Lattice Boltzmann Method for Fluid Simulations

Yuanxun Bill Bao & Justin Meskas

Simon Fraser University

April 7, 2011

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Ludwig Boltzmann and His Kinetic Theory of Gases

The Boltzmann Transport Equation

∂f

∂t+ ~v · ∇f = Ω

(1) f(~x, t) is the particledistribution function

(2) ~v is the particle velocity

(3) Ω is the collision operator

Figure 1: Ludwig Boltzmann

I Gases/Fluids contain a large number of small particles with random motion

I Interchange of energy through particle streaming and collision

I Microscopic distribution function ←→ Macroscopic gases/fluids variables(pressure, velocity)

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Lattice Boltzmann Method

fi(~x+ c~ei∆t, t+ ∆t)− fi(~x, t)︸ ︷︷ ︸Streaming

= − [fi(~x, t)− feqi (~x, t)]

τ︸ ︷︷ ︸Collision

I c =∆x

∆t, lattice speed,

I τ is the relaxation parameter, τ =1

c2∆t

(3ν +

1

2

),

ν is the kinematic viscosity

I fi is the discrete distribution function, i = 1...9Figure 2: D2Q9 lattice

I ~ei =

(0, 0) i = 1(cos[(i− 2)π

2], sin[(i− 2)π

2]) i = 2, 3, 4, 5√

2(cos[(i− 6)π2

+ π4

], sin[(i− 6)π2

+ π4

]) i = 6, 7, 8, 9

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Lattice Boltzmann Method

The Streaming Step

Figure 3: Streaming Process

The Collision Step (BGK collision operator)

feqi (~x, t) = wiρ(~x)

[1 + 3

~ei · ~uc2

+9

2

(~ei · ~u)2

c4− 3

2

~u · ~uc2

],

where wi is the weights,

wi =

4/9 i = 11/9 i = 2, 3, 4, 51/36 i = 6, 7, 8, 9

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Lattice Boltzmann Method

I to recover the macroscopic density and velocity,

ρ(~x, t) =9∑i=1

fi(~x, t), ~u(~x, t) =1

ρ

9∑i=1

fi~ei

Finite Difference Perspective

fi(~x, t+ ∆t)− fi(~x, t)∆t

+fi(~x+ ~ei∆x, t+ ∆t)− fi(~x, t+ ∆t)

∆x

= −fi(~x, t)− feqi (~x, t)

τ

I In our case ∆t = ∆x = 1. This recovers the Lattice Boltzmann Method.

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Boundary Conditions: Bounce-back

Figure 4: Illustration of on-grid bounce-back

Figure 5: Illustration of mid-grid bounce-back

I Equivalent to no-slipboundary condtions

I On-grid — 1st orderMid-grid — 2nd order

I Easy to implement forcomplex geometries

I Applicable to flows withimpermeable walls

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Boundary Conditions: Zou-He

Figure 6: Zou-He velocity boundary

condition

Given the velocity ~uL = (u, v) on the leftboundary,

ρ =1

1− u [f1 + f3 + f5 + 2(f4 + f7 + f8)]

f2 = f4 +2

3ρv

f6 = f8 −1

2(f3 − f5) +

1

6ρu+

1

2ρv

f9 = f7 +1

2(f3 − f5) +

1

6ρu− 1

2ρv

I Other boundary conditons: periodic, free-slip, frictional-slip, sliding walls,the Inamuro method . . . etc.

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Simulation 1: Plane Poiseuille flow

Figure 7: Illustration of a Poiseuille flow

I Time independent flowdriven by a pressuregradient ∆P = P1 − P0

I Periodic BCs at the inletand outlet of the flow

I No-slip BCs on the solidwalls

0 1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

30

35

u(y)

y

parabolic velocity profile

LBMAnalytical

100 101 10210 5

10 4

10 3

10 2

10 1

N

err

or

convergence of bounce back boundary conditions

mid gridon grid2nd order1st order

Figure 8: Parabolic velocity profile ∆P = 0.0125, H = 32, ν = 0.05

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Simulation 2: Lid Driven Cavity

I 2D fluid flow driven by a top moving lid

I No-slip (bounce-back) BCs on the otherthree stationary walls

I Zou-He BCs on the moving lid

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

y

Stream Trace for Re = 400

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 Stream Trace for Re = 1000

y

x

Figure 9: Stream traces for Re = 400 and 1000. The Vd = 0.0868 and 0.2170 respectively.

Other parameters: ν = 1/18, τ = 2/3, 256 × 256 lattice9 / 16

Simulation 3: Flow past a Cylinder

I No-slip BCs on the solidwalls and cylinder

I Zou-He velocity anddensity BCs at the inletand outlet

Regimes of the Flow

I Re < 5: Laminar flow, no separation of streamlines

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Simulation 3: Flow past a Cylinder

I 5 < Re < 40: A fixed pair of symmetric vortices

I 40 < Re < 400: Vortex street

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Simulation 3: Flow past a Cylinder

Figure 10: Vorticity plot of flow past a cylinder at Re = 150, a Karman vortex street is generated

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Simulation 4: Rayleigh-Benard Convection

Nondimensional Boussinesq Equations

∇ · ~u = 0

∂~u

∂t+ ~u · ∇~u = Pr∆~u+Ra · PrT z −∇p

∂T

∂t+ ~u · ∇T = ∆T Figure 11: Illustration of

Rayleigh-Benard convection

I Ra: Rayleigh number , Pr: Prandtl number

I A D2Q9 model for ~u and a D2Q5 model for T , and the two models arecombined into one coupled model for the whole system

I BCs on ~u: No-slip (bounce-back) BCs on the top/bottom walls, periodicBCs on the two vertical walls

I BCs on T : Zou-He BCs on the top/bottom walls, periodic BCs on the twovertical walls

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Convection cells

20 40 60 80 100 120 140 160 180 200

10

20

30

40

50 Streamlines (Ra = 20000, t = 8100)

x axis y

axis

20 40 60 80 100 120 140 160 180 200

10

20

30

40

50 Streamlines (Ra = 2000000, t = 5800)

x axis

yax

is

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Summary

Features of Lattice Boltzmann Method

I A celluar automata model, as well as a special FD method for Boltzmannequation

I Errors are 2nd order in space

I Very successful for simulating multiphase/multicomponent flows

I Simulating flows with complex boundary conditions are much easier usingLBM (porous media flow)

I LBM can be easily parallelized

A Controversy

I The compressible Navier-Stokes equations (NSEs) can be recovered fromLBM through Chapman-Enskog expansions

I A method with artificial-compressibilty for the incompressible NSEs

I Some other LBMs have been developed for modelling the incompressibleNSEs in the incompressible limit

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References

1. S. Chen, D. Martınez, and R. Mei, On boundary conditions in latticeBoltzmann methods, J. Phys. Fluids 8, 2527-2536 (1996)

2. Q. Zou, and X. He, pressure and velocity boundary conditions for thelattice Boltzmann, J. Phys. Fluids 9, 1591-1598 (1997)

3. R. Begum, and M.A. Basit, Lattice Boltzmann Method and itsApplications to Fluid Flow Problems, Euro. J. Sci. Research 22, 216-231(2008)

4. Z. Guo, B. Shi, and N. Wang, Lattice BGK Model for IncompressibleNavier-Stokes Equation, J. Comput. Phys. 165, 288-306 (2000)

5. Z. Guo, B. Shi, and C. Zheng, A coupled lattice BGK model for theBoussinesq equations, Int. J. Numer. Meth. Fluids 39, 325-342 (2002)

6. S. Succi, The Lattice Boltzmann Equation for Fluid Dynamics andBeyond. Oxford University Press, Oxford. (2001)

7. M. Sukop and D.T. Thorne, Lattice Botlzmann Modeling: an introductionfor geoscientists and engineers. Springer Verlag, 1st edition. (2006)

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