The Kelvin-Helmholtz (KH) model is based on linear stability analysis of a liquid jet [46] and is used to model the jet’s primary breakup [77] region. From the linear stability analysis any perturbation applied to a liquid-gas interface can be expanded as a Fourier series, with its fastest growing mode contributing to the eventual breakup and generation of new droplets. The growth rate and wave length of the fastest growing mode was found numerically [77] to be:

(6–20)

(6–21)

Here is the wave length of the fastest growing wave, is its growth rate, r _p is the jet radius, is surface tension, the dimensionless gas Weber number We _g is defined as:

(6–22)

with U _rel being the magnitude of the liquid-gas relative velocity:

(6–23)

Here is the CFD gas-phase velocity, is the local gas-phase turbulent fluctuating velocity vector, and is the droplet velocity vector. Similarly, the liquid Weber number is defined as:

(6–24)

and the dimensionless Ohnesorge number is:

(6–25)

The Reynolds number is defined as:

(6–26)

where is the dynamic viscosity of the liquid. The last dimensionless number is the Taylor number:

(6–27)

The radius of new droplets (denoted as ), which are created during the primary breakup process, is related to the wave length as follows:

(6–28)

where is the Size Constant of the KH breakup model.

Ansys Forte employs the concept of "blob injection" [78] , in which the liquid-jet injection is represented with parcels of blobs that have their initial characteristic sizes equal to the effective nozzle diameter. Using this concept, primary breakup is modeled as a process where new droplets (with radius ) are formed from parent droplets (with radius ). The radius change of the parent droplet due to mass lost to its "child" droplets is described by the rate equation [78] :

(6–29)

where is less than or equal to . Here, the breakup time scale is calculated as:

(6–30)

In Equation 6–30 , is the Time Constant of KH breakup. This is a user-controlled input for which a general-purpose default value is provided in Ansys Forte.

In general it is impractical to track each droplet formed from the parent droplet. Using the "droplets in parcel" concept, new droplets cannot leave their parent parcel until their mass reaches or exceeds 3% of the average injected parcel mass. The new droplets are then grouped into a "shed parcel" (or child parcel) that is separated from its parent. This parcel-shedding process helps to predict a realistic droplet size distribution in the primary breakup region. Also, as the number of parcels is increased, the local statistical resolution is improved. The value 3% is provided as a default model constant in Ansys Forte, which may be adjusted to change the size-distribution resolution.

Therefore, the KH breakup model involves two steps, as depicted in Figure 6.3: Two-step approach in implementing the KH breakup model: the first step gradually reduces the parent droplet size () by the rate equation (Equation 6–29 ); the second step creates a new child parcel from the parent, in which the child droplet size () is predicted by Equation 6–28. The number of droplets in the parent parcel is assumed to remain the same before and after the breakup, (that is: ).

Figure 6.3: Two-step approach in implementing the KH breakup model

The problem remains to determine the droplet number in the child parcel (), as well as the droplet size of the parent parcel (). Two different options can be used:

Beyond the Breakup Length from the nozzle exit, the Rayleigh-Taylor (RT) model is used together with the KH model to predict secondary breakup of spray droplets [[90] , [11] ]. The "breakup length" is defined in Levich's theory [46] as:

(6–35)

in which is a constant with value 10.29. The breakup length is implemented in Ansys Forte as

(6–36)

in which is nozzle diameter and is nozzle cross-sectional area, is set to 10.29, and is the RT Distance Constant exposed in the Ansys Forte interface. Compared to Levich’s original definition Equation 6–35 , in Equation 6–36 serves two purposes: one is a conversion factor from to ; the other is to serve as a knob for model calibration. The recommended value for is approximately 1.9. The RT model is applied beyond the Breakup Length from the nozzle exit.

The RT model considers the instability generated when two fluids with different densities accelerate in a direction that is perpendicular to their interface. The frequency and wavelength of the fastest-growing wave are given by Bellman [12] as:

(6–37)

(6–38)

In the case of a high-speed droplet moving in air, a is the deceleration due to drag. As in the KH model, this fastest growing wave is assumed to lead to the breakup of droplets. The radius of the newly-formed droplet and the time of breakup are predicted as:

(6–39)

(6–40)

in which , are two constants. is the Size Constant of RT Breakup. is the Time Constant of RT Breakup. Unlike the KH model, the standard implementation of the RT model does not strip small child droplets off from their parent; instead, a catastrophic breakup of the parent droplet into small droplets is assumed, and the liquid column is eventually disintegrated and dispersed into the ambient gas.

To reduce the time step dependency of the RT model, the RT breakup process is modeled by a rate equation similar to that of the KH model, Equation 6–29 :

(6–41)

in which is predicted by Equation 6–39 . The parent droplet breaks up continuously at each time step, and the time-step dependency is avoided. Assuming r _c and does not change with time, Equation 6–41 can be solved analytically as:

(6–42)

Here r _p,0 is the radius of parent droplet at the start of break-up. Thus, is the characteristic time required to break the parent droplet.

In practice, neither rate equation (Equation 6–29 ) nor (Equation 6–41 ) can be solved analytically, since both r _c and and (or ) change with time, and can change significantly. Therefore, a sub-cycle approach is used to solve these rate equations to eliminate time-step dependency.