DPM Coupling -- One-Way vs Two-Way, Source Terms, and Decision Logic

#+CATEGORY: cfd

#+TAGS: DPM coupling one-way two-way source-term volume-fraction mass-loading

#+MATURITY: growing

DPM Coupling :: One-Way vs Two-Way and Source Term Computation

Coupling describes the direction of information flow between the continuous phase (fluid) and the discrete phase (particles). One-way: fluid affects particles but not vice versa. Two-way: particles return source terms to the fluid momentum, mass, and energy equations. The correct choice depends on the volume fraction and mass loading. Getting this wrong wastes compute time (two-way when one-way suffices) or produces incorrect results (one-way when two-way is needed).

Coupling Regimes -- Definitions

Physical Meaning

One-way coupling:

The fluid is solved first (RANS on the Eulerian mesh). The resulting velocity, pressure, temperature, and turbulence fields are read by each particle. The particle integrates its trajectory. But the fluid solution does not know the particles exist. This is valid when the particles occupy a negligible volume and exert negligible forces on the fluid.

Two-way coupling:

After tracking all particles, Fluent computes source terms for the fluid equations:

• Momentum source: each particle exerts a drag force on the fluid (Newton's third law -- the reaction to the drag the fluid exerts on the particle).
• Mass source: evaporating or combusting particles release or absorb mass.
• Energy source: heating, cooling, or reacting particles exchange energy with the fluid.
• Turbulence source: particles can generate or suppress turbulence.

These source terms are fed back into the fluid equations, and the fluid is re-solved. The process iterates until both phases converge simultaneously.

Four-way coupling:

In addition to two-way, particles can collide with each other. This requires the Dense Discrete Phase Model (DDPM) or a Discrete Element Method (DEM). Particle-particle collisions matter only when the volume fraction exceeds $\sim 10^{-3}$.

Decision Criterion -- Volume Fraction and Mass Loading

The two key parameters are:

Volume fraction $\alpha_p$:

where $C_m$ is the mass concentration [kg/m^3].

Mass loading ratio $\Phi_m$:

These boundaries are approximate guidelines, not hard limits. The true test is whether the source terms are significant compared to the fluid equation terms.

Volume Fraction Check for Shisha PM2.5

Given:**

• Smoke concentration: $C_m \approx 10$ mg/m^3 (typical shisha exhaust)
• Particle density: $\rho_p = 1200$ kg/m^3
• Air density: $\rho = 1.2$ kg/m^3

Volume fraction:**

Mass loading:**

Both values are well below the one-way thresholds ($\alpha_p < 10^{-6}$, $\Phi_m < 0.1$). The volume fraction is slightly above $10^{-6}$ but the mass loading is far below $0.1$. One-way coupling is the correct choice.

Margin of safety:** Even if the smoke concentration is 100x higher ($C_m = 1$ g/m^3, a dense chamber fill), $\alpha_p = 8.3 \times 10^{-4}$ and $\Phi_m = 0.83$. This is still in the one-way to early-two-way range. For typical shisha operation, one-way is robustly correct.

How Two-Way Coupling Works in Fluent

Activation:

In the DPM model panel: "Interaction" = "Continuous Phase" (or "Update DPM Sources Every Flow Iteration" for tight coupling).

Algorithm:

1. Solve the continuous phase (fluid) equations. 2. Track all particles through the converged flow field. 3. For each particle, compute the source terms:

• Momentum source per cell: sum of drag forces from all particles in that cell.
• Mass source per cell: sum of evaporation/boiling rates.
• Energy source per cell: sum of convective, radiative, and reaction heat.

4. Map source terms from Lagrangian particles to the Eulerian mesh (volume-weighted averaging into each cell). 5. Add source terms to the fluid equations. 6. Re-solve the fluid equations. 7. Repeat steps 2--6 until both phases converge (typically 5--20 DPM iterations per coupling cycle).

Under-relaxation:

The DPM source terms are under-relaxed to prevent divergence:

where $\alpha_{\text{DPM}}$ is the DPM under-relaxation factor (default 0.5). If the coupling diverges, reduce $\alpha_{\text{DPM}}$ (0.1--0.3) or increase the number of coupling iterations.

TUI commands:

• solve/set/dpm-iterations -- number of DPM tracking iterations per coupling cycle
• solve/set/dpm-under-relaxation -- under-relaxation factor for DPM source terms
• define/models/dpm/interaction -- toggle coupling mode

Two-Way Turbulence Coupling

Separate from the momentum/mass/energy coupling, particles can also modify turbulence. The mechanism is:

• Turbulence generation: particle wakes create additional velocity fluctuations.
• Turbulence suppression: particle drag damps fluid velocity fluctuations.

Toggle "Two-way turbulence coupling" in the Physical Models tab. This adds source terms to the $k$ and $\epsilon$ (or $\omega$) equations.

For shisha PM2.5: leave off. The smoke mass loading ($\Phi_m \sim 10^{-2}$) is too low for particles to affect turbulence.

Four-Way Coupling -- DDPM and DEM

Four-way coupling is available through the Dense Discrete Phase Model (DDPM) or DEM collision model. It adds:

• Particle-particle collision detection (hard-sphere or soft-sphere)
• Collision force computation
• Momentum exchange from collisions

This is only needed for:

• Fluidized beds ($\alpha_p > 0.1$)
• Sediment transport with high concentration
• Pneumatic conveying with dense particle streams

For shisha PM2.5: not applicable. The volume fraction is far too low for inter-particle collisions.

See Dense Discrete Phase Model for details.

Source Term Magnitude Estimate

For two-way coupling, the momentum source per unit volume is:

where $F_{\text{drag},p}$ is the drag force on particle $p$ and the sum is over all particles in the cell.

Order of magnitude for shisha PM2.5:**

• Drag on one PM2.5 particle: $F_D \sim 3\pi\mu d_p U \sim 3\pi \times 1.8\times 10^{-5} \times 10^{-6} \times 0.5 \approx 8.5 \times 10^{-11}$ N
• Number density of particles at $C_m = 10$ mg/m^3: $n = C_m / (\rho_p \frac{\pi}{6} d_p^3) \sim 10^{-2} / (1200 \times 5.2\times 10^{-19}) \sim 1.6 \times 10^{13}$ particles/m^3
• Source per unit volume: $S_u \sim n \times F_D \sim 1.6 \times 10^{13} \times 8.5 \times 10^{-11} \sim 1360$ N/m^3

Compare to pressure gradient:**

The fluid momentum equation has a pressure gradient term $\nabla p \sim \rho U^2 / L \sim 1.2 \times 0.5^2 / 0.1 \sim 3$ Pa/m = 3 N/m^3.

So $S_u / |\nabla p| \sim 1360 / 3 \sim 450$.

Wait -- this suggests the source is large! But this estimate is wrong because it uses the peak number density. In reality, the number density is highly non-uniform: high near the injection and nearly zero elsewhere. The volume-averaged source over the full chamber is:

This is small compared to the pressure gradient. Confirmation: one-way coupling is valid.

Coupling Decision Flowchart

Step 1: Compute volume fraction
        alpha_p = C_m / rho_p

Step 2: Compute mass loading
        Phi_m = C_m / rho

Step 3: Decision
        if alpha_p < 1e-6 AND Phi_m < 0.1:
            --> One-way coupling (skip to Step 7)
        elif alpha_p < 0.1 AND Phi_m < 10:
            --> Two-way coupling (go to Step 4)
        else:
            --> Four-way coupling (DDPM/DEM) (go to Step 5)

Step 4: For two-way:
        - Enable "Interaction with Continuous Phase"
        - Set DPM iterations per flow iteration (5--20)
        - Set DPM under-relaxation (default 0.5)
        - If particles modify turbulence: enable two-way turbulence coupling

Step 5: For four-way:
        - Switch to DDPM model (not standard DPM)
        - Enable collision model
        - Set restitution coefficient
        - Ensure mesh resolves particle mean free path

Step 6: Validate
        - Run initial case with chosen coupling
        - Check source term magnitude relative to fluid terms
        - If source terms are negligible: can switch to simpler coupling
        - If source terms dominate: may need denser coupling or DDPM

Step 7: For shisha PM2.5:
        - One-way coupling is correct
        - Do not enable "Interaction with Continuous Phase"
        - Do not enable "Two-way turbulence coupling"
        - Run DPM after fluid convergence

Common Mistakes

Troubleshooting

Summary

1. One-way coupling: fluid affects particles only. Two-way: bidirectional with source terms. Four-way: two-way + collisions (DDPM). 2. The decision is based on volume fraction ($\alpha_p$) and mass loading ratio ($\Phi_m$). 3. For shisha PM2.5: $\alpha_p \sim 8 \times 10^{-6}$, $\Phi_m \sim 8 \times 10^{-3}$. One-way coupling is correct. 4. Two-way coupling adds source terms for momentum, mass, energy, and (optionally) turbulence to the fluid equations. 5. Two-way requires iteration: track particles, compute sources, re-solve fluid, repeat. 6. Under-relaxation of DPM sources (default 0.5) prevents divergence. 7. For dilute flows, one-way coupling is faster, simpler, and correct.

See Also

• DPM Theory -- Governing Equations (parent theory, coupling overview)
• DPM Turbulence Interaction -- DRW (turbulence coupling detail)
• Shisha PM2.5 Case Study (coupling configuration for shisha)
• Dense Discrete Phase Model (four-way coupling, collisions)
• DPM Physical Models -- Fluent Panel Walkthrough
• DPM Setup and Activation in Fluent
• ANSYS Fluent Multiphase Modeling