Modelling of the evolution of a droplet cloud in a turbulent flow

The effects of droplet inertia and turbulent mixing on the droplet number density distribution in a turbulent flow field are studied. A formulation of the turbulent convective diffusion equation for the droplet number density, based on the modified Fully Lagrangian Approach, is proposed. The Fully Lagrangian Approach for the dispersed phase is extended to account for the Hessian of transformation from Eulerian to Lagrangian variables. Droplets with moderate inertia are assumed to be transported and dispersed by large scale structures of a filtered field in the Large Eddy Simulation (LES) framework. Turbulent fluctuations, not visible in the filtered solution for the droplet velocity field, induce an additional diffusion mass flux and hence additional dispersion of the droplets. The Lagrangian formulation of the transport equation for the droplet number density and the modified Fully Lagrangian Approach (FLA) make it possible to resolve the flow regions with intersecting droplet trajectories in the filtered flow field. Thus, we can cope successfully with the problems of multivalued filtered droplet velocity regions and caustic formation. The spatial derivatives for the droplet number density are calculated by projecting the FLA solution on the Eulerian mesh, resulting in a hybrid Lagrangian-Eulerian approach to the problem. The main approximations for the method are supported by the calculation of droplet mixing in an unsteady one-dimensional flow field formed by large-scale oscillations with an imposed small-scale modulation. The results of the calculations for droplet mixing in decaying homogeneous and isotropic turbulence are validated by the results of Direct Numerical Simulations (DNS) for several values of the Stokes number.