Mixed flows in closed water pipes A kinetic approach Christian - PowerPoint PPT Presentation

Mixed flows in closed water pipes A kinetic approach Christian Bourdarias, Mehmet Ersoy and Stéphane Gerbi LAMA, Université de Savoie, Chambéry, France French-Chinese Summer Research Institute Project “Stress tensor effects on compressible flows”, Morningside Center of Mathematics of the Chinese Academy of Sciences Beijing, 2-23 janvier 2010. S. Gerbi (LAMA, UdS, Chambéry) Mixed flows in closed pipes Beijing 2010 1 / 33

Table of contents Modelisation: the pressurised and free surface flows model 1 The free surface model The pressurised model The PFS-model : a natural coupling The kinetic approach 2 The Kinetic Formulation The kinetic scheme : the case of a non transition point The case of a transition point Numerical experiments 3 Conclusion and perspectives 4 S. Gerbi (LAMA, UdS, Chambéry) Mixed flows in closed pipes Beijing 2010 2 / 33

What is a transient mixed flow in closed pipes Free surface (FS) area : only a part of the section is filled. S. Gerbi (LAMA, UdS, Chambéry) Mixed flows in closed pipes Beijing 2010 3 / 33

What is a transient mixed flow in closed pipes Free surface (FS) area : only a part of the section is filled. Pressurized (P) area : the section is completely filled. S. Gerbi (LAMA, UdS, Chambéry) Mixed flows in closed pipes Beijing 2010 3 / 33

Some closed pipes a forced pipe a sewer in Paris The Orange-Fish Tunnel (in Canada) S. Gerbi (LAMA, UdS, Chambéry) Mixed flows in closed pipes Beijing 2010 4 / 33

Outline Modelisation: the pressurised and free surface flows model 1 The free surface model The pressurised model The PFS-model : a natural coupling The kinetic approach 2 The Kinetic Formulation The kinetic scheme : the case of a non transition point The case of a transition point Numerical experiments 3 Conclusion and perspectives 4 S. Gerbi (LAMA, UdS, Chambéry) Mixed flows in closed pipes Beijing 2010 5 / 33

Euleur Incompressible equations div ( ρ 0 U ) = 0 ∂ t ( ρ 0 U ) + ρ 0 U · ∇ ( ρ 0 U ) + ∇ P = ρ 0 F S. Gerbi (LAMA, UdS, Chambéry) Mixed flows in closed pipes Beijing 2010 6 / 33

The framework The domain Ω F ( t ) of the flow at time t : the union of sections Ω( t , x ) orthogonal to some plane curve C lying in ( O , i , k ) following main flow axis. ω = ( x , 0 , b ( x )) in the cartesian reference frame ( O , i , j , k ) where k follows the vertical direction; b ( x ) is then the elevation of the point ω ( x , 0 , b ( x )) over the plane ( O , i , j ) Curvilinear variable defined by: � x � X = 1 + ( b ′ ( ξ )) 2 d ξ x 0 where x 0 is an arbitrary abscissa. Y = y and we denote by Z the B -coordinate of any fluid particle M in the Serret-Frenet reference frame ( T , N , B ) at point ω ( x , 0 , b ( x )) . S. Gerbi (LAMA, UdS, Chambéry) Mixed flows in closed pipes Beijing 2010 7 / 33

The derivation of the FS model write the Euler equations in a curvilinear reference frame, 1 ǫ = H / L with H (the height) and L (the length) and take ǫ = 0 in the Euler 2 curvilinear equations, the conservative variables A ( t , X ) : the wet area, Q ( t , X ) the discharge 3 defined by � A ( t , X ) = dYdZ , Q ( t , X ) = A ( t , X ) U Ω( t , X ) � 1 U ( t , X ) = U ( t , X ) dYdZ . A ( t , X ) Ω( t , X ) approximation : U 2 ≈ U U and U V ≈ U V . 4 S. Gerbi (LAMA, UdS, Chambéry) Mixed flows in closed pipes Beijing 2010 8 / 33

The derivation of the FS model write the Euler equations in a curvilinear reference frame, 1 ǫ = H / L with H (the height) and L (the length) and take ǫ = 0 in the Euler 2 curvilinear equations, the conservative variables A ( t , X ) : the wet area, Q ( t , X ) the discharge 3 defined by � A ( t , X ) = dYdZ , Q ( t , X ) = A ( t , X ) U Ω( t , X ) � 1 U ( t , X ) = U ( t , X ) dYdZ . A ( t , X ) Ω( t , X ) approximation : U 2 ≈ U U and U V ≈ U V . 4 S. Gerbi (LAMA, UdS, Chambéry) Mixed flows in closed pipes Beijing 2010 8 / 33

The derivation of the FS model write the Euler equations in a curvilinear reference frame, 1 ǫ = H / L with H (the height) and L (the length) and take ǫ = 0 in the Euler 2 curvilinear equations, the conservative variables A ( t , X ) : the wet area, Q ( t , X ) the discharge 3 defined by � A ( t , X ) = dYdZ , Q ( t , X ) = A ( t , X ) U Ω( t , X ) � 1 U ( t , X ) = U ( t , X ) dYdZ . A ( t , X ) Ω( t , X ) approximation : U 2 ≈ U U and U V ≈ U V . 4 S. Gerbi (LAMA, UdS, Chambéry) Mixed flows in closed pipes Beijing 2010 8 / 33

The derivation of the FS model write the Euler equations in a curvilinear reference frame, 1 ǫ = H / L with H (the height) and L (the length) and take ǫ = 0 in the Euler 2 curvilinear equations, the conservative variables A ( t , X ) : the wet area, Q ( t , X ) the discharge 3 defined by � A ( t , X ) = dYdZ , Q ( t , X ) = A ( t , X ) U Ω( t , X ) � 1 U ( t , X ) = U ( t , X ) dYdZ . A ( t , X ) Ω( t , X ) approximation : U 2 ≈ U U and U V ≈ U V . 4 S. Gerbi (LAMA, UdS, Chambéry) Mixed flows in closed pipes Beijing 2010 8 / 33

The FS-Model  ∂ t A + ∂ X Q = 0   � Q 2 �  ∂ t Q + ∂ X A + gI 1 ( X , A ) cos θ = gI 2 ( X , A ) cos θ − gA sin θ (1)    − gAZ ( X , A )( cos θ ) ′ � h I 1 ( X , A ) = ( h − Z ) σ dZ : the hydrostatic pressure term − R � h I 2 ( X , A ) = ( h − Z ) ∂ X σ dZ : the pressure source term − R � p = ρ 0 ( h ( t , X ) − Z ) cos θ : the hydrostatic pressure. � Z = Z dY dZ : the center of mass Ω( t , X ) We add the Manning-Strickler friction term of the form S f ( A , U ) = K ( A ) U | U | . S. Gerbi (LAMA, UdS, Chambéry) Mixed flows in closed pipes Beijing 2010 9 / 33

Outline Modelisation: the pressurised and free surface flows model 1 The free surface model The pressurised model The PFS-model : a natural coupling The kinetic approach 2 The Kinetic Formulation The kinetic scheme : the case of a non transition point The case of a transition point Numerical experiments 3 Conclusion and perspectives 4 S. Gerbi (LAMA, UdS, Chambéry) Mixed flows in closed pipes Beijing 2010 10 / 33

Euleur compressible equations ∂ t ρ + div ( ρ U ) = 0 , (2) ∂ t ( ρ U ) + div ( ρ U ⊗ U ) + ∇ p = F , (3) Linearized pressure law: p = p a + ρ − ρ 0 βρ 0 1 c = √ βρ 0 S. Gerbi (LAMA, UdS, Chambéry) Mixed flows in closed pipes Beijing 2010 11 / 33

The derivation of the P-Model write the Euler equations in a curvilinear reference frame, 1 ǫ = H / L with H (the height) and L (the length) and takes ǫ = 0 in the 2 Euler curvilinear equations, the conservative variables A ( t , X ) : the wet equivalent area, Q ( t , X ) the 3 equivalent discharge defined by A = ρ S , Q = AU ρ 0 � 1 U ( t , X ) = U ( t , X ) dYdZ . S ( , X ) S ( X ) Approximation : ρ U 2 ≈ ρ U U and ρ U ≈ ρ U . 4 S. Gerbi (LAMA, UdS, Chambéry) Mixed flows in closed pipes Beijing 2010 12 / 33

The derivation of the P-Model write the Euler equations in a curvilinear reference frame, 1 ǫ = H / L with H (the height) and L (the length) and takes ǫ = 0 in the 2 Euler curvilinear equations, the conservative variables A ( t , X ) : the wet equivalent area, Q ( t , X ) the 3 equivalent discharge defined by A = ρ S , Q = AU ρ 0 � 1 U ( t , X ) = U ( t , X ) dYdZ . S ( , X ) S ( X ) Approximation : ρ U 2 ≈ ρ U U and ρ U ≈ ρ U . 4 S. Gerbi (LAMA, UdS, Chambéry) Mixed flows in closed pipes Beijing 2010 12 / 33

The derivation of the P-Model write the Euler equations in a curvilinear reference frame, 1 ǫ = H / L with H (the height) and L (the length) and takes ǫ = 0 in the 2 Euler curvilinear equations, the conservative variables A ( t , X ) : the wet equivalent area, Q ( t , X ) the 3 equivalent discharge defined by A = ρ S , Q = AU ρ 0 � 1 U ( t , X ) = U ( t , X ) dYdZ . S ( , X ) S ( X ) Approximation : ρ U 2 ≈ ρ U U and ρ U ≈ ρ U . 4 S. Gerbi (LAMA, UdS, Chambéry) Mixed flows in closed pipes Beijing 2010 12 / 33

The derivation of the P-Model write the Euler equations in a curvilinear reference frame, 1 ǫ = H / L with H (the height) and L (the length) and takes ǫ = 0 in the 2 Euler curvilinear equations, the conservative variables A ( t , X ) : the wet equivalent area, Q ( t , X ) the 3 equivalent discharge defined by A = ρ S , Q = AU ρ 0 � 1 U ( t , X ) = U ( t , X ) dYdZ . S ( , X ) S ( X ) Approximation : ρ U 2 ≈ ρ U U and ρ U ≈ ρ U . 4 S. Gerbi (LAMA, UdS, Chambéry) Mixed flows in closed pipes Beijing 2010 12 / 33

The P-Model  ∂ t ( A ) + ∂ X ( Q ) = 0      � Q 2 �  A + c 2 A − gA sin θ − gAZ ( X , S )( cos θ ) ′ (4) ∂ t ( Q ) + ∂ X =    + c 2 AS ′    S c 2 A : the pressure term c 2 A S ′ S : the pressure source term due to geometry changes gAZ ( X , S )( cos θ ) ′ : the pressure source term due to the curvature Z : the center of mass. We add the Manning-Strickler friction term of the form S f ( A , U ) = K ( A ) U | U | . S. Gerbi (LAMA, UdS, Chambéry) Mixed flows in closed pipes Beijing 2010 13 / 33

Outline Modelisation: the pressurised and free surface flows model 1 The free surface model The pressurised model The PFS-model : a natural coupling The kinetic approach 2 The Kinetic Formulation The kinetic scheme : the case of a non transition point The case of a transition point Numerical experiments 3 Conclusion and perspectives 4 S. Gerbi (LAMA, UdS, Chambéry) Mixed flows in closed pipes Beijing 2010 14 / 33