SPARSE Model: Closed!

The direct Eulerian-Lagrangian simulation of turbulent, particle-laden flow through the Navier-Stokes equations combined with the tracing of a large number of particles is computationally expensive for large-scale problems. To reduce computational cost, small scale turbulence is often modeled and groups of physical particles are amalgamated into clouds, whose average location is traced. Typical Lagrangian models (such as Particle-Source-In-Cell and Cloud-In-Cell models) assume that the average motion of the cloud is governed by only the average interphase momentum difference between the carrier and disperse phases, neglecting subscale perturbations.  In [Davis, Jacobs, Sen Udaykumar, “SPARSE—A subgrid particle averaged Reynolds stress equivalent model: testing with a priori closure”, Proc. Roy. Soc. A, 473, 2017] we presented a Cloud-In-Cell (CIC) formulation for particle-laden flows that accounts for cloud size and subgrid-scale stresses using averaging techniques, and for cloud deformation using methods from continuum mechanics. By expanding the particle drag correction factor to include fluctuating terms and Reynolds averaging the full particle momentum equation, the so-called SPARSE model accounts for the effect of particle variances within the cloud. Here, we propose a closure of the SPARSE model using expansion techniques at the averaged center location of the particle  cloud. The model is shown to accurately predict particle dispersion of clouds for a number of test cases, including a stagnation flow, an ABC flow and an isotropic turbulence flow.