Lattice Boltzmann Method and its Application to Flow Analysis in Porous Media

Transcription

1 Specal Issue Multscale Smulatons for Materals 7 Research Report Lattce Boltzmann Method and ts Applcaton to Flow Analyss n Porous Meda Hdemtsu Hayash Abstract Under the exstence of an external force, a lattce Boltzmann method (LBM) s derved by dscretzng the Boltzmann equaton wth respect to velocty, space and tme. The LBM s appled to smulatons of flow through three-dmensonal porous structures of Nafon polymer membranes. Geometry data of Nafon are constructed based on the result of a dsspatve partcle dynamcs smulaton for three values of water content, %, %, and 3%, and are used as the geometry nput for the LBM. Usng Darcy's law, the permeablty of the porous structure s extracted from the results obtaned by two knds of LBM, the LBM under an external force and the LBM under a pressure gradent. The two types of LBM are found to produce permeabltes that are n good agreement wth each other. Keywords Computer smulaton, Flud dynamcs, Porous meda, Lattce Boltzmann method R&D Revew of Toyota CRDL Vol. 3 No.

2 . Introducton, ) has The Lattce Boltzmann method (LBM) recently attracted consderable attenton n the antcpaton of smulatng flud flows n porous meda and n multphase states, whch are dffcult problems to solve by conventonal computatonal flud dynamcs technques. Ths method and ts predecessor, the Lattce- Gas method (LGM), were found to be easly appled to flud flows n porous meda mmedately after ther dscovery. 3) Buckles et al. ) and Auzeras et al. 5) nvestgated flow through porous rocks usng the LBM and calculated the permeablty of these rocks ),.e. the fundamental physcal quantty of porous meda. They found the calculated value of permeablty to be n good agreement wth that obtaned expermentally. In the present study, we apply the LBM to flow analyss of Nafon polymer membranes, the bestknown proton conductng electrolyte of fuel cells. The greatest problem n applyng the LBM s determnaton of the structure of the porous meda. Two possble approaches to ths problem are: () constructng a three-dmensonal dgtal mage of the pore space usng X-ray computer tomography (CT); () mmckng by smulaton the manufacturng process used to create the actual porous materal. We cannot apply the frst approach to Nafon polymer membranes, because the resoluton of X-ray CT, a few µm at best, s nsuffcent to resolve the nanoscale pore structure of Nafon. Hence, we adopt the second approach, and fabrcate the morphology of Nafon polymer membranes usng a dsspatve partcle dynamcs (DPD) technque. 7, ). Lattce Boltzmann method Hstorcally, the Lattce Boltzmann method developed from the Lattce-Gas method. 9) Recently, the LBM s proved to be a specal dscretzed form of the contnuous Boltzmann equaton (CBE)., ) We derve an LBM wth an external force from the CBE n Sec... We formulate the method of applyng the LBM to porous meda n Sec.... From Boltzmann equaton to LBM For the case n whch an external force f exsts, the contnuous Boltzmann equaton wth the BGK approxmaton ) for the collson term s wrtten as f t + e r f + f e f = τ (f ), () where f s the dstrbuton functon of a spatal coordnate r and a velocty e, and τ s the relaxaton tme. The local equlbrum dstrbuton functon s descrbed by the Maxwell dstrbuton: ] ρ (e u) = exp. () (πrt) D RT Here, R s the gas constant; T s the temperature; D s the spatal dmenson. The local densty ρ and the local velocty u are calculated as the moments of the dstrbuton functon: ρ = fde, (3) ρu = ef de. () Snce n ths study we are consderng the case n whch devaton from the local equlbrum s small, the dervatve of the dstrbuton functon wth respect to the velocty n Eq.() can be approxmated as 3, ) e f e (e u) = RT. (5) Consequently, we obtan the followng equaton as the startng pont for dervng the LBM: f t +e f (e u) f = r RT τ (f ). () The LBM wth an external force s derved through the dscretzaton of Eq.() wth respect to the velocty, spatal and tme coordnates.,, 5 7) We frst execute the dscretzaton n the velocty space: f t +e r f f (e u) RT = τ (f ), (7) R&D Revew of Toyota CRDL Vol.3 No.

3 9 where e s the dscrete veloctes, the dstrbuton functon f and the local equlbrum dstrbuton functon are defned as f w (πrt) D exp e RT ] f (r, e ), () ] w (πrt) D e exp (u, e ), RT (9) ( u e u ) ] = ρw exp, () RT ] ρw + e u RT + (e u) (RT ) u ; RT () and w are the weght coeffcents. The dscrete veloctes e and the weght coeffcents w are determned so as to recover the Naver-Stokes equaton from the LBM n the longwage-length and low-frequency lmt. Therefore, the necessary condtons mposed on e and w are as follows: ) w =, () w e α e β δ α,β, (3) w e α e β e γ e δ () e = (δ α,β δ γ,δ + δ α,γ δ β,δ + δ α,δ δ β,γ ). the nne- For the two-dmensonal problem, velocty model (DQ9): (, ), ( =) w = (±, )c, (, ±)c, ( =,..., ) (±, ±)c, ( =5,..., ) /9, ( =) /9, ( =,..., ) /3, ( =5,..., ) (5) () s frequently used. For the three-dmensonal problem, several velocty models have been proposed, ncludng the ffteen-velocty model (D3Q5), the nneteenvelocty model (D3Q9), and the twenty-sevenvelocty model (D3Q7). 9) We wll use the D3Q5 model n a latter secton, where e and w are gven by (,, ), ( =) (±,, )c, (, ±, )c, e = (7) (,, ±)c, ( =,.., ) (±, ±, ±)c, ( =7,..., ) w = /9, ( =) /9, ( =,.., ) /7, ( =7,..., ). In Eqs. (5,7), c s defned by () c 3RT. (9) We can easly prove that the followng relatons are satsfed for every par of e and w n Eqs.(5,) and Eqs.(7,): w =, () w e α e β = 3 c δ α,β, () w e α e β e γ e δ () = 9 c (δ α,β δ γ,δ + δ α,γ δ β,δ + δ α,δ δ β,γ ). Thus, we have confrmed that these pars of e and w obey the condton of Eqs.(,3,). We requre that the Boltzmann equaton dscretzed wth respect to the velocty coordnate, Eq.(7), satsfes conservaton of mass and momentum at each spatal pont: f = = ρ, (3) e f = e = ρu. () We have used Eqs.(,,) to derve the sum rules for n Eqs.(3,). Recently, the Boltzmann equaton dscretzed wth respect to the velocty coordnate has been solved drectly by a fnte dfference method. ) In the next step, we perform the dscretzaton of Eq.(7) wth respect to the spatal and tme coordnates. Let us rewrte Eq.(7) as follows: f t + e r f = τ (f ) F, (5) F f (e u). () RT R&D Revew of Toyota CRDL Vol.3 No.

4 Integratng both sde of Eq.(5) from t to t + δ t, the left-hand sde s reduced to t+δ t ( f t + e ) r f dt (7) t = f (x + e δ t,t+ δ t ) f (x,t), and the rght-hand sde becomes t+δ t t ( τ (f ) F )dt = δ t + δ t τ (f ] ) F () t+δt ] τ (f t ) F + O(δt 3 ) n the second order approxmaton wth respect to δ t. We have used the trapezodal rule to derve Eq.() and then obtan f (x + e δ t,t+ δ t ) f (x,t) τ (f = δ t + δ t ] ) F (9) t+δt ] τ (f t ) F + O(δt 3 ). To translate Eq.(9) nto the explct form, we ntroduce f defned by f f + δ t τ (f )+ δ t F. (3) Usng f, Eq.(9) s expressed as f (x + e δ t,t+ δ t ) f (x,t) = δ t ] τ (f t ) F (3) δ t ( = f τ +.5δ t ) τδ t F. τ +.5δ t (3) Introducng the dmensonless relaxaton tme τ through τ = τ δ t +.5, (33) Eq.(3) becomes f (x + e δ t,t+ δ t ) f (x,t) = τ ( ) f ( τ.5) f (e u) + δ t. τ RT (3) Through the use of Eqs.(,,3,,,3), we confrm that f satsfes the followng relatons: f = f = = ρ, (35) e f = e f + δ t e F = e + δ t e F = ρu δ t ρf. (3) The lattce Boltzmann method wth an external force f s composed of Eqs.(3,35,3) and Eq.(). From the defnton of τ, Eq.(33), we fnd that τ must satsfy the constrant τ >.5. Pror to the works of He et al.,, 3), the LBM wth an external force had been proposed based on emprcal studes, and recently a few papers on ths subject have been publshed., ) However, as we have examned n ths secton, the LBM wth an external force can be derved consstently from the contnuous Boltzmann equaton n the deductve method.. Applcaton of LBM to porous meda The LBM derved n Sec.. takes the densty and the velocty as ndependent varables. To smulate flud flow n porous meda, we utlze an LBM for ncompressble flud, 3) n whch pressure and velocty are ndependent varables. Ths LBM s convenent for confrmng the conservaton of flow late, whch, for an ncompressble flud, must be constant over a porous meda. The LBM takng the velocty u and the pressure p as ndependent varables s: f (x + e δ t,t+ δ t ) f (x,t) = τ (f ) τ.5 δ t F, (37) τ where we have expressed the dmensonless relaxaton tme ntroduced n Eq.(33) by τ. The local equlbrum dstrbuton functon and the ex- R&D Revew of Toyota CRDL Vol.3 No.

5 ternal force term F are: = w p + e u RT + (e u) (RT) ] u, RT (3) F = f (e u), (39) RT p = c s ρ p = f, () u = e f +.5fδ t, () where p s the pressure, p s the dmensonless pressure, c s s the sound velocty, and ρ s the densty of the ncompressble flud. Usng the Chapman-Enskog expanson, ) we can derve the macroscopc hydrodynamc equatons from the LBM for an ncompressble flud n the regon of low Knudsen number and low Mach number: 3) p c s ρ + u =, () t u t + u u = p + u + f. (3) ρ In all of the velocty models (DQ9, D3Q5, D3Q9, D3Q7) mentoned n Sec.., the knematc vscosty and the sound velocty c s are gven by = τ c δ t ; c s = c. () 3 For the case n whch the external force f s conservatve, the external force s gven usng the potental Ω: f = Ω. (5) Then, Eq.(3) becomes u t + u u = (p + ρ Ω) + u. () ρ Equatons (3,) ndcate that for the flud flow generated by a constant pressure gradent p, such as the Poseulle flow, the flud velocty u s the same as that drven by the followng constant external force f: f = p = c ρ s p = c p. (7) 3 Permeablty, the fundamental physcal quantty of porous meda, s defned usng Darcy s law: ) u = K µ ( p ρ f), () where u s the flud velocty, s the average over the porous meda, p s the pressure gradent, ρ f s the external force operatng on the unt volume of the flud, and µ s the vscosty related to the knematc vscosty through = µ/ρ. As we wll see n the next secton, the pressure n porous meda shows a complex poston dependence and ts gradent s not constant. Nevertheless, Darcy s law as gven n Eq.() ndcates that the averaged flud velocty generated by the averaged pressure gradent p s the same as that derved usng the constant external force f defned n Eq.(7). We wll verfy the dentty between the effect of the pressure gradent and that of the external force stated n the Darcy s law through calculatons of permeablty n Sec.3. For the purpose of numercal calculatons n the next secton, t s convenent to ntroduce the dmensonless permeablty, whch s related to the permeablty K for porous meda havng the shape of a cube wth sde length L c : = L K = c 3ũ Re p, (9) where the dmensonless velocty ũ, the Reynolds number Re, and the dmensonless pressure dfference p are defned by ũ = u c, Re = ul c, p = L c p. (5) Hereafter, we use the lattce unt defned through c = δ t =. 3. Flow analyss n porous meda In ths secton, we apply the LBM developed n Sec. to three-dmensonal porous structures of Nafon polymer membranes. The structures of Nafon constructed as a result of a dsspatve partcle dynamcs smulaton ) are shown n Fg. for three values of water content, %, %, and 3%. Polymer molecules of Nafon 7 occupy gray R&D Revew of Toyota CRDL Vol.3 No.

6 regons n Fg.. A cube surroundng the gray regon, whose edge length s 5nm, s dvded nto a voxel lattce, and the vertces of the voxels are used as the lattce ponts of the LBM. (a) % (b) % (c) 3% (a) (b) (c) (a ) % (b ) % (c ) 3% Fg. Structure of Nafon obtaned by the DPD smulaton for three values of water content, vol % (a), vol % (b), and 3 vol % (c). We mpose the bounce back boundary condton (BBBC) 5) at the lattce ponts occuped by the polymer molecules. At the nlet and outlet, we prepare a runway havng a wdth of lattce ponts along the pressure gradent or the external force. The nflow and outflow boundary condtons are constant pressure on the nlet and outlet plane 5) for the flow generated by the pressure gradent, and the nlet and the outlet are connected by the perodc boundary condton for the flow drven by the external force. In ths study, we adopted the ffteen-velocty model (D3Q5) defned n Sec... Hereafter, the flow generated by the pressure gradent and the flow drven by the external force are desgnated as FGPG and FDEF, respectvely. The calculated flud velocty vectors are shown n Fg.. The magntude of the velocty vector s represented by color, where the hghest velocty s red and the lowest velocty s blue. Fgures (a,b,c) are the results for FGPG, and Fgs. (a,b,c ) are those for FDEF. In these fgures, a quarter of the regon occuped by polymer molecules s removed to dsplay the flud velocty. In ths calculaton, the knematc vscosty s.5; the pressure dfference between the nlet and outlet, p, s.5 for FGPG; the magntude of the external force for FDEF s estmated from the relaton: f =3Lf = p, where L(= ) s the number of lattce ponts along the flow drecton. From these fgures, we can see that the result for FGPG and that for FDEF are n good agreement. Fgure 3 shows the calculated pressure dstr- Fg. Calculated velocty vectors n Nafon. The magntude of the velocty vector s represented by color, where the largest velocty s red and the lowest velocty s blue. Results are compared for the flow generated by the pressure gradent (a,b,c) and the flow drven by the external force (a,b,c ). =.5, p = f =.5. buton n the porous meda for FGPG, whch s averaged on the plane perpendcular to the flow drecton. The knematc vscosty s.5 and the pressure dfference between the nlet and outlet, p, s.5. From ths fgure, we fnd that: () almost all of the pressure change (the pressure-loss) occurs nsde the porous meda; () the pressure can be consdered as constant at the runway; (3) the pressure dstrbuton shows a complex change along the flow drecton nsde the porous meda. Fg % % 3% 3 5 Dstance from the nlet Calculated pressure dstrbuton averaged on the plane perpendcular to the flow drecton n Nafon for three values of water content. =.5, p =.5. Fgure shows the velocty dstrbuton averaged on the plane perpendcular to the flow drecton. In ths fgure, results are compared for FGPG (a) and FDEF (b). Parameters of the flow R&D Revew of Toyota CRDL Vol.3 No.

7 3 are as follows: =.5 n both Fg. (a) and Fg. (b); p =.5 n Fg. (a) where p s the dfference n pressure between the nlet and the outlet; f =3Lf = p n Fg. (b). Fgure shows that the result for FGPG s n good agreement wth that for FDEF and the average flow velocty s constant n the regon between the nlet and the outlet. The constant velocty ndcates conservaton of flow rate, and the LBM satsfes one of the necessary constrants on the ncompressble flow. -3 (a) -3 (b) 5 % % 5 % % 3% 3% u~ ~ 3 u Dstance from the nlet Dstance from the nlet Fg. Calculated velocty dstrbuton averaged on the plane perpendcular to the flow drecton. Results are compared for the flow generated by the pressure gradent (a) and the flow drven by the external force (b). =.5, p = f =.5. Usng the calculated velocty ũ, we can estmate the permeablty from Eq.(9). Fgures 5(a) and 5(b) show the calculaton results for the dmensonless permeablty. Fgure 5(a) shows the dependence of on the pressure dfference between the nlet and the outlet. Fgure 5(b) shows the dependence of on the magntude of the external force. In ths fgure, the abscssa f s defned through the relaton: f =3Lf(= p). In these fgures, the knematc vscosty s fxed at =.. Two results for FGPG and FDEF are n good agreement. The permeablty must be ndependent of the pressure gradent or the external force used n the calculaton thereof, because, accordng to Darcy s law, the permeablty s an ntrnsc quantty of each porous meda. As shown n Fg. 5(a) and Fg. 5(b), the LBM produces the correct result for ths requrement for the calculated permeablty. Changng the magntude of the knematc vscosty, we have calculated the dmensonless permeablty, and the result s gven n Fg.. The -5 (a) -5 (b) % 5 % 5 3% 3 Fg p ~ 3 % % 3%... ~ f Dependence of dmensonless permeablty on the pressure gradent and the external force. Results are compared for the flow generated by the pressure gradent (a) and the flow drven by the external force (b). =.. dmensonless pressure dfference p n Fg. (a) and the magntude of the external force n Fg. (b) are fxed under the relatonshp: p = f =.. Here, the results for FGPG n Fg. (a) and for FDEF n Fg. (b) are n good agreement, and the calculated permeablty depends strongly on the knematc vscosty of the flud. Accordng to Darcy s law, the second tem s unphyscal, because the permeablty s an ntrnsc qualty of the porous meda and must be ndependent of the knematc vscosty of the flud used n the calculaton. -5 (a) -5 (b) % % 3% Fg % % 3% Dependence of dmensonless permeablty on the flud vscosty. Results are compared for the flow generated by the pressure gradent (a) and the flow drven by the external force (b). p = f =.. The problem wheren the permeablty vares wth the flud vscosty has been nvestgated by Ferréol et al. ), and has been nterpreted to orgnate from nsuffcent resoluton of the underlng lattce of the LBM. In order to confrm ths nterpretaton, we ncreased the resoluton by preparng a fne grd, n whch each gray regon n Fg. s dvded nto an voxel lattce, and cal- R&D Revew of Toyota CRDL Vol.3 No.

8 culated the permeablty usng the fne grd. The result s shown n Fg. 7. We understand from ths fgure that the vscosty dependence decreases wth the ncrease n the grd resoluton, and the nterpretaton mentoned above s confrmed. In addton, Fg. 7 ndcates that the dependence of the permeablty on the grd resoluton decreases as the vscosty decreases. Fg. 7 x -5 x -5 (a) % xx xx x -5 (b) % xx xx (c) 3% xx xx Dependence of dmensonless permeablty on the flud vscosty and the grd resoluton. The water content of Nafon s % for (a), % for (b), 3% for (c). p =... Conclusons We have developed the LBM wth an external force for ncompressble flud, n whch the ndependent varables are pressure and velocty. Usng ths LBM, we can mpose the perodc boundary condton on the nlet and outlet of the flow drven by the external force. Ths s an advantage of the LBM wth an external force, because we can easly code the perodc boundary condton to be applcable to any velocty model, whle the flud mechancal boundary condton must be prepared for each velocty model. In addton, we have performed smulatons of flow through threedmensonal porous structures of Nafon polymer membranes usng two knds of LBM,.e. LBM wth and wthout an external force. The two LBMs produce permeabltes that are n good agreement wth each other, and the LBM s confrmed to reproduce Darcy s law. References ) Chen, S. and Doolen, G.D.: Annu. Rev. Flud Mech., 3(99), 39 ) Succ, S.: The Lattce Boltzmann Equaton for Flud Dynamcs and Beyond, (), Oxford Unversty Press 3) Rothman, D.H. and Zalesk, S.: Rev. Mod. Phys., (99), 7 ) Buckles, J.J., et al.: Los Alamos Scence, (99), 3 5) Auzeras, F.M., et al.: Geophys. Res. Lett., 3(99), 75 ) See for example, Bear, J.: Dynamcs of Fluds n Porous Meda, (9), Dover Publcatons; Sahm, M.: Rev. Mod. Phys., 5(993), 393 7) Frenkel, D. and Smt, B.: Understandng Molecular Smulaton, (), 5, Academc Press ) Yamamoto, S. and Hyodo, S.: Kobunsh-Gakka Yokoshu (n Japanese), 5-3 (), 37, Jpn. Soc. Polymer Sc. 9) McNamara, G.R. and Zanett, G.: Phys. Rev. Lett., (9), 33 ) He, X. and Luo, L.-S.: Phys. Rev., E55(997), R333 ) He, X. and Luo, L.-S.: Phys. Rev., E5 (997), ) Bhatnagar, P.L., et al.: Phys. Rev., 9(95), 5 3) He, X., et al.: Phys. Rev., E57(99), R3 ) Martys, N.S., et al.: Phys. Rev., E5(99), 55 5) Sterlng, J.D. and Chen, S.: J. Compt. Phys., 3(99), 9 ) Abe, T.: J. Compt. Phys., 3(997), 7) Shan, X. and He, X.: Phys. Rev. Lett., (99), 5 ) Wolf-Gladrow, D.A.: Lattce-Gas Cellular Automata and Lattce Boltzmann Models An Introducton, (), Sprnger-Verlag, New York R&D Revew of Toyota CRDL Vol.3 No.

9 5 9) Qan, Y.H., et al.: Europhys. Lett. 7(99), 79 ) Me, R.W. and Shyy, W.: J. Comput. Phys., 3(99), ) Buck, J.M. and Created, C.A.: Phys. Rev., E(), 537 ) Guo, Z., et al.: Phys. Rev., E5(), 3 3) He, X. and Luo, L.-S.: J. Stat. Phys., (997), 97 ) See for example, Chapman, S.and Cowlng, T.G.: The Mathematcal Theory of Non-Unform Gases, Thrd Edton, (97), Chap.7, Cambrdge Unversty Press, or, Cercgnan, C.: The Boltzmann Equaton and Its Applcatons, (9), Chap.V., Sprnger-Verlag, New York 5) Zou, Q. and He, X.: Phys. Fluds, 9(997), 59 ) Ferréol, B. and Rothman, D.H.: Transport n Porous Meda, (995), 3 (Report receved on December, ) Hdemtsu Hayash Year of brth : 95 Dvson : Dgtal Engneerng Lab. Researc h felds : Computatonal flud mechancs Academc degree : Dr. Sc. Academc socety : Ph ys. Soc. Jpn. R&D Revew of Toyota CRDL Vol.3 No.