DPM Turbulence Interaction -- Discrete Random Walk, Stochastic Tracking, and DRW Deep Dive

#+CATEGORY: cfd

#+TAGS: DPM DRW turbulence stochastic-tracking turbulent-dispersion particle-eddy

#+MATURITY: growing

DPM Turbulence Interaction :: Discrete Random Walk and Stochastic Tracking

Turbulence scatters particles away from mean streamlines. Without a turbulence model for the discrete phase, every parcel follows the same deterministic path, and the computed concentration field is a collection of delta-function trajectories -- physically wrong for fine particles. Fluent's Discrete Random Walk (DRW) model adds stochastic turbulent dispersion by sampling random fluid-velocity fluctuations at each particle-eddy interaction. This page gives the full mathematics, the algorithm step-by-step, convergence guidance, and the shisha PM2.5 application.

Why Turbulence Matters for Particles

In a laminar flow, a particle's trajectory is uniquely determined by the initial conditions and the force balance. In a turbulent flow, the fluid velocity at the particle location has two components:

where $\bar{\mathbf{u}}$ is the Reynolds-averaged (mean) velocity and $\mathbf{u}'$ is the turbulent fluctuation. RANS solves only for $\bar{\mathbf{u}}$. The fluctuation $\mathbf{u}'$ is unknown but its statistical properties (variance, integral time scale) come from the turbulence model.

Without $\mathbf{u}'$, the particle feels only the mean flow. Its trajectory is smooth and deterministic. In reality, the particle bounces randomly between eddies. For PM2.5 ($\text{Stk} \ll 1$), the particle follows the instantaneous fluid velocity closely, so the random fluctuation translates directly into random trajectory deviations. The result is turbulent dispersion -- particles spread outward from the mean path.

The physical picture:

• A smoke filament entering the shisha chamber does not stay coherent. It spreads and fills the chamber volume.
• The spreading rate depends on the local turbulence intensity, not on molecular diffusion (which is negligible for PM2.5).
• The DRW model captures this spreading by sampling $\mathbf{u}'$ from a Gaussian distribution at each eddy encounter.

The DRW Model -- Full Algorithm

The DRW model works as a sequence of particle-eddy interactions:

Step 0: Initialization

The solver reads the local turbulence quantities from the RANS solution:

• Turbulent kinetic energy $k$ [m^2/s^2]
• Dissipation rate $\epsilon$ [m^2/s^3] (for $k$-$\epsilon$) or specific dissipation $\omega$ (for $k$-$\omega$)

Step 1: Compute RMS velocity fluctuation

The RMS of the isotropic turbulent velocity fluctuation is:

Step 2: Generate random velocity fluctuation

For each spatial direction $i \in \{x, y, z\}$, sample a Gaussian random number $\zeta_i$ with zero mean and unit variance. The instantaneous fluid velocity seen by the particle is:

Fluent uses the Box-Muller algorithm (or a built-in Gaussian RNG) to generate $\zeta_i$.

Step 3: Compute eddy length scale

The turbulent length scale (eddy size) is:

where $C_\mu = 0.09$ is the standard $k$-$\epsilon$ model constant. For $k$-$\omega$ SST, the dissipation rate is recovered as $\epsilon = C_\mu\,\omega\,k$.

Step 4: Compute eddy lifetime

The characteristic lifetime of the eddy is:

This is the time scale over which the eddy's velocity fluctuation is correlated.

Step 5: Compute eddy crossing time

The time for the particle to cross the eddy is:

where $\tau_p$ is the particle relaxation time:

For $Re_p \ll 0.1$ (Stokes regime), this simplifies to:

If the argument of the logarithm is non-positive (the particle cannot cross the eddy within its lifetime), the crossing time is set to infinity and the interaction time is $t_e$.

Step 6: Determine interaction time

The particle interacts with the current eddy for:

The particle integrates its equations of motion over $t_{\text{int}}$ with a constant fluid velocity $\mathbf{\tilde{u}}$.

Step 7: Meet the next eddy

After $t_{\text{int}}$, the particle encounters a new eddy. Go to Step 2 and sample a new random fluctuation.

Step 8: Repeat

Continue until the particle exits the domain, hits a wall, or reaches the maximum tracking time.

The Complete DRW Picture

The particle trajectory is a sequence of straight-line segments (or small integration arcs) through a series of eddies, each with its own random velocity perturbation. The result is a random walk superimposed on the mean flow.

Eddy 1        Eddy 2        Eddy 3        Eddy 4
+------+      +------+      +------+      +------+
|  u1~ |----->|  u2~ |----->|  u3~ |----->|  u4~ |
+------+      +------+      +------+      +------+
  t_int_1        t_int_2       t_int_3       ...

Over many parcels and many "tries," the ensemble of trajectories builds a smooth concentration field that captures turbulent dispersion.

Particle Relaxation Time

The particle relaxation time is the characteristic time for a particle to adjust its velocity to the surrounding fluid:

Stokes number: The Stokes number is the ratio of particle response time to flow time scale:

For shisha PM2.5: $\tau_p \sim 10^{-6}$ s, $\tau_f \sim L/U \sim 0.1$ s, so $\text{Stk} \sim 10^{-5} \ll 1$. The particles are excellent tracers. They follow the instantaneous fluid velocity, including the random fluctuation.

Stokes Number -- Consequence for DRW

When $\text{Stk} \ll 1$, the particle velocity is approximately:

The particle velocity equals the instantaneous fluid velocity. The random walk is directly inherited from the turbulence. In this limit, the DRW model is essentially a Lagrangian realization of the turbulent diffusivity.

When $\text{Stk} \sim 1$, the particle partially filters the fluctuation. The crossing time $t_c$ becomes important because the particle may not respond fast enough to fully sample the eddy before being ejected.

For PM2.5, $\text{Stk} \ll 1$ and $t_c \gg t_e$ (the particle crosses the eddy almost instantly). The interaction time is always $t_e$, the eddy lifetime.

Stochastic Tracking -- Number of Tries

Each "try" is one complete trajectory realization with a unique sequence of random numbers. Fluent runs Number of Tries trajectories per parcel and superposes them. The concentration field from each parcel is an average over all tries.

Why multiple tries?**

A single try with random Gaussian fluctuations produces one noisy trajectory. The ensemble average over many tries converges to the true probability density function of particle locations. For concentration fields:

$$C(\mathbf{x}) = \frac{1}{N_{\text{tries}}} \sum_{n=1}^{N_{\text{tries}}} C_n(\mathbf{x})$$

Convergence guidance:

Practical rule:** Plot the concentration at a monitoring point against the number of tries. When successive points change by less than 5%, convergence is reached. For the shisha case, 10 tries is the minimum; 20 is recommended for the breathing-zone concentration.

Effect of Turbulence Model Choice

The DRW model reads $k$ and $\epsilon$ (or $\omega$) from the continuous-phase solution. The accuracy of turbulent dispersion depends directly on the accuracy of these fields.

For the shisha simulation, $k$-$\omega$ SST is recommended because:

• It resolves the near-wall region where particle deposition occurs.
• It handles the adverse pressure gradient in the chamber.
• The $k$ and $\omega$ fields near walls are well-bounded, giving physical $u_{\text{rms}}$ for DRW.

One-Way vs Two-Way Turbulence Coupling

By default, particles do not modify the turbulence field (one-way coupling in the turbulence sense). This is correct when the particle mass loading is low.

For dilute shisha smoke (mass loading $\ll 1$), the particles do not generate or suppress turbulence. One-way coupling is correct.

When mass loading exceeds $\sim 10$, particles modify turbulence through:

• Turbulence generation by particle wakes
• Turbulence suppression by added effective viscosity

Enable "Two-way turbulence coupling" in the Physical Models tab to account for this. For PM2.5 at typical shisha concentrations, leave it off.

DRW Limitations

The DRW model makes several assumptions:

1. Isotropic turbulence: The fluctuation is Gaussian with equal variance in all directions ($u_{\text{rms}}' = v_{\text{rms}}' = w_{\text{rms}}'$). Real turbulence near walls is anisotropic. The model still gives reasonable dispersion but not exact near-wall statistics.

2. Constant eddy properties during interaction: The fluctuation is constant over $t_{\text{int}}$. In reality, the eddy's velocity changes continuously. For fine particles ($\text{Stk} \ll 1$), this is a good approximation because the crossing time is short.

3. $k$-$\epsilon$-style isotropic $k$: The model assumes $u_{\text{rms}} = \sqrt{2k/3}$, which is exact only for isotropic turbulence. For anisotropic flows, the actual component-wise fluctuations differ.

4. No spectral information: DRW does not know the energy spectrum. It uses a single time scale and length scale. This is adequate for RANS but cannot capture multi-scale dispersion.

When DRW is not sufficient:

• For LES, use the subgrid-scale stochastic model (Fluent's LES-DPM coupling resolves large eddies directly).
• For highly anisotropic near-wall dispersion, consider a custom UDF that reads Reynolds stresses.

Application to Shisha PM2.5

Configuration for shisha:

Why DRW is essential:

Without stochastic tracking, all parcels from the same injection follow identical paths. The concentration field is a set of thin filaments (one per parcel). This is physically wrong -- real smoke disperses and fills the chamber. DRW adds the turbulent spreading that produces a smooth, physically correct concentration field.

Expected behavior:

• Parcels from the hose tip spread radially due to turbulent dispersion.
• The concentration field peaks along the mean path and decays Gaussian-like outward.
• Near walls, turbulence intensity is lower, so dispersion is reduced (particles deposit rather than spread).
• In the high-turbulence core of the chamber, dispersion is strong; the smoke cloud fills the volume.

Validation check:

Compare the simulated concentration profile at a monitoring plane against experimental probe data. The shape should be a smooth Gaussian-like distribution. If the profile is too narrow, increase the number of tries. If too wide, reduce the turbulence intensity (check the $k$ field).

Advanced: Spectral Dispersion Model

Fluent 2024 R1 offers an alternative to DRW: the Spectral Dispersion Model (SDM). Instead of sampling discrete eddy interactions, the SDM generates a continuous stochastic process from the turbulence energy spectrum.

The SDM is more physically rigorous for:

• Highly non-isotropic turbulence
• Flows with a wide range of eddy scales
• LES-based DPM

For RANS-based PM2.5 in shisha, the DRW model is sufficient and computationally cheaper. Use SDM only if you switch to LES.

Troubleshooting

Summary

1. The DRW model adds stochastic turbulent dispersion by sampling random velocity fluctuations at each particle-eddy interaction. 2. The algorithm: compute $u_{\text{rms}}$ from $k$, sample Gaussian fluctuation, determine eddy lifetime and crossing time, integrate over the minimum, repeat. 3. For PM2.5 ($\text{Stk} \ll 1$), particles track the instantaneous fluid velocity closely. The DRW fluctuation directly becomes trajectory spread. 4. Use 20 tries for statistical convergence of concentration fields. 10 is the minimum. 5. The $k$-$\omega$ SST turbulence model gives the best near-wall $k$ and $\omega$ fields for DRW in the shisha geometry. 6. One-way turbulence coupling is correct for dilute smoke. 7. If DRW is insufficient (anisotropic, LES, etc.), consider the Spectral Dispersion Model.

See Also

• DPM Theory -- Governing Equations (parent theory, DRW overview)
• DPM Drag Models (drag law detail; determines $\tau_p$)
• Shisha PM2.5 Case Study (DRW configuration for shisha)
• DPM Coupling -- One-Way vs Two-Way
• CFD Turbulence Models (continuous-phase turbulence)
• DPM Setup and Activation in Fluent