Droplet Splitting Model
Droplet Splitting: Theoretical Foundations
Overview
Professor, what is the droplet breakup model used for?
It's a model for the process where droplets break up (secondary breakup) due to aerodynamic forces, such as in fuel injection, spray painting, and fire extinguisher sprays. It describes the process where large droplets immediately after injection break up into finer droplets.
What mechanisms are involved in droplet breakup?
The breakup mode (regime) changes depending on the Weber number $We$.
| Regime | Weber Number Range | Characteristics |
|---|---|---|
| Vibrational | $We < 12$ | Vibration only, no breakup |
| Bag breakup | $12 < We < 50$ | Inflates into a thin film and ruptures |
| Multimode | $50 < We < 100$ | Bag + Stripping |
| Sheet stripping | $100 < We < 350$ | Thin film strips from the surface |
| Catastrophic | $We > 350$ | Explosive breakup |
The Ohnesorge number $Oh = \mu_d / \sqrt{\rho_d \sigma d}$ is also important; high viscosity delays breakup.
Typical Breakup Models
Please tell me about the models used in CFD.
| Model | Overview | Applicable Range |
|---|---|---|
| TAB (Taylor Analogy Breakup) | Analogy to a spring-mass-damper system | $We < 100$, low-speed sprays |
| KHRT (Kelvin-Helmholtz / Rayleigh-Taylor) | Competition between KH instability and RT instability | High-speed diesel injection |
| SSD (Stochastic Secondary Droplet) | Generates size distribution stochastically | General purpose |
| ETAB (Enhanced TAB) | Improved TAB, better child droplet distribution after breakup | Medium-speed sprays |
The TAB model describes droplet deformation using a forced vibration equation.
$y$ is a dimensionless parameter for droplet deformation, and breakup occurs when $y = 1$. $C_F$, $C_b$, $C_k$, $C_d$ are constants from O'Rourke & Amsden (1987).
What is the concept behind the KHRT model?
It pits Kelvin-Helmholtz instability (growth of waves on the droplet surface) against Rayleigh-Taylor instability (interface instability due to acceleration). In high-speed injection (diesel engines), KH instability dominates, while RT instability becomes important in deceleration regions.
Weber Number DominatesโWhen Does a Droplet Break?
The dimensionless number governing droplet breakup is the Weber number We = ฯ_g u_rel^2 d / ฯ. If We is below 12, surface tension acts as a restoring force and the droplet maintains a spherical shape. However, when We exceeds 100, "Catastrophic Breakup" occurs, and the droplet instantly disperses into a fine mist. This critical We value is almost identical to the one Hinze determined experimentally in the 1940s, and it is still used today, 75 years later, as a benchmark standard for CFD droplet breakup models. In engine fuel injection design, the prediction accuracy of spray droplet size directly impacts fuel efficiency and emissions, making the choice of breakup model a technical decision with business implications.
Computational Methods for Droplet Splitting
Details of Numerical Methods
How is the droplet breakup model incorporated into CFD?
Within the Lagrangian particle tracking method (DPM), the breakup condition is evaluated for each computational particle (parcel) at every time step. When breakup occurs, the size, velocity, and number of child droplets are calculated, and new parcels are generated.
TAB Model Implementation
In the TAB model, the deformation amount $y$ and deformation rate $\dot{y}$ are tracked for each droplet. Breakup occurs when $y \geq 1$, and the child droplet radius is determined from energy conservation.
How many child droplets are produced?
The number of child droplets is determined from mass conservation. In practice, the parcel concept is used, so the number of droplets within a parcel is updated, and the representative droplet diameter of each parcel changes accordingly.
KHRT Model Implementation
In the KHRT model, the growth rate $\Omega$ and wavelength $\Lambda$ of surface waves due to KH instability are obtained from the dispersion relation.
Here, $T = Oh \sqrt{We}$ is the Taylor number. The child droplet radius generated by KH breakup is $r_{child} = B_0 \Lambda$, with $B_0 = 0.61$ as the standard value.
RT instability depends on the droplet deceleration $a_{decel}$, and the child droplet radius is determined from the fastest growing wave number. KH breakup and RT breakup compete, and whichever condition is met first is applied.
Settings in Fluent and OpenFOAM
How do you set it up in actual software?
Fluent's Wave model is only the KH part; KHRT (KH + RT) is recommended. KHRT is most commonly used for diesel injection. OpenFOAM's sprayFoam solver is specialized for Lagrangian spray calculations, allowing separate selection of breakupModel and atomizationModel.
TAB Model and KH-RT ModelโThe Two Major Breakup Models Supporting Spray CFD
The dominant breakup models in engine spray simulation are the two families: TAB (Taylor Analogy Breakup) and KH-RT (Kelvin-Helmholtz / Rayleigh-Taylor). TAB solves for a droplet as an elastic sphere using a vibration equation and determines breakup when the oscillation amplitude exceeds a critical value. While computationally light, it has the weakness of being poor at reproducing "stripping breakup" of large droplets. KH-RT is derived from hydrodynamic instability theory and has high accuracy in the high We number range, but requires calibration of model constants. In commercial engine CFD, a "KH-RT hybrid" that combines both models is mainstream.
Droplet Splitting in Practice
Practical Guide
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