Homogenization of a heat transfer problem 1 G. Allaire
HOMOGENIZATION OF A CONVECTIVE,
CONDUCTIVE AND RADIATIVE
HEAT TRANSFER PROBLEM
Work partially supported by CEA
Grégoire ALLAIRE, Ecole Polytechnique
Zakaria HABIBI, CEA Saclay.
1. Introduction and model
2. Homogenization
3. Numerical results
Multiscale Simulation & Analysis in Energy and the Environment,
December 12-16, 2011, Linz
Homogenization of a heat transfer problem 2 G. Allaire
-I- INTRODUCTION
✃ Motivation: gas cooled nuclear reactor core.
✃ Heat transfer by convection, conduction and radiation.
✃ Very heterogenous periodic porous medium Ω: fluid part ΩFǫ , solid part ΩSǫ .
✃ Small parameter ǫ = ratio between period and macroscopic size.
✃ Interface Γǫ between solid and fluid where the radiative operator applies.
Goals: define a macroscopic or effective model (not obvious), propose a
multiscale numerical algorithm.
Homogenization of a heat transfer problem 3 G. Allaire
Homogenization of a heat transfer problem 4 G. Allaire
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�Model of radiative transfer
✃ Radiative transfer takes place only in the gas (assumed to be transparent).
✃ Model = non-linear and non-local boundary condition on the interface Γ.
✃ For simplicity we assume black walls (emissivity e = 1).
✃ Single radiation frequency.
On Γ, continuity of the temperature and of the total heat flux
TS = TF and −KS∇TS · n = −KF∇TF · n+ σG((TF )4
)on Γ
with σ the Stefan-Boltzmann constant and F (s, x) the view factor
G(T 4) = (Id− ζ)(T 4) and ζ(T 4)(s) =
∫
Γ
T 4(x)F (s, x)dx
Homogenization of a heat transfer problem 5 G. Allaire
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�Formula for the view factor
F 3D(s, x) =nx · (s− x)ns · (x− s)
π|x− s|4, F 2D(s′, x′) =
n′x · (s′ − x′)n′s · (x
′ − s′)
2|x′ − s′|3
Homogenization of a heat transfer problem 6 G. Allaire
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�Properties of the radiative operator
✄ The view factor F (s, x) satisfies (for a closed surface Γ)
F (s, x) ≥ 0, F (s, x) = F (x, s),
∫
Γ
F (s, x)ds = 1
✄ The kernel of G = (Id− ζ) is made of all constant functions
ker(Id− ζ) = R
✄ As an operator from L2 into itself, ‖ζ‖ ≤ 1
✄ The radiative operator G is self-adjoint on L2(Γ) and non-negative in the
sense that ∫
Γ
G(f) f ds ≥ 0 ∀ f ∈ L2(Γ)
Homogenization of a heat transfer problem 7 G. Allaire
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�Scaled model
− div(KSǫ ∇TSǫ ) = f in Ω
Sǫ
− div(ǫKFǫ ∇TFǫ ) + Vǫ · ∇T
Fǫ = 0 in Ω
Fǫ
−KSǫ ∇TSǫ · n = −ǫK
Fǫ ∇T
Fǫ · n+
σ
ǫGǫ(T
Fǫ )
4 on Γǫ
TSǫ = TFǫ on Γǫ
Tǫ = 0 on ∂Ω.
f is the source term (due to nuclear fission, only in the solid part).
Vǫ is the (given) incompressible fluid velocity.
KSǫ , ǫKFǫ are the thermal conductivities.
Homogenization of a heat transfer problem 8 G. Allaire
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�Modelling issues
✍ The solid part ΩSǫ is a connected domain.
✍ The fluid part ΩFǫ is the union of parallel cylinders.
✍ The cylinders boudaries Γǫ,i are disjoint and are not closed surfaces
Gǫ(Tǫ)(s) = Tǫ(s)−
∫
Γǫ,i
Tǫ(x)F (s, x)dx = (Id− ζǫ)Tǫ(s) ∀ s ∈ Γǫ,i
and
∫
Γǫ,i
F (s, x)dx < 1
Some radiations are escaping at the top and bottom of the cylinders.
✍ The fluid thermal conductivity is very small so it is scaled like ǫ (this is not
crucial).
✍ The radiative operator is scaled like 1/ǫ to ensure a perfect balance between
conduction and radiation at the microscopic scale y.
Homogenization of a heat transfer problem 9 G. Allaire
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�Geometry of Ω
Vertical fluid cylinders.
x = (x′, x3) with x′ ∈ R2.
Homogenization of a heat transfer problem 10 G. Allaire
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�Geometry of the unit cell
2-D unit cell !
Microscopic variable y′ ∈ Λ = ΛS ∪ ΛF .
Homogenization of a heat transfer problem 11 G. Allaire
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�Assumptions on the coefficients
Given fluid velocity
Vǫ(x) = V (x,x′
ǫ) in ΩFǫ ,
with a smooth vector field V (x, y′), defined in Ω×ΛF , periodic with respect to y′
and satisfying the two incompressibility constraints
divxV = 0 and divy′V = 0 in ΛF , and V · n = 0 on γ.
A typical example is V = (0, 0, V3).
Conductivities
KSǫ (x) = KS(x,
x′
ǫ) in ΩSǫ , ǫK
Fǫ (x) = ǫK
F (x,x′
ǫ) in ΩFǫ ,
where KS(x, y′), KF (x, y′) are periodic symmetric positive definite tensors
defined in Ω× Λ.
Homogenization of a heat transfer problem 12 G. Allaire
-II- HOMOGENIZATION RESULT
By the method of formal two-scale asymptotic expansions
Tǫ = T0(x) + ǫ T1(x,x′
ǫ) + ǫ2 T2(x,
x′
ǫ) +O(ǫ3)
we can obtain the homogenized and cell problems (in the non-linear case).
A rigorous justification by the method of two-scale convergence has been
obtained in the linear case (upon linearization of the radiative operator).
Homogenization of a heat transfer problem 13 G. Allaire
Theorem. T0 is the solution of a non-linear homogenized problem
− div(K∗(x, T 30 )∇T0(x)) + V∗(x) · ∇T0(x) = θ f(x) in Ω
T0(x) = 0 on ∂Ω
with the porosity factor θ = |ΛS | / |Λ| and the homogenized velocity
V ∗ =1
|Λ|
∫
ΛF
V (x, y′) dy′.
The corrector term T1 is given by
T1(x, y′) =
3∑
j=1
ωj(x, T3
0 (x), y′)∂T0∂xj
(x)
where(ωj(x, T
30 (x), y
′))
1≤j≤3are the solutions of the cell problems.
Homogenization of a heat transfer problem 14 G. Allaire
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�Cell problems
(ωj(x, T
30 (x), y
′))
1≤j≤3are the solutions of the 2-D cell problems
− divy′(KS(x, y′)(ej +∇yω
Sj (y
′)))= 0 in ΛS
−KS(y′, x3)(ej +∇yωSj (y
′)) · n = 4σT 30 (x)G(ωSj (y
′) + yj) on γ
− divy′(KF (x, y′)(ej +∇yω
Fj (y
′)))+ V (x, y′) · (ej +∇yω
Fj (y
′)) = 0 in ΛF
ωFj (y′) = ωSj (y
′) on γ
y′ 7→ ωj(y′) is Λ-periodic,
First we solve for ωSj in the solid part with a linearized radiative boundary
condition.
Second we solve for ωFj in the fluid part with a Dirichlet boundary condition.
Homogenization of a heat transfer problem 15 G. Allaire
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�Homogenized conductivity coefficients
The homogenized conductivity is given by its entries, for j, k = 1, 2, 3,
K∗j,k(x, T3
0 ) =1
|Λ|
[∫
ΛS
KS(x, y′)(ej +∇yωj(y′)) · (ek +∇yωk(y
′))dy′
+ 4σT 30 (x)
∫
γ
G(ωk(y′) + yk)(ωj(y
′) + yj)
+ 2σT 30 (x)
∫
γ
∫
γ
F 2D(s′, y′)|s′ − y′|2dy′ds′ δj3δk3
]
The above last term is due to radiation losses at both end of the cylinders.
Note that the cell solutions ωj and the effective conductivity depend on T30 .
Homogenization of a heat transfer problem 16 G. Allaire
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�Remarks
✗ Radiative transfer appears only in the cell problems.
✗ Space dimension reduction (3-D to 2-D): the cell problems are 2-D.
✗ Additional vertical diffusivity due to radiation losses.
✗ The 2-D case was simpler (A. and El Ganaoui, SIAM MMS 2008).
✗ Even the formal method of two-scale ansatz is not obvious because the
radiative operator has a singular ǫ-scaling.
✗ A naive method of volume averaging does not work.
✗ Numerical multiscale approximation
Tǫ ≈ T0(x) + ǫ3∑
j=1
ωj(x, T3
0 (x),x′
ǫ)∂T0∂xj
(x)
Big CPU gain because of the 3-D to 2-D reduction of the integral operator.
Homogenization of a heat transfer problem 17 G. Allaire
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�Key ideas of the proof
1. Do not plug the ansatz in the strong form of the equations !
2. Rather use the variational formulation (following an idea of J.-L. Lions).
3. Periodic oscillations occur only in the horizontal variables x′/ǫ.
4. Perform a 3-D to 2-D limit in the radiative operator.
5. Transform a Riemann sum over the periodic surfaces Γǫ,i into a volume
integral over Ω.
Homogenization of a heat transfer problem 18 G. Allaire
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�Variational two-scale ansatz
∫
ΩSǫ
KSǫ (x)∇Tǫ(x) · ∇φǫ(x)dx+ ǫ
∫
ΩFǫ
KFǫ (x)∇Tǫ(x) · ∇φǫ(x)dx
+
∫
ΩFǫ
Vǫ(x) · ∇Tǫ(x)φǫ(x)dx+σ
ǫ
∫
Γǫ
Gǫ(Tǫ)(x)φǫ(x)ds
=
∫
ΩSǫ
f(x)φǫ(x)dx ∀φǫ ∈ H1
0 (Ω)
Take
φǫ(x) = φ0(x) + ǫ φ1(x,x′
ǫ) + ǫ2 φ2(x,
x′
ǫ)
and assume
Tǫ = T0(x) + ǫ T1(x,x′
ǫ) + ǫ2 T2(x,
x′
ǫ) +O(ǫ3)
Homogenization of a heat transfer problem 19 G. Allaire
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�Singular radiative term
The radiative term seems to blow up
σ
ǫ
∫
Γǫ
Gǫ(Tǫ)(x)φǫ(x)ds
because convergence takes place for
limǫ→0
ǫ
∫
Γǫ
ψ(x,x′
ǫ)ds =
1
|Λ|
∫
Ω
∫
γ
ψ(x, y′) dx dsy′
However, using the fact that kerGǫ = R and performing a Taylor expansion of
the test function around the center of each cylinder Γǫ,i, one can gain a ǫ2 factor.
Homogenization of a heat transfer problem 20 G. Allaire
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�3-D to 2-D asymptotic of the view factor
Lemma. For any given s3 ∈ (0, L),
∫ L
0
F 3D(s, x)dx3 = F2D(s′, x′) +O(
ǫ2
L3)
For any function g ∈ C3(0, L) with compact support in (0, L),
∫ L
0
g(x3)F3D(s, x)dx3 = F
2D(s′, x′)(
g(s3) +|x′ − s′|2
2g′′(s3) +O(ǫ
3| log ǫ|))
,
where g′′ denotes the second derivative of g.
✗ Note that |x′ − s′|2 = O(ǫ2).
✗ The corrector term, proportional to g′′(s3), is the cause of the additional
vertical diffusion.
Homogenization of a heat transfer problem 21 G. Allaire
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�Corrector
To obtain the corrector term in the fluid part, we perform an ansatz of the
variational formulation
aǫ(Tǫ, φǫ) = Lǫ(φǫ) ∀φǫ ∈ H1
0 (Ωǫ)
as
a0(T0, T1, φ0, φ1)+ ǫa1(T0, T1, T2, φ0, φ1, φ2) = L
0(φ0, φ1)+ ǫL1(φ0, φ1, φ2)+O(ǫ
2)
The zero-order equality a0(T0, T1, φ0, φ1) = L0(φ0, φ1) gives the homogenized
equation and the cell problem in the solid part.
The first-order equality a1(T0, T1, T2, φ0, φ1, φ2) = L1(φ0, φ1, φ2) yields the cell
problem in the fluid part.
Homogenization of a heat transfer problem 22 G. Allaire
-III- NUMERICAL RESULTS
Geometry of a typical fuel assembly for a gas-cooled nuclear reactor
Ω =∏3
j=1(0, Lj) with L3 = 0.025m and, for j = 1, 2, Lj = Njℓjǫ where N1 = 3,
N2 = 4 and ℓ1 = 0.04m, ℓ2 = 0.07m.
Homogenization of a heat transfer problem 23 G. Allaire
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�Numerical parameters
✍ Reference computation for ǫ = ǫ0 =1
4.
✍ Each (2-D) periodicity cell contains 2 circular holes with radius 0.0035m.
✍ No source term, f = 0.
✍ Periodic boundary conditions in the x1 direction and non-homogeneous
Dirichlet boundary conditions in the other directions
Tǫ = 3200x1 + 400x2 + 800
✍ Conductivities KS = 30Wm−1K−1 and ǫ0KF = 0.3Wm−1K−1.
✍ Constant vertical velocity V = 80 e3ms−1.
Homogenization of a heat transfer problem 24 G. Allaire
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�Standard homogenization in a fixed domain Ω
Homogenization of a heat transfer problem 25 G. Allaire
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�Rescaled process of homogenization with a constant periodicity cell
The domain is increasing as ǫ−1Ω.
Homogenization of a heat transfer problem 26 G. Allaire
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�Solutions of the cell problems for T0 = 800K
Homogenization of a heat transfer problem 27 G. Allaire
Homogenized conductivities
K∗︸︷︷︸
T0=50K
=
25.90 0. 0.
0. 25.91 0.
0. 0. 30.05
, K∗
︸︷︷︸
T0=20000K
=
49.80 0. 0.
0. 49.71 0.
0. 0. 3680.
.
Homogenized velocity
V ∗ =
0
0
15.13
ms−1.
Homogenization of a heat transfer problem 28 G. Allaire
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�Homogenized conductivities as a function of T0
Homogenization of a heat transfer problem 29 G. Allaire
Homogenization of a heat transfer problem 30 G. Allaire
Homogenization of a heat transfer problem 31 G. Allaire
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�To avoid boundary layers: smaller domain
Homogenization of a heat transfer problem 32 G. Allaire
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�To avoid boundary layers: smaller domain
Homogenization of a heat transfer problem 33 G. Allaire
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�Relative error as a function of ǫ
Homogenization of a heat transfer problem 34 G. Allaire
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�Relative error as a function of ǫ
Homogenization of a heat transfer problem 35 G. Allaire
The same approach works for unsteady problems too.