Multiphase Flow Analysis — Overview
When two or more phases share a flow domain, single-phase CFD breaks down. This guide covers the three dominant modeling strategies — VOF, Euler-Euler, and Euler-Lagrange — and when to use each one.
By NovaSolver Contributors (Anonymous Engineers & AI) | CFD / Fluid Analysis | 日本語版 →
1. What is Multiphase Flow?
I understand single-phase CFD for air or water, but what exactly counts as "multiphase flow"? Is it any mixture of fluids?
Multiphase flow means two or more distinct phases — with different densities, viscosities, and possibly chemistry — coexist and interact in the same flow domain. The phases can be gas-liquid (water waves, bubble columns), liquid-liquid (oil-water separation), or solid-fluid (sand transport, slurry pipelines). The key challenge is that the interface between phases moves, deforms, and possibly breaks up or merges — none of which standard single-fluid Navier-Stokes equations handle directly.
So there must be different CFD methods depending on how the interface behaves? Like whether the bubbles are huge or tiny compared to the mesh?
Exactly right. If you have a sharp, large interface — like a dam break or a ship's bow wave — you track it explicitly with VOF. If you have millions of tiny bubbles smaller than your mesh cells, you treat both phases as interpenetrating continua with Euler-Euler. And if you have discrete particles few enough to track individually, Euler-Lagrange DPM is the right tool. Each method has a specific sweet spot determined by the size and number density of the dispersed phase.
2. Classification of Multiphase Flow Regimes
| Flow Regime | Description | Typical Application | Recommended Method |
|---|---|---|---|
| Free surface / stratified | Sharp, large-scale interface between two immiscible fluids | Ship waves, dam break, sloshing | VOF |
| Bubbly flow | Small bubbles dispersed in continuous liquid; void fraction α < ~30% | Bubble columns, aeration tanks | Euler-Euler |
| Slug / churn flow | Irregular large structures; intermediate regime | Offshore pipeline flow | VOF or coupled |
| Spray / mist | Dispersed droplets in gas; low volume fraction | Fuel injectors, spray drying | DPM (Euler-Lagrange) |
| Particle-laden flow | Solid particles in fluid | Sand erosion, pneumatic conveying | DPM or Euler-Euler |
| Fluidized bed | Dense particle suspension; high void fraction | Chemical reactors, powder drying | Euler-Euler (TFM) |
3. Volume of Fluid (VOF) — Free Surface Tracking
How does VOF actually track the interface? I keep seeing "phase fraction" α mentioned but I don't understand what value it takes at the interface.
Each computational cell holds a phase fraction α between 0 and 1. α = 1 means fully liquid, α = 0 means fully gas, and intermediate values indicate the interface is inside that cell. You evolve α with a transport equation, then compute mixture density and viscosity as weighted averages. The momentum equation is solved once for the mixture — you do not solve separate equations for each phase. That is what makes VOF memory-efficient for large free surface problems like dam breaks or ship wakes.
The VOF transport equation (with interface compression term as used in OpenFOAM's interFoam):
The third term keeps the interface sharp. The compression velocity magnitude is $|\mathbf{u}_c| = C_\alpha |\mathbf{u}|$ with $C_\alpha = 1$ by default.
Mixture fluid properties:
$$\rho = \alpha \rho_1 + (1-\alpha)\rho_2, \quad \mu = \alpha \mu_1 + (1-\alpha)\mu_2$$Surface tension via the Continuum Surface Force (CSF) model:
$$\mathbf{F}_{st} = \sigma \kappa \nabla \alpha, \quad \kappa = -\nabla \cdot \left(\frac{\nabla \alpha}{|\nabla \alpha|}\right)$$VOF Strengths and Limitations
Best for: Large-scale free surfaces, dam break, sloshing, ship hydrodynamics, wave-structure interaction
Works well when: Interface is sharp and the two phases have similar velocities at the interface
Struggles with: Heavily dispersed flows with many small bubbles; large density ratios (>1000:1) can cause pressure-velocity instabilities
OpenFOAM solvers: interFoam (isothermal), interIsoFoam (isoAdvector, sharper interface), interPhaseChangeFoam (cavitation)
4. Euler-Euler (Two-Fluid) Model
For a bubble column reactor with millions of 3 mm bubbles, VOF seems impossible — you'd need a mesh finer than 3 mm everywhere. What does Euler-Euler do differently?
In Euler-Euler you treat both liquid and gas as separate interpenetrating continua — each with its own velocity field and volume fraction α. You never see individual bubbles; instead you have a gas fraction field α_g(x,t) that says what fraction of space is gas at each location. You solve two momentum equations, one per phase, coupled via drag forces, lift, and virtual mass. For a bubble column this gives you the mixing patterns, gas holdup profiles, and mass transfer rates without resolving individual bubble surfaces at all.
The two-fluid continuity equation for phase $k$ (with $\alpha_l + \alpha_g = 1$):
$$\frac{\partial (\alpha_k \rho_k)}{\partial t} + \nabla \cdot (\alpha_k \rho_k \mathbf{u}_k) = \dot{m}_{kj}$$The interfacial momentum transfer is dominated by drag:
$$\mathbf{M}_{drag} = \frac{3}{4} \frac{C_D}{d_b} \alpha_g \rho_l |\mathbf{u}_g - \mathbf{u}_l|(\mathbf{u}_g - \mathbf{u}_l)$$Common drag correlations: Schiller-Naumann for rigid spheres, Grace model for deformable bubbles at higher Reynolds number.
5. Euler-Lagrange (DPM) — Tracking Individual Particles
And for diesel spray injection — the fuel jet breaks into droplets. Is that where DPM comes in? How does it relate to what the gas phase is doing?
DPM tracks each parcel of droplets as Lagrangian particles moving through the Eulerian gas background. You solve Newton's second law for each parcel — drag from surrounding gas, gravity, Saffman lift, and evaporation force. The gas solver sees source terms wherever particles exchange momentum and mass. It works brilliantly for diesel injection: you inject thousands of parcels at the nozzle, each representing many physical droplets, and watch them disperse, break up (Kelvin-Helmholtz, Rayleigh-Taylor models), and evaporate. The key constraint is dilute flow — if the dispersed phase exceeds ~10% volume fraction, particle-particle collisions become important and Euler-Euler is more appropriate.
Lagrangian equation of motion for a single droplet parcel:
$$m_p \frac{d\mathbf{u}_p}{dt} = \frac{1}{2} C_D \rho_f A_p |\mathbf{u}_f - \mathbf{u}_p|(\mathbf{u}_f - \mathbf{u}_p) + m_p \mathbf{g}\left(1 - \frac{\rho_f}{\rho_p}\right) + \mathbf{F}_{other}$$6. Method Selection Summary
| Criterion | VOF | Euler-Euler | DPM (Lagrangian) |
|---|---|---|---|
| Interface character | Sharp, large-scale | Dispersed (unresolved) | Discrete particles/droplets |
| Volume fraction of dispersed phase | Any (large structures) | 0–100% (dense OK) | < ~10% (dilute) |
| Two distinct velocity fields? | No (shared velocity field) | Yes (each phase has own velocity) | Yes (Lagrangian + Eulerian) |
| Coalescence / breakup | Naturally tracked at large scale | Modeled via population balance | Secondary breakup models (KH-RT) |
| Evaporation / mass transfer | Possible (phase change) | Full species transport | D2 law evaporation model |
| OpenFOAM solver | interFoam, interIsoFoam | reactingTwoPhaseEulerFoam | DPMFoam, sprayFoam |
7. Application Examples
| Application | Method | OpenFOAM Solver | Key Physics |
|---|---|---|---|
| Dam break / sloshing | VOF | interFoam | Inertia-dominated free surface, impact loads |
| Ship bow wave / hull resistance | VOF | interFoam + waves2Foam | Free surface deformation, Froude number effects |
| Bubble column bioreactor | Euler-Euler | reactingTwoPhaseEulerFoam | Bubble-liquid drag, gas holdup, mass transfer |
| Diesel spray injection | DPM (Lagrangian) | sprayFoam | Atomization, secondary breakup, evaporation |
| Fluidized bed dryer | Euler-Euler TFM | reactingTwoPhaseEulerFoam | Granular pressure, inter-particle collisions |
| Oil-water pipeline separator | VOF or Euler-Euler | interFoam or two-phase | Stratification, settling, droplet coalescence |