Eulerian-Lagrangian Model
Also known as: ELM, two-fluid model, multiphase Eulerian-Lagrangian
The Eulerian-Lagrangian Model (ELM) is a framework for simulating multiphase flows by treating the continuous phase (liquid or gas) using Eulerian descriptions (fixed grid) and discrete dispersed phases (particles, droplets, bubbles) using Lagrangian tracking. Developed by Crowe and collaborators in 1977, this approach exploits the strengths of both perspectives: Eulerian methods for the bulk continuous phase and Lagrangian methods for individual dispersed elements. ELM is widely used in industrial applications including spray combustion, pneumatic conveying, and particle-laden flows.
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When to use it
Use ELM for particle-laden flows, sprays, fluidized beds, and pneumatic conveying where particle volume fraction is <30%. ELM is ideal when individual particle tracking provides insight (residence time, wall impact) and continuous phase detail (turbulence, combustion) is important. Avoid ELM when particles are so numerous that grid resolution becomes prohibitive or when particle-particle interactions dominate (>30% volume fraction; use Eulerian multiphase instead). ELM is excellent for engineering design of spray systems, particle separators, and industrial reactors.
Strengths & limitations
- Combines grid efficiency of Eulerian methods with individual tracking of Lagrangian particles
- Naturally includes two-way coupling: particles affect continuous phase momentum and heat
- Leverages existing continuous phase solvers (RANS, LES) without extensive modification
- Provides detailed particle information: trajectories, residence times, temperature, evaporation
- Computationally efficient for dilute to moderately dense flows (volume fraction <30%)
- Particle-particle interactions are difficult to model; typically ignored in dilute flows
- Coupling becomes expensive at high particle concentrations; quadratic scaling in particle count affects compute time
- Drag models and heat transfer correlations are empirical, designed for specific shape/size ranges
- Grid resolution must accommodate continuous phase features; fine grids increase cost
- Non-spherical particles require orientation tracking, dramatically increasing complexity
Frequently asked
What is the difference between one-way and two-way coupling in ELM?
One-way coupling: continuous phase is solved, then particle equations use resulting velocity field; particles do not affect the flow (valid for volume fraction <5%). Two-way coupling: particles are tracked and their momentum/heat transfers back to the continuous phase, modifying the flow field. Two-way coupling is necessary for volume fractions >5% and is computationally more expensive but physically essential.
What drag models are appropriate for different particle Reynolds numbers?
Stokes drag (Fd ∝ d·u) applies for Re_p < 1 (very small particles). Morsi-Alexander correlation covers Re_p ~ 0.1-1000. Haider-Levenspiel extends to Re_p ~ 5000. At very high Re_p (>5000), drag coefficient plateaus near 0.4. Always check particle Reynolds number Re_p = ρ_c |u_rel| d / μ_c and select appropriate correlation. Applying Stokes law at Re_p = 100 introduces 50% error.
How do I model particle-particle collisions and agglomeration?
Simple models: track particle pairs within interaction radius and apply collision force or coagulation kernel. Advanced: use collision detection algorithms and soft-sphere or hard-sphere models. For agglomeration, track mass fraction in each size bin. Full particle-particle simulation with 10^8+ particles is computationally prohibitive; use phenomenological agglomeration rates instead.
When should I switch from ELM to Eulerian multiphase?
ELM is efficient for volume fraction φ < 30%. Above 30%, particle-particle interactions dominate and must be included; Eulerian multiphase with kinetic theory treats all particles as a continuum field. Also switch if particles are so numerous that tracking each one individually is infeasible (>10^7 particles). Eulerian multiphase models the dispersed phase as a continuum with its own conservation equations, trading individual particle detail for computational efficiency.
How do I account for turbulence effects on particle dispersion?
Use a stochastic turbulence model: add random velocity fluctuations sampled from the continuous phase turbulent kinetic energy and dissipation rate. Common approach: Lagrangian turbulent dispersion (LTFD) or stochastic turbulence model (STM). Without it, particles follow deterministic paths and underpredict dispersion compared to experiments. Accuracy depends on continuous phase turbulence model quality (RANS vs LES).
Sources
- Crowe, C., Sommerfeld, M., & Tsuji, Y. (2011). Multiphase Flows with Droplets and Particles (2nd ed.). CRC Press. ISBN: 978-1439840474
- Elghobashi, S. (1994). On predicting particles-laden turbulent flows. Applied Scientific Research, 52(4), 309-329. DOI: 10.1007/BF00936835 ↗
- Sanders, R. S., & Loeffler, A. L. (1998). Modeling the effects of bubble interactions on the viscosity of bubbly flows. International Journal of Multiphase Flow, 24(3), 345-357. link ↗
How to cite this page
ScholarGate. (2026, June 3). Eulerian-Lagrangian Model. ScholarGate. https://scholargate.app/en/fluid-dynamics/eulerian-lagrangian-model
Which method?
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