CFD Lab Course The Lattice Boltzmann Method Philipp Neumann - - PowerPoint PPT Presentation

Technische Universit¨ at M¨ unchen

CFD Lab Course

The Lattice Boltzmann Method

Philipp Neumann

20.5.2011

• P. Neumann: CFD Lab Course

The Lattice Boltzmann Method, 20.5.2011 1

Technische Universit¨ at M¨ unchen

Review: The story so far

So far: Navier–Stokes

• Compute pressure and velocity of the fluid
• Nonlinear equations
• Equation system to be solved in each timestep

However:

• Only valid in the continuum regime, that is for

Kn := lmfp Lc << 1 Kn Knudsen number, lmfp mean free path of fluid molecules, Lc characteristic length of flow system

• Not that trivial to bring Navier–Stokes to supercomputers
• P. Neumann: CFD Lab Course

The Lattice Boltzmann Method, 20.5.2011 2

Technische Universit¨ at M¨ unchen

Lattice What???

Lattice Boltzmann Method: Derived from Boltzmann equation (→ microscopic description)

• Describes fluid by means of statistics

→ Find probability f( x, v, t) that fluid molecules are in a small surrounding of x at time t with a certain velocity v

• Macroscopic quantities are obtained by integration over the

velocity space: ρ(x, t) =

∞

• −∞

f( x, v, t)dv ρ(x, t) u =

∞

• −∞

f( x, v, t) v dv (1) with fluid density ρ and fluid velocity u

• Assumption “Ideal gas” ⇒ Pressure p = c2

sρ, cs speed of sound

• P. Neumann: CFD Lab Course

The Lattice Boltzmann Method, 20.5.2011 3

Technische Universit¨ at M¨ unchen

Lattice What???

Problems:

• Probabilities live in a 2D + 1-dimensional space

→ “Curse of dimensionality”

• Integration over infinite domains can be quite hard
• P. Neumann: CFD Lab Course

The Lattice Boltzmann Method, 20.5.2011 4

Technische Universit¨ at M¨ unchen

Lattice What???

Problems:

• Probabilities live in a 2D + 1-dimensional space

→ “Curse of dimensionality”

• Integration over infinite domains can be quite hard

How can we solve these issues?

• P. Neumann: CFD Lab Course

The Lattice Boltzmann Method, 20.5.2011 4

Technische Universit¨ at M¨ unchen

Welcome to the dark ages: Castle invasion

Let’s find a simple model for a castle invasion: Assume Q bowmen on a castle tower, defending the castle. Assume, we have ‘fair” battle, i.e. there are only Q invaders

• P. Neumann: CFD Lab Course

The Lattice Boltzmann Method, 20.5.2011 5

Technische Universit¨ at M¨ unchen

Welcome to the dark ages: Castle invasion

Let’s find a simple model for a castle invasion: Assume Q bowmen on a castle tower, defending the castle. Assume, we have ‘fair” battle, i.e. there are only Q invaders

• Each bowman can shoot arrows only

into the direction of one enemy (and vice versa)

• P. Neumann: CFD Lab Course

The Lattice Boltzmann Method, 20.5.2011 5

Technische Universit¨ at M¨ unchen

Welcome to the dark ages: Castle invasion

Let’s find a simple model for a castle invasion: Assume Q bowmen on a castle tower, defending the castle. Assume, we have ‘fair” battle, i.e. there are only Q invaders

• Each bowman can shoot arrows only

into the direction of one enemy (and vice versa)

• If a bowman does not have any more

arrows, he gets new arrows from the

• ther bowmen
• P. Neumann: CFD Lab Course

The Lattice Boltzmann Method, 20.5.2011 5

Technische Universit¨ at M¨ unchen

Welcome to the dark ages: Castle invasion

Let’s find a simple model for a castle invasion: Assume Q bowmen on a castle tower, defending the castle. Assume, we have ‘fair” battle, i.e. there are only Q invaders

• Each bowman can shoot arrows only

into the direction of one enemy (and vice versa)

• If a bowman does not have any more

arrows, he gets new arrows from the

• ther bowmen
• The arrows coming from the enemies

are collected and re-used by the bowmen

• P. Neumann: CFD Lab Course

The Lattice Boltzmann Method, 20.5.2011 5

Technische Universit¨ at M¨ unchen

Dark ages ↔ Lattice Boltzmann

Q Bowmen on a tower Probability densities fi (i = 1, ..., Q) in a (cubic) lattice cell Q directions for the arrows Q lattice velocities ci for fluid molecules Re-distribution of arrows Collision process between populations fi within a cell Incoming arrows from Q attackers Molecules streamed from Q neighboured cells

• P. Neumann: CFD Lab Course

The Lattice Boltzmann Method, 20.5.2011 6

Technische Universit¨ at M¨ unchen

Collision and Streaming

1 collision + 1 streaming = 1 timestep

• P. Neumann: CFD Lab Course

The Lattice Boltzmann Method, 20.5.2011 7

Technische Universit¨ at M¨ unchen

Reduce the effort...

Problems:

• Probabilities live in a 2D + 1-dimensional space → “Curse of

dimensionality”

• Integration over infinite domains can be quite hard

Our discrete formulation...

• P. Neumann: CFD Lab Course

The Lattice Boltzmann Method, 20.5.2011 8

Technische Universit¨ at M¨ unchen

Reduce the effort...

Problems:

• Probabilities live in a 2D + 1-dimensional space → “Curse of

dimensionality”

• Integration over infinite domains can be quite hard

Our discrete formulation...

• ... lives in a D + 1-dimensional space, having Q unknowns per

lattice cell

• P. Neumann: CFD Lab Course

The Lattice Boltzmann Method, 20.5.2011 8

Technische Universit¨ at M¨ unchen

Reduce the effort...

Problems:

• Probabilities live in a 2D + 1-dimensional space → “Curse of

dimensionality”

• Integration over infinite domains can be quite hard

Our discrete formulation...

• ... lives in a D + 1-dimensional space, having Q unknowns per

lattice cell

• ... can easily (and locally!) handle the integration procedure:

ρ =

Q

• i=1

fi ρ u =

Q

• i=1

fi ci

• P. Neumann: CFD Lab Course

The Lattice Boltzmann Method, 20.5.2011 8

Technische Universit¨ at M¨ unchen

Collision

• Interaction of local populations fi inside each lattice cell
• Denote the post-collision state by f ∗

i

• Assume, our system is close to an equilibrium state
• If f eq

i

represents the equilibrium state of the fluid system, the molecular collisions should drive our system even closer towards this equilibrium BGK model (Bhatnagar – Gross – Krook): f ∗

i (x, t) = fi(x, t) − 1

τ

• fi(x, t) − f eq

i

• with relaxation time τ ∈ (0.5, 2)
• P. Neumann: CFD Lab Course

The Lattice Boltzmann Method, 20.5.2011 9

Technische Universit¨ at M¨ unchen

The equilibrium distribution

• Can be derived from the Maxwell-Boltzmann distribution
• Equilibrium state only depends on macroscopic quantities (ρ,

u) f eq

i

(ρ, u) = wiρ

• 1 +

ci · u c2

s

+

• ci ·

u 2 2c4

s

− u · u 2c2

s

• with cs :=

1 √ 3 speed of sound on the lattice

• P. Neumann: CFD Lab Course

The Lattice Boltzmann Method, 20.5.2011 10

Technische Universit¨ at M¨ unchen

The Lattice Boltzmann Algorithm

Given: fi(x, tstart) everywhere in our computational domain for (t = tstart; t < tend; t = t + dt) ; do done

• P. Neumann: CFD Lab Course

The Lattice Boltzmann Method, 20.5.2011 11

Technische Universit¨ at M¨ unchen

The Lattice Boltzmann Algorithm

Given: fi(x, tstart) everywhere in our computational domain for (t = tstart; t < tend; t = t + dt) ; do Compute density and velocity inside a fluid cell: ρ =

i

fi, ρ u =

i

fi ci done

• P. Neumann: CFD Lab Course

The Lattice Boltzmann Method, 20.5.2011 11

Technische Universit¨ at M¨ unchen

The Lattice Boltzmann Algorithm

Given: fi(x, tstart) everywhere in our computational domain for (t = tstart; t < tend; t = t + dt) ; do Compute density and velocity inside a fluid cell: ρ =

i

fi, ρ u =

i

fi ci Local collision: f ∗

i (x, t) = fi(x, t) − 1 τ

• fi(x, t) − f eq

i

(ρ, u)

• done
• P. Neumann: CFD Lab Course

The Lattice Boltzmann Method, 20.5.2011 11

Technische Universit¨ at M¨ unchen

The Lattice Boltzmann Algorithm

Given: fi(x, tstart) everywhere in our computational domain for (t = tstart; t < tend; t = t + dt) ; do Compute density and velocity inside a fluid cell: ρ =

i

fi, ρ u =

i

fi ci Local collision: f ∗

i (x, t) = fi(x, t) − 1 τ

• fi(x, t) − f eq

i

(ρ, u)

• Streaming to/ from neighbours: fi(x +

cidt, t + dt) = f ∗

i (x, t)

done

• P. Neumann: CFD Lab Course

The Lattice Boltzmann Method, 20.5.2011 11

Technische Universit¨ at M¨ unchen

The Lattice Boltzmann Algorithm

Given: fi(x, tstart) everywhere in our computational domain for (t = tstart; t < tend; t = t + dt) ; do Compute density and velocity inside a fluid cell: ρ =

i

fi, ρ u =

i

fi ci Local collision: f ∗

i (x, t) = fi(x, t) − 1 τ

• fi(x, t) − f eq

i

(ρ, u)

• Streaming to/ from neighbours: fi(x +

cidt, t + dt) = f ∗

i (x, t)

done fi(x + cidt, t + dt) = fi(x, t) − 1 τ

• fi − f eq

i

• P. Neumann: CFD Lab Course

The Lattice Boltzmann Method, 20.5.2011 11

Technische Universit¨ at M¨ unchen

Remarks

• P. Neumann: CFD Lab Course

The Lattice Boltzmann Method, 20.5.2011 12

Technische Universit¨ at M¨ unchen

Remarks

• Explicit method, no equation system that needs to be solved!
• P. Neumann: CFD Lab Course

The Lattice Boltzmann Method, 20.5.2011 12

Technische Universit¨ at M¨ unchen

Remarks

• Explicit method, no equation system that needs to be solved!
• Asymptotic theory: In the continuum limit (Kn→0):

Lattice Boltzmann → (incompr.) Navier-Stokes

• P. Neumann: CFD Lab Course

The Lattice Boltzmann Method, 20.5.2011 12

Technische Universit¨ at M¨ unchen

Remarks

• Explicit method, no equation system that needs to be solved!
• Asymptotic theory: In the continuum limit (Kn→0):

Lattice Boltzmann → (incompr.) Navier-Stokes

• Lattice Boltzmann: Simulation of weakly compressible flows

→ ρ ≈ 1.0,

• u << cs
• P. Neumann: CFD Lab Course

The Lattice Boltzmann Method, 20.5.2011 12

Technische Universit¨ at M¨ unchen

Boundary Conditions: Back to the dark ages

Bowman vs. Bowman What should a bow- man do when there’s no enemy?

• P. Neumann: CFD Lab Course

The Lattice Boltzmann Method, 20.5.2011 13

Technische Universit¨ at M¨ unchen

Boundary conditions - No-slip wall

• No-slip wall ⇔

u = 0 at wall

• P. Neumann: CFD Lab Course

The Lattice Boltzmann Method, 20.5.2011 14

Technische Universit¨ at M¨ unchen

Boundary conditions - No-slip wall

• No-slip wall ⇔

u = 0 at wall

• In “Dark Ages”-words:

If an arrow (loaded with particles) leaves the domain ⇔ An arrow with the same amount of particles needs to enter the domain

• P. Neumann: CFD Lab Course

The Lattice Boltzmann Method, 20.5.2011 14

Technische Universit¨ at M¨ unchen

Boundary conditions - No-slip wall

• No-slip wall ⇔

u = 0 at wall

• In “Dark Ages”-words:

If an arrow (loaded with particles) leaves the domain ⇔ An arrow with the same amount of particles needs to enter the domain

• Write f ∗

i (x, t) to f ∗ inv(i)(x +

cidt, t)

• Similar modeling of moving walls (see worksheet)
• P. Neumann: CFD Lab Course

The Lattice Boltzmann Method, 20.5.2011 14

Technische Universit¨ at M¨ unchen

Worksheet 2: 3D Cavity flow

• Program a 3D cavity based on the D3Q19 Lattice Boltzmann

model (→ 3D, 19 lattice velocities)

• P. Neumann: CFD Lab Course

The Lattice Boltzmann Method, 20.5.2011 15

Technische Universit¨ at M¨ unchen

Worksheet 2: 3D Cavity flow

• Program a 3D cavity based on the D3Q19 Lattice Boltzmann

model (→ 3D, 19 lattice velocities)

• Simulate cavities at different Reynolds numbers

→ kinematic viscosity ν is defined by ν = c2

s

• τ − 1

2

• → viscosity limited to the range
• 0, 1

2

• → velocity needs to be (much) smaller than the speed of sound
• P. Neumann: CFD Lab Course

The Lattice Boltzmann Method, 20.5.2011 15