  A Lagrangian particle cloud model is proposed that accounts for the effects
of Reynolds-averaged particle and turbulent stresses and the averaged
carrier-phase velocity of the sub-particle-cloud scale on the averaged motion
and velocity of the cloud. The SPARSE (Subgrid Particle Average Reynolds Stress
Equivalent) model is based on a combination of a truncated Taylor expansion of
a drag correction function and Reynolds averaging. It reduces the required
number of computational parcels to trace a cloud of particles in
Eulerian-Lagrangian methods for the simulation of particle-laden flow. Closure
is performed in an a priori manner using a reference simulation where all
particles in the cloud are traced individually with a point particle model.
Comparison of a first-order model and SPARSE with the reference simulation in
one-dimension shows that both the stress and the averaging of the carrier-phase
velocity on the cloud subscale affect the averaged motion of the particle. A
three-dimensional isotropic turbulence computation shows that only one
computational parcel is sufficient to accurately trace a cloud of tens of
thousand of particles.
