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

Contents Introduction Flow model Flow solver Flow problems Conclusions

Introduction

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

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

Flow Model

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

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

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

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

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

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

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

Flow Solver

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

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

Flow Problems

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

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

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

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

Shock-bubble interaction problem Introduction Flow Model Flow Solver Flow Problems Conclusions Shock-bubble interaction problem R22 – 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

Introduction Flow Model Flow Solver Flow Problems Conclusions R22 – density

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

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

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

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, 6220-6242 Room for further extensions and applications

Thank you for your interest