KHRT Breakup Model

The KHRT Breakup model combines two submodels, one based on Kelvin-Helmholtz (KH) theory and one based on Rayleigh-Taylor (RT) theory. Both breakup submodels consider the growth of instabilities on a droplet and provide expressions for their wavelength and frequency.

Kelvin-Helmholtz instabilities are due to the slip velocity of the droplet, which eventually shears small child droplets off the parent [688], corresponding to the stripping regime. Rayleigh-Taylor instabilities are due to the acceleration of the droplet and tend to shatter the droplet completely [683], corresponding to the catastrophic regime. The KH and RT submodels compete: instabilities due to both can grow simultaneously; if they grow for long enough, breakup occurs due to the RT instabilities. Otherwise KH breakup occurs.

Because there is a tendency for the KH submodel to generate a bi-modal size distribution, the smaller child droplets are accumulated into a new parcel; the parent parcel retains the larger parent droplets. However, to prevent too many child parcels being created, these parcels are only released when their mass reaches a predefined proportion of the original mass of the parent. Child parcels are given a lateral velocity proportional to the kinetic energy of the droplet oscillation at the instant of breakup. This velocity tends to generate a spreading effect from an injector which, to some extent, makes a cone injector unnecessary.

The KHRT model was developed as a replacement for the TAB model [683]. Although the latter can still be preferred at low Weber numbers, the KHRT model is preferred at higher Weber numbers. The principal difficulty in using the model is tuning the coefficients to the application under consideration. This difficulty reflects the complexity of the physics and is not due to a flaw in the model.

Kelvin-Helmholtz instabilities are postulated as the cause of stripping breakup.

The model schematics and key equations are illustrated below.


KH Model	RT Model

$Child drop diameter = 2 B_{0} Λ_{K H}$	$Child drop diameter = C_{3} Λ_{R T}$
$Breakup time constant = \frac{1.894 B_{1} D_{p}}{{Λ_{K H} Ω}_{K H}}$	$Breakup time constant = \frac{C_{τ}}{Ω_{R T}}$

Reitz [688] proposed expressions for the wavelength and growth-rate of these instabilities:

Figure 1. EQUATION_DISPLAY

Λ_{K H} = \frac{4.51 D_{p} (1 + 0.45 {Oh}_{r}^{0.5}) (1 + 0.4 {Ta}_{r}^{0.7})}{{(1 + 0.865 {We}_{r}^{1.67})}^{0.6}}

(3095)

Figure 2. EQUATION_DISPLAY

Ω_{K H} = \frac{0.34 + 0.385 {We}_{r}^{1.5}}{(1 + {Oh}_{r}) (1 + 1.4 {Ta}_{r}^{0.6})} \sqrt{\frac{4 π σ}{3 m_{p}}}

(3096)

where ${We}_{r} = 0.5 We$ is the Weber number, which is based on the droplet radius,

Figure 3. EQUATION_DISPLAY

{Oh}_{r} = μ_{l} \sqrt{\frac{2}{ρ_{l} D_{p} σ}}

(3097)

is the Ohnesorge number based on the droplet radius and ${Ta}_{r} = {Oh}_{r} \sqrt{{We}_{r}}$ is the Taylor number that is based on the droplet radius. The wavelength and growth-rate are used to formulate a characteristic time-scale and length-scale for breakup caused by these instabilities:

Figure 4. EQUATION_DISPLAY

τ_{K H} = \frac{1.894 B_{1} D_{p}}{Λ_{KH} Ω_{KH}}

(3098)

Figure 5. EQUATION_DISPLAY

D_{K H} = 2 B_{0} Λ_{K H}

(3099)

The coefficient $B_{0}$ takes the value 0.61. The coefficient $B_{1}$ can be tuned to account for uncertainties in, for example, the droplet initial conditions. A wide range of values have been reported in the literature, from 1.73 to 60 or higher.

If $D_{p} > D_{K H}$ , breakup due to KH instabilities is possible, in which case the parent droplet diameter decreases according to:

Figure 6. EQUATION_DISPLAY

\frac{d D_{p}}{d t} = \frac{D_{K H} - D_{p}}{τ_{K H}}

(3100)

However, this is allowed to occur only if breakup due to RT instabilities (see below) does not occur.

The mass shed during KH breakup is accumulated [683] until it reaches a user-specified proportion of the parent parcel’s original mass, default 3%. One or more child parcels are created, with child droplets having diameter $D_{K H}$ . The diameter of the droplets in the parent parcel remains unchanged; the number of droplets is calculated to conserve mass. Child parcels are also given a velocity normal to the original velocity of the parent parcel:

Figure 7. EQUATION_DISPLAY

v_{⊥} = A_{1} Λ_{K H} Ω_{KH}

(3101)

The normal velocity coefficient $A_{1}$ can be tuned to produce a known spray angle; alternatively the spray angle may be specified directly at a cone injector, in which case $A_{1}$ should be set to zero.

Rayleigh-Taylor instabilities are postulated as the cause of “catastrophic” breakup. Following Senecal et al [702], Simcenter STAR-CCM+ incorporates the effect of viscosity on these instabilities. The relationship between their wavenumber and growth-rate as a function of the droplet acceleration $a$ is:

Figure 8. EQUATION_DISPLAY

ω_{R T} = - k_{R T}^{2} (\frac{μ_{l} + μ_{g}}{ρ_{l} + ρ_{g}}) + \sqrt{k_{R T} (\frac{ρ_{l} - ρ_{g}}{ρ_{l} + ρ_{g}}) a - \frac{k_{R T}^{3} σ}{ρ_{l} + ρ_{g}} + k_{R T}^{4} {(\frac{μ_{l} + μ_{g}}{ρ_{l} + ρ_{g}})}^{2}}

(3102)

Eqn. (3102) is solved numerically to give the wavelength $Λ_{R T}$ corresponding to the maximum growth-rate $Ω_{R T}$ . This is assumed to be the critical mode. In the limit of negligible viscosity, this method reproduces the inviscid RT formulation that is used by [683], for example. The justification for including viscosity is that its effects can be significant; implementations which neglect it typically require additional modifications such as a user-defined breakup length [691].

The wavelength and growth-rate in turn are used to formulate a characteristic time-scale and length-scale for breakup that is caused by these instabilities:

Figure 9. EQUATION_DISPLAY

τ_{R T} = \frac{C_{τ}}{Ω_{R T}}

(3103)

Figure 10. EQUATION_DISPLAY

D_{R T} = C_{3} Λ_{R T}

(3104)

The coefficient $C_{τ}$ takes the default value 1. The coefficient $C_{3}$ takes the default value 0.1, with values in the range 0.1-5.33 being reported in the literature.

If $D_{p} > D_{R T}$ , a critical Rayleigh-Taylor instability is assumed to be growing on the droplet. If the instability persists for a duration $τ_{R T}$ , a breakup event occurs. The parent droplets shatter completely, producing child droplets with new diameter $D_{R T}$ .