Closure modelling of inertial particle-pair behaviour using the kinetic PDF approach

The fully Lagrangian approach (FLA) is a promising means of modelling gas-droplet flows, through its representationof the droplet phase as a continuum in combination with a seeding of Lagrangian trajectories which retain the mesoscopic detail of individual droplet motion. Despite the advantages in both accuracy and computational economythat are afforded by such a procedure, to date the FLA has only been applied to unsteady problems within laminarflows. In reality, most gas-droplet flows which are of interest in the context of industrial spray systems are highlyturbulent, and require additional modelling to accurately capture the effects of mixing over the range of scalesinvolved. This motivates the need for extending the FLA methodology to turbulent flows, which in addition to thecomputational efficiency advantage also offers high resolution of the droplet number density field as it undergoesclustering and segregation in response to the structural behaviour of the carrier flow. The present work addresses thisby developing a formulation of the FLA in the framework of large eddy simulation. In keeping with the continuumrepresentation of the FLA, droplets are tracked in the filtered turbulent velocity field, and the sub-grid scale (SGS) fluidvelocity is formally accounted for through the Lagrangian from of the droplet phase continuity equation. Thiscontribution takes the form of an additional unclosed mass flux, and is modelled using a kinetic PDF approach, withthe correlations between the droplet number density and SGS fluid velocity fluctuations being closed via a correlationsplitting procedure that incorporates the non-local turbulent drift effects on the droplet phase. The evolution of thenumber density field calculated using this procedure is examined, and compared to the number density based on onlythe filtered fluid velocity field across a range of values of droplet inertia.