1. Two-Fluid Effective-Field Equations

2. Mathematical Issues Non-conservative: –Uniqueness of Discontinuous solution? –Pressure oscillations Non-hyperbolic system: Ill-posedness? –Stability –Uniqueness How to sort it out?

3. Remedy for hyperbolicity: Interfacial pressure correction term and virtual mass term

4. Modeling – Interfacial Pressure (IP) Stuhmiller (1977):

5. Here, we have

6. Faucet Problem: Ransom (1992) Hyperbolicity insures non-increase of overshoot, but suffering from smearing Location and strength of void discontinuity is converged, not affected by non- conservative form Effect of hyperbolicity Solution convergence

8. Modeling – Virtual Mass (VM) Drew et al (1979)

9.  VM is necessary if IP is not present, the coefficients are unreasonably high for droplet flows.  Requirement of VM can be reduced with IP.

11. Numerical Method Extended from single-phase AUSM + -up (2003). Implemented in the All Regime Multiphase Simulator (ARMS). -Cartesian. -Structured adaptive mesh refinement. -Parallelization.

12. A case with 40% liquid fraction U gas =1km/s  L =0.4, liquid mass =400kg V L =150m/s(in radial) Liquid area: l=2m, r=0.4m L=60m R=12m Axis ( Grid size 10cm, calculation time :0-150ms Calculation domain:,L=60m,R=12m )

13. Liquid fraction, pressure and velocity contours of particle cloud for time 0-150 ms. Lquid fraction (Min:10 -8 -Max:10 -3 ) Pressure (Min:1bar-Max:7bar) Gas Velocity (Min:0m/s -Max:1,000m/s)

14. Droplet radius R = 3.2mm, incoming shock speed M = 1.509

16. Current and future works Complete the hyperbolicity work on the multi-fluid system. Complete the adaptive mesh refinement into our solver – ARMS Expand Music-ARMS to solve 3D problems. Introduce physical models: Surface tension model Turbulence model Verification and validation. Real world applications.