Extension of the Fully Lagrangian Approach for the Integration of the Droplet Number Density on Caustic Formations

Abstract

The spatial structure of the agglomeration regions (caustics) formed by inertia droplets and particles in dispersed flows is studied. The Fully Lagrangian Approach (FLA) is extended to a second order FLA that accounts for the Hessian of the deformed dispersed phase continuum. The calculation of the Hessian matrix along the droplet trajectory is also presented. The second order FLA adresses the singularities of the standard FLA and demonstrates the integrability of the point-wise number density of the standard FLA on caustic formations. Furthermore, with the approach presented here the FLA number density can be related to a defined finite length scale needed for the introduction of the FLA to turbulent flows in the LES context.