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_i_i:_:_:_.... Applications of the Lattice Boltzmann. Method to. Complex and Turbulent Flows. Li-Shi Luo. ICASE, Hampton, Virginia. Dewei Qi. Western Michigan.
NASA/CR-2002-211659 ICASE

Report

:...

No. 2002-19

.._i_i:_:_:_ ....

Applications of the Lattice Boltzmann Complex and Turbulent Flows Li-Shi

Luo

ICASE, Dewei

Hampton,

Virginia

Qi

Western

Michigan

Lian-Ping

Wang

University

of Delaware,

July 2002

University,

Kalamazoo,

Newark,

Delaware

Michigan

Method

to

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NASA/CR-2002-21165 ICASE

Report

:...

9

No.

2002-19

.._i_i:_:_:_ ....

Applications of the Lattice Boltzmann Complex and Turbulent Flows Li-Shi

Method

to

Luo

ICASE,

Hampton,

Dewei

Virginia

Qi

Western

Michigan

University,

Lian-Ping

Wang

University

of Delaware,

Kalamazoo,

Newark,

Michigan

Delaware

ICASE NASA

Langley

Research

Hampton,

Virginia

Operated

by Universities

Center

Space

Research

Association

Prepared for Langley Research under Contract NAS 1-97046

July

2002

Center

Available

fi'om the following:

NASA Center 7121 Standard Hanover,

for AeroSpace Drive

MD 21076

(301) 621 0390

1320

hlfomlation

(CASI)

National

Technical

5285 Pol_ Royal

hlfomlation Road

Springfield, VA 22161 (703) 487 4650

2171

Service

(NTIS)

APPLICATIONS

OF

THE

LATTICE

BOLTZMANN

TURBULENT LI-SHI LUO*, DEWEI

Abstract.

We briefly

dimensional tation

LBE

in fluid.

agree

lattice

non-spherical

Subject 1.

in the

physics the

particularly

their

(LBE)

fluids,

between

two fluids,

systems,

particulate

crystallization,

such

and

f(0)

macroscopic temperature,

a decade

the

and

find

turbulence that

the

the pseudo-spectral

two

method

turbulence,

Boltzmann

of the lattice

structures,

media,

reactive

(see recent

approximation

[1]. The

lattice

spectral

and

fluids,

reviews

[3, 16] and

evolved

discretized

from

Boltzmann

Boltzmann

boundaries

demonstrate

BGK

equation

instability in flow

therein).

the lattice-gas

with

or/and

free boundaries

references

equation

been

magnetohydrodynamics,

form of the continuous

we shall

started

have

the Rayleigh-Taylor

viscoelastic

interest

methods

methods

lattice

dynamics

much

LBE

complicated

the

fluid

attracted

Boltzmann

flows and combustions,

equation

continuous

have

LGA

flow over complicated porous

and

the

involving

loss of generality,

the

that

applications

Boltzmann

from

methods

flow

the LBE is a special without

recently

[5, 24, 6] and

for computational

lattice-gas

of fluid

systems

(LGA)

The

fluids through

complex

that

isotropic

automata

as alternatives

lattice

in fluid, chemical

equation

automata

[5, 24, 6].

Boltzmann

equation

an a priori the

single

derivation

relaxation

can be written

in the

of time

form

of

equation:

Ot + _" V,

Dtf

+ _f

_ f(o)

f -

f(w,

_, t) is the

Boltzmann

density

lattice-gas

it was only very

external

sake of simplicity

is the

and

community.

suspensions

differential

flow, 3D homogeneous

ago, the

lattice-gas

multi-component

models

turbulent

than

as turbulent

Boltzmann

-

than

the threein sedimen-

isotropic

simulation,

dissipative

We show

16 particles

homogeneous

pseudo-spectral is more

(LBE).

flow and

suspensions

in simulations

(Bhatnagar-Gross-Krook)

Dt

method,

CFD

it has been shown

[8, 9]. For the

where

the

but the LBE method

However,

from

and other

Historically,

an ordinary

equation

in Couette

of the three-dimensional

[17, 12, 2, 22] were proposed

community.

attention

complex

lattice

AND

WANG*

Boltzmann

particle

1283 with

Mechanics

inception,

successful

Recently,

size

Fluid More

equation Since

to gain

of the

particulate

Introduction.

(CFD).

QIt, AND LIAN-PING

of the lattice

LBE simulation

COMPLEX

FLOWS

for a non-spherical

Boltzmann

classification.

Boltzmann

the

box

method

TO

as expected.

words,

method,

the

well with each other

in small scales, Key

cubic

the

results

We compare

flow in a periodic results

review

simulation

METHOD

of mass,

the Boltzmann

distribution the

single

function

velocity,

constant,

f(o)

and

particle

distribution

in D-dimensions, the

and particle

(1.1)

= (27r_) D/2 exp

normalized mass.

function,

in which temperature,

The macroscopic

/_ is the

p, u and respectively, variables

relaxation

0 -- kBT/rn T, kB and

are the moments

time, are the rn are of the

*ICASE, Mail Stop 132C, NASA Langley Research Center, 3 West Reid Street, Building 1152, Hampton, Virginia 236812199 (email address: [email protected]). This research was supported by the National Aeronautics and Space Administration under NASA Contract No. NAS1-97046 while the author was in residence at ICASE, NASA Langley Research Center, Hampton, Virginia 23681-2199. tDepartment of Paper and Printing Science and Engineering, Western Michigan University, Kalamazoo, Michigan 49008. *Department of Mechanical Engineering, University of Delaware, Newark, Delaware 19716.

distributionfunctionf

with

respect

to the molecular

(1.2a)

pu = f _ f d_ = / _ f(°) d_ ,

(1.2b)

(1.1) can be formally

(_ - u) 2 f d_ =

integrated

O((i_)

that

(it is small

or smaller

in the Taylor

f(x where series

7- -

enough

A/(it is the

the

be preserved

and

f(0)

is smooth

of the

dimensionless

enough

right

+ _(it, _, t + (it) - f(x,

hand

/) (27tO)D

Navier-Stokes in finite

2

time.

(1.3)

and neglecting

side of (1.3),

The

equations,

discretized

--_

exp

the

the

terms

of the

order

we obtain

_, t) - f(0)(x,

equilibrium

1+

f(0)

_, t)], can

(1.4)

be expanded

S_

---- Z

space

fa(eq)

as a Taylor

{(_la

momentum

We can derive

space

the

f_(xi the

_,

(_

f(eq)

- u) 2 f_ = 1_

by using

equilibrium

+ e_(it,

f(eq),

Gaussian-type

athermal

the discrete

velocity

(0,+1)c,

(4-1, 4-1) v_c,

c - (ix/(it.

neighbors.

Equation

Therefore

(1.7)

their

fluxes

b}, i.e.,

involves

(1.6b)

Z(_o

out that

model

_ __ U)

2 fa(eq)

these

moments

(1.6C)

set {e_},

and the

t)

it is easy to implement

calculations

in

in two-dimensions

-

(xi,

f(eq)

t)],

coefficients

(1.7) {w_}

are given

by

(1.Sa)

w_=

and natural

lattice

weight

a = 5, 6, 7, 8, local

exactly

3U22c 2 } ,

a=1,2,3,4,

only

can be evaluated

[8, 9, 23].

on a square

1 t) = -_[f_(xi,

+ 9(e_.2c4u)2

{,00, (4-1,0)c,

pO) and

,

quadrature

LBE

t + (it) - f_(xi,

f(eq) = w_ p { 1 + 3(e_.c_u)

e_ =

= 1, 2, ...,

and

C_

t) [8, 9]. It turns

nine-velocity

(p, pu,

(1.6a)

C_

discretized

moments

(1.5)

'

(_ f_ = Z

pO = _1 Z(_

t) -- W_ f(x,

.

C_

pu = Z

f_ -- f_(x,

202

+

hydrodynamic

momentum

C_

and

locally,

_, t) = - l[f(x,

relaxation

P ---- Z

where

(it:

in u up to u 2

To obtain

where

(1.2c)

1Ae-5_/x foS_et'/xf(°)(x+_t',_,t+t')dt'.

expansion

f(eq)_

must

(_ _ u)2 f(o) d_.

over a time interval

f(x+_6t,_,t+6t)=e-5_/Xf(x,_,t)+ Assuming

_:

p = f f d_ = / f(°) d_ ,

pO = _ Equation

velocity

and

1/9,

a=1,2,3,4,

1/36,

a = 5, 6, 7, 8,

uniform

to massively

parallel

communications computers.

(1.8b)

to the

nearest

The(incompressible) Navier-Stokes equationderivedfromthe aboveLBEmodelis: pcgtu q- pu.Vu with the

isothermal

ideal

gas equation

of state,

P=c_p, It should

be noted

to the second order

order

accurate.

the

and

ings of the

BGK

follows

LBE

suspensions

in fluid is difficult

been

some

such

as the

finite

remains

as a challenge

applied

to simulate method

interfaces

to the

the

to this

conventional

flow of non-spherical

problem

spherical

simulate

relies

Particle

include

to Jeffery's

at zero Reynolds

reaches to the

theory

an equilibrium vorticity

vector

determined

by the

fifty discs

in Couette

orientation

Figure The

demonstrates experimentally.

number

in which

flow, respectively.

effect, the

method

and

maximum

in some

cases.

Yet

such as fluidized

beds.

There

have

flow.

conventional

method

CFD

methods,

suspensions

has been

[19, 20].

The

handle

still

successfully

success

the

of the

particle-fluid

state.

interactions

states

Reynolds

We show that of the suspension,

number,

perpendicular

of either

significantly

than

for discs.

particles

falling

contrary particle

and

dynamically

up to

the inertial

a non-spherical

is unique, Systems

in a non-

number

motion

are aligned

state

dissipation

equilibrium

and discs.

Reynolds axes

for cylinders

The

for a particle

rotational

shortest

Multi-particle

effect is stronger

The flow of particulate

can easily

cylinders

the

equilibrium

energy

box.

due to the fluid flow [18].

[13]. At a non-zero

This

the flow of non-

numerically

the LBE

Couette

spheroids, changes

its longest

a 3D LBE simulation the

formation To the

numerical

LBE

and

parallel

stable,

fully

fifty cylinders

change

The details

or

the equilibrium of this

work will

[21].

illustrates the

the

method

references).

in a periodic

in three-dimensions

in the

and oblate

state

The

Recently

inter-

calculation

to simulate

by using

flow are simulated

qualitatively

flow are also simulated.

1 shows

that

particle

of the

inertial

elsewhere

left figure

LBE direct

number

of solid particle.

be reported

fact

element

state

(i)

shortcom-

[10, 7] with

of the flow of non-spherical

the force on the particle

prolate

Reynolds

suspensions

simulation

methods.

in a 3D Couette

effect at any finite

applications

in

method:

some

the finite

Suspensions.

and to simulate

suspensions

non-spherical

suspension

geometries

on the

evaluate

a single

particulate

the

CFD

method

turbulence

Particulate

of the flow of spherical

mesh

for steady

second

[9].

overcomes

by using

due

lattice

Boltzmann

[16] for further

isotropic

to industrial

However,

lattice

which

scheme

of the LBE

experimentally

is important

method.

[15], and accurately

We first

320.

to quantify

simulations

element

applications

of Non-Spherical

suspensions

successful

the

on a triangular

body-fitted

grid

review

in fluid and homogeneous

of Flows

the flow of particulate

LBE

demonstrate

(see a recent

the

[4] and

of implicit

viscosity

in three-dimensions

times

of unstructured

for the numerical

models

lattice

refinement

by

the LGA and LBE methods

to improve

relaxation

(iv) application

convergence

suspensions

Simulation

with multiple

(iii) adaptation

makes

on a cubic recently

given

(1.10)

for p accounts

and seven-velocity

models made

speed

1 _c.

cs=

in p formally

six-velocity

progress

method;

we shall

formula

[14]; (ii) use of grid

to accelerate

particulate

2.

the

equation

equation

Galerkin

technique

In what

in the above

(1.9)

and the sound

c_(_t,

This correction

significant

techniques;

characteristic

spherical

of f_.

Boltzmann

polation/extrapolation

multi-grid

-1/2

+ ppV2u,

viscosity,

(_) T-

the twenty-seven-velocity

lattice

lattice

the

u=

we can derive

have been some

the generalized

or the

factor

derivatives

Similarly,

two-dimensions, There

that

= -VP

best

time

of sixteen

evolution

of inverted

of the entire

T configurations

of our knowledge,

simulation

[20].

cylindrical

this

system in the

phenomenon

of sixteen

under particles,

sedimentation, was first

the influence while

which

reproduced

has

the

of gravity. right

been

numerically

figure

observed by the

H

FIG.

1.

Nx

× Ny

× Nz

top

to bottom).

3D

LBE

simulation

= 140 × 150 (right)

of particles

× 35.

The

Formation

averaged

of inverted

sedimentation

in )fluid.

single-particle

Re _

T configurations

which

TABLE Parameters the

viscosity;

the

Taylor

3. lence

in lattice u _ is the

microscale

LBE

Boltzmann

RMS Reynolds

be easily benchmark

velocity

field;

and

M is the

Mach

number.

t_

Spectral

2u

1283

0.01189

LBE

128

1283

0.009869

of turbulence.

standard.

of 3D

periodic

the

flow.

cubic

Because

Here the LBE simulation of the simulation

are given

L dt

observed

is the

is the

12

and

L

of 16 particles in

24.

System left

size

to right

is and

experiment.

length

time

of box

step

size;

T

side;

N a is the

is total

Re_

M

0.993311

0.002

2

35.0

0

0.040471

1

1000

35.0

0.0687

remains

Turbulence.

as a stand

boundary

of its accuracy,

problem

the

3.1.

The

in the field of direct

initial

result

is a random

_ is

Re_

is

turbu-

numerical method

is often

with the pseudo-spectral condition

size;

time,

isotropic

the pseudo-spectral

pseudo-spectral

of the flow is compared in Table

Homogeneous

conditions,

system

integration

T

of the

=

(from

dt

Isotropic

box

simplicity

also

Ut

Homogeneous

Due to the

to simulate

The parameters

initial

N 3

Evolution

3.1

of the

number;

size

(left)

are

simulations:

L

Simulation

used

)fluctuation

pseudo-spectral

Method

in a three-dimensional

simulation

and

is D =

Particle

16.9.

used

can as a

simulation. velocity

field

(a)

t=o.o4

Energy

Spectrum

1.5

10 -1

10

(_Meankinetic energy K and dissipation rate e 10

0.5

10 .4

--

--

K

....

E Speclral K Lalllce e Lattice

s peclral 0

Laltice

0

Bollznmnn ....

I

.......

(

....

10"

I

0

,

,

Speclral-_.

,

,

10

Boltzmann Boltzmann

I 0.5

,

,

,

,

I 1

k

FIG. energy

2.

LBE

vs.

spectrum

from

the

Pseudo-spectral

E(k)

LBE

as

simulation

with a Gaussian

DNS

of

a function

of time.

are

according

scaled

distribution

3D homogeneous (b)

The

decay

to the

conditions

and a compact

are periodic

isotropic of the

dimensions

energy

E(k) The boundary

,

,

,

,

I 1.5

,

,

,

,

I 2

t

used

kinetic

in the

System

energy

spectral

K

size and

is 1283.

Rex

dissipation

= 35.

rate

e.

(a)

The

The

results

simulation.

spectrum

o( ko

in three

turbulence.

mean

exp

-k0

dimensions.

"

The Taylor microscale

Reynolds

number

is defined

as

P

where

K(t

Gaussian

= O) = (ug/2)v velocity

= (3u_Ms/2)v

field u0 with RMS

is the

component

volume

P

averaged

uaMs), and

kinetic

energy

A is the transverse

(of the

Taylor

initial

zero-mean

microscopic

scale:

A=V/15pu_s/e, where

e is the dissipation

Figure energy

2 shows

K and

spectral

rate.

the energy

dissipation

results

rate

(lines).

The

method

is more

dissipative,

number

of mesh

nodes

space

and 4.

time and

Conclusions

Pentium

CPUs.

Boltzmann the

method

MD part

CPUs

of the

up to 32 CPUs

a system

as function

of time,

lattice

Boltzmann

results

results

especially

more

and For the

E(k)

e. The LBE

in each

thus

spectrum

at high

than The

of the

fluid and

the system particles

(symbols)

above

k

>

the LBE

simulations suspension,

dynamics

(MD)

the

of the

speed

size is 643 and

on our Beowulf

1 _kmax,

of the mean

compared results.

with

kmax

5N,

----

is only second

kinetic

the pseudo-

Obviously 1

where

method

the

and

order

LBE

N is the

accurate

in

cluster

of

method. were

performed

the for the

on

a Beowulf

code consists solid

two

particles

code still scales

with fifty particles.

system.

evolution

are

pseudo-spectral

the pseudo-spectral

particulate

molecular

the

numbers

dissipative

code is not yet parallelized,

of a few hundred

wave

This is because

simulation

when

well with

direction.

Discussion.

for the

agree

and the time

Presently

part:

[19].

well with

the Even

the

we can easily

lattice though

number simulate

of

Asfor thesimulationofthe3Dhomogeneous isotropicturbulence, theLBE codewithoutoptimization hasthe samespeedasthe spectralcodewith a Beowulfclusterof eightCPUs(aboutls pertimestep). However, wedoexpecttheLBEcodewill scalelinearlywiththenumberofCPUs,butnotthespectralcode. Ourcurrentresearch includes particulatesuspension in fluidwith highvolumefractionofparticles, viscoelastic andnon-Newtonian fluids,andforcedor free-decay homogeneous isotropicturbulence in a periodic cubeby usingthelatticeBoltzmannmethodonmassively parallelcomputers. REFERENCES [1]P. L. BHATNAGAR, E. P. GROSS, ANDM. KROOK, A model amplitude

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L Small

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525. [2] g.

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gas Boltzmann

AND W. method,

g.

MATTHAEUS,

Phys.

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Phys.

AND D.

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Grid refinement

HANEL,

B. HASSLACHER,

Rev.

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Rev.

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471-526.

Form

REPORT

DOCUMENTATION

PAGE

Approved

OMB No. 0704-0188

Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington, VA 22202-4302, and to the Office of Management and Budget, Paperwork Reduction Project (0704-0188), Washington, DC 20503. 1. AGENCY

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2.

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July 4. TITLE

AND

DATE

3.

REPORT

2002

TYPE

Contractor

AND

DATES

SUBTITLE

5.

APPLICATIONS

OF THE

COMPLEX

TURBULENT

AND

LATTICE

BOLTZMANN

METHOD

COVERED

Report FUNDING

NUMBERS

TO C NAS1-97046 WU 505-90-52-01

FLOWS

6. AUTHOR(S) Li-Shi

Luo,

Dewei

Qi, and

Lian-Ping

Wang

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) ICASE Mail

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NASA Hampton,

Research

Center

National

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Hampton,

AGENCY NAME(S) AND ADDRESS(ES)

and

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NUMBER

ICASE

Report

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2002-19

SUPPLEMENTARY

SPONSORING/MONITORING AGENCY REPORT NUMBER

NASA/CR-2002-211659 ICASE Report No.

VA 23681-2199

2002-19

NOTES

Langley Technical Monitor: Dennis M. Bushnell Final Report To appear in the Lecture Notes in Computational 12a.

ORGANIZATION

REPORT

VA 23681-2199

9. SPONSORING/MONITORING

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Science

and

Engineering,

Vol. 21, 2002. 12b.

STATEMENT

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CODE

Unclassified-Unlimited Subject Category 34 Distribution: Nonstandard Availability: 13.

ABSTRACT

NASA-CASI (Maximum

200

(301)

621-0390

words)

We briefly review the method of the lattice Boltzmann equation (LBE). We show the three-dimensional LBE simulation results for a non-spherical particle in Couette flow and 16 particles in sedimentation in fluid. We compare the LBE simulation of the three-dimensional homogeneous isotropic turbulence flow in a periodic cubic box of the size 1283 with the pseudo-spectral simulation, and find that the two results agree well with each other but the LBE method is more dissipative than the pseudo-spectral method in small scales, as expected.

14.

SUBJECT

lattice

TERMS

Boltzmann

spectral

method,

15.

method,

turbulent

non-spherical

flow, 3D homogeneous

particulate

isotropic

NUMBER

suspensions

OF

PAGES

12

turbulence, 16.

PRICE

CODE

A03 17.

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REPORT

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7540-01-280-5500

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ABSTRACT

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Form 298(Rev. 2-89) by ANSI Std. Z39-18