A non-perturbative analysis of the Bhatnagar-Gross-Krook (BGK) model kinetic
equation for finite values of the Knudsen number is presented. This analysis
indicates why discrete kinetic versions of the BGK equation, and notably the
Lattice Boltzmann method, can provide semi-quantitative results also in the
non-hydrodynamic, finite-Knudsen regime, up to $Kn\sim {\cal O}(1)$. This may
help the interpretation of recent Lattice Boltzmann simulations of microflows,
which show satisfactory agreement with continuum kinetic theory in the
moderate-Knudsen regime.Comment: 7 PAGES, 1 FIGUR

Benzi R.

Cercignani C.

Chapman S.

D' Humières D.

Higuera F.

Holdych D. J.

Karniadakis G.

M Sbragaglia

Qian Y.

S Succi

Shan X.

Sone Y.

Succi S.

Wolfe-Gladrow D.

English

arXiv

A non-perturbative study of the BGK  kinetic equation for finite values of the Knudsen number is presented

Sbragaglia, M

Succi, S

Archivio della Ricerca - Università di Roma Tor vergata

A note on the lattice Boltzmann method beyond the Chapman-Enskog limits

A non-perturbative analysis of the Bhatnagar-Gross-Krook (BGK) model kinetic equation for finite values of the Knudsen number is presented. This analysis indicates why discrete kinetic versions of the BGK equation, and notably the lattice Boltzmann method, can provide semi-quantitative results also in the non-hydrodynamic, finite-Knudsen regime, up to Kn ~ (1). This challenges the pessimistic stance, according to which the lattice Boltzmann method should only be used for strictly hydrodynamic purposes. It may also help the interpretation of recent simulations of microflows, which show satisfactory agreement with continuum kinetic theory in the moderate-Knudsen regime

Sbragaglia, M.

Succi, S.

University of Twente Research Information

Europhys. Lett., 73 (3), pp. 370–376 (2006)DOI: 10.1209/epl/i2005-10404-8EUROPHYSICS LETTERS 1 February 2006A note on the lattice Boltzmann methodbeyond the Chapman-Enskog limitsM. Sbragaglia1 and S. Succi21 Dipartimento di Fisica, Università “Tor Vergata”, and INFNVia della Ricerca Scientifica 1, I-00133 Roma, Italy2 CNR, IAC - Viale del Policlinico 137, I-00161 Roma, Italyreceived 8 September 2005; accepted in final form 2 December 2005published online 23 December 2005PACS. 47.11+j – Computational methods in fluid dynamics.PACS. 47.45-n – Rarefied gas dynamics.Abstract. – A non-perturbative analysis of the Bhatnagar-Gross-Krook (BGK) model kineticequation for finite values of the Knudsen number is presented. This analysis indicates whydiscrete kinetic versions of the BGK equation, and notably the lattice Boltzmann method,can provide semi-quantitative results also in the non-hydrodynamic, finite-Knudsen regime,up to Kn ∼ O(1). This challenges the pessimistic stance, according to which the latticeBoltzmann method should only be used for strictly hydrodynamic purposes. It may also helpthe interpretation of recent simulations of microflows, which show satisfactory agreement withcontinuum kinetic theory in the moderate-Knudsen regime.In the last decade, the lattice Boltzmann (LB) method has developed into a very flexibleand effective numerical method for the simulation of a large variety of complex flows, mostly inthe macroscopic domain [1–3]. Fueled by relentless progress in micro, nano and bio-sciences,the recent years have witnessed a growing interest in exploring the possibility to enrich LB inthe direction of describing micro-structured flows [4, 5].The LB method is based on a stylized stream-and-collide microscopic dynamics of fictitiousparticles, located on the nodes of discrete lattices and interacting according to local collisionrules that drive the system towards a local equilibrium [2,3]. Mathematically:fi(x + ci∆t, t + ∆t) − fi(x, t) = −ω∆t[fi(x, t) − f (eq)i (x, t)],where fi(x, t) is the probability to find a particle at position x and time t, moving along thelattice direction defined by the discrete speeds ci (i = 1, ..., b). The second term at the r.h.s.of the above equations denotes relaxation towards a local equilibrium, the lattice analogue ofa Maxwellian distribution in continuum kinetic theory:feqi (x, t) = nwi(1 + βci · u + β22[(ci · u)2 − u2]).In the above, n(x, t) is the fluid density, u the flow speed, β = 1/c2s is the inverse square speedof sound, and wi a discrete set of weights normalized to unity. Finally, ω indicates a typicalc© EDP SciencesArticle published by EDP Sciences and available at http://www.edpsciences.org/epl or http://dx.doi.org/10.1209/epl/i2005-10404-8M. Sbragaglia et al.: Lattice BGK beyond Chapman-Enskog 371time-scale relaxation (frequency relaxation) to local equilibrium, and governs the kinematicviscosity of the LB macroscopic fluid [2,6,7]. More general versions account for multiple-timerelaxation [8–10], but the simpler single-time (BGK) relaxation form will be sufficient for ourpresent purposes.By taking suitable averages over molecular speeds, it can be shown that the resultingmacroscopic quantities obey the Navier-Stokes equation of continuum mechanics. The fluidand current density are given by a linear superposition of the discrete distributions fi:n(x, t) =b∑i=1fi(x, t), J(x, t) =b∑i=1cifi(x, t). (1)In order for those quantities to satisfy the exact hydrodynamic equations, it is required thatthe macroscopic fields do not vary appreciably on the scale of the mean free path λ and thelattice spacing ∆x. Based on the consolidated Chapman-Enskog background [11], one mightbe led to conclude that the range of applicability of LB methods is bounded by the domain ofvalidity of the Chapman-Enskog method [12], the implication being that LB can only be usedfor strictly hydrodynamic purposes. Yet, such a restrictive stance is challenged by a number ofrecent numerical simulations [13–20] which clearly show that, by using appropriate boundaryconditions, LB can reproduce some salient features of flows beyond the hydrodynamic regime,such as the onset of slip flow at finite-Knudsen numbers.In this letter, we wish to propose a theoretical explanation for this rather unexpected valid-ity of LB in the beyond-Chapman-Enskog region. Our analysis is based on a non-perturbativesolution of the lattice BGK (LBGK) equation, which overcomes the restrictions imposed tothe standard Chapman-Enskog multiple scale analysis and extensions thereof [21, 22]. Thisnon-perturbative character sets our work apart also from very recent and systematic work onthe finite-Knudsen behaviour of lattice-BGK models [23].Our analysis is confined to the bulk region of the flow, giving for granted that the use ofproper boundary conditions is crucial to obtain correct results in actual LB simulations ofmicroflows [24].Let us refer for simplicity to the 1d-continuum Boltzmann equation, written in BGK [25]form:∂tf(x, v, t) + v∂xf(x, v, t) = −ω[f(x, v, t) − f (eq)(x, v, t)],where the local equilibrium f (eq)(x, v, t) depends on x and t via the local velocity and densityfields. Let us now consider the exact solution as provided, for each given velocity v (seealso [26]), by an integration along the particle trajectory, from t to t + ∆t:f(x + v∆t, t + ∆t) = e−ω∆tf(x, t) + ωe−ω∆t∫ ∆t0esωf (eq)(x + vs, t + s)ds, (2)being ∆t a generic time increment corresponding to the lattice time step. The above expressionis exact, but purely formal, until one specifies a concrete procedure to compute the integralat the right-hand side. Even though the quadratic dependence of f (eq) on f itself prevents anexact analytical solution, one can formally expand the integrand asf (eq)(x + vs, t + s) =∞∑n=0snDnn!f (eq)(x, t), (3)where D ≡ ∂t + v∂x denotes the streaming operator. By inserting this expansion into (2), we372 EUROPHYSICS LETTERScan formally solve the integral exactly, and obtain:ft+∆t = e−ω∆tft +eD∆t − e−ω∆t1 + D/ωf(eq)t , (4)where we have used the short-hand notation ft+s = f(x + vs, t + s). This expression invitesa number of useful considerations. Let us first recast it in the symbolic propagator form:ft+∆t = P (ω∆t)ft + Q(ω∆t,D∆t)f(eq)t , (5)whereP (ω∆t) = e−ω∆t (6)is the propagator from time t to time t + ∆t, andQ(ω∆t,D∆t) =eD∆t − e−ω∆t1 + D/ω(7)co-propagates the influence of the equilibrium at time t on the solution at time t + ∆t.By neglecting the equilibrium variations on a scale ∆t, i.e.f(eq)t+t′  f (eq)t , 0 ≤ t′ ≤ ∆t (8)the propagator eD∆t acting on f (eq) is such thateD∆tf(eq)t = f(eq)t . (9)This leads to a much simplified version of the integral equation in the limit Df (eq) → 0:ft+∆t = e−ω∆tft + (1 − e−ω∆t)f (eq)t (10)or, equivalently,ft+∆t = P (ω∆t)ft + QLB(ω∆t)f(eq)t , (11)where we have defined the LB co-propagator asQLB(ω∆t) ≡ (1 − e−ω∆t). (12)The stick-to-equilibrium approximation (9) is tantamount to retaining only the zeroth orderterm in the Taylor expansion (3), and consequently it is expected to work only for small valuesof ∆t in the integral of (2).It is interesting to note that, already at this zeroth order level, and without any furtherassumption on ω, it is possible to identify a fully explicit discrete Boltzmann equation in BGKform. This reads:ft+∆t − ft = −ωLB∆t(ft − f (eq)t ) (13)with relaxation frequency ωLB∆t = (1 − e−ω∆t).It is now instructive to analyze the two extreme limits ω∆t  1 and ω∆t  1, in theexact solution (5) first, and then in the discrete lattice BGK solution (11). Let us begin withthe former. Provided that the kinetic operator D∆t remains bounded, we can identify theenslaving limit (ω∆t  1) characterized byP (ω∆t) = e−ω∆t −→ 0,M. Sbragaglia et al.: Lattice BGK beyond Chapman-Enskog 373Q(ω∆t,D∆t) =eD∆t − e−ω∆t1 + D/ω−→ eD∆t.This is equivalent to enslave the solution to the local equilibrium, i.e. ft+∆t = f(eq)t+∆t. Theopposite situation (ω∆t  1) represents the free-molecular limit, in which one hasP (ω∆t) = e−ω∆t −→ 1Q(ω∆t,D∆t) =eD∆t − e−ω∆t1 + D/ω−→ 0corresponding to a free-flow (collisionless) solution ft+∆t = ft.A natural question arises on the nature of the same limits in the Lattice BGK equation (11).The effect on the streaming propagator is the same, since the streaming term is integratedexactly in the lattice version. There remains to inspect the behaviour of QLB(ω∆t) in thetwo aforementioned limits. The enslaving limit (ω∆t  1) yieldsQLB(ω∆t) = 1 − e−ω∆t −→ 1 (ωLB∆t ∼ O(1))that is ft+∆t = f(eq)t . In view of the relation (8), this is equivalent to state ft+∆t = f(eq)t+∆t.The opposite situation (ω∆t  1) yieldsQLB(ω∆t) = 1 − e−ω∆t −→ 0 (ωLB∆t → 0).Thus, the two limits, full-enslaving and free-flow, are recovered by the LBGK, provided thatthe relaxation frequency is turned from bare ω to ωLB . The question remains: what happensin between? A restrictive tenet is that lattice BGK cannot work properly because the interme-diate regime involves all-order tensors, through the powers Dn, which cannot be reproducedcorrectly in the discrete lattice because of lack of symmetry. On a more optimistic vein, onecould counter-argue that since both extreme limits are correctly recovered, there might behope that even in between the LB method could continue to provide a reasonable agreementwith continuum kinetic theory.In order to analyze this point, we introduce the fine-scale Knudsen number Kn, defined asKn = λ/δ, (14)where λ = v/ω is the kinetic mean-free path and δ is the smallest macroscopic length.Let us consider the most critical situation δ = ∆x = v∆t, i.e. the macroscopic fieldsshow appreciable variation on the scale of a single lattice unit/length. Under these specificconditions, we obtainω∆t = 1/Kn (15)from which it follows thatP (ω∆t) = P (Kn) = e−1/Kn, (16)Q(ω∆t,D∆t) = Q(Kn,D∆t) =eD∆t − e−1/Kn1 + KnD∆t, (17)QLB(ω∆t,D∆t) = QLB(Kn,D∆t) = 1 − e−1/Kn. (18)Our Knudsen-number–dependent approach is consistent with the fact that the full enslav-ing limit is recovered at Kn → 0, which is equivalent to use a LBGK approach (13) withωLB∆t = (1−e−ω∆t) ∼ O(1). Starting from this Chapman-Enskog limit (Kn → 0), it is then374 EUROPHYSICS LETTERS 0.001 0.01 0.1 1 0.001  0.01  0.1  1E(Kn)KnHydrodynamicSlip flowTransitionFig. 1 – Plot of the percentage error E, as defined in (20), as a function of the Knudsen number upto the transition regime. Note that the error between the exact Boltzmann solution and the discreteform reaches a maximum of about ten percent at the top of the slip-flow regime (Kn ∼ 0.1).interesting to see whether the LB approach can be extended to the slip flow and transitionregimes (up to Kn ∼ O(1)). In these regimes, the macroscopic fields fluctuate, in the mostpessimistic case, on the same scale of the kinetic fields. Therefore, we may consistently assumethat Df/f ∼ v/δ, and make the identificationD∆t ∼ Kn. (19)This is the zeroth order approximation relating D∆t to Kn. More rigorously, one shouldconsider a full series in Kn. However, to any practical purpose, this series can be truncatedat the O(Kn) without hampering the results up to Kn ∼ O(1). In this approximation, thepropagators becomeQ(Kn) =eKn − e−1/Kn1 + Kn2,QLB(Kn) = 1 − e−1/Kn,where Q(Kn) and QLB(Kn) represent the exact physical co-propagator effects and the latticeones, in the range Kn < 1, respectively. The relative departure:E(Kn) =Q(Kn) − QLB(Kn)Q(Kn)(20)is a quantitative measure of the spurious lattice-induced effects on the co-propagator. Fromfig. 1, where we report E(Kn) as a function of Kn, it is recognized that, up to Kn ∼ 0.1, i.e.in the slip-flow regime, the lattice Boltzmann approach does not differ from the exact solutionM. Sbragaglia et al.: Lattice BGK beyond Chapman-Enskog 375for more than to within a ten percent. Only as Kn exceeds 0.1, and the transition regime isentered, significant error build-up is observed.It is important to emphasize that the above figures apply to the worst-case scenario, inwhich macroscopic fields vary on the scale of a single lattice spacing (an extreme beyond-Chapman-Enskog situation). One may soften this assumption and assume that macroscopicfields exhibit significant variations on scales larger that a single grid spacing ∆x, sayδ = h∆x, h ≥ 1, (21)h = 1 reproducing the previous worst-case scenario. This means that the Knudsen number isreduced by a factor h, Kn → Kn/h, leading to a corresponding reduction in the error E(Kn).In addition, we observe that E(Kn) measures the error of the propagator, whereas physicalobservables result from the summation of discrete populations fi over the discrete speeds(see eqs. (1)) and, consequently, E(kn) as defined in (20) may well represent a pessimisticbound. In any event, the point of the present analysis is not to state the case for the accuracyof LB in the non-hydrodynamic regime, but only to point out that, even at finite-Knudsennumbers, discreteness effects remain within fairly tolerable limits, at least for semi-quantitativepurposes. Whether or not LB should be used instead of more accurate, and much moreexpensive, methods, such as Direct Simulation Monte Carlo, remains to be decided on acase-by-case basis.Summarizing, a non-perturbative analysis of the Boltzmann equation in BGK form, indi-cates that the LB method may continue to provide semi-quantitative agreement beyond thelimits of the Chapman-Enskog theoretical framework. This could be of interest for the in-terpretation of lattice Boltzmann simulations in the finite-Knudsen regime, including kineticmodeling of fluid turbulence [27]. Clearly, in order to realize the bulk properties highlightedby the present analysis, proper kinetic boundary conditions [18,20] must be used in actual LBsimulations of finite-Knudsen flows.∗ ∗ ∗We warmly thank R. Benzi, L. Biferale and F. Toschi for useful comments and criticalreading of this manuscript.REFERENCES[1] Benzi R., Succi S. and Vergassola M., Phys. Rep., 222 (1992) 145.[2] Wolfe-Gladrow D., Lattice Gas Cellular Automata and Lattice Boltzmann Model (SpringerVerlag, Berlin) 2000.[3] Succi S., The Lattice Boltzmann Equation (Oxford University Press, Oxford) 2001.[4] Ho C. M. and Tai Y. C., Annu. Rev. Fluid Mech., 30 (1998) 579.[5] Karniadakis G. and Beskok A., Microflows (Springer Verlag, Berlin) 2002.[6] Qian Y., d’Humières D. and Lallemand P., Europhys. Lett., 17 (1992) 479.[7] Chen H., Chen S. and Mattheus W., Phys. Rev. A, 45 (1992) 5339.[8] Higuera F., Succi S. and Benzi R., Europhys. Lett., 9 (1989) 345.[9] Higuera F. and Jimenez J., Europhys. Lett., 9 (1989) 663.[10] D’ Humières D., Prog. Astronaut. Aeronaut., 159 (1992) 450.[11] Chapman S. and Cowling T. G., The Mathematical Theory of Non Uniform Gases (CambridgeUniversity Press, Cambridge) 1991.[12] Cercignani C., Theory and Application of the Boltzmann Equation (Scottish Academic Press,New York) 1975.376 EUROPHYSICS LETTERS[13] Succi S., Phys. Rev. Lett., 89 (2002) 064502.[14] Nie X., Doolen G. and Chen S., J. Stat. Phys., 107 (2002) 29.[15] Lim C., Shu C., Niu X. and Chew Y., Phys. Fluids, 14 (2002) 2299.[16] Zhang Y., Qin R. and Emerson D., Phys. Rev. E, 71 (2005) 047702.[17] Toschi F. and Succi S., Europhys. Lett., 69 (2005) 549.[18] Benzi R., Biferale L., Sbragaglia M., Succi S. and Toschi F., to be published in J. Fluid.Mech.[19] Zhang J. and Kwok D., Phys. Rev. E, 70 (2004) 056701.[20] Ansumali S. and Karlin I., Phys. Rev. E, 66 (2002) 026311.[21] Inamuro T., Yoshino M. and Ogino F., Phys. Fluids, 9 (1997) 3535.[22] Sone Y., Rarefied Gas Dynamics, Vol. 2 (Editrice tecnico scientifica, Pisa) 1971, p. 737.[23] Shan X., Yuan X. and Chen H., to be published in J. Fluid. Mech.[24] This point was analyzed in Holdych D. J. et al., J. Comput. Phys., 193 (2004) 594. However,the authors did not seem to focus on finite-Knudsen issues, hence did not consider kineticboundary conditions.[25] Bhatnagar L., Gross E. and Krook M., Phys. Rev., 94 (1954) 511.[26] He X. and Luo L. S., Phys. Rev. E, 56 (1997) 6811.[27] Chen H., Kandasamy S., Orszag S., Shock R., Succi S. and Yakhot V., Science, 301(2003) 633.

A note on the Lattice Boltzmann Method Beyond the Chapman Enskog Limits

Crossref

NARCIS 

SBRAGAGLIA, MAURO

A non-perturbative analysis of the Bhatnagar-Gross-Krook (BGK) 
model kinetic equation for finite values of the Knudsen
number is presented.
This analysis indicates why discrete kinetic versions of the BGK 
equation, and notably the lattice Boltzmann method, can provide 
semi-quantitative results also in the non-hydrodynamic, 
finite-Knudsen regime, up to $Kn\sim {\cal O}(1)$. This challenges the pessimistic stance, according to which the lattice Boltzmann method should only be used for strictly hydrodynamic purposes.
It may also help the interpretation of recent simulations 
of microflows, which show satisfactory agreement with continuum kinetic
theory in the moderate-Knudsen regime

A note on the lattice Boltzmann method beyond the Chapman Enskog limits

Abstract

Similar works

Full text

Available Versions

Archivio della Ricerca - Università di Roma Tor vergata

University of Twente Research Information

Crossref

NARCIS

ART

Crossref

EDP Sciences OAI-PMH repository (1.2.0)