Post on 31-Dec-2015
description
IPAM Plasma Tutorials 2012
Kinetic Theory for Gases and PlasmasLecture 1. Gas Kinetics
Russel CaflischIPAM
Mathematics Department, UCLA
IPAM Plasma Tutorials 2012
Outline• Gas kinetics
– Macroscopic variables ρ,u, T– Velocity distribution function f(x,v,t) – Hard sphere collisions (short range)– Boltzmann equation
• Plasma kinetics– Debye length λD
– Coulomb interactions (long-range)– Landau-Fokker-Planck equation
• Simulation methods– Direct Simulation Monte Carlo (DSMC) for gases– Binary collision methods for plasmas
IPAM Plasma Tutorials 2012
Plasmas
• Plasma– gas of ionized
particles
– 99% of visible matter
• Examples– fluorescent lights
– sun
– fusion energy plasmas
IPAM Plasma Tutorials 2012
Gas or Plasma Flow: Kinetic vs. Fluid
Kinetic description• Discrete particles• Motion by particle velocity• Interact through collisions
Fluid (continuum) description• Density, velocity, temperature• Evolution following fluid eqtns (Euler or Navier-Stokes or MHD)
When does continuum description fail?
IPAM Plasma Tutorials 2012
When Does the Continuum Description Fail?
• Knudsen number Kn=ε– ε = (mean free path)/(characteristic distance)– measures significance of collisions– mean free path = distance traveled by a particle
between collisions
• Rarefied gases and plasmas– Ε not negligibly small
IPAM Plasma Tutorials 2012
Rarefied vs. Continuum Flow:Knudsen number Kn
IPAM Plasma Tutorials 2012
Collisional Effects in the Atmosphere
IPAM Plasma Tutorials 2012
Collisional Effects in MEMS and NEMS
Gas kinetics
IPAM Plasma Tutorials 2012
IPAM Plasma Tutorials 2012
Velocity Distribution Function
• A typical gas or plasma consists of too many particles for a tractable direct description– typical number of 1020 particles
• Statistical description– Probability density function f=f(x,v,t) – phase space (position x, velocity v)– Maxwell & Boltzmann
IPAM Plasma Tutorials 2012
Macroscopic Variables and Equilibrium
• Macroscopic variables– Density ρ, velocity u, temperature T
• Maxwellian equilibrium3 2 2( ) (2 ) exp( ( ) 2 )M T T v v u
2
1
( )
1
3
u v f v dv
Tv u
IPAM Plasma Tutorials 2012
Collisions
• Velocities -v,w before collision-v’, w’ after collision
• Momentum and energy conservation- v + w = v’ + w’- |v|2 + |w|2 = |v’|2 + |w’|2
-Two free parameters- ω = (ε,θ) on sphere- θ = scattering angle- ε = angle of collision plane
-Explicit formulas
vv’ w’
w
' ( )
' ( )
v v v w
w w v w
v-w
2θ
IPAM Plasma Tutorials 2012
Collision Cross-Section
•The differential collision cross section (per particle)
- the area (per angle and per particle) governing the collision rate -for particles with velocity difference v
• The rate of collisions between velocities v and w with collision parameters in dω is
-Factor of |v-w| accounts for flux-f2 is two particle distribution function
-Hard spheres of radius r
( , )v
2( , ) ( , )v w v w f v w d dv
2 cosr v-w
2θ
Molecular Chaos • Boltzmann assumed particles are
independent before a collision– Before collision of v and w
– Particles not independent after collision
– Proof by Lanford (1976) as n→∞
– Origin of irreversibility
• For a given velocity v– Rate of collisions v → v’
– Rate of collisions v’ → v
IPAM Plasma Tutorials 2012
2 ( , ) ( ) ( )f v w f v f w
( , ) ( ) ( )v w v w f v f w dw
( , ) ( ') ( ')v w v w f v f w dw
Boltzmann’s grave
Boltzmann Equation
• Evolution of velocity distribution function f(x,v,t) due to– Convection of particles by velocity– Collisions between particles
IPAM Plasma Tutorials 2012
( , )( ) ( , ) ( ') ( ') ( ) ( )
( , ) ( , ) sin
Q f f v B v w f v f w f v f w d d dw
B v v v
( , )x
fv f Q f f
t
Collision InvariantsBy symmetries of collision operator Q
in which
IPAM Plasma Tutorials 2012
1
1 1
1 1 1
1 1
1 1
1
1
1 1
( ) ( , )
( , ) ( ) ( ') ( ') ( ) ( )
( ) ( )
1' ' ' '
' '
' '
' ' '
' ' '
4
v Q f f dv
B v w v f v f w f v f w d d dwdv
B f f f f d d dwdv
B f f
v f v dvt
B f f f f d d d
f f d d dwdv
B f f f f d d dwdv
B f f f f d d dw
w v
dv
d
1 1, , ', ' ( ), ( ), ( '), ( ')f f f f f v f w f v f w
Collision Invariants
• By symmetries of collision operator Q
• This is 0 for all f and g iff
for all v, w, θ, ε, which implies
• Corresponds to conservation of mass, momentum and energy
IPAM Plasma Tutorials 2012
1 1 1
0 ( ) ( , ) 0
( ' ') ' '
v Q f g dv
B f f fg d d dwdv
1 1' ' 0
1 2 3
2
0 4 ( ,) , )(v c c v c v c c c c
Entropy
• Let φ=log f, then
• It follows that
Boltzmann’s H-theorem
IPAM Plasma Tutorials 2012
1 1 1 1
1
1
( ' ') log log log ' log '
log' '
f f f f
f f
f f
11 1
1
( ) ( , )
1' ' log
4 '
og
'
l
0
t v Q f g dv
f ff
f f d
f f f d d dwdvf f
v
Entropy and Equilibration
• Since
with equality iff ; i.e.,
Then
IPAM Plasma Tutorials 2012
log 0t f f dv
3 2 2( ) ( ) (2 ) exp( ( ) 2 )f v M T T v v u
f M as t
2
0 4log ( )f v c c v c v
Fluid Dynamic Limit of Boltzmann
• The limit of small mean free path ε
• Hilbert expansion
• Leading order ε-1
IPAM Plasma Tutorials 2012
1( , )x
fv f Q f f
t
0 1f f f
0( , ) , , )0 ( ;fQ M T vf uf
Fluid Dynamic Limit of Boltzmann• Order ε0
• Conservation laws
• Use to get
IPAM Plasma Tutorials 2012
0 0 12 ( , )t xv f Q f f
0 ( , , ; )f M u T v
0
2 2
1 1
( , ) 0 0tv Q f f dv v v f dv
v v
( ) 0
0
( ) 0
t
t
t
u
u u u p
E uE up
21( 3 )
2E u T
p T
IPAM Plasma Tutorials 2012
Dominant numerical method for dilute flows
• DSMC = Direct Simulation Monte Carlo– Invented by Graeme Bird, early 1970’s
– Represents density f as collection of particles
• γ = number of physical particles per numerical particle
>>1
– Perform randomly chosen collisions
• Velocity pairs vk and vℓ and collision angles (ε,θ)
• vk , vℓ → vk’ , vℓ’
– Particle advection
1
( ) ( ( )) ( ( ))N
k kk
f v v v t x x t
/k kdx dt v
IPAM Plasma Tutorials 2012
DSMC Features
• Fidelity to physics– Each collision is a (randomly chosen) physical collision
– Each particle has physically correct number of collisions• Probability of collision (vk ,vℓ , ε, θ) proportional to B(θ, | vk - vℓ |)
– Total numerical collision rate
= physical collision rate for N particles
= γ-1 (total physical collision rate)
• Convergence (Wagner 1992)
,
( , )kk
c B v v d
Simulation for Gases:Test Cases
• Relaxation to equilibrium
• Shock
• Flow past a plate
IPAM Plasma Tutorials 2012
HKUST, 9 Dec 2007
Relxation to Equilibrium• Spatially homogeneous, Kac model• Similarity solution (Krook & Wu, 1976)
Comparison of exact solution (-), DSMC(+) and IFMC(◊)At time t=1.5 (left) and t=3.0 (right).
v v
f f
HKUST, 9 Dec 2007
Comparison of DSMC (blue) and IFMC (red) for a shock with Mach=1.4 and Kn=0.019
Direct convection of Maxwellians
ρ u
T
HKUST, 9 Dec 2007
Comparison of DSMC (contours with num values) and IFMC (contours w/o num values)
for the leading edge problem.
ρ T
u v
IPAM Plasma Tutorials 2012
Conclusions and Prospects• Velocity distribution function• Molecular chaos• Boltzmann equation• H-theorem (entropy)• Maxwellian equilibrium• Fluid dynamic limit• DSMC
– DSMC becomes computationally intractable near fluid regime, since collision time-scale becomes small
– math needed to design methods that overcome this difficulty!– Multiscale method combining fluid & particle descriptions
• RC, Ricketson, Rosen, Yann (UCLA), Cohen , Dimits (LLNL)