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
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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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6. AUTHOR(S) Li-Shi
Luo,
Dewei
Qi, and
Lian-Ping
Wang
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ABSTRACT
NASA-CASI (Maximum
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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
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suspensions
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