Outline Semi-implicit multiresolution for multiphase flows Marie - - PowerPoint PPT Presentation

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Outline Semi-implicit multiresolution for multiphase flows Marie - - PowerPoint PPT Presentation

Mar 22, 2024 337 likes •714 views

Outline Semi-implicit multiresolution for multiphase flows Marie Postel Laboratoire Jacques-Louis Lions Universit e Pierre et Marie Curie, Paris 6 Work in collaboration with Fr eric Coquel (UPMC-CNRS), ed Quang-Huy Tran (IFP), Nikolay

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SLIDE 1

Outline Semi-implicit multiresolution for multiphase flows

Marie Postel Laboratoire Jacques-Louis Lions Universit´ e Pierre et Marie Curie, Paris 6 Work in collaboration with Fr´ ed´ eric Coquel (UPMC-CNRS), Quang-Huy Tran (IFP), Nikolay Andrianov (Schlumberger), Quang-Long Nguyen (IFP)

1

Physical problem

2

Numerical treatment

3

Local time stepping

  • M. Postel (LJLL)

Semi-implicit multiresolution for multiphase flows HYP2006 1 / 21

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SLIDE 2

Outline

1

Physical problem

2

Numerical treatment

3

Local time stepping

  • M. Postel (LJLL)

Semi-implicit multiresolution for multiphase flows HYP2006 2 / 21

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SLIDE 3

Physical problem: flows in pipeline.

  • 100 to 300 km

Transient Simulation

  • 1000 to 5000 m

Gas 20 to 1000 m Liquid

  • M. Postel (LJLL)

Semi-implicit multiresolution for multiphase flows HYP2006 3 / 21

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SLIDE 4

technological bottleneck: slugging.

(Loading MaestroIFP .avi) Slugging video courtesy IFP Solaize

  • M. Postel (LJLL)

Semi-implicit multiresolution for multiphase flows HYP2006 4 / 21

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SLIDE 5

technological bottleneck: severe slugging.

0.3 0.6 0.9 200 400 600 800 1000 Y t Gas massic fraction at riser exit 180000 200000 220000 200 400 600 800 1000 P t Pressure at riser foot

  • M. Postel (LJLL)

Semi-implicit multiresolution for multiphase flows HYP2006 5 / 21

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SLIDE 6

PDEs system

Density gas massic fraction velocity    ∂t(ρ) + ∂x(ρv) = ∂t(ρY) + ∂x(ρYv − σ) = ∂t(ρv) + ∂x(ρv2 + P) = S

  • M. Postel (LJLL)

Semi-implicit multiresolution for multiphase flows HYP2006 6 / 21

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SLIDE 7

PDEs system

Density gas massic fraction velocity    ∂t(ρ) + ∂x(ρv) = ∂t(ρY) + ∂x(ρYv − σ) = ∂t(ρv) + ∂x(ρv2 + P) = S drift φ= vg − vℓ, σ= ρY(1 − Y)φ pressure P= p + ρY(1 − Y)φ2 with p(ρ, ρY) = a2

g

ρℓρY ρℓ − ρ(1 − Y) source term S = −ρg sin θ − Cρv|v|. conditionally hyperbolic system with eigenvalues : v − c << v << v + c

  • M. Postel (LJLL)

Semi-implicit multiresolution for multiphase flows HYP2006 6 / 21

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SLIDE 8

Modeling difficulties.

Quantity of interest: gas mass fraction (slow wave CFL ≃ 1) Fast acoustic waves of less interest CFL ≃ 20 → semi-implicit scheme with large time step (Faille,Heintze,’99, Baudin,Coquel,Tran,’05,) Expensive hydrodynamical closure laws → AMR (Cohen,Dyn,Kaber,P .’00 , Cohen,Kaber,Mueller,P .’03 ) Novelty: coupling AMR with large time step methods

  • M. Postel (LJLL)

Semi-implicit multiresolution for multiphase flows HYP2006 7 / 21

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SLIDE 9

Modeling difficulties.

Quantity of interest: gas mass fraction (slow wave CFL ≃ 1) Fast acoustic waves of less interest CFL ≃ 20 → semi-implicit scheme with large time step (Faille,Heintze,’99, Baudin,Coquel,Tran,’05,) Expensive hydrodynamical closure laws → AMR (Cohen,Dyn,Kaber,P .’00 , Cohen,Kaber,Mueller,P .’03 ) Novelty: coupling AMR with large time step methods

  • M. Postel (LJLL)

Semi-implicit multiresolution for multiphase flows HYP2006 7 / 21

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SLIDE 10

Modeling difficulties.

Quantity of interest: gas mass fraction (slow wave CFL ≃ 1) Fast acoustic waves of less interest CFL ≃ 20 → semi-implicit scheme with large time step (Faille,Heintze,’99, Baudin,Coquel,Tran,’05,) Expensive hydrodynamical closure laws → AMR (Cohen,Dyn,Kaber,P .’00 , Cohen,Kaber,Mueller,P .’03 ) Novelty: coupling AMR with large time step methods

  • M. Postel (LJLL)

Semi-implicit multiresolution for multiphase flows HYP2006 7 / 21

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SLIDE 11

Modeling difficulties.

Quantity of interest: gas mass fraction (slow wave CFL ≃ 1) Fast acoustic waves of less interest CFL ≃ 20 → semi-implicit scheme with large time step (Faille,Heintze,’99, Baudin,Coquel,Tran,’05,) Expensive hydrodynamical closure laws → AMR (Cohen,Dyn,Kaber,P .’00 , Cohen,Kaber,Mueller,P .’03 ) Novelty: coupling AMR with large time step methods

  • M. Postel (LJLL)

Semi-implicit multiresolution for multiphase flows HYP2006 7 / 21

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SLIDE 12

Outline

1

Physical problem

2

Numerical treatment

3

Local time stepping

  • M. Postel (LJLL)

Semi-implicit multiresolution for multiphase flows HYP2006 8 / 21

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SLIDE 13

Numerical treatment

The complicated closure laws are treated by relaxation → 5 × 5 system of PDEs. ∂t       ρ ρY ρv ρΠ ρΣ       + ∂x       ρv ρYv − Σ ρv2 + Π ρΠv + a2v ρΣv − b2Y       =       S λρ[P(ρ, Y, v) − Π] λρ[σ(ρ, Y, v) − Σ]       with Π and Σ reset at equilibrium P and σ at each time step. Implicitation of the Roe scheme, formulation in increments, with the Roe matrix frozen at time n Semi-implicitation : the terms corresponding to the slow waves are set to zero in the diagonal matrices.

  • M. Postel (LJLL)

Semi-implicit multiresolution for multiphase flows HYP2006 9 / 21

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SLIDE 14

0.2 0.3 0.4 200 400 600 800 u x Y at time 1. 0.2 0.3 0.4 200 400 600 800 u x Y at time 3.

  • 10
  • 5

5 10 200 400 600 800 u x Velocity at time 1.

  • 10
  • 5

5 10 200 400 600 800 u x Velocity at time 3. 300 400 500 200 400 600 800 u x ρat time 1. initial semi exp 300 400 500 200 400 600 800 u x ρ at time 3. initial semi exp

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SLIDE 15

Multiresolution adaptive scheme

Local grid refinement near the transport wave discontinuities Smoothness analysis of the solution by wavelet decomposition (Harten, ’89, Cohen, ’03) Discretization of the solution on a time-varying nonuniform grid Adaptation of the numerical scheme

  • M. Postel (LJLL)

Semi-implicit multiresolution for multiphase flows HYP2006 11 / 21

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SLIDE 16

Example on v-shaped pipe

200 400 600 800 1000 2000 4000 x Density 5e+06 1e+07 2000 4000 x Pressure 0.003 0.005 0.007 2000 4000 x Gas mass fraction 0.02 0.04 2000 4000 x Debit gaz

  • M. Postel (LJLL)

Semi-implicit multiresolution for multiphase flows HYP2006 12 / 21

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SLIDE 17

Example on v-shaped pipe

200 400 600 800 1000 2000 4000 x Density 5e+06 1e+07 2000 4000 x Pressure 0.003 0.005 0.007 2000 4000 x Gas mass fraction 0.02 0.04 2000 4000 x Debit gaz

  • M. Postel (LJLL)

Semi-implicit multiresolution for multiphase flows HYP2006 12 / 21

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SLIDE 18

Example on v-shaped pipe

200 400 600 800 1000 2000 4000 x Density 5e+06 1e+07 2000 4000 x Pressure 0.003 0.005 0.007 2000 4000 x Gas mass fraction 0.02 0.04 2000 4000 x Debit gaz

  • M. Postel (LJLL)

Semi-implicit multiresolution for multiphase flows HYP2006 12 / 21

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SLIDE 19

Example on v-shaped pipe

200 400 600 800 1000 2000 4000 x Density 5e+06 1e+07 2000 4000 x Pressure 0.003 0.005 0.007 2000 4000 x Gas mass fraction 0.02 0.04 2000 4000 x Debit gaz

  • M. Postel (LJLL)

Semi-implicit multiresolution for multiphase flows HYP2006 12 / 21

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SLIDE 20

Example on v-shaped pipe

200 400 600 800 1000 2000 4000 x Density 5e+06 1e+07 2000 4000 x Pressure 0.003 0.005 0.007 2000 4000 x Gas mass fraction 0.02 0.04 2000 4000 x Debit gaz

  • M. Postel (LJLL)

Semi-implicit multiresolution for multiphase flows HYP2006 12 / 21

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SLIDE 21

Multiresolution performances

0.05 0.1 0.15 0.2 0.25 0.3 1 2 3 4 5 6 7 8 9 10 error cpu gain Total % error (ref 8192) mr1024-4levels mr2048-4levels error 1024 unif error 2048 unif 1024/2048

  • M. Postel (LJLL)

Semi-implicit multiresolution for multiphase flows HYP2006 13 / 21

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SLIDE 22

Lagrange-Projection

Two properties required for the implicit scheme Positivity principle Locally conservative scheme Lagrangian step dx = vdt + τdy with τ = 1/ρ ⇒            ∂tτ − ∂yv = ∂tv + ∂yΠ = ∂tΠ + a2∂yv = ∂tY − ∂yΣ = ∂tΣ − b2∂yY = Euler projection step ∂t   ρ ρY ρv   + v ∂x   ρ ρY ρv   =    

  • M. Postel (LJLL)

Semi-implicit multiresolution for multiphase flows HYP2006 14 / 21

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Numerical treatment of Lagrange Projection

Advantages of the new algorithm Lagrangian step - fast acoustic non linear waves - implicit scheme Explicit CFL condition ensuring positivity Euler projection step - slow transport wave - explicit scheme Flux scheme ensuring local conservation

  • M. Postel (LJLL)

Semi-implicit multiresolution for multiphase flows HYP2006 15 / 21

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SLIDE 24

Numerical treatment of Lagrange Projection

Advantages of the new algorithm Lagrangian step - fast acoustic non linear waves - implicit scheme Explicit CFL condition ensuring positivity Euler projection step - slow transport wave - explicit scheme Flux scheme ensuring local conservation Numerical implementation Explicit scheme with boundary conditions Implicit scheme for Riemann problem Boundary conditions Source terms

  • M. Postel (LJLL)

Semi-implicit multiresolution for multiphase flows HYP2006 15 / 21

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SLIDE 25

Riemann problem on v-shaped pipe

400 500 600 700 400 800 uniform order 1 implicit/explicit exp uni o1 imp uni o1 400 500 600 700 400 800 uniform order 2 implicit/explicit exp uni o2 imp uni o2 400 500 600 700 400 800 1 2 3 4 5 Explicit order 2 - cpu amr= 28 cpu uni=172 exp uni o2 exp amr o2 levels 400 500 600 700 400 800 1 2 3 4 5 Implicit order 2 - cpu amr= 7 cpu uni=42 imp uni o2 imp amr o2 levels

  • M. Postel (LJLL)

Semi-implicit multiresolution for multiphase flows HYP2006 16 / 21

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SLIDE 26

Outline

1

Physical problem

2

Numerical treatment

3

Local time stepping

  • M. Postel (LJLL)

Semi-implicit multiresolution for multiphase flows HYP2006 17 / 21

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SLIDE 27

From multiresolution to local time stepping

(Mueller, Stiriba ’06) So far, the time step is bounded by λ = δt

δx < CFL max |VP|, with δx the

smallest cell width. Goal: use a time-step well adapted to the local size of the cell

  • M. Postel (LJLL)

Semi-implicit multiresolution for multiphase flows HYP2006 18 / 21

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SLIDE 28

From multiresolution to local time stepping

(Mueller, Stiriba ’06) So far, the time step is bounded by λ = δt

δx < CFL max |VP|, with δx the

smallest cell width. Goal: use a time-step well adapted to the local size of the cell

δ δ δ 2δ x x t t 2

  • M. Postel (LJLL)

Semi-implicit multiresolution for multiphase flows HYP2006 18 / 21

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SLIDE 29

From multiresolution to local time stepping

(Mueller, Stiriba ’06) So far, the time step is bounded by λ = δt

δx < CFL max |VP|, with δx the

smallest cell width. Goal: use a time-step well adapted to the local size of the cell

λ λ λ λ λ λ λ λ λ λ δ δ δ x x 2 t

?

  • M. Postel (LJLL)

Semi-implicit multiresolution for multiphase flows HYP2006 18 / 21

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SLIDE 30

Intermediate time stepping on coarse level boundary cell

Rely on flux conservation to synchronize the different levels solution.

Ui−1 U2i n,k n,k+1 U2i n+1/2,k+1 U 2i n+1,k+1 Ui−1 2i−1 Fn,k+1 F2i−1 n+1/2,k+1 Fi−2 n,k n+1,k Ui−1 n+1/2,k

Un+1,k

i−1

= Un,k

i−1 − λ

  • F n,k

i−1 − F n,k i−2

  • =

Un,k

i−1 − λ

2

  • F n,k+1

2i−1

− F n,k

i−2

  • +
  • F n+1/2,k+1

2i−1

− F n,k

i−2

  • M. Postel (LJLL)

Semi-implicit multiresolution for multiphase flows HYP2006 19 / 21

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SLIDE 31

Local time stepping on multilevel grid

3 levels: 0 1 and 2

x t

  • M. Postel (LJLL)

Semi-implicit multiresolution for multiphase flows HYP2006 20 / 21

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SLIDE 32

Local time stepping on multilevel grid

Fluxes evaluation at all cells interfaces First intermediate time step on level-2-cells

λ λ λ λ λ/2 x t

  • M. Postel (LJLL)

Semi-implicit multiresolution for multiphase flows HYP2006 20 / 21

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SLIDE 33

Local time stepping on multilevel grid

Actualization of fluxes Second intermediate time step on level-2-cells Synchronization with level-1 cells

λ λ λ λ λ λ λ λ λ/2 λ/2 λ λ λ/2 x t

  • M. Postel (LJLL)

Semi-implicit multiresolution for multiphase flows HYP2006 20 / 21

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SLIDE 34

Local time stepping on multilevel grid

Actualization of fluxes Third intermediate time step on level-2-cells

λ/2 λ/2 λ/2 λ λ λ λ λ λ λ λ λ λ λ/2 λ λ λ λ λ x t

  • M. Postel (LJLL)

Semi-implicit multiresolution for multiphase flows HYP2006 20 / 21

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SLIDE 35

Local time stepping on multilevel grid

Actualization of fluxes Fourth intermediate time step on level-2-cells Synchronization with level-1 and level-0 cells

λ λ/2 λ/2 λ λ λ λ λ/2 λ/2 λ/2 λ/2 λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ x t

  • M. Postel (LJLL)

Semi-implicit multiresolution for multiphase flows HYP2006 20 / 21

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SLIDE 36

Local time stepping example on 5 levels

ε = 10−3, global ∆t=32 δt uniform MR LTT CPU 2701 424 104 Calls to p(y, ρ) 107 5.105 105

400 500 600 2000 6000 10000 14000 2 4 6 x Density 0.2 0.4 2000 6000 10000 14000 2 4 6 x Gas mass fraction

  • M. Postel (LJLL)

Semi-implicit multiresolution for multiphase flows HYP2006 21 / 21