A Lagrangian probability-density-function model for collisional turbulent fluidâ•fiparticle flows

Inertial particles in turbulent flows are characterised by preferential concentration and segregation and, at sufficient mass loading, dense particle clusters may spontaneously arise due to momentum coupling between the phases. These clusters, in turn, can generate and sustain turbulence in the fluid phase, which we refer to as cluster-induced turbulence (CIT). In the present work, we tackle the problem of developing a framework for the stochastic modelling of moderately dense particle-laden flows, based on a Lagrangian probability-density-function formalism. This framework includes the Eulerian approach, and hence can be useful also for the development of two-fluid models. A rigorous formalism and a general model have been put forward focusing, in particular, on the two ingredients that are key in moderately dense flows, namely, two-way coupling in the carrier phase, and the decomposition of the particle-phase velocity into its spatially correlated and uncorrelated components. Specifically, this last contribution allows us to identify in the stochastic model the contributions due to the correlated fluctuating energy and to the granular temperature of the particle phase, which determine the time scale for particle–particle collisions. The model is then validated and assessed against direct-numerical-simulation data for homogeneous configurations of increasing difficulty: (i) homogeneous isotropic turbulence, (ii) decaying and shear turbulence and (iii) CIT. Disciplines Complex Fluids | Dynamics and Dynamical Systems Comments This is a manuscript of an article published as Innocenti, A. , R. O. Fox, M. V. Salvetti, and S. Chibbaro. "A Lagrangian probability-density-function model for collisional turbulent fluid–particle flows. " Journal of Fluid Mechanics 862 (2019): 449-489. doi: 10.1017/jfm.2018. 895. Posted with permission. Authors Alessio Innocenti, Rodney O. Fox, Maria Vittoria Salvetti, and Sergio Chibbaro This article is available at Iowa State University Digital Repository: https://lib.dr.iastate.edu/cbe_pubs/357 This draft was prepared using the LaTeX style file belonging to the Journal of Fluid Mechanics 1 A Lagrangian probability-density-function model for collisional turbulent fluid–particle flows. II. Application to homogeneous flows A. Innocenti12, R. O. Fox3 , M. V. Salvetti2 and S. Chibbaro1† Sorbonne Université, UPMC Univ Paris 06, CNRS, UMR 7190, Institut Jean Le Rond d’Alembert, F-75005 Paris, France Dipartimento di Ingegneria Civile e Industriale, Università di Pisa, Via G. Caruso 8, 56122 Pisa, Italia Department of Chemical and Biological Engineering, 618 Bissell Road, Iowa State University, Ames, IA 50011-1098, USA (Received xx; revised xx; accepted xx) The Lagrangian probability-density-function model, proposed in Part I for dense particleladen turbulent flows, is validated here against Eulerian-Lagrangian direct numerical simulation (EL) data for different homogeneous flows, namely statistically steady and decaying homogeneous isotropic turbulence, homogeneous-shear flow and cluster-induced turbulence (CIT). We consider the general model developed in Part I adapted to the homogeneous case together with a simplified version in which the decomposition of the phase-averaged (PA) particle-phase fluctuating energy into the spatially correlated and uncorrelated components is not used, and only total exchange of kinetic energy between phases is allowed. The simplified model employs the standard two-way coupling approach. The comparison between EL simulations and the two stochastic models in homogeneous and isotropic turbulence and in homogeneous-shear flow shows that in all cases both models are capable to reproduce rather well the flow behaviour, notably for dilute flows. The analysis of the CIT gives more insights on the physical nature of such systems and about the quality of the models. Results elucidate the fact that simple two-way coupling is sufficient to induce turbulence, even though the granular energy is not considered. Furthermore, first-order moments including velocity of the fluid seen by particles can be fairly well represented with such a simplified stochastic model. However, the decomposition into spatially correlated and uncorrelated components is found to be necessary to account for anisotropic energy exchanges. When these factors are properly accounted for as in the complete model, the agreement with the EL statistics is satisfactory up to second order.