Modelisation Kinetic approach Tests Conclusion A kinetic scheme for transient mixed flows in non uniform closed pipes: a global manner to upwind all the source terms C. B OURDARIAS M. E RSOY S. G ERBI LAMA-Université de Savoie de Chambéry, France September 7 2009,Castro-Urdiales, Cantabria, Spain
Modelisation Kinetic approach Tests Conclusion Table of contents Modelisation: the pressurised and free surface flows model 1 The kinetic approach 2 The Kinetic Formulation The kinetic scheme A way to upwind the source terms Numerical Tests 3 Conclusion and perspectives 4
Modelisation Kinetic approach Tests Conclusion Table of contents Modelisation: the pressurised and free surface flows model 1 The kinetic approach 2 The Kinetic Formulation The kinetic scheme A way to upwind the source terms Numerical Tests 3 Conclusion and perspectives 4
Modelisation Kinetic approach Tests Conclusion Definition of the mixed flow Free surface (FS) area : only a part of the section is filled.
Modelisation Kinetic approach Tests Conclusion Definition of the mixed flow Free surface (FS) area : only a part of the section is filled. Pressurized (P) area : the section is completely filled.
Modelisation Kinetic approach Tests Conclusion PFS-model [BEG09a] ∂ t ( A ) + ∂ x ( Q ) = 0 � Q 2 � = − g A d ∂ t ( Q ) + ∂ x A + p ( x , A , S ) dx Z ( x ) . + Pr ( x , A , S ) − G ( x , A , S ) − g A K ( x , S ) u | u | A = ρ S : wet equivalent area, ρ 0 Q = A u : discharge, S the physical wet area. The pressure is p ( x , A , S ) = c 2 ( A − S ) + g I 1 ( x , S ) cos θ.
Modelisation Kinetic approach Tests Conclusion Source terms The pressure source term: � d � c 2 ( A / S − 1 ) Pr ( x , A , S ) = dx S + g I 2 ( x , S ) cos θ, the z − coordinate of the center of mass term: G ( x , A , S ) = g A Z ( x , S ) d dx cos θ, the friction term: 1 K ( x , S ) = s R h ( S ) 4 / 3 . K 2 K s > 0 is the Strickler coefficient, R h ( S ) is the hydraulic radius. [BEG09a] C. Bourdarias and M. Ersoy and S. Gerbi. A model for unsteady mixed flows in non uniform closed water pipes and a well-balanced finite volume scheme. Submitted. Available on arXiv http://arxiv.org/abs/0812.0057, 2009.
Modelisation Kinetic approach Tests Conclusion Summarize of notations � H ( S ) I 1 ( x , S ) = ( H ( S ) − z ) σ dz : the pressure and − R � H ( S ) I 2 ( x , S ) = ( H ( S ) − z ) ∂ x σ dz : the pressure source − R term with: R ( x ) the radius, σ ( x , z ) the width of the cross-section, H ( S ) the z − coordinate of the free surface. 1 √ βρ 0 c = : the sound of speed in the P zones with: ρ 0 the density at atmospheric pressure p 0 , β the water compressibility coefficient. Z ( x , S ) = ( H ( S ) − I 1 ( x , S ) / S ) : the z − coordinate of the center of the mass.
Modelisation Kinetic approach Tests Conclusion Some Properties The PFS system is strictly hyperbolic for A ( t , x ) > 0. For smooth solutions, the mean velocity u = Q / A satisfies � 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 u = 0 reads: c 2 ln ( A / S ) + g H ( S ) cos θ + g Z = 0 . It admits 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 the entropy inequality ∂ t E + ∂ x ( E u + p ( x , A , S ) u ) = − g A K ( x , S ) u 2 | u | � 0
Modelisation Kinetic approach Tests Conclusion Table of contents Modelisation: the pressurised and free surface flows model 1 The kinetic approach 2 The Kinetic Formulation The kinetic scheme A way to upwind the source terms Numerical Tests 3 Conclusion and perspectives 4
Modelisation Kinetic approach Tests Conclusion The Kinetic Formulation The Kinetic Formulation (KF) [P02] With � � ω 2 χ ( ω ) d ω = 1 , χ ( ω ) = χ ( − ω ) ≥ 0 , χ ( ω ) d ω = 1 , R R
Modelisation Kinetic approach Tests Conclusion The Kinetic Formulation The Kinetic Formulation (KF) [P02] With � � ω 2 χ ( ω ) d ω = 1 , χ ( ω ) = χ ( − ω ) ≥ 0 , χ ( ω ) d ω = 1 , R R we define the Gibbs equilibrium � ξ − u ( t , x ) � A M ( t , x , ξ ) = c ( A ) χ c ( A ) with � g I 1 ( x , A ) c ( A ) = cos θ in the FS zones and, A � g I 1 ( x , S ) cos θ + c 2 in the P zones. c ( S ) = S
Modelisation Kinetic approach Tests Conclusion The Kinetic Formulation The Kinetic Formulation (KF) [P02] We have the macroscopic-microscopic relations: � A = M ( t , x , ξ ) d ξ R � Q = ξ M ( t , x , ξ ) d ξ R � Q 2 A + Ac ( A ) 2 = ξ 2 M ( t , x , ξ ) d ξ R
Modelisation Kinetic approach Tests Conclusion The Kinetic Formulation The Kinetic Formulation (KF) [P02] The Kinetic Formulation ( A , Q ) is a strong solution of PFS-System if and only if M satisfies the kinetic transport equation: ∂ t M + ξ · ∂ x M − g Φ( x , A , S ) ∂ ξ M = K ( t , x , ξ ) for some collision term K ( t , x , ξ ) which satisfies for a.e. ( t , x ) � � K d ξ = 0 , ξ Kd ξ = 0, and Φ which take into account all R R the source terms. [P02] B. Perthame . Kinetic formulation of conservation laws. Oxford University Press. Oxford Lecture Series in Mathematics and its Applications, Vol 21, 2002.
Modelisation Kinetic approach Tests Conclusion The Kinetic Formulation If , Φ reads: Conservative Non conservative product � �� � � �� � dx Z − c 2 d d Z ( x , S ) d dx ln ( S ) + dx cos θ g � + d K ( x , S ) u | u | dx dx x If , Φ reads: Non conservative product Conservative � �� � � �� � d γ ( x , A ) cos θ dx ln ( A ) + Z ( x , A ) d d dx Z + dx cos θ A � + d K ( x , S ) u | u | dx dx x Back
Modelisation Kinetic approach Tests Conclusion The kinetic scheme Geometric terms and unknowns are piecewise constant appro- ximations on the cell m i at time t n : Geometric terms S i , cos θ i Macroscopic unknowns i = Q n W n i = ( A n i , Q n i ) , u n i A n i Microscopic unknown � ξ − u n � i ( ξ ) = A n M n i i χ c n c n i i
Modelisation Kinetic approach Tests Conclusion The kinetic scheme Consequently Φ n i is null on m i . Indeed, we have: d dx ( 1 m i Z ) = 0, d dx ( ln ( 1 m i S )) = 0, d dx ( 1 m i cos θ ) = 0, and we forget the friction term temporarly (friction splitting). Go [PS01] B. Perthame and C. Simeoni . A kinetic scheme for the Saint-Venant system with a source term. Calcolo, Vol 38(4) 201–231, 2001
Modelisation Kinetic approach Tests Conclusion The kinetic scheme Discretisation of the kinetic transport equation Neglecting the collision term, the transport equation reads on [ t n , t n + 1 [ × m i : ∂ t f + ξ · ∂ ∂ ∂ x f = 0 with f ( t n , x , ξ ) = M n i ( ξ ) for x ∈ m i and thus it is discretised on m i as: � � i ( ξ ) + ∆ t n f n + 1 ( ξ ) = M n M − 2 ( ξ ) − M + ∆ x ξ 2 ( ξ ) , i i + 1 i − 1
Modelisation Kinetic approach Tests Conclusion The kinetic scheme Although f n + 1 is not a Gibbs equilibrium, we have : i � A n + 1 � 1 � � � def W n + 1 f n + 1 = i := ( ξ ) d ξ i Q n + 1 i ξ R i → M n + 1 − defined without using the collision kernel : it is a way i to perform all collisions at once
Modelisation Kinetic approach Tests Conclusion The kinetic scheme Finally the kinetic scheme reads: i + ∆ t n W n + 1 = W n ∆ x ( F − 2 − F + 2 ) i i + 1 i − 1 with the interface fluxes � 1 � � F ± M ± 2 = ξ 2 ( ξ ) d ξ i + 1 i + 1 ξ R where the microscopic fluxes are defined following e.g. [BEG09b, PS01]: [BEG09b] C. Bourdarias and M. Ersoy and S. Gerbi . A kinetic scheme for pressurised flows in non uniform closed water pipes. Monografias de la Real Academia de Ciencias de Zaragoza, Vol 31 1–20, 2009.
Modelisation Kinetic approach Tests Conclusion The kinetic scheme The microscopic fluxes and physical interpretation positive transmission reflection � �� � � �� � 1 ξ> 0 M n i + 1 / 2 < 0 M n M − i + 1 / 2 ( ξ ) = i ( ξ ) + i ( − ξ ) 1 ξ< 0 , ξ 2 − 2 g ∆Φ n � � � 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 is the jump condition for a particle with the kinetic speed ξ . So, ∆Φ n can also seen as a space and time dependent slope.
Modelisation Kinetic approach Tests Conclusion A way to upwind the source terms Geometric source terms The friction term. � x i + 1 / 2 � x i + 1 � � � � 1 Q | Q | 1 Q | Q | dx + dx K 2 K 2 A 2 R h ( S ) 4 / 3 A 2 R h ( S ) 4 / 3 x i s x i + 1 / 2 s � � Q i + 1 | Q i + 1 | Q i | Q i | 1 ≈ i + 1 R h ( S i + 1 ) 4 / 3 + := FR i + 1 / 2 . 2 ∆ x K 2 A 2 A 2 i R h ( S i ) 4 / 3 s Geometric terms. ∂ x Z and ∂ x ln ( S ) are easily upwinded � S i + 1 � Z i + 1 − Z i ([ PS01 ]) and ln . S i
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