On Higher Order Dynamics in Lattice-based Models Using Chapman-Enskog Method

Transcription

1 NASA/CR ICASE Report No On Hgher Order Dynamcs n Lattce-based Models Usng Chapman-Enskog Method Yue-Hong Qan Columba Unversty, New York, New York Ye Zhou IBM, Yorktown Heghts, New York and ICASE, Hampton, Vrgna Insttute for Computer Applcatons n Scence and Engneerng NASA Langley Research Center Hampton, VA Operated by Unverstes Space Research Assocaton Natonal Aeronautcs and Space Admnstraton Langley Research Center Hampton, Vrgna Prepared for Langley Research Center under Contract NAS June 1999

2 ON HIGHER ORDER DYNAMICS IN LATTICE-BASED MODELS USING CHAPMAN-ENSKOG METHOD YUE-HONG QIAN AND YE ZHOU Abstract. In ths paper, we nvestgate the exstence of hgher order dynamcs n lattce-based models. We have dentfed two condtons that determne whether a model would allow some Burnett-lke equatons when the Chapman-Enskog expanson s used. These two condtons are the number of the conserved quanttes as well as the space and tme dscretzaton. We shall demonstrate these condtons by dscussng (1) pure dffuson equaton, and (2) hydrodynamc equatons. Whle the fact that dffuson equaton allows the hgher order dynamcs can be shown easly, we wll llustrate that care must be taken when dervng Burnett-lke equatons for lattce-based hydrodynamcs models usng the Chapman-Enskog method. Key words. Boltzmann equaton, lattce-based hydrodynamcs models, Naver-Stokes equaton Subject classfcaton. Flud Mechancs 1. Introducton. Compared to tradtonal methods n computatonal flud dynamcs (CFD), the lattcebased models are smple and easy to mplement on computers. The advantages and dsadvantages of the orgnal lattce gas automata (LGA) have been well documented [1-7]. The lattce Boltzmann equaton (LBE) was later ntroduced to remove some of the drawbacks [8-10]. A further smplfcaton to the LBE s acheved usng the BGK procedure (LBGK) [11-14]. In lattce-based models, t s well establshed that the Naver-Stokes equaton can be deduced at low order expanson of Chapman-Enskog expanson [15]. Many authors further asserted that the Burnett-lke equaton could be obtaned by performng hgher order usng Chapman-Enskog expanson [4,6,7]. The motvaton of ths paper s to carry out these hgher order Chapman-Enskog expanson to nvestgate whether t s consstent to do so. We wll frst study the lattce-based model for pure dffuson model [16,17]; and demonstrate that hgher order dynamcs s allowed n ths case. We wll then pont out that the Burnett-lke equatons could be derved for lattce-based hydrodynamcs models. Attenton should be pad, however, when the classc Chapman-Enskog expanson s appled because of the non-commutatve feature of cross dervatves of two tme scales, these dervatves do not exst n the contnuous tme and space whle do exst n dscrete velocty models [18]. The number of conserved quanttes s also crtcal for the exstence of hgher order equatons. 2. Hgh Order Dynamcs: Pure Dffuson. We now consder the lattce BGK models for pure dffuson problems where the only quantty conserved durng the redstrbuton s the total mass. The propagaton step s the same as lattce gas models whle the collson step s just a redstrbuton of mass n all possble drectons. We start wth the followng evoluton equaton [12], (2.1) f ( x + c,t+1)=f ( x, t)+ω( ( x, t) f ( x, t)) Department of Appled Physcs and Appled Mathematcs, Columba Unversty, New York, NY Insttute for Computer Applcatons n Scence and Engneerng, NASA Langley Research Center, Hampton, VA and IBM Research Dvson, T.J. Watson Research Center, P.O. Box, 218, Yorktown Heghts, NY Ths research was supported by the Natonal Aeronautcs and Space Admnstraton under NASA Contract No. NAS whle the second author was n resdence at the Insttute for Computer Applcatons n Scence and Engneerng (ICASE), NASA Langley Research Center, Hampton, VA

3 where f s the average populaton of partcles wth velocty c ( =1, 2,..., B) whch belongs to a predetermned fnte set and ω the relaxaton parameter whch satsfes 0 ω 2. The local equlbrum populaton ( x, t) s chosen as [17], (2.2) ( x, t) =w ρ( x, t), w = 1 B. B s the number of partcles dscrete veloctes. Ths s a homogeneous equlbrum populaton n all velocty drectons. The macroscopc densty, denoted by ρ, s defned by: (2.3) ρ( x, t) = f ( x, t) = ( x, t). The weghtng factor w satsfes the normalzaton constrant: B w = 1. The choce (2.2) for the equlbrum populaton, when used together wth (2.1) and (2.3), wll be shown to lead to the dffuson equaton. We consder models wth the partcle velocty set n D dmenson (D =1, 2 and 3). The smplest models take the velocty set of 2D elements: D drectons along axs and D opposte drectons. The rest partcles can also be ncluded. We assume a weak devaton from the local equlbrum ( x, t), (2.4) f ( x, t) = ( x, t)+ɛf (1) ( x, t)+ɛ 2 f (2) ( x, t)+ where ɛ s the approprate Knudsen number. The space and tme dervatves are expressed n terms of multple-scale varables up to the fourth order n tme (see, for example, Huang [19]), (2.5) (2.6) α = ɛ α t = ɛ t1 + ɛ 2 t2 + ɛ 3 t3 + ɛ 4 t4. When the total mass s conserved, t follows from (2.1), (2.2), (2.3) and (2.4) that, (2.7) f (j) =0, j > 0. Usng the classc Chapman-Enskog expanson and takng nto account of the dscreteness of lattce model, we obtan the frst order equaton n ɛ, (2.8) t1 ρ =0. The second order equaton s, (2.9) t2 ρ c2 2D ( 2 ω 1) ααρ =0. The equatons (2.8) and (2.9),.e., the dynamcal equatons from the two separated tme scales 1/ɛ and 1/ɛ 2, are now reconsttuted to obtan the macro-dynamcal equatons for the model. The equaton of dffuson equaton s obtaned from (2.8) and (2.9) (2.10) t ρ = κ 2 αα ρ where the dffusvty κ 2 s gven by (2.11) κ 2 = c2 2D ( 2 ω 1). 2

4 κ Fg The dsperson relaton (up to fourth order) versus k for the D3Q6 model, The open trangles, sold κ 2 trangles, open squares, sold squares and open crcles are numercal smulatons correspondng to ω =0.75, 1.0, 1.25, 1.5 and 1.75, respectvely. The crtcal value ω cr s 1.0 for ths model. We can also obtan hgher order equatons by carryng the Chapman-Enskog expanson further. We derve the thrd order equaton, (2.12) t3 ρ =0 and the fourth order equaton, (2.13) t4 ρ = A 1 ααββ ρ A 2 αααα ρ. The coeffcents A 1,A 2 and κ 2 n (2.10) for models ncludng rest partcles are obtaned after some algebrac calculatons, (2.14) (2.15) κ 2 = c2 2D ( 2 ω 1) A 1 = c2 D ( 2 ω 2 2 ω )κ 2 (2.16) A 2 = c 2 ( 1 ω ω 1 12 )κ 2. The fnal fourth order equaton s the followng [17], (2.17) t ρ = κ 2 αα ρ A 1 ααββ ρ A 2 αααα ρ. We note that Equaton 2.17 s ansotropc due to the last term. Applyng the Fourer transform exp( Ωt kx) (k s the wavenumber and Ω the frequency) to the above equaton n one-dmensonal space, we get the dsperson relaton whch reads as, κ =1+ κ 4 (2.18) k 2, κ 2 κ 2 3

5 where κ 4 = A 1 + A 2 and κ = Ω k. 2 Numercal result s gven by the Fgure 2.1. The curves correspond to theoretcal results κ/κ 2 whle the ponts correspond to numercal smulatons of the lattce model presented above. Satsfactory agreements n all cases are acheved. The fourth order correctons may have effects n the regme of large Knudsen number,.e., large k and small ω. Equaton 2.18 s vald only for wavevector along x (or y, z) axs, so s the crtcal value ω cr = 1 for the D3Q6 numercal model [12] used for Equaton Hgh Order Dynamcs: Hydrodynamcs. We now turn our attenton to lattce-based hydrodynamcs models. In the LGA, LBE, and LBGK models, both the mass and momentum are conserved. The common features n these models are dscrete velocty space of partcles, evoluton steps of local nteractons and neghbor-to-neghbor propagaton of movng partcles. Snce the prncple of dervng large-scale equatons s the same and outlned n the prevous secton. For the sake of smplcty, we use lattce BGK models to llustrate the exstence of hgh order dynamcs: Burnett-lke equatons. In classc knetc theory, Euler, Naver-Stokes, Burnett and Super-Burnett equatons consttute the successve approxmatons of the Boltzmann equaton n the order of Knudsen number. Lke n classc knetc theory, the lattce-based models for hydrodynamcs use the Chapman-Enskog expanson n order to derve the Naver-Stokes equatons. We outlne the basc ngredents of the dervaton. The tme evoluton equaton s the same as secton 2, except that the equlbrum dstrbuton contans not only mass, but also momentum, (3.1) = t p ρ(1 + c αu α c 2 s + (c αc β c 2 s δ αβ)u α u β 2c 4 ) s where c s s a constant. The densty ρ and velocty u are defned by, (3.2) f = = ρ, c f = c = ρ u whch leads to the constrants on hgh order correctons f (j), (3.3) f (j) =0, c f (j) =0, j > 0. The leadng order on ɛ yelds the nvscd flud equatons, (3.4) (3.5) t1 ρ + α (ρu α )=0 t1 (ρu α )+ β (ρu α u β )= c 2 s αρ and the second order ɛ 2 results n the dsspatve terms, (3.6) (3.7) t2 ρ =0 t2 (ρu α )=ν( ββ (ρu α )+ αβ (ρu β )) where ν s the shear vscosty (ν = c 2 s (1/ω 1/2)). Now, n order to obtan hgh order hydrodynamcal equatons of the lattce-based models, let us look at the thrd order ɛ 3, the Taylor expanson gves the followng equaton, 4

6 (3.8) t3 + c α α f (2) + t1 f (2) + t2 f (1) ( t 1t 2 + t2t 1 +2c α t2α) ( t 1t 1 +2c α t1α + c α c β αβ )f (1) ( t 1t 1t 1 +3c α t1t 1α +3c α c β t1αβ + c α c β c γ αβγ ) = ωf (3). Summng the underlned cross dervatve t1t 2 n the above equaton over, we get a term, t1t 2 (ρ). Usng the frst and second order Equatons , we obtan two dfferent results, (1). f we frst take the dervatve over t 2 then t 1,wehave, (2). Reversely, we have, t2t 1 (ρ) =0. t1t 2 (ρ) = ν α ( ββ (ρu α )+ αβ (ρu β )) It means that the operators are not commutatve, t1t 2 ( ) t2t 1 ( ) where s ether ρ or ρu α. Note that 1 the thrd order macroscopc equatons can be also obtaned by the wavevector expanson (see for example, van Coervorden et al. [20]). Even though the above-mentoned operators are not commutatve, the essental pont n the Equaton 3.8 s the sum of the two terms. After a tedous algebrac calculaton, we get the thrd order equatons, t3 ρ = c2 s (3.9) 6 αββ(ρu α ) t3 (ρu α )= c4 s (3.10) 6 ( 12 ω 2 12 ω +1) αββ(ρ). We check the dsperson relaton up to the thrd order numercally n Fgure 3.1 (the curves are theoretcal predctons wth Equatons and ponts numercal smulatons). Good agreement s obtaned. Even hgher order (fourth and up) dynamcs can be obtaned whle tremendous care has to be taken snce more non-commutatve operators are nvolved and results wll be publshed elsewhere. 4. Concludng Remarks. In ths paper, we ponted out that two condtons determne whether the lattce-based models could or could not have hgher order dynamcs when classcal Chapman-Enskog expanson s used. These condtons are number of conservaton laws and the space and tme dscretzaton. The pure dffuson model, a system wth only one conserved quantty, s frst presented to llustrate that the hgher order dynamcs s allowed. We then turned our attenton to the lattce-based hydrodynamcs equatons. Wth more than one conserved quanttes, we note that specal care must be taken to derve governng equatons for hgher order dynamcs. After notng the feature of no-commutatve cross tme dervatve, we demonstrate how Burnett-lke equatons could be obtaned for lattce-based hydrodynamcs models usng the classc Chapman-Enskog expanson method. The results reported n ths paper can be used to analyze theoretcally systems where hydrodynamc descrpton may break down, a typcal example s smulatons of the mcro-electronc mechancal systems (MEMS) [21,22]. 1 The authors are very grateful to the referee of the Phys. Rev. E for ths and several other mportant observatons. 5

7 Fg The dsperson relaton (up to thrd order): The speed of sound versus k for the D1Q5 model, The open trangles, sold trangles, open squares, sold squares and open crcles are numercal smulatons correspondng to ω =0.75, 1.00, and 1.50 whle the curves are theoretcal predctons. Acknowledgments. Our specal thanks goes to Dr. S.Y. Chen of CNLS at Los Alamos Natonal Laboratory. Part of the work was accomplshed durng a vst of Qan s at the Hong Kong Unversty of Scence and Technology. REFERENCES [1] U. Frsch, B. Hasslacher, and Y. Pomeau, Phys. Rev. Lett. 56 (1986), pp [2] U. Frsch, D. d Humères, B. Hasslacher, P. Lallemand, Y. Pomeau, and J.-P. Rvet, Complex Systems 1 (1987), pp [3] G.D. Doolen, edtor, Lattce Gas Methods for Partal Dfferental Equatons, Addson-Wesley Publshng Company, [4] R. Benz, S. Succ, and M. Vergassola, Phys. Reports 222, No. 3 (1992), pp [5] Y.H. Qan, S. Succ, and S.A. Orszag, Annual Revew of Comp. Phys. Vol. III (1995), pp [6] D. Rothman and S. Zalesk, Lattce Gas Automata, Cambrdge Unversty Press, [7] S.Y. Chen and G.D. Doolen, Annual Revew of Flud Mech. 30 (1998), pp [8] G.R. McNamara and G. Zanett, Phys. Rev. Lett. 61 (1988), p [9] F.J. Hguera and J. Jmenez, Europhys. Lett. 9, No. 7 (1989), pp [10] Y.H. Qan, Lattce Gas and Lattce Knetc Theory Appled to the Naver-Stokes Equaton, PhD thess, Ecole Normale Supéreure and Unversty of Pars 6, [11] P. Bhatnagar, E.P. Gross, and M.K. Krook, Phys. Rev. 94 (1954), p

8 [12] Y.H. Qan, D. d Humères, and P. Lallemand, Europhys. Lett. 17, No. 6 (1992), pp [13] H.D. Chen, S.Y. Chen, and W. Matthaeus, Phys. Rev. A 45 (1992), p. R5339. [14] Y.H. Qan and S.A. Orszag, Europhys. Lett. 21, No. 3 (1993), p [15] S. Chapman and T.G. Cowlng, The Mathematcal Theory of Nonunform Gases. Cambrdge Unversty Press, 3rd edton, [16] B. Hasslacher, R. Kapral, and A. Lawnczak, Chaos 3, No. 1 (1993), p. 7. [17] Y.H. Qan and S.A. Orszag, J. Stat. Phys. 81, No. 1/2 (1995). [18] R. Gatgnol, Théore Cnétque des Gaz àrépartton dscrète devtesses, Volume 36 of Lectures Notes n Physcs, Sprnger-Verlag, [19] K. Huang, Statstcal Mechancs, John Wley, New York, Second Edton, [20] D.V. van Coevorden, M.H. Ernst, R. Brto, and J.A. Somers, J. Stat. Phys. 74 (1994), pp [21] J. Huang, D.H. Feng, and Y.H. Qan, submtted to Phys. Fluds, [22] X.B. Ne, G.D. Doolen, and S.Y. Chen, submtted to Phys. Fluds,