Physical Model

Droplet module (v1.4p only)

Version 1.4p contains a module for the Lagrangian tracking of droplets or inertial particles. The module tracks the motion of single droplets by solving the relevant momentum equations (Stokes drag & gravity):

Here, x_k is the droplet position and v_k is its velocity, while u is the fluid velocity in correspondence of the droplet.

Moreover:

Droplets are assumed to be spherical. The radius R_k of each droplet is considered to vary as a function of the local relative humidity φ in correspondence with the droplet position.

The coefficient C is given by

where L is the latent heat of evaporation/condensation, κ is the thermal diffusivity of air (Pr about 0.7), D is the diffusivity of vapour in the air (Sc about 0.5), ρ_w is the water density and p_sat is the saturation pressure, approximated by the Clapeyron-Clausius law:

• where p_R is the saturation pressure at the reference temperature T_R
• Droplets can collide. At each step, the module checks the occurrence of collisions among the droplets. The collision detection method is schematically described here, and carried out at each step.

A momentum and (water) mass feedback of the droplets on the flow is considered (see below).

Navier-Stokes (Boussinesq) equations

The code solves the Navier-Stokes equations together with up to 6 additional scalar equations. In versions 1.4strat and 1.4p the first two scalars represent temperature and water vapour density:

here L is the latent heat of evaporation/condensation, C_v is the condensation rate per unit volume, and f_p is the particle force per unit volume on the fluid phase. The last two terms represent the momentum and mass feedback of the droplets to the fluid phase, and are linked to the equations for the droplets: