Fog formation

The flow solver computes the fog formation in a gas mixture. The gas mixture constitutes the continuum phase, and the airborne condensate that forms the dispersed phase.

The Quadrature Method of Moments (QMOM) scheme, proposed in [53], is used for computing the droplet distribution in the mixture. In addition to the mixture mass, momentum, and energy conservation equations, the flow solver solves the conservation equations for the water vapor and the first 2k moments of the droplet number density function, where k represents the k^th moment of the number density function. By default, only the first and second moments are solved.

The conservation equation for the water vapor is as follows:

where:

ρ is the density of the fluid mixture.
ϕ_v = ρ_v/ρ is the vapor mass fraction, where ρ_v is the vapor density.
D_{ϕ_v} is the effective diffusion coefficient of the vapor into air. It includes both laminar and turbulent diffusions. The turbulent diffusion is computed from the turbulent viscosity and the Schmidt number.
S_{ϕ_v} is the source term. It includes the water droplet formation and the droplet growth/evaporation effects.

The droplet distribution in the domain is described by a number density function f(x_i,r,t), which is the function of Cartesian coordinates, x_i, the droplet radius, r, and time, t.

The k^th moments of the number density function is given by:

where:

D_f is the effective diffusion coefficient for the droplet number density function. It includes both laminar and turbulent diffusions. The laminar diffusion coefficient is computed using the mixture dynamic viscosity and the assumption of a Schmidt number of 0.61. The turbulence contribution is obtained using the turbulence viscosity and a turbulence Schmidt number of unity.
is the droplet growth rate.

J is the number of droplets at critical radius, r*, generated per unit volume of the mixture per unit time, proposed in [54].

The solver uses the nucleation model that corresponds to the homogeneous nucleation where droplets are formed as a result of random collisions of water vapor molecules. The homogeneous nucleation source term is given by [54]:

where

q_c is the condensation coefficient.
m is the mass of one molecule of water.
σ is the surface tension.
k_B is the Boltzmann constant.
η is the correction factor.
T_g is the temperature of the saturated vapor.
ρ_v and ρ_l are the vapor and liquid density, respectively.

The heterogeneous nucleation where water droplets are formed as a result of the condensation around small suspended aerosol particles is the dominant mode of the nucleation. A model for the heterogeneous nucleation is provided in [55] as follow:

where:

R is the average radius of aerosol particles.
n_p is the number of particles per unit mass of the mixture.
J₀ = 10²⁵ (cm^-2s^-1) is the nucleation pre-factor.
f is the factor that depends on the contact angle between the water and the nuclei and given by:

where m, z and κ are calculated as:

Where θ is the contact angle between water and the aerosol particles.

The following table lists the advanced parameters to model fog formation in your simulation.


Advanced parameter	Default	Description
`SOLVE EXTRA EQUATIONS TO MODEL FOG FORMATION`	False	When you set it to True, it activates the QMOM scheme to model fog formation.
`NUMBER OF MOMENT QUADRATURES TO SOLVE IN THE QMOM SCHEME`	2	Sets the number of moments of the number density function to be solved in the QMOM scheme. This number must be even.
`HETEROGENEOUS NUCLEATION NUCLEI AVARAGE RADIUS`	-1.0	Sets the average radius of aerosol nuclei, r, used for modeling the heterogeneous nucleation.
`HETEROGENEOUS NUCLEATION NUCLEI NUMBER DENSITY`	-1.0	Sets the concentration of aerosol nuclei, n_p, used for modeling the heterogeneous nucleation.
`HETEROGENEOUS NUCLEATION CONSTANT ANGLE`	80.0	Sets the contact angle, θ, between water and the aerosol particles used for modeling the heterogeneous nucleation.