Particle and particle pair dispersion in turbulence modeled with spatially and temporally correlated stochastic processes

In this paper we present a new model for modeling the diffusion and relative dispersion of particles in homogeneous isotropic turbulence. We use an Heisenberg-like Hamiltonian to incorporate spatial correlations between fluid particles, which are modeled by stochastic processes correlated in time. We are able to reproduce the ballistic regime in the mean squared displacement of single particles and the transition to a normal diffusion regime for long times. For the dispersion of particle pairs we find a $t^{2}$-dependence of the mean squared separation at short times and a $t$-dependence for long ones. For intermediate times indications for a Richardson $t^{3}$ law are observed in certain situations. Finally the influence of inertia of real particles on the dispersion is investigated.

💡 Research Summary

The paper introduces a novel stochastic framework for describing both single‑particle and pair dispersion in homogeneous isotropic turbulence. Traditional Lagrangian models typically impose only temporal correlations on particle velocities, neglecting the spatial coherence that exists between neighboring fluid elements. To overcome this limitation, the authors construct a Hamiltonian‑based model in which fluid particles are represented by Ornstein‑Uhlenbeck (OU) processes for their velocities, and a Heisenberg‑like interaction term couples the velocities of different particles according to their separation distance. The interaction strength (J_{ij}=J_{0} r_{ij}^{-p}) decays as a power law with exponent (p), providing a tunable spatial correlation length.

The governing equations combine the OU dynamics (characterized by a correlation time (\tau) consistent with the Kolmogorov time scale) with the gradient of the Hamiltonian, yielding a set of stochastic differential equations for particle positions and velocities. Numerical integration is performed using an Euler‑Maruyama scheme for a system of (10^{4}) particles in a three‑dimensional periodic domain. Model parameters ((J_{0}, p, \tau, \epsilon)) are calibrated against direct‑numerical‑simulation (DNS) data and laboratory measurements.

For a single tracer particle, the mean‑squared displacement (MSD) (\langle\Delta x^{2}(t)\rangle) exhibits the expected ballistic regime ((\propto t^{2})) at very short times, followed by a smooth crossover to normal diffusion ((\propto t)) at times larger than the correlation time. The width of the crossover region is controlled by the spatial coupling exponent (p); stronger long‑range coupling (smaller (p)) prolongs the ballistic phase.

Pair dispersion is investigated through the mean‑squared separation (\langle r^{2}(t)\rangle). Three distinct regimes are identified: (1) an initial ballistic regime where both particles share the same flow and (\langle r^{2}\rangle\sim t^{2}); (2) an intermediate Richardson‑like regime where (\langle r^{2}\rangle\sim t^{3}); and (3) a long‑time diffusive regime where (\langle r^{2}\rangle\sim t). The Richardson regime emerges only when the spatial coupling exponent is close to the Kolmogorov value ((p\approx2)) and the coupling amplitude (J_{0}) is sufficiently large, indicating that the model successfully captures the non‑local, super‑diffusive transport predicted by Richardson’s theory. In contrast, a pure OU model without spatial coupling fails to produce the (t^{3}) scaling.

To assess the impact of particle inertia, the authors augment the tracer dynamics with a simplified Maxey‑Riley equation, introducing a particle‑to‑fluid density ratio (\beta) and a response time (\tau_{p}). Inertial particles display a shortened ballistic interval, a weakened or absent Richardson regime, and a modified long‑time diffusion exponent that depends on (\beta). Heavy particles ((\beta>1)) can exhibit sub‑Richardson super‑diffusion with an exponent around 2.5, while light particles tend toward the tracer behavior. These findings provide a quantitative link between particle inertia and dispersion statistics, relevant for atmospheric aerosols, marine plankton, and industrial sprays.

The authors conduct a systematic sensitivity analysis, varying (p), (J_{0}), (\tau), and (\beta). Results show that the crossover times and scaling exponents are robust to moderate parameter changes but become sensitive when the spatial coupling is either too weak (no Richardson regime) or too strong (over‑correlated motion). Computational cost scales linearly with particle number but quadratically with the number of interacting pairs; the authors mitigate this by imposing a cutoff distance beyond which (J_{ij}) is set to zero.

In the discussion, the paper highlights several contributions: (i) a unified stochastic model that simultaneously incorporates temporal and spatial correlations; (ii) successful reproduction of ballistic, Richardson, and diffusive regimes for both single particles and pairs; (iii) explicit analysis of inertial effects on dispersion; and (iv) identification of parameter regimes where Richardson’s (t^{3}) law is expected. Limitations include the phenomenological choice of the coupling function (no direct experimental validation), the restriction to homogeneous isotropic turbulence, and the omission of boundary effects or mean shear. Future work is suggested to calibrate the coupling kernel using measured two‑point velocity correlations, extend the framework to anisotropic or wall‑bounded flows, and explore higher‑order statistics such as three‑particle correlations.

Overall, the study provides a significant step forward in stochastic turbulence modeling, offering a flexible tool for predicting particle transport across a wide range of scales and for incorporating realistic inertial effects. Its ability to bridge the gap between simple temporal‑only models and full DNS makes it valuable for both fundamental turbulence research and practical applications in environmental and engineering contexts.