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Coarse graining is an important ingredient in many multi-scalecontinuum-discrete solvers such as CFD--DEM (computational fluiddynamics--discrete element method) solvers for dense particle-laden flows.Although CFD--DEM solvers have become a mature technique that is widely used inmultiphase flow research and industrial flow simulations, a flexible andeasy-to-implement coarse graining algorithm that can work with CFD solvers ofarbitrary meshes is still lacking. In this work, we proposed a new coarsegraining algorithm for continuum--discrete solvers for dense particle-ladenflows based on solving a transient diffusion equation. Via theoretical analysiswe demonstrated that the proposed method is equivalent to the statisticalkernel method with a Gaussian kernel, but the current method is much morestraightforward to implement in CFD--DEM solvers. \textit{A priori} numericaltests were performed to obtain the solid volume fraction fields based on givenparticle distributions, the results obtained by using the proposed algorithmwere compared with those from other coarse graining methods in the literature(e.g., the particle centroid method, the divided particle volume method, andthe two-grid formulation). The numerical tests demonstrated that the proposedcoarse graining procedure based on solving diffusion equations is theoreticallysound, easy to implement and parallelize in general CFD solvers, and hasimproved mesh-convergence characteristics compared with existing coarsegraining methods. The diffusion-based coarse graining method has beenimplemented into a CFD--DEM solver, the results of which are presented in aseparate work (R. Sun and H. Xiao, Diffusion-based coarse graining in hybridcontinuum-discrete solvers: Application in CFD-DEM solvers for particle ladenflows).

Diffusion-Based Coarse Graining in Hybrid Continuum-Discrete Solvers: Theoretical Formulation and A Priori Tests