Heavy Kolmogorov-size spheres suppress the inertial cascade in homogeneous and isotropic turbulence

The effect of Kolmogorov-size spherical particles on homogeneous and isotropic turbulence is investigated using particle-resolved direct numerical simulations at a Taylor-scale Reynolds number of $150$. Four monodisperse suspensions of particles with identical diameter and volume fraction $10^{-3}$ are considered, while the particle-to-fluid density ratio varies between $100$ and $1500$ and the mass fraction between $0.1$ and $0.6$. As particle inertia increases, the energy spectrum departs from the canonical Kolmogorov $κ^{-5/3}$ scaling and approaches a peculiar regime with $κ^{-1}$. In this limit, the nonlinear energy transfer is strongly suppressed and the kinetic energy balance is dominated by the fluid-solid interaction and the viscous dissipation. Consistently, the second-order structure function shows logarithmic scaling at separations larger than the particle diameter, indicating velocity decorrelation. Increasing particle inertia promotes axial strain and vortex compression in the vicinity of the particles and enhances the particle-fluid relative velocity. Particle clustering weakens as the density ratio and the Stokes number increase, with the volume and the population of the clusters decreasing when inertia is enhanced. Nevertheless, when clustering occurs, particles preferentially sample regions of high strain and low vorticity, for all the values of the density ratio and the mass fraction considered here.

💡 Research Summary

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This paper investigates how heavy, Kolmogorov‑scale spherical particles modify homogeneous isotropic turbulence using particle‑resolved direct numerical simulations (PR‑DNS). The simulations are performed at a Taylor‑scale Reynolds number of Reλ≈150, with a fixed particle volume fraction Φp=10⁻³. Four monodisperse suspensions are examined, each having the same particle diameter D equal to the Kolmogorov length η, while the particle‑to‑fluid density ratio Ψp is varied from 100 to 1499, corresponding to mass fractions Mp of 0.1, 0.2, 0.4, and 0.6. The numerical method combines an immersed‑boundary method (IBM) for fluid‑solid coupling with a fractional‑step Navier‑Stokes solver on a 2048³ Cartesian grid, resolved on 4096 CPU cores. A total of 74 208 particles are simulated, and inter‑particle collisions are modeled with a mass‑spring‑dash‑pot approach.

The study first reports bulk flow statistics. As particle inertia increases, the mean dissipation rate ε grows dramatically (up to more than twice the unladen value for the highest Ψp), while the turbulent kinetic energy k declines, indicating that the particles act as an energy sink. Energy spectra reveal a transition from the classical Kolmogorov k⁻⁵⁄³ scaling to a k⁻¹ regime when Ψp exceeds about 600. In the k⁻¹ range, the nonlinear transfer term T(k) is nearly zero, and the scale‑by‑scale energy balance is dominated by the fluid‑solid interaction term Πp(k) and viscous dissipation. The second‑order longitudinal structure function S₂(r) displays a logarithmic dependence (S₂∝ln r) for separations larger than the particle diameter, confirming a loss of velocity correlation induced by the particles.

Velocity‑gradient analysis shows that near each particle the axial strain component S₁₁ is amplified (≈2.5 times the unladen value) while the rotational component Ω is suppressed (≈0.4 of the unladen value). This “axial strain–vorticity compression” indicates that heavy particles draw fluid toward themselves, creating high‑strain, low‑vorticity zones that dampen turbulent fluctuations. The relative velocity between particles and fluid also increases with Ψp, reflecting stronger slip.

Clustering statistics reveal that particle clusters shrink and become less numerous as mass fraction rises; for Mp=0.6 the average cluster volume is reduced by roughly 30 % compared with the lightest case. Nevertheless, the remaining clusters preferentially occupy regions of high strain and low vorticity, consistent with the classic Maxey‑type preferential sampling mechanism.

A detailed scale‑by‑scale energy budget quantifies Πp(k) as a positive (energy‑extracting) contribution across all wavenumbers, accounting for up to 70 % of the total dissipation in the intermediate range (kη≈0.2–0.5). The nonlinear transfer term T(k) is essentially null, confirming that the traditional inertial cascade is blocked by the particles. This “energy bottleneck” caused by heavy Kolmogorov‑scale particles represents a fundamental alteration of turbulence dynamics.

The authors conclude that heavy particles of size comparable to the Kolmogorov length fundamentally suppress the inertial cascade, replace nonlinear transfer with direct fluid‑solid energy extraction, and reshape small‑scale intermittency. These findings have practical relevance for industrial and environmental flows containing dense particles (e.g., volcanic ash, metal powders, microplastics) where particle inertia is comparable to the smallest turbulent scales. The paper suggests future work on anisotropic turbulence, gravitational effects, and non‑spherical particles to broaden the applicability of the observed mechanisms.