A multilayer Saint-Venant system Derivation, Hyperbolicity, - PowerPoint PPT Presentation

A multilayer Saint-Venant system Derivation, Hyperbolicity, Discretization Emmanuel Audusse Marie-Odile Bristeau, Astrid Decoene LAGA - University Paris 13 BANG Project - INRIA Rocquencourt Laboratoire National d’Hydraulique et d’Environnement - EDF HYP 2006 - 21/07/2006 – p. 1

Outline of the talk Why a Multilayer Saint-Venant model ? Derivation of a non-conservative multilayer SW system Analysis of the non-conservative multilayer SW system Introduction of a conservative multilayer SW system Analysis of the conservative multilayer SW system Discretization of the conservative multilayer SW system Some numerical results HYP 2006 - 21/07/2006 – p. 2

Saint-Venant system vs. Navier-Stokes equations INRIA-MODULEF INRIA-MODULEF • Nature of equations : Hyperbolic (SW) vs. Parabolic (NS) • Computational Domain : Fixed 2d (SW) vs. Moving 3d (NS) • CPU Time : 1500 s (SW) vs. 17000 s (NS) • Accuracy ??? 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 SAINT-VENANT NAVIER-STOKES 0.1 0.1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 . Vertical profile of the horizontal velocity . SW (left) vs. NS (right) INRIA-MODULEF INRIA-MODULEF Is it possible to combine accuracy of NS and efficiency of SW ??? HYP 2006 - 21/07/2006 – p. 3

Bibliography : The bi-fluid SW model Castro M., Macias J. and Pares C., A Q -scheme for a class of systems of coupled conservation laws with source term. Application to a two-layer 1-D shallow water system , M2AN Math. Model. Numer. Anal. (1) 35 (2001), 107–127. ∂h 1 8 ∂t + ∇ · ( h 1 u 1 ) = 0 , > > > > > > > > > ∂h 1 u 1 + ∇ · ( h 1 u 1 ⊗ u 1 ) + ∇ ( g 1 ) = − ρ 2 > 2 h 2 > ρ 1 gh 1 ∇ h 2 > > ∂t > > > > < ( BF − SW ) > > > ∂h 2 > > ∂t + ∇ · ( h 2 u 2 ) = 0 , > > > > > > > > > ∂h 2 u 2 + ∇ · ( h 2 u 2 ⊗ u 2 ) + ∇ ( g > > 2 h 2 2 ) = − gh 2 ∇ h 1 > : ∂t Why do not use the same multilayer approach for a single fluid ? HYP 2006 - 21/07/2006 – p. 4

Derivation of multilayer SW model from NS equations Derivation of the classical SW model Gerbeau J.F. and Perthame B., Derivation of viscous Saint-Venant system for laminar shallow water Discrete Cont. Dyn. Syst. Ser. B (1) 1 (2001), 89–102. See also Ferrari et al. (2002), Rodriguez et al.(2004), Marche (2006) Derivation of the multilayer SW model 1. Formal asymptotic analysis of NS equations under the shallow water assumption � Hydrostatic NS equations 2. Vertical discretization of the fluid into an arbitrary number of layers 3. Vertical integration of the hydrostatic NS equations on each layer Consequences Each layer has its own velocity Coupling between the layers through the pressure term (global coupling) the viscous effect (local coupling) HYP 2006 - 21/07/2006 – p. 5

Multilayer approach Free surface h 4 ( t, x ) u 4 ( t, x ) h 3 ( t, x ) H 3 ( t, x ) u 3 ( t, x ) h 2 ( t, x ) u 2 ( t, x ) h 1 ( t, x ) u 1 ( t, x ) River bottom z ( x ) HYP 2006 - 21/07/2006 – p. 6

Non conservative multilayer Saint-Venant system ∂th α + ∂ ∂ ∂xh α ¯ u α = 0 , Λ ∂ u α + ∂ α + gh α ∂ h β = 2 µ ¯ u α +1 − ¯ h α +1 + h α − 2 µ ¯ u α u α − ¯ u α − 1 X u 2 ∂th α ¯ ∂xh α ¯ ∂x h α + h α − 1 β =1 HYP 2006 - 21/07/2006 – p. 7

Non conservative multilayer Saint-Venant system ∂th α + ∂ ∂ ∂xh α ¯ u α = 0 , Λ ∂ u α + ∂ α + gh α ∂ h β = 2 µ ¯ h α +1 + h α − 2 µ ¯ u α +1 − ¯ u α u α − ¯ u α − 1 X u 2 ∂th α ¯ ∂xh α ¯ ∂x h α + h α − 1 β =1 To compare with ∂th α + ∂ ∂ ∂xh α ¯ u α = 0 , h 2 ∂ u α + ∂ α + g ∂ u 2 α ∂th α ¯ ∂xh α ¯ 2 = − κu α ∂x . Classical SW system HYP 2006 - 21/07/2006 – p. 7

Non conservative multilayer Saint-Venant system ∂th α + ∂ ∂ ∂xh α ¯ u α = 0 , Λ ∂ u α + ∂ α + gh α ∂ h β = 2 µ ¯ u α +1 − ¯ h α +1 + h α − 2 µ ¯ u α u α − ¯ u α − 1 X u 2 ∂th α ¯ ∂xh α ¯ ∂x h α + h α − 1 β =1 To compare with ∂th α + ∂ ∂ ∂xh α ¯ u α = 0 , „ « ∂ u α + ∂ α + gh α ∂ h α + ρ β u 2 ∂th α ¯ ∂xh α ¯ ρ α h β = 0 ∂x . Bi-fluid SW system HYP 2006 - 21/07/2006 – p. 8

Non conservative multilayer Saint-Venant system ∂th α + ∂ ∂ ∂xh α ¯ u α = 0 , Λ ∂ u α + ∂ α + gh α ∂ h β = 2 µ ¯ h α +1 + h α − 2 µ ¯ u α +1 − ¯ u α u α − ¯ u α − 1 X u 2 ∂th α ¯ ∂xh α ¯ ∂x h α + h α − 1 β =1 The multilayer SW system ensures the conservation of the total water height admits an invariant domain ( h α ≥ 0 ) admits an energy E and an associated entropy inequality X „ h α U 2 « + gh α X α E = E α = 2 H 2 If no friction a solution of the multilayer SW system can be deduced from the solution of the classical SW system. HYP 2006 - 21/07/2006 – p. 9

Non conservative bilayer system ∂U ∂t + A ( U ) ∂U ∂x = S ( U ) 2 3 0 1 0 0 u 2 − ¯ 1 + gh 1 2¯ u 1 gh 1 0 6 7 A ( U ) = 5 . 6 7 6 0 0 0 1 7 4 u 2 gh 2 0 − ¯ 2 + gh 2 2¯ u 2 0 1 0 0 1 h 1 2 µ ¯ u 2 − ¯ u 1 B h 2 + h 1 − κ ¯ u 1 C h 1 ¯ u 1 B C B C U = A , S ( U ) = , B C B C B h 2 C B 0 C @ B C 2 µ ¯ u 1 − ¯ u 2 @ A h 2 ¯ u 2 h 2 + h 1 HYP 2006 - 21/07/2006 – p. 10

Non-conservative bilayer system ∂U ∂t + A ( U ) ∂U ∂x = S ( U ) 2 3 0 1 0 0 u 2 − ¯ 1 + gh 1 2¯ u 1 gh 1 0 6 7 A ( U ) = 5 . 6 7 6 0 0 0 1 7 4 u 2 gh 2 0 − ¯ 2 + gh 2 2¯ u 2 u 1 , ¯ ¯ u 2 = ¯ u + O ( ǫ ) p → 2 real eigenvalues : ¯ u m ± g ( h 1 + h 2 ) + o ( ǫ ) Barotropic eigenvalues related to the free surface waves r ” 2 “ h 1 − h 2 → 2 complex eigenvalues : ¯ u c ± iǫ 1 − + o ( ǫ ) h 1 + h 2 Baroclinic eigenvalues related to the internal interface waves Non hyperbolic system = ⇒ Instabilities (cf Castro, Macias, Pares [2001]) HYP 2006 - 21/07/2006 – p. 11

Conservative multilayer Saint-Venant system ∂th α + ∂ ∂ ∂xh α ¯ u α = 0 , “P Λ ” 0 1 h α β =1 h β ∂ u α + ∂ u 2 ∂th α ¯ @ h α ¯ α + g A ∂x 2 ” 2 “P Λ g β =1 h β ! ∂ h α + 2 µ ¯ h α +1 + h α − 2 µ ¯ u α +1 − ¯ u α u α − ¯ u α − 1 = P Λ 2 ∂x h α + h α − 1 β =1 h β (Non-conservative SV system ” u α +1 − ¯ ¯ u α − ¯ ¯ u α u α − 1 u 2 P Λ ∂ ∂ α + gh α ∂ ∂t h α ¯ u α + ∂x h α ¯ β =1 h β = 2 µ h α +1 + h α − 2 µ ∂x h α + h α − 1 HYP 2006 - 21/07/2006 – p. 12

Conservative bilayer system ∂U ∂t + ∂ ∂xF ( U ) = S ( U ) 2 3 0 1 0 0 1 + gh 1 + gh 2 gh 1 u 2 6 7 − ¯ 2¯ u 1 0 6 7 2 2 DF ( U ) = . 6 7 6 0 0 0 1 7 6 7 gh 2 2 + gh 2 + gh 1 4 5 u 2 0 − ¯ 2¯ u 2 2 2 0 1 0 „ « g ( h 1 + h 2 ) 2 ∂ h 1 + 2 µ ¯ u 2 − ¯ u 1 B C h 2 + h 1 − κ ¯ u 1 B C ∂x h 1 + h 2 B C S ( U ) = , B C 0 B C B C „ « g ( h 1 + h 2 ) 2 ∂ h 2 + 2 µ ¯ u 1 − ¯ u 2 @ A ∂x h 1 + h 2 h 2 + h 1 HYP 2006 - 21/07/2006 – p. 13

Conservative bilayer system ∂U ∂t + ∂ ∂xF ( U ) = S ( U ) 2 3 0 1 0 0 1 + gh 1 + gh 2 gh 1 u 2 6 7 − ¯ 2¯ u 1 0 6 7 2 2 DF ( U ) = . 6 7 6 0 0 0 1 7 6 7 gh 2 2 + gh 2 + gh 1 4 5 u 2 0 − ¯ 2¯ u 2 2 2 u 1 , ¯ ¯ u 2 = ¯ u + O ( ǫ ) → 4 real (and distinct) eigenvalues g ( h 1 + h 2 ) + o ( ǫ ) : Consistent barotropic eigenvalues p u m ± ¯ q g ( h 1 + h 2 ) : Artificial baroclinic eigenvalues u c ± ¯ + o ( ǫ ) 2 HYP 2006 - 21/07/2006 – p. 14

Numerical discretization of the multilayer SW system Conservative part : Finite volume solver → Also for non conservative pressure source terms Extension of FV schemes developped for classical SW system ??? Roe scheme : For each layer, Roe average states are equal to Roe average states of the classical SW system Exact computation of the eigenvalues of the Roe matrix is not possible Need to deal with the approximations of the eigenvalues when layer velocities are closed Kinetic scheme : Need a kinetic interpretation of the multilayer SW system... If possible, how is it related to the kinetic interpretation of the classical SW system ? Viscous terms : Implicit solver → Well-posed tridiagonal linear system HYP 2006 - 21/07/2006 – p. 15

Kinetic formulation Equivalency between the multilayer Saint-Venant system and a set of kinetic equations ∂M α + ξ ∂M α = Q ( t, x, ξ ) ∂t ∂x Classical kinetic formulation M ( t, x, ξ ) = h ( t, x ) χ ( ξ − u ( t, x ) ) p p gh/ 2 gh/ 2 Multilayer kinetic formulation M α ( t, x, ξ ) = h α ( t, x ) χ ( ξ − u ( t, x ) ) p p gh/ 2 gh/ 2 HYP 2006 - 21/07/2006 – p. 16

BC and Vertical velocity Boundary conditions Boundary conditions are imposed for the global flow. Then boudary conditions for each layer are deduced (parabolic or constant profile for the velocity). Vertical velocity Impermeability condition at the bottom, U . n = 0 Incompressibility condition Z z ∂u w ( x, z ) = w ( x, z m ) − ∂xdz z m We define w at the nodes of the 2D “mesh”. HYP 2006 - 21/07/2006 – p. 17

1D numerical results Dam break on flat bottom Friction κ = 0 . 1 Viscosity µ = 0 . 01 Ten layers (Navier-Stokes and multilayer SV system) HYP 2006 - 21/07/2006 – p. 18