Multiphase Chapter 4: Eulerian Flows
These notes provide a unified conceptual and computational foundation for engineering Eulerian multiphase flows. They integrate gas–liquid, gas–solid (granular), thin wall-film, and population balance modeling into a single coherent framework. The focus is on physical intuition, conservation principles, interfacial forces, turbulence coupling, mass/heat transfer, granular rheology, film transport physics, and numerical best practices. The goal is to understand how dispersed structures, bubbles, droplets, and particles, interact with carrier phases and evolve through breakup, coalescence, collisions, friction, and film processes.
Context & Motivation
Eulerian methods treat each phase as a continuum with its own conservation equations and its own velocity field. This makes Eulerian modeling the natural choice for:
High volume fraction dispersed phases
Strong two-way or four-way coupling (e.g., dense solids, bubbly columns)
Multiphase turbulence with interfacial exchange
Systems where droplet or bubble size evolves (PBM coupling)
Film processes involving spreading, evaporation, and re-atomization
Unlike Lagrangian particle tracking, Eulerian methods scale well for:
Dense gas–solid flows (fluidized beds)
Bubble columns or airlift reactors
Wall-film dominated systems (icing, combustor sprays)
Multiscale dispersed systems needing PSD (Power Spectral Density) prediction via PBM (Population Balance Modeling)
The shared challenge: interfacial physics is complex and often empirical; accuracy depends heavily on choosing appropriate drag, lift, turbulent dispersion, breakup, coalescence, and granular stress closures.
Main Concepts
3.1 Volume Fractions and Effective Density
Each phase occupies a fractional volume; all fractions sum to one.
Eulerian equations scale physical densities by volume fraction, yielding effective densities that represent the mass of each phase in a control volume. This naturally embeds local holdup, void fraction, and mixture loading.
3.2 Phase-specific Conservation Equations
For each phase qqq:
Mass: includes phase change sources.
Momentum: includes drag, lift, wall lubrication, virtual mass, turbulent dispersion, interphase momentum transfer coefficients.
Energy: includes interphase heat transfer and enthalpy transfer during phase change.
3.3 Interphase Forces
Forces govern relative motion and ultimately regime behavior:
Drag dominates slip and controls bubble/particle acceleration.
Lift arises from velocity gradients; relevant in shear layers.
Wall lubrication pushes bubbles away from walls, preventing unphysical accumulation.
Virtual mass captures added inertia when an accelerating dispersed phase pushes surrounding fluid.
Turbulent dispersion represents turbulent-induced mixing of phases.
These forces are configurable per phase pair. Multiple empirical drag and lift models exist (e.g., Tomiyama, Grace, Schiller–Naumann).
3.4 Dispersed Phase Diameter Models
Dispersed phases require a representative diameter:
Constant (known or assumed)
User-defined correlation
Interfacial Area Concentration (IAC) models (Sauter diameter approximation)
Population Balance Model (PBM) for dynamic PSDs (most comprehensive)
3.5 Multiphase Turbulence
Only k–ε and k–ω supported in Eulerian systems; turbulence can transfer between phases through additional interfacial source terms or via turbulent dispersion. High gas fractions or rapid bubbles modify turbulence production significantly.
Modeling Framework / Formulations
4.1 Eulerian Flow Equations
Mass conservation
Tracks phase accumulation, convection, and mass exchange during boiling/condensation. Phase-change terms differentiate donor-phase and receiver-phase velocities.
Momentum conservation
Physically, this expresses:
Pressure forces (shared across phases)
Viscous stresses (per phase)
Gravity
Interphase momentum drag
Lift, wall lubrication, virtual mass, turbulent dispersion
Momentum carried with phase-change mass transfer
The interphase momentum coefficient K_{pq} is central: it lumps drag correlations, interfacial area, and particle relaxation time. Most closures assume symmetric exchange: K_{pq} = K_{qp}.
Energy conservation
Tracks enthalpy transport, pressure work, viscous dissipation, heat conduction, and interphase heat exchange. Essential for boiling, condensation, evaporating films, and reacting solids.
Interfacial Forces and Phase Interaction Models
5.1 Drag
Drag is the dominant term in nearly all multiphase simulations. It controls slip velocity, residence time, bubble rise behavior, and stability of regimes. Available correlations include:
Schiller–Naumann (simple, widely used)
Morsi–Alexander (polynomial fit across Re ranges)
Grace et al. and Tomiyama (non-spherical bubble behavior)
Ishii model (for broader flow regimes)
For particles, additional models include Gidaspow, Wen–Yu, Syamlal–O’Brien - all relevant when solids volume fraction is high.
5.2 Lift
Typically small for spherical bubbles/particles unless strong shear exists. Tomiyama’s model captures sign reversal with bubble size, a key phenomenon in churn-turbulent bubble columns.
5.3 Wall Lubrication
Prevents bubbles hugging walls in vertical columns. Several forms exist (Antal, Tomiyama, Frank). Important for slug and churn regimes.
5.4 Virtual Mass
Captures added inertia due to acceleration of displaced fluid. Important when density ratios are large (e.g., bubbles in liquid).
5.5 Turbulent Dispersion
Represents spread of dispersed phases by turbulence. Models include Lopez de Bertodano, Simonin–Viollet, and Burns. Turbulent dispersion is crucial for homogenizing volume fraction fields.
Gas–Liquid Flows
6.1 Typical Applications & Regimes
Bubble columns, absorption units, distillation, boiling, rain/hail, sprays.
Regimes include bubbly, slug, churn-turbulent, annular. Maps based on gas and liquid flux help identify expected structure. Bubbles < 6 mm behave nearly uniformly; larger bubbles (>15 mm) in churn regime rise faster and deform.
6.2 Relevant Physics
Key phenomena to represent:
Buoyancy-driven motion
Coalescence/breakup (via PBM or empirical diameter models)
Surface tension (affects small bubbles)
Turbulence modification
Mass transfer (e.g., absorption, boiling, evaporation/condensation)
Interfacial area estimation
6.3 Multiphase Turbulence
Turbulence can be:
Phase-specific (each solves its own k–ε)
Coupled through interfacial turbulence transfer either in drag or dedicated source terms
Bubble-induced turbulence can dominate in high-gas-flux flows.
6.4 Boiling (RPI Model)
The wall boiling (RPI) model decomposes heat flux into:
Evaporation
Quenching
Convective components
Requires local wall superheat, departure diameter, nucleation site density. This is essential in subcooled boiling and nuclear thermal–hydraulics.
6.5 Numerical Practices
Recommendations include:
Use second-order schemes for volume fraction transport
Ensure high-quality mesh near inlets
Use implicit body force for buoyancy stability
Initialize volume fractions close to expected regime (reduces instability)
Limit under-relaxation for interphase forces in transient churn flows
Gas–Solid / Granular Flows
7.1 Why Granular?
Classical Eulerian–Eulerian fails in dense particulate flows because it ignores particle–particle collisions. The Eulerian–Granular Model (EGM) uses KTGF to account for collisional and frictional stresses.
7.2 Flow Regimes
Solids volume fraction determines coupling:
Dilute: one-way coupling
Moderate: two-way coupling
Dense: four-way coupling (fluid ↔ particles + particle ↔ particle)
7.3 Kinetic Theory of Granular Flow (KTGF)
KTGF introduces granular temperature, a measure of fluctuating kinetic energy of particles. Collisions dissipate energy, so granular temperature continuously evolves.
Two mechanisms:
Kinetic transport: migration during free flight
Collisional transport: momentum exchange during collisions
7.4 Constitutive Relations
KTGF provides expressions for:
Solids pressure (from particle agitation)
Solids shear viscosity
Solids bulk viscosity
Dissipation rate (due to inelastic collisions)
Frictional stresses dominate near the packing limit; inertial flows dominate at moderate volume fractions; quasi-static flows dominate near walls and hopper corners.
7.5 Applications
Fluidized beds, risers, pneumatic conveying, hopper discharge, mixing. Eulerian–granular captures bubbling, channeling, and cluster formation in fluidized systems.
Eulerian Wall-Film Model
8.1 Purpose
Captures thin-film behavior where thickness is much smaller than characteristic flow dimensions. Occurs in icing, combustor liners, defogging, engine cooling, coating, and annular pipe flow.
8.2 Mass, Momentum, and Energy
Film equations are depth-averaged:
Mass: tracks film height growth due to impingement, evaporation, stripping, or splashing
Momentum: includes pressure gradient along the surface, gravity spreading, interfacial shear, and film viscosity
Energy: accounts for conduction to wall and gas, impingement heating, and latent heat of vaporization/condensation
Assumptions:
Film velocity is parallel to wall
Parabolic velocity profile
Bilinear temperature profile through film depth
8.3 Interactions with Other Phases
Film can receive droplets from DPM, exchange mass with gas, evaporate/condense, and shed droplets via re-atomization. This links wall-film to both Lagrangian and Eulerian core flow.
8.4 Engineering Relevance
Allows modeling:
Runback water on wings
Icing film evolution
Combustor wall wetting
Cooling-oil films
Coating uniformity
Population Balance Models (PBM)
9.1 Importance
PBM is essential when droplet/bubble/particle sizes evolve via breakup, coalescence, nucleation, or growth. Provides dynamic PSDs that strongly affect interfacial area, mass transfer, and reaction rates.
9.2 Coordinates
PBM solves for number density as a function of:
External coordinates: spatial location
Internal coordinates: particle size, composition, surface area, etc.
9.3 Governing Structure
The PBM contains:
Transient term
Convective transport in physical space
Growth in internal space
Birth and death via breakup & aggregation
Breakup produces smaller daughter particles; aggregation produces larger clusters.
9.4 Coupling with Eulerian Fields
PBM interacts with:
Interfacial area concentration
Drag (via diameter)
Mass and heat transfer coefficients
Turbulence (in breakup kernels)
Fluent supports multiple numerical approaches (method of moments, discretized size classes).
Physical Interpretation & Engineering Intuition
Drag dominates regime behavior. Large slip ⇒ dispersed regime; small slip ⇒ more homogeneous behavior.
Granular temperature indicates collisional agitation. High granular temperature → intense mixing; low → frictional, near-static behavior.
Turbulence redistributes phases. Bubble-induced turbulence enhances mixing; particles often damp turbulence.
Film behavior is gravity/shear/tension balanced. Thin films accelerate quickly under shear and spread under gravity; thick films retain inertia.
PBM enables predictive multiphase modeling. Interfacial area and PSD determine mass transfer, heat transfer, and reaction rates.
Applications
Bubble columns (mass transfer, chemical reactors)
Fluidized beds (combustion, gasification)
Nuclear thermal–hydraulics (boiling, condensation)
Aerospace icing (runback films, freezing)
Combustion systems (spray-wall interaction)
Process engineering (granulation, crystallization, emulsification)
Limitations & Assumptions
Continuum assumption breaks down at very low dispersed volume fractions.
Empirical closures can limit fidelity (drag, breakup, lift).
Turbulence modeling is limited to k–ε/k–ω.
Wall-film assumes parabolic profile and no normal velocity.
PBM accuracy depends heavily on breakup/aggregation kernels.
Study Priorities
If time is limited, the most important concepts to look into:
Eulerian conservation principles (mass, momentum, energy)
Key interfacial force models (drag first, then others)
Gas–liquid turbulence and regime behavior
KTGF granular constitutive relations
Wall-film mass/momentum/energy intuition
PBM birth/death and growth concepts
Key Takeaways
Eulerian multiphase modeling treats each phase as an interpenetrating continuum with its own velocity field.
Interphase momentum closure (especially drag) is central to accuracy.
KTGF extends Eulerian modeling to dense solids through granular temperature and rheology.
Wall-film modeling captures thin-film transport critical in icing, combustion, and coating.
PBM provides dynamic PSDs, enabling realistic breakup/coalescence predictions.
Turbulence, mass transfer, and interfacial area are tightly coupled across all regimes.