Multiphase Chapter 5: Discrete Phase Flows
These notes present a unified treatment of the Discrete Phase Model (DPM) for gas–particle and gas–droplet flows, including particle dynamics, turbulence interaction, heat and mass transfer, sprays, wall interaction, and dense regimes. The framework is extended to multiscale atomization problems through the coupling of Volume of Fluid (VOF) with DPM, enabling predictive transition from resolved liquid structures to Lagrangian particles. Emphasis is placed on physical interpretation, coupling regimes, and practical CFD modeling decisions
Context & Motivation
Many engineering flows contain discrete objects embedded in a carrier fluid:
Solid particles (dust, coal, sand, catalyst)
Liquid droplets (sprays, fuel injection, aerosols)
Evaporating or reacting particles
When the dispersed phase volume fraction is low (typically < 10%), tracking each particle individually is more accurate and efficient than Eulerian continuum approaches. This motivates the Eulerian–Lagrangian framework of the Discrete Phase Model (DPM).
However, real systems often span multiple length scales:
Near injectors or nozzles, liquid interfaces must be resolved (VOF)
Far downstream, droplets are dilute and spherical (DPM)
The VOF–to–DPM transition model bridges this gap, allowing physically consistent atomization modeling without prohibitive mesh cost.
Main Concepts
3.1 Eulerian–Lagrangian Philosophy
Continuous phase (gas or liquid): solved in Eulerian form
Discrete phase (particles/droplets): tracked in Lagrangian form
Coupling occurs via source terms for mass, momentum, and energy
Particles are treated as finite objects, not volume fractions. This enables accurate prediction of trajectories, residence times, wall impact, evaporation, and breakup.
3.2 Validity Range of DPM
Key assumptions:
Dispersed phase volume fraction < 10%
Particle–particle interactions negligible (unless DEM/DDPM used)
Particle diameter much smaller than flow scales
Important distinction:
Mass loading can be very high (>100%)
Volume loading must remain low
3.3 Coupling Regimes
DPM supports different coupling levels depending on particle concentration and inertia:
One-way coupling
Fluid affects particles
No feedback from particles to flow
Two-way coupling
Momentum, heat, mass exchanged both ways
Four-way coupling
Particle–particle collisions important
Requires DEM or dense modeling
This classification aligns with turbulence modulation maps proposed by Elghobashi.
Modeling Framework / Formulations
4.1 Particle Equation of Motion (Physical Meaning)
Each particle follows Newton’s second law:
Acceleration arises from imbalance of forces
Motion is evaluated through the instantaneous Eulerian flow field
Particle properties evolve as it crosses cells
Key forces include:
Drag (dominant in most flows)
Gravity and buoyancy
Added mass
Lift (shear-induced)
Pressure gradient effects
Physically, particles act as inertial filters of the flow: they respond only to flow structures with time scales comparable to their relaxation time.
4.2 Particle Relaxation Time & Stokes Number
The particle relaxation time quantifies how fast a particle adjusts to fluid velocity.
The Stokes number compares particle inertia to fluid time scales:
St ≪ 1 → particles follow flow closely (tracers)
St ≫ 1 → particles decouple and cross streamlines
This explains why:
Small aerosols follow turbulence
Large droplets impact walls and separate from flow
Turbulence–Particle Interaction
5.1 Turbulence Modulation
Particles modify turbulence through:
Displacement of fluid (no eddies inside particles)
Wake generation behind large particles
Additional dissipation due to slip velocity
Rules of thumb:
Particles smaller than Kolmogorov scale → weak interaction
Larger particles → turbulence attenuation or enhancement
5.2 Turbulent Dispersion Models
Since Eulerian turbulence models only provide mean velocity, stochastic models are required:
Discrete Random Walk (DRW)
Particles sample turbulent eddies randomly
Requires many stochastic tries
Accurate but noisy source terms
Particle Cloud Model
Averages turbulence over particle clouds
Produces smoother coupling
Each diameter class tracked separately
Heat and Mass Transfer
6.1 Particle Thermal Evolution
Particles can:
Heat up or cool down
Evaporate or boil
Undergo devolatilization or surface reactions
Heat transfer depends on relative velocity and particle Reynolds number.
6.2 Particle Types
DPM supports:
Massless particles (residence time studies)
Inert particles
Droplets (evaporation/boiling)
Multicomponent droplets
Combusting particles
Mass transfer alters particle size, density, and momentum coupling continuously.
Sprays and Droplet Breakup
7.1 Spray Physics
Sprays involve:
Primary breakup (near injector)
Secondary breakup (downstream)
Evaporation and possible combustion
DPM handles secondary breakup, assuming droplets are already formed.
7.2 Secondary Breakup Models
Common models:
TAB: droplet behaves like spring–mass system
Wave: surface instabilities strip droplets
KH–RT: shear-driven breakup
Stochastic models
Choice depends on Weber number, injection velocity, and regime.
Particle–Wall Interaction
When particles hit boundaries, outcomes include:
Escape (leave domain)
Trap (deposit on wall)
Reflect (bounce with restitution)
Wall jet (high-energy impact)
Wall film formation
Droplet impact regimes depend on impact energy and wall temperature: stick, rebound, spread, splash.
Dense Regimes and Extensions
9.1 Dense DPM (DDPM)
DDPM introduces volume blockage effects while still tracking particles Lagrangianly. Useful for moderately dense flows where collisions are not dominant.
9.2 DEM Coupling
For fully dense granular flows:
Particle–particle collisions dominate
DEM resolves contact forces explicitly
Much higher computational cost
Coupling VOF with DPM
10.1 Motivation
VOF is required to resolve:
Primary breakup
Liquid sheets and ligaments
Near-nozzle atomization
But VOF becomes impractical once structures are smaller than the mesh. DPM is ideal downstream.
10.2 Transition Concept
Large interface structures → VOF
Small, spherical liquid lumps → DPM
Transition region bridges both frameworks
Liquid “lumps” are evaluated for conversion based on:
Size
Volume fraction
Sphericity
10.3 Conversion Criteria
A lump is converted if:
Its volume corresponds to a valid equivalent sphere diameter
Its shape is sufficiently spherical
It lies in a region where interface is under-resolved
Mass, momentum, and energy are conserved during conversion.
10.4 Parcel Splitting and Stability
Large lumps may be split into multiple parcels:
Prevents excessive source terms
Improves numerical stability
Particularly important with evaporation/boiling
Numerical and Solver Considerations
DPM can be steady or transient
Particle tracking is explicit in time
Source term stiffness increases with two-way coupling
Excessive parcel mass leads to instability
Turbulence dispersion requires adequate stochastic sampling
VOF–DPM coupling benefits from adaptive mesh refinement near atomization regions.
Physical Interpretation and Engineering Intuition
Particles respond only to flow structures they “have time to feel”
Stokes number controls deviation from streamlines
Drag dominates unless particles are very small or very heavy
Turbulence spreads particles laterally
Wall interaction often governs deposition efficiency
VOF–DPM coupling enables predictive sprays, not correlation-based ones
Applications
Spray combustion (engines, burners)
Cyclone separators
Pneumatic conveying
Spray drying
Aerosol transport
Fuel injection systems
Icing and wall wetting
Limitations & Assumptions
Low volume fraction required
Empirical drag and breakup models
Turbulence closure limits accuracy
DEM required for dense collisions
VOF–DPM transition sensitive to mesh and thresholds
Study Priorities
If time is limited, the most important concepts to look into:
Eulerian–Lagrangian concept
Particle relaxation time & Stokes number
One- vs two-way coupling
Turbulent dispersion models
Spray breakup logic
VOF–to–DPM transition philosophy
Key Takeaways
DPM tracks individual particles with high physical fidelity
Coupling strength depends on particle inertia and loading
Turbulence strongly affects dispersion and deposition
Wall interaction is often dominant in real systems
VOF–DPM coupling enables multiscale atomization modeling
Correct regime identification is more important than model complexity