Introduction to the lattice Boltzmann method

Transcription

1 Introducton to LB Introducton to the lattce Boltzmann method Burkhard Dünweg Max Planck Insttute for Polymer Research Ackermannweg 10, D Manz, Germany Introducton Naver-Stokes Buldng an LB Equlbrum populatons Lnear collson Statstcal mechancs B. D. & A. J. C. Ladd, arxv: v2, Advances n Polymer Scence 221, 89 (2009

2 Lattce Boltzmann c small set of veloctes c h connects two stes lnearzed Boltzmann equaton (knetc theory of gases fully dscretzed stes r, lattce spacng a tme t, tme step h n ( r,t: real number, mass densty on ste r correspondng to velocty c Introducton to LB Introducton Naver-Stokes Buldng an LB Equlbrum populatons Lnear collson Statstcal mechancs n ( r + c h,t + h = n ( r,t = n ( r,t + {n ( r,t}

3 Lattce Boltzmann algorthm Introducton to LB ρ = n j = n c, u = j/ρ Π= n c c n eq neq (ρ, u = w ρ = ρ ( 1 + u c c 2 s + ( u c 2 2c 4 s neq c = j neq c c = ρcs 2 + ρ u u lnear relaxaton: = j L j(n j n eq j streamng u2, such that 2cs 2 Introducton Naver-Stokes Buldng an LB Equlbrum populatons Lnear collson Statstcal mechancs remarks: Π relaxaton rates determne vscostes thermal nose: Langevn stress term n Π

4 Goal of ths lecture Introducton to LB Introducton What s the connecton between lattce Boltzmann and Naver Stokes? Why 19 veloctes? Where does the equlbrum dstrbuton come from? What are the magc numbers w? Relaxaton rates vscostes? How to put n nose? Naver-Stokes Buldng an LB Equlbrum populatons Lnear collson Statstcal mechancs

5 Some phlosophcal remarks Introducton to LB n ( r + c h,t + h = n ( r,t = n ( r,t + {n ( r,t} ρ = n j = ρ u = n c = c = 0 Introducton Naver-Stokes Buldng an LB Equlbrum populatons Lnear collson Statstcal mechancs mass conservaton momentum conservaton localty rotatonal symmetry (lattce! Galle nvarance (fnte number of veloctes

6 only u c only u c s Ma = u/c s 1 low Mach number compressblty does not matter equaton of state does not matter choose deal gas! m p partcle mass: p = ρ m p k B T Introducton to LB Introducton Naver-Stokes Buldng an LB Equlbrum populatons Lnear collson Statstcal mechancs c 2 s = p ρ = 1 m p k B T p = ρc 2 s k B T = m p c 2 s

7 Necessty of thermal fluctuatons Ideal gas, temp. T, partcle mass m p, sound speed c s : k B T = m p c 2 s c s a/h (a lattce spacng, h tme step c s as small as possble Example (water: mass densty ρ = 10 3 kg/m 3 sound speed realstc: m/s sound speed artfcal: c s = 10 2 m/s temperature T 300K, k B T = J partcle mass: m p = kg macroscopc scale molecular scale lattce spacng a = 1mm a = 1nm tme step h = 10 5 s h = s mass of a ste m a = 10 6 kg m a = kg Boltzmann Bo = (m p /m a 1/2 Bo = (m p /m a 1/2 number = = 0.6 Introducton to LB Introducton Naver-Stokes Buldng an LB Equlbrum populatons Lnear collson Statstcal mechancs

8 Where we want to get Introducton to LB Introducton t ρ + α j α = 0 Naver-Stokes η αβγδ = t j α + β ( ρc 2 s δ αβ + ρu α u β = β σ αβ σ αβ = η αβγδ γ u δ (ζ 23 η δ αβ δ γδ + η (δ αγ δ βδ + δ αδ δ βγ Buldng an LB Equlbrum populatons Lnear collson Statstcal mechancs η shear vscosty ζ bulk vscosty

9 expanson Mult tme scale analyss: ε 1 (e. g. ε = 10 3 : r 1 = ε r nterpretaton: coarse graned ruler : 1µm nstead of 978nm t 1 = εt nterpretaton: coarse graned clock : 1ns nstead of 837ps t 2 = ε 2 t nterpretaton: yet more coarse graned clock : 1µs nstead of 976ns572ps t 1 to capture wave lke phenomena t 2 to capture dffusve phenomena Introducton to LB Introducton Naver-Stokes Buldng an LB Equlbrum populatons Lnear collson Statstcal mechancs locaton n space and tme: read off r 1, t 1, t 2

10 Introducton to LB r 1 fxed r vares wth ε LBE: r = ε 1 r 1 n = n (0 + εn (1 + O(ε 2 = (0 + ε (1 + ε 2 (2 + O(ε 3 (k = (k c = 0 Introducton Naver-Stokes Buldng an LB Equlbrum populatons Lnear collson Statstcal mechancs n ( r 1 + ε c h,t 1 + εh,t 2 + ε 2 h n ( r 1,t 1,t 2 = ( r 1,t 1,t 2 Taylor expanson wrt ε ε 0 corresponds to macroscopc lmt

11 ε orders Introducton to LB ε 0 : ε 1 : ε 2 : (0 = 0 ( t1 + c r1 n (0 = h 1 (1 t2 n (0 + h 2 ( t 1 + c r1 2 n (0 + ( t1 + c r1 n (1 = h 1 (2 Introducton Naver-Stokes Buldng an LB Equlbrum populatons Lnear collson Statstcal mechancs or t2 n (0 + 1 ( 2 ( t 1 + c r1 n (1 + n (1 = h 1 (2

12 Zeroth order Introducton to LB 0 = (0 = ({n (0 } {n (0 } collsonal nvarant, {n (0 } = n eq no spurous conservaton laws n (0 = n (0 (ρ, j ρ (0 = ρ, j (0 = j n eq = ρ n eq c = j hgher orders: take moments..., c... Introducton Naver-Stokes Buldng an LB Equlbrum populatons Lnear collson Statstcal mechancs

13 Zeroth moment: Mass conservaton Introducton to LB Now, t1 ρ + 1α j α = 0 t2 ρ = 0 α = ε 1α t = ε t1 + ε 2 t2 Introducton Naver-Stokes Buldng an LB Equlbrum populatons Lnear collson Statstcal mechancs Hence, contnuty equaton OK!!!

14 Frst moment: Momentum conservaton Introducton to LB wth t2 j α β t1 j α + 1β Π (0 αβ = 0 Π αβ = ( Π (1 αβ + Π(1 αβ = 0 n c α c β Introducton Naver-Stokes Buldng an LB Equlbrum populatons together: t j α + β Π eq αβ + 1 ( 2 β Π neq αβ + Π neq αβ = 0 Lnear collson Statstcal mechancs wth n neq = εn (1 comparson wth Naver Stokes: Π eq αβ = ρc2 s δ 1 αβ + ρu α u β 2 ( Π neq αβ + Π neq αβ = σ αβ

15 Second moment: A useful relaton wth or ( t1 Π (0 αβ + 1γΦ (0 αβγ = h 1 Π (1 αβ Π(1 αβ Π neq αβ later (from explct form of n eq : Φ αβγ = n c α c β c γ ( Π neq αβ = h t Π eq αβ + γφ eq αβγ Φ eq αβγ = ρc2 s (u αδ βγ + u β δ αγ + u γ δ αβ use contnuty and Euler ( t1!!! for t Π eq αβ = t(ρc 2 s δ αβ + ρu α u β result (neglectng terms O(u 3 : Π neq αβ Π neq αβ = hρc2 s ( α u β + β u α Introducton to LB Introducton Naver-Stokes Buldng an LB Equlbrum populatons Lnear collson Statstcal mechancs

16 Lnear stress relaxaton Introducton to LB later: together wth Π neq αβ Π αβ = Π αβ δ αβπ γγ Π neq αβ Π neq αα = γ s Πneq αβ = γ bπ neq αα Π neq αβ = hρc2 s ( αu β + β u α two equatons wth two unknowns Π neq, Π neq result: 1 ( Π neq 2 αβ + Π neq αβ = σ αβ wth η = hρc2 s γ s ζ = hρc2 s 1 γ s γ b 1 γ b Introducton Naver-Stokes Buldng an LB Equlbrum populatons Lnear collson Statstcal mechancs

17 So, what do we need? 1. n eq such that: n eq (and hopefully = ρ n eq c = j n eq c α c β = ρc 2 s δ αβ + ρu α u β n eq c α c β c γ = ρc 2 s (u α δ βγ + u β δ αγ + u γ δ αβ 2. such that n eq nvarant, and = 0 Π neq αβ = γ neq s Π αβ c = 0 Π neq αα = γ bπ neq αα Introducton to LB Introducton Naver-Stokes Buldng an LB Equlbrum populatons Lnear collson Statstcal mechancs

18 D3Q19 I: Equlbrum populatons ansatz: n eq (ρ, u = w ρ ( 1 + A u c + B( u c 2 + Cu 2 cubc symmetry: w = 1 w c α = 0 w c α c β = σ 2 δ αβ w c α c β c γ = 0 w c α c β c γ c δ = κ 4 δ αβγδ +σ 4 (δ αβ δ γδ + δ αγ δ βδ + δ αδ δ βγ Introducton to LB Introducton Naver-Stokes Buldng an LB Equlbrum populatons Lnear collson Statstcal mechancs

19 ρ, j, Π: n eq soluton: (ρ, u = w ρ ( 1 + u c c 2 s c 2 s = σ 2 + ( u c 2 2c 4 s u2 2cs 2 w = 1 σ 4 = σ2 2 κ 4 = 0 w = 1/3 (zero velocty w = 1/18 (6 nearest neghbors w = 1/36 (12 next nearest neghbors Introducton to LB Introducton Naver-Stokes Buldng an LB Equlbrum populatons Lnear collson Statstcal mechancs c 2 s = (1/3(a/h2 3rd moment OK

20 D3Q19 II: Lnear collson ansatz (n eq nvarant: n neq = j Γ j n neq j modes dagonal representaton: m k = e k n m 0 ρ (m 1,m 2,m 3 j m 4 span(ρ,π αα span(m 5,...,m 9 = span( Π αβ knetc modes m 10,...,m 19 m neq k conserved modes: egenvalue rrelevant: γ 0 =... = γ 3 = 1 bulk stress: γ 4 = γ b shear stresses: γ 5 =...γ 9 = γ s knetc modes: γ k = 0 (easest choce = γ k m neq k Introducton to LB Introducton Naver-Stokes Buldng an LB Equlbrum populatons Lnear collson Statstcal mechancs

21 Statstcal mechancs: Theory of fluctuatons B. D., U. Schller, A. J. C. Ladd, PRE 76, (2007 µ = m p /a 3, ν = n /µ, ν = w ρ/µ P ({ν } ( ν ν e ν δ µ ( ν ρ δ µ ν! Strlng: ν c j Introducton to LB Introducton Naver-Stokes Buldng an LB Equlbrum populatons S ({ν } = (ν lnν ν ν ln ν + ν ( ( Lnear collson Statstcal mechancs P exp[s ({ν }] δ µ ν ρ δ µ ν c j P = max S = max wth constrants soluton (neglectng terms O(u 3 : n eq as before

22 Fluctuaton statstcs Gaussan approxmaton around the mean: n eq P exp ( ρw ( P exp ( n neq 2 2µn eq ( n neq 2µρw 2 recall: unnormalzed modes: δ δ ( ( n neq n neq δ δ ( ( c n neq c n neq Introducton to LB Introducton Naver-Stokes Buldng an LB Equlbrum populatons Lnear collson Statstcal mechancs m k = e k n normalzed modes ˆm k m k : ˆm k = ê k n w µρ wth orthogonal matrx ê k

23 Fluctuaton statstcs n mode space, Monte Carlo Introducton to LB P r k Gaussan wth neq} ({ˆm k exp 1 2 k 4 ˆm neq k = γ k ˆm neq k + ϕ k r k ˆm neq 2 k Introducton Naver-Stokes Buldng an LB Equlbrum populatons Lnear collson r k = 0 r 2 k = 1 Statstcal mechancs detaled balance holds for ϕ k = ( 1 γk 2 1/2

24 Fluctuatng hydrodynamcs Introducton to LB appled to the stochastc term equatons of fluctuatng hydrodynamcs (Landau / Lfshtz 1957 t ρ + α j α = 0 t j α + β ( ρc 2 s δ αβ + ρu α u β = β η αβγδ γ u δ + β Q αβ Introducton Naver-Stokes Buldng an LB Equlbrum populatons Lnear collson Statstcal mechancs Q αβ = 0 Qαβ ( r,t Q γδ ( r,t = 2k B Tη αβγδ δ ( r r δ ( t t