Coarse graining is an important ingredient in many multi-scale
continuum-discrete solvers such as CFD--DEM (computational fluid
dynamics--discrete element method) solvers for dense particle-laden flows.
Although CFD--DEM solvers have become a mature technique that is widely used in
multiphase flow research and industrial flow simulations, a flexible and
easy-to-implement coarse graining algorithm that can work with CFD solvers of
arbitrary meshes is still lacking. In this work, we proposed a new coarse
graining algorithm for continuum--discrete solvers for dense particle-laden
flows based on solving a transient diffusion equation. Via theoretical analysis
we demonstrated that the proposed method is equivalent to the statistical
kernel method with a Gaussian kernel, but the current method is much more
straightforward to implement in CFD--DEM solvers. \textit{A priori} numerical
tests were performed to obtain the solid volume fraction fields based on given
particle distributions, the results obtained by using the proposed algorithm
were compared with those from other coarse graining methods in the literature
(e.g., the particle centroid method, the divided particle volume method, and
the two-grid formulation). The numerical tests demonstrated that the proposed
coarse graining procedure based on solving diffusion equations is theoretically
sound, easy to implement and parallelize in general CFD solvers, and has
improved mesh-convergence characteristics compared with existing coarse
graining methods. The diffusion-based coarse graining method has been
implemented into a CFD--DEM solver, the results of which are presented in a
separate work (R. Sun and H. Xiao, Diffusion-based coarse graining in hybrid
continuum-discrete solvers: Application in CFD-DEM solvers for particle laden
flows)