Grid-averaged Lagrangian equations of dispersed phase in dilute two-phase flow

A set of grid-averaged Lagrangian transport equations of the dispersed-phase turbulence kinetic energy <u_pi^k′u_pi^k′> and the turbulence modulation quantity <u′_iu_pi^k′> is derived and demonstrated through a well-defined test problem of droplet-loading mixing layer. It is shown that accurate predictions of the turbulence kinetic energy of the droplets can be achieved by solving from this set of grid-averaged Lagrangian equations with significantly less computational droplets than those of a purely stochastic model. The information of the turbulence modulation quantity, which is solved from the grid-averaged Lagrangian transport equation of <u′_iu_pi^k′>, can also be used to evaluate accurately the source term accounting for the two-phase interaction in the turbulence kinetic energy k transport equation of the carrier fluid, which is important for determining the turbulence characteristics of the carrier fluid.