NUMERICAL SIMULATION OF COMPRESSIBLE TWO-PHASE FLUID FLOW : Ghost fluid method vs Saurel Abgrall approach 1 Mathieu Bachmann Institut für Geometrie und Praktische Mathematik, RWTH Aachen University Joint work with : Josef Ballmann, Siegfried Müller, RWTH Aachen University. Mohsen Alizadeh, Dennis Kröninger, Thomas Kurz, Werner Lauterborn, Universität Göttingen. Philippe Helluy, Hélène Mathis, Université de Louis Pasteur Strasbourg. 1 DFG-CNRS FOR 563: Micro-Macro Modelling and Simulation of Liquid-Vapor Flows RWTH Aachen IGPM Mathieu Bachmann 1
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NUMERICAL SIMULATION OF COMPRESSIBLE
TWO-PHASE FLUID FLOW :
Ghost fluid method vs Saurel Abgrall approach 1
Mathieu Bachmann
Institut für Geometrie und Praktische Mathematik,RWTH Aachen University
Joint work with :
Josef Ballmann, Siegfried Müller, RWTH Aachen University.
Mohsen Alizadeh, Dennis Kröninger, Thomas Kurz, Werner Lauterborn, Universität Göttingen.
Philippe Helluy, Hélène Mathis, Université de Louis Pasteur Strasbourg.
1DFG-CNRS FOR 563: Micro-Macro Modelling and Simulation of Liquid-Vapor Flows
• Level Set Function : Φ < 0 (liquid), Φ = 0 (interface), Φ > 0 (gas),
• Evolution equation : (no mass transfer)
D ΦD t
≡ ∂ Φ∂ t
+ v∂ Φ∂ x
= 0
• Initialization : Distance function
Φ(0, x) = sgn(x− xI) |x− xI|
• Reinitialization : (Sussman et al.)
∂Φ∂τ
= sgn(Φ)(
1−∣∣∣∣∂ Φ∂ x
∣∣∣∣) resp.
∂ Φ∂ τ
+ a(Φ)∂ Φ∂ x
= sgn(Φ), a = sgn(Φ) ∂ Φ∂ x
/ ∣∣∣∂ Φ∂ x
∣∣∣RWTH Aachen IGPM Mathieu Bachmann 6
Numerical Discretization
I. Saurel-Abgrall Approach
II. "Real" Ghost Fluid Method (Wang, Liu, Khoo)
RWTH Aachen IGPM Mathieu Bachmann 7
Numerical Discretization
Saurel-Abgrall Approach
• Finite Volume Discretization :
– Time Evolution of conserved quantities:
vn+1i = vn
i −∆t
∆x
(F n
i+12− F n
i−12
)– Numerical Flux at the cell interface xi+1
2:
∗ Compute 2nd order reconstruction of the primitive variables and thegas fraction: W
±i+1
2= (ρ±, v±, p±, ϕ±)T
i+12
∗ Solve two-phase Riemann problem: W i+12
= W(ξ = 0, W
−i+1
2, W
+
i+12
)• Non-conservative Upwind Discretization :
– Idea: Preserve contact discontinuity avoid pressure oscillations– Time Evolution of gas fraction:
ϕn+1i = ϕn
i −∆t
∆x
(v n
i+12(ϕn
i+12− ϕn
i )− v ni−1
2(ϕn
i−12− ϕn
i ))
RWTH Aachen IGPM Mathieu Bachmann 8
Numerical Discretization
"Real" Ghost Fluid Method (Wang, Liu, Khoo)
• Finite Volume Discretization :
– Time Evolution of conserved quantities:
vn+1i = vn
i −∆t
∆x
(F n,−
i+12− F n,+
i−12
)– Numerical Flux at the cell interface xi+1
2with ϕi ϕi+1 > 0:
∗ Compute 2nd order reconstruction of primitive variables: W±i+1
2
∗ Solve single-phase Riemann problem: Wi+1
2= W
“ξ = 0, W
−i+1
2, W
+
i+12
”∗ Evaluate flux: F n,−
i+12
= F n,+
i+12
= F (Wi+1
2)
– Numerical Flux at the cell interface xi+12
with ϕi ϕi+1 < 0:∗ solve a two-phase Riemann problem interfacial states at the phase boundary∗ determine states in the ghost cells and modify real fluid by interfacial states∗ compute the reconstruction of the primitive variables∗ solve two single-phase Riemann problems at the cell interface
RWTH Aachen IGPM Mathieu Bachmann 9
Numerical Discretization
Numerical Flux at Phase Boundary
• Solve a two-phase Riemann problem with the states UL:=U i−1 and UR:=U i+2
interfacial states: U IL:=(ρIL, uI , pI )T and U IR:=(ρIR,uI , pI)T
• Redefine real fluid in cells i and i + 1 by interfacial states: U i=U IL and U i+1=U IR
• Define states in ghost cells (boundary cells of fluid A and B) by interfacial states:ghost fluid A: UA
j :=U IL, j > i + 1
ghost fluid B: UBj :=U IR, j < i− 1
• Solve two single-phase Riemann problems for fluid A and B two numerical fluxes F n,±
i+12
at the cell interface next to the phase boundary
RWTH Aachen IGPM Mathieu Bachmann 10
Numerical Results
I. Shock-Interface Interaction (Convergence Study, 1D)
• In case of rGFM: interface position is always shifted by 2-5 cells
RWTH Aachen IGPM Mathieu Bachmann 16
Shock-Interface Interaction
Influence of Shock-Interface Interaction for S-A Approach
RWTH Aachen IGPM Mathieu Bachmann 17
Shock-Interface Interaction
Conclusion
• Numerical order of convergence is 0.5 for both schemes
• Influence of shock-interface interaction:
– S-A approach: oscillations in water and wrong shock position in air– rGFM: interface velocity slightly over-predicted– Perturbations become weaker under grid refinement