A relaxation framework for morphodynamics modelling E. Audusse . - - PowerPoint PPT Presentation

From SW flows to Morphodynamics Relaxation framework for SW-Exner model Towards new models (and other perspectives)

A relaxation framework for morphodynamics modelling

• E. Audusse

. LAGA, UMR 7569, Univ. Paris 13 BANG project-team, INRIA Paris-Rocquencourt . HYP 2012 - Padova June 26, 2012

• E. Audusse

A relaxation framework for morphodynamics modelling

From SW flows to Morphodynamics Relaxation framework for SW-Exner model Towards new models (and other perspectives)

Joint work with

◮ BANG project - ANGE group

M.O. Bristeau, J. Sainte-Marie

◮ EDF LNHE – Saint-Venant Lab.

• N. Goutal, M. Jodeau

◮ C. Berthon, C. Chalons, O. Delestre, S. Cordier

• E. Audusse

A relaxation framework for morphodynamics modelling

From SW flows to Morphodynamics Relaxation framework for SW-Exner model Towards new models (and other perspectives)

From SW flows to Morphodynamics Shallow Water Flows Morphodynamic processes Relaxation framework for SW-Exner model Relaxation model Relaxation scheme Numerical results Towards new models (and other perspectives)

• E. Audusse

A relaxation framework for morphodynamics modelling

From SW flows to Morphodynamics Relaxation framework for SW-Exner model Towards new models (and other perspectives) Shallow Water Flows Morphodynamic processes

Shallow water flows

◮ Shallow water equations

∂th + ∇ · (hu) = 0, ∂t(hu) + ∇ ·

• hu ⊗ u + gh2

2 I

• =

−gh∇z −2Ω × hu −κ(h, u)u

◮ Applications

• E. Audusse

A relaxation framework for morphodynamics modelling

From SW flows to Morphodynamics Relaxation framework for SW-Exner model Towards new models (and other perspectives) Shallow Water Flows Morphodynamic processes

Shallow water flows

◮ Shallow water equations

∂th + ∇ · (hu) = 0, ∂t(hu) + ∇ ·

• hu ⊗ u + gh2

2 I

• =

−gh∇z −2Ω × hu −κ(h, u)u

◮ Positive and well-balanced numerical schemes

◮ Extended Godunov schemes (Greenberg-Leroux) ◮ Kinetic interpretation of source terms (Pertame-Simeoni) ◮ Extended Suliciu relaxation schemes (Bouchut) ◮ Hydrostatic reconstruction (ABBKP) ◮ Hydrostatic upwind (Berthon-Foucher) ◮ ...

• E. Audusse

A relaxation framework for morphodynamics modelling

From SW flows to Morphodynamics Relaxation framework for SW-Exner model Towards new models (and other perspectives) Shallow Water Flows Morphodynamic processes

From SW flows to coupled problems

◮ Pollutant processes

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

4 6.5 9 11.5 14 4 6.5 9 11.5 14

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

.0325 .065 .0975 .13 .0325 .065 .0975 .13

◮ Hydrobiological processes (A.C. Boulanger)

500 1000 1500 2000 2500 3000 3500 −0.45 −0.4 −0.35 −0.3 −0.25 −0.2 −0.15 −0.1 −0.05 0.05 Time(s) Depth (m) Depth of particles through time. Omega = 1.33. Water height Particle 1 Particle 2 Particle 3

• E. Audusse

A relaxation framework for morphodynamics modelling

From SW flows to Morphodynamics Relaxation framework for SW-Exner model Towards new models (and other perspectives) Shallow Water Flows Morphodynamic processes

Morphodynamic processes

◮ Dune formation ◮ Coastal erosion ◮ Impact on industrial building

(harbour, dam, nuclear plant...)

◮ River morphodynamic ◮ Strong events

(tsunami, dam drain or break...)

• E. Audusse

A relaxation framework for morphodynamics modelling

From SW flows to Morphodynamics Relaxation framework for SW-Exner model Towards new models (and other perspectives) Shallow Water Flows Morphodynamic processes

Hyperbolic models in Morphodynamics

◮ Suspended sediment model

◮ Applications : High coupling and light sediments ◮ SW equations + Transport + Bottom evolution (ODE) ◮ Closure : Erosion and deposition source terms

◮ Bedload transport model

◮ Applications : Low coupling or heavy sediments ◮ SW equations + Bottom evolution ◮ Closure : Sediment flux ◮ Empirical formula : Grass, Meyer-Peter-M¨

uller, Einstein...

• E. Audusse

A relaxation framework for morphodynamics modelling

From SW flows to Morphodynamics Relaxation framework for SW-Exner model Towards new models (and other perspectives) Shallow Water Flows Morphodynamic processes

Numerical strategies for bedload transport

◮ Steady state strategy

◮ Hydrodynamic computation on fixed topography

Steady state

◮ Evolution of topography forced by hydrodynamic steady state ◮ Efficient for low coupling and different time scales (dune

formation)

◮ External coupling

◮ Use of two different softwares for hydro- and morphodynamics ◮ Allow to use existing solvers and different numerical strategies ◮ Actual strategy at EDF (MASCARET-COURLIS) ◮ Efficient for low coupling (slow river morphodynamics)

◮ Internal coupling

◮ Solution of the whole system at once ◮ Need for a new solver ◮ Efficient for high coupling (dam drain, tsunami)

• E. Audusse

A relaxation framework for morphodynamics modelling

From SW flows to Morphodynamics Relaxation framework for SW-Exner model Towards new models (and other perspectives) Shallow Water Flows Morphodynamic processes

Saint-Venant – Exner model

∂H ∂t + ∂Q ∂x = 0, ∂Q ∂t + ∂ ∂x Q2 H + g 2 H2

• =

−gH ∂Z ∂x , ρs(1 − p)∂Z ∂t + ∂Qs ∂x = 0,

◮ Hyperbolic for classical choices of Qs(h, u) ◮ Eigenvalues hard to compute except for special choices of Qs ◮ No dynamic effects in the solid phase ◮ No transport in the fluid phase ◮ Numerical strategies : Hudson, Nieto, Morales, Benkhaldoun...

• E. Audusse

A relaxation framework for morphodynamics modelling

From SW flows to Morphodynamics Relaxation framework for SW-Exner model Towards new models (and other perspectives) Relaxation model Relaxation scheme Numerical results

Relaxation model

∂tH + ∂xHu = ∂tHu + ∂x

• Hu2 + Π
• =

−gH∂xZ ∂tΠ + u∂xΠ + a2 H ∂xu = 1 λ gH2 2 − Π

• ∂tZ + ∂xΩ

= ∂tΩ + b2 H2 − u2

• ∂xZ + 2u∂xΩ

= 1 λ(Qs − Ω)

◮ (Π, Ω) : Auxiliary variables (fluid pressure, sediment flux) ◮ λ > 0 : (Small) relaxation parameter ◮ (a, b) > 0 : Have to be fixed to ensure stability

• E. Audusse

A relaxation framework for morphodynamics modelling

From SW flows to Morphodynamics Relaxation framework for SW-Exner model Towards new models (and other perspectives) Relaxation model Relaxation scheme Numerical results

Relaxation model

∂tH + ∂xHu = ∂tHu + ∂x

• Hu2 + Π
• =

−gH∂xZ ∂tΠ + u∂xΠ + a2 H ∂xu = 1 λ gH2 2 − Π

• ∂tZ + ∂xΩ

= ∂tΩ + b2 H2 − u2

• ∂xZ + 2u∂xΩ

= 1 λ(Qs − Ω)

◮ (Π, Ω) : Auxiliary variables (fluid pressure, sediment flux) ◮ λ > 0 : (Small) relaxation parameter ◮ (a, b) > 0 : Have to be fixed to ensure stability

• E. Audusse

A relaxation framework for morphodynamics modelling

From SW flows to Morphodynamics Relaxation framework for SW-Exner model Towards new models (and other perspectives) Relaxation model Relaxation scheme Numerical results

Main properties

◮ Formally tends to SW-Exner model when λ tends to 0 ◮ No explicit dependency on sediment flux QS ◮ Always hyperbolic (H = 0) ◮ Eigenvalues easy to compute (case a < b)

u − b H < u − a H < u < u + a H < u + b H

◮ Linearly degenerate system

Exact (homogeneous) Riemann problem ”easy” to solve

• E. Audusse

A relaxation framework for morphodynamics modelling

From SW flows to Morphodynamics Relaxation framework for SW-Exner model Towards new models (and other perspectives) Relaxation model Relaxation scheme Numerical results

Stability of the relaxation model

◮ Chapman-Enskog expansion

Π = p(h) + λΠ1 + ... Ω = Qs(h, u) + λΩ1 + ...

◮ Insert in the auxiliary equations

−Π1 = 1 h

• a2 − h2p′(h)
• ∂xu + O(λ)

−Ω1 =

• u∂hQs − p′(h)

h ∂uQs

• ∂xh + (−h∂hQs + u∂uQs) ∂xu

+ b2 h2 − u2 − g∂uQs

• ∂xz + O(λ)

◮ Insert in the physical equations

• E. Audusse

A relaxation framework for morphodynamics modelling

From SW flows to Morphodynamics Relaxation framework for SW-Exner model Towards new models (and other perspectives) Relaxation model Relaxation scheme Numerical results

Stability of the relaxation model

◮ Diffusive physical system W = (h, u, Z)T

∂tW + A(W )∂x(W ) = λ∂x (D(W )∂xW ) + O(λ2)

◮ Diffusion matrix

D(W ) =   

1 h

• a2 − h2p′(h)
• ×

×

• b2

h2 − u2 − g∂uQs

• 

 

◮ Stability requirement

a2 > h2p′(h), b2 > (hu)2 + gh2∂uQs

• E. Audusse

A relaxation framework for morphodynamics modelling

From SW flows to Morphodynamics Relaxation framework for SW-Exner model Towards new models (and other perspectives) Relaxation model Relaxation scheme Numerical results

Relaxation scheme with time splitting

◮ Start from (Hn, un, Z n) ◮ Computation of auxiliary variables

Πn Ωn

• =
• p(hn)

Qs(hn, un)

• ⇔
• ∂tΠ

=

1 λ

• gH2

2

− Π

• ∂tΩ

=

1 λ(Qs − Ω)

”λ = 0”

◮ Solution of homogeneous Riemann problems

XR = X(Xl, Xr, x, t)

◮ Computation of new physical variables

• Hn+1, un+1, Z n+1
• E. Audusse

A relaxation framework for morphodynamics modelling

From SW flows to Morphodynamics Relaxation framework for SW-Exner model Towards new models (and other perspectives) Relaxation model Relaxation scheme Numerical results

Solution of Riemann problem

• E. Audusse

A relaxation framework for morphodynamics modelling

From SW flows to Morphodynamics Relaxation framework for SW-Exner model Towards new models (and other perspectives) Relaxation model Relaxation scheme Numerical results

Solution of Riemann problem

◮ Continuity of the Riemann invariants (k-wave) ◮ Computation of the intermediate states ◮ Computation of the new variables

W n+1

i

= xi

xi−1/2

WR(Ui−1, Ui, x, ∆tn)+ xi+1/2

xi

WR(Ui, Ui+1, x, ∆tn)

◮ CFL condition ensures that Riemann pbs do not interact ◮ Definition of intermediate states

Positivity of the intermediate water heights Additional requirements on parameters a and b

• E. Audusse

A relaxation framework for morphodynamics modelling

From SW flows to Morphodynamics Relaxation framework for SW-Exner model Towards new models (and other perspectives) Relaxation model Relaxation scheme Numerical results

Intermediate states

Z ∗ =

• ur + b

Hr

• Zr −
• ul − b

Hl

• Zl
• ur + b

Hr

• −
• ul − b

Hl

• −

1

• ur + b

Hr

• −
• ul − b

Hl

(Ωr − Ωl) , 1 ¯ Hl = 1 H2

l

− 2g b2 − a2 (Z ∗ − Zl) 1

2

, 1 ¯ Hr = 1 H2

r

− 2g b2 − a2 (Z ∗ − Zr)

• 1

H∗

l

= 1 ¯ Hl − 1 a2 (Π∗ − ¯ Πl), 1 H∗

r

= 1 ¯ Hr − 1 a2 (Π∗ − ¯ Πr)

• E. Audusse

A relaxation framework for morphodynamics modelling

From SW flows to Morphodynamics Relaxation framework for SW-Exner model Towards new models (and other perspectives) Relaxation model Relaxation scheme Numerical results

”Steady” flow over a movable bump

Free surface and bottom evolution (O. Delestre)

• E. Audusse

A relaxation framework for morphodynamics modelling

From SW flows to Morphodynamics Relaxation framework for SW-Exner model Towards new models (and other perspectives) Relaxation model Relaxation scheme Numerical results

Dam break : Comparison for external and internal coupling

0.5 1 1.5 2 2 4 6 8 10 zb+Hf [m] x [m] Free surface at t=1 s - J=4000 zb+Hf at t=1 s

• 0.1
• 0.05

0.05 0.1 0.15 2 4 6 8 10 zb [m] x [m] Bottom topography at t=1 s - J=4000 zb+Hf at t=1 s

Internal coupling (O. Delestre)

• E. Audusse

A relaxation framework for morphodynamics modelling

From SW flows to Morphodynamics Relaxation framework for SW-Exner model Towards new models (and other perspectives) Relaxation model Relaxation scheme Numerical results

Dam break : Comparison for external and internal coupling

External coupling (M. Jodeau)

• E. Audusse

A relaxation framework for morphodynamics modelling

From SW flows to Morphodynamics Relaxation framework for SW-Exner model Towards new models (and other perspectives)

Three-layer model

◮ Introduction of three layers in the model

◮ NS Fluid layer (density ρf ) ◮ NS Mixed layer (density ρm(Cs)) + Transport of Cs ◮ Solid layer (no transport, but erosion and deposition)

◮ From Navier-Stokes to Saint-Venant in fluid and mixed layers

Two-layer shallow water flow with variable densities

◮ Closure laws at the interfaces : model for exchange terms

Solution of a 5 × 5 non linear system (constant density case)

◮ Relaxation model

Solution of two ”linear” 5 × 5 uncoupled systems

• E. Audusse

A relaxation framework for morphodynamics modelling

From SW flows to Morphodynamics Relaxation framework for SW-Exner model Towards new models (and other perspectives)

Perspectives

◮ Relaxation scheme for SW-Exner model

◮ Complete definition of parameters a and b ◮ Well-balancing ◮ Computation of wet-dry area ◮ Second order extension ◮ Improved relaxation models

◮ Modelization for Morphodynamics

◮ Closure laws and numerical tests for three-layer model ◮ Fluid-structure interaction ◮ Multilayer models with transport, erosion and deposition

• E. Audusse

A relaxation framework for morphodynamics modelling