A CASCADE MODEL FOR NEUTRALLY BUOYANT DISPERSED 2-PHASE HOMOGENEOUS TURBULENCE .2. NUMERICAL-SOLUTION AND RESULTS

The two-phase cascade model for homogeneous, neutrally buoyant turbulent dispersions proposed earlier (Jairazbhoy et al. 1995 Int. J. Multiphase Flow 21, 467-483) results in a system of partial integrodifferential equations. These are the energy, intermittency and population balance equations, and they account for the effects of drop-eddy interactions. A semi-discretization technique is developed for the solution. The drop number densities are discretized non-uniformly, the integrals approximated by Gaussian quadrature, and the resulting transient ODEs solved numerically to steady state using an integrator package. The results represent steady, isotropic turbulence with constant power input in the large eddies. The effects of phase fraction, drop size, Reynolds number and the model parameter on the turbulent spectrum and drop populations are examined. It is observed that the energy contained in large eddies is unaffected by drops, while small eddies are damped considerably. At low phase fractions, the energy spectrum is essentially unchanged by the dispersion. At high phase fractions, however, selective damping of the smaller eddies results in a sharp drop-off in the spectrum, tantamount to an increase in the apparent viscosity. The relative energy loss contributions from drop-eddy terms vary from about 8% for some cases with a phase fraction of 0.01 to over 82% in some instances with a phase fraction of 0.2. It is also found that, when the drops are too large to reside in a small eddy, the drop distribution in these eddies is substantially different from the overall distribution. Computation results comparing the energy spectra are in agreement with the model of Al Taweel & Landau at smaller and intermediate wave numbers, over which range comparisons are valid. These results suggest the model displays potential for the description of dense two-phase flows of breaking and coalescing drops, while accounting for drop interactions.