The COVID-19 pandemic has inspired several studies on the fluid dynamics of respiratory events. Here, we propose a computational approach in which respiratory droplets are coarse-grained into an Eulerian liquid field advected by the fluid streamlines. A direct numerical simulation is carried out for a moist cough using a closure model for space-time dependence of the evaporation time scale. Stokes-number estimates are provided, for the initial droplet size of 10 mu m, which are found to be MUCH LESS-THAN1, thereby justifying the neglect of droplet inertia, over the duration of the simulation. Several important features of the moist-cough flow reported in the literature using Lagrangian tracking methods have been accurately captured using our scheme. Some new results are presented, including the evaporation time for a 'mild' cough, a saturation-temperature diagram and a favourable correlation between the vorticity and liquid fields. The present approach can be extended for studying the long-range transmission of virus-laden droplets.