Lattice Boltzmann for - PowerPoint Presentation

Slide1

Lattice Boltzmann for Fluids

Sauro Succi

1

Slide2

LB For fluids2

The general idea of LB is

to write down a set ofhyperbolic equations for a discrete set of movers

(“

propagons

”

)

obeying

a

Propagation

-Relaxation

dynamics

a

round

prescribed

local

equilibria

.

Suitable

choices

of the

e

quilibria

lead

to

a

broad

variety

of

linear and

non-linear

PDE’s

.

A

major

advantage

of the

Prop

-Relax

dynamics

is

that

it

a

lways

proceeds

along

straight

lines

, no

matter

how

c

omplex

the

physics

of the

PDE’s

.

The

method

is

especially

useful

for the

Navier-Stokes

fluid

equations

for quasi

-

incompressible

fluids

in d=

1,2,3,

w

hich

we

now

proceed

to illustrate.

Slide3

Navier

-Stokes

equations

Basic

equations

are

known for nearly 2 centuries Reynolds number

(L. Da

Vinci)

Slide4

Turbulence!

Slide5

Turbulent

scales

Kolmogorov

length

Faucet, Re=10^4, DOF=10^9 Car/Airpl, Re=10^6-8, DOF=10^14-18 Geo/Astro, Re=10^10, DOF=10^22

Slide6

Energy spectrum: broad and

gapless!

Slide7

Navier

-Stokes in d dimensions

Advection

Pressure

Dissipation

# of

M

acrofields

= Constraints:

7Coordinate representation

:

Slide8

Lattice

Boltzmann

:

Platonic

hydrodynamics

Exact sampling of frequent events

M

agic

speeds!

Triple infinity to just

19!

8

Slide9

LB in d dimensions

How to

find

:

c_i

,

Omega_ij

, f_i^eq?

9Lattice BGK (single-time relaxation)

Slide10

Navier

-Stokes in d=2 dimensions

YES!

NO!

D2Q4=HPP (1976)

10

D2Q6=FHP (1986)

Natural: 4

speeds

versus 6

macrofields

: no way!

Natural: 6 speeds versus 6 macrofields

: ok!

Slide11

Navier

-Stokes in d=3 dimensions

YES!

6

speeds

vs 10

fields

: NO WAY!

D3Q19: YES!10 hydrofields + 9 “ghosts”11

Slide12

12Moment matching

proceeds exactly as in d=1, with

algebraic aggravation due to the tensorial structure.A crucial

step

is

to

recognize that the expansion is thelattice analogue of the Hermite expansion in continuumk

inetic theory. This simplifies enormously moment-matchingprocedure and puts it on a systematic basis.Let us take a close lok at the procedure.Moment matching

Slide13

Moment matching:

Mass

Mass, sum over

all

discrete

speeds

:

By

imposing

:

Start from LB in

differential BGK form

: 13

Slide14

Moment matching:

Momentum

Momentum

:

multiply

by

c_ia

and sum over

all discrete directions:

By imposing

:

Is it Navier-Stokes? NOT YET!

Constraints must be imposed on P_ab14

Slide15

Moment matching: momentum

Advection

+ Pressure

Dissipation

is

the

macroscopic Manifestation of non-equilibrium!15The Navier-Stokes pressure tensor without Dissipation ( inviscid Euler regime)

Slide16

Moment matching:

Momflux

Momentum

flux

,

multiply

by c_{ia}c_{ib} and sum over i:Now

because momflux is not a

micro-invariant!

16

How do we close this equation?

Slide17

Boltzmann to Navier-Stokes

17

1. Scale separation:The molecular mean free path must be

much

smaller

than any hydrodynamic length-scale (small Knudsen number)2

. Weak departure from local equilibrium:The Boltzmann probability distribution function must be close to a local Maxwell-Boltzmann

T

Slide18

Momflux

: enslaving

Momentum flux equation:

Enslaving

: eliminate time derivative on a

timescale

tau=1/omega:

Dissipation

Close to local equil:Inertia+Pressure

18

Slide19

Momflux

: enslaving

Injecting

This

must

now

match

exactly

the Navier-Stokes equations:DissipationInertia+Pressure19into the momentum equation, we obtain:Which

implies the following tensorial constraints:

Slide20

List of hydrodynamic constraints

20

Scalar

: 1

constraint

Vector

:

d

constraints2-Tensor: d*(d+1)/2 constraints3-Tensor: d*(d+1)*(d+2)/6 constraintsQ: Can we match the full list? How many discrete velocities?

Slide21

Lattice

equilibria: constraints

21

Continuity

equation

Momentum

EquationPressure+AdvectionNewtonian (linear) Dissipation

Slide22

Lattice

E

quilibria

Expand

the

local

Maxwell-

Boltzmann to second

order in Mach number:Question: How do we find the weights?Projection of the flow field over the i-th discrete speed where:22

is a lattice

projector

Slide23

Lattice

E

quilibria

By

inserting

4th

order isotropy2nd order isotropy0th order isotropy in the list of hydrodynamic constraints, we obtain:

23

Slide24

Lattice

equilibria: Mass

Insert

local

equils

:

Order 2: defines the lattice sound speedOrder 1: Guaranteed by mirror symmetry (Parity invariance)24

Order 0: Normalization

into Mass Conservation:

Slide25

Higher Order Lattice

equilibria

Expand

the

local

Maxwell-

Boltzmann

to third order in Mach number:

Hexapole:25Dipole

Quadrupole

Slide26

26

D1 lattice

equilibria

D1Q5

Slide27

D2Q9

equilibria

Quadratic

polynomials

(

Check

the algebra!):

27

Slide28

Standard 3d LB lattices

28

D3Q27=(D1Q3)^3

D3Q19= D3Q27-8

vertices

Slide29

Higher order lattices

29

D3Q39:

order

6

isotropy

D3Q93:

order 8 isotropy

Slide30

Very Higher

O

rder Lattices

30

D=2, up to

order

16!

D=3, relativistic

Slide31

Galilean invariance requires infinite series in the Mach number

Infinite rank isotropic tensors, i.e. infinite connectivity!!!

Scaling

invariance v/

vthermal

: global to local very hard

Local

Global

Lattice

vs

continuum

equilibria

Why

not

just take:

31

Slide32

Realizability

(positivity

): Low

M

ach

32

Slide33

33Summarizing: by

choosing a suitable set of discretevelocities and

associated weights, ensuring fourth order isotropy

, the LBE

reproduces

the

Navier-Stokes

equations for a fluid with:

Eq. of state:Viscosity:The negative contribution (propagation viscosity) stems fromsecond order expansion of the lattice streaming operator…

Summary

Slide34

+

Streaming

is

linear and

exact

(no )

LBE assets

+

Easy handling of complex geometries (straight lines)

+

Outstanding for parallel computing

+

NOT limited to dilute gases !

+

Emergent complexity nearly for free

+

Non-linearity is local (round-off conservative)

Laplacian-free dissipation

+

Pressure is local (weakly compressible)

34

Slide35

LB across scales: from turbulence to biopolymers to quark-gluon plasma

Au

35

Slide36

Boundary Conditions

36

Slide37

No-slip flow via bounce-back

FLUID

BUFFER

WALL

B

F

F

F

37

Slide38

B

ounce-Back

FLUID

BUFFER

WALL

B

F

F

F

38

Slide39

Write a D2Q9 code for channel flow and test the Poiseuille

flow at different viscosities (

see lb2.f) Same with a cylinder within the channel

Same

with a

random

porous media

Assignements39

Slide40

End of Lecture(for the detail thirsty,

see my book!

40

End of the Lecture

Slide41

41Left over material

Slide42

Equilibrium constraints

1

d

d

(d+1)/2

We

need

at least 1 + d + d*(d+1)/2 discrete velocities b> d*(1+(d+1)/2) = 5 movers in d=2 and 9 in d=3.42

Slide43

Dissipation-Viscosity

Expand

in

powers

of Knudsen

43

Slide44

Viscosity bounds

Vanishing

viscosity

with

dt

=O(tau)=O(1)

Stable

up to 44

Slide45

Lattice BGK

equilibria

Expand

the

local

Maxwell-

Boltzmann

(at least) to second order in Mach number

:45Dipole

Quadrupole

Slide46

Lattice BGK

equilibria: Mom

Insert

local

equils

:

Mirror symmetryDeja’ vu46Deja’ vu: nothing new

Slide47

Lattice BGK

equilibria: Momflux

Insert

local

equils:

Deja’ vu47Mirror symmetryOrder 4: pressure and inertia

Slide48

Lattice BGK

equilibria: Dissipation

Insert

local

equils

:

48Mirror symmetryStill order 4 isotropy: deja’ vu.Dissipation does not impose any high order constraint wrt pressure and inertia

Mirror symmetry

Slide49

Equation of State

49

Slide50

LB Dissipation

Usually

:

But

LBGK

has

:

hence

50

Slide51

Boltzmann to NSE

*

Take kinetic moments: open hierarchy

* Closure: Weak departure from local equilibrium (Kn<<1)

T

u=u(

r,t

)

T=T(

r,t

)

51

Slide52

Dissipation

This

requires

:

52

Slide53

Discrete streaming

Discrete streaming operator

contains

derivatives

at

all orders:

For consistency with diffusion must be expanded to second order Propagation viscosity = -Delta t/2 < 0: CRUCIAL! 53

Slide54

Viscosity

Vanishing

viscosity

with

dt

=O(tau)=O(1)

Stable

up to Beware: Large viscosities violate enslaving principle!Viscosity should be well below 1 (in LB units)54