Droplet Splitting Model
Theory and Physics
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 Dominates—When Does a Droplet Break?
The dimensionless number governing droplet breakup is the Weber number We = ρ_g u_rel^2 d / σ. 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.
Physical Meaning of Each Term
- Temporal term $\partial(\rho\phi)/\partial t$: Think of the moment you turn on a faucet. At first, water comes out in an unstable, spluttering manner, but after a while, it becomes a steady flow, right? This "period of change" is described by the temporal term. The pulsation of blood flow from a heartbeat, or the flow fluctuation each time an engine valve opens and closes—all are unsteady phenomena. So what is steady-state analysis? It looks only at "after sufficient time has passed and the flow has settled down"—meaning this term is set to zero. Since computational cost drops significantly, the basic CFD strategy is to first solve it as steady-state.
- Convection term $\nabla \cdot (\rho \mathbf{u} \phi)$: What happens if you drop a leaf into a river? It gets carried downstream by the flow, right? This is "convection"—the effect where fluid motion transports things. The warm air from a heater reaching the far corner of a room is also because the "carrier," air, transports heat via convection. Here's the interesting part—this term contains "velocity × velocity," making it nonlinear. That is, as the flow speed increases, this term rapidly strengthens, making control difficult. This is the root cause of turbulence. A common misconception: "Convection and conduction are similar things" → They are completely different! Convection is carried by flow, conduction is transmitted by molecules. There is an order of magnitude difference in efficiency.
- Diffusion term $\nabla \cdot (\Gamma \nabla \phi)$: Have you ever put milk in coffee and left it? Even without stirring, after a while, it naturally mixes. That's molecular diffusion. Now a question—honey and water, which flows more easily? Obviously water, right? Honey has high viscosity ($\mu$), so it flows poorly. When viscosity is high, the diffusion term becomes strong, and the fluid moves in a "thick" manner. In low Reynolds number flows (slow, viscous), diffusion dominates. Conversely, in high Re number flows, convection overwhelmingly dominates, and diffusion plays a supporting role.
- Pressure term $-\nabla p$: When you push the plunger of a syringe, liquid shoots out forcefully from the needle tip, right? Why? Because the piston side is high pressure, and the needle tip is low pressure—this pressure difference provides the force that pushes the fluid. Dam discharge works on the same principle. On a weather map, where isobars are tightly packed? That's right, strong winds blow. "Flow is generated where there is a pressure difference"—this is the physical meaning of the pressure term in the Navier-Stokes equations. A point of confusion here: "Pressure" in CFD is often gauge pressure, not absolute pressure. When you switch to compressible analysis and suddenly get strange results, it might be due to mixing up absolute/gauge pressure.
- Source term $S_\phi$: Warmed air rises—why? Because it becomes lighter (lower density) than its surroundings, so it is pushed upward by buoyancy. This buoyancy is added to the equation as a source term. Other examples: chemical reaction heat from a gas stove flame, Lorentz force acting on molten metal in a factory's electromagnetic pump... These are all actions that "inject energy or force into the fluid from the outside," expressed by the source term. What happens if you forget the source term? In natural convection analysis, if you forget to include buoyancy, the fluid doesn't move at all—a physically impossible result where warm air doesn't rise in a heated room in winter.
Assumptions and Applicability Limits
- Continuum assumption: Valid for Knudsen number Kn < 0.01 (molecular mean free path ≪ characteristic length)
- Newtonian fluid assumption: Shear stress and strain rate have a linear relationship (non-Newtonian fluids require viscosity models)
- Incompressible assumption (for Ma < 0.3): Treat density as constant. For Mach number 0.3 and above, consider compressibility effects
- Boussinesq approximation (Natural convection): Density variation considered only in the buoyancy term, using constant density in other terms
- Non-applicable cases: Rarefied gas (Kn > 0.1), supersonic/hypersonic flow (shock capturing required), free surface flow (VOF/Level Set, etc. required)
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Velocity $u$ | m/s | When converting from volumetric flow rate for inlet conditions, pay attention to cross-sectional area units |
| Pressure $p$ | Pa | Distinguish between gauge pressure and absolute pressure. Use absolute pressure for compressible analysis |
| Density $\rho$ | kg/m³ | Air: approx. 1.225 kg/m³@20°C, Water: approx. 998 kg/m³@20°C |
| Viscosity coefficient $\mu$ | Pa·s | Be careful not to confuse with kinematic viscosity coefficient $\nu = \mu/\rho$ [m²/s] |
| Reynolds number $Re$ | Dimensionless | $Re = \rho u L / \mu$. Judgment indicator for laminar/turbulent transition |
| CFL number | Dimensionless | $CFL = u \Delta t / \Delta x$. Directly related to time step stability |
Numerical Methods and Implementation
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 Model—The 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.
Upwind Differencing (Upwind)
First-order upwind: Large numerical diffusion but stable. Second-order upwind: Improved accuracy but risk of oscillations. Essential for high Reynolds number flows.
Central Differencing
Second-order accurate, but numerical oscillations occur for Pe number > 2. Suitable for low Reynolds number, diffusion-dominated flows.
TVD Schemes (MUSCL, QUICK, etc.)
Suppress numerical oscillations while maintaining high accuracy using limiter functions. Effective for capturing shock waves and steep gradients.
Finite Volume Method vs Finite Element Method
FVM: Naturally satisfies conservation laws. Mainstream in CFD. FEM: Advantageous for complex shapes and multiphysics. Mesh-free methods like SPH are also developing.
CFL Condition (Courant Number)
Explicit method: CFL ≤ 1 is the stability condition. Implicit method: Stable even for CFL > 1, but affects accuracy and iteration count. LES: CFL ≈ 1 recommended. Physical meaning: Information should not travel more than one cell per time step.
Residual Monitoring
Convergence is judged when the residuals for each of the continuity equation, momentum, and energy drop by 3-4 orders of magnitude. The mass conservation residual is particularly important.
Relaxation Factors
Pressure: 0.2–0.3, Velocity: 0.5–0.7 are typical initial values. If diverging, lower the relaxation factor. After convergence, increase to accelerate.
Internal Iterations for Unsteady Calculations
Iterate within each time step until a steady solution converges. Internal iteration count: 5–20 times is a guideline. If residuals fluctuate between time steps, review the time step size.
Analogy for the SIMPLE Method
The SIMPLE method is an "alternating adjustment" technique. First, velocity is tentatively determined (predictor step), then pressure is corrected so that mass conservation is satisfied with that velocity (corrector step), and velocity is revised using the corrected pressure—this catchball is repeated to approach the correct solution. It resembles two people leveling a shelf: one adjusts the height, the other balances it, and they repeat this alternately.
Analogy for Upwind Differencing
Upwind differencing is a method that "stands in the river flow and prioritizes upstream information." A person in the river cannot tell where the water comes from by looking downstream—it's a discretization method reflecting the physics that upstream information determines downstream. Although it's first-order accurate, it is highly stable because it correctly captures the flow direction.
Practical Guide
Practical Guide
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