Comparison between Lagrangian and mesoscopic Eulerian modelling approaches for inertial particles suspended in decaying isotropic turbulence

The purpose of this paper is to evaluate the accuracy of the mesoscopic approach proposed by Février et al. [P. Février, O. Simonin, K.D. Squires, Partitioning of particle velocities in gas–solid turbulent flows into a continuous field and a spatially uncorrelated random distribution: theoretical formalism and numerical study, J. Fluid Mech. 533 (2005) 1–46] by comparison against the Lagrangian approach for the simulation of an ensemble of non-colliding particles suspended in a decaying homogeneous isotropic turbulence given by DNS. The mesoscopic Eulerian approach involves to solve equations for a few particle PDF moments: number density, mesoscopic velocity, and random uncorrelated kinetic energy (RUE), derived from particle flow ensemble averaging conditioned by the turbulent fluid flow realization. In addition, viscosity and diffusivity closure assumptions are used to compute the unknown higher order moments which represent the mesoscopic velocity and RUE transport by the uncorrelated velocity component. A detailed comparison between the two approaches is carried out for two different values of the Stokes number based on the initial fluid Kolmogorov time scale, StK=0.17and2.2. In order to perform reliable comparisons for the RUE local instantaneous distribution and for the mesoscopic kinetic energy spectrum, the error due to the computation method of mesoscopic quantities from Lagrangian simulation results is evaluated and minimized. A very good agreement is found between the mesoscopic Eulerian and Lagrangian predictions for the small particle Stokes number case corresponding to the smallest particle inertia. For larger particle inertia, a bulk viscous term is included in the mesoscopic velocity governing equation to avoid spurious spatial oscillation that may arise due to the inability of the numerical scheme to resolve sharp number density gradients. As a consequence, for StK=2.2StK=2.2, particle number density and RUE spatial distribution predicted by the mesoscopic Eulerian approach are more smooth with respect to the ones measured from the Lagrangian simulations results. Similarly, the Eulerian approach underestimates the mesoscopic kinetic energy for the high wavenumber modes while the agreement remains very good for the low wavenumber modes. For both cases, the mesoscopic Eulerian approach provides a good prediction of the time dependent particle and fluid-particle velocity correlations measured by spatial averaging in the whole computational domain.