Equilibrium Eulerian approach for predicting the thermal field of a dispersion of small particles

The equilibrium Eulerian method [J. Ferry, S. Balachandar, A fast Eulerian method for disperse two-phase flow, Int. J. Multiphase Flow 27 (7) (2001) 1199–1226] provides an accurate approximation to the velocity field of sufficiently small dispersed particles in a turbulent fluid. In particular, it captures the important physics of particle response to turbulent flow, such as preferential concentration and turbophoresis. It is therefore employed as an efficient alternative to solving a PDE to determine the particle velocity field. Here we explore two possible extensions of this method to determine the particle temperature field accurately and efficiently, as functions of the underlying fluid velocity and temperature fields. Both extensions are theoretically shown to be highly accurate for asymptotically small particles. Their behavior for finite-size particles is assessed in a DNS of turbulent channel flow (Reτ = 150) with a passive temperature field (Pr = 1). Here it is found that although the order of accuracy of the two extensions is the same, the constant factor by which one is superior to the other can be quite large, so the less accurate extension is appropriate only in the case of a very small mechanical-to-thermal response time ratio.