1. A New Model and Numerical Method for Compressible Two-Fluid Euler Flow
HYP2012, Padova
June 28, 2012
Barry Koren, Jasper Kreeft, Jeroen Wackers

2. Contents Introduction Flow model Flow solver Flow problems Conclusions

3. Introduction

4. Two-fluid interface Separates two fluids
Introduction Flow Model Flow Solver Flow Problems Conclusions
Two-fluid interface
Separates two fluids
Divide domain in small volumes

5. Interface Capturing Not applicable: single-fluid flow models only
Introduction Flow Model Flow Solver Flow Problems Conclusions
Interface Capturing
Not applicable: single-fluid flow models only
Not directly imposable: boundary conditions at interface

6. Flow Model

7. Euler equations Mass Momentum Energy Mass transport across boundary
Introduction Flow Model Flow Solver Flow Problems Conclusions
Euler equations
Mass transport
across boundary
Rate of change
of mass
Mass
Momentum
Energy

8. Interface-capturing model
Introduction Flow Model Flow Solver Flow Problems Conclusions
Interface-capturing model
Assumptions
Equal velocities
Equal pressures

9. Interface-capturing model
Introduction Flow Model Flow Solver Flow Problems Conclusions
Interface-capturing model
Assumptions
Equal velocities
Equal pressures

10. Interface-capturing model
Introduction Flow Model Flow Solver Flow Problems Conclusions
Interface-capturing model
Volume fraction:
Bulk mass
Bulk momentum
Bulk energy
Mass fluid 1
Energy fluid 1
2 equations of state

11. Energy-exchange terms
Introduction Flow Model Flow Solver Flow Problems Conclusions
Energy-exchange terms
Quasi-1D channel flow D two-fluid flow
Pressure force due to change in volume fraction

12. Energy-exchange terms
Introduction Flow Model Flow Solver Flow Problems Conclusions
Energy-exchange terms
Friction force to keep velocities equal

13. Energy-exchange terms
Introduction Flow Model Flow Solver Flow Problems Conclusions
Energy-exchange terms
Compression or expansion
Isentropic compressibility
Energy exchange to keep pressures equal

14. Flow Solver

15. Finite-volume discretization
Introduction Flow Model Flow Solver Flow Problems Conclusions
Finite-volume discretization
Integral form: ?
Time stepping:
three-stage explicit Runge-Kutta
Monotone second-order accurate spatial discretization: limiter BK
Flux vector evaluation: Approximate Riemann solver

16. Energy-exchange-term evaluation
Introduction Flow Model Flow Solver Flow Problems Conclusions
Energy-exchange-term evaluation
In solution space:

17. Flow Problems

18. Shock-tube problems Exact solutions known Perfect gases
Introduction Flow Model Flow Solver Flow Problems Conclusions
Shock-tube problems
Exact solutions known
Perfect gases

19. Translating-interface problem
Introduction Flow Model Flow Solver Flow Problems Conclusions
Translating-interface problem
Density
Pressure
Volume fraction
Pressure-oscillation-free without special precaution

20. No-reflection problem
Introduction Flow Model Flow Solver Flow Problems Conclusions
No-reflection problem
Shock hitting interface
Density distributions
Influence of energy-exchange term
Without exchange term
With exchange term

21. Water-air mixture problem
Introduction Flow Model Flow Solver Flow Problems Conclusions
Water-air mixture problem

22. Shock-bubble interaction problem
Introduction Flow Model Flow Solver Flow Problems Conclusions
Shock-bubble interaction problem
R – Higher density and lower ratio of specific heats than air lower speed of sound
Helium – Lower density and higher ratio of specific heats than air higher speed of sound

23. Introduction Flow Model Flow Solver Flow Problems Conclusions
R22 – density

24. Comparison with experiment
Introduction Flow Model Flow Solver Flow Problems Conclusions
Comparison with experiment

25. Helium bubble – density
Introduction Flow Model Flow Solver Flow Problems Conclusions
Helium bubble – density

26. Comparison with experiment
Introduction Flow Model Flow Solver Flow Problems Conclusions
Comparison with experiment

27. Conclusions New five-equation model (improvement to Kapila’s model);
Introduction Flow Model Flow Solver Flow Problems Conclusions
Conclusions
New five-equation model (improvement to Kapila’s model);
with energy-exchange laws
Approximate Riemann solver used for both flux and
energy-exchange evaluation
Mixture flows can also be computed
Physically correct solutions without tuning or post-processing
J.J. Kreeft and BK, J. Comput. Phys., 229,
Room for further extensions and applications

28. Thank you for your interest