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

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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

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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

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Saint-Venant system vs. Navier-Stokes equations

INRIA-MODULEF INRIA-MODULEF 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 ???

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 SAINT-VENANT 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 NAVIER-STOKES

. Vertical profile of the horizontal velocity . SW (left) vs. NS (right) Is it possible to combine accuracy of NS and efficiency of SW ???

HYP 2006 - 21/07/2006 – p. 3

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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. (BF − SW) 8 > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > : ∂h1 ∂t + ∇ · (h1u1) = 0, ∂h1u1 ∂t + ∇ · (h1u1 ⊗ u1) + ∇(g 2h2

1) = −ρ2

ρ1 gh1∇h2 ∂h2 ∂t + ∇ · (h2u2) = 0, ∂h2u2 ∂t + ∇ · (h2u2 ⊗ u2) + ∇(g 2h2

2) = −gh2∇h1

Why do not use the same multilayer approach for a single fluid ?

HYP 2006 - 21/07/2006 – p. 4

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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

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Multilayer approach

Free surface River bottom

h2(t, x) h1(t, x) h4(t, x) h3(t, x) u4(t, x) u3(t, x) u2(t, x) u1(t, x) H3(t, x) z(x)

HYP 2006 - 21/07/2006 – p. 6

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Non conservative multilayer Saint-Venant system

∂ ∂thα + ∂ ∂xhα¯ uα = 0, ∂ ∂thα¯ uα + ∂ ∂xhα¯ u2

α + ghα ∂

∂x

Λ

X

β=1

hβ = 2µ ¯ uα+1 − ¯ uα hα+1 + hα − 2µ ¯ uα − ¯ uα−1 hα + hα−1

HYP 2006 - 21/07/2006 – p. 7

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Non conservative multilayer Saint-Venant system

∂ ∂thα + ∂ ∂xhα¯ uα = 0, ∂ ∂thα¯ uα + ∂ ∂xhα¯ u2

α + ghα ∂

∂x

Λ

X

β=1

hβ = 2µ ¯ uα+1 − ¯ uα hα+1 + hα − 2µ ¯ uα − ¯ uα−1 hα + hα−1 To compare with ∂ ∂thα + ∂ ∂xhα¯ uα = 0, ∂ ∂thα¯ uα + ∂ ∂xhα¯ u2

α + g ∂

∂x h2

α

2 = −κuα . Classical SW system

HYP 2006 - 21/07/2006 – p. 7

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Non conservative multilayer Saint-Venant system

∂ ∂thα + ∂ ∂xhα¯ uα = 0, ∂ ∂thα¯ uα + ∂ ∂xhα¯ u2

α + ghα ∂

∂x

Λ

X

β=1

hβ = 2µ ¯ uα+1 − ¯ uα hα+1 + hα − 2µ ¯ uα − ¯ uα−1 hα + hα−1 To compare with ∂ ∂thα + ∂ ∂xhα¯ uα = 0, ∂ ∂thα¯ uα + ∂ ∂xhα¯ u2

α + ghα ∂

∂x „ hα + ρβ ρα hβ « = 0 . Bi-fluid SW system

HYP 2006 - 21/07/2006 – p. 8

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Non conservative multilayer Saint-Venant system

∂ ∂thα + ∂ ∂xhα¯ uα = 0, ∂ ∂thα¯ uα + ∂ ∂xhα¯ u2

α + ghα ∂

∂x

Λ

X

β=1

hβ = 2µ ¯ uα+1 − ¯ uα hα+1 + hα − 2µ ¯ uα − ¯ uα−1 hα + hα−1 The multilayer SW system ensures the conservation of the total water height admits an invariant domain (hα ≥ 0) admits an energy E and an associated entropy inequality E = X Eα = X „hαU 2

α

2H + ghα 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

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Non conservative bilayer system

∂U ∂t + A(U)∂U ∂x = S(U) A(U) = 2 6 6 6 4 1 −¯ u2

1 + gh1

2¯ u1 gh1 1 gh2 −¯ u2

2 + gh2

2¯ u2 3 7 7 7 5. U = B B B @ h1 h1¯ u1 h2 h2¯ u2 1 C C C A , S(U) = B B B B B @ 2µ ¯ u2 − ¯ u1 h2 + h1 − κ¯ u1 2µ ¯ u1 − ¯ u2 h2 + h1 1 C C C C C A ,

HYP 2006 - 21/07/2006 – p. 10

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Non-conservative bilayer system

∂U ∂t + A(U)∂U ∂x = S(U) A(U) = 2 6 6 6 4 1 −¯ u2

1 + gh1

2¯ u1 gh1 1 gh2 −¯ u2

2 + gh2

2¯ u2 3 7 7 7 5. ¯ u1, ¯ u2 = ¯ u + O(ǫ) → 2 real eigenvalues : ¯ um ± p g(h1 + h2) + o(ǫ) Barotropic eigenvalues related to the free surface waves → 2 complex eigenvalues : ¯ uc ± iǫ r 1 − “

h1−h2 h1+h2

”2 + o(ǫ) Baroclinic eigenvalues related to the internal interface waves Non hyperbolic system = ⇒ Instabilities (cf Castro, Macias, Pares [2001])

HYP 2006 - 21/07/2006 – p. 11

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Conservative multilayer Saint-Venant system

∂ ∂thα + ∂ ∂xhα¯ uα = 0, ∂ ∂thα¯ uα + ∂ ∂x @hα¯ u2

α + g

hα “PΛ

β=1 hβ

” 2 1 A = g “PΛ

β=1 hβ

”2 2 ∂ ∂x hα PΛ

β=1 hβ

! + 2µ ¯ uα+1 − ¯ uα hα+1 + hα − 2µ ¯ uα − ¯ uα−1 hα + hα−1 (Non-conservative SV system

∂ ∂thα¯

uα +

∂ ∂xhα¯

u2

α + ghα ∂ ∂x

PΛ

β=1 hβ = 2µ ¯ uα+1−¯ uα hα+1+hα − 2µ ¯ uα−¯ uα−1 hα+hα−1

”

HYP 2006 - 21/07/2006 – p. 12

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Conservative bilayer system

∂U ∂t + ∂ ∂xF(U) = S(U) DF(U) = 2 6 6 6 6 6 4 1 −¯ u2

1 + gh1 + gh2

2 2¯ u1 gh1 2 1 gh2 2 −¯ u2

2 + gh2 + gh1

2 2¯ u2 3 7 7 7 7 7 5 . S(U) = B B B B B B @ g (h1 + h2)2 ∂ ∂x „ h1 h1 + h2 « + 2µ ¯ u2 − ¯ u1 h2 + h1 − κ¯ u1 g (h1 + h2)2 ∂ ∂x „ h2 h1 + h2 « + 2µ ¯ u1 − ¯ u2 h2 + h1 1 C C C C C C A ,

HYP 2006 - 21/07/2006 – p. 13

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Conservative bilayer system

∂U ∂t + ∂ ∂xF(U) = S(U) DF(U) = 2 6 6 6 6 6 4 1 −¯ u2

1 + gh1 + gh2

2 2¯ u1 gh1 2 1 gh2 2 −¯ u2

2 + gh2 + gh1

2 2¯ u2 3 7 7 7 7 7 5 . ¯ u1, ¯ u2 = ¯ u + O(ǫ) → 4 real (and distinct) eigenvalues ¯ um ± p g(h1 + h2) + o(ǫ) : Consistent barotropic eigenvalues ¯ uc ± q

g(h1+h2) 2

+ o(ǫ) : Artificial baroclinic eigenvalues

HYP 2006 - 21/07/2006 – p. 14

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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

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Kinetic formulation

Equivalency between the multilayer Saint-Venant system and a set of kinetic equations ∂Mα ∂t + ξ ∂Mα ∂x = Q(t, x, ξ) Classical kinetic formulation M(t, x, ξ) = h(t, x) p gh/2 χ(ξ − u(t, x) p gh/2 ) Multilayer kinetic formulation Mα(t, x, ξ) = hα(t, x) p gh/2 χ(ξ − u(t, x) p gh/2 )

HYP 2006 - 21/07/2006 – p. 16

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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 w(x, z) = w(x, zm) − Z z

zm

∂u ∂xdz We define w at the nodes of the 2D “mesh”.

HYP 2006 - 21/07/2006 – p. 17

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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

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Water height - Longitudinal profile

1 1.2 1.4 1.6 1.8 2

60
40
20

20 40 60 NAVIER-STOKES MULTILAYER SAINT-VENANT

HYP 2006 - 21/07/2006 – p. 19

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Velocity - Longitudinal profile

0.1 0.2 0.3 0.4 0.5 0.6 0.7

60
40
20

20 40 60

HYP 2006 - 21/07/2006 – p. 20

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Velocity - Vertical profile

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.2 0.4 0.6 0.8 1 1.2 1.4 MULTILAYER SAINT-VENANT NAVIER-STOKES PARABOLIC PROFILE

HYP 2006 - 21/07/2006 – p. 21

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Velocity - Vertical profile - No slip

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 ’Multilayer Saint-Venant’ ’Navier-Stokes’

HYP 2006 - 21/07/2006 – p. 22

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1D numerical results

Dam break Friction κ = 0.1 Viscosity µ = 0.01 Ten layers (Navier-Stokes and multilayer SV system) CPU time 2D Navier-Stokes equations (ALE, implicit solver...) → CPU Time = 23.77 seconds (35 time steps) 1D Saint-Venant system → CPU Time = 0.07 second (73 time steps) Multilayer Saint-Venant system → CPU Time = 1.19 second (76 time steps)

HYP 2006 - 21/07/2006 – p. 23

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2D numerical results

Stationary flow over a bump Geometrical data

channel length ≈ 21 m
bump length ≈ 5.75 m
bump height ≈ 0.2 m

Transcritical case

given inflow discharge : 2.m3/s
outflow water depth : 0.6 m

Friction : Strickler = 30 Viscosity µ = 0.01 Six layers (2D mesh: 1452 nodes, 2620 triangles)

HYP 2006 - 21/07/2006 – p. 24

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Horizontal velocity

Multilayer Saint-Venant Model Hydrostatic Navier-Stokes Model Non-hydrostatic Navier-Stokes Model

HYP 2006 - 21/07/2006 – p. 25

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Vertical velocity

Multilayer Saint-Venant Model Hydrostatic Navier-Stokes Model Non-hydrostatic Navier-Stokes model

HYP 2006 - 21/07/2006 – p. 26

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Vertical profile of the horizontal velocity

0.2
0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 5 10 15 20 25 X svml hydro nonhydro

0.5 1 1.5 2 2.5 0.1 0.2 0.3 0.4 0.5 0.6 svml hydro 0.5 1 1.5 2 2.5 0.1 0.2 0.3 0.4 0.5 0.6 svml hydro 0.5 1 1.5 2 2.5 0.1 0.2 0.3 0.4 0.5 0.6 svml hydro

HYP 2006 - 21/07/2006 – p. 27

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2D numerical results

CPU Time 3D Navier-Stokes equations (FEM, ALE, semi-implicit solver, ∆t = 0.01) → CPU Time = 64 minutes (10000 time steps) 3D hydrostatic Navier-Stokes equations → CPU Time = 33 minutes (10000 time steps) 2D Saint-Venant system → CPU Time = 1,5 minute (23600 time steps) Multilayer Saint-Venant system → CPU Time = 13 minutes (23600 time steps)

HYP 2006 - 21/07/2006 – p. 28

SLIDE 30

Malpasset Dam Break

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.05 25.21350.47575.738

101

.05 25.21350.47575.738

101

INRIA-MODULEF INRIA-MODULEF INRIA-MODULEF INRIA-MODULEF

.05 25.21350.47575.738

101

.05 25.21350.47575.738

101

INRIA-MODULEF INRIA-MODULEF INRIA-MODULEF INRIA-MODULEF

.05

2.463 4.975 7.488 10

.05

2.463 4.975 7.488 10

INRIA-MODULEF INRIA-MODULEF INRIA-MODULEF INRIA-MODULEF

.05

2.463 4.975 7.488 10

.05

2.463 4.975 7.488 10

HYP 2006 - 21/07/2006 – p. 29

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Malpasset Dam Break

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.05

2.463 4.975 7.488 10

.05

2.463 4.975 7.488 10

INRIA-MODULEF INRIA-MODULEF INRIA-MODULEF INRIA-MODULEF

.05

2.463 4.975 7.488 10

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2.463 4.975 7.488 10

HYP 2006 - 21/07/2006 – p. 30

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CPU Time

Malpasset Dam Break Four layers Friction : Strickler = 30 Viscosity µ = 0.01 3D Navier-Stokes equations (FEM, ALE, semi-implicit solver) → CPU Time = 17000 seconds Multilayer Saint-Venant system → CPU Time = 8000 seconds

HYP 2006 - 21/07/2006 – p. 31

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Some Open questions

Interface condition (here : free surface condition) Viscous coupling (estimation of the vertical derivative of the horizontal velocity at the interfaces) Choice of the conservative multilayer SW system (hyperbolicity, preservation of the barotropic eigenvalues...) Numerical treatment of the non conservative pressure term Other solvers and high order methods Extension of the multilayer approach to the multifluid case ???

HYP 2006 - 21/07/2006 – p. 32