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

TL;DR

This study uses particle-resolved DNS to investigate how increasing inertia of Kolmogorov-size particles (with $D \cong \eta$) in a dilute turbulent suspension ($\Phi_p = 10^{-3}$) modulates homogeneous isotropic turbulence at $Re_\lambda \approx 150$. Four density ratios up to $\Psi_p = 1499$ (corresponding to $M_p = 0.1,0.2,0.4,0.6$) reveal a progressive suppression of the inertial cascade: the energy spectrum departs from the Kolmogorov $\kappa^{-5/3}$ scaling, developing a $\kappa^{-1}$ regime at intermediate scales and a $\kappa^{-4}$ tail at high wavenumbers, while the second-order structure function scales as $S_2(r) \sim \log(r/\eta)$ for $r > D$. The scale-by-scale energy budget shows nonlinear flux $\Pi(\kappa)$ diminishing and fluid-particle interaction $\Pi_{fs}(\kappa)$ dominating as inertia increases, with the heaviest particles driving energy balance primarily through $\Pi_{fs}$ and viscous dissipation $D_v$. Velocity-gradient statistics shift toward strain-dominated dynamics, and particle dynamics exhibit Gaussian speed distributions with ballistic short-time dispersion and increasingly Brownian-like long-time behavior for heavier particles; clustering weakens with inertia and eventually collapses to a random Poisson process at $M_p = 0.6$. Collectively, these results demonstrate that dense, Kolmogorov-scale particles can effectively suppress the inertial cascade and locally decorrelate velocities, while preserving significant small-scale activity in particle wakes and preferential sampling in high-strain regions at intermediate inertia.

Abstract

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.

Figures (12)

• Figure 1: Effect of increasing particle-to-fluid density ratio and mass fraction on the turbulent dissipation rate, $\varepsilon$. Each panel shows the dissipation rate with a logarithmic color scale in a particle-laden domain with different mass fractions: $M_p = 0.1$ ($a$), $0.2$ ($b$), $0.4$ ($c$), and $0.6$ ($d$). Insets in panels (a) and (d) provide a zoomed view of the dissipation in the vicinity of the particles (green spheres).
• Figure 2: Effect of increasing particle-to-fluid density ratio and mass fraction on the enstrophy, $\mathcal{E}$. Each panel shows the enstrophy with a logarithmic color scale in a particle-laden domain with different mass fractions: $M_p = 0.1$ (a), $0.2$ (b), $0.4$ (c), and $0.6$ (d). Insets in panels (a) and (d) provide a zoomed view of the enstrophy in the vicinity of the particles (green spheres).
• Figure 3: ($a$) Kinetic energy spectra $\hat{E}(\kappa)$ and ($b$) compensated energy spectra $(\kappa/\kappa_L)^4\langle\hat{E}(\kappa)\rangle$, for different mass fraction $M_p$ of the suspension. All wavenumbers are scaled by $\kappa_L=2\pi/L$. The black dashed line, the brown dash-dot line, and the black dotted line represent the $(\kappa/\kappa_L)^{-5/3}$, $(\kappa/\kappa_L)^{-1}$, and $(\kappa/\kappa_L)^{-4}$ scaling, respectively. The black solid vertical line in ($a$) denotes the wavenumber of the particle diameter $\kappa/\kappa_L = L/D$, while the blue vertical lines in ($b$) represent the minima of the high-wavenumber oscillations $\kappa_{min,m} = m\pi/D$, $m\in\mathbb{N}$.
• Figure 4: Structure functions of the fluid velocity $\left\langle{S_n}\right\rangle$ with $n=2,4,6$. The thick solid curves represent the laden cases for $M_p=0.2$ ($a$) and $M_p=0.6$ ($b$), while the structure functions of the unladen flow field are also plotted for comparison with thin, dashed lines. The vertical line denotes the particle diameter normalised with the Kolmogorov length $D/\eta$. The inset in ($b$) compares the semi-logarithmic plot of $S_2$ (thick blue curve) along with the plot of $\log(r/\eta)$, for $M_p=0.6$ (thin blue line).
• Figure 5: ($a$) Skewness $\mathcal{S}$ and ($b$) kurtosis $\mathcal{K}$ of the velocity field for increasing particle density $\rho_p$. The vertical line denotes the particle diameter normalised with the Kolmogorov length $D/\eta$. The horizontal dashed line represent the values $\mathcal{S}=0$ in plot ($a$) and $\mathcal{K}=3$ in plot ($b$).