Lattice Boltzmann Method for Fluid Simulations

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1 1 / 16 Lattice Boltzmann Method for Fluid Simulations Yuanxun Bill Bao & Justin Meskas Simon Fraser University April 7, 2011

2 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 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 = (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, Figure 2: D2Q9 lattice

4 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 ( 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 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 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 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 ρv f 6 = f (f3 f5) ρu ρv Figure 6: Zou-He velocity boundary condition f 9 = f (f3 f5) ρu 1 2 ρv Other boundary conditons: periodic, free-slip, frictional-slip, sliding walls, the Inamuro method... etc.

8 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 parabolic velocity profile LBM Analytical 10 1 convergence of bounce back boundary conditions mid grid on grid 2nd order 1st order error u(y) N Figure 8: Parabolic velocity profile P = , H = 32, ν = 0.05

9 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 = Stream Trace for Re = x x Figure 9: Stream traces for Re = 400 and The V d = and respectively. Other parameters: ν = 1/18, τ = 2/3, lattice 9 / 16

10 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 11 / 16 Simulation 3: Flow past a Cylinder 5 < Re < 40: A fixed pair of symmetric vortices 40 < Re < 400: Vortex street

12 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 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

14 Convection cells Streamlines (Ra = 20000, t = 8100) 50 yïaxis xïaxis Streamlines (Ra = , t = 5800) 50 yïaxis xïaxis / 16

15 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 / 16 References 1. S. Chen, D. Martínez, and R. Mei, On boundary conditions in lattice Boltzmann methods, J. Phys. Fluids 8, (1996) 2. Q. Zou, and X. He, pressure and velocity boundary conditions for the lattice Boltzmann, J. Phys. Fluids 9, (1997) 3. R. Begum, and M.A. Basit, Lattice Boltzmann Method and its Applications to Fluid Flow Problems, Euro. J. Sci. Research 22, (2008) 4. Z. Guo, B. Shi, and N. Wang, Lattice BGK Model for Incompressible Navier-Stokes Equation, J. Comput. Phys. 165, (2000) 5. Z. Guo, B. Shi, and C. Zheng, A coupled lattice BGK model for the Boussinesq equations, Int. J. Numer. Meth. Fluids 39, (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)

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