An economical simulation method for droplet motions in turbulent flows

A economical method is described for simulating droplet motions and collisions in turbulent flows in the EMPM (Explicit Mixing Parcel Model).

The EMPM predicts the evolving in-cloud variability due to entrainment and finite-rate turbulent mixing using a 1D representation of a cloudy parcel. The 1D formulation allows the model to resolve temperature and water vapor variability down to the smallest turbulent scales (about 1 mm). The EMPM calculates the growth of individual cloud droplets based on each droplet's local environment.

In the EMPM, turbulent advection of fluid is implemented by permuting the fluid cells comprising the 1D EMPM domain. Each permutation represents the effect of an individual turbulent eddy, and is called a "triplet map." This Eulerian implementation of the triplet map captures flow processes as small as the smallest turbulent eddy (Kolmogorov microscale), but the response of small droplets to turbulence has important features at smaller scales, in fact as small as the droplet radius. Namely, droplet slip at scales less than the Kolmogorov microscale induces droplet clustering that is estimated to increase droplet collision rates by one or more orders of magnitude.

The strategy is to track droplet locations in a Lagrangian manner so that the Eulerian mesh need only resolve fluid properties down to the Kolmogorov microscale. Droplets are displaced by amounts (1+S)D, where D is the displacement due to the fluid triplet map, and the additional displacement SD represents droplet motion relative to the the fluid. Here S is analogous to the droplet Stokes number. Despite the simplicity, and consequent computational efficiency, of this representation of droplet slip compared to the droplet momentum equation, it captures important features of droplet motions.