SLIDE 1
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Unique moment set from the order of magnitude method
Henning Struchtrup
University of Victoria, Canada
SLIDE 2 Goal: Find approximation to kinetic equation for Knudsen order O ¡ ελ¢ Chapman-Enskog expansion:
- gives classical transport laws for λ = 1
- stability problems for λ ≥ 2
- full summation not feasible
Grad moment method:
- no relation to Knudsen number
- (mostly) hyperbolic equations
- unphysical subshocks
- rder of magnitude method [HS2004]:
- combines ideas of Grad and CE methods
- moments and their equations directly related to Knudsen order
- regularizing terms remove subshocks, damp high frequencies
SLIDE 3
Moment method in kinetic theory
kinetic equation: Knudsen number ε ∂f ∂t + ck ∂f ∂xk = 1 εS (f) moment method: replace kinetic equation with equations for moments uA = Z ψA (ci) fdc A = 1, . . . , N moment equations: multiply kinetic equation with ψA (ci) and integrate ∂uA ∂t + ∂FAk ∂xk = 1 εPA
with FAk = R ψA (ci) ckfdc , PA= R ψA (ci) S (f) dc
Question 1: which moments, ψA (ci) = ?? Question 2: how many moments, N = ?? Question 3: how to close the equations, FAk (uB) = ??, PA (uB) = ?? Q3 answered by Grad [1949], Q1 and Q2 today
SLIDE 4 Order of magnitude method
[HS 2004]
Step 1:
- set up moment system for arbitrary number of moments N
- use infinite system (if possible), or close with Grad method
- must use arbitrary complete function system for moments
Step 2:
- Chapman-Enskog expansion to find leading ε−order of moments
- linear combination of moments such that number of moments at given
ε−order is minimal
- repeat for next order of magnitude
Step 3:
- use ε−orders to rescale equations for new moments
- use scaling for model reduction to a given order of accuracy
SLIDE 5 Step 0: A simple kinetic model and its properties
kinetic model for 1-D heat transfer: simplified phonon/photon model with scattering
∂f ∂t + μ∂f ∂x = − 1 ετ κ (μ) ∙ f − R κ (μ) fdμ R κ (μ) dμ ¸
f (x, t, μ) - distribution function, ε- Knudsen number, μ = cos ϑ - direction cosine anisotropic scattering probability (γ = 0 for isotropic scattering) κ (μ) = 1 + γμ2 energy density and heat flux are moments u0 (x, t) = Z 1
−1
f (x, t, μ) dμ , w1 (x, t) = Z 1
−1
μf (x, t, μ) dμ energy is conserved ∂u0 ∂t + ∂w1 ∂x = 0 H-theorem ∂η ∂t + ∂φ ∂x = σ ≥ 0 entropy density, flux, generation η = − Z 1
−1
f2dμ , φ = − Z 1
−1
μf2dμ , σ = 1 τε Z κ (μ) (f − f0)2 dμ ≥ 0
SLIDE 6 Step 1: Grad closure for monomials
(2N + 1) monomials: uα = Z 1
−1
μ2αfdμ , α = 0, 1, . . . , N wα = Z 1
−1
μ2α−1fdμ , α = 1, . . . , N nested moment equations:
φα =
1 2α+1+ γ 2α+3
1+γ
3
∂u0 ∂t + ∂w1 ∂x = 0 ∂uα ∂t + ∂wα+1 ∂x = − 1 τε [uα + γuα+1 − φα (u0 + γu1)] ∂wα ∂t + ∂uα ∂x = − 1 τε [wα + γwα+1] needs closure for uN+1 and wN+1
SLIDE 7 Step 1: Grad closure for monomials
Grad-type distribution: just a polynomial [e.g., from maximizing entropy...] fG =
N
X
β=0
υβμ2β +
N
X
β=1
ωβμ2β−1 , where υα =
N
X
β=0
A−1
αβuβ
, ωα =
N
X
β=1
B−1
αβwβ
Aαβ = 2 2 (α + β) + 1 , Bαβ = 2 2 (α + β) − 1 closure for higher moments: uN+1 =
N
X
α=0
ξαuα , wN+1 =
N
X
α=1
ζαwα
with coefficients ξα =
N
X
β=0
2 2 (N + β) + 3A−1
βα
, ζα =
N
X
β=1
2 2 (N + β) + 1B
SLIDE 8 Step 1: Grad closure for monomials
Closed equations for (2N + 1) monomials ∂u0 ∂t + ∂w1 ∂x = 0 ∂uα ∂t +
N
X
β=1
Rαβ ∂wβ ∂x = − 1 τε
N
X
β=1
Uαβuβ + 1 τε (φα − γξ0δαN) u0 ∂wα ∂t + ∂uα ∂x = − 1 τε
N
X
β=1
Wαβwβ
Rαβ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 1 ... ... ... 1 ζ1 ζ2 ζ3 · · · · · · ζN ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , Uαβ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 − γφ1 γ −γφ2 1 γ . . . ... ... ... γ γξ1 − γφN γξ2 γξ3 · · · 1 + γξN ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , Wαβ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 γ 1 γ ... ... ... γ γζ1 γζ2 γζ3 · · · · · · 1 + γζN ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
Order of Magnitude Method, next steps:
- Chapman-Enskog expansions to determine leading order of moments as O
¡ ελ¢
- reconstruct moments so that minimum number of variables at given order λ
SLIDE 9
Step 2: O of M method, isotropic scattering (γ = 0)
moment equations for γ = 0: ∂u0 ∂t + ∂w1 ∂x = 0 ∂uα ∂t + ∂wα+1 ∂x = − 1 τε µ uα − u0 2α + 1 ¶ ∂wα ∂t + ∂uα ∂x = − 1 τεwα equilibrium: limit ε → 0 uα|E = u0 2α + 1 , wα|E = 0 = ⇒ all uα are O ¡ ε0¢ = ⇒ all wα are at least O ¡ ε1¢
SLIDE 10
Step 2: O of M method, isotropic scattering (γ = 0)
first non-equilibrium moments: subtract equilibrium values from higher moments u(1)
α = uα − uα|E = uα −
u0 2α + 1
(α=1,2,...)
w(1)
α = wα − wα|E = wα (α=1,2,...)
= ⇒ u0 is O ¡ ε0¢ = ⇒ u(1)
α , w(1) α are at least O
¡ ε1¢ new moment equations: variables u0, u(1)
α , w(1) α
∂u0 ∂t + ∂w(1)
1
∂x = 0 ∂u(1)
α
∂t − 1 2α + 1 ∂w(1)
1
∂x + ∂w(1)
α+1
∂x = − 1 τεu(1)
α (α=1,2,...)
∂w(1)
α
∂t + 1 2α + 1 ∂u0 ∂x + ∂u(1)
α
∂x = − 1 τεw(1)
α (α=1,2,...)
Remark: balance laws for new variables, i.e., only the variable in time derivative
SLIDE 11
Step 2: O of M method, isotropic scattering (γ = 0)
expand high order variables in Chapman-Enskog series: u(1)
α = εu(1) α,1 + ε2u(1) α,2 + . . .
, w(1)
α = εw(1) α,1 + ε2w(1) α,2 + . . .
leading terms only: 0 = u(1)
α,1
, − 1 2α + 1τ ∂u0 ∂x = w(1)
α,1 (α=1,2,...)
the w(1)
α are linearly dependent
w(1)
α,1 =
3 2α + 1w(1)
1,1 (α=2,3,...)
second non-equilibrium moments: use above u(2)
α = u(1) α (α=1,2,...)
w(2)
α = w(1) α −
3 2α + 1w(1)
1 (α=2,3,...)
= ⇒ u0 is O ¡ ε0¢ = ⇒ w(1)
1
is O ¡ ε1¢ = ⇒ u(2)
α , w(2) α are at least O
¡ ε2¢
SLIDE 12
Step 2: O of M method, isotropic scattering (γ = 0)
equations for u0 = ˆ u0, w(1)
1
= ε ˆ w(1)
1 , u(2) α , w(2) α
∂ˆ u0 ∂t + ε∂ ˆ w(1)
1
∂x = 0 ε∂ ˆ w(1)
1
∂t + ∂u(2)
1
∂x = −1 τ ∙ ˆ w(1)
1 + τ
3 ∂ˆ u0 ∂x ¸ ∂u(2)
α
∂t + 4α (2α + 3) (2α + 1)ε∂ ˆ w(1)
1
∂x + ∂w(2)
α+1
∂x = − 1 τεu(2)
α (α=1,2,...)
∂w(2)
α
∂t − 3 2α + 1 ∂u(2)
1
∂x + ∂u(2)
α
∂x = − 1 τεw(2)
α (α=2,3,...)
SLIDE 13
Step 2: O of M method, isotropic scattering (γ = 0)
expand high order variables in Chapman-Enskog series: u(2)
α = ε2u(2) α,2 + ε3u(2) α,3 + . . .
, w(2)
α = ε2w(2) α,3 + ε3w(2) α,3 + . . .
leading terms only: − 4α (2α + 3) (2α + 1)τ ∂ ˆ w(1)
1
∂x = u(2)
α,2 (α=1,2,...)
0 = w(2)
α,2 (α=2,3,...)
the u(2)
α,2 are linearly dependent
u(2)
α,2 =
15α (2α + 3) (2α + 1)u(2)
1,2 (α=1,2,...)
third non-equilibrium moments: use above u(3)
α = u(2) α −
15α (2α + 3) (2α + 1)u(2)
1 (α=2,3,...)
w(3)
α = w(2) α (α=2,3,...)
= ⇒ u0 is O ¡ ε0¢ = ⇒ w(1)
1
is O ¡ ε1¢ = ⇒ u(2)
1
is O ¡ ε2¢ = ⇒ u(3)
α , w(3) α are at least O
¡ ε3¢
SLIDE 14
Step 2: O of M method, isotropic scattering (γ = 0)
equations for u0 = ˆ u0, w(1)
1
= ε ˆ w(1)
1 , u(2) 1
= ε2ˆ u(2)
1
and u(3)
α , w(3) α
∂ˆ u0 ∂t + ε∂ ˆ w(1)
1
∂x = 0 ε∂ ˆ w(1)
1
∂t + ε2∂ˆ u(2)
1
∂x = −1 τ ∙ ˆ w(1)
1 + τ
3 ∂ˆ u0 ∂x ¸ ε2∂ˆ u(2)
1
∂t + ∂w(3)
2
∂x = −1 τ ε " ˆ u(2)
1 − 4
15τ ∂ ˆ w(1)
1
∂x # ∂u(3)
α
∂t − 15α (2α + 3) (2α + 1) ∂w(3)
2
∂x + ∂w(3)
α+1
∂x = − 1 τεu(3)
α (α=2,3,...)
∂w(3)
α
∂t + 9 (α − 1) (2α + 3) (2α + 1)ε2∂ˆ u(2)
1
∂x + ∂u(3)
α
∂x = − 1 τεw(3)
α (α=2,3,...)
SLIDE 15
Step 2: O of M method, isotropic scattering (γ = 0)
expand high order variables in Chapman-Enskog series: u(3)
α = ε3u(3) α,3 + ε4u(3) α,4 + . . .
, w(3)
α = ε3w(3) α,3 + ε4w(3) α,4 + . . .
leading terms only: 0 = u(3)
α,3 (α=2,3,...)
− 9 (α − 1) (2α + 3) (2α + 1)τ ∂ˆ u(2)
1
∂x = w(3)
α,3 (α=2,3,...)
the w(3)
α,3 are linearly dependent
w(3)
α,3 =
35 (α − 1) (2α + 3) (2α + 1)w(3)
2,3 (α=3,4,...)
fourth non-equilibrium moments: use above u(4)
α = u(3) α (α=2,3,...)
w(4)
α = w(3) α −
35 (α − 1) (2α + 3) (2α + 1)w(3)
2 (α=3,4,...)
= ⇒ u0 is O ¡ ε0¢ , w(1)
1
is O ¡ ε1¢ , u(2)
1
is O ¡ ε2¢ , w(3)
2
is O ¡ ε3¢ = ⇒ u(4)
α , w(4) α are at least O
¡ ε4¢
SLIDE 16 Step 2: O of M method, isotropic scattering (γ = 0)
equations for u0 = ˆ u0, w(1)
1
= ε ˆ w(1)
1 , u(2) 1
= ε2ˆ u(2)
1 , w(3) 2
= ε3 ˆ w(3)
2
and u(4)
α , w(4) α
∂ˆ u0 ∂t + ε∂ ˆ w(1)
1
∂x = 0 ε∂ ˆ w(1)
1
∂t + ε2∂ˆ u(2)
1
∂x = −1 τ ∙ ˆ w(1)
1 + τ
3 ∂ˆ u0 ∂x ¸ ε2∂ˆ u(2)
1
∂t + ε3∂ ˆ w(3)
2
∂x = −1 τ ε " ˆ u(2)
1 + 4
15τ ∂ ˆ w(1)
1
∂x # ε3∂ ˆ w(3)
2
∂t + ∂u(4)
2
∂x = −1 τ ε2 " ˆ w(3)
2 + 9
35τ ∂ˆ u(2)
1
∂x # ∂u(4)
α
∂t + 40 (α − 1) α (2α + 5) (2α + 3) (2α + 1)ε3∂ ˆ w(3)
2
∂x + ∂w(4)
α+1
∂x = − 1 τεu(4)
α (α=2,3,...)
∂w(4)
α
∂t − 35 (α − 1) (2α + 3) (2α + 1) ∂u(4)
2
∂x + ∂u(4)
α
∂x = − 1 τεw(4)
α (α=3,4,...)
SLIDE 17
Step 2: O of M method, isotropic scattering (γ = 0)
can be continued ad infinitum, to produce the variables: up to fifth order u0 = Z fdμ w(1)
1
= Z μfdμ u(2)
1
= u1 − u0 3 = 2 3 Z 1 2 ¡ 3μ2 − 1 ¢ fdμ w(3)
2
= w2 − 3 5w1 = 2 5 Z 1 2 ¡ 5μ3 − 3μ ¢ fdμ u(4)
2
= u2 − 6 7u1 + 3 35u0 = 8 35 Z 1 8 ¡ 35μ4 − 30μ2 + 3 ¢ fdμ w(5)
3
= w3 − 70 63w2 + 15 63w1 = 8 63 Z 1 8 ¡ 63μ5 − 70μ3 + 15μ ¢ fdμ these are the Legendre polynomials Pα times factors (1, 1, 2
3, 2 5, 8 35, . . .)
= ⇒ method constructs Pα as optimum moments ...for isotropic scattering [HS 2007]
SLIDE 18
Step 2: O of M method, anisotropic scattering (γ 6= 0)
equations: ∂u0 ∂t + ∂w1 ∂x = 0 ∂uα ∂t +
N
X
β=1
Rαβ ∂wβ ∂x = − 1 τε
N
X
β=1
Uαβuβ + 1 τε (φα − γξ0δαN) u0 ∂wα ∂t + ∂uα ∂x = − 1 τε
N
X
β=1
Wαβwβ equilibrium: limit ε → 0 ... same as before = ⇒ consistency of Grad closure uα|E =
N
X
β=1
U−1
αβ (φβ − γξ0δβN) u0 =
u0 2α + 1 = λ(1)
α u0
wα|E = 0 = ⇒ all uα are O ¡ ε0¢ , all wα are at least O ¡ ε1¢ first non-equilibrium moments: subtract equilibrium values from higher moments u(1)
α = uα − uα|E = uα − λ(1) α u0 (α=1,...,N)
w(1)
α = wα − wα|E = wα (α=1,...,N)
SLIDE 19 Step 2: O of M method, anisotropic scattering (γ 6= 0)
equations for u0 = ˆ u0, and u(1)
α , w(1) α
∂ˆ u0 ∂t + ∂w(1)
1
∂x = 0 ∂u(1)
α
∂t − λ(1)
α
∂w(1)
1
∂x +
N
X
β=1
Rαβ ∂w(1)
β
∂x = − 1 τε
N
X
β=1
Uαβu(1)
β (α=1,...,N)
∂w(1)
α
∂t + λ(1)
α
∂u0 ∂x + ∂u(1)
α
∂x = − 1 τε
N
X
β=1
Wαβw(1)
β (α=1,...,N)
expand high order variables in Chapman-Enskog series: u(1)
α = εu(1) α,1 + ε2u(1) α,2 + . . .
, w(1)
α = εw(1) α,1 + ε2w(1) α,2 + . . .
leading terms only: u(1)
β,1 = 0
, w(1)
α,1 = −τ N
X
β=1
W −1
αβ λ(1) β
∂u0 ∂x
(α=1,...,N)
the w(1)
α are linearly dependent κ(1) = PN
β=1 W −1 1β λ(1) β
w(1)
α,1 =
PN
β=1 W −1 αβ λ(1) β
PN
β=1 W −1 1β λ(1) β
w(1)
1,1 =
PN
β=1 W −1 αβ λ(1) β
κ(1) w(1)
1,1 = θ(1) α w(1) 1,1
SLIDE 20 Step 2: O of M method, anisotropic scattering (γ 6= 0)
second non-equilibrium moments: use above u(2)
α = u(1) α (α=1,...,N)
w(2)
α = w(1) α − θ(1) α w(1) 1 (α=2,...,N)
equations for u0 = ˆ u0, w(1)
1
= ε ˆ w(1)
1 , and u(2) α , w(2) α
∂ˆ u0 ∂t + ε∂ ˆ w(1)
1
∂x = 0 ε∂ ˆ w(1)
1
∂t + ∂u(2)
1
∂x = −λ(1)
1
" ˆ w(1)
1
τκ(1) + ∂u0 ∂x # − 1 τε
N
X
β=2
W1βw(2)
β
∂u(2)
α
∂t + χ(1)
α ε∂ ˆ
w(1)
1
∂x +
N
X
β=2
Rαβ ∂w(2)
β
∂x = − 1 τε
N
X
β=1
Uαβu(2)
β
∂w(2)
α
∂t − θ(1)
α
∂u(2)
1
∂x + ∂u(2)
α
∂x = −ψ(1)
α
" ˆ w(1)
1
τκ(1) + ∂ˆ u0 ∂x # − 1 τε
N
X
β=2
W (2)
αβ w(2) β
with W (2)
αβ =
³ Wαβ − θ(1)
α W1β
´ , ψ(1)
α =
³ λ(1)
α − θ(1) α λ(1) 1
´
(α,β=2,...,N)
χ(1)
α = N
X
β=1
Rαβθ(1)
β − λ(1) α (α=1,...,N)
SLIDE 21 Step 2: O of M method, anisotropic scattering (γ 6= 0)
expand high order variables in Chapman-Enskog series: u(2)
α = ε2u(2) α,2 + ε3u(2) α,3 + . . .
, w(2)
α = ε2w(2) α,2 + ε3w(2) α,3 + . . .
leading terms only: u(2)
α,2 = −
⎡ ⎣
N
X
β=1
U−1
αβ χ(1) β
⎤ ⎦ τ ∂ ˆ w(1)
1
∂x
(α=1,...,N)
w(2)
α,2 = −
⎡ ⎣
N
X
β=2
W (2)−1
αβ
ψ(1)
β
⎤ ⎦ τ *" ˆ w(1)
1
τκ(1) + ∂ˆ u0 ∂x #+
(α=2,...,N)
the u(2)
α,2 , w(2) α,2 are linearly dependent
u(2)
α,2 =
PN
β=1 U−1 αβ χ(1) β
PN
β=1 U−1 1β χ(1) β
u(2)
1,2 = λ(2) α u(2) 1,2 (α=1,...,N)
w(2)
α,2 =
PN
β=2 W (2)−1 αβ
ψ(1)
β
PN
β=2 W (2)−1 2β
ψ(1)
β
w(2)
2,2 = θ(2) α w(2) 2,2 (α=2,...,N)
third non-equilibrium moments: use above u(3)
α = u(2) α − λ(2) α u(2) 1 (α=2,...,N)
w(3)
α = w(2) α − θ(2) α w(2) 2 (α=3,...,N)
SLIDE 22 Step 2: O of M method, anisotropic scattering (γ 6= 0)
equations for variables ˆ u0, ε ˆ w(1)
1 , ε2ˆ
u(2)
1 , ε3ˆ
u(3)
2 , ε3 ˆ
w(3)
2
and u(4)
α , w(4) α ∂ˆ u0 ∂t + ε∂ ˆ w(1)
1
∂x = 0 ε∂ ˆ w(1)
1
∂t + ε2∂ˆ u(2)
1
∂x = −1 3 " ˆ w(1)
1
τκ(1) + ∂ˆ u0 ∂x # − 1 τ εγ ˆ w(2)
2
ε2∂ˆ u(2)
1
∂t + ε2∂ ˆ w(2)
2
∂x = −χ(1)
1 ε
" ˆ u(2)
1
τμ(2) + ∂ ˆ w(1)
1
∂x # − 1 τεγu(3)
2
ε2∂ ˆ w(2)
2
∂t + ³ λ(2)
2 − θ(1) 2
´ ε2∂ˆ u(2)
1
∂x + ∂u(3)
2
∂x = −ψ(1)
2 ε
"*" ˆ w(1)
1
τκ(1) + ∂ˆ u0 ∂x #+ + ˆ w(2)
2
τκ(2) # − 1 τε
N
X
β=3
W (2)
2β w(3) β
∂u(3)
α
∂t + ⎡ ⎣
N
X
β=2
R(2)
αβθ(2) β
⎤ ⎦ ε2∂w(2)
2
∂x +
N
X
β=3
R(2)
αβ
∂w(3)
β
∂x = − ³ χ(1)
α − λ(2) α χ(1) 1
´ ε2 *" ˆ u(2)
1
τμ(2) + ∂ ˆ w(1)
1
∂x #+ − 1 τε
N
X
β=2
U(2)
αβ u(3) β
∂w(3)
α
∂t + ³ λ(2)
α − θ(1) α − θ(2) α
³ λ(2)
2 − θ(1) 2
´´ ε2∂ˆ u(2)
1
∂x − θ(2)
α
∂u(3)
2
∂x + ∂u(3)
α
∂x = − ³ ψ(1)
α − θ(2) α ψ(1) 2
´ ε2 *"*" ˆ w(1)
1
τκ(1) + ∂ˆ u0 ∂x #+ + ˆ w(2)
2
τκ(2) #+ − 1 τε
N
X
β=3
W (3)
αβ w(3) β
SLIDE 23 Step 2: O of M method, anisotropic scattering (γ 6= 0)
expand high order variables in Chapman-Enskog series: u(3)
α = ε3u(3) α,3 + ε4u(3) α,4 + . . .
, w(3)
α = ε3w(3) α,3 + ε4w(3) α,4 + . . .
leading terms only: in terms of lower moments ˆ u0, ˆ w(1)
1 , ˆ
u(2)
1 ,
ˆ w(2)
2
u(3)
α,3 = −λ(3) α τ
*" ˆ u(2)
1
τμ(2) + ∂ ˆ w(1)
1
∂x #+ − ¯ λ(3)
α τ ∂w(2) 2
∂x
(α=2,...,N)
w(3)
α,3 = −θ(3) α τ
*"*" ˆ w(1)
1
τκ(1) + ∂ˆ u0 ∂x #+ + ˆ w(2)
2
τκ(2) #+ − ¯ θ(3)
α τ ∂ˆ
u(2)
1
∂x
(α=3,...,N)
the u(3)
α,3 , w(3) α,3 are linearly dependent
u(3)
α,3 =
¯ λ(3)
α λ(3) 3 − λ(3) α ¯
λ(3)
3
¯ λ(3)
2 λ(3) 3 − λ(3) 2 ¯
λ(3)
3
u(3)
2,3 + λ(3) α ¯
λ(3)
2 − ¯
λ(3)
α λ(3) 2
¯ λ(3)
2 λ(3) 3 − λ(3) 2 ¯
λ(3)
3
u(3)
3,3 (α=2,...,N)
w(3)
α,3 =
¯ θ(3)
α θ(3) 4 − θ(3) α ¯
θ(3)
4
¯ θ(3)
3 θ(3) 4 − θ(3) 3 ¯
θ(3)
4
w(3)
3,3 + θ(3) α ¯
θ(3)
3 − ¯
θ(3)
α θ(3) 3
¯ θ(3)
3 θ(3) 4 − θ(3) 3 ¯
θ(3)
4
w(3)
4,3 (α=3,...,N)
fourth non-equilibrium moments: use above u(4)
α = u(3) α −
¯ λ(3)
α λ(3) 3 − λ(3) α ¯
λ(3)
3
¯ λ(3)
2 λ(3) 3 − λ(3) 2 ¯
λ(3)
3
u(3)
2 − λ(3) α ¯
λ(3)
2 − ¯
λ(3)
α λ(3) 2
¯ λ(3)
2 λ(3) 3 − λ(3) 2 ¯
λ(3)
3
u(3)
3 (α=4,...,N)
w(4)
α = w(3) α −
¯ θ(3)
α θ(3) 4 − θ(3) α ¯
θ(3)
4
¯ θ(3)
3 θ(3) 4 − θ(3) 3 ¯
θ(3)
4
w(3)
3 − θ(3) α ¯
θ(3)
3 − ¯
θ(3)
α θ(3) 3
¯ θ(3)
3 θ(3) 4 − θ(3) 3 ¯
θ(3)
4
w(3)
4 (α=5,...,N)
... and go round and round and round in the circle game ...
SLIDE 24
Step 2: O of M method, anisotropic scattering (γ 6= 0)
Approximation for 3rd order variables . . . use leading order for closure: u(3)
α = ε3ˆ
u(3)
α = ε3u(3) α,3
, w(3)
α = ε3 ˆ
w(3)
α = ε3w(3) α,3 .
leading terms: depend on lower moments ˆ u0, ˆ w(1)
1 , ˆ
u(2)
1 ,
ˆ w(2)
2
ˆ u(3)
α = u(3) α,3 = −λ(3) α τ
*" ˆ u(2)
1
τμ(2) + ∂ ˆ w(1)
1
∂x #+ − ¯ λ(3)
α τ ∂w(2) 2
∂x
(α=2,...,N)
ˆ w(3)
α = w(3) α,3 = −θ(3) α τ
*"*" ˆ w(1)
1
τκ(1) + ∂ˆ u0 ∂x #+ + ˆ w(2)
2
τκ(2) #+ − ¯ θ(3)
α τ ∂ˆ
u(2)
1
∂x
(α=3,...,N)
with many new coefficients that depend on γ, N
SLIDE 25 Step 2: O of M method, anisotropic scattering (γ 6= 0)
equations for variables ˆ u0, ε ˆ w(1)
1 , ε2ˆ
u(2)
1 , ε3ˆ
u(3)
2 , ε3 ˆ
w(3)
2
∂ˆ u0 ∂t + ε∂ ˆ w(1)
1
∂x = 0 ε∂ ˆ w(1)
1
∂t + ε2∂ˆ u(2)
1
∂x = −1 3 " ˆ w(1)
1
τκ(1) + ∂ˆ u0 ∂x # − 1 τ εγ ˆ w(2)
2
ε2∂ˆ u(2)
1
∂t + α1ε2∂ ˆ w(2)
2
∂x = −α2ε " ˆ u(2)
1
τμ(2) + ∂ ˆ w(1)
1
∂x # ε2∂ ˆ w(2)
2
∂t + β1ε2∂ˆ u(2)
1
∂x − β2τε2∂2 ˆ w(1)
1
∂x2 − β3τε3∂2 ˆ w(2)
2
∂x2 = −β4ε *" ˆ w(1)
1
τκ(1) + ∂ˆ u0 ∂x #+ − β5ε ˆ w(2)
2
τ numerical values of coefficients
γ = 0 γ = 1, N = 1 γ = 1, N = 2 γ = 1, N = 4 γ = 1, N = 10 γ = 5, N = 10 κ(1)
1 3 = 0.333
0.2083 0.2144 0.2146 0.2146 0.09712 α1 1 n/a 0.7843 0.8263 0.8266 0.6029 α2
4 15 = 0.2667
n/a 0.2160 0.2135 0.2135 0.1316 μ(2)
4 15 = 0.2667
0.1830 0.1619 0.1606 0.1606 0.08338 β1
9 35 = 0.2571
n/a 0.1790 0.1764 0.1766 0.1232 β2 n/a 0.0053 0.0065 0.0065 0.0043 β3
16 63 = 0.2440
1 0.2157 0.1737 0.1734 0.07943 β4 n/a 0.0151 0.0161 0.0163 0.03190 β5 1 n/a 1.5565 1.5223 1.539 2.5678
SLIDE 26
Step 3: Model reduction by order of accuracy
definition of order of accuray: A set of equations is accurate of order λ, when the energy flux w1 = w(1)
1
is known within the order O ¡ ελ¢ . zeroth order accuracy energy flux is of leading order ε, w1 = w(1)
1
= ε ˆ w(1)
1 :
∂u0 ∂t = 0 first order accuracy conservation law and leading term for w(1)
1
∂ˆ u0 ∂t + ε∂ ˆ w(1)
1
∂x = 0 , 0 = −1 3 " ˆ w(1)
1
τκ(1) + ∂ˆ u0 ∂x # diffusion equation (ε scaling removed) ∂u0 ∂t − τκ(1)∂2u0 ∂x2 = 0
diffusion coefficient τκ(1) is computed from the full set of (2N + 1) moment equations
SLIDE 27
Step 3: Model reduction by order of accuracy
equations for variables ˆ u0, ε ˆ w(1)
1 , ε2ˆ
u(2)
1 , ε3ˆ
u(3)
2 , ε3 ˆ
w(3)
2
∂ˆ u0 ∂t + ε∂ ˆ w(1)
1
∂x = 0 ε∂ ˆ w(1)
1
∂t + ε2∂ˆ u(2)
1
∂x = −1 3 " ˆ w(1)
1
τκ(1) + ∂ˆ u0 ∂x # − 1 τ εγ ˆ w(2)
2
ε2∂ˆ u(2)
1
∂t + α1ε2∂ ˆ w(2)
2
∂x = −α2ε " ˆ u(2)
1
τμ(2) + ∂ ˆ w(1)
1
∂x # ε2∂ ˆ w(2)
2
∂t +β1ε2∂ˆ u(2)
1
∂x −β2τε2∂2 ˆ w(1)
1
∂x2 −β3τε3∂2 ˆ w(2)
2
∂x2 = −β4ε *" ˆ w(1)
1
τκ(1) + ∂ˆ u0 ∂x #+ −β5ε ˆ w(2)
2
τ
SLIDE 28
Step 3: Model reduction by order of accuracy
second order accuracy conservation law and the two leading terms for w(1)
1
∂ˆ u0 ∂t + ε∂ ˆ w(1)
1
∂x = 0 , ε∂ ˆ w(1)
1
∂t = −1 3 " ˆ w(1)
1
τκ(1) + ∂ˆ u0 ∂x # − 1 τ γε ˆ w(2)
2
. leading order of ˆ w(2)
2
required as well: ˆ w(2)
2
= −β4 β5 τ *" ˆ w(1)
1
τκ(1) + ∂ˆ u0 ∂x #+ Grad-type hyperbolic system (ε scaling removed) ∂u0 ∂t + ∂w(1)
1
∂x = 0 , ∂w(1)
1
∂t + µ1 3 − γκ(2) ¶ ∂u0 ∂x = − µ1 3 − γκ(2) ¶ w(1)
1
τκ(1)
SLIDE 29
Step 3: Model reduction by order of accuracy
equations for variables ˆ u0, ε ˆ w(1)
1 , ε2ˆ
u(2)
1 , ε3ˆ
u(3)
2 , ε3 ˆ
w(3)
2
∂ˆ u0 ∂t + ε∂ ˆ w(1)
1
∂x = 0 ε∂ ˆ w(1)
1
∂t + ε2∂ˆ u(2)
1
∂x = −1 3 " ˆ w(1)
1
τκ(1) + ∂ˆ u0 ∂x # − 1 τ εγ ˆ w(2)
2
ε2∂ˆ u(2)
1
∂t + α1ε2∂ ˆ w(2)
2
∂x = −α2ε " ˆ u(2)
1
τμ(2) + ∂ ˆ w(1)
1
∂x # ε2∂ ˆ w(2)
2
∂t +β1ε2∂ˆ u(2)
1
∂x −β2τε2∂2 ˆ w(1)
1
∂x2 −β3τε3∂2 ˆ w(2)
2
∂x2 = −β4ε *" ˆ w(1)
1
τκ(1) + ∂ˆ u0 ∂x #+ −β5ε ˆ w(2)
2
τ
SLIDE 30 Step 3: Model reduction by order of accuracy
third order accuracy conservation law, full equation for w(1)
1
∂ˆ u0 ∂t + ε∂ ˆ w(1)
1
∂x = 0 ε∂ ˆ w(1)
1
∂t + ε2∂ˆ u(2)
1
∂x = −1 3 " ˆ w(1)
1
τκ(1) + ∂ˆ u0 ∂x # − 1 τ εγ ˆ w(2)
2
leading term for ˆ u(2)
1 , leading term + first correction for ˆ
w(2)
2
0 = −α2 " ˆ u(2)
1
τμ(2) + ∂ ˆ w(1)
1
∂x # ε2∂ ˆ w(2)
2
∂t + β1ε2∂ˆ u(2)
1
∂x − β2τε2∂2 ˆ w(1)
1
∂x2 = −β4ε *" ˆ w(1)
1
τκ(1) + ∂u0 ∂x #+ − β5ε ˆ w(2)
2
τ
mixed type: hyperbolic + regularization (ε scaling removed)
∂u0 ∂t + ∂w(1)
1
∂x = 0 ∂w(1)
1
∂t − τμ(2)∂2w(1)
1
∂x2 = −1 3 " w(1)
1
τκ(1) + ∂u0 ∂x # − 1 τ γw(2)
2
∂w(2)
2
∂t − ³ μ(2)β1 + β2 ´ τ ∂2w(1)
1
∂x2 = −β4 " w(1)
1
τκ(1) + ∂u0 ∂x # − β5 w(2)
2
τ regularization ≡ diffusive closure, appears automatically
SLIDE 31
Step 3: Model reduction by order of accuracy
fourth order accuracy more terms, hyperbolic + regularization (ε scaling removed) ∂u0 ∂t + ∂w(1)
1
∂x = 0 ∂w(1)
1
∂t + ∂u(2)
1
∂x = −1 3 " w(1)
1
τκ(1) + ∂u0 ∂x # − 1 τ εγw(2)
2
∂u(2)
1
∂t + α1 ∂w(2)
2
∂x = −α2 " u(2)
1
τμ(2) + ∂w(1)
1
∂x # ∂w(2)
2
∂t + β1 ∂u(2)
1
∂x − β2τ ∂2 ˆ w(1)
1
∂x2 − β3τ ∂2w(2)
2
∂x2 = −β4 " w(1)
1
τκ(1) + ∂u0 ∂x # − β5 w(2)
2
τ
SLIDE 32 Order of magnitude method
[HS 2004]
Step 1:
- set up moment system for arbitrary number of moments N
- use infinite system (if possible), or close with Grad method
- must use arbitrary complete function system for moments
Step 2:
- Chapman-Enskog expansion to find leading ε−order of moments
- linear combination of moments such that number of moments at given
ε−order is minimal
- repeat for next order of magnitude
Step 3:
- use ε−orders to rescale equations for new moments
- use scaling for model reduction to a given order of accuracy
SLIDE 33 Order of magnitude method
[HS 2004]
- produces a hierarchy of systems for increasing order O
¡ ελ¢
- pde‘s with one time derivative, one or two space derivatives
- equations are either hyperbolic, or hyperbolic with regularization
- regularization (diffusive closure) damps unwanted high frequencies
- all equations are stable
- quite different from higher order Chapman-Enskog expansions!!
- Grad closure to order N ≡ Sonine expansion in CE method
- Grad closure can be used to generate boundary conditions
- higher order equations describe Knudsen layers
- method can be applied to all kinetic equations
- Example: Generalized Grad 13 moment equations
- Example: Regularized 13 moment equations (R13)
SLIDE 34
Generalized 13 moment equations [HS 2004]
2nd order for arbitrary interaction potentials matched to Burnett by CE expansion
Dσij Dt + σij ∂vk ∂xk + 2σkhi ∂vji ∂xk + 4 5 Pr $3 $2 µ ∂qhi ∂xji − ωqhi ∂ ln θ ∂xji ¶ + 4 5 Pr $4 $2 qhi ∂ ln p ∂xji + 4 5 Pr $5 $2 qhi ∂ ln θ ∂xji + $6 $2 σkhiSjik = − 2 $2 p μ ∙ σij + 2μ ∂vhi ∂xji ¸ Dqi Dt + qk ∂vi ∂xk + 5 3qi ∂vk ∂xk − 5 2 1 Prσik ∂θ ∂xk + 5 4 1 Pr θ3 θ2 θσik ∂ ln p ∂xk + 5 4 1 Pr θ4 θ2 θ µ∂σik ∂xk − ωσik ∂ ln θ ∂xk ¶ + 5 2 1 Pr 3 2 θ5 θ2 σik ∂θ ∂xk = − 1 θ2 5 2 1 Pr p μ ∙ qi + 5 2 μ Pr ∂θ ∂xi ¸
γ ω $1 $2 $3 $4 $5 $6 θ1 θ2 θ3 θ4 θ5 5 1 3.33 2 3 3 8 9.375 5.625 −3 3 9.75 7 0.833 3.561 2.003 2.793 0.217 1.942 7.781 10.038 5.647 −3.010 2.793 9.113 7.66 0.8 3.600 2.004 2.761 0.254 1.784 7.748 10.160 5.656 −3.014 2.761 9.019 9 0.75 3.679 2.007 2.695 0.328 1.466 7.681 10.402 5.674 −3.023 2.695 8.829 17 0.625 3.863 2.016 2.553 0.500 0.814 7.543 10.995 5.736 −3.053 2.553 8.442 ∞ 0.5 4.056 2.028 2.418 0.681 0.219 7.424 11.644 5.822 −3.09 2.418 8.286 Burnett coefficients for power potentials with exponent γ (Maxwell molecules: γ = 5, hard spheres: γ = ∞)
SLIDE 35
R13 equations (non-linear) [HS & Torrilhon 2003, HS 2004]
(Euler / NSF / Grad13 / R13)
Dρ Dt + ρ∂vk ∂xk = 0 ρDvi Dt + ρ ∂θ ∂xi + θ ∂ρ ∂xi + ∙∂σik ∂xk ¸ = ρGi 3 2ρDθ Dt + ρθ∂vk ∂xk + ∙∂qk ∂xk + σkl ∂vk ∂xl ¸ = 0 ∙Dσij Dt + 4 5 ∂qhi ∂xji + 2σkhi ∂vji ∂xk + σij ∂vk ∂xk ¸ + ∙∂mijk ∂xk ¸ = −ρθ ∙σij μ + 2 ∂vhi ∂xji ¸ ∙Dqi Dt + 5 2σik ∂θ ∂xk − σikθ∂ ln ρ ∂xk + θ∂σik ∂xk + 7 5qi ∂vk ∂xk + 7 5qk ∂vi ∂xk + 2 5qk ∂vk ∂xi ¸ + ∙ −σij % ∂σjk ∂xk + 1 2 ∂Rik ∂xk + 1 6 ∂∆ ∂xi + mijk ∂vj ∂xk ¸ = −5 2ρθ ∙qi κ + ∂θ ∂xi ¸ ∆ = −σijσij ρ − 12μ p ∙ θ ∂qk ∂xk + θσkl ∂vk ∂xl + 7 2qk ∂θ ∂xk − qk θ p ∂p ∂xk ¸ Rij = −4 7 1 ρσkhiσjik − 24 5 μ p ∙ θ ∂qhi ∂xji + 2qhi ∂θ ∂xji + 5 7θ µ σkhi ∂vji ∂xk + σkhi ∂vk ∂xji − 2 3σij ∂vk ∂xk ¶ − θ pqhi ∂p ∂xji ¸ mijk = −2μ p ∙ θ∂σhij ∂xki + σhij ∂θ ∂xki + 4 5qhi ∂vj ∂xki − σhij θ p ∂p ∂xki ¸
Chapman-Enskog expansion of R13 ⇒ Euler / NSF / Burnett / super-Burnett
SLIDE 36
Euler / NSF / Grad13 / R13 (linearized) [HS & Torrilhon 2003] ∂tρ + ρ0∇ · v = 0 ρ0∂tv + ∇p + ∇ · σ = ρ0G 3 2ρ0∂tθ + p0∇ · v + ∇ · q = 0 ∂tσ + 4 5 h∇qi + 2p0 h∇vi = −p0 μ0 σ + 2 3 μ0 p0 ∙ 4σ + 6 5 h∇ (∇ · σ)i ¸ ∂tq + ∇ · σ + 5 2∇θ = −2 3 p0 μ0 q + 6 5 μ0 p0 h 4q + 2∇ (∇ · q) i
hφi — symmetric tracefree tensors
SLIDE 37 Boundary conditions for moments [Torrilhon & HS 2008]
derived from Maxwell boundary conditions for Boltzmann eq. kinetic BC for odd fluxes (at left and right boundary)
slip σtn = − χ 2 − χ r 2 πθ ∙ P ¡ vt − vW
t
¢ + 1 5qt + 1 2mtnn ¸ nn jump qn = − χ 2 − χ r 2 πθ ∙ 2P (θ − θW) − 1 2PV 2 + 1 2θσnn + 1 15∆ + 5 28Rnn ¸ nn mttn = − χ 2 − χ r 2 πθ ∙ 1 14Rtt + θσtt − 1 5θσnn + 1 5P (θ − θW) − 4 5PV 2 + 1 150∆ ¸ nn mnnn = + χ 2 − χ r 2 πθ ∙2 5P (θ − θW) − 3 5PV 2 − 7 5θσnn + 1 75∆ − 1 14Rnn ¸ nn Rtn = + χ 2 − χ r 2 πθ ∙ Pθ ¡ vt − vW
t
¢ − 11 5 qt − 1 2θmtnn − PV 3 + 6PV (θ − θW) ¸ nn Vt = vt − vW
t
, vn = 0 , P = ρθ + 1
2σnn − 1 120 ∆ θ − 1 28 Rnn θ
χ accommodation coefficient indices n, t: normal/tangential components = ⇒ purely local BC, well-posed problem! H-Theorem at wall in linear case [HS & Torrilhon 2007] 2nd order BC for NSF in limit O ¡ Kn2¢
[HS & Torrilhon 2009] [Gu&Emerson 2007]: kinetic BC for R13, but too many BC lead to spurious wall layers
SLIDE 38 Force driven Poiseuille flow
[Taheri, Torrilhon & HS 2008]
R13 equations exhibit temperature dip [Tij & Santos 1994/98, Xu 2003]
θ = C4 − G2
1
Kn2 ∙y4 45 − 488 525 Kn2 y2 ¸ − C3 2 5 cosh h √
5y √ 6Kn
i + C2 956 375 G1 Kn cosh h√
5y 3Kn
i + C2 32 35 √ 5 σ12 sinh h√
5y 3Kn
i superposition of bulk solution Knudsen layers Kn = 0.072, 0.15, 0.4, 1.0
SLIDE 39 Lid-driven cavity flow [Rana & HS (subm.)]
velocity streamlines and stress contours Kn = 0.1, vlid = 30m
s
numerical solution at steady state, 70 × 70 cells, matlab, not optimized = ⇒ < 5 minutes
Drag δ =
1 Kn
NSF Varoutis et al. 2008 R13 10 0.452 0.412 0.418 5 0.609 0.500 0.504 2 0.796 0.584 0.563 1 0.899 0.625 0.573
SLIDE 40
Lid-driven cavity flow [Rana & HS (subm.)]
temperature contours and heat flux streamlines Kn = 0.1, vlid = 30m
s
numerical solution at steady state, 70 × 70 cells, matlab, not optimized = ⇒ < 5 minutes
SLIDE 41
2D transpiration flow: Heated bottom plate [Rana & HS (subm.)]
qualitative comparison to DSMC for N2, Kn = 0.1/0.2
