Irina Ginzburg and Dominique dHumires Cemagref, Antony, France, - PowerPoint PPT Presentation

� � Irina Ginzburg and Dominique d’Humières Cemagref, Antony, France, Water and Environmental Engineering École Normale Supérieure, Paris, France Laboratory of Statistical Physics Some elements of Lattice Boltzmann method for hydrodynamic www.cemagref.fr and anisotropic advection-diffusion problems Paris, 20 December, 2006 Slide 1

✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✠ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✠ ✠ ✠ ✡ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✡ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✡ ✠ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✠ ✠ ✡ ☞ ☛ ☛ ☛ ☛ ☛ ☛ ☞ ☞ ✁ ☞ ☞ ☞ ☞ ☞ ✌ ✌ ✌ � ☛ ✠ � ✠ ✠ ✆ ☎ ☎ ✡ ☎ ✝ ☎ ✟ ✁ ✁ ☛ ☎ ✝ ✆ ☎ ✡ ✡ ✠ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✠ ✡ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✡ ✡ ✡ ✡ ✠ ✠ ✠ ✠ ✠ ✡ ✡ ✡ ✡ ✡ ✡ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✡ ✡ ✠ ✠ ✠ ✠ ✠ ✠ ✡ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ U. Frisch, D. d’Humières, B. Hasslacher, P. Lallemand, – Y. Pomeau, and J.P. Rivet, Lattice gas hydrodynamics in two and three dimensions. , Complex Sys. , 1 , 1987 F. J. Higuera and J. Jiménez , Boltzmann approach to lattice – gas simulations. Europhys. Lett. , 9 , 1989 D. d’Humières , Generalized Lattice-Boltzmann Equations , – AIAA Rarefied Gas Dynamics: Theory and Simulations , 59 , 1992 D. d’Humières, I. Ginzburg, M. Krafczyk, P. Lallemand and – ☎✄✞ ✁✄✂ L.-S. Luo , Multiple-relaxation-time lattice Boltzmann models in three dimensions , Phil. Trans. R. Soc. Lond. A 360 , 2005 ☎✄✞ I. Ginzburg, Equilibrium-type and Link-type Lattice – Boltzmann models for generic advection and anisotropic-dispersion equation, Adv Water Resour , 28 , 2005 Slide 2

Lattice Boltzmann two phase calculations in porous media, 2002 Fraunhofer Institut for Industrial Mathematics (ITWM), Kaiserslautern Oil distribution in an anisotropic fibrous material Fleece oil is wetting oil is non-wetting Slide 3

Free surface Lattice Boltzmann method for Newtonian and Bingham fluid I. Ginzburg and K. Steiner , Lattice Bolzmann model for free-surface flow and its application to filling process in casting, J.Comp.Phys. , 185 , 2003 Slide 4

Key points Basic LB method: (1) Linear collision operators (2) Chapman-Enskog expansion Applications: (3) Boundary schemes – Permeability computations in porous media (4) Finite-difference type recurrence equations – Richard’s equations for (5) Knudsen layers variably saturated flow in heterogeneous anisotropic (6) Stability conditions aquifers (7) Interface analysis Slide 5

✎ ✎ ✎ ✍ ✒ ✍ ✎ ✒ ✏ ✏ ☞ ✏ ✎ ✍ ✎ ✒ ✏ ☞ ✎ � ✎ ✒ ✎ ✏ ✒ ✏ ✍ ✒ ✌ ✍ ✎ ☞ ✒ ✏ ✎ ✑ ☞ ✎ ✏ ✌ ✎ ✒ ✎ ✎ ✏ ✏ ✍ ✒ ✎ ✍ ✌ ✏ ✎ ✍ ✍ ✎ ✒ ✏ ✎ ✒ ✎ ✒ ✎ ✍ ✒ ✎ ✎ ✍ ✎ ✒ ✏ ✏ ✎ ✏ ✎ ✍ ✎ ✎ ✒ ✏ ✎ ✎ ✏ ✍ ✎ ✎ ☞ ✒ ✒ ✏ ✏ ✒ ✎ ☞ ✒ ✎ ✏ ✎ ✒ ☞ ✎ ✏ ☎ ✠ ✠ ✠ ✠ ☎ ☎ ☎ ☎ ☎ ☎ ✠ ☎ ✆ ✁ ✁ ✁ ✁ ☞ ✁ ✁ ✠ ✠ ✁ ✄ ✁ ✂ ✄ ☎ ☎ ✂ ✝ ✞ ✁ � ✠ ✂ ✄ ✂ ☎ ☎ ☎ ✂ ✞ ✟ ✁ ✁ ✁ ☞ ☎ ☎ ✡ ☛ � ☞ ✔ ☞ ✏ ☞ ☎ ✒ ☞ ✎ ☞ ☞ ☞ ✎ ☞ ✒ ✍ ☎ ✝ ☎ ✁ ✁ ✁ ✁ ✁ ✟ ☎ ☎ c q q 0 Q 1 Cubic velocity sets ✂✆☎ α c q c q α 1 d – d2Q5 : 0 and d2Q9 1 0 , 0 1 ✍✓✒ c 6 c 2 c 5 ☎✄✞ ✁✄✂ – d3Q7 : 0 and , , 1 0 0 0 1 0 0 0 1 ✍✓✒ – d3Q13 : 0 and c 3 c 0 c 1 1 1 0 , 0 1 1 , 1 0 1 ✍✓✒ – 0 and d3Q15 : , , , 1 0 0 0 1 0 0 0 1 1 1 1 c 7 c 4 c 8 ✍✓✒ ✍✓✒ – d3Q19 : 0 and one rest (immobile): 1 0 0 , 0 1 0 , 0 0 1 , 1 1 0 , 0 1 1 , 1 0 1 c 0 0 0 0 ✍✓✒ ✍✓✒ 1 moving: Q – d3Q27 = d3Q19 d3Q15 , , c q 1 0 0 1 1 1 ✍✓✒ Slide 6

☎ ✞ ✡ ✏ ✄ ✑ ☞ ✍ ✠ ✌ ✟ ✌ ✎ ✁ ✁ ✌ ✝ ✎ ✆ ✌ ✑ � ☞ � ✁ ☎ ✂ ✌ ✌ ✂ ✄ ✝ ✄ ✄ ☞ ✄ ✁ ✂ ✌ ✄ ☛ ☞ ☛ ✌ ✌ ✂ ✎ ✁ ✏ ✁ � ✑ ✎ � ✍ ✑ ☎ ☞ ✁ ✂ ✌ ☎ ✌ ✒ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✂ � ✒ ☞ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✄ ☞ � ✁ ☞ ✂ ✌ ✂ ☞ ✍ ✌ ✎ ☞ ✝ ✌ ✑ ✏ ✎ ✁ ✁ ✁ ✎ ✍ ✑ ☞ ✏ ☞ ✁ ✂ ✌ ✂ ✄ ☞ ✂ ✁ ✄ ✂ ✂ ✂ Multiple-relaxation-time MRT-model Velocity space Moment space ∆ r α 1 , α f 0 c 0 f 0 b 1 1 d Grid space : – ✂✆☎ ∆ t f k c k f k b k 1 (1 update) Time : – f Q c Q f Q b Q 1 1 1 1 f r t f q Population vector: – , q 0 Q 1 Moment (physical) space: – basis vectors b k e r t e q Equilibrium function: – , q 0 Q 1 and eigenvalues λ k , 1 . k 0 Q A q , A f e – Collision q r t Q Q -matrix Projection into moment – ˜ f q r t f q r t Collision q space: ✌✓✒ ∑ Q 1 f k b k , . f f k f b k k 0 ˜ Propagation: f q r c q t 1 f q r t – Collision in moment space: – A f e q 0 λ k ∑ Q 1 b kq . f k e k k Linear stability: – λ k 0 . 2 Slide 7

✑ ✌ ✄ ✑ ✌ ✑ ☞ ✑ ✄ ✑ ✑ ☞ ☞ ✑ ✑ ✝ � ✂ ✄ ✂ ✌ ✑ ✑ ☞ ✑ ✑ ✌ ✑ ☞ ✌ ✑ ✝ ✑ ☞ ✑ ✑ ✄ ✑ ✌ ✑ ☞ ✌ ✑ ✝ � � ☞ ✑ ✁ ✌ ✂ ☞ ✝ ✄ ✒ ✌ ☞ ✂ ✌ ✂ ☎ ☎ ☎ ✑ ✄ ✑ ✂ ✄ ☞ � ✄ ✒ ✂ ✂ ✄ � ✁ ✂ ✂ ✝ ✂ ✁ ✄ ✄ ✌ ✂ ☞ ✁ ✄ ✂ ✌ ✄ MRT basis of d 2 Q 9 model d 2 Q 9 : b k k 1 9 ✂✆☎ b 1 b 2 b 3 b 4 b 5 b 6 b 7 b 8 b 9 1 0 0 4 0 0 0 0 4 1 1 0 1 1 0 2 0 2 b 1 1 q 1 1 1 2 0 1 1 1 1 b 2 c qx 1 0 1 1 1 0 0 2 2 q 1 1 1 2 0 1 1 1 1 b 3 c qy q 1 1 0 1 1 0 2 0 2 3 c 2 c 2 c 2 c 2 1 1 1 2 0 1 1 1 1 b 4 4 q q q qx qy 1 0 1 1 1 0 0 2 2 2 c 2 c 2 1 1 1 2 0 1 1 1 1 b 5 q qx q Eigenvalues λ λ λ λ λ λ λ λ λ b 6 c qx c qy q 0 1 2 1 2 3 3 4 4 3 c 2 b 7 5 c qx q q λ 0 , λ 0 , λ 0 , 3 c 2 0 1 2 b 8 5 c qy q λ ν ξ , λ ν , λ ν . q 1 2 3 1 9 c 4 21 c 2 b 9 8 q q q 2 Slide 8

✄ ☞ ✝ ✁ ✒ ✄ ✁ ☞ ✁ ✝ ✁ ✌ ✁ ☞ ✄ ☞ ✄ ☞ ☞ ☞ ✄ ☞ ✄ ✞ ☞ ✄ ☞ ✌ ☞ ✄ � ✁ ✄ ✄ ✄ ✁ ✁ ☞ ✒ ✝ ✁ ✒ ✄ ✌ � ✁ ✂ ✝ ✄ ✒ ✡ ✠ ✠ ✠ ✠ ✂ ✡ ✡ ✡ ✡ ✡ ✠ ✡ ✡ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✠ ✠ ☞ ✡ ✡ ✠ ✟ ✞ ✌ ✡ ✡ ✡ ✡ ✡ ✄ ✡ ✆ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✌ Link-model LM, (2005) c 6 c 2 c 5 ✡ ☎✝ ✠ ☎✄ c q c ¯ , c q c ¯ Link: – q q f 2 f 6 f 5 – Collision q = p q m q f ¯ f 1 f 0 c 3 c 1 1 f ¯ f ¯ 5 6 f ¯ 2 λ p q f e Symmetric collision part : – q q q c 7 c 4 c 8 Decomposition: – λ Antisymmetric collision part : m q f e – q q q f q f f q q f e q e e q , e q e q Local equilibrium : – q Symmetric part : – 1 f f q f ¯ q q 2 Antisymmetric – 1 part : f f q f ¯ q q 2 Slide 9