Benzi, S. Succi, and M. Vergassola, Phys. Rep. 222,

145 (1992).[9] S. Chen and G. D. Doolen, Annu. Rev. Fluid Mech. 30,

329 (1998).[10] D. Yu, R. Mei, W. Shyy, and L. Luo, Prog. Aerosp. Sci. 39,

329 (2003).[11] P. Degond and M. Lemou, J. Math. Fluid Mech. 4, 257

(2002).[12] H. Chen, S. Kandasamy, S. Orszag, R. Shock, S. Succi,

and V. Yakhot, Science 301, 633 (2003).[13] H. Chen, S. Succi, and S. Orszag, Phys. Rev. E 59, R2527

(1999).[14] S. Succi, O. Filippova, H. Chen, and S. Orszag, J. Stat.

Phys. 107, 261 (2002).[15] S. Ansumali, V. Karlin, and S. Succi, Physica

(Amsterdam) 338A, 379 (2004).

t'

k

10-3 10-2 10-1 1000

0.5

1

DNS-1283

LES-323-NewLES-323-Old

FIG. 1. Kinetic energy decay.

t’=0.24

κ

E

100 101 10210-7

10-6

10-5

10-4

10-3

10-2

10-1

100

DNS-1283

LES-643-NewLES-643-OldLES-323-NewLES-323-Old

FIG. 2. Energy spectrum.

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