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Modeling, Mathematical and Numerical Analysis for some Compressible and Incompressible Equations in Thin Layer. M. Ersoy 15 october 2010 Outline of the talk Outline of the talk 1 Introduction Atmosphere dynamic Sedimentation Unsteady mixed

  1. Unsteady mixed flows in closed water pipes Dynamic : ◮ Incompressible or compressible fluid following the area ◮ Small vertical extension with respect to horizontal ◮ Principally horizontal movements : unidirectional Modeling : A nice coupling of Saint-Venant like equations ◮ free surface part − → usual Saint-Venant equations ◮ pressurized part − → Saint-Venant like equations M. Ersoy (BCAM) PhD Works 15 october 2010 10 / 59

  2. Unsteady mixed flows in closed water pipes Dynamic : ◮ Incompressible or compressible fluid following the area ◮ Small vertical extension with respect to horizontal ◮ Principally horizontal movements : unidirectional Modeling : A nice coupling of Saint-Venant like equations ◮ free surface part − → usual Saint-Venant equations ∂ t A fs + ∂ x Q fs = 0 ,   � � Q 2  = − gA fs d Z   fs ∂ t Q fs + ∂ x A fs + p fs ( x, A fs ) dx + Pr fs ( x, A fs ) − G ( x, A f   − K ( x, A fs ) Q fs | Q fs |     A fs ◮ pressurized part − → Saint-Venant like equations M. Ersoy (BCAM) PhD Works 15 october 2010 10 / 59

  3. Unsteady mixed flows in closed water pipes Dynamic : ◮ Incompressible or compressible fluid following the area ◮ Small vertical extension with respect to horizontal ◮ Principally horizontal movements : unidirectional Modeling : A nice coupling of Saint-Venant like equations ◮ free surface part − → usual Saint-Venant equations ∂ t A fs + ∂ x Q fs = 0 ,   � � Q 2  = − gA fs d Z   fs ∂ t Q fs + ∂ x A fs + p fs ( x, A fs ) dx + Pr fs ( x, A fs ) − G ( x, A f   − K ( x, A fs ) Q fs | Q fs |     A fs ◮ pressurized part − → Saint-Venant like equations  ∂ t A p + ∂ x Q p = 0 ,   � Q 2 �  − gA p d Z  p ∂ t Q p + ∂ x A p + p p ( x, A p ) = dx + Pr p ( x, A p ) − G ( x, A p )  − K ( x, A p ) Q p | Q p |      A p M. Ersoy (BCAM) PhD Works 15 october 2010 10 / 59

  4. Unsteady mixed flows in closed water pipes Dynamic : ◮ Incompressible or compressible fluid following the area ◮ Small vertical extension with respect to horizontal ◮ Principally horizontal movements : unidirectional Modeling : A nice coupling : The PFS model ◮ from the coupling : � A fs if FS A = : the mixed variable A p if P Q = Au : the discharge ↓  ∂ t ( A ) + ∂ x ( Q ) = 0      � Q 2 �  = − g A d   ∂ t ( Q ) + ∂ x A + p ( x, A, E ) dxZ ( x )   + Pr ( x, A, E )    − G ( x, A, E )    − g K ( x, S ) Q | Q |     A where E is a state indicator and appropriate p and Pr M. Ersoy (BCAM) PhD Works 15 october 2010 10 / 59

  5. Outline Outline 1 Introduction Atmosphere dynamic Sedimentation Unsteady mixed flows in closed water pipes 2 Mathematical results on CPEs An intermediate model Toward an existence result for the 2D-CPEs Toward a stability result for the 3D-CPEs Perspectives 3 An upwinded kinetic scheme for the PFS equations Finite Volume method Kinetic Formulation and numerical scheme Numerical results Perspectives 4 Formal derivation of a SVEs like model A nice coupling : Vlasov and Anisotropic Navier-Stokes equations Hydrodynamic limit, toward a“mixed model” A Viscous Saint-Venant-Exner like model Perspective M. Ersoy (BCAM) PhD Works 15 october 2010 11 / 59

  6. Energy estimates ? CPEs :  ∂ t ρ + div x ( ρ u ) + ∂ y ( ρv ) = 0 ,    ∂ t ( ρ u ) + div x ( ρ u ⊗ u ) + ∂ y ( ρ v u ) + ∇ x p ( ρ ) = 2 div x ( ν 1 D x ( u )) + ∂ y ( ν 2 ∂ y u ) , ∂ y p ( ρ ) = − gρ    p ( ρ ) = c 2 ρ M. Ersoy (BCAM) PhD Works 15 october 2010 12 / 59

  7. Energy estimates ? CPEs :  ∂ t ρ + div x ( ρ u ) + ∂ y ( ρv ) = 0 ,    ∂ t ( ρ u ) + div x ( ρ u ⊗ u ) + ∂ y ( ρ v u ) + ∇ x p ( ρ ) = 2 div x ( ν 1 D x ( u )) + ∂ y ( ν 2 ∂ y u ) , ∂ y p ( ρ ) = − gρ    p ( ρ ) = c 2 ρ Problem : How to obtain energy estimates since : the sign of � ρgv dxdy Ω � � � d ρ | u | 2 + ρ ln ρ − ρ +1 dxdy + 2 ν 1 | D x ( u ) | 2 + ν 2 | ∂ 2 y u | dxdy + ρgv dxdy = 0 dt Ω Ω Ω is unknown ! M. Ersoy (BCAM) PhD Works 15 october 2010 12 / 59

  8. Energy estimates ? CPEs :  ∂ t ρ + div x ( ρ u ) + ∂ y ( ρv ) = 0 ,    ∂ t ( ρ u ) + div x ( ρ u ⊗ u ) + ∂ y ( ρ v u ) + ∇ x p ( ρ ) = 2 div x ( ν 1 D x ( u )) + ∂ y ( ν 2 ∂ y u ) , ∂ y p ( ρ ) = − gρ    p ( ρ ) = c 2 ρ Problem : How to obtain energy estimates since : the sign of � ρgv dxdy Ω � � � d ρ | u | 2 + ρ ln ρ − ρ +1 dxdy + 2 ν 1 | D x ( u ) | 2 + ν 2 | ∂ 2 y u | dxdy + ρgv dxdy = 0 dt Ω Ω Ω is unknown ! Consequently standard techniques fails M. Ersoy (BCAM) PhD Works 15 october 2010 12 / 59

  9. Outline Outline 1 Introduction Atmosphere dynamic Sedimentation Unsteady mixed flows in closed water pipes 2 Mathematical results on CPEs An intermediate model Toward an existence result for the 2D-CPEs Toward a stability result for the 3D-CPEs Perspectives 3 An upwinded kinetic scheme for the PFS equations Finite Volume method Kinetic Formulation and numerical scheme Numerical results Perspectives 4 Formal derivation of a SVEs like model A nice coupling : Vlasov and Anisotropic Navier-Stokes equations Hydrodynamic limit, toward a“mixed model” A Viscous Saint-Venant-Exner like model Perspective M. Ersoy (BCAM) PhD Works 15 october 2010 13 / 59

  10. The key point : the hydrostatic equation Using the hydrostatic equation, we obviously have : ρ ( t, x, y ) = ξ ( t, x ) e − g/c 2 y for some function ξ ( t, x ) : ρ is stratified M. Ersoy (BCAM) PhD Works 15 october 2010 14 / 59

  11. The key point : the hydrostatic equation Using the hydrostatic equation, we obviously have : ρ ( t, x, y ) = ξ ( t, x ) e − g/c 2 y for some function ξ ( t, x ) : ρ is stratified Problem : find equations satisfied by ξ M. Ersoy (BCAM) PhD Works 15 october 2010 14 / 59

  12. The key point : the hydrostatic equation Using the hydrostatic equation, we obviously have : ρ ( t, x, y ) = ξ ( t, x ) e − g/c 2 y for some function ξ ( t, x ) : ρ is stratified Problem : find equations satisfied by ξ An intermediate model : replace ρ by ξe − g/c 2 y in CPEs � � � �  ξe − g/c 2 y u ∂ t ( ξe − g/c 2 y ) + div x ξe − g/c 2 y v  + ∂ y = 0 ,   � � � � � �   ξe − g/c 2 y u ξe − g/c 2 y u ⊗ u ξe − g/c 2 y v u ∂ t + div x + ∂ y   + ∇ x c 2 ∇ x ( ξe − g/c 2 y ) = 2 div x ( ν 1 D x ( u )) + ∂ y ( ν 2 ∂ y u ) ,    ρ = ξe − g/c 2 y multiply CPEs by e + g/c 2 y M. Ersoy (BCAM) PhD Works 15 october 2010 14 / 59

  13. The key point : the hydrostatic equation Using the hydrostatic equation, we obviously have : ρ ( t, x, y ) = ξ ( t, x ) e − g/c 2 y for some function ξ ( t, x ) : ρ is stratified Problem : find equations satisfied by ξ An intermediate model : replace ρ by ξe − g/c 2 y in CPEs multiply CPEs by e + g/c 2 y � �  ∂ t ( ξ ) + div x ( ξ u ) + e g/c 2 y ∂ y ξe − g/c 2 y v  = 0 ,   � �   ξe − g/c 2 y v u ∂ t ( ξ u ) + div x ( ξ u ⊗ u ) + e g/c 2 y ∂ y + c 2 ∇ x ξ =  2 e g/c 2 y div x ( ν 1 D x ( u )) + e g/c 2 y ∂ y ( ν 2 ∂ y u ) ,     ρ = ξe − g/c 2 y set z = 1 − e − g/c 2 y such that e g/c 2 y ∂ y = ∂ z and w = e − g/c 2 y v under suitable choice of viscosities. M. Ersoy (BCAM) PhD Works 15 october 2010 14 / 59

  14. The key point : the hydrostatic equation Using the hydrostatic equation, we obviously have : ρ ( t, x, y ) = ξ ( t, x ) e − g/c 2 y for some function ξ ( t, x ) : ρ is stratified Problem : find equations satisfied by ξ An intermediate model :  ∂ t ξ + div x ( ξ u ) + ξ∂ z w = 0 ,  ∂ t ( ξ u ) + div x ( ξ u ⊗ u ) + ∂ z ( ξ w u ) + c 2 ∇ x ( ξ ) = 2 div x ( ν 1 D x ( u )) + ∂ z ( ν 2 ∂ z u ) ,  ∂ z ξ = 0 M. Ersoy (BCAM) PhD Works 15 october 2010 14 / 59

  15. The key point : the hydrostatic equation Using the hydrostatic equation, we obviously have : ρ ( t, x, y ) = ξ ( t, x ) e − g/c 2 y for some function ξ ( t, x ) : ρ is stratified Problem : find equations satisfied by ξ An intermediate model :  ∂ t ξ + div x ( ξ u ) + ξ∂ z w = 0 ,  ∂ t ( ξ u ) + div x ( ξ u ⊗ u ) + ∂ z ( ξ w u ) + c 2 ∇ x ( ξ ) = 2 div x ( ν 1 D x ( u )) + ∂ z ( ν 2 ∂ z u ) ,  ∂ z ξ = 0 � � d ξ | u | 2 + ξ ln ξ − ξ + 1 dxdz + 2 ν 1 | D x ( u ) | 2 + ν 2 | ∂ 2 z u | dxdz = 0 dt Ω Ω M. Ersoy (BCAM) PhD Works 15 october 2010 14 / 59

  16. Outline Outline 1 Introduction Atmosphere dynamic Sedimentation Unsteady mixed flows in closed water pipes 2 Mathematical results on CPEs An intermediate model Toward an existence result for the 2D-CPEs Toward a stability result for the 3D-CPEs Perspectives 3 An upwinded kinetic scheme for the PFS equations Finite Volume method Kinetic Formulation and numerical scheme Numerical results Perspectives 4 Formal derivation of a SVEs like model A nice coupling : Vlasov and Anisotropic Navier-Stokes equations Hydrodynamic limit, toward a“mixed model” A Viscous Saint-Venant-Exner like model Perspective M. Ersoy (BCAM) PhD Works 15 october 2010 15 / 59

  17. The 2D-CPEs We set � ν 1 ( t, x, y ) = ν 0 e − g/c 2 y for some given positive constant ν 0 , ν 2 ( t, x, y ) = ν 1 e g/c 2 y for some given positive constant ν 1 . the boundary conditions (BC)  u | x =0 = u | x = l = 0 ,  v | y =0 = v | y = h = 0 ,  ∂ y u | y =0 = ∂ y u | y = h = 0 and the initial conditions (IC) : � u | t =0 = u 0 ( x, y ) , ρ | t =0 = ξ 0 ( x ) e − g/c 2 y where ξ 0 : 0 < m � ξ 0 � M < ∞ . M. Ersoy (BCAM) PhD Works 15 october 2010 16 / 59

  18. Theorem ([EN2010]) Suppose that initial data ( ξ 0 , u 0 ) have the properties : ( ξ 0 , u 0 ) ∈ W 1 , 2 (Ω) , u 0 | x =0 = u 0 | x = l = 0 . Then ρ ( t, x, y ) is a bounded strictly positive function and the 2D-CPEs with BC has a weak solution in the following sense : a weak solution of 2D-CPEs with BC is a collection ( ρ, u, v ) of functions such that ρ � 0 and ρ ∈ L ∞ (0 , T ; W 1 , 2 (Ω)) , ∂ t ρ ∈ L 2 (0 , T ; L 2 (Ω)) , u ∈ L 2 (0 , T ; W 2 , 2 (Ω)) ∩ W 1 , 2 (0 , T ; L 2 (Ω)) , v ∈ L 2 (0 , T ; L 2 (Ω)) which satisfies the 2D-CPEs in the distribution sense ; in particular, the integral identity holds for all φ | t = T = 0 with compact support : � T � ρu∂ t φ + ρu 2 ∂ x φ + ρuv∂ z φ + ρ∂ x φ + ρvφ dxdydt 0 Ω � T � � = − ν 1 ∂ x u∂ x φ + ν 2 ∂ y u∂ y φ dxdydt + u 0 ρ 0 φ | t =0 dxdy 0 Ω Ω M. Ersoy and T. Ngom Existence of a global weak solution to one model of Compressible Primitive Equations. submitted to Applied Mathematics Letters , 2010. M. Ersoy (BCAM) PhD Works 15 october 2010 17 / 59

  19. the proof The intermediate model (IM) is exactly the model studied by Gatapov et al [GK05], derived from Equations 2D-CPEs by neglecting some terms, for which they provide the following global existence result : Theorem (B. Gatapov and A.V. Kazhikhov 2005) Suppose that initial data ( ξ 0 , u 0 ) have the properties : ( ξ 0 , u 0 ) ∈ W 1 , 2 (Ω) , u 0 | x =0 = u 0 | x =1 = 0 . Then ξ ( t, x ) is a bounded strictly positive function and the IM has a weak solution in the following sense : a weak solution of the IM satisfying the BC is a collection ( ξ, u, w ) of functions such that ξ � 0 and ξ ∈ L ∞ (0 , T ; W 1 , 2 (0 , 1)) , ∂ t ξ ∈ L 2 (0 , T ; L 2 (0 , 1)) , u ∈ L 2 (0 , T ; W 2 , 2 (Ω)) ∩ W 1 , 2 (0 , T ; L 2 (Ω)) , w ∈ L 2 (0 , T ; L 2 (Ω)) which satisfy the IM in the distribution sense. B. V. Gatapov and A. V. Kazhikhov Existence of a global solution to one model problem of atmosphere dynamics. Sibirsk. Mat. Zh. , pages 1011 :1020–722, 2005. M. Ersoy (BCAM) PhD Works 15 october 2010 18 / 59

  20. the proof By the simple change of variables z = 1 − e − y in the integrals, we get : � ρ � L 2 (Ω) = α � ξ � L 2 (Ω) , � ∇ x ρ � L 2 (Ω) = α � ∇ x ξ � L 2 (Ω) , � ∂ y ρ � L 2 (Ω) = α � ξ � L 2 (Ω) � 1 − e − 1 where α = (1 − z ) dz < + ∞ . We deduce then, 0 � ρ � W 1 , 2 (Ω) = α � ξ � W 1 , 2 (Ω) which provides ρ ∈ L ∞ (0 , T ; W 1 , 2 (Ω)) and ∂ t ρ ∈ L 2 (0 , T ; L 2 (Ω)) . v ∈ L 2 (0 , T ; L 2 (Ω)) since the inequality holds : � 1 � 1 | v ( t, x, y ) | 2 dy dx � v � L 2 (Ω) = 0 0 � 1 � 1 − e − 1 � � 3 1 | w ( t, x, z ) | 2 dz dx = 1 − z 0 0 e 3 � w � L 2 (Ω) . < Finally, all estimates on u remain true. � M. Ersoy (BCAM) PhD Works 15 october 2010 18 / 59

  21. Outline Outline 1 Introduction Atmosphere dynamic Sedimentation Unsteady mixed flows in closed water pipes 2 Mathematical results on CPEs An intermediate model Toward an existence result for the 2D-CPEs Toward a stability result for the 3D-CPEs Perspectives 3 An upwinded kinetic scheme for the PFS equations Finite Volume method Kinetic Formulation and numerical scheme Numerical results Perspectives 4 Formal derivation of a SVEs like model A nice coupling : Vlasov and Anisotropic Navier-Stokes equations Hydrodynamic limit, toward a“mixed model” A Viscous Saint-Venant-Exner like model Perspective M. Ersoy (BCAM) PhD Works 15 october 2010 19 / 59

  22. The 3D-CPEs We set ν 2 ρ ( t, x, y ) e 2 y . ν 1 ( t, x, y ) = ¯ ν 1 ρ ( t, x, y ) and ν 2 = ¯ for some positive constant ¯ ν 1 and ¯ ν 2 . We consider the IC and BC’ where we prescribe periodic conditions on the spatiale domain with respect to x . We define the set of function ρ ∈ PE ( u , v ; y, ρ 0 ) such that √ ρ ∈ L ∞ (0 , T ; H 1 (Ω)) , ρ ∈ L ∞ (0 , T ; L 3 (Ω)) , √ ρ u ∈ L 2 (0 , T ; ( L 2 (Ω)) 2 ) , √ ρv ∈ L ∞ (0 , T ; L 2 (Ω)) , √ ρD x ( u ) ∈ L 2 (0 , T ; ( L 2 (Ω)) 2 × 2 ) , √ ρ∂ y v ∈ L 2 (0 , T ; L 2 (Ω)) , ∇√ ρ ∈ L 2 (0 , T ; ( L 2 (Ω)) 3 ) with ρ � 0 and where ( ρ, √ ρ u , √ ρv ) satisfies : � ∂ t ρ + div x ( √ ρ √ ρ u ) + ∂ y ( √ ρ √ ρv ) = 0 , ρ t =0 = ρ 0 . M. Ersoy (BCAM) PhD Works 15 october 2010 20 / 59

  23. The 3D-CPEs We define, for any smooth test function ϕ with compact support such as ϕ ( T, x, y ) = 0 and ϕ 0 = ϕ t =0 , the operators : � T � A ( ρ, u , v ; ϕ, dy ) = − ρ u ∂ t ϕ dxdydt 0 Ω � T � + (2 ν 1 ( t, x, y ) ρD x ( u ) − ρ u ⊗ u ) : ∇ x ϕ dxdydt 0 Ω � T � � T � + rρ | u | u ϕ dxdydt − ρ div ( ϕ ) dxdydt 0 Ω 0 Ω � T � − u ∂ y ( ν 2 ( t, x, y ) ∂ y ϕ ) dxdydt 0 Ω � T � − ρv u ∂ y ϕ dxdydt 0 Ω � T � B ( ρ, u , v ; ϕ, dy ) = ρvϕ dxdydt 0 Ω and � C ( ρ, u ; ϕ, dy ) = ρ | t =0 u | t =0 ϕ 0 dxdy Ω M. Ersoy (BCAM) PhD Works 15 october 2010 20 / 59

  24. A weak solution Definition A weak solution of System 3D-CPEs on [0 , T ] × Ω , with BC and IC, is a collection of functions ( ρ, u , v ) such as ρ ∈ PE ( u , v ; y, ρ 0 ) and the following equality holds for all smooth test function ϕ with compact support such as ϕ ( T, x, y ) = 0 and ϕ 0 = ϕ t =0 : A ( ρ, u , v ; ϕ, dy ) + B ( ρ, u , v ; ϕ, dy ) = C ( ρ, u ; ϕ, dy ) . M. Ersoy, T. Ngom, M. Sy Compressible primitive equations : formal derivation and stability of weak solutions. submitted to NonLinearity , 2010. M. Ersoy (BCAM) PhD Works 15 october 2010 21 / 59

  25. A weak solution Theorem ([ENS2010]) Let ( ρ n , u n , v n ) be a sequence of weak solutions of System 3D-CPEs, with BC and IC, satisfying an entropy and energy inequality (EEI) such as ρ n 0 → ρ 0 in L 1 (Ω) , ρ n 0 u n 0 → ρ 0 u 0 in L 1 (Ω) . ρ n � 0 , Then, up to a subsequence, ρ n converges strongly in C 0 (0 , T ; L 3 / 2 (Ω)) , √ ρ n u n converges strongly in L 2 (0 , T ; ( L 3 / 2 (Ω)) 2 ) , ρ n u n converges strongly in L 1 (0 , T ; ( L 1 (Ω)) 2 ) for all T > 0 , ( ρ n , √ ρ n u n , √ ρ n v n ) converges to a weak solution of System 3D-CPEs, ( ρ n , u n , v n ) satisfies the EEI and converges to a weak solution of 3D-CPEs-BC. M. Ersoy, T. Ngom, M. Sy Compressible primitive equations : formal derivation and stability of weak solutions. submitted to NonLinearity , 2010. M. Ersoy (BCAM) PhD Works 15 october 2010 21 / 59

  26. Sketch of the proof-step 1 Prove first the stability for the IM’ with IC and BC’,  ∂ t ξ + div x ( ξ u ) + ∂ z ( ξ w ) = 0 ,    ∂ t ( ξ u ) + div x ( ξ u ⊗ u ) + ∂ z ( ξ u w ) + ∇ x ξ + rξ | u | u = 2¯ ν 1 div x ( ξD x ( u )) + ¯ ν 2 ∂ z ( ξ∂ z u ) ,    ∂ z ξ = 0 and by the reverse change of variables“transport”the result to the 3D-CPEs. M. Ersoy (BCAM) PhD Works 15 october 2010 22 / 59

  27. Sketch of the proof-step 1 Prove first the stability for the IM’ with IC and BC’,  ∂ t ξ + div x ( ξ u ) + ∂ z ( ξ w ) = 0 ,    ∂ t ( ξ u ) + div x ( ξ u ⊗ u ) + ∂ z ( ξ u w ) + ∇ x ξ + rξ | u | u = 2¯ ν 1 div x ( ξD x ( u )) + ¯ ν 2 ∂ z ( ξ∂ z u ) ,    ∂ z ξ = 0 and by the reverse change of variables“transport”the result to the 3D-CPEs. So, Definition ′ , with BC’ and IC, is a collection of A weak solution of System IM’ on [0 , T ] × Ω functions ( ξ, u , w ) , if ξ ∈ PE ( u , w ; z, ξ 0 ) and the following equality holds for all smooth test function ϕ with compact support such as ϕ ( T, x, y ) = 0 and ϕ 0 = ϕ t =0 : A ( ξ, u , w ; ϕ, dz ) = C ( ξ, u ; ϕ, dz ) . M. Ersoy (BCAM) PhD Works 15 october 2010 22 / 59

  28. Sketch of the proof-step 1 Prove first the stability for the IM’ with IC and BC’,  ∂ t ξ + div x ( ξ u ) + ∂ z ( ξ w ) = 0 ,    ∂ t ( ξ u ) + div x ( ξ u ⊗ u ) + ∂ z ( ξ u w ) + ∇ x ξ + rξ | u | u = 2¯ ν 1 div x ( ξD x ( u )) + ¯ ν 2 ∂ z ( ξ∂ z u ) ,    ∂ z ξ = 0 and by the reverse change of variables“transport”the result to the 3D-CPEs. So, Definition ′ , with BC’ and IC, is a collection of A weak solution of System IM’ on [0 , T ] × Ω functions ( ξ, u , w ) , if ξ ∈ PE ( u , w ; z, ξ 0 ) and the following equality holds for all smooth test function ϕ with compact support such as ϕ ( T, x, y ) = 0 and ϕ 0 = ϕ t =0 : A ( ξ, u , w ; ϕ, dz ) = C ( ξ, u ; ϕ, dz ) . Difficulty : show that under suitable sequence of weak solutions, we can pass to the � limit in the non-linear term ξ u ⊗ u : typically ξ u requires strong convergence. M. Ersoy (BCAM) PhD Works 15 october 2010 22 / 59

  29. Theorem Let ( ξ n , u n , w n ) be a sequence of weak solutions of the IM’ with BC’ and IC satisfying an energy and entropy inequality (EEI) such as ξ n ′ ) , ξ n 0 u n ′ ) . 0 → ξ 0 in L 1 (Ω 0 → ξ 0 u 0 in L 1 (Ω ξ n � 0 , Then, up to a subsequence, ′ )) , ξ n converges strongly in C 0 (0 , T ; L 3 / 2 (Ω ′ )) 2 ) , � ξ n u n converges strongly in L 2 (0 , T ; ( L 3 / 2 (Ω ′ )) 2 ) for all T > 0 , ξ n u n converges strongly in L 1 (0 , T ; ( L 1 (Ω � � ( ξ n , ξ n u n , ξ n w n ) converges to a weak solution of the IM’, ( ξ n , u n , w n ) satisfies the EEI and converges to a weak solution of the IM’ with BC’. The energy inequality : ξ u 2 d � � ν 1 | D x ( u ) | 2 + ¯ ν 2 | ∂ z u | 2 ) dxdz � � 2 + ( ξ ln ξ − ξ + 1) dxdz + Ω ′ ξ (2¯ dt Ω ′ � Ω ′ ξ | u | 3 dxdz � 0 + r M. Ersoy (BCAM) PhD Works 15 october 2010 23 / 59

  30. Theorem Let ( ξ n , u n , w n ) be a sequence of weak solutions of the IM’ with BC’ and IC satisfying an energy and entropy inequality (EEI) such as ′ ) , ′ ) . ξ n 0 → ξ 0 in L 1 (Ω ξ n 0 u n 0 → ξ 0 u 0 in L 1 (Ω ξ n � 0 , Then, up to a subsequence, ξ n converges strongly in C 0 (0 , T ; L 3 / 2 (Ω ′ )) , ′ )) 2 ) , � ξ n u n converges strongly in L 2 (0 , T ; ( L 3 / 2 (Ω ′ )) 2 ) for all T > 0 , ξ n u n converges strongly in L 1 (0 , T ; ( L 1 (Ω � � ( ξ n , ξ n u n , ξ n w n ) converges to a weak solution of the IM’, ( ξ n , u n , w n ) satisfies the EEI and converges to a weak solution of the IM’ with BC’. The entropy inequality : 1 d � ν 1 ∇ x ln ξ | 2 + 2( ξ log ξ − ξ + 1) � � ξ | u + 2¯ dxdz 2 dt Ω ′ � � ν 1 ξ | ∂ z w | 2 + 2¯ ν 1 ξ | A x ( u ) | 2 + ¯ ν 2 ξ | ∂ z u | 2 dxdz + Ω ′ rξ | u | 3 + 2¯ + Ω ′ 2¯ ν 1 r | u | u ∇ x ξ dxdz � ξ | 2 dxdz = 0 . � + Ω ′ 8¯ ν 1 |∇ x M. Ersoy (BCAM) PhD Works 15 october 2010 23 / 59

  31. Sketch of the proof-step 2 To prove the stability result on IM’, we proceed as follows : 1 we obtain suitable a priori bounds on ( ξ, u , w ) , we get estimates from the energy inequality, 1 we get estimates from the BD-entropy inequality, i.e. : a kind of energy with 2 the muliplier u + 2¯ ν 1 ∇ x ξ . 2 we show the compactness of sequences ( ξ n , u n , w n ) in appropriate space function, � we show the convergence of the sequence ξ n , 1 � � we seek bounds of ξ n u n and ξ n w n , 2 we prove the convergence of ξ n u n , 3 � we prove the convergence of ξ n u n . 4 3 we prove that we can pass to the limit in all terms of the IM’, 4 We“transport”this result with the reverse change of variable to the 3D-CPEs. � M. Ersoy (BCAM) PhD Works 15 october 2010 24 / 59

  32. Outline Outline 1 Introduction Atmosphere dynamic Sedimentation Unsteady mixed flows in closed water pipes 2 Mathematical results on CPEs An intermediate model Toward an existence result for the 2D-CPEs Toward a stability result for the 3D-CPEs Perspectives 3 An upwinded kinetic scheme for the PFS equations Finite Volume method Kinetic Formulation and numerical scheme Numerical results Perspectives 4 Formal derivation of a SVEs like model A nice coupling : Vlasov and Anisotropic Navier-Stokes equations Hydrodynamic limit, toward a“mixed model” A Viscous Saint-Venant-Exner like model Perspective M. Ersoy (BCAM) PhD Works 15 october 2010 25 / 59

  33. 1 Prove the existence of weak solutions of the 3D-CPEs 2 Generalize to any anisotropic pair of viscosities 3 Deal with the case of p = kρ γ , γ � = 1 , k = cte (also the case k = k ( t, x, y ) ) M. Ersoy (BCAM) PhD Works 15 october 2010 26 / 59

  34. Outline Outline 1 Introduction Atmosphere dynamic Sedimentation Unsteady mixed flows in closed water pipes 2 Mathematical results on CPEs An intermediate model Toward an existence result for the 2D-CPEs Toward a stability result for the 3D-CPEs Perspectives 3 An upwinded kinetic scheme for the PFS equations Finite Volume method Kinetic Formulation and numerical scheme Numerical results Perspectives 4 Formal derivation of a SVEs like model A nice coupling : Vlasov and Anisotropic Navier-Stokes equations Hydrodynamic limit, toward a“mixed model” A Viscous Saint-Venant-Exner like model Perspective M. Ersoy (BCAM) PhD Works 15 october 2010 27 / 59

  35. The PFS Equation are :  ∂ t ( A ) + ∂ x ( Q ) = 0      � Q 2 �    = − g A d   ∂ t ( Q ) + ∂ x A + p ( x, A, E ) dxZ ( x )  + Pr ( x, A, E )     − G ( x, A, E )     − g K ( x, S ) Q | Q |  A � A fs if FS with A = A p if P M. Ersoy (BCAM) PhD Works 15 october 2010 28 / 59

  36. Outline Outline 1 Introduction Atmosphere dynamic Sedimentation Unsteady mixed flows in closed water pipes 2 Mathematical results on CPEs An intermediate model Toward an existence result for the 2D-CPEs Toward a stability result for the 3D-CPEs Perspectives 3 An upwinded kinetic scheme for the PFS equations Finite Volume method Kinetic Formulation and numerical scheme Numerical results Perspectives 4 Formal derivation of a SVEs like model A nice coupling : Vlasov and Anisotropic Navier-Stokes equations Hydrodynamic limit, toward a“mixed model” A Viscous Saint-Venant-Exner like model Perspective M. Ersoy (BCAM) PhD Works 15 october 2010 29 / 59

  37. Finite Volume (VF) numerical scheme of order 1 Cell-centered numerical scheme PFS equations under vectorial form : ∂ t U ( t, x ) + ∂ x F ( x, U ) = S ( t, x ) M. Ersoy (BCAM) PhD Works 15 october 2010 30 / 59

  38. Finite Volume (VF) numerical scheme of order 1 Cell-centered numerical scheme PFS equations under vectorial form : ∂ t U ( t, x ) + ∂ x F ( x, U ) = S ( t, x ) � 1 cte per mesh with U n ≈ U ( t n , x ) dx and S ( t, x ) constant per mesh, i ∆ x m i M. Ersoy (BCAM) PhD Works 15 october 2010 30 / 59

  39. Finite Volume (VF) numerical scheme of order 1 Cell-centered numerical scheme PFS equations under vectorial form : ∂ t U ( t, x ) + ∂ x F ( x, U ) = S ( t, x ) � 1 cte per mesh with U n ≈ U ( t n , x ) dx and S ( t, x ) constant per mesh, i ∆ x m i Cell-centered numerical scheme : i − ∆ t n � � U n +1 = U n + ∆ t n S ( U n F i +1 / 2 − F i − 1 / 2 i ) i ∆ x where � t n +1 � ∆ t n S n i ≈ S ( t, x ) dx dt t n m i M. Ersoy (BCAM) PhD Works 15 october 2010 30 / 59

  40. Finite Volume (VF) numerical scheme of order 1 Upwinded numerical scheme PFS equations under vectorial form : ∂ t U ( t, x ) + ∂ x F ( x, U ) = S ( t, x ) � 1 cte per mesh with U n ≈ U ( t n , x ) dx and S ( t, x ) constant per mesh, i ∆ x m i Upwinded numerical scheme : � � i − ∆ t n U n +1 = U n F i +1 / 2 − � � F i − 1 / 2 i ∆ x M. Ersoy (BCAM) PhD Works 15 october 2010 30 / 59

  41. Choice of the numerical fluxes Our goal : define F i +1 / 2 in order to preserve continuous properties of the PFS-model Positivity of A , conservativity of A , discrete equilibrium, discrete entropy inequality M. Ersoy (BCAM) PhD Works 15 october 2010 31 / 59

  42. Choice of the numerical fluxes Our goal : define F i +1 / 2 in order to preserve continuous properties of the PFS-model Positivity of A , conservativity of A , discrete equilibrium, discrete entropy inequality M. Ersoy (BCAM) PhD Works 15 october 2010 31 / 59

  43. Choice of the numerical fluxes Our goal : define F i +1 / 2 in order to preserve continuous properties of the PFS-model Positivity of A , conservativity of A , discrete equilibrium, discrete entropy inequality M. Ersoy (BCAM) PhD Works 15 october 2010 31 / 59

  44. Choice of the numerical fluxes Our goal : define F i +1 / 2 in order to preserve continuous properties of the PFS-model Positivity of A , conservativity of A , discrete equilibrium, discrete entropy inequality Our choice VFRoe solver[BEGVF] Kinetic solver[BEG10] C. Bourdarias, M. Ersoy and S. Gerbi. A model for unsteady mixed flows in non uniform closed water pipes and a well-balanced finite volume scheme. International Journal On Finite Volumes , Vol 6(2) 1–47, 2009. C. Bourdarias, M. Ersoy and S. Gerbi. A kinetic scheme for transient mixed flows in non uniform closed pipes : a global manner to upwind all the source terms. To appear in J. Sci. Comp., 2010. M. Ersoy (BCAM) PhD Works 15 october 2010 31 / 59

  45. Outline Outline 1 Introduction Atmosphere dynamic Sedimentation Unsteady mixed flows in closed water pipes 2 Mathematical results on CPEs An intermediate model Toward an existence result for the 2D-CPEs Toward a stability result for the 3D-CPEs Perspectives 3 An upwinded kinetic scheme for the PFS equations Finite Volume method Kinetic Formulation and numerical scheme Numerical results Perspectives 4 Formal derivation of a SVEs like model A nice coupling : Vlasov and Anisotropic Navier-Stokes equations Hydrodynamic limit, toward a“mixed model” A Viscous Saint-Venant-Exner like model Perspective M. Ersoy (BCAM) PhD Works 15 october 2010 32 / 59

  46. Principle Density function We introduce � � ω 2 χ ( ω ) dω = 1 , χ ( ω ) = χ ( − ω ) ≥ 0 , χ ( ω ) dω = 1 , R R M. Ersoy (BCAM) PhD Works 15 october 2010 33 / 59

  47. Principle Gibbs Equilibrium or Maxwellian We introduce � � ω 2 χ ( ω ) dω = 1 , χ ( ω ) = χ ( − ω ) ≥ 0 , χ ( ω ) dω = 1 , R R then we define the Gibbs equilibrium by � ξ − u ( t, x ) � M ( t, x, ξ ) = A ( t, x ) b ( t, x ) χ b ( t, x ) with � p ( t, x ) b ( t, x ) = A ( t, x ) M. Ersoy (BCAM) PhD Works 15 october 2010 33 / 59

  48. Principle micro-macroscopic relations Since � � ω 2 χ ( ω ) dω = 1 , χ ( ω ) = χ ( − ω ) ≥ 0 , χ ( ω ) dω = 1 , R R and � ξ − u ( t, x ) � M ( t, x, ξ ) = A ( t, x ) b ( t, x ) χ b ( t, x ) then � A = M ( t, x, ξ ) dξ � R Q = ξ M ( t, x, ξ ) dξ � R Q 2 A + A b 2 ξ 2 M ( t, x, ξ ) dξ = ���� R p M. Ersoy (BCAM) PhD Works 15 october 2010 33 / 59

  49. Principle [P02] The kinetic formulation ( A, Q ) is solution of the PFS system if and only if M satisfy the transport equation : ∂ t M + ξ · ∂ x M − g Φ ∂ ξ M = K ( t, x, ξ ) where K ( t, x, ξ ) is a collision kernel satisfying a.e. ( t, x ) � � K dξ = 0 , ξ K d ξ = 0 R R and Φ are the source terms. B. Perthame . Kinetic formulation of conservation laws. Oxford University Press. Oxford Lecture Series in Mathematics and its Applications, Vol 21, 2002. M. Ersoy (BCAM) PhD Works 15 october 2010 34 / 59

  50. Principe The kinetic formulation ( A, Q ) is solution of the PFS system if and only if M satisfy the transport equation : ∂ t M + ξ · ∂ x M − g Φ ∂ ξ M = K ( t, x, ξ ) where K ( t, x, ξ ) is a collision kernel satisfying a.e. ( t, x ) � � K dξ = 0 , ξ K d ξ = 0 R R and Φ are the source terms. General form of the source terms : conservative non conservative friction � �� � ���� � �� � d B · d K Q | Q | Φ = dxZ + dx W + A 2 with W = ( Z, S, cos θ ) M. Ersoy (BCAM) PhD Works 15 october 2010 34 / 59

  51. Discretization of source terms Recalling that A, Q and Z, S, cos θ constant per mesh forgetting the friction : « splitting » . . . M. Ersoy (BCAM) PhD Works 15 october 2010 35 / 59

  52. Discretization of source terms Recalling that A, Q and Z, S, cos θ constant per mesh forgetting the friction : « splitting » . . . ◦ Then ∀ ( t, x ) ∈ [ t n , t n +1 [ × m i Φ( t, x ) = 0 since Φ = d dxZ + B · d dx W M. Ersoy (BCAM) PhD Works 15 october 2010 35 / 59

  53. Simplification of the transport equation Recalling that A, Q and Z, S, cos θ constant per mesh forgetting the friction : « splitting » . . . ◦ Then ∀ ( t, x ) ∈ [ t n , t n +1 [ × m i Φ( t, x ) = 0 since Φ = d dxZ + B · d dx W = ⇒ ∂ t M + ξ · ∂ x M = K ( t, x, ξ ) M. Ersoy (BCAM) PhD Works 15 october 2010 35 / 59

  54. Simplification of the transport equation Recalling that A, Q and Z, S, cos θ constant per mesh forgetting the friction : « splitting » . . . ◦ Then ∀ ( t, x ) ∈ [ t n , t n +1 [ × m i Φ( t, x ) = 0 since Φ = d dxZ + B · d dx W = ⇒  ∂ t f + ξ · ∂ x f = 0  � ξ − u ( t n , x, ξ ) � := A ( t n , x, ξ ) def f ( t n , x, ξ ) = M ( t n , x, ξ ) b ( t n , x, ξ ) χ  b ( t n , x, ξ ) by neglecting the collision kernel M. Ersoy (BCAM) PhD Works 15 october 2010 35 / 59

  55. Discretization of source terms On [ t n , t n +1 [ × m i , we have : � ∂ t f + ξ · ∂ x f = 0 M n f ( t n , x, ξ ) = i ( ξ ) M. Ersoy (BCAM) PhD Works 15 october 2010 36 / 59

  56. Discretization of source terms On [ t n , t n +1 [ × m i , we have : � ∂ t f + ξ · ∂ x f = 0 M n f ( t n , x, ξ ) = i ( ξ ) i.e. � � i ( ξ ) + ξ ∆ t n f n +1 M − 2 ( ξ ) − M + ( ξ ) = M n 2 ( ξ ) i i + 1 i − 1 ∆ x M. Ersoy (BCAM) PhD Works 15 october 2010 36 / 59

  57. Discretization of source terms On [ t n , t n +1 [ × m i , we have : � ∂ t f + ξ · ∂ x f = 0 M n f ( t n , x, ξ ) = i ( ξ ) i.e. � � i ( ξ ) + ξ ∆ t n f n +1 M − 2 ( ξ ) − M + ( ξ ) = M n 2 ( ξ ) i i + 1 i − 1 ∆ x where � � � � � A n +1 1 def U n +1 f n +1 i = := ( ξ ) dξ i Q n +1 i ξ i R M. Ersoy (BCAM) PhD Works 15 october 2010 36 / 59

  58. Discretization of source terms On [ t n , t n +1 [ × m i , we have : � ∂ t f + ξ · ∂ x f = 0 M n f ( t n , x, ξ ) = i ( ξ ) i.e. � � i ( ξ ) + ξ ∆ t n f n +1 M − 2 ( ξ ) − M + ( ξ ) = M n 2 ( ξ ) i i + 1 i − 1 ∆ x or � � i − ∆ t n � i +1 / 2 − � U n +1 = U n F − F + i i − 1 / 2 ∆ x with � 1 � � F ± � M ± 2 = ξ 2 ( ξ ) dξ. i ± 1 i ± 1 ξ R M. Ersoy (BCAM) PhD Works 15 october 2010 36 / 59

  59. The microscopic fluxes Interpretation : potential bareer positive transmission � �� � M − 1 { ξ> 0 } M n i +1 / 2 ( ξ ) = i ( ξ ) � � � ξ 2 − 2 g ∆Φ n i +1 / 2 > 0 } M n + 1 { ξ< 0 , ξ 2 − 2 g ∆Φ n − i +1 i +1 / 2 � �� � negative transmission M. Ersoy (BCAM) PhD Works 15 october 2010 37 / 59

  60. The microscopic fluxes Interpretation : potential bareer positive transmission reflection � �� � � �� � M − 1 { ξ> 0 } M n i +1 / 2 < 0 } M n i +1 / 2 ( ξ ) = i ( ξ ) + 1 { ξ< 0 , ξ 2 − 2 g ∆Φ n i ( − ξ ) � � � ξ 2 − 2 g ∆Φ n i +1 / 2 > 0 } M n + 1 { ξ< 0 , ξ 2 − 2 g ∆Φ n − i +1 i +1 / 2 � �� � negative transmission M. Ersoy (BCAM) PhD Works 15 october 2010 37 / 59

  61. The microscopic fluxes Interpretation : potential bareer positive transmission reflection � �� � � �� � M − 1 { ξ> 0 } M n i +1 / 2 < 0 } M n i +1 / 2 ( ξ ) = i ( ξ ) + 1 { ξ< 0 , ξ 2 − 2 g ∆Φ n i ( − ξ ) � � � ξ 2 − 2 g ∆Φ n i +1 / 2 > 0 } M n + 1 { ξ< 0 , ξ 2 − 2 g ∆Φ n − i +1 i +1 / 2 � �� � negative transmission ∆Φ n i +1 / 2 may be interpreted as a time-dependant slope ! M. Ersoy (BCAM) PhD Works 15 october 2010 37 / 59

  62. The microscopic fluxes ⇒ d´ Interpretation : pente dynamique = ecentrement de la friction positive transmission reflection � �� � � �� � M − 1 { ξ> 0 } M n i +1 / 2 < 0 } M n i +1 / 2 ( ξ ) = i ( ξ ) + 1 { ξ< 0 , ξ 2 − 2 g ∆Φ n i ( − ξ ) � � � ξ 2 − 2 g ∆Φ n i +1 / 2 > 0 } M n + 1 { ξ< 0 , ξ 2 − 2 g ∆Φ n − i +1 i +1 / 2 � �� � negative transmission ∆Φ n i +1 / 2 may be interpreted as a time-dependant slope ! . . . we reintegrate the friction . . . M. Ersoy (BCAM) PhD Works 15 october 2010 37 / 59

  63. Upwinding of the source terms conservative ∂ x W : W i +1 − W i non-conservative B ∂ x W : B ( W i +1 − W i ) where � 1 B = B ( s, φ ( s, W i , W i +1 )) ds 0 for the « straight lines paths » , i.e. φ ( s, W i , W i +1 ) = s W i +1 + (1 − s ) W i , s ∈ [0 , 1] G. Dal Maso, P. G. Lefloch and F. Murat. Definition and weak stability of nonconservative products. J. Math. Pures Appl. , Vol 74(6) 483–548, 1995. M. Ersoy (BCAM) PhD Works 15 october 2010 38 / 59

  64. Numerical properties With [ABP00] 1 χ ( ω ) = √ √ √ 3] ( ω ) 3 1 [ − 3 , 2 we have : Positivity of A (under a CFL condition), Conservativity of A , Natural treatment of drying and flooding area. for example E. Audusse and M-0. Bristeau and B. Perthame . Kinetic schemes for Saint-Venant equations with source terms on unstructured grids. INRIA Report RR3989, 2000. M. Ersoy (BCAM) PhD Works 15 october 2010 39 / 59

  65. Numerical properties With [ABP00] 1 χ ( ω ) = √ √ √ 3] ( ω ) 3 1 [ − 3 , 2 we have : Positivity of A (under a CFL condition), Conservativity of A , Natural treatment of drying and flooding area. for example − → non well-balanced scheme with such a χ − → but easy computation of the numerical fluxes E. Audusse and M-0. Bristeau and B. Perthame . Kinetic schemes for Saint-Venant equations with source terms on unstructured grids. INRIA Report RR3989, 2000. M. Ersoy (BCAM) PhD Works 15 october 2010 39 / 59

  66. Outline Outline 1 Introduction Atmosphere dynamic Sedimentation Unsteady mixed flows in closed water pipes 2 Mathematical results on CPEs An intermediate model Toward an existence result for the 2D-CPEs Toward a stability result for the 3D-CPEs Perspectives 3 An upwinded kinetic scheme for the PFS equations Finite Volume method Kinetic Formulation and numerical scheme Numerical results Perspectives 4 Formal derivation of a SVEs like model A nice coupling : Vlasov and Anisotropic Navier-Stokes equations Hydrodynamic limit, toward a“mixed model” A Viscous Saint-Venant-Exner like model Perspective M. Ersoy (BCAM) PhD Works 15 october 2010 40 / 59

  67. Upwinding of the friction 2.15 the « double dam break » 2.1 Hauteur piezometrique (m) 2.05 • horizontal pipe : L = 100 m , R = 1 m . 2 • initial state : Q = 0 m 3 /s , y = 1 . 8 m . 1.95 • Symmetric boundary conditions : 1.9 1.85 1.8 0 20 40 60 80 100 Temps (s) downstream and upstream M. Ersoy (BCAM) PhD Works 15 october 2010 41 / 59

  68. Qualitative analysis of convergence T = 0.000 Eau Ligne piezometrique 104 103 m d’eau 102 101 100 99 0 100 200 300 400 500 600 700 800 900 m upstream piezometric head 104 m Niveau piezometrique aval 103.2 103 102.8 102.6 m d’eau 102.4 102.2 102 101.8 101.6 Hauteur piezo haut du tuyau 101.4 0 2 4 6 8 10 12 14 downstream piezometric head : Temps (s) M. Ersoy (BCAM) PhD Works 15 october 2010 42 / 59

  69. Convergence During unsteady flows t = 100 s Erreur L2 : Ligne piezometrique au temps t = 100 s 0.8 Ordre VFRoe (polyfit) = 0.91301 VFRoe (sans polyfit) Ordre FKA (polyfit) = 0.88039 0.6 FKA (sans polyfit) 0.4 0.2 0 � y � L 2 -0.2 -0.4 -0.6 -0.8 -1 ln(∆ x ) 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 M. Ersoy (BCAM) PhD Works 15 october 2010 42 / 59

  70. Convergence Stationary t = 500 s Erreur L2 : Ligne piezometrique au temps t = 500 s 0 Ordre VFRoe (polyfit) = 1.0742 VFRoe (sans polyfit) Ordre FKA (polyfit) = 1.0371 -0.2 FKA (sans polyfit) -0.4 -0.6 -0.8 � y � L 2 -1 -1.2 -1.4 -1.6 -1.8 ln(∆ x ) 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 M. Ersoy (BCAM) PhD Works 15 october 2010 42 / 59

  71. Outline Outline 1 Introduction Atmosphere dynamic Sedimentation Unsteady mixed flows in closed water pipes 2 Mathematical results on CPEs An intermediate model Toward an existence result for the 2D-CPEs Toward a stability result for the 3D-CPEs Perspectives 3 An upwinded kinetic scheme for the PFS equations Finite Volume method Kinetic Formulation and numerical scheme Numerical results Perspectives 4 Formal derivation of a SVEs like model A nice coupling : Vlasov and Anisotropic Navier-Stokes equations Hydrodynamic limit, toward a“mixed model” A Viscous Saint-Venant-Exner like model Perspective M. Ersoy (BCAM) PhD Works 15 october 2010 43 / 59

  72. 1 Study of the convergence with respect to the χ function 2 Study of the convergence with respect to the paths used to define the non-conservative product M. Ersoy (BCAM) PhD Works 15 october 2010 44 / 59

  73. Outline Outline 1 Introduction Atmosphere dynamic Sedimentation Unsteady mixed flows in closed water pipes 2 Mathematical results on CPEs An intermediate model Toward an existence result for the 2D-CPEs Toward a stability result for the 3D-CPEs Perspectives 3 An upwinded kinetic scheme for the PFS equations Finite Volume method Kinetic Formulation and numerical scheme Numerical results Perspectives 4 Formal derivation of a SVEs like model A nice coupling : Vlasov and Anisotropic Navier-Stokes equations Hydrodynamic limit, toward a“mixed model” A Viscous Saint-Venant-Exner like model Perspective M. Ersoy (BCAM) PhD Works 15 october 2010 45 / 59

  74. A Saint-Venant-Exner model Saint-Venant equations for the hydrodynamic part :  ∂ t h + div ( q ) = 0 ,  � q ⊗ q � � � g h 2  ∂ t q + div + ∇ = − gh ∇ b h 2 + a bedload transport equation for the morphodynamic part : ∂ t b + ξ div ( q b ( h, q )) = 0 M. Ersoy (BCAM) PhD Works 15 october 2010 46 / 59

  75. A Saint-Venant-Exner model Saint-Venant equations for the hydrodynamic part :  ∂ t h + div ( q ) = 0 ,  � q ⊗ q � � � g h 2  ∂ t q + div + ∇ = − gh ∇ b h 2 + a bedload transport equation for the morphodynamic part : ∂ t b + ξ div ( q b ( h, q )) = 0 with h : water height, q = hu : water discharge, q b : sediment discharge (empirical law : [MPM48], [G81]), ξ = 1 / (1 − ψ ) : porosity coefficient. E. Meyer-Peter and R. M¨ uller , Formula for bed-load transport, Rep. 2nd Meet. Int. Assoc. Hydraul. Struct. Res., 39–64 , 1948. A.J. Grass , Sediment transport by waves and currents, SERC London Cent. Mar. Technol. Report No. FL29 , 1981. M. Ersoy (BCAM) PhD Works 15 october 2010 46 / 59

  76. A Saint-Venant-Exner model Saint-Venant equations for the hydrodynamic part :  ∂ t h + div ( q ) = 0 ,  � q ⊗ q � � � g h 2  ∂ t q + div + ∇ = − gh ∇ b h 2 + a bedload transport equation for the morphodynamic part : ∂ t b + ξ div ( q b ( h, q )) = 0 with h : water height, q = hu : water discharge, q b : sediment discharge (empirical law : [MPM48], [G81]), ξ = 1 / (1 − ψ ) : porosity coefficient. Our goal : derive formally this type of equation from a non classical way M. Ersoy (BCAM) PhD Works 15 october 2010 46 / 59

  77. Outline Outline 1 Introduction Atmosphere dynamic Sedimentation Unsteady mixed flows in closed water pipes 2 Mathematical results on CPEs An intermediate model Toward an existence result for the 2D-CPEs Toward a stability result for the 3D-CPEs Perspectives 3 An upwinded kinetic scheme for the PFS equations Finite Volume method Kinetic Formulation and numerical scheme Numerical results Perspectives 4 Formal derivation of a SVEs like model A nice coupling : Vlasov and Anisotropic Navier-Stokes equations Hydrodynamic limit, toward a“mixed model” A Viscous Saint-Venant-Exner like model Perspective M. Ersoy (BCAM) PhD Works 15 october 2010 47 / 59

  78. The morphodynamic part is governed by the Vlasov equation : ∂ t f + div x ( vf ) + div v (( F + � g ) f ) = r ∆ v f where : f ( t, x, v ) density function of particles g = (0 , 0 , − g ) t , � F = 6 πµa ( u − v ) Stokes drag force with M ◮ a radius of a particle (assumed constant) ◮ M = ρ p 4 3 πa 3 mass of a particle (assumed constant) with ρ p density of a particle (assumed constant) u fluid velocity µ characteristic viscosity of the fluid (assumed constant) r ∆ v f brownian motion of particles where r is the velocity diffusivity M. Ersoy (BCAM) PhD Works 15 october 2010 48 / 59

  79. The hydrodynamic part is governed by the Compressible Navier-Stokes equations  ∂ t ρ w + div ( ρ w u ) = 0 , ,   ∂ t ( ρ w u ) + div ( ρ w u ⊗ u ) = div σ ( ρ w , u ) + F ,   p = p ( t, x ) (1) where σ ( ρ w , u ) is the anisotropic total stress tensor : − pI 3 + 2Σ( ρ w ) .D ( u ) + λ ( ρ w ) div ( u ) I 3 The matrix Σ( ρ w ) is anisotropic   µ 1 ( ρ w ) µ 1 ( ρ w ) µ 2 ( ρ w )   µ 1 ( ρ w ) µ 1 ( ρ w ) µ 2 ( ρ w ) µ 3 ( ρ w ) µ 3 ( ρ w ) µ 3 ( ρ w ) with µ i � = µ j for i � = j and i, j = 1 , 2 , 3 . M. Ersoy (BCAM) PhD Works 15 october 2010 49 / 59

  80. The coupling As the medium may be heterogeneous, we propose the following inhomogeneous pressure law as : k ( t, x 1 , x 2 ) = gh ( t, x 1 , x 2 ) p ( t, x ) = k ( t, x 1 , x 2 ) ρ ( t, x ) 2 with 4 ρ f where ρ := ρ w + ρ s is called mixed density We set ρ s , the macroscopic density of sediments : � ρ s = R 3 f dv The last term F on the right hand side of CNEs is the effect of the particles motion on the fluid obtained by summing the contribution of all particles : � � 9 µ F = − R 3 Ffdv + ρ w � g = R 3 ( v − u ) fdv + ρ w � g. 2 a 2 ρ p M. Ersoy (BCAM) PhD Works 15 october 2010 50 / 59

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