modeling mathematical and numerical analysis for some
play

Modeling, Mathematical and Numerical Analysis for some Compressible - PowerPoint PPT Presentation

Jul 20, 2023 •99 likes •1.22k views

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

Download Presentation
Download Policy: The content available on the website is offered to you 'AS IS' for your personal information and use only. It cannot be commercialized, licensed, or distributed on other websites without prior consent from the author. To download a presentation, simply click this link. If you encounter any difficulties during the download process, it's possible that the publisher has removed the file from their server.

Recommend

Mathematical modeling and analysis of tumor-immune interactions 6 aot 2019 Atsou, Goudon

Mathematical modeling and analysis of tumor-immune interactions 6 aot 2019 Atsou, Goudon

1 tumors and efgector T-cells interactions : Biological context Mathematical modeling : Earlier stages of tumor-growth Numerical simulations Mathematical modeling and analysis of tumor-immune interactions 6 aot 2019 Atsou, Goudon

788 views • 41 slides
Electrochemical processes and porous media: mathematical and numerical modeling Jrgen Fuhrmann

Electrochemical processes and porous media: mathematical and numerical modeling Jrgen Fuhrmann

Weierstrass Institute for Applied Analysis and Stochastics Electrochemical processes and porous media: mathematical and numerical modeling Jrgen Fuhrmann Alfonso Caiazzo, Klaus Grtner, Hartmut Langmach, Alexander Linke, Hong Zhao

855 views • 59 slides
Source Modeling, Numerical Simulations, and Data Analysis Joan Centrella Laboratory for High

Source Modeling, Numerical Simulations, and Data Analysis Joan Centrella Laboratory for High

Source Modeling, Numerical Simulations, and Data Analysis Joan Centrella Laboratory for High Energy Astrophysics NASA/Goddard Space Flight Center Source Modeling guide data analysis strategies provide templates identify sources

322 views • 15 slides
Coupling free and porous-media flows: modeling, analysis and numerical approximation Marco

Coupling free and porous-media flows: modeling, analysis and numerical approximation Marco

Coupling free and porous-media flows: modeling, analysis and numerical approximation Marco Discacciati Special Semester on Multiscale Simulation &amp; Analysis in Energy and the Environment RICAM, Linz , October 5, 2011 Partial support of the

595 views • 48 slides
Numerical Differentiation &amp; Integration Numerical Differentiation I Numerical Analysis (9th

Numerical Differentiation &amp; Integration Numerical Differentiation I Numerical Analysis (9th

Numerical Differentiation &amp; Integration Numerical Differentiation I Numerical Analysis (9th Edition) R L Burden &amp; J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University c 2011 Brooks/Cole, Cengage

774 views • 64 slides
Numerical Differentiation &amp; Integration Numerical Differentiation II Numerical Analysis (9th

Numerical Differentiation &amp; Integration Numerical Differentiation II Numerical Analysis (9th

Numerical Differentiation &amp; Integration Numerical Differentiation II Numerical Analysis (9th Edition) R L Burden &amp; J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University c 2011 Brooks/Cole, Cengage

825 views • 43 slides
Numerical Differentiation &amp; Integration Composite Numerical Integration I Numerical Analysis

Numerical Differentiation &amp; Integration Composite Numerical Integration I Numerical Analysis

Numerical Differentiation &amp; Integration Composite Numerical Integration I Numerical Analysis (9th Edition) R L Burden &amp; J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University c 2011 Brooks/Cole,

1.14k views • 93 slides
Numerical Analysis of Coupled Circuit and Device Models Caren Tischendorf Humboldt University of

Numerical Analysis of Coupled Circuit and Device Models Caren Tischendorf Humboldt University of

Numerical Analysis of Coupled Circuit and Device Models Caren Tischendorf Humboldt University of Berlin ETH Z urich, MACSI-NET Workshop, 2.-3. May 2003 Overview 1. motivation 2. network modeling 3. device modeling 4. coupling of both

597 views • 30 slides
Numerical Solutions to Partial Differential Equations Zhiping Li LMAM and School of Mathematical

Numerical Solutions to Partial Differential Equations Zhiping Li LMAM and School of Mathematical

Numerical Solutions to Partial Differential Equations Zhiping Li LMAM and School of Mathematical Sciences Peking University Finite Difference Methods for Hyperbolic Equations Fourier Analysis of the Upwind Scheme for the Advection Equation

380 views • 37 slides
Operator analysis of geometric data structures Wojciech Czaja Reduced Order Modeling in General

Operator analysis of geometric data structures Wojciech Czaja Reduced Order Modeling in General

Mathematical Techniques Numerical Techniques Operator analysis of geometric data structures Wojciech Czaja Reduced Order Modeling in General Relativity Pasadena, June 6, 2013 Wojciech Czaja Operator analysis of geometric data structures

665 views • 35 slides
C0002M Numerical analysis, Lecture 11 Ove Edlund Ove Edlund C0002M Numerical analysis,

C0002M Numerical analysis, Lecture 11 Ove Edlund Ove Edlund C0002M Numerical analysis,

C0002M Numerical analysis, Lecture 11 Ove Edlund Ove Edlund C0002M Numerical analysis, Lecture 11 Numerical integration (quadrature) b When the definite integral a f ( x ) dx , is approximated, we make use of an evenly spaced

311 views • 13 slides
Chapter 1 Mathematical Preliminaries and Error Analysis Per-Olof Persson persson@berkeley.edu

Chapter 1 Mathematical Preliminaries and Error Analysis Per-Olof Persson persson@berkeley.edu

Chapter 1 Mathematical Preliminaries and Error Analysis Per-Olof Persson persson@berkeley.edu Department of Mathematics University of California, Berkeley Math 128A Numerical Analysis Limits and Continuity Definition A function f defined on

394 views • 16 slides
What scaffolds the early development of numerical and mathematical competencies? Daniel Ansari

What scaffolds the early development of numerical and mathematical competencies? Daniel Ansari

What scaffolds the early development of numerical and mathematical competencies? Daniel Ansari Numerical Cognition Laboratory Department of Psychology &amp; Brain and Mind Institute University of Western Ontario A Forum For Action, Effective

746 views • 49 slides
Can numerical analysis help in understanding piezoviscous hydrodynamic lubrication? M. Lanzend

Can numerical analysis help in understanding piezoviscous hydrodynamic lubrication? M. Lanzend

Can numerical analysis help in understanding piezoviscous hydrodynamic lubrication? M. Lanzend orfer Mathematical Institute, Charles University in Prague Institute of Computer Science, AS CR M odelling Math A nalysis C omputing M. Lanzend

569 views • 32 slides
Numerical Solutions to Partial Differential Equations Zhiping Li LMAM and School of Mathematical

Numerical Solutions to Partial Differential Equations Zhiping Li LMAM and School of Mathematical

Numerical Solutions to Partial Differential Equations Zhiping Li LMAM and School of Mathematical Sciences Peking University More on Consistency, Stability and Convergence Modified Equation Analysis Modified Equation of a Difference Scheme

467 views • 29 slides
Motion of thin droplets over surfaces numerical and modeling techniques for moving contact lines

Motion of thin droplets over surfaces numerical and modeling techniques for moving contact lines

Weierstrass Institute for Applied Analysis and Stochastics Motion of thin droplets over surfaces numerical and modeling techniques for moving contact lines Dirk Peschka, Sebastian Jachalski, Barbara Wagner Stefan Bommer, Ralf Seemann, Georgy

740 views • 48 slides
CS 543 - Computer Graphics: 2D Viewing, Part 2 by Robert W. Lindeman gogo@wpi.edu (with help

CS 543 - Computer Graphics: 2D Viewing, Part 2 by Robert W. Lindeman gogo@wpi.edu (with help

CS 543 - Computer Graphics: 2D Viewing, Part 2 by Robert W. Lindeman gogo@wpi.edu (with help from Emmanuel Agu ;-) Window-to-Viewport Mapping Applications Zooming in on a portion of object Tiling W-to-V in loop Multiple

389 views • 11 slides
Towards New International Cryptographic Standards Designing and Breaking Cryptography Lo Perrin

Towards New International Cryptographic Standards Designing and Breaking Cryptography Lo Perrin

Towards New International Cryptographic Standards Designing and Breaking Cryptography Lo Perrin Cosmiq TEAM Inria, Paris, France FIC 2020, Lille We (the Cosmiq team) are working on the foundations of cryptography. What kind of algorithms do

444 views • 40 slides
FPGAs 2 Some of the Slides picked from Xilinx Educational Resources FPGA Design Flow

FPGAs 2 Some of the Slides picked from Xilinx Educational Resources FPGA Design Flow

FPGAs 2 Some of the Slides picked from Xilinx Educational Resources FPGA Design Flow Functional Simulation Verify syntax and functionality Perform Separate Simulations With larger hierarchical Hardware Description Language (HDL)

402 views • 18 slides
CSE217 INTRODUCTION TO DATA SCIENCE COURSE WEBSITE, SYLLABUS, ACADEMIC INTEGRITY Spring 2019

CSE217 INTRODUCTION TO DATA SCIENCE COURSE WEBSITE, SYLLABUS, ACADEMIC INTEGRITY Spring 2019

CSE217 INTRODUCTION TO DATA SCIENCE COURSE WEBSITE, SYLLABUS, ACADEMIC INTEGRITY Spring 2019 Marion Neumann ABOUT Marion Neumann office: Jolley Hall 222 You are a real person! office hours: THU 3-4pm contact: use Piazza

559 views • 8 slides
Obligatory Presuppositions Pascal Amsili Universit e Paris Diderot &amp; LLF

Obligatory Presuppositions Pascal Amsili Universit e Paris Diderot &amp; LLF

Obligatory Presuppositions Obligatory Presuppositions Pascal Amsili Universit e Paris Diderot &amp; LLF amsili@linguist.jussieu.fr 3rd CAuLD Meeting, Nancy, December 14-15 2009 On going joint research with Claire Beyssade 1 / 46

490 views • 47 slides
Pillar: one format to rule them all! D. Cassou and S. Ducasse

Pillar: one format to rule them all! D. Cassou and S. Ducasse

Pillar: one format to rule them all! D. Cassou and S. Ducasse http://www.smalltalkhub.com/#!/~Pier/Pillar Problem We want more doc! We want you to write doc on your great software! But we want pdf html epub. Current strategies LaTeX

314 views • 10 slides
B STREET / BROADWAY PIERS DOWNTOWN ANCHORAGE SWITZER CREEK 21 APRIL 2003 WORKSHOP Craig L.

B STREET / BROADWAY PIERS DOWNTOWN ANCHORAGE SWITZER CREEK 21 APRIL 2003 WORKSHOP Craig L.

CA Regional Water Quality Control Board San Diego Region B STREET / BROADWAY PIERS DOWNTOWN ANCHORAGE SWITZER CREEK 21 APRIL 2003 WORKSHOP Craig L. Carlisle I NTRODUCTI ONS CONTAMI NATED SEDI MENTS - OVERVI EW TMDLs: A

513 views • 12 slides
Status of the MICE Online Status of the MICE Online Systems Systems Pierrick Hanlet Pierrick

Status of the MICE Online Status of the MICE Online Systems Systems Pierrick Hanlet Pierrick

Status of the MICE Online Status of the MICE Online Systems Systems Pierrick Hanlet Pierrick Hanlet 6 March 2012 6 March 2012 Outline I. Introduction II. Data Acquisition III.Online Reconstruction IV.Controls and Monitoring V.

822 views • 35 slides

More recommend