SSD Breakup Model

In the Stochastic Secondary Droplet (SSD) breakup model, breakup is a random process unrelated to original droplet size. This results in a log-normal distribution of droplet diameters in the long time limit.

The particles interact with the continuous fluid phase as they travel. Particles break up when they meet three criteria:

The particle radius is larger than the critical radius, $r_{c}$ .
The Weber number is larger than a critical Weber number, ${We}_{c r}$ .
The particle's accumulated residence time is larger than the breakup time $τ_{bu}$ (Eqn. (3115)). The model tracks parcels as they cross the domain and compares the residence time with a locally evaluated breakup time. The residence time is only calculated on parcels with a radius larger than a critical radius and a Weber number larger than a critical Weber number.

The first two criteria are related according to:

Figure 1. EQUATION_DISPLAY

r_{c} = \frac{{We}_{c r} σ_{l}}{{2 ρ}_{g} {v_{s}}^{2}}

(3114)

where:

$r_{c}$ is the critical radius
${We}_{c r}$ is the critical Weber number specified in the interface. The default value is 12.
$σ_{l}$ is the surface tension of the liquid.
$ρ_{g}$ is the density of the continuous phase.
$v_{s}$ , slip velocity, is the relative velocity of the particle through the continuous phase.

Particles cease to exist when they break up into child particles. In turn, these child particles can also break up, if they meet the criteria above. The higher the critical Weber number, the larger the particles when breakup stops.

Particle break up only if they have a residence time greater than the breakup time. The breakup time scale is defined as:

Figure 2. EQUATION_DISPLAY

τ_{bu} = B_{1} \sqrt{\frac{ρ_{l}}{ρ_{g}}} \frac{r_{p}}{v_{s}}

(3115)

where:

$τ_{bu}$ is the breakup time.
$B_{1}$ is a parameter regulating speed of breakup, set in the interface; larger values mean breakup occurs later. The default value is $\sqrt{3}$ .
$ρ_{l}$ is the density of the liquid.
$ρ_{g}$ is the density of the continuous phase.
$r_{p}$ is the radius of the particle.
$v_{s}$ , slip velocity, is the relative velocity of the droplet through the continuous phase.

When breakup happens, the parent parcel is replaced by a number of child parcels, each containing a number of identical particles. The particle diameters in a given parcel follow a log-normal distribution:

Figure 3. EQUATION_DISPLAY

Φ (r) = \frac{1}{2} (1 + erf [\frac{\log (\frac{r}{r_{0}}) - 〈 ξ 〉}{\sqrt{2 〈 ξ^{2} 〉}}])

(3116)

where:

$Φ (r)$ is the breakup distribution function, that is, the probability density as a function of $r$ .
$r$ is the radius of the child droplet.
$r_{0}$ is the radius of the parent droplet.
$〈 ξ 〉$ is the first moment of $Φ (r)$ , equal to a constant, $K_{0}$ , set in the interface with default value -0.1. More negative values of $K_{0}$ give a smaller mean distribution of child diameters. Positive values of $K_{0}$ give larger mean child diameters.
$〈 ξ^{2} 〉$ is the second moment of $Φ (r)$ ,
Figure 4. EQUATION_DISPLAY

$〈 ξ^{2} 〉 = K_{1} ln (\frac{We}{{We}_{cr}})$

(3117)

where:
- $K_{1}$ is a constant set in the interface, default value 0.1. $K_{1}$ must be positive.
- $We$ is the Weber number of the particle.
- ${We}_{cr}$ is the critical Weber number, a constant set in the interface, default value 12.
- $〈 ξ^{2} 〉$ is the variance of the distribution, and $K_{1}$ is directly proportional to it. The larger they are, the broader the distribution of diameters.

Child parcels receive the velocity of the parent parcel multiplied by a factor, $| W_{bu} |$ :

Figure 5. EQUATION_DISPLAY

| W_{bu} | = K_{3} r_{p, c} ν = \frac{K_{3} r_{p, c}}{τ_{bu}}

(3118)

where:

$K_{3}$ is a constant set in the interface, Normal Velocity Coefficient, default value 0.1, limited to values between 0 and 1. The larger $K_{3}$ is, the more rapid the radial diffusion of the spray. Use a value of 0 if the spray angle is known and fixed at the injector.
$r_{p, c}$ is the radius of the child particle.
$ν = \frac{1}{τ_{bu}}$ is the breakup frequency, (the reciprocal of the breakup time).

This model differs from other breakup models in three ways:

The parent drop no longer exists after the breakup event. (The particle numbering system re-numbers after the breakup, so a particle before the event with the same number as a particle after the event is not the same particle.)
Because the model works with a user-specified particle count, you seldom have very large particle counts that are difficult to handle numerically.
The breakup distribution function, $Φ r$ in Eqn. (3116), is a function of local conditions and depends only weakly on the diameter of the parent. Though the distribution is limited to the range between the current radius and half the critical radius, the function depends more on the Weber number, which is a strong function of local flow.