1 / 16 Lattice Boltzmann Method for Fluid Simulations Yuanxun Bill Bao & Justin Meskas Simon Fraser University April 7, 2011
2 / 16 Ludwig Boltzmann and His Kinetic Theory of Gases The Boltzmann Transport Equation f t + v f = Ω (1) f( x, t) is the particle distribution function (2) v is the particle velocity (3) Ω is the collision operator Figure 1: Ludwig Boltzmann Gases/Fluids contain a large number of small particles with random motion Interchange of energy through particle streaming and collision Microscopic distribution function Macroscopic gases/fluids variables (pressure, velocity)
3 / 16 Lattice Boltzmann Method f i( x + c e i t, t + t) f i( x, t) }{{} Streaming = eq [fi( x, t) fi ( x, t)] }{{ τ } Collision c = x, lattice speed, t τ is the relaxation parameter, τ = 1 c 2 t ν is the kinematic viscosity ( 3ν + 1 ), 2 f i is the discrete distribution function, i = 1...9 (0, 0) i = 1 e i = (cos[(i 2) π 2 ], sin[(i 2) π ]) i = 2, 3, 4, 5 2 2(cos[(i 6) π + π ], sin[(i 6) π + π ]) i = 6, 7, 8, 9 2 4 2 4 Figure 2: D2Q9 lattice
4 / 16 Lattice Boltzmann Method The Streaming Step Figure 3: Streaming Process The Collision Step (BGK collision operator) f eq i ( x, t) = w iρ( x) [ ei u 1 + 3 + 9 ( e i u) 2 3 c 2 2 c 4 2 where w i is the weights, 4/9 i = 1 w i = 1/9 i = 2, 3, 4, 5 1/36 i = 6, 7, 8, 9 ] u u, c 2
5 / 16 Lattice Boltzmann Method to recover the macroscopic density and velocity, 9 ρ( x, t) = f i( x, t), i=1 u( x, t) = 1 ρ 9 f i e i i=1 Finite Difference Perspective f i( x, t + t) f i( x, t) t = fi( x + ei x, t + t) fi( x, t + t) + x eq fi( x, t) fi ( x, t) τ In our case t = x = 1. This recovers the Lattice Boltzmann Method.
6 / 16 Boundary Conditions: Bounce-back Equivalent to no-slip boundary condtions Figure 4: Illustration of on-grid bounce-back On-grid 1st order Mid-grid 2nd order Easy to implement for complex geometries Applicable to flows with impermeable walls Figure 5: Illustration of mid-grid bounce-back
7 / 16 Boundary Conditions: Zou-He Given the velocity u L = (u, v) on the left boundary, ρ = 1 [f1 + f3 + f5 + 2(f4 + f7 + f8)] 1 u f 2 = f 4 + 2 3 ρv f 6 = f 8 1 2 (f3 f5) + 1 6 ρu + 1 2 ρv Figure 6: Zou-He velocity boundary condition f 9 = f 7 + 1 2 (f3 f5) + 1 6 ρu 1 2 ρv Other boundary conditons: periodic, free-slip, frictional-slip, sliding walls, the Inamuro method... etc.
y 8 / 16 Simulation 1: Plane Poiseuille flow Figure 7: Illustration of a Poiseuille flow Time independent flow driven by a pressure gradient P = P 1 P 0 Periodic BCs at the inlet and outlet of the flow No-slip BCs on the solid walls 35 30 parabolic velocity profile LBM Analytical 10 1 convergence of bounce back boundary conditions mid grid on grid 2nd order 1st order 25 10 2 20 error 10 3 15 10 10 4 5 0 0 1 2 3 4 5 6 7 8 9 10 u(y) 10 5 10 0 10 1 10 2 N Figure 8: Parabolic velocity profile P = 0.0125, H = 32, ν = 0.05
y y Simulation 2: Lid Driven Cavity 2D fluid flow driven by a top moving lid No-slip (bounce-back) BCs on the other three stationary walls Zou-He BCs on the moving lid 1 Stream Trace for Re = 400 1 Stream Trace for Re = 1000 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 0.2 0.4 0.6 0.8 1 x 0 0 0.2 0.4 0.6 0.8 1 x Figure 9: Stream traces for Re = 400 and 1000. The V d = 0.0868 and 0.2170 respectively. Other parameters: ν = 1/18, τ = 2/3, 256 256 lattice 9 / 16
10 / 16 Simulation 3: Flow past a Cylinder No-slip BCs on the solid walls and cylinder Zou-He velocity and density BCs at the inlet and outlet Regimes of the Flow Re < 5: Laminar flow, no separation of streamlines
11 / 16 Simulation 3: Flow past a Cylinder 5 < Re < 40: A fixed pair of symmetric vortices 40 < Re < 400: Vortex street
12 / 16 Simulation 3: Flow past a Cylinder Figure 10: Vorticity plot of flow past a cylinder at Re = 150, a Karman vortex street is generated
13 / 16 Simulation 4: Rayleigh-Bénard Convection Nondimensional Boussinesq Equations u = 0 u + u u = P r u + Ra P rt ẑ p t T + u T = T Figure 11: Illustration of t Rayleigh-Bénard convection Ra: Rayleigh number, P r: Prandtl number A D2Q9 model for u and a D2Q5 model for T, and the two models are combined into one coupled model for the whole system BCs on u: No-slip (bounce-back) BCs on the top/bottom walls, periodic BCs on the two vertical walls BCs on T : Zou-He BCs on the top/bottom walls, periodic BCs on the two vertical walls
Convection cells Streamlines (Ra = 20000, t = 8100) 50 yïaxis 40 30 20 10 20 40 60 80 100 xïaxis 120 140 160 180 200 140 160 180 200 Streamlines (Ra = 2000000, t = 5800) 50 yïaxis 40 30 20 10 20 40 60 80 100 xïaxis 120 14 / 16
15 / 16 Summary Features of Lattice Boltzmann Method A celluar automata model, as well as a special FD method for Boltzmann equation Errors are 2nd order in space Very successful for simulating multiphase/multicomponent flows Simulating flows with complex boundary conditions are much easier using LBM (porous media flow) LBM can be easily parallelized A Controversy The compressible Navier-Stokes equations (NSEs) can be recovered from LBM through Chapman-Enskog expansions A method with artificial-compressibilty for the incompressible NSEs Some other LBMs have been developed for modelling the incompressible NSEs in the incompressible limit
16 / 16 References 1. S. Chen, D. Martínez, and R. Mei, On boundary conditions in lattice Boltzmann methods, J. Phys. Fluids 8, 2527-2536 (1996) 2. Q. Zou, and X. He, pressure and velocity boundary conditions for the lattice Boltzmann, J. Phys. Fluids 9, 1591-1598 (1997) 3. R. Begum, and M.A. Basit, Lattice Boltzmann Method and its Applications to Fluid Flow Problems, Euro. J. Sci. Research 22, 216-231 (2008) 4. Z. Guo, B. Shi, and N. Wang, Lattice BGK Model for Incompressible Navier-Stokes Equation, J. Comput. Phys. 165, 288-306 (2000) 5. Z. Guo, B. Shi, and C. Zheng, A coupled lattice BGK model for the Boussinesq equations, Int. J. Numer. Meth. Fluids 39, 325-342 (2002) 6. S. Succi, The Lattice Boltzmann Equation for Fluid Dynamics and Beyond. Oxford University Press, Oxford. (2001) 7. M. Sukop and D.T. Thorne, Lattice Botlzmann Modeling: an introduction for geoscientists and engineers. Springer Verlag, 1st edition. (2006)