Lagrangian粒子追跡（DPM）
Theory and Physics
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.
Physical Meaning of Each Term
- Temporal Term $\partial(\rho\phi)/\partial t$: Think of the moment you turn on a faucet. At first, water comes out erratically and unstably, but after a while, the flow becomes steady, right? This term describes that "period of change." The pulsation of blood flow from a heartbeat, or the flow fluctuation each time an engine valve opens/closes—all are unsteady phenomena. So what is steady-state analysis? It's looking only at "after sufficient time has passed and the flow has settled down"—meaning setting this term to zero. This significantly reduces computational cost, so solving first in steady-state is a basic CFD strategy.
- Convection Term $\nabla \cdot (\rho \mathbf{u} \phi)$: What happens if you drop a leaf into a river? It gets carried downstream by the flow, right? This is "convection"—the effect where fluid motion transports things. Warm air from a heater reaching the far side of a room is also because the "carrier," air, transports heat via convection. Here's the interesting part—this term contains "velocity × velocity," making it nonlinear. That is, as flow speed increases, this term rapidly strengthens, making control difficult. This is the root cause of turbulence. A common misconception: "Convection and conduction are similar" → They are completely different! Convection is carried by flow, conduction is transmitted by molecules. There is an order of magnitude difference in efficiency.
- Diffusion Term $\nabla \cdot (\Gamma \nabla \phi)$: Have you ever put milk in coffee and left it? Even without stirring, after a while they naturally mix. That's molecular diffusion. Now a question—honey and water, which flows more easily? Obviously water, right? Honey has high viscosity ($\mu$), so it flows poorly. When viscosity is high, the diffusion term becomes strong, and the fluid moves in a "thick" manner. In low Reynolds number flow (slow, viscous), diffusion is dominant. Conversely, in high Re number flow, convection overwhelms and diffusion plays a supporting role.
- Pressure Term $-\nabla p$: When you push a syringe plunger, liquid shoots out forcefully from the needle tip, right? Why? Because the plunger side is high pressure, the needle tip is low pressure—this pressure difference becomes the force pushing the fluid. Dam discharge works on the same principle. On a weather map, where isobars are tightly packed? That's right, strong winds blow. "Where there is a pressure difference, flow is generated"—this is the physical meaning of the pressure term in the Navier-Stokes equations. A point of confusion here: "Pressure" in CFD is often gauge pressure, not absolute pressure. When switching to compressible analysis, if results become strange, it might be due to mixing up absolute/gauge pressure.
- Source Term $S_\phi$: Heated air rises—why? Because it becomes lighter (lower density) than its surroundings, so buoyancy pushes it up. This buoyancy is added to the equation as a source term. Other examples: chemical reaction heat from a gas stove flame, Lorentz force acting on molten metal in a factory electromagnetic pump... These are all actions that "inject energy or force into the fluid from outside," expressed by the source term. What happens if you forget the source term? In natural convection analysis, forgetting buoyancy means the fluid doesn't move at all—a physically impossible result where warm air doesn't rise in a room with the heater on in winter.
Assumptions and Applicability Limits
- Continuum Assumption: Valid for Knudsen number Kn < 0.01 (molecular mean free path ≪ characteristic length)
- Newtonian Fluid Assumption: Linear relationship between shear stress and strain rate (viscosity model needed for non-Newtonian fluids)
- Incompressibility Assumption (for Ma < 0.3): Treat density as constant. Consider compressibility effects for Mach number ≥ 0.3
- Boussinesq Approximation (Natural Convection): Consider density change only in the buoyancy term, using constant density in other terms
- Non-applicable Cases: Rarefied gas (Kn > 0.1), supersonic/hypersonic flow (requires shock capturing), free surface flow (requires VOF/Level Set, etc.)
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Velocity $u$ | m/s | When converting from volumetric flow rate for inlet conditions, pay attention to cross-sectional area units |
| Pressure $p$ | Pa | Distinguish between gauge and absolute pressure. Use absolute pressure for compressible analysis |
| Density $\rho$ | kg/m³ | Air: ~1.225 kg/m³ @20°C, Water: ~998 kg/m³ @20°C |
| Viscosity Coefficient $\mu$ | Pa·s | Be careful not to confuse with kinematic viscosity coefficient $\nu = \mu/\rho$ [m²/s] |
| Reynolds Number $Re$ | Dimensionless | $Re = \rho u L / \mu$. Criterion for laminar/turbulent transition |
| CFL Number | Dimensionless | $CFL = u \Delta t / \Delta x$. Directly related to time step stability |
Numerical Methods and Implementation
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.
Upwind Differencing (Upwind)
1st-order upwind: Large numerical diffusion but stable. 2nd-order upwind: Improved accuracy but risk of oscillations. Essential for high Reynolds number flow.
Central Differencing (Central Differencing)
2nd-order accuracy, but numerical oscillations occur for Pe number > 2. Suitable for low Reynolds number diffusion-dominated flow.
TVD Scheme (MUSCL, QUICK, etc.)
Maintains high accuracy while suppressing numerical oscillations via limiter functions. Effective for capturing shock waves or steep gradients.
Finite Volume Method vs Finite Element Method
FVM: Naturally satisfies conservation laws. Mainstream in CFD. FEM: Advantageous for complex shapes and multiphysics. Mesh-free methods like SPH are also developing.
CFL Condition (Courant Number)
Explicit method: CFL ≤ 1 is the stability condition. Implicit method: Stable even for CFL > 1, but affects accuracy and iteration count. LES: CFL ≈ 1 recommended. Physical meaning: Information should not travel more than one cell per time step.
Residual Monitoring
Convergence is judged when residuals for the continuity equation, momentum, and energy decrease by 3-4 orders of magnitude. The mass conservation residual is particularly important.
Relaxation Factor
Typical initial values: Pressure: 0.2–0.3, Velocity: 0.5–0.7. Reduce the factor if diverging. Increase after convergence to accelerate.
Internal Iterations for Unsteady Calculations
Iterate within each time step until a steady solution converges. Internal iteration count: 5–20 times as a guideline. If residuals fluctuate between time steps, review the time step size.
Analogy for SIMPLE Method
The SIMPLE method is an "alternating adjustment" technique. First, velocity is tentatively determined (predictor step), then pressure is corrected so that mass conservation is satisfied with that velocity (corrector step), and velocity is revised using the corrected pressure—this back-and-forth is repeated to approach the correct solution. It resembles two people leveling a shelf: one adjusts the height, the other balances it, and they repeat this alternately.
Analogy for Upwind Differencing
Upwind differencing is a method that "stands in the river flow and prioritizes upstream information." A person in the river cannot tell where the water comes from by looking downstream—this discretization method reflects the physics that upstream information determines downstream. Although first-order in accuracy, it is highly stable because it correctly captures flow direction.
Practical Guide
Practical Guide
Please tell me the procedure for DPM analysis.
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