Lattice Gas Automata

Transcription

1 a historical overview Jean Pierre Boon Université Libre de Bruxelles DSFD Rome 2010

2 Outline 1 A little bit of history

3 Bénard-von Kàrmànn street Wake behind a plate at Re = 70 (D. d Humières and P. Lallemand, 1985)

4 Discovery in Flow Dynamics May Aid Car, Plane Design By Philip J. Hilts Washington Post Staff Writer The Washington Post (1974-Current file); Nov 19, 1985; ProQuest Historical Newspapers The Washington Post ( ) pg. A1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

5 A new paradigm A virtual simplified micro-world is constructed as an automaton universe based not on a realistic description of interacting particles (as in molecular dynamics simulations) but merely on the conservation laws of microscopic physics and the laws of symmetry of macroscopic physics of fluids. Frisch, Haslacher, Pomeau, 1986 ; Wolfram, 1986 And Feynman...

6 ... told us to explain it like this : We have noticed in nature that the behavior of a fluid depends very little on the nature of the individual particles in that fluid. For example, the flow of sand is very similar to the flow of water or the flow of a pile of ball bearings (*). We have therefore taken advantage of this fact to invent a type of imaginary particle that is especially simple for us to simulate. This particle is a perfect ball bearing that can move at a single speed in one of six directions. The flow of these particles on a large enough scale is very similar to the flow of natural fluids. Physics Today (February, 1989) (*) Reynolds similarity law

7 Hydrodynamics von Kàrmàn vortex street Streamlines at Re = 200 : (a) L. Prandtl, 1934 ; (b) LGA, J.P. Rivet, 1987

8 LGA : reservoir of excitations Hydrodynamic fluctuations : S(k, ω) S(k,!) GHz GHz S(k,!) ! (GHz) ! Liquid Ar, Fleury & Boon, LGA, Grosfils, Boon & Lallemand, 1992.

9 Long time behavior of the velocity autocorrelation function ψ(t) D 1 v 0b 1 d (4π(ν + D D d s)t) D/2 t 1 for D = 2 mode coupling theory and LGA, van der Hoef, Frenkel, Ernst, 1990.

10 The lattice gas automaton Set of particles moving in a space-time discretized universe : a regular D-dimensional lattice L at discrete time steps, t = m t (m N) L : V nodes labeled by D-dimensional position vectors r L each node has b channels (i, j,... = [0, b 1] or [1, b]) At given time t, a channel can be empty or occupied : occupation variable n i (r, t) = 0 or n i (r, t) = 1 If channel i at node r is occupied, there is one and only one particle at node r with velocity c i = Exclusion principle. The set of velocities is such that r + c i t L.

11 LGA dynamics Propagation phase (deterministic and reversible) : particles move according to c i : ni (r + c i t, t + t) = ni (r, t) Collision phase (local) : particles are redistributed according to collision rules = n i (r + c i t, t + t) = n i (r, t) + Ω i ({n i })

12 Conservation laws (collisional invariants) Mass : i n i (r, t) BC = i n i (r, t) AC Momentum : i c i n i (r, t) BC = i c i n i (r, t) AC Energy : i c i 2 n i (r, t) BC = i c i 2 n i (r, t) AC Energy is trivially conserved in single speed LGA (e.g. c i = 1) = non-thermal LGA.

13 2-D square lattice : HPP (*) (*) Hardy, de Pazzis, Pomeau, 1973

14 2-D square lattice gas : (i) spurious invariant (ii) non-isotropic hydrodynamics = 2-D triangular lattice (*) (*) Frisch, Haslacher, Pomeau, 1986

15 2-D triangular lattice : FHP (*) Basic 2- and 3-particle collisions (*) Frisch, Haslacher, Pomeau, 1986

16 FHP-2 Collisions

17 Thermal LGA - GBL (*) model Y C9 C 15 C 8 C 16 C3 C 2 C 14 C 10 C 4 C 0 C 1 C 7 C 17 C 5 C 6 C 13 C C C 18 X (*) Grosfils, Boon, Lallemand, 1992

18 3-D LGA : FCHC (*) : 3-D projection of the F4 lattice (*) d Humières, Lallemand, Frisch, 1986

19 Microdynamics ( Newton s equations for LGA) State of a node (Boolean field) n(r, t) = { ni (r, t) ; r L ; i = 0,..., b 1} ni (r, t) : occupation number of channel i at node r at time t 2 step updating rule : Collision + Propagation = Microdynamical Equation n i (r + c i, t + 1) = n i (r, t) + Ω i ({n j (t)}) Ex : Ω i = n i n i+1 n i+2 n i+3 n i n i+1 n i+2 n i+3

20 Probabilistic automaton : Ω i = s,s (s i s i ) ξs,s j n j ξ s,s : state s s transition selection Boolean variable s j n s j j ;

21 Probabilistic automaton : Ω i = s,s (s i s i ) ξs,s j n j ξ s,s : state s s transition selection Boolean variable s j n s j j ; Mean field dynamics ni (r + c i, t + 1) = ni (r, t) + Ω i ({n j (r, t)}) Def : f i (r, t) = n i (r, t) ne Boltzmann factorization ansatz : Ω i ( n j ) ne = Ω i ( f j ) Correlations before collision are neglected ( Molecular chaos) Transition probabilities : A s,s = ξ s,s ne = Semi-detailed balance : s A s s = s A s s

22 Microdynamics : n i (r + c i, t + 1) = n i (r, t) + Ω i ({n j (t)}) Mean field : n i (r + c i, t + 1) = n i (r, t) + Ω i ({n j (r, t)}) Def : f i (r, t) = n i (r, t) ne ; Factorization : Ω i ( n j ) ne = Ω i ( f j ) = Lattice Boltzmann Equation ( tf i + c i rf i = Coll ) f i (r + c i, t + 1) f i (r, t) = Ω i (f ) Collision term : Ω i = s,s (s i s i ) A s,s j f s j j (1 f j ) 1 s j = H theorem = f eq i = [1 + exp( α β c i 2 )] 1 Fermi-Dirac distribution α chemical potential ; β reciprocal Temperature

23 H-Theorem Entropy h(t * ) t * FHP-1 : h(t ) = P i h i f i (t ) ln f i (t ) + f i (t ) ln f i (t ) (Tribel, Boon, 1997)

24 Macrodynamics Space and time behavior of slow variables Density : ρ(r, t) = i f i(r, t) = i n i (r, t) (per channel : d = ρ/b) Current density : j(r, t) = i c if i (r, t) Energy density : ε(r, t) = 1 2 (mean velocity u = j/ρ) i c i 2 f i (r, t) (+ P i U i)

25 Long-wavelength (kl 0), long-time (ωτ 0) dynamics microscopic characteristic scales : l : lattice unit length ; τ = l c : lattice time scale ; (c = c ) Macroscopic characteristic scales : Macroscopic length scale : Λ l with l Λ = ɛ 1 Macroscopic time scales : (i) Propagation : t s = 1 c s k = Λ c s (ii) Dissipation : t D = 1 ν k 2 = Λ2 ν Λ c = Λ l c = ɛ 1 τ Λ2 lc s l = Λ2 l 2 = multiscale expansion = Macrodynamics l c = ɛ 2 τ

26 Hydrodynamic equations t ρ + (ρ u) = 0 t (ρ u) + (g(ρ) ρ uu) = p + S Pressure : p = c2 D ρ (1 O(u2 /c 2 )) When u c compressibility equation : 1 ρ sound velocity : c s = c D ρ p = D ρ c 2 ; Viscous stress tensor S Symmetry : S αβ = ν s α (ρ u β ) + ( D 2 D ν s + ν B ) α (ρ u α ) ν s = kinematic viscosity ; ν B = bulk viscosity. non-gallilean invariance factor : g(ρ) = D 1 2d D+2 1 d

27 dρ dt = 0 u = 0 Incompressible fluid Scaling : r = 1 L H r ; t = g(ρ 0)u H L H t = t T H ; ũ = 1 1 u H u ; p = p ρ 0 g(ρ 0 ) uh 2 Re = u H L H ν R = u H L H ν(ρ 0 )/g(ρ 0 ) = R L H M a = Ma K n ũ = 0 t ũ + ũ ũ = p + 1 R ũ Turbulent flow Limitations : ν, L H

28 3-D von Kàrmàn street a c U U Three-dimensional incompressible flow around a cylinder in wind tunnel Re = 1.57 R c (a) 74 T H ; (c) 112 T H (FCHC, Rivet, 1993)

29 Hydrodynamic fluctuations LGA : (i) Large number of degrees of freedom (ii) Boolean microscopic nature + Stochastic dynamics = LGA : Reservoir of excitations ( real fluid) δf i (r, t) = f i (r, t) f eq = Linearized Boltzmann Equation δf i (r + c i, t + 1) = δf i (r, t) + j Ω ij δf j (r, t) = δ f i (k, s) = j e s+ık c j e s+ık c j 1 Ω ij δf j (k, t = 0) Resolvent = Eigen vectors and values = LGA modes

30 = Dynamic structure factor S(k, ω) = 2 Re i,j δ f i (k, s = ıω)δf j (k, t = 0) + S 0 (k) S 0 (k) = Pj f eq (1 f eq ) j j Pj f eq j (static structure factor) S(k,!) ! Thermal LGA (19 velocities)

31 Statistical Thermodynamics Equation of state : p = β 1 ρ (1 + B 2 ρ +...) ρ = density per node ; B 2 = 1 2 ρ 2 i f 2 i ; Density fluctuation correlations : Def : δf i (k, t) = r e ık r [f i (r, c i ; t) f ] compressibility : χ = 1 ρ ρ p Static structure factor : ρ S(k) = 1 T T t=1 i,j δf i(k, t)δf j (k, t) LGA (with local collisions) : ρ S 0 (k) = j f eq j (1 f eq j ) j κ(2) j S 0 (k) = ρ β 1 χ 0 (continuous systems : lim k 0 S(k) = ρ β 1 χ 0 )

32 LGA with non-local collisions (*) (Long distance momentum transfer) i+1 6 Q i i+5 R i i+5 i+1 S 6 (b) i i+3 i+1 Q i+5 i R i i+5 i+3 i+1 (a) S i T i i (*) Appert, Zaleski, 1993 ; Tribel, Boon, 1995

33 "pair potential" ( rate of momentum transfer) u(r) = ± γ f 2 (1 f ) 2 F (r) with distribution F(r)(e.g. r z ) u(r) r S(k) = ρ β 1 χ 0 1 γ r κ (3) ; κ (3) f (1 f )(1 2f ) ; γ r NLI interactions

34 Static structure factor : S(k) = 1 + f h(k) ; h(k) = F{(g(r) 1)} S 0 (k) g(r) = exp[ βφ(r)] : pair correlation function ; φ(r) : potential of mean force

35 Static structure factor : S(k) = 1 + f h(k) ; h(k) = F{(g(r) 1)} S 0 (k) g(r) = exp[ βφ(r)] : pair correlation function ; φ(r) : potential of mean force r max = 6, µ= 0.0 r max = 8, µ = 0.0 r max = 10, µ = r max = 6, µ = 0.0 r max = 8, µ = 0.0 r max = 10, µ = g(r) 1.003!(r) r r

36 cs 2 = p ρ = ( ) c2 0 1 γ r κ (3) r cs 2 < 0 = Acoustic modes become unstable = Spinodal decomposition Distribution time steps Density

37 Violation of detailed balance Entropy h est t!t * h(t t ) = P i h i f i (t t ) ln f i (t t ) + f i (t t ) ln f i (t t ) (Tribel, Boon, 1997)

38 Beyond basics Multiphase Fluids Multispecies Fluids Reactive (R-D) systems Turbulent diffusion LGA with dynamical geometry Lattice Boltzmann Equation method Applications

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