Barry Koren, Jasper Kreeft, Jeroen Wackers

A New Model and Numerical Method for Compressible Two-Fluid Euler Flow

HYP2012, PadovaJune 28, 2012

Contents

• Introduction

• Flow model

• Flow solver

• Flow problems

• Conclusions

Introduction Flow Model Flow Solver Flow Problems Conclusions

Introduction

Two-fluid interface

Introduction Flow Model Flow Solver Flow Problems Conclusions

Separates two fluids

Divide domain in small volumes

Interface Capturing

Introduction Flow Model Flow Solver Flow Problems Conclusions

• Not applicable: single-fluid flow models only

• Not directly imposable: boundary conditions at interface

Flow Model

Euler equations

Mass

Momentum

Energy

Rate of changeof mass

Mass transport across boundary

Introduction Flow Model Flow Solver Flow Problems Conclusions

Interface-capturing model

Assumptions

• Equal velocities

• Equal pressures

Introduction Flow Model Flow Solver Flow Problems Conclusions

Interface-capturing model

Assumptions• Equal velocities• Equal

pressures

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

Introduction Flow Model Flow Solver Flow Problems Conclusions

2 equations of state

Energy-exchange terms

Quasi-1D channel flow 1D two-fluid flow

Introduction Flow Model Flow Solver Flow Problems Conclusions

Pressure force due to change in volume fraction

Energy-exchange terms

Friction force to keep velocities equal

Introduction Flow Model Flow Solver Flow Problems Conclusions

Energy-exchange terms

Compression or expansion

Isentropic compressibility

Energy exchange to keep pressures equal

Introduction Flow Model Flow Solver Flow Problems Conclusions

Flow Solver

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

Introduction Flow Model Flow Solver Flow Problems Conclusions

Energy-exchange-term evaluation

Introduction Flow Model Flow Solver Flow Problems Conclusions

In solution space:

Flow Problems

Shock-tube problems

• Exact solutions known

• Perfect gases

Introduction Flow Model Flow Solver Flow Problems Conclusions

Translating-interface problem

Pressure Volume fractionDensity

Introduction Flow Model Flow Solver Flow Problems Conclusions

Pressure-oscillation-free without special precaution

No-reflection problemIntroduction Flow Model Flow Solver Flow Problems Conclusions

• 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

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

Introduction Flow Model Flow Solver Flow Problems Conclusions

Comparison with experiment

Introduction Flow Model Flow Solver Flow Problems Conclusions

Introduction Flow Model Flow Solver Flow Problems Conclusions

Helium bubble – density

Comparison with experiment

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

Introduction Flow Model Flow Solver Flow Problems Conclusions

Thank you for your interest