basic boltzmann gas concepts. kinetic theory complete set of position (x) and momentum (p)...
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Basic Boltzmann Gas Concepts
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Kinetic Theory
• Complete set of position (x) and momentum (p) coordinates for all individual particles gives exact dynamical state of system
• Together with classical mechanics, allows exact prediction of future states
• However, this level of description is essentially not possible
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Statistical Mechanics
• Represent system by ensemble of many copies
• Describe by distribution function f(N)
(xN,pN,t); N is number of particles
• Changes in f(N)(xN,pN,t) with time given by Liouville equation (6N variables)
• Usually interested in low order distribution functions (N = 1, 2)
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First Order Distribution Function
• f(1)(x,p,t) gives probability of finding a particular molecule with given position and momentum; positions and momenta of remaining N-1 molecules unspecified
• No experiment can distinguish between molecules, so the choice of which molecule doesn’t matter
• ‘Single particle’ distribution function• f(1) adequate for describing all gas properties that
don’t depend on relative positions of molecules (dilute gas with long mfp)
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• External force Xi (small relative to intermolecular forces)
• For each component i there is an fi(1)(x,pi,t)
such that probable number of type i molecules with position coordinates in the range x ±dx and momentum coordinates pi ±dpi is fi
(1)(x,pi,t) dxdpi
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Expected Evolution w/o Collisions(streaming)
• If no collisions, then at time t + dt, the new positions of molecules starting at x are [x + (pi/mi)dt]
• New momenta are pi = pi +Xidt
• Thus,
iiiiiii
ii ddtfdddttdtdt
mf pxpxpxXp
px ,,,, )1()1(
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Collisions
• But there are collisions that result in some phase points starting at (x,pi) not arriving at (x + pi/mi dt, pi+Xi dt) and some not starting at (x,pi) arriving there too
• Set ij(-)dxdpidt equal to the number of molecules
that do not arrive in the expected portion of phase space due to collisions with type j particles during time dt
• Similarly, set ij(+)dxdpidt equal to the number of
molecules that start somewhere other than (x, pi) and arrive in the portion of phase space due to collisions with type j particles during time dt
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Account for collisions
dtddddtf
dddttdtdtm
rf
ij
ijijiii
iiii
ii
prprpr
prXpp
,,
,,
)1(
)1(
iiiiiii
ii ddtfdddttdtdt
mf pxpxpxXp
px ,,,, )1()1(
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Taylor Series Expansion of lhs 0
0
0)(
!)( xx
k
xfxf
k
k
k
iiii
ii dddttdtdt
mrf prXp
p
,,)1(
ii
i
ii
ii
iii dddt
t
fdt
fdt
f
mtf pr
pX
rppr
)1()1()1()1( 1
,,
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Boltzmann Equation
dtddddtf
dddtt
fdt
fdt
f
mtf
ij
ijijiii
ii
i
ii
ii
iii
prprpr
prp
Xr
ppr
,,
1,,
)1(
)1()1()1()1(
jijij
i
i
ii
ii
i t
fff
m
)1()1()1(1
pX
rp
Time rate of change of f
Hirschfelder, Curtiss, and Bird, 1954. Molecular Theory of Gases and Liquids, John Wiley and Sons, Inc. New York.
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Lattice Boltzmann Model
e1
e3
e2
e5
e6
e4
Unit Vectors ea
FHP D2Q9
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LBM Basics ILBM Basics I
Lattice Unit, lu
0 11
22
33
44
5566
77 88
D2Q9
e1
e2
e3
e4
e5e6
e7e8
f1
f2
f3
f4
f5f6
f7 f8
f0
Histogram view of the distribution function, f.
f1f5
1 2 5 a
f
f2f3
f4
3 4 6 7 8
f6 f7 f8
Notes:• Based on statistical mechanics and kinetic theory.• Solute and buoyancy will be covered in later slides.
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Lattice Boltzmann Model
f6
f5
f4f3
f1f2
e1
e3
e2
e5
e6
e4
Unit Vectors ea
Direction-specific particle densities fa
a
af
a
aaf eu
Density Velocity
f7 (rest)
Macroscopic flows
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Single Relaxation Time BGK (Bhatnagar-Gross-Krook) Approximation
2242
0)6,...,2,1( 22
)2(1)( uuueeuexx
bc
D
bc
DD
bc
D
b
df aaa
eqa
tftftfttf
eqaa
aaa
,,,1,
xxxex
Streaming Collision (i.e., relaxation towards local equilibrium)
• relaxation time
• d0 fraction of rest particles
• b number of unit velocity directions
2
2
0)7( )(c
df eqa
uxx
• D dimension of space
• c maximum speed on lattice (1 lu /time step)
Collision and streaming steps must be separated if solid boundaries present (bounce back boundary is a separate collision)
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D2Q9 feq
2
2
4
2
2 2
3
2
931)(
cccwf a
eqa
uueuexx aa
wa are 4/9 for the rest particles (a = 0), 1/9 for a = 1, 2, 3, 4, and 1/36 for a = 5, 6, 7, 8.