Equilibrium expansion for the Eulerian velocity of small particles

We present a method for evaluating the velocity of particles in a two-phase flow that is valid for small particles. It is more efficient than the traditional Eulerian–Eulerian method, particularly for polydisperse systems, and therefore can be termed the fast Eulerian method. The methodʹs efficiency is based on the fact that under certain well-defined conditions, there exists a unique, equilibrium velocity field ve(x, t) for particles, independent of past particle behavior. Particles whose velocities differ from ve(x, t) readjust exponentially fast, so after a brief transient period, particle velocities are determined uniquely by the fluid velocity field and its derivatives. The need to solve PDEs for the particle velocity field is thereby avoided. ablish that the first-order approximation to the equilibrium velocity is sufficient to produce an accurate numerical method. This approximate velocity may be used with either a discrete or a continuum treatment of particles. We justify the approximation for the discrete case by comparing the statistics of particles evolved exactly with those evolved via the approximate velocity. We then evolve an Eulerian concentration field of particles moving with the approximate velocity, and show that it is consistent with the distribution of an ensemble of Lagrangian particles. These tests were performed in a DNS of turbulent channel flow.