Statistical analysis of DNS data on clustering in homogeneous particle-laden turbulent flow for stochastic model development

Recent developments in stochastic Euler-Lagrange (EL) models for particle velocity and acceleration are successful in capturing particle dispersion and particle velocity variance\footnote{Lattanzi, A., Tavanashad, V., Subramaniam, S., Capecelatro, J., 2020. Stochastic models for capturing dispersion in particle-laden flows. Jounral of Fluid Mechanics 903.}. However, existing models do not take account of particle clustering that can significantly affect the fluid particle system. Consequently, advanced stochastic models for particle clustering that capture spatial correlation of particles need to be developed. Murphy\footnote{Murphy, E., 2017. Analysis and modeling of structure formation in granular and fluid-solid flows, Ph.D. thesis, Iowa State University.} established a theoretical framework to represent particle clustering and preferential concentration in terms of two-particle statistics. The key statistical quantities that need to be accurately modeled in this framework include the mean relative velocity $\langle w_{i}|r \rangle$ and the covariance of the relative velocity $\langle w_i w_j |r \rangle$ derived from particle pairs conditional on the pair separation $r$. Accurate prediction of the covariance requires modeling the correlation of particle-pair relative acceleration with particle-pair relative velocity that appears as a source or sink term in the evolution equation for the velocity-acceleration covariance. Point-particle direct numerical simulation (PP-DNS) data are analyzed to extract these statistics that can be used to develop advanced stochastic closure models for preferential concentration and particle clustering.