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

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 Andrianov (Schlumberger), Quang-Long Nguyen (IFP) Physical problem 1 Numerical treatment 2 Local time stepping 3 M. Postel (LJLL) Semi-implicit multiresolution for multiphase flows HYP2006 1 / 21

Outline Physical problem 1 Numerical treatment 2 Local time stepping 3 M. Postel (LJLL) Semi-implicit multiresolution for multiphase flows HYP2006 2 / 21

Physical problem: flows in pipeline. ������������� ������������� Transient Simulation ������������� ������������� ������������� ������������� �������� �������� 100 to 300 km �������� �������� ����� ����� 20 to 1000 m ����� ����� ������ ������ ������ ������ 1000 to 5000 m ������ ������ ��� ��� Gas ��� ��� Liquid M. Postel (LJLL) Semi-implicit multiresolution for multiphase flows HYP2006 3 / 21 ��� ��� ��� ��� ��� ���

technological bottleneck: slugging. Slugging video courtesy IFP Solaize (Loading MaestroIFP .avi) M. Postel (LJLL) Semi-implicit multiresolution for multiphase flows HYP2006 4 / 21

technological bottleneck: severe slugging. Gas massic fraction at riser exit Pressure at riser foot 0.9 220000 0.6 200000 Y P 0.3 180000 0 0 200 400 600 800 1000 0 200 400 600 800 1000 t t M. Postel (LJLL) Semi-implicit multiresolution for multiphase flows HYP2006 5 / 21

PDEs system Density gas massic fraction velocity ∂ x ( ρ v )  ∂ t ( ρ ) + = 0  ∂ t ( ρ Y ) ∂ x ( ρ Yv − σ ) + = 0 ∂ t ( ρ v ) ∂ x ( ρ v 2 + P ) S + =  M. Postel (LJLL) Semi-implicit multiresolution for multiphase flows HYP2006 6 / 21

PDEs system Density gas massic fraction velocity ∂ x ( ρ v )  ∂ t ( ρ ) + = 0  ∂ t ( ρ Y ) ∂ x ( ρ Yv − σ ) + = 0 ∂ x ( ρ v 2 + P ) ∂ t ( ρ v ) S + =  drift φ = v g − v ℓ , σ = ρ Y ( 1 − Y ) φ ρ ℓ ρ Y pressure P = p + ρ Y ( 1 − Y ) φ 2 with p ( ρ, ρ Y ) = a 2 g ρ ℓ − ρ ( 1 − Y ) source term S = − ρ g sin θ − C ρ v | v | . conditionally hyperbolic system with eigenvalues : v − c << v << v + c M. Postel (LJLL) Semi-implicit multiresolution for multiphase flows HYP2006 6 / 21

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

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

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

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

Outline Physical problem 1 Numerical treatment 2 Local time stepping 3 M. Postel (LJLL) Semi-implicit multiresolution for multiphase flows HYP2006 8 / 21

Numerical treatment The complicated closure laws are treated by relaxation → 5 × 5 system of PDEs. ρ v  ρ     0  ρ Y ρ Yv − Σ 0       ρ v 2 + Π ρ v S       ∂ t + ∂ x =       ρ Π v + a 2 v λρ [ P ( ρ, Y , v ) − Π]       ρ Π       ρ Σ v − b 2 Y λρ [ σ ( ρ, Y , v ) − Σ] ρ 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

Y at time 1. Y at time 3. 0.4 0.4 u 0.3 u 0.3 0.2 0.2 200 400 600 800 200 400 600 800 x x Velocity at time 1. Velocity at time 3. 10 10 5 5 u 0 u 0 -5 -5 -10 -10 200 400 600 800 200 400 600 800 x x ρ at time 1. ρ at time 3. 500 500 initial initial semi semi exp exp u u 400 400 300 300 200 400 600 800 200 400 600 800 x x

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

Example on v-shaped pipe Density Pressure 1000 1e+07 800 600 5e+06 400 200 0 0 2000 4000 0 2000 4000 x x Gas mass fraction Debit gaz 0.007 0.04 0.005 0.003 0.02 0 2000 4000 0 2000 4000 x x M. Postel (LJLL) Semi-implicit multiresolution for multiphase flows HYP2006 12 / 21

Example on v-shaped pipe Density Pressure 1000 1e+07 800 600 5e+06 400 200 0 0 2000 4000 0 2000 4000 x x Gas mass fraction Debit gaz 0.007 0.04 0.005 0.003 0.02 0 2000 4000 0 2000 4000 x x M. Postel (LJLL) Semi-implicit multiresolution for multiphase flows HYP2006 12 / 21

Example on v-shaped pipe Density Pressure 1000 1e+07 800 600 5e+06 400 200 0 0 2000 4000 0 2000 4000 x x Gas mass fraction Debit gaz 0.007 0.04 0.005 0.003 0.02 0 2000 4000 0 2000 4000 x x M. Postel (LJLL) Semi-implicit multiresolution for multiphase flows HYP2006 12 / 21

Example on v-shaped pipe Density Pressure 1000 1e+07 800 600 5e+06 400 200 0 0 2000 4000 0 2000 4000 x x Gas mass fraction Debit gaz 0.007 0.04 0.005 0.003 0.02 0 2000 4000 0 2000 4000 x x M. Postel (LJLL) Semi-implicit multiresolution for multiphase flows HYP2006 12 / 21

Example on v-shaped pipe Density Pressure 1000 1e+07 800 600 5e+06 400 200 0 0 2000 4000 0 2000 4000 x x Gas mass fraction Debit gaz 0.007 0.04 0.005 0.003 0.02 0 2000 4000 0 2000 4000 x x M. Postel (LJLL) Semi-implicit multiresolution for multiphase flows HYP2006 12 / 21

Multiresolution performances Total % error (ref 8192) 0.3 mr1024-4levels mr2048-4levels 0.25 error 1024 unif error 2048 unif 1024/2048 0.2 error 0.15 0.1 0.05 0 0 1 2 3 4 5 6 7 8 9 10 cpu gain M. Postel (LJLL) Semi-implicit multiresolution for multiphase flows HYP2006 13 / 21

Lagrange-Projection Positivity principle Two properties required for the implicit scheme Locally conservative scheme Lagrangian step ∂ t τ − ∂ y v  = 0  dx = vdt + τ dy ∂ t v + ∂ y Π  = 0    ∂ t Π + a 2 ∂ y v with ⇒ = 0 ∂ t Y − ∂ y Σ τ = 1 /ρ = 0    ∂ t Σ − b 2 ∂ y Y  = 0  Euler projection step       ρ ρ 0 ρ Y  + v ∂ x ρ Y  = ∂ t 0     ρ v ρ v 0 M. Postel (LJLL) Semi-implicit multiresolution for multiphase flows HYP2006 14 / 21

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