SPARSE: A Subgrid Particle Averaged Reynolds Stress Equivalent Model:   Testing with A Priori Closure
  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. 
Sean Davis, Gustaaf Jacobs, Oishik Sen, H.S. Udaykumar
physics.flu-dyn