A Finite Volume Kinetic (FVK) framework for unsteady mixed flows in non uniform closed water pipes. Mehmet Ersoy 1, Christian Bourdarias 2 and St´ ephane Gerbi 3 Euskadi - Kyushu 2011 1. BCAM, Spain, mersoy@bcamath.org 2. LAMA–Savoie, France, christian.bourdarias@univ-savoie.fr 3. LAMA–Savoie, France, stephane.gerbi@univ-savoie.fr
Outline of the talk Outline of the talk 1 Unsteady mixed flows : PFS equations (Pressurized and Free Surface) Definitions and examples The PFS model 2 A FVK Framework Kinetic Formulation and numerical scheme Numerical results 3 Conclusion and perspectives M. Ersoy (BCAM) FVK framework for PFS -model Euskadi - Kyushu 2011 2 / 26
Outline Outline 1 Unsteady mixed flows : PFS equations (Pressurized and Free Surface) Definitions and examples The PFS model 2 A FVK Framework Kinetic Formulation and numerical scheme Numerical results 3 Conclusion and perspectives M. Ersoy (BCAM) FVK framework for PFS -model Euskadi - Kyushu 2011 3 / 26
Outline Outline 1 Unsteady mixed flows : PFS equations (Pressurized and Free Surface) Definitions and examples The PFS model 2 A FVK Framework Kinetic Formulation and numerical scheme Numerical results 3 Conclusion and perspectives M. Ersoy (BCAM) FVK framework for PFS -model Euskadi - Kyushu 2011 4 / 26
Unsteady mixed flows in closed water pipes ? Free surface area (SL) sections are not completely filled and the flow is incompressible. . . M. Ersoy (BCAM) FVK framework for PFS -model Euskadi - Kyushu 2011 5 / 26
Unsteady mixed flows in closed water pipes ? Free surface area (SL) sections are not completely filled and the flow is incompressible. . . Pressurized area (CH) sections are non completely filled and the flow is compressible. . . M. Ersoy (BCAM) FVK framework for PFS -model Euskadi - Kyushu 2011 5 / 26
Unsteady mixed flows in closed water pipes ? Free surface area (SL) sections are not completely filled and the flow is incompressible. . . Pressurized area (CH) sections are non completely filled and the flow is compressible. . . Transition point • M. Ersoy (BCAM) FVK framework for PFS -model Euskadi - Kyushu 2011 5 / 26
Examples of pipes Orange-Fish tunnel Sewers . . . in Paris Forced pipe problems . . . at Minnesota http://www.sewerhistory.org/grfx/ misc/disaster.htm M. Ersoy (BCAM) FVK framework for PFS -model Euskadi - Kyushu 2011 6 / 26
Outline Outline 1 Unsteady mixed flows : PFS equations (Pressurized and Free Surface) Definitions and examples The PFS model 2 A FVK Framework Kinetic Formulation and numerical scheme Numerical results 3 Conclusion and perspectives M. Ersoy (BCAM) FVK framework for PFS -model Euskadi - Kyushu 2011 7 / 26
The PFS model ∂ 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 sl � A sl if E = 0 (FS state) , if E = 0 , with A = S = A ch if E = 1 (P state) , S = | Ω( x ) | if E = 1 , and ... 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. Int. J. On Finite Volumes , 6(2) :1–47, 2009. M. Ersoy (BCAM) FVK framework for PFS -model Euskadi - Kyushu 2011 8 / 26
Pressure and source terms c 2 ( A − S ) + gI 1 ( x, S ) cos θ p = : mixed pressure law � A ch � d S c 2 Pr = − 1 dx + gI 2 ( S ) cos θ : pressure source term S gAz d G = dx cos θ : curvature source term 1 K = : Manning-Strickler law K 2 s R h ( S ) 4 / 3 M. Ersoy (BCAM) FVK framework for PFS -model Euskadi - Kyushu 2011 9 / 26
The PFS model Mathematical properties The PFS system is strictly hyperbolic for A ( t, x ) > 0 . For regular solutions, the mean speed u = Q/A verifies � u 2 � 2 + c 2 ln( A/S ) + g H ( S ) cos θ + g Z ∂ t u + ∂ x = − g K ( x, S ) u | u | and for u = 0 , we have : c 2 ln( A/ S ) + g H ( S ) cos θ + g Z = cte where H ( S ) is the physical water height. There exists a mathematical entropy E ( A, Q, S ) = Q 2 2 A + c 2 A ln( A/ S ) + c 2 S + gz ( x, S ) cos θ + gAZ which satisfies ∂ t E + ∂ x ( E u + p ( x, A, E ) u ) = − g A K ( x, S ) u 2 | u | � 0 M. Ersoy (BCAM) FVK framework for PFS -model Euskadi - Kyushu 2011 10 / 26
Outline Outline 1 Unsteady mixed flows : PFS equations (Pressurized and Free Surface) Definitions and examples The PFS model 2 A FVK Framework Kinetic Formulation and numerical scheme Numerical results 3 Conclusion and perspectives M. Ersoy (BCAM) FVK framework for PFS -model Euskadi - Kyushu 2011 11 / 26
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) FVK framework for PFS -model Euskadi - Kyushu 2011 12 / 26
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) FVK framework for PFS -model Euskadi - Kyushu 2011 12 / 26
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 ( U i , U i − 1 ) − F i − 1 / 2 ( U i − 1 , U i ) i ) i ∆ x where � t n +1 � ∆ t n S n i ≈ S ( t, x ) dx dt t n m i M. Ersoy (BCAM) FVK framework for PFS -model Euskadi - Kyushu 2011 12 / 26
Finite Volume (VF) numerical scheme of order 1 Upwinded numerical scheme : the principle 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 ( U i , U i − 1 , S i,i +1 ) − � � F i − 1 / 2 ( U i − 1 , U i , S i − 1 ,i ) i ∆ x M. Ersoy (BCAM) FVK framework for PFS -model Euskadi - Kyushu 2011 12 / 26
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) FVK framework for PFS -model Euskadi - Kyushu 2011 13 / 26
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) FVK framework for PFS -model Euskadi - Kyushu 2011 13 / 26
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) FVK framework for PFS -model Euskadi - Kyushu 2011 13 / 26
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. J. Sci. Comp.,pp 1-16, 10.1007/s10915-010-9456-0, 2011. M. Ersoy (BCAM) FVK framework for PFS -model Euskadi - Kyushu 2011 13 / 26
Outline Outline 1 Unsteady mixed flows : PFS equations (Pressurized and Free Surface) Definitions and examples The PFS model 2 A FVK Framework Kinetic Formulation and numerical scheme Numerical results 3 Conclusion and perspectives M. Ersoy (BCAM) FVK framework for PFS -model Euskadi - Kyushu 2011 14 / 26
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) FVK framework for PFS -model Euskadi - Kyushu 2011 15 / 26
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) FVK framework for PFS -model Euskadi - Kyushu 2011 15 / 26
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) FVK framework for PFS -model Euskadi - Kyushu 2011 15 / 26
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