The modification of decaying homogeneous turbulence due to its interaction with dispersed small solid particles (d/v < l), at a volumetric loading ratio $,<5 X lo-", is studied using direct numerical simulation. The results show that the particles increase the fluid turbulence energy at high wave numbers. This increase of energy is accompanied by an increase of the viscous dissipation rate, and, hence, an increase in the rate of energy transfer T(k) from the large-scale motion. Thus, depending on the conditions at particle injection, the fluid turbulence kinetic energy may increase initially. But, in the absence of external sources [shear or buoyancy), the turbulence energy eventually decays faster than in the particle-free turbulence. In gravitational environment, particles transfer their momentum to the small-scale motion but in an anisotropic manner. The pressure-strain correlation acts to remove this anisotropy by transferring energy from the direction of gravity to the other two directions, but at the same wave number, i.e., to the small-scale motion in directions normal to gravity. This input of energy in the two directions with lowest energy content causes a reverse cascade. This reverse cascade tends to build up the energy level at lower wave numbers, thus reducing the decay rate of energy as compared to that of either the particle-free turbulence or the zero-gravity particle-laden flow.

Dispersion of solid particles in decaying isotropic turbulence is studied numerically. The three-dimensional, time-dependent velocity field of a homogeneous, non-stationary turbulence was computed using the method of direct numerical simulation (DNS). A numerical grid containing 963 points was sufficient to resolve the turbulent motion at the Kolmogorov lengthscale for a range of microscale Reynolds numbers starting from Rλ = 25 and decaying to Rλ = 16. The dispersion characteristics of three different solid particles (corn, copper and glass) injected in the flow, were obtained by integrating the complete equation of particle motion along the instantaneous trajectories of 223 particles for each particle type, and then performing ensemble averaging. The three different particles are those used by Snyder & Lumley (1971), referred to throughout the paper as SL, in their pioneering wind-tunnel experiment. Good agreement was achieved between our DNS results and the measured time development of the mean-square displacement of the particles.The simulation results also include the time development of the mean-square relative velocity of the particles, the Lagrangian velocity autocorrelation and the turbulent diffusivity of the particles and fluid points. The Lagrangian velocity frequency spectra of the particles and their surrounding fluid, as well as the time development of all the forces acting on one particle are also presented. In order to distinguish between the effects of inertia and gravity on the dispersion statistics we compare the results of simulations made with and without the buoyancy force included in the particle motion equation. A summary of the significant results is provided in §7 of the paper.The main objective of the paper is to enhance the understanding of the physics of particle dispersion in a simple turbulent flow by examining the simulation results described above and answering the questions of how and why the dispersion statistics of a solid particle differ from those of its corresponding fluid point and surrounding fluid and what influences inertia and gravity have on these statistics.

Part I of this paper [Elghobashi and Truesdell, Phys. Fluids A 5, 1790 (1993)] examined the modulation of turbulence by the particles. Here the effects of the two-way interaction on particle dispersion are discussed. In zero gravity, the two-way coupling enhances the alignment of the surrounding fluid velocity vector with the direction of the solid particle trajectory. This alignment reduces the mean-square relative velocity and increases the Lagrangian velocity autocorrelation coefficient of the solid particle, the fluid point and the surrounding fluid, and the mean-square displacement of the solid particles. However, the fluid point mean-square displacement decreases because the larger inertia of the solid particles increases the decay rate of turbulence energy. In gravity environment, the particles augment the component of turbulence energy in the gravity direction, and thus increase the mean-square displacement of the solid particles and fluid points in that direction. However, their dispersion in the lateral directions is reduced due to the crossing trajectories effect [Yudine, Adv. Geophys. 6, 185 (1959)].