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Discrete Kinetic Theory of Gases
Renée Gatignol
Institut Jean le Rond d'Alembert
Université Pierre et Marie Curie & CNRS (UMR 7190)
Lattice Boltzmann Scheme Tutorial, Paris, 19 January 2010
Discrete Kinetic Theory of Gases: Outline
1. Introduction
2. Discrete kinetic theory
3. Hydrodynamical description for regular discrete models
4. Boundary conditions
5. Applications
6. Conclusion
1. Introduction
Introduction
Balance laws
Entropy inequality
Constitutive laws
Boltzmann equation
)f,f(f JD =
Dimensionless
Boltzmann equation
1≅ε
1<<ε
L/
)f,f()/1(f
λ=εε= JD
Euler, Navier-Stokes, Burnett, … equations
Free molecular flows
Transition flows: Waveshock structure, Knudsenlayer, …
1>>ε
Need models
AuthorsVelocitynumber
Introduction: The first works
Study of the H-function4Harris1967
Discrete kinetic theorypR.G.1970
Kinetic and hydrodynamical descriptions6Godunov & Sultangazin
1971
Lattice gases4Hardy & Pomeau
1972
Ternary collisions and H-Theorem6Harris1966
Shock wave structure6R. G.1965
Couette and Rayleigh problems8Broadwell« «
Shock wave structure6Broadwell1964
The velocity discretization is emphasizedGross1960
H-Theorem2Carleman1957
Subject
R. G.
Discrete kinetic theory
4 theses, about 20 papers and 20 proceedings1965 -…
Boundary conditionsPhysics of Fluids1977
Physics Fluids1975
Lecture Notes in Physics (Vol. 39)1975
« Théorie cinétique des gaz àrépartition discrète de vitesses »
Zeitschrift fürFlugwissenschaften
1970
Shock structure, H-Theorem, General kinetic equations, Chapman-Enskog expansion, …
6 CRAS1965 -1972
H. Cabannes
Lecture notes, Berkeley University, « The discrete Boltzmann Equation »
1980
On the solutions of the discrete kinetic equations(existence theorems, exact solutions)
1977 -…
Shock structure (14 velocities)J. de Mécanique1975
2. Discrete Kinetic Theory of Gases
Discrete Kinetic Theory of Gases
� In discrete kinetic theory, the main idea is that thevelocities of the molecules belong to a given set of vectors
� The Boltzmann equation is replaced by a system ofpartial differential equations
� This system has an interesting mathematical structure (H. Cabannes, Bellomo, Cercignani, Kawashima, …)
� The discrete models, by their simplicity, help to understand the fundamental problems of rarefied gasdynamics
� The hydrodynamic description of discrete gases isobtained via the Chapman-Enskog expansion
The particles are identicalThe particle velocities belong to a given set of vectors:
denotes the number of particles with velocity(i.e. particle « k ») per unit of volume
Macroscopic quantities
Discrete kinetic theory: Binary collisions
kur
)t,x(Nk
r
p...,,2,1k,u k =r
kkNn ∑=
kkkuNunrr
∑=
2kkk
)uu(N2
men
rr −= ∑
)uu()uu(NmP kkkk
rrrrrr
−−= ∑
)uu()uu(N2
mq k
2kkk
rrrrr−−= ∑
lkijA
ijk
kij AA
l
l =
Transition probability
Binary collision
Microreversibility property
jur
kur
l
ru
iur
In the collision, the
mass, momentum and
energy are conserved
ji u,urr
Before the collision
l
rru,u k
After the collision
Coplanar models
Discrete kinetic theory: Examples
Spatial models with 6 velocities or with 8 velocities
(Broadwell, 1964)
OO
Kinetic equations (binary collisions)
( )p21 N...,,N,N=N
p...,,2,1k,LGNuNt
kkkkk =−=∇⋅+∂∂ rr
( ) p...,,2,1k,NNANNA2
1NuN
tk
jikji
kjijikkk =−=∇⋅+
∂∂
∑ ll
l
l
rr
Kinetic equations
Notations
∑=><k kk VU,V U
),( V UFFFF
),(t
N NNN FFFFAAAA =+∂∂
Linear mapping of intopp RR × pR
Symmetry property
( ) ll
l ,k,j,i0A jikkij ∀=ϕ−ϕ−ϕ+ϕ
( ) pp21 R,...,, ∈ϕϕϕ=φ
F
Kinetic densities
Summational invariants: such as
( ) ( )ijjijikijkijk
VUVUA8
1),(, +ϕ−ϕ−ϕ+ϕ−=> ∑φ<
lllV UFFFF
q...,21: VV,VFBase in
)qofensiondimR( p =⊂ FF
pR∈φ
Linear subspace
p...,
1q,
q...,21:R p WWVV,V
+Base in
β+α= β
=β
+=βα
=α
=α∑∑ WbVaNp
1
q
1 q
Macrocospic variable Microscopic variable
Equations for the and the variables
p...,,2q,1q,,),(t
++=β>β<=>β<+∂
∂ βWN NW,N
bFFFFAAAA
><= ),(lnH N N,N FFFF
q...,,2,1,0x
)...,,,(
t
)...,,,( q212q
1
q212q
1
=α=∂∂
∂∂∂
+∂
∂∂∂
∂ δ
δα
=δ
=δ
δ
δα
=δ
=δ∑∑
c
cc
cccc
cc
ccc ML
conservation lawsq...,,2,1,0t
=α=>α<+∂
∂ α V,Na
AAAA
),(t
N NNN FFFFAAAA =+∂∂
βbαa
H – Theorem: H is decreasing with
Maxwellian state:
Euler equations associated with the model: Equations for the
variables or equivalently for the variables
αα
=α
=α∑=⇔=⇔∈ VcNN NFN
q
1
ln0),(ln FFFF
αa αc
Two problems are present in discrete kinetic theory:
In order to reduce and possibly to eliminate them,
multiple collisions are introduced and some symmetry
properties on the models are adopted
1. The existence of macrocospic variables
other than mass, momentum and energy
2. The anisotropic character generally
related to the discrete models
r
r
J
IATransition probability:
The multiple collisions
is the algebraic number of particles « k » created
in the r – collision
A r – collision is a collision between r particles:
)J,I,k( rrδrr JI →
is the algebraic number of
particles « k » created in all the r – collisions (per
unit time)
r21
r
r
rr
iiiJ
Irr
JI
N....NNA)J,I,k(δ∑
Before the collision After the collisionr21 iii u,...,u,u
rrr
)i,...,i,i(I r21r =
r21 jjj u,...,u,urrr
)j,...,j,j(J r21r =
Kinetic equations with r – collisions (r = 2,3,…,R)
r,J,I0)J,I,k(A rrkrrk
J
Ir
r∀=ϕδ∑
p...,,2,1k
N....NNA)J,I,k(2
1NuN
t r21
r
r
rr
iiiJ
Irr
JIR,...,3,2rkkk
=
δ=∇⋅+∂∂
∑∑ =
rr
F
Summational invariants
)(t
NNN CCCCAAAA =+∂∂
)R( p⊂F
pp21 R)...,,,( ∈ϕϕϕ=φ
Linear subspace
Two remarks
� By taking into account multiple collisions, the dimension of
is decreasing
� By taking into account all the r – collisions, it is possible to find
the dimension of , without explicitly determining all the
collisions between the particles (Ph. Chauvat)
F
F
Examples: Dimension of is 4 or 5
Generalizations
Spatial models related to
the cubeCoplanar models related to
the hexagonal lattice
F
O
Velocity number
6
8
14
26 5dim
Z)c,b,a(
,KcJbIau
3kkk
kkkk
F =∈
++=rrrr
Navier – Stokes
equations for
Maxwellian densities
Euler equations for
Chapman – Enskog expansion
ε
)0(N )1()0( NN ε+
Linearized collision operator
But: To obtain balance laws for the variables
⋅⋅⋅+ε+ε+= )2(2)1()0( NNNN
),(ε
1
tN NNN CCCCAAAA =+
∂∂
Chapman – Enskog expansion
( Knudsen number)1<<ε
0),( )0()0( =N NCCCC
)0()0()1(
t)( NNN AAAAJJJJ +
∂∂=
q...,,2,1,0t
=α=>α<+∂
∂ α V,Na
AAAA
αa
with , the variables depending on the),( βα= ba NN βb αa
αaαa
3. Hydrodynamical description for
regular discrete models
{ }ll
ll
l
rrp,...,2,1k,cu,u kk ===U
{ }l
l
l
rUU L
1k p,...,2,1k,u ==∪===
Mean properties
Regular discrete models
Examples: Coplanar models related to the hexagonal lattice,
Spatial models related to the cubic lattice
" Quasi – isotropic " models (Chauvat, Coulouvrat, R.G.)
The successful simulations undertaken with the lattice gas
method introduced by Frisch, Hasslacher and Pomeau, …
have provided a new light on the discrete models of gas
2DFdim +=
G UU ∈∀= g)(g
G
(The multiple collisions are introduced)
: Isometry group in RD
Quasi - homogeneous Maxwellian state 1a/u,1a/e 2 <<<<∆ r
Hydrodynamical description of the gas
Maxwellian state ( )( )22kk
)0(k auuexpN −γ+⋅β+α= rrr
,Nn)0(
kk∑= ,uNun k)0(
kk
rr∑= 2)0(
kk kuN
2
1en
r∑=
e,u,n,,rr
⇔γβα Bijection
aup
1a
2/12
k k1 ≡
= ∑
rr2/1
r2
k kr up
1a
= ∑
rNotations:
Homogeneous Maxwellian state )a2
1e,0u(
p
nN 2)0(
k === r
( )
−
−
−−
+
−+
−
−−
++=
)uu(2
ae
a
1
aa
au
a
D2
uuID
uuu
a2
D
2
ae
aa
au2uu
a
D1
p
nN
k
2
2442
22k
2
2
2k
kk4
22
442
22k
k2
)0(k
.
. :
rrr
rrrr
rrr
rrr
ρ−ρ⋅∇ϕ=
ϖ+ρ+ρ⋅∇η+ρ+ρ
∂∂
ρ⋅∇+
ρ∇−η=ϖ∇+ρ⋅∇η+
∂ρ∂
=ρ⋅∇+∂
ρ∂
u2
aeu)
2
ue()
2
ue(
t
MD
u)1()uu(
t
)u(
0)u(t
222
an
2
rrrr
rr
rrrr
rrrrrr
rr
Euler equations associated with the model
η and ϕ depend on the discrete models
Continuous fluids:
Remarks: We can provide equivalent forms of the Euler
equations; for example with a pressure tensor not necessarily
spherical or a vector heat flux not necessarily zero
2D
D
a
a4
42
+=η
01 =ϕ=η
( )442
4
63
242
482
aaa
aaaaa2
−−−
=ϕ
( )eu
2
aeu)
2
ue()
2
ue(
t
BI)D(TrD2
MD
u)1()uu(
t
)u(
0)u(t
222
anD
an
2
∇κ⋅∇+
ρ−ρ⋅∇ϕ=
ϖ+ρ+ρ⋅∇η+ρ+ρ
∂∂
∇+
λ+µ∇+
ρ⋅∇+
ρ∇−η=ϖ∇+ρ⋅∇η+
∂ρ∂
=ρ⋅∇+∂
ρ∂
⋅⋅
rr
rrrr
rr
rrrrrrrrrr
rrrr
rrrrrr
rr
Navier -Stokes equations associated with the model
Remarks: isotropickkkk k uuuurrrr
∑
( )4
L...,,2,12
4
L...,,2,12a
L
1
aD,0D2,a
L
1
a2Dllllll
δρ≅κ=λ+µβ+ρ≅µ ∑∑ ==
0B0M anan ==rrrr
Numerical results for the transport coefficients
0.250.560.14Chahine (1967)**
0.280.500.14Chauvat (1989)*
0.610.410.25IV
0.920.240.22III
0.730.330.24II
0.620.240.15I
PrModel
Models: I (12) II (12) III (18) IV (18)
κµ
κ=κ2S
ma
κµ=Pr
µ=µ2S
ma
* model I (η = 0.67, ϕ = 0
** coplanar continuous model
4. Boundary Conditions
(as n) is unknown; is known when the problem is solved
Particular case of the diffuse reflexion:
The densities are Maxwellian densities associated with the
macroscopic variables of the wall
)Rr( ∈
)Ii( ∈iur
: probability for a particle of velocity impinging the wall,
to be reflected with the velocity
Boundary conditions on an impermeable wall (R.G., 1975)
H – Theorem in a vessel
riB
rur
RrNn)uu(BNn)uu( iwiri
Ii
rwr ∈∀⋅−=⋅− ∑∈
rrrrrr
gaswallEmerging particles
Incident particles
rur iu
r
nr
wrr NN λ=wrN
...,T,u,1n www
r=λλ
ww Tur
: saturation density of the vapor at the temperature
Remark: This boundary condition is valid only when the models
are symmetrical about the normal . This condition is similar to
that of the continuous kinetic theory
Boundary conditions for the vapor
Boundary conditions on an interface
Gas in Maxwellian equilibrium with the condensed phase
( )2
kkwk uuexpNrrr
γ+⋅β+α=
Rr,NnN wrsatr ∈∀=
nr
wTsatn
)Rr( ∈
)Ii( ∈iur
rurvapor
condensed
phase
nr
ww T,0u =r
,1Nkwk=∑ ,0uN kkwk
=∑r
m
Tk
2
3)uu(N
2
1 w2kkk
=−∑rr
5. Applications
Applications
� Shock wave structure
� Unsteady and steady Couette flows (Knudsen layer, initial layer, …)
� Flow and heat transfer between two parallel plates
� Evaporation / condensation between two interfacex(temperature inversion)
� Evaporation or condensation on a liquid interface
� Flow in a microchannel
� …
Flow and heat transfer between two parallel plates (d’Almeida)
6.2T
5.0u
w
w
=
=−
−
+
+
w
w
T
u
5.0− 5.0
0.5
Red line: test case
Blue line: Thermophoresis phenomenon
Temperature2.0u,2T ww == ++
Longitudinal velocity
5.0u,6.2T ww == ++
1
−wT/T
5.0Kn =
Evaporation / condensation between two interfaces (d’Almeida)
2Tw =− 67.2Tw =+
5.0− 5.0Cold phase:
condensation
Hot phase:
evaporation
1.0Kn =
−+
−
−−
ww
w
TT
TT15.0n =
08.0n =
1.0n =
06.0n =Temperature
inversion
These problems depend on 3 parameters :
The results obtained with a very simple model (16 velocities only) are in
very good agreement with those of Sone, Aoki and their collaborators with
continuous theory
Evaporation or condensation on a liquid interface (Nicodin)
Condensed
phase
0θ0<ωνθ ∞∞∞
Condensation
Condensed
phase
0θ0>ωνθ ∞∞∞
Evaporation
0M,,0
>θ
ω=ν
θθ
∞
∞∞∞
∞
Condensation problem (Nicodin)
414.1M,12,10
==ν=θθ
∞∞∞
707.0M,4,10
==ν=θθ
∞∞∞
The vapor is compressed on the
condensed phase (η = 0). The
steady state is obtained for
about 30 mean free times
There is a Knudsen layer near
the condensed phase (η = 0). A
compression wave (shock
wave) propagates to infinity
Evaporation problem (Nicodin)
058.0M,8.0,10
==ν=θθ
∞∞∞ 316.0M,1,1
0
==ν=θθ
∞∞∞
1M,01.0,10
==ν=θθ
∞∞∞ 73.1M,25.0,1
0
==ν=θθ
∞∞∞
6. Conclusion
Many generalizations
Gas mixtures (Cercignani, Cornille, …)
Chemical reactions (Pandolfi, …)
Numerical approaches (Leguillon, Teman, Golstein, …)
Semi discrete Boltzmann equation (Cabannes, Toscani, …)
…
Many mathematical papers
Existence theorems, exact solutions, asymptotic analysis, …
Cabannes, Bardos, Beale, Bellomo, Bobylev, Bony,
Cercignani, Cornille, Godunov, Golse, Hamdache, Illner,
Kawashimha, Levermore, Nishida, Platkowski, Sultangazin,
Tartar, Vedenyapin, …
Conclusion
Special thanks to:
Alain Fanget (Thesis, 1980)
Philippe Chauvat (Thesis, 1991)
Amah d’Almeida (Thesis, 1994)
Ioana Nicodin (Thesis, 2001)
François Coulouvrat (DEA, 1987)
Mohammed Kane (DEA, 1977)
Ayaovi Bodjrenou (DEA, 2000)
Yassine Benbouali (DEA, 2003)
………
Thank you for your attention
