HOMOGENIZATION OF A CONVECTIVE, CONDUCTIVE AND RADIATIVE HEAT - PowerPoint PPT Presentation

Homogenization of a heat transfer problem 1 G. Allaire HOMOGENIZATION OF A CONVECTIVE, CONDUCTIVE AND RADIATIVE HEAT TRANSFER PROBLEM Work partially supported by CEA Gr´ egoire 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 Ω F ǫ , solid part Ω S ǫ . ✃ Small parameter ǫ = ratio between period and macroscopic size. ✃ Interface Γ ǫ 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 ✄ � 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 T S = T F − K S ∇ T S · n = − K F ∇ T F · n + σG ( T F ) 4 � � and on Γ with σ the Stefan-Boltzmann constant and F ( s, x ) the view factor � G ( T 4 ) = ( Id − ζ )( T 4 ) ζ ( T 4 )( s ) = T 4 ( x ) F ( s, x ) dx and Γ

Homogenization of a heat transfer problem 5 G. Allaire ✄ � Formula for the view factor ✂ ✁ x · ( s ′ − x ′ ) n ′ s · ( x ′ − s ′ ) F 2 D ( s ′ , x ′ ) = n ′ F 3 D ( s, x ) = n x · ( s − x ) n s · ( x − s ) , 2 | x ′ − s ′ | 3 π | x − s | 4

Homogenization of a heat transfer problem 6 G. Allaire ✞ ☎ 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 ) , F ( s, x ) ds = 1 Γ ✄ The kernel of G = ( Id − ζ ) is made of all constant functions ker ( Id − ζ ) = R ✄ As an operator from L 2 into itself, � ζ � ≤ 1 ✄ The radiative operator G is self-adjoint on L 2 (Γ) and non-negative in the sense that � ∀ f ∈ L 2 (Γ) G ( f ) f ds ≥ 0 Γ

Homogenization of a heat transfer problem 7 G. Allaire ✄ � Scaled model ✂ ✁  − div( K S ǫ ∇ T S in Ω S ǫ ) = f   ǫ     − div( ǫK F ǫ ∇ T F ǫ ) + V ǫ · ∇ T F in Ω F  = 0  ǫ ǫ    ǫ · n + σ = − ǫK F ǫ ∇ T F ǫ G ǫ ( T F ǫ ) 4 − K S ǫ ∇ T S ǫ · n on Γ ǫ     T S = T F on Γ ǫ   ǫ ǫ      T ǫ = 0 on ∂ Ω . f is the source term (due to nuclear fission, only in the solid part). V ǫ is the (given) incompressible fluid velocity. K S ǫ , ǫK F ǫ are the thermal conductivities.

Homogenization of a heat transfer problem 8 G. Allaire ✞ ☎ Modelling issues ✝ ✆ ✍ The solid part Ω S ǫ is a connected domain. ✍ The fluid part Ω F ǫ is the union of parallel cylinders. ✍ The cylinders boudaries Γ ǫ,i are disjoint and are not closed surfaces � G ǫ ( T ǫ )( s ) = T ǫ ( s ) − T ǫ ( x ) F ( s, x ) dx = ( Id − ζ ǫ ) T ǫ ( s ) ∀ s ∈ Γ ǫ,i Γ ǫ,i � and F ( s, x ) dx < 1 Γ ǫ,i 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 ✞ ☎ Geometry of Ω ✝ ✆ Vertical fluid cylinders. x = ( x ′ , x 3 ) with x ′ ∈ R 2 .

Homogenization of a heat transfer problem 10 G. Allaire ✞ ☎ Geometry of the unit cell ✝ ✆ 2-D unit cell ! Microscopic variable y ′ ∈ Λ = Λ S ∪ Λ F .

Homogenization of a heat transfer problem 11 G. Allaire ✞ ☎ Assumptions on the coefficients ✝ ✆ Given fluid velocity V ǫ ( x ) = V ( x, x ′ in Ω F ǫ ) ǫ , with a smooth vector field V ( x, y ′ ), defined in Ω × Λ F , periodic with respect to y ′ and satisfying the two incompressibility constraints div x V = 0 and div y ′ V = 0 in Λ F , and V · n = 0 on γ. A typical example is V = (0 , 0 , V 3 ). Conductivities ǫ ( x ) = K S ( x, x ′ ǫ ( x ) = ǫK F ( x, x ′ K S in Ω S ǫK F in Ω F ǫ ) ǫ , ǫ ) ǫ , where K S ( x, y ′ ) , K F ( 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 T ǫ = T 0 ( x ) + ǫ T 1 ( x, x ′ ǫ ) + ǫ 2 T 2 ( x, x ′ ǫ ) + O ( ǫ 3 ) 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. T 0 is the solution of a non-linear homogenized problem  − div( K ∗ ( x, T 3 0 ) ∇ T 0 ( x )) + V ∗ ( x ) · ∇ T 0 ( x ) = θ f ( x ) in Ω   T 0 ( x ) = 0 on ∂ Ω with the porosity factor θ = | Λ S | / | Λ | and the homogenized velocity � V ∗ = 1 Λ F V ( x, y ′ ) dy ′ . | Λ | The corrector term T 1 is given by 3 0 ( x ) , y ′ ) ∂T 0 � ω j ( x, T 3 T 1 ( x, y ′ ) = ( x ) ∂x j j =1 � � ω j ( x, T 3 0 ( x ) , y ′ ) where 1 ≤ j ≤ 3 are the solutions of the cell problems.

Homogenization of a heat transfer problem 14 G. Allaire ✞ ☎ Cell problems ✝ ✆ � � ω j ( x, T 3 0 ( x ) , y ′ ) 1 ≤ j ≤ 3 are the solutions of the 2-D cell problems  − div y ′ � � K S ( x, y ′ )( e j + ∇ y ω S in Λ S j ( y ′ )) = 0       − K S ( y ′ , x 3 )( e j + ∇ y ω S j ( y ′ )) · n = 4 σT 3 0 ( x ) G ( ω S j ( y ′ ) + y j )  on γ     − div y ′ � � K F ( x, y ′ )( e j + ∇ y ω F + V ( x, y ′ ) · ( e j + ∇ y ω F in Λ F j ( y ′ )) j ( y ′ )) = 0     ω F j ( y ′ ) = ω S j ( y ′ ) on γ      y ′ �→ ω j ( y ′ ) is Λ-periodic,   First we solve for ω S j in the solid part with a linearized radiative boundary condition. Second we solve for ω F j in the fluid part with a Dirichlet boundary condition.

Homogenization of a heat transfer problem 15 G. Allaire ✞ ☎ Homogenized conductivity coefficients ✝ ✆ The homogenized conductivity is given by its entries, for j, k = 1 , 2 , 3, �� 0 ) = 1 K ∗ j,k ( x, T 3 Λ S K S ( x, y ′ )( e j + ∇ y ω j ( y ′ )) · ( e k + ∇ y ω k ( y ′ )) dy ′ | Λ | � + 4 σT 3 G ( ω k ( y ′ ) + y k )( ω j ( y ′ ) + y j ) 0 ( x ) γ � � � F 2 D ( s ′ , y ′ ) | s ′ − y ′ | 2 dy ′ ds ′ δ j 3 δ k 3 + 2 σT 3 0 ( x ) γ γ The above last term is due to radiation losses at both end of the cylinders. Note that the cell solutions ω j and the effective conductivity depend on T 3 0 .

Homogenization of a heat transfer problem 16 G. Allaire ✄ � 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 3 0 ( x ) , x ′ ǫ ) ∂T 0 � ω j ( x, T 3 T ǫ ≈ T 0 ( x ) + ǫ ( x ) ∂x j j =1 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 ✞ ☎ 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 Γ ǫ,i into a volume integral over Ω.

Homogenization of a heat transfer problem 18 G. Allaire ✄ � Variational two-scale ansatz ✂ ✁ � � K S K F ǫ ( x ) ∇ T ǫ ( x ) · ∇ φ ǫ ( x ) dx + ǫ ǫ ( x ) ∇ T ǫ ( x ) · ∇ φ ǫ ( x ) dx Ω S Ω F ǫ ǫ � � V ǫ ( x ) · ∇ T ǫ ( x ) φ ǫ ( x ) dx + σ + G ǫ ( T ǫ )( x ) φ ǫ ( x ) ds ǫ Ω F Γ ǫ ǫ � ∀ φ ǫ ∈ H 1 = f ( x ) φ ǫ ( x ) dx 0 (Ω) Ω S ǫ Take φ ǫ ( x ) = φ 0 ( x ) + ǫ φ 1 ( x, x ′ ǫ ) + ǫ 2 φ 2 ( x, x ′ ǫ ) and assume T ǫ = T 0 ( x ) + ǫ T 1 ( x, x ′ ǫ ) + ǫ 2 T 2 ( x, x ′ ǫ ) + O ( ǫ 3 )