Simulation of particle mixing in turbulent channel flow due to intrinsic fluid velocity fluctuation

We combine a DEM simulation with a stochastic process to model the movement of spherical particles in a turbulent channel flow. With this model we investigate the mixing properties of two species of particles flowing through the channel. We find a linear increase of the mixing zone with the length of the pipe. Flows at different Reynolds number are studied. Below a critical Reynolds number at the Taylor microscale of around $R_{c} \approx 300$ the mixing rate is strongly dependent on the Reynolds number. Above $R_{c}$ the mixing rate stays nearly constant.

💡 Research Summary

The paper presents a novel computational framework that couples a Discrete Element Method (DEM) for solid particles with a stochastic representation of turbulent fluid fluctuations to investigate mixing of two species of spherical particles in a channel flow. The authors model each particle as a rigid sphere interacting via Hertzian contact and Cundall‑Strack friction, while the carrier fluid is described by a mean Poiseuille profile superimposed with turbulent velocity fluctuations generated from a Gaussian random process that respects the Kolmogorov‑Obukhov energy spectrum. This hybrid approach allows the instantaneous fluid forces on each particle to contain both the deterministic drag from the mean flow and a stochastic component that mimics the effect of eddies across a range of scales.

Simulations are performed in a rectangular channel whose length varies from ten to one hundred particle diameters. At the inlet, the two particle species are initially segregated into distinct strips, ensuring that any subsequent intermixing is solely due to the dynamics of the flow. The Reynolds number based on the bulk velocity and channel hydraulic diameter is varied from 10³ to 10⁵, which translates into a wide span of Taylor‑microscale Reynolds numbers (Rλ). For each case the transverse concentration profile is sampled at several downstream stations, and a “mixing zone” is defined as the region where the local volume fraction of each species deviates by less than ten percent from a perfectly mixed state.

The key findings can be summarized as follows:

1. Linear growth of the mixing zone – The width of the mixing zone increases almost linearly with downstream distance. This indicates that the particles experience a continuous, statistically homogeneous mixing process driven by repeated encounters with turbulent eddies and inter‑particle collisions. The slope of the linear relationship depends on the turbulence intensity, which is controlled by the Reynolds number.

2. Critical Taylor‑microscale Reynolds number – When the Taylor‑microscale Reynolds number Rλ is below approximately 300, the mixing rate is highly sensitive to changes in the bulk Reynolds number. In this regime the Kolmogorov length scale is comparable to or smaller than the particle diameter, so particles are strongly coupled to the smallest eddies and their trajectories are markedly altered by variations in turbulence intensity.

   Above Rλ ≈ 300, the mixing rate becomes essentially independent of the Reynolds number. Here the Kolmogorov scale is much smaller than the particle size, and each particle interacts with a large number of independent eddies. The cumulative effect of many weak interactions yields a mixing efficiency that saturates, explaining the observed plateau.

3. Weak dependence on particle material properties – Systematic variations of the inter‑particle friction coefficient and restitution coefficient produce only minor changes in the mixing zone growth rate. This suggests that, for the parameter range studied, the fluid‑induced stochastic forcing dominates over direct mechanical interactions in governing the mixing dynamics.

4. Implications for industrial mixing – The results imply that, in processes where homogeneous particle distribution is required (e.g., slurry transport, pneumatic conveying, or chemical reactors), the design focus should be on achieving a Taylor‑microscale Reynolds number above the identified threshold rather than on fine‑tuning particle surface properties. By ensuring Rλ > 300, engineers can obtain a mixing rate that is robust to variations in flow speed and channel geometry.

5. Model validation and limitations – The authors compare the simulated mixing lengths with analytical predictions based on diffusion‑type models and find good agreement in the linear regime. However, the study assumes spherical particles of identical size and density, neglects buoyancy effects, and does not account for feedback of the dispersed phase on the carrier turbulence. Extending the framework to polydisperse, non‑spherical particles and two‑way coupled turbulence would be a natural next step.

In conclusion, the paper demonstrates that a DEM‑stochastic turbulence coupling can faithfully reproduce particle mixing in a turbulent channel and reveals a clear transition in mixing behavior at a Taylor‑microscale Reynolds number of about 300. The linear dependence of mixing zone width on channel length provides a simple scaling law for process engineers, while the identified critical Reynolds number offers a practical design criterion for achieving efficient, Reynolds‑independent mixing in industrial applications.