Introduction Analyse Numerical analysis Fluid/solid coupled convection/diffusion in unidirectional flows. Charles Pierre 1 , Franck Plourabou´ e 2 1 Laboratoire de Math´ ematiques et de leurs Applications, CNRS. Universit´ e de Pau. 2 IMFT, Institut de M´ ecanique des Fluides de Toulouse, CNRS. Universit´ e Paul Sabatier. October 15 th , 2008
Introduction Analyse Numerical analysis 1- The problem : physical configuration Ω Ω × R n Ω 1 v 2 > 0 Ω 2 n 3 , 1 v 1 = 0 n 2 , 1 Ω 3 v 3 > 0 y + ∞ x −∞ z Heat transfer in an infinite cylinder with cross-section Ω. • 3 sub-domains : Ω 1 (solid), Ω 2 , 3 (fluid). → → • Laminar steady flow : v = v ( x , y ) e z . • v i = v | Ω i , here : v 1 = 0 (solid), v 2 , v 3 � = 0 (fluid). • Heterogeneous conductivities k : k i = k | Ω i , k i � = k j . • Γ i , j interface between Ω i and Ω j , n i , j normal to Γ i , j .
Introduction Analyse Numerical analysis 1- The problem : physical configuration Ω Ω × R n Ω 1 v 2 > 0 Ω 2 n 3 , 1 v 1 = 0 n 2 , 1 Ω 3 v 3 > 0 y + ∞ x −∞ z This settlement both include : • co-current flows, •
Introduction Analyse Numerical analysis 1- The problem : physical configuration Ω Ω × R n Ω 1 v 2 < 0 Ω 2 n 3 , 1 v 1 = 0 n 2 , 1 Ω 3 v 3 > 0 y + ∞ x −∞ z This settlement both include : • co-current flows, • counter-current flows.
Introduction Analyse Numerical analysis 1- The problem : physical configuration Ω Ω × R n Ω 1 v 2 < 0 Ω 2 n 3 , 1 v 1 = 0 n 2 , 1 Ω 3 v 3 > 0 y + ∞ x −∞ z This settlement both include : • co-current flows, • counter-current flows. It has extension to : • planar configurations (unbounded in x ), • periodic configurations.
Introduction Analyse Numerical analysis 2- The problem : mathematical formulation Ω Ω × R n Ω 1 v 2 < 0 Ω 2 n 3 , 1 v 1 = 0 n 2 , 1 Ω 3 v 3 > 0 y + ∞ x −∞ z Energy equation on the temperature T : v = v ( x , y ), k = k ( x , y ), div( k ∇ T ) + k ∂ 2 z T = v ∂ z T , + Continuity coupling conditions between the sub-domains : T i = T j , k i ∇ T i · n i , j = k j ∇ T j · n i , j on Γ i , j ,
Introduction Analyse Numerical analysis 2- The problem : mathematical formulation Ω Ω × R n Ω 1 v 2 < 0 Ω 2 n 3 , 1 v 1 = 0 n 2 , 1 Ω 3 v 3 > 0 y + ∞ x −∞ z Energy equation on the temperature T : v = v ( x , y ), k = k ( x , y ), div( k ∇ T ) + k ∂ 2 z T = v ∂ z T , + Boundary conditions on ∂ Ω : Dirichlet with jump at z = 0, + Limit conditions at ±∞ : � T = 1 , z < 0 � T → 1 , z → −∞ on ∂ Ω : . and T = 0 , z > 0 T → 0 , z → + ∞
Introduction Analyse Numerical analysis 3- Objectives Ω Ω × R n Ω 1 v 2 < 0 Ω 2 n 3 , 1 v 1 = 0 n 2 , 1 Ω 3 v 3 > 0 y + ∞ x −∞ z 1. Macroscopic description of an average temperature T ⋆ ( z ) : T ⋆ ( z ) ≃ C 1 e λ 1 z + C 2 e λ 2 z + . . . 2. Exchanges between sub-domains description : � k i ∇ T i · n i , j ds . Γ i , j
Introduction Analyse Numerical analysis 4- Pending questions Ω Ω × R n Ω 1 v 2 < 0 Ω 2 n 3 , 1 v 1 = 0 n 2 , 1 Ω 3 v 3 > 0 y + ∞ x −∞ z � c λ t λ ( x , y ) e λ z 1. Does T read : T ( x , y , z ) = ? λ ∈ Λ 2. Location of the “ spectrum ” Λ, get a computation method for the eigenvalues/eigenfunctions λ , t λ ( x , y ). 3. Computation of the constants c λ : searching an orthogonality property for the t λ .
Introduction Analyse Numerical analysis Introductory exemple : the Graetz problem r r = 1 . . . Semi-infinite tube (radius 1) 1 fluid phase v ( r ) 0 z 0 Axi-symmetry . . . High P´ eclet : Pe ≫ 1 T 0 ( r ) ∂ 2 Taylor approximation → axial diffusion z T neglected “ Directional ” problem → entry condition T 0 ( r ) given 1 r ∂ r ( r ∂ r T ) = v ( r ) ∂ z T , T ( r , 0) = T 0 ( r ) , T (1 , z ) = 0 . T = t ( r ) e λ z Separate variable → Eigenvalue problem → λ , t ( r ) read : 1 r ∂ r ( r ∂ r t ) = λ Pe v ( r ) t , t (1) = 0 .
Introduction Analyse Numerical analysis Introductory exemple : the Graetz problem r r = 1 . . . Semi-infinite tube (radius 1) 1 fluid phase v ( r ) 0 z 0 Axi-symmetry . . . High P´ eclet : Pe ≫ 1 T 0 ( r ) Self-adjoint, negative and compact problem : ⇒ Complete orthogonal system of eigenfunctions ( t i ( r )) i , with eigenvalues 0 > λ 1 ≥ λ 2 ≥ · · · → −∞ . ⇒ Analytical solution : � 1 c i t i ( r ) e λ i z , � T ( r , z ) = c i = t i T 0 r dr . 0 i ∈ N
Introduction Analyse Numerical analysis Generalisation 1 : extended Graetz Axial diffusion ∂ 2 z T is no longer neglected. 1. Separate variable → T = t ( r ) e λ z → do not provide an eigenvalue problem 1 λ v ( r ) − λ 2 � � r ∂ r ( r ∂ r t ) = t . 2. No symmetry property available : → problem for the spectrum location : Λ ∈ R ? → computational problem for the c λ . 3. The problem is not directional any more : entry condition T 0 ( r ) not relevant. → switch to limit conditions in ±∞ .
Introduction Analyse Numerical analysis Generalisation 2 : conjugated Graetz R R r 1 v ( r ) z Coupling with a solid wall where diffusion occurs. 1 < r < R : 1 r ∂ r ( r ∂ r T ) + ∂ 2 z T = 0 r = 1 : T (1 + , z ) = T (1 − , z ) , ∂ r T (1 + , z ) = k ∂ r T (1 + , z ) . ⇒ Same difficulties as before : 1. no real eigenvalue problem, 2. no symmetry property , 3. problem not directional .
Introduction Analyse Numerical analysis Mixed reformulation : statement One reformulate the initial problem div( k ∇ T ) + k ∂ 2 z T = v ∂ z T , adding a vectorial unknown X = X ( x , y , z ) : k ∂ z T = v T − div( X ) ∂ z X = k ∇ T → Introducing the operator A : � v k − 1 − k − 1 div � � � T T � � ∂ z = A , A = � � k ∇ X X � �
Introduction Analyse Numerical analysis Mixed reformulation : analysis Theorem 1. The unbounded operator A : D ( A ) ⊂ H �→ H , H = L 2 (Ω) × L 2 (Ω) 2 , D ( A ) = H 1 0 (Ω) × H ( div , Ω) : 1. is self adjoint, 2. is diagonal on an eigenfunctions orthogonal system, 3. λ 0 = 0 excepted, all eigenvalues have finite order. Its spectrum Λ reads Λ = { λ 0 } ∪ Λ + ∪ Λ − : • Λ + = downstream modes : 0 > λ + 1 ≥ λ + 2 ≥ · · · → −∞ → related to the z > 0 region. • Λ − = upstream modes : 0 < λ + 1 ≤ λ + 2 ≤ · · · → + ∞ → related to the z < 0 region.
Introduction Analyse Numerical analysis Mixed reformulation : solution definition Analytical solution : defined from • Downstream eigenvalues / eigenfunctions : λ + n , t + n ( x , y ) eigenvalues / eigenfunctions : λ − n , t − • Upstream n ( x , y ) • The coefficients α n α n := 1 � k ∇ t n · n ds , λ 2 ∂ Ω n Corollary . The sought temperature field reads : � α − n t − n ( x , y ) e λ − n z 1 + z ≤ 0 n T ( x , y , z ) = n ( x , y ) e λ + � α + n t + n z − z ≥ 0 n
Introduction Analyse Numerical analysis Some numerical analysis The following eigen-pronlem has to be solved : � � � T T T � � � find λ ∈ R , ∈ D ( A ) : A = λ . (1) � � � X X X � � � Theorem 2. Eigen-problem (1) is equivalent to the following variational problem : find λ ∈ R and ( T , X ) ∈ L 2 (Ω) × H (div , Ω), such that ∀ ( u , Y ) ∈ L 2 (Ω) × H (div , Ω) : � � � T u v dx − u div( X ) dx = λ T u k dx Ω Ω Ω � � X · Y k − 1 dx . − T div( Y ) dx = λ Ω Ω
Introduction Analyse Numerical analysis Axi-symmetric convergence analysis Discretisation using mixed finite element spaces : e. g. T h ∈ P 0 , X h ∈ RT 0 . Evaluation of the method on the conjugated Graetz problem : → Reduction to a 1D numerical problem (axi-symmetry). → Comparison with analytical reference solutions. Relative error on the first eigenfunctions : T + 1 , T + 2 and T − 1 , with respect to the nodes number. Dashed line : slope = -1 Pe = 0 . 1 Pe = 10 Convergence rate on the eigenvalues → same order 1.
Introduction Analyse Numerical analysis Conclusion • Nice mathematical framework : orthogonality properties, problem analysis on a compete orthogonal base. • Natural mixed numerical formulation. • From 3D to 2D problem reduction, Only smallest modulus eigenvalues to be computed (principal modes), Numerical validation on a test case.
Introduction Analyse Numerical analysis Conclusion and perspectives • Nice mathematical framework : orthogonality properties, problem analysis on a compete orthogonal base. • Natural mixed numerical formulation. • From 3D to 2D problem reduction, Only smallest modulus eigenvalues to be computed (principal modes), Numerical validation on a test case. • General 2D implementation, • heat exchanger shape optimisation.
Recommend
More recommend
