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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