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・ Department of Aerospace Engineering・
LATTICE BOLTZMANNMETHOD for CFD
More @ LBE
Dr. Jacques C. Richard, [email protected]. Sharath Girimaji, [email protected]
Dr. Dazhi Yu, [email protected]. Huidan Yu, [email protected]
12/1/2003
・ Department of Aerospace Engineering・
What makes up fluids Fluid made of small, individual molecules.
Molecules are in a state of constant motion.
Molecules are at continuous collision witheach other.
Molecule has internal structure.
There are intermolecular forces.
Macroscopic variables, such as pressure,temperature, and internal energy, are determinedby the mass, velocity, and internal structure ofmolecules.
・ Department of Aerospace Engineering・
Modeling FluidsKnudsen number
Microscopic Method : Molecular dynamics Kn<∞Mesoscopic Method : Boltzmann Equation Kn<∞Macroscopic Method: Navier-Stokes Equations Kn <0.1A Novel Method : Lattice Boltzmann Method Kn<0.1
�
Kn ! Mean free pathCharacteristic hydrodynamic length
・ Department of Aerospace Engineering・
Modeling of Fluids
BoltzmannEquation
MomentEquations
DiscreteVelocity model
Direct simulationMonte Carlo
Lattice BoltzmannEquation
Discretized phase-space & time
Kinetic Theory, Chapman-Enskog analysis
Navier Stokes Equations
microscopic
macroscopic
Taylor expansion in space and time
・ Department of Aerospace Engineering・
Velocity distribution function
Velocity distribution function: F(ci)
cdVPhysical space : ( x1, x2, x3 ) Velocity space: ( c1, c2, c3 )
ic
m : mass of molecule!N : number of molecules in !Vx
n(xi) : local number density
x2
x3
x1
xi
c2
ci
c3
c1
�
! = lim"Vx #0
m "N"Vx
= mn(xi)
�
F(ci) = lim!Vc "0
!N!Vc
・ Department of Aerospace Engineering・
Velocity Distribution Function
Normalized Velocity distribution Functionf(ci) = F(ci) / NN: Number of molecules in system
Characteristics of Distribution Function f 1.
2.
is the average value of Q for all moleculesMacrosopic velocity:
�
Nf (ci)dVc = N!"
"# $ f (ci)dVc =1!"
"#
�
Q =QdN
N!N
=Q(ci)Nf (ci)dVc"#
#!N
= QfdVc"#
#!
�
Q
�
r u = u = c = ci!"
"# fdVc
・ Department of Aerospace Engineering・
Velocity Distribution Function underEquilibrium State
Equilibrium State In equilibrium state, the number of molecules in ci is constant.
Distribution function under equilibrium state is theMaxwellian distribution:
T: temperature;k : Boltzmann constant (1.38054"10-16 erg-K)m: mass of moleculeu : macroscopic velocity
ic
�
f eq (ci) = m2!kT" # $
% & ' 3 / 2
e( m2kT
(c1(u1 )2 +(c2(u2 )
2 +(c3(u3 )2[ ]
・ Department of Aerospace Engineering・
Boltzmann Equation
Under non-equilibrium state distributionfunctionf = f(xi, ci, t) = f(x, c, t)
The molecular velocity distribution functionhas a rate of change, with respect to positionand time, that is described by
�
!!t
nf (ci)[ ] + c j!!x j
nf (ci)[ ] = !!t
nf (ci)[ ]" # $
% & ' collision
・ Department of Aerospace Engineering・
Collisions in fluids
Molecules constantly collide with each other.
Collision will change the velocity of molecules.
In collision, the translational energy of molecule may transferto internal energy, if molecule has internal structure.
In general, the rate of collision depends on the moleculevelocity, number density n(x), and temperature T.
zi
ci
z´i
c´i
・ Department of Aerospace Engineering・
Collision Operator in Boltzmann Equation Elastic-sphere molecule model
1. no internal structure of molecule,2. the molecule is treated as a rigid ball,3. so no rotation and vibration,4. only translation.
no attraction force between molecules.
Rigid elastic spheres
Repulsion
Attraction
True representation
d
r
∞
d: diameter of molecule; r: distance between molecules
・ Department of Aerospace Engineering・
Collision operator Rigid ball model gives:
g: relative velocity between two collided molecules
f(c´i)f(z´i): replenishing of the molecules of class ci
f(ci)f(zi) : depleting of the molecules of class ci
Integration limits: # from 0 to "/2 ; $ from 0 to 2".
Original collisionDepletion of ci and zi
zi
ci
z´i
c´i
zi
ci
z´i
c´i
Inverse collisionReplenishing of ci and zi
�
!!t
nf (ci)[ ]" # $
% & ' collision
= n2d2 f c'i( ) f z'i( ) ( f ci( ) f zi( )[ ]0
) / 2
* gsin+ cos+d+d,dVx0
2)
*(-
-
*
・ Department of Aerospace Engineering・
Macroscopic variables
Quantities of interest
Density
Momentum
Translational energy
�
! = mfdVc"#
#$
�
!u = mcfdVc"#
#$
�
etr = 12
c ! c ( )2 fdVc!"
"#
・ Department of Aerospace Engineering・
Moments of the Boltzmann Equation Let Q(ci) be a function of ci but not of position
and time, the equation of transfer of Q(ci) is
if Q=m, mci, mc2/2, the change in Q for bothmolecules must be zero in collision. We have
Let Q = m, we have the continuity equation in NS. We recover themomentum and energy equations in NS when Q=mci, mc2/2
�
Q(ci)!!t
nf (ci)[ ] + c j!!x j
nf (ci)[ ]" # $
% & ' dVc
()
)
* = Q(ci)!!t
nf (ci)[ ]" # $
% & ' collision
dVc()
)
*
�
!!t
nQ [ ] + !!x j
nQci[ ] = " Q[ ]
�
!!t
nQ [ ] + !!x j
nQci[ ] = 0
・ Department of Aerospace Engineering・
The Conservation Equations from theBoltzmann Equation
New definition: %ij = -[& CiCj - p 'ij] qj = & Cj C2
C = c - c
Kinetic theory: e = etr = C2/2 h = htr = etr + p/& = 5RT/2
& = nm p = &[C12 + C2
2 + C32 ]/3
tr: translational
Q = m
Q = mci
Q = mc2/2�
!"!t
+ !!x j
"c j( ) = 0
�
!!t
"c j( ) + !!xi
"c jc i( ) = # !p!x j
+!$ ij
!xi
�
!!t
" e + c 2( )[ ] + !!xi
"c i h + 12
c 2# $ %
& ' (
) * +
, - . =
!!xi
/ ijc k 0 qi( )
・ Department of Aerospace Engineering・
The Chapmen-Enskog Solution ofBoltzmann Equation
Non-dimensional form
cr: reference molecule speed ; L: characteristic length
vr: reference collision frequency
( = cr / L vr is proportional to the Knudsen #, the ratio of the meanfree path to a characteristic length, thus it is a very small value. Chapman-Enskog expansion (simplifying definitions ( ):
f(x, c, t) = f(eq)(x, c, t) + ( f(1)(x, c, t) + …Solve the f(l) , then we get the solution for f �
f = ˆ n ̂ f �
! ""ˆ t
ˆ n ̂ f [ ] + ˆ c j""ˆ x j
ˆ n ̂ f [ ]# $ %
& ' (
= ""ˆ t
ˆ n ̂ f [ ]# $ %
& ' ( collision
・ Department of Aerospace Engineering・
Chapman-Enskog Procedure
Taking the 1st order departure of f from theMaxwellian distribution as:f(x, c, t) = f(eq)(x, c, t) + ( f(1)(x, c, t) + …
Substituting into the Boltzmann equation:
�
!f!t
" # $
% & ' c
= f (1)'(x,c',t) f (eq )'(x,z',t) + f (1)'(x,z',t) f (eq )'(x,c',t)[ ] z ( c n) *( )dVzd*++( f (1)(x,c,t) f (eq )(x,z,t) + f (1)(x,z,t) f (eq )(x,c,t)[ ] z ( c n) *( )dVzd*++
・ Department of Aerospace Engineering・
Chapman-Enskog Procedure Keeping only 1st order terms in the expansion:
Whereo The average relative velocity btwn particles iso The total collision cross-section is )tot
The collision frequency is
�
!f!t
" # $
% & '
c
= f (1)(x,c,t) f (eq )'(x,z,t)[ ] z ( c n) *( )dVzd*++ , ( f (1)(x,c,t)n) totc rel
�
c rel
�
! r = ! c = n" totc rel
・ Department of Aerospace Engineering・
The “1st Order” Boltzmann Equation
So the Boltzmann equation becomes, to1st order:
Or
On the characteristic line c = dx/dt Where * = 1 / +r
�
!f!t
+ c " #f = $ 1%
f $ f (eq )( )
�
dfdt
+ 1!f = 1
!f (eq )
・ Department of Aerospace Engineering・
Integrating the “1st Order” BE
Integrating the “1st Order” BE over a timestep 't:
Assuming 't is small enough & f(eq) issmooth enough locally, then for 0≤t’≤'t:
�
f (x + c!t ,c,t + !t ) = 1"e#! t /" et ' /" f (eq )(x + ct',c,t + t')dt'
0
! t$ + e#! t /" f (x,c,t)
�
f (eq )(x + ct',c,t + t') = 1! t'"t
#
$ %
&
' ( f (eq )(x,c,t) + t'
"tf (eq )(x + c"t ,c,t + "t ) + O "t
2( )
・ Department of Aerospace Engineering・
Integrating the “1st Order” BE
Putting these last 2 eqs. together:
Expanding in a Taylor series whileneglecting terms of O('t
2) and also defining% = */'t:
�
f (x + c!t ,c,t + !t ) " f (x,c,t) = e"! t /# "1( ) f (x,c,t) " f (eq )(x,c,t)[ ]+ 1+ #
!te"! t /# "1( )
$
% &
'
( ) f (eq )(x + c!t ,c,t + !t ) " f (eq )(x,c,t)[ ]
�
e!" t /#
�
f (x + c!t ,c,t + !t ) " f (x,c,t) = " 1#
f (x,c,t) " f (eq )(x,c,t)[ ]
・ Department of Aerospace Engineering・
Low Mach Number Approximation In LBE, the equilibrium distribution
is obtained from a truncated small velocityexpansion or low-Mach-number approximation
D = number of dimensions (e.g., D = 3 for 3D)
�
f (eq ) = m2!kT
" # $
% & ' D / 2
e( m2kT
c(u( )2= m2!kT" # $
% & ' D / 2
e( m2kT
c 2e( m2kT
2c•u(u•u( )
�
f (eq ) = m2!kT" # $
% & ' D / 2
e( m2kT
c 21+ c •u
kT+ 12c •ukT
" # $
% & ' 2
( u•u2kT
)
* +
,
- . + O u3( )
・ Department of Aerospace Engineering・
Discretization of Phase Space Discretization of momentum space is
coupled to that of configuration space suchthat a lattice structure is obtained
This is a special characteristic of LBE Quadrature must be accurate enough to
• Preserve conservation constraints exactly• Retain necessary symmetries of Navier-Stokes
・ Department of Aerospace Engineering・
Discretization of Phase Space The first 2 order approximations of the distribution
function (f(eq), f(1)) are used to derive Navier-Stokes So quadrature used must evaluate hydrodynamic
moments w.r.t f(eq) exactly:&: 1, ci, cicj,u: ci, cicj, cicjck,T: cicj, cicjck, cicjckcl,
Assuming particle has linear d.o.f. only, d.o.f=D
・ Department of Aerospace Engineering・
Discretization of Phase Space
To obtain Navier-Stokes, must evaluate momentsof 1, c, …, c6, w.r.t. wt. fnctn e-mc•c/2kT exactly
Hydro-dynamic moments of f(eq): I = ∫ #(c) f(eq) dc
Use Gaussian-type quadrature to evaluate
�
I = m2!kT" # $
% & ' D / 2
((c)e) m2kT
c 21+ c •u
kT+ 12c •ukT
" # $
% & ' 2
) u•u2kT
*
+ ,
-
. / 0 dc
�
!(x)e"x2dx#
・ Department of Aerospace Engineering・
Square Lattice-Boltzmann Model 9-bit 2D
In Cartesian coordinates: Then
Where
�
!mn (c) = cxmcy
n
�
I = !mn (c) f (eq )dc"#
#
$ = m%
2kT( )m+n&
1" u2
2kT'
( )
*
+ , ImIn +
2 uxIm+1In + uyImIn+1( )2kT
+ux2Im+2In + 2uxuyIm+1In+1 + ux
2ImIn+2
RT
- . /
0 /
1 2 /
3 /
�
Im = e!"2" md" ,
!#
#
$ " = c/ 2kT
・ Department of Aerospace Engineering・
Square Lattice-Boltzmann Model 9-bit 2D Use 3rd order Hermite formula to evaluate
Where the 3 abscissas of the quadrature are:
And the corresponding weight coefficientsare:
�
Im = e!"2" md"
!#
#
$ = % j" jm
j=1
3
&
�
!1 = " 3/2, ! 2 = 0, ! 3 = 3/2
�
!1 = " /6, !2 = 2 " /3, !3 = " /6
・ Department of Aerospace Engineering・
Moment integral becomes
From which, parts of the equilibriumdistribution function are identified as
�
Im = m!
" i" j# ci, j( ) 1+ci, j •ukT
+ci, j •u( )22 kT( )2
$ u2
2kT
% & '
( '
) * '
+ ' i, j=1
3
,
�
fi, j(eq ) = m
!" i" j 1+
ci, j •ukT
+ci, j •u( )22 kT( )2
# u2
2kT
$ % &
' &
( ) &
* &
2D Square Lattice-Boltzmann Model 9-bit
・ Department of Aerospace Engineering・
Select possible molecular velocities on 2D squarelattice to go in as many directions of such square
The chosen velocities have a certain symmetry toaccount for molecules moving in any and alldirections independent of directions (isotropy)
For square this means sides & corners This means 9 possible velocities To go w/at least 9 terms in #mn(c) 1
2 3 4
7
e2 e 3
e1
e4
e5
e6e7 e8
5
6 8
2D Square Lattice-Boltzmann Model 9-bit
2
3
4
56
7
・ Department of Aerospace Engineering・
The 9 possible velocities are:
�
e! =(0,0),cos"! ,sin"!( )c,2 cos"! ,sin"!( )c,
"! = (! #1)$ /2,"! = (! # 5)$ /2 + $ /4,
! = 0,! =1,2,3,4,! = 5,6,7,8
%
& '
( '
2D Square Lattice-Boltzmann Model 9-bit
1
2 3 4
7
e2 e 3
e1
e4
e5
e6e7 e8
5
6 8
2
3
4
56
7
・ Department of Aerospace Engineering・
2D Square Lattice-Boltzmann Model 9-bit Then parts of the equilibrium distribution function are
identified as
Where the corresponding weight coefficients are now
And RT = cs2 = c2/3 or
�
f!(eq ) = mw! 1+ e! •u
c 2+e! •u( )22c 4
" u2
2c 2# $ %
& %
' ( %
) %
�
w! =" i" j
#=4 /9,1/9,1/36,
i = j = 2,i =1, j = 2,...,i = j =1,
! = 0,! =1,2,3,4,! = 5,6,7,8
$ % &
' &
�
RTr ! = 3RT = c
・ Department of Aerospace Engineering・
This is a straight-forward extension of 2D:
where
�
Im = m! 3 / 2 " i" j"k# ci, j ,k( ) 1+
ci, j,k •ukT
+ci, j ,k •u( )22 kT( )2
$ u2
2kT
% & '
( '
) * '
+ ' i, j ,k=1
3
,
�
e! =
(0,0,0)±1,0,0( )c, 0,±1,0( )c, 0,0,±1( )c,±1,±1,0( )c, ±1,0,±1( )c, 0,±1,±1( )c,±1,±1,±1( )c,
! = 0,! =1,2,...,6,! = 7,8,...,18,! =19,20,...,26
"
# $ $
% $ $
3D Cube Lattice-Boltzmann Model 27-bit
・ Department of Aerospace Engineering・
3D Cube Lattice-Boltzmann Model 27-bit
Then parts of the equilibrium distributionfunction are identified as
where
�
f!(eq ) = mw! 1+ 3e! •u
c 2+9 e! •u( )22c 4
" 3u2
2c 2# $ %
& %
' ( %
) %
�
w! =
8 /272 /27,1/54,1/216,
i = j = k = 2i = j = 2,k =1,...,i = j =1,k = 2,...,i = j = k =1,....,
! = 0,! =1,2,...,6,! = 7,8,...,18,! =19,20,...,26
"
# $ $
% $ $
・ Department of Aerospace Engineering・
• Equilibrium distribution function for f,
! 9-velocity model(2D):
! 15-velocity model:
! 19-velocity model:
!"
!#$
=%=%=%
=%.14,...8,7,72/1
6,...2,1,9/1
0,9/2
w
]2
3)(
2
931[
22
42)( uuueue !!+!+=
c-
ccwf eq """" #
!"
!#$
=%=%=%
=%.18,...8,7,36/1
6,...2,1,18/1
0,3/1
w
Lattice Boltzmann Equation
Qian YH, d'Humieres D, Lallemand P. Lattice BGK Models for Navier Stokes Equation.Europhys Lett 1992;17:479-484.
!"
!#$
===
=8,6,4,2 ,36/1
7,5,3,1 ,9/1
0 ,9/4
%%%
%w
・ Department of Aerospace Engineering・
Examples of 3-D lattice models
2'x
6
8
7
9
1011
12
2 1
4
3
5
14
13
Q15D3 lattice
19-velocity 3-D lattice model
5
6
8
7
9
10
11
1215
1617
2 1
4
3
13
18
14
Q19D3 lattice
15-velocity 3-D lattice model
2'x
・ Department of Aerospace Engineering・
Macroscopic variables is obtained from:
Chapman-Enskog analysis (multi-scale expansion) -NS Eqs. recovered in near incompressible flow limit.
Equation of state:
What are the advantages? What are potential benefits comparing with the standard CFD
methods for the Navier-Stokes equations? Let’s look at the actual implementation:
LBGK scheme.
,)(!=!="=#
#=#
#N
0
eqN
0ff !=!="
=###
=###
N
1
)eq(N
1ff eeu
3/2 !! == scp
Lattice Boltzmann Equation
・ Department of Aerospace Engineering・
LBGK Scheme: discretization in time & space - fa(xi + eadt, t + dt) - fa(xi, t) = -
Viscosity: + = (%-1/2) Order of accuracy: 2nd in x & 1st in t. Computation:
collision step:streaming step: fa(xi + eadt, t + dt) =
Advantages:• collision step is local; streaming step takes no computation.• explicit in form, easy to implement, and natural to parallelize.• Pressure is obtained simply as:
)],(),([1 )( tftf i
eqi xx !!"
#
tcs!2
�
˜ f ! (xi, t) " f! (xi, t) = ")(
~, tf ix!
3/2 !! == scp
Lattice Boltzmann Equation
)],(),([1 )( tftf i
eqi xx !!"
#
・ Department of Aerospace Engineering・
Collision:
Computational Procedure
Streaming:
t=t+'t
Calculate physical variables
�
˜ f !(t ) = f!
(t ) " 1#
f!( t ) " f!
( t,eq )[ ]
�
f!(t+"t ) xi + ei"t,t + "t( ) = f!
(t ) xi,t( )
�
!(t+"t ) = f#( t+"t )
#= 0
N
$
�
!(t+"t )u( t+"t ) = e# f#(t+"t )
#= 0
N
$
�
f! = f!(eq ) = "w![1+ 3
c 2e! # u + 9
2c 4(e! # u)2- 3
2c 2u # u]
Initialization:
Assign f(eq)
・ Department of Aerospace Engineering・
Boltzmann's H-theorem
Generally, macroscopic processes are irreversible. The relaxation to a Maxwellian distribution as a result
of collisions, is an irreversible process. H-theorem states that if the distribution function
evolves according to the Boltzmann equation, then fora uniform gas in the absence of external forces H cannever increase:
if we begin with a uniform gas having a non-equilibrium distribution function, H decreases untilthe gas relaxes to the equilibrium distribution when Hattains a minimum value�
dHdt
< 0
�
H = f log fdVc!"
"#
・ Department of Aerospace Engineering・
Applications of LBE Simulation of incompressible flows
Fully compressible and thermal flows
Multi-phase and multi-component flows
Particulate Suspensions
Turbulent Flows
Micro Flows
・ Department of Aerospace Engineering・
Streamlines in the cavity flow atRe=100
X
Y
0 50 1000
10
20
30
40
50
60
70
80
90
100
110
120
130
・ Department of Aerospace Engineering・
Instantaneous streamlines for channel flow overan asymetrically placed cylinder at Re=100
X
Y
50 10 0 15 0 2 00
10
20
30
40
50
・ Department of Aerospace Engineering・
Block and lattice layout for flow overNACA 0012
The lattice spacing is reduced by a factor 32 for graphicalclarity
X
Y
0 50 0 1 00 0 1 50 0 20 0 0 25 0 0
-3 0 0
-2 0 0
-1 0 0
0
1 0 0
2 0 0
3 0 0
4 0 0
5 0 0
6 0 0
・ Department of Aerospace Engineering・
Streamlines, pressure contour, velocity vector foruniform flow over NACA 0012 airfoil at Re=2000.
X
Y
55 0 6 0 0 6 50 7 0 0 7 5 0 80 0 8 5 0 9 00 9 5 0 10 0 00
2 5
5 0
7 5
1 0 0
1 2 5
1 5 0
1 7 5
2 0 0
・ Department of Aerospace Engineering・
Comparing Cd between present simulation andXfoil calculation vs. Re for NACA0012 flow.
10 410 310 210 -2
10 -1
10 0
Present
1/Re fitting
X-foil calculation
Re
Cd
1
2
1/2
The straight line is the slope according tothe laminar boundary layer theory
・ Department of Aerospace Engineering・
Rayleigh-Taylor Instability
Single mode perturbation
W = channel width
&h= heavy fluid&l = lighter fluid
�
Re = W Wg!
= 2048
�
A = !h " !l
!h + !l
= 0.5
・ Department of Aerospace Engineering・
Implementation of complex LBE model.
For complicated problems, new LBEmodels may need to be designed.
The number of particle velocities innew models can vary.
Hybrid modes which incorporatefinite different method in LBE needto be considered.
・ Department of Aerospace Engineering・
References for LBEMcNamara G, Zanetti G. Use of the Boltzmann equation to simulate lattice-gas automata. PhysRev Lett 1988; 61: 2332 –2335.
Higuera FJ, Jemenez J. Boltzmann approach to lattice gas simulations. Europhys Lett1989;9:663-668.
Koelman JMVA. A simple lattice Boltzmann scheme for Navier-Stokes fluid flow. EurophysLett 1991;15:603-607.
Qian YH, d'Humieres D, Lallemand P. Lattice BGK Models for Navier Stokes Equation.Europhys Lett 1992;17:479-484.
Chen H, Chen S, Matthaeus WH. Recovery of the Navier-Stokes equations using a lattice-gasBoltzmann method. Phys Rev A 1992;45:R5339-R5342.
d’Humieres D. Generalized lattice Boltzmann equations, In Rarefied Gas Dynamics: Theory andSimulations, ed. by D. Shizgal and D.P. Weaver. Prog. in Astro. Aero. 1992;159:450-458.
Bhatnagar PL, Gross EP, Krook M. A model for collision processes in gases, I. small amplitudeprocesses in charged and neutral one-component system. Phys Rev 1954;94:511-525.
He X, Luo L-S. A priori derivation of the lattice Boltzmann equation. Phys Rev E1997;55:R6333-R6336.
He X, Luo L-S. Theory of the lattice Boltzmann equation: From Boltzmann equation to latticeBoltzmann equation. Phys Rev E 1997;56:6811-6817.
・ Department of Aerospace Engineering・
Boltzmann Equation
Studying molecules (class ci) inside dVx, wesee total number of molecules whose velocityis between ci and ci + dVc isnf(ci) dVx dVc
Change of number of molecules in class cimust result from convection of moleculesacross the surface of dVc and dVx or fromintermolecular collision within
・ Department of Aerospace Engineering・
Number increased by convection thru dVx
1dx
3dxThe total number increase byconvection through surface
of dVx in unit time is:
ci is taken out of the differentiationbecause of it is independent of xi
x2
x1
x3
dx1
dx3
dx2
c3n f(ci)dVcdx1dx2
c1n f(ci)dVcdx2dx3
�
c2nf (ci) + !!x2
c2nf (ci)[ ]dx2" # $
% & ' dVcdx1dx2
�
c1nf (ci) + !!x1
c1nf (ci)[ ]dx1" # $
% & ' dVcdx2dx3
�
c3nf (ci) + !!x3
c3nf (ci)[ ]dx3" # $
% & ' dVcdx1dx2
c2n f(ci)dVcdx1dx3
�
!c j""x j
nf (ci)[ ]dVcdVx
・ Department of Aerospace Engineering・
Boltzmann Equation
If there is no external force, the convection ofmolecules through the surface of dVc is zero.
The rate of increase of number of molecules inclass ci results from collision is:
The total rate of increase of number ofmolecules of class ci is:
ic
�
!!t
nf (ci)[ ]" # $
% & ' collision
dVcdVx
�
!!t
nf (ci)[ ]dVcdVx
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Boltzmann Equation
The rate of increase of the number of molecules ofclass ci in the volume element is equal to the rateof increase by convection through surface of dVxplus the rate of increase by collision.
This gives the Boltzmann Equation:
How to determine the collision operator?
xdV
�
!!t
nf (ci)[ ] + c j!!x j
nf (ci)[ ] = !!t
nf (ci)[ ]" # $
% & ' collision
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Solid wall boundary
Ghost point
wall
x1
x0
Fluid pointAfter collision, set:
bounce back idea
If the wall has a velocity u, momentum flux can beadded:
�
˜ f 7 x1( )
�
˜ f 3 x0( )
�
˜ f 3 x0( ) = ˜ f 7 x1( )
�
˜ f 3 x0( ) = ˜ f 7 x1( ) + 6!w3 e3 "u( )
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Implementation in moving boundaryproblems
p0 p1
p2
p3
Particles are moving; they may belong to differentprocessors at different time.
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Multi-block method in LBE
Different grid resolutionin different block.
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Implementation of adaptive grid method inLBE
Around the corner, we put fine grids.We may also need to increase gridresolution during computation.
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Flows in porous media
In solid region,computation andmemory are notrequired.
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Boltzmann’s H Theorem Generally, entropy, S, is defined to be related to the
# of possible arrangements of molecules In other words, entropy is a function of # of
possible micro-states, . The probability distribution function provides a
way of determining the # of possible macro-states, Because . for combined systems of certain micro-
states is a product of each, .AB = .A .B And because total entropy is SAB = SA + SB
S = - k log . = - k ∫ f log f dVc,
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Boltzmann’s H Theorem Differentiating H, using Boltzmann’s eq., & combining
terms:
H theorem shows why entropy is automatically satisfied ifBoltzmann’s eq. is satisfied
Many CFD schemes do not easily guarantee that they willnot violate entropy
Through solving BE,• entropy is inherently & automatically satisfied and• fluid field is found from f rather than using f to find properties to
use in Navier-Stokes eqs. derived from BE
�
dHdt
= ! 14
f (c') f (z') ! f (c) f (z)[ ] log f (c) f (z)f (c') f (z')
c ! z"d#dVz$$ % 0