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Two-Fluid Effective-Field Equations

Two-Fluid Effective-Field Equations. Mathematical Issues. Non-conservative: Uniqueness of Discontinuous solution? Pressure oscillations. Non-hyperbolic system: Ill-posedness? Stability Uniqueness. How to sort it out?. Remedy for hyperbolicity: Interfacial pressure correction term and

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Two-Fluid Effective-Field Equations

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  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. Effect of hyperbolicity Solution convergence 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

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

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

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

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

  11. 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)

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

  13. 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.

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