A fast Eulerian method for disperse two-phase flow

Abstract

We propose a variant of the Eulerian method for two-phase flow that is valid for small particle response time τ. For small τ, the particle velocity field v ( x ,t) approaches a unique, equilibrium field, independent of initial conditions. A precise inequality is derived specifying how small τ must be for this to occur. When it does, v ( x ,t) depends only on local fluid quantities (velocity and its spatial and temporal derivatives), and may be expressed as an expansion in τ. We derive an expansion which generalizes those of previous researchers. The first-order truncation of this expansion may be computed efficiently, so by using it to approximate v, the method avoids the need to solve additional partial differential equations, and therefore is much faster than the standard Eulerian method. Results from a direct numerical simulation of turbulent channel flow indicate that this first-order approximation of v is sufficiently accurate. Static tests performed at one time-instance show the actual velocities of particles evolved in a Lagrangian fashion are estimated well by evaluating the first-order approximation of v at the particles' positions. In particular, turbophoresis is represented accurately. Dynamic tests examine the effect of using the first-order approximation of v to evolve particles. The distribution of particles evolved in this way differs little from that of particles evolved using the standard Lagrangian method, indicating that static errors do not accumulate over time. In particular, the approximate method accurately captures preferential concentration in regions of high strain and low vorticity. Analogous results hold for bubbles. Therefore, for sufficiently small particles of any density, the first-order approximation to v is accurate, so the proposed variant of the Eulerian method is both accurate and fast.