Non-equilibrium formulation for inertial particles in turbulent swirling flows

TL;DR

This work develops a scale-space, Markovian description of inertial-particle dynamics in turbulent swirling flows by deriving a Fokker-Planck equation for velocity increments $u_\tau$ through the Friedrich–Peinke approach. It demonstrates that the drift is approximately linear in $u_\tau$ and the diffusion scales as a quadratic form in $u_\tau$, with coefficients that depend on the cascade scale and Stokes number, revealing inertial filtering and intermittency. The framework reproduces increment PDFs across scales via a short-scale propagator and shows that trajectory entropy production obeys the Integral Fluctuation Theorem, linking stochastic thermodynamics to turbulent transport. These results validate data-driven stochastic modeling of particle-laden turbulence and offer a path toward thermodynamically consistent, reduced descriptions of turbulent transport in complex flows.

Abstract

We study the dynamics of inertial particles in turbulence using datasets obtained from both direct numerical simulations and laboratory experiments of turbulent swirling flows. By analyzing time series of particle velocity increments at different scales, we show that their evolution is consistent with a Markov process across the inertial range. This Markovian character enables a coarse-grained description of particle dynamics through a Fokker-Planck equation, from which we can extract drift and diffusion coefficients directly from the data. The inferred coefficients reveal scale-dependent relaxation and noise amplitudes, indicative of inertial filtering and intermittency effects. Beyond the kinematic description, we analyze the thermodynamic properties of particle trajectories by computing the trajectory-dependent entropy production. We show that the statistics of entropy fluctuations satisfy both the Integral Fluctuation Theorem and, under certain conditions, the Detailed Fluctuation Theorem. These results establish a quantitative bridge between stochastic thermodynamics and particle-laden flows, and open the door to modeling turbulent transport using effective stochastic theories constrained by data and physical consistency.

Figures (9)

• Figure 1: Contour plots comparing the single-conditioned probability (black contours) and the double-conditioned probability (red contours) of velocity increments of particles at three time scales, $\tau_2 < \tau_1 < \tau_0$. As a representative example, the time scales are chosen relative to the Einstein--Markov time of each dataset: $\tau_2 = \Delta_\text{EM}$, $\tau_1 = 2\tau_2$, and $\tau_0 = 3\tau_2$. The left panel shows results for the TG-3.2 dataset, and the right panel for VK-2.0. Velocity increments are normalized using $\sigma_\infty = 2\sigma$, where $\sigma$ is the standard deviation of the increments. Note that the choice $u_{\tau_0} = 0$ is arbitrary; changing this value shifts the center of the conditioned distributions but still yields comparable results.
• Figure 2: Standardized Wilcoxon test statistic as a function of the time lag $\Delta \tau$ for all datasets analyzed. The time lag is given in units of the dissipation time $\tau_\eta$ of each flow. The left panels correspond to simulations, and the right panels to experimental data. The top row shows results for the $x$ velocity component, while the bottom row corresponds to the $z$ component. The horizontal dashed black line marks the threshold value of $1$, which corresponds to the expected result when the Markovian assumption holds.
• Figure 3: Drift coefficients $D^{(1)}(u_\tau, \tau)$ (top panels) and diffusion coefficients $D^{(2)}(u_\tau, \tau)$ (bottom panels) for the $x$ velocity component. Solid curves correspond to the coefficients at $\tau = T/2$. The dashed yellow lines indicate the same coefficients for the TG-3.2 and VK-2.0 datasets at $\tau = T/5$; the solid and dashed yellow lines allow for a comparison of the coefficients at different $\tau$. The 3D plots on the right display these coefficients as functions of $u_\tau$ and $\tau$ for the TG-3.2 simulation, used as a representative example (all datasets exhibit similar features). The overlaid solid and dashed yellow curves indicate the cross-section at $\tau = T/2$ and at $\tau = T/5$, respectively. In all cases, time lags are normalized by $T$, and increments of the velocity by the asymptotic standard deviation $\sigma_\infty$.
• Figure 4: PDFs of velocity increments $u_\tau$ for different time lags (see labels), for the $x$ component of the particles' velocities in the TG-3.2 (left panel) and VK-2.0 datasets (right panel). Black circles show the PDFs computed directly from the data, yellow solid lines show the PDFs obtained from the Fokker--Planck formulation using the short-scale propagator, and dashed yellow lines indicate the Gaussian distribution that best fits the data. Velocity increments are normalized by the asymptotic standard deviation $\sigma_\infty$. For clarity, the curves are vertically shifted by arbitrary offsets.
• Figure 5: PDFs of total entropy production $\Delta S$ for all datasets. Left panels correspond to simulations, and right panels to experimental data. The top row shows results for the $x$ velocity component, while the bottom row shows the $z$ component.