Lagrangian Particle Tracking (DPM)
Lagrangian Particle Tracking (DPM): Theoretical Foundations
Overview
Professor, what is Lagrangian particle tracking?
The Lagrangian particle tracking method (DPM: Discrete Phase Model) is a technique that individually tracks the trajectories of discrete particles, droplets, or bubbles. It is used in systems where the volume fraction of the dispersed phase is low (typically below 10%), such as spray droplets, powder transport, aerosol dispersion, cyclone separation, and particle capture in exhaust gas.
How do we decide when to use it versus the Euler-Euler method?
The basic rule is DPM for low dispersed-phase volume fractions and Euler-Euler for high fractions. DPM has the advantage of naturally tracking particle size distribution and individual particle history (temperature change, evaporation, reaction).
Governing Equations
Please tell me the equation of motion for particles.
The motion of each particle follows Newton's second law.
The most important force is the drag force $\mathbf{F}_D$, with the Stokes/Schiller-Naumann correlation being standard.
What other forces exist besides drag?
| Force | Equation | Important Situations |
|---|---|---|
| Gravity/Buoyancy | $m_p(1 - \rho_g/\rho_p)\mathbf{g}$ | Settling/Rising |
| Saffman Lift | $C_{LS} \rho_g \nu_g^{1/2} d_{ij} (\mathbf{u}_g - \mathbf{v}_p)$ | Lateral movement in shear flow |
| Pressure Gradient Force | $m_p \frac{\rho_g}{\rho_p} \frac{D\mathbf{u}_g}{Dt}$ | Systems where density ratio is near 1 |
| Virtual Mass Force | $C_{VM} m_p \frac{\rho_g}{\rho_p} \frac{d}{dt}(\mathbf{u}_g - \mathbf{v}_p)$ | Bubble tracking, rapid acceleration |
| Thermophoretic Force | $-\frac{6\pi d_p \mu_g^2 C_s}{\rho_g} \frac{\nabla T}{T}$ | Fine particles near high-temperature walls |
| Brownian Force | Stochastic external force | Submicron particles |
In solid particle-air systems ($\rho_p / \rho_g \gg 1$), drag and gravity are dominant, and other forces can often be omitted. Additional forces become important for bubble tracking or fine particles.
Basset Force—A Memory Effect Ignored for Over 100 Years
When particles are subjected to unsteady flow, the "Basset force (history force)" arises, where past acceleration history affects the current fluid drag force. Derived by Basset in 1888, this force is computationally expensive due to its integral form and has long been ignored in practical Lagrangian particle tracking. However, in particle transport involving rapid velocity changes (valve opening/closing, shock wave passage), the Basset force can reach 10-30% of the inertial force, and ignoring it can cause large errors in particle concentration distribution. Recent GPU computing has made the integral calculation of the Basset force practical, and it is now actively incorporated in analyses such as cleanroom contamination particle analysis for semiconductor manufacturing equipment.
Computational Methods for Lagrangian Particle Tracking (DPM)
Details of Numerical Methods
Please tell me the numerical points of DPM.
Explicit Euler method or analytical integration is used for particle trajectory integration. Particle position updates are performed via sub-stepping within the gas-phase CFD time step.
1-way vs 2-way vs 4-way coupling
The coupling level changes according to the degree of interaction between particles and the gas phase.
| Coupling | Condition | Particle→Gas Phase | Particle-Particle Collision |
|---|---|---|---|
| 1-way | $\alpha_p < 10^{-6}$ | None | None |
| 2-way | $10^{-6} < \alpha_p < 10^{-3}$ | Momentum/Heat/Mass Source | None |
| 4-way | $\alpha_p > 10^{-3}$ | Source + Particle-Particle Collision | Yes |
How is the effect on the gas phase calculated in 2-way coupling?
For CFD cells that particles pass through, momentum sources (reaction force of drag), energy sources (heat exchange), and mass sources (evaporation) are accumulated. This is called the Particle Source in Cell (PSI-Cell) method.
Turbulent Dispersion Model
How do particles diffuse in turbulence?
The DRW (Discrete Random Walk) model is standard. Turbulent velocity fluctuations are generated stochastically from a Gaussian distribution and added to the particle.
$\zeta$ is a standard normal random number, $k$ is turbulent kinetic energy. The persistence time of the fluctuating velocity is controlled by the turbulent time scale $\tau_e = C_L k/\varepsilon$.
Implementation by Tool
| Tool | DPM Model Name | Turbulent Dispersion | Wall Interaction |
|---|---|---|---|
| Ansys Fluent | Discrete Phase Model | DRW, CRW | Reflect, Trap, Escape, Wall Film |
| STAR-CCM+ | Lagrangian Multiphase | Stochastic | Rich wall models |
| OpenFOAM | icoUncoupledKinematicParcelFoam, etc. | Supported | Customizable |
| Ansys CFX | Particle Transport | Stochastic | Basic models |
Fluent's DPM is the most feature-rich, with integrated sub-models for evaporation, combustion, droplet breakup, and wall interaction.
One-Way vs Four-Way Coupling—Stages of Particle-Fluid Coupling
The "degree of coupling" in Lagrangian particle methods determines the trade-off between computational accuracy and cost. One-Way Coupling is the cheapest approximation where the fluid affects particles but particles do not affect the fluid, valid only for dilute systems with particle volume fraction α_p < 10^-6. Two-Way Coupling includes feedback of particle momentum/energy to the fluid, essential for systems with α_p > 10^-6. Four-Way Coupling further handles particle-particle collisions, important for high-density systems with α_p > 10^-3. In industrial dust collector particle deposition prediction, there have been cases where pressure loss prediction changed by over 50% depending on the presence of two-way coupling.
Lagrangian Particle Tracking (DPM) in Practice
Practical Guide
Please tell me the procedure for DPM analysis.
- Set up the fluid domain — Compute the gas-phase flow field first. For dilute systems (1-way coupling), you can compute this independently.
- Define particle properties — Input diameter, density, material, initial velocity, temperature, and any additional properties.
- Set initial and boundary conditions for particles — Specify particle injection location, velocity, temperature. Decide on wall interaction (bounce, stick, escape).
- Select coupling mode — Choose 1-way, 2-way, or 4-way coupling based on particle volume fraction.
- Set up turbulent dispersion model — Apply DRW with appropriate time scale correlation if particles are in a turbulent flow.
- Choose sub-time step strategy — For 2-way coupling, coordinate particle time steps with CFD iteration.
- Run simulation and monitor convergence — Track total particle mass, momentum, and energy conservation.
- Post-process — Visualize particle trajectories, residence time, concentration distributions, deposition patterns.
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