Effect of particle velocity fluctuations in particle-fluid flows

For many purposes, the flow of a particle-fluid mixture can be described by equations of balance of mass and momentum for each material, i.e., for the particles and the fluid. These equations must be supplemented by constitutive equations for the stresses and the momentum interaction. The stress in the particle component due to the random motions of the particles can be significant in some flows. In addition, the velocity fluctuations of the particles induce velocity fluctuations in the fluid, resulting in a stress there. Moreover, the momentum interaction term is also affected, resulting in a stress-like term. In this paper, the terms resulting from the particle velocity fluctuations are calculated by averaging the flow around a single sphere. The stresses and momentum interaction terms are given in terms of the average velocity and volume fraction fields, and the fluctuation stress for the particles. The particle stress must be given by a separate constitutive assumption. A framework for the particle velocity fluctuations that accounts for the evolution of the particle temperature is given. Constitutive equations for the stresses in the components, the Reynolds stress in the fluid, the momentum interaction and the interaction work term are calculated by averaging the appropriate quantity using the inviscid flow around a single sphere whose velocity is the mean particle velocity plus a velocity perturbation. The averaging process used is a version of ensemble averaging wherein the particle position is assumed to be distributed over the interior of a spherical “cell”, and the particle velocities are distributed so that the average of the fluctuation is zero, and the average of the negative of the fluctuation dyad is the particle Reynolds stress.