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Knudsen Layers from Moment Equations
Henning Struchtrup
University of Victoria, Canada / ETH Zürich, Switzerland
Mathematics of Model Reduction, Research Workshop University of Leicester, UK, 2007
Knudsen layers in microscale transporte.g., Couette flow in ideal gases
black: DSMC blue: Navier-Stokes-Fourier red: R13 equations
Knudsen layers: dominate in linear flows (for not too small Knudsen numbers)
not too important in strongly non-linear flows
Knudsen layers and moments?
Moments are superposition of many Knudsen layers [HS 2002]
Question: How many Knudsen layers / moment equations required?
Answer: use simple linear kinetic model
⇛ analytical calculations for all moment numbers
The kinetic model and its properties
kinetic model for 1-D heat transfer (simplified phonon/photon model)
∂f
∂t+ µ
∂f
∂x= −
1
ε
(f −
1
2λ0
)
f (x, t, µ) - distribution function, ε- Knudsen number, µ = cosϑ - direction cosine
energy density and heat flux are moments
λ0 (x, t) =
∫ 1
−1
f (x, t, µ) dµ , λ1 (x, t) =
∫ 1
−1
µf (x, t, µ) dµ
energy is conserved∂λ0∂t+
∂λ1∂x
= 0
H-theorem∂η
∂t+
∂φ
∂x= σ ≥ 0
entropy density, flux, generation
η = −1
2
∫ 1
−1
f 2dµ , φ = −1
2
∫ 1
−1
µf 2dµ , σ =1
ε
∫ 1
−1
(f −
λ02
)2dµ
Moments and their equations
N moments of Legendre polynomials
λn =
∫ 1
−1
Pn (µ) fdµ (n = 0, 1, . . . , N)
PN - approximation of distribution
f (N) =N∑
n=0
(n +
1
2
)Pn (µ)λn
moment equations from f (N) and kinetic equation
∂λ0∂t+
∂λ1∂x
= 0
∂λn
∂t+
n
2n + 1
∂λn−1
∂x+
n + 1
2n + 1
∂λn+1
∂x= −
1
ελn
∂λN
∂t+
N
2N + 1
∂λN−1
∂x= −
1
ελN
Question: what value of N for Knudsen number ε ??
H-theorem for moments
PN approximation gives second law
∂η
∂t+
∂φ
∂x= σ ≥ 0
with quadratic expressions
η = −1
2
N∑
n=0
(n+
1
2
)λ2n
φ = −N−1∑
n=0
n + 1
2λnλn+1
σ =1
ε
N∑
n=1
(n+
1
2
)λnλn
Remark: Quadratic entropy for R13 eqs. [HS & MT 2007]
Boundary conditions
Maxwell boundary conditions for distribution function (fW = 12λW )
f̄ =
χfW + (1− χ) f (−γµ) γµ > 0
f (γµ) γµ < 0
χ - accommodation coefficient, γ = ±1 at x = ∓1/2
boundary conditions for moments: use odd moments only! [Grad 1949]
λ̄n = −γΨn0
[λ̄0 − λW
]− γ
N∑
m=2
Ψnmλ̄m (n odd, m even)
Ψnm =2χ
2− χ
(m+
1
2
)∫1
0
Pn (µ)Pm (µ) dµ
Remark: H-theorem at walls fulfilled, computation via entropy fluxes
Asymptotics: Chapman-Enskog expansion
expansion in Knudsen number only non-conserved moments
λn =∑
α=0
εαλ(α)n (n ≥ 1)
zeroth order contributions vanish
λ(0)n = 0 (n ≥ 1)
only heat flux has first order contribution (Fourier’s law)
λ(1)1 = −
1
3
∂λ0∂x
, λ(1)n = 0 (n ≥ 2) .
n-th moment has order n
λ(α−1)n = 0 for α ≤ n
e.g., third order transport eq.
∂λ0∂t−1
3ε∂2λ0∂x2
+ ε31
45
∂3λ0∂x4
= 0
expansion only valid in bulk — not in Knudsen layer
Asymptotics: order of magnitude method
based on CE order of magnitude
λn = εnλ̃n
step by step reduction to order O(ε2N)yields truncated set [Leicester 2005]
∂λ0∂t+
∂λ1∂x
= 0
∂λn
∂t+
n
2n + 1
∂λn−1
∂x+
n + 1
2n + 1
∂λn+1
∂x= −
1
ελn (n = 1, . . . , N − 1)
∂λN
∂t+
N
2N + 1
∂λN−1
∂x= −
1
ελN
expansion only valid in bulk — not in Knudsen layer
Moment system as discrete velocity model
N-moment equations in matrix form
∂λm
∂t+Amn
∂λn
∂x= Cmnλn
diagonalize∂γl
∂t+ g(l)
∂γl
∂x= −
1
εγl +
1
εθ−1l0 θ0rγr
g(l) - eigenvalues, θmn - matrix of eigenvectors, γn - population numbers, λn = θnrγr - org. moments
eigenvalues correspond to discrete angles evenly distributed in [0, π]
g(l) = µ(l) = cosϑ(l)
Question: Is there similar analogy for 3-D moment eqs??
Knudsen layer solutions (steady state)
bulk moments
λ0 = K −3
ελ1x− 2λ2 , λ1 = const.
Knudsen layer moments
λn =∑
m=2
ΦnmΓ0m exp
[−
x
εb(m)
](n ≥ 2)
b(m), Φnm - eigenvalues/vectors of Bnm =n+12n+3δn,m−1 +
n+22n+3δn,m+1
constants of integration K, λ1, Γ0m from jump boundary conditions with λW
(±12
)= ±1
Results for ε = 0.1 (N = 1, 3, 5, 21)
energy density
eigenvalues/amplitudes and moments
jumps, invisible Knudsen layers, no visible differences with N
Results for ε = 1 (N = 1, 3, 7, 21)
N = {1, 3, 7, 21}: heat flux:λ1 = {−0.2857,−0.2779, 0.2770,−0.2767}
energy density
eigenvalues/amplitudes and moments
marked Knudsen layers, already N = 3 gives good agreement!!
Results for ε = 10 (N = 1, 11, 31)
Energy density, second moment
Eigenvalues/amplitudes and moments
marked linear Knudsen layers, jumps; N must be large (N ≥ 31)
Results for 0 < ε <∞ (N = 31)
energy density at wall, heat flux
deviation [%] for N = 1, 3 [corr: with Knudsen layer correction]
error less than 5% requires ε < 0.3 (N = 1) and ε < 1 (N = 3)
Results for 0 < ε <∞ (N = 31)
entropy generation: total/bulk/boundary
Asymptotics with boundary conditions
evaluation of bulk eqs and boundary conditions shows
heat flux: λ1 = O (ε)
energy jump λ̄0 − λW = O (ε)
Knudsen layer moments (n ≥ 2) λn = O (ε)
energy first order
λ0 = K −3
ελ1x− 2λ2
Knudsen layer correction: assume λn ∝ λ1, correction factor ζ of order unity
λ̄0 − λW = −γ2χ
2− χ
λ1 +N∑
m=2, m even
Ψ1mλm
≃ −ζγ2χ
2− χλ1
improves energy jump, worsens heat flux ζ = 0.869
Asymptotics with boundary conditions
Knudsen layer moments are O (ε), contribute to O(ε2)
⇛ higher order theories must include Knudsen layers of width εb(α)
Γ0α exp[−
x
εb(α)
]
N=1: no Knudsen layer
N=3: b(a) = {±0.5071}
N=5 b(l) = {±0.8162,±0.3122}
N=7: b(l) = {±0.9065,±0.6282,±0.2243}
for ε < 1, one (two) Knudsen layer(s) is not too bad =⇒ N = 3 (5)
for ε > 1 need more, criterion presently unclear
ε = 1 N = 7
ε = 2 N = 9
ε = 4 N = 11
e = 10 N = 31
Conclusions
• linear moment equations with quadratic entropy (H-theorem)
• equivalent to discrete velocity model (DVM)
• boundary conditions from kinetic model
• Knudsen layers (from eigenvalue problem)
• Chapman-Enskog etc valid only in bulk
• Knudsen layers are second order effect
• include at least some Knudsen layers (more is better, but expensive)
Conjecture
• other (linear) moment systems should behave similarly