An exponential distribution scheme for the two-way coupling in transported PDF method for dilute spray combustion

Abstract

When the dual-Lagrangian formulation is employed for simulating dilute spray flames, the subtle mass and energy coupling between the gas and liquid phases could potentially lead to issues such as unrealistic low gas phase particle temperature, numerical instability and grid convergence not ensured. In this study, an exponential distribution (ExpD) scheme is formulated for the non-local distribution of mass and energy source terms from droplet evaporation to adjacent cells, in which the sources are distributed following exponential decay with respect to the distance between the droplet and the cell centre and therefore the mass and energy sources of a computational cell may have contribution from the evaporation of non-local droplets in the domain. Furthermore, a two-step procedure is proposed to avoid unrealistic low temperatures for gas phase particles, in which energy source from evaporation is first applied to all existing gas phase particles in a cell, followed by a substep that handles the enthalpy of the fuel vapour being consistent with the mass coupling model employed. The ExpD scheme together with the new energy coupling strategy have been validated in the single droplet evaporation and combustion simulations. For mass coupling, three existing models, i.e. the NEW, EQUAL and SAT model have been investigated in conjunction with the IEM mixing model. Grid convergence test becomes possible with the ExpD and energy coupling scheme, whose robustness is validated with different mass coupling models. Specifically, the EQUAL model approaches second-order accuracy in space, whereas the NEW and SAT models are first-order accurate. For the steady evaporation of methanol droplet in convective flow, with the finest grid, all the mass coupling models converge to the envelop flame as observed in experiment, but fuel is consumed more completely with the EQUAL model since the fuel and oxidiser are relatively well mixed at particle level.