| A methodology for computing spectral statistics of particulate flows, equivalent to single-phase flows, is developed. As an example, the density autocorrelation is related to the particle radial distribution function (RDF) and a geometric quantity, referred to as the Overlap function. The density spectrum is found to be a strong function of particle size and concentration. The effect of relative motion between the two phases is probed by adding a small, attractive, hydrodynamic potential which induced a shift of the spectrum toward larger scales. Behavior in the limit of uniform motion under the imposition of an asymptotically large pressure gradient is also studied. Exact relationships between various moments of interest, in this regime, are derived. The formalism is subsequently generalized to include a polydisperse particle-fluid system. The roles of variance, symmetry and tail behavior of the particle size distribution are investigated. The limiting regime of asymptotically large pressure gradient is revisited. However, it is noted that moments of different order in velocity can no longer be related exactly.; Direct Numerical Simulations of a turbulent fluid laden with finite sized particles were performed using 128{dollar}sp3{dollar} grid and up to 262,144 {dollar}(Nsb{lcub}p{rcub}){dollar} particles. An algorithm which reduces calculation of hard-sphere collisions from {dollar}O(Nsbsp{lcub}p{rcub}{lcub}2{rcub}){dollar} to {dollar}O(Nsb{lcub}p{rcub}){dollar} is presented. Particle feedback or "reverse" coupling is accomplished in a manner ensuring correct discrete energy conservation. The two-field formalism just developed was extended to include velocity correlations and simplified for the present case of dilute suspensions of small and heavy particles. The simplified relations are then employed to calculate two-point correlations and equivalent spectral densities. The appropriate basis, when considering energy behavior of particle-laden flows, is equal total energies of the systems being compared. Using such a carefully initialized set of runs, particle inertia was observed to increase both the viscous and drag dissipations. It also, caused particle velocities to correlate for longer distances. Like previous investigators, "pivoting" or crossover of the fluid energy spectra was observed. A possible new scaling for this phenomenon is suggested. Furthermore, investigations of the influence of particle mass and number densities on turbulence modulation were also carried out. |
