  A large amount of published data show that particles with diameter above 10\%
of the turbulence integral length scale ($D/l >0.1$) tend to increase the
turbulent kinetic energy of the carrier fluid above the single-phase value, and
smaller particles tend to suppress it. A revised phenomenological model of the
$k-\epsilon$ type was developed to reproduce these effects with the correct
asymptotic limit of no turbulence modulation for small particles, and
augmentation for larger diameter solids. Particle-kinetic theory was used to
derive the work exchanged between the particles and the fluid due to both drag
and added mass forces to accommodate any particle/fluid density ratios
including bubbles, droplets and heavy solids. For the larger particles, we
devised a new model for vortex shedding induced by the slip between the
particles and the turbulent flow, due to particle inertia. Simple approximate
formulae for the turbulence modulation were obtained through asymptotic
analysis.
  The overall effect for solid particles is that augmentation for large
diameter solids is due to vortex shedding, and turbulence suppression for small
diameters is due to mainly to turbulent drag forces and extra fluid
dissipation. The transition from suppression to augmentation around $D/l = 0.1$
is a robust feature for a wide range of particle Reynolds and Stokes numbers,
but we could not prove this to be a general relation on a theoretical basis.
Indeed, bubbles and droplets may not display turbulence augmentation at all for
the larger diameters due to moderate turbulence levels needed to prevent
breakup, and the velocity difference between particles and fluid may therefore
be too low for vortex shedding to occur. On the basis of the model we find that
some data for solids in vertical gas flow show very large turbulence
augmentation that can only be due to gravitational settling.
