Stochastic resonance of rotating particles in turbulence

TL;DR

This work demonstrates stochastic resonance in the rotational dynamics of magnetic particles embedded in turbulence, showing that turbulent vorticity can nonlinearly enhance particle response to a rotating magnetic field. A combined experimental–DNS–theoretical framework derives a reduced phase equation for the phase lag $\beta$, revealing phase-locked, back-and-forth, and turbulence-dominated regimes, with SR emerging at the transition between phase-locked and turbulent-dynamics states. The study establishes a magnetic-resonance–based method to probe small-scale vorticity and shows that nanoscale and optically inaccessible flows could be quantified via emitted magnetic signals from spinning particles, enabling turbulence microscopy and active flow control possibilities. The results highlight symmetry-breaking effects when applying zero-mean, time-varying magnetic forcing, whereby turbulence can drive net rotation and create measurable, tunable responses in complex flows. Key findings include a predicted boundary shift $\omega_a\geq 2\omega_H+\omega_\eta$ for phase locking, SR peaks at $\omega_\eta/\omega_a\approx1$ (light), $1.5$ (neutral), and $2$ (heavy), and a zero-mean driving protocol that yields net lab-frame rotation through turbulent assistance.

Abstract

The chaotic dynamics of small-scale vorticity plays a key role in understanding and controlling turbulence, with direct implications for energy transfer, mixing, and coherent structure evolution. However, measuring or controlling its dynamics remains a major conceptual and experimental challenge due to its transient and chaotic nature. Here we use a combination of experiments, theory and simulations to show that small magnetic particles of different densities, exploring flow regions of distinct vorticity statistics, can act as effective probes for measuring and forcing turbulence at its smallest scale. The interplay between the magnetic torque, from an externally controllable magnetic field, and hydrodynamic stresses, from small-scale turbulent vorticity, reveals an extremely rich phenomenology. Notably, we present the first observation of stochastic resonance for particles in turbulence: turbulent fluctuations, effectively acting as noise, counterintuitively enhance the particle rotational response to external forcing. We identify a pronounced resonant peak in particle rotational phase-lag when the applied magnetic field matches the characteristic intensity of small-scale vortices. Furthermore, we uncover a novel symmetry-breaking mechanism: an oscillating magnetic field with zero-mean angular velocity remarkably induces net particle rotation in turbulence with zero-mean vorticity, as turbulent fluctuations aid the particle in "surfing" the magnetic field. Our findings offer insights into flexible particle manipulation in complex flows and open up a novel magnetic resonance-based approach for measuring vorticity: magnetic particles as probes emit detectable magnetic fields, enabling turbulence quantification even under optically-inaccessible conditions.

Figures (10)

• Figure 1: Rotational dynamics of magnetic particles in a rotating magnetic field: "phase-locked" versus "back-and-forth" regimes.a, Magnetic particles in turbulence under a rotating magnetic field: The experimental setup consists of a turbulence generator, a magnetic field generator, and magnetic particles. A Von Kármán-type turbulent flow is generated in an octagonal water-filled container with an internal diameter of $2R = 150$ mm and a height of 220 mm, driven by two counter-rotating bladed disks. A uniform rotating magnetic field $\bm{H} = H \bm{h}$, with angular frequency $\omega_H$, is generated using a system of two pairs of perpendicular Helmholtz coils. b, Magnetic particles consist of a Styrofoam core coated with magnetic paint. Surface patterns enable tracking rotational motion, as illustrated in the stroboscopic time-lapse trajectories (c-d). e, Photograph of the setup. f-i, In a quiescent fluid, particles exhibit two rotational regimes: "phase-locked" (low $\omega_H$) and "back-and-forth" (high $\omega_H$). f (Supplementary Movie 1) and h (Supplementary Movie 2) show rotational trajectories within one rotation period, viewed in a plane perpendicular to the rotation plane of the magnetic field. The particle initial preferred magnetization direction $\bm{n}_0$ (dashed arrows) and instantaneous orientation $\bm{n}$ (solid arrows) are marked over time. During rotation within a quiescent fluid, $\bm{n}$ and $\bm{h}$ remain within the same plane tierno2009overdamped. The corresponding normalized particle angular velocity, $\omega_{p,i}/\omega_H$ ($i=x,y,z$), is shown in g and i. Detailed information about the experiments and simulations can be found in Methods.
• Figure 1: Comparison between experimental and numerical results.a and b: The probability density distribution of particle angular velocity $\omega_{p,i}$ with $i=x,y,z$ representing the three components. a, In the absence of a magnetic field, the particle's rotation is purely driven by turbulence, exhibiting isotropic behavior. The simulation results (squares) are flattened over all directions (i.e., reshape the angular velocity vector into a one-dimensional (1D) array while preserving the original data order) for direct comparison with the experimental results (circles). b, With an applied magnetic field ($1.6~\text{mT}$) and increased turbulence (here $Re_{\lambda}\approx 360$, while the turbulence used in Fig. 2 of the main text is Re$_{\lambda}\approx 320$), there is a good agreement indicating the numerical simulation can indeed capture the main physics of the experiments when the physical parameters are normalized by the Kolmogrov scales, i.e, $\omega_\eta$. c, Phase diagram of particle rotation dynamic regimes (reproduced from Fig. 3m of the main text). The experimental setting of b is marked as a green triangle.
• Figure 2: Experimental validation of the theoretical model for particle rotational dynamics with turbulence.a, Schematic of a magnetic particle in turbulence under a rotating magnetic field. The particle with preferred magnetization direction $\bm{n}$ is subjected to a magnetic field $\bm{h}$ in the $xoy$-plane, rotating around the $z$-axis with frequency $\omega_H$, and experiences turbulent vorticity $\bm{\omega}_f$ in homogeneous isotropic turbulence. The phase lag $\beta$ (defined as the angle between the projection of $\bm{n}$, i.e., $\bm{n}_{xoy}$, and $\bm{h}$) and the rotation angle $\psi$ (angle of $\bm{n}_{xoy}$ relative to the $x$-axis) describe the particle dynamics. The dynamical evolution of $\beta$ is governed by the interplay of the magnetic field rotation drag, magnetic torque, and turbulence torque. b-d, Comparison of the probability density distribution function (PDF) of particle angular velocity in experimental (circles) and numerical (squares) studies. The experimental results (circles) are shown for varying magnetic field strengths with fixed turbulence intensity. b weak (1.2mT, Supplementary Movie 3), c intermediate (1.4mT, Supplementary Movie 4), and d strong (1.6mT, Supplementary Movie 5), with constant rotational frequency of the magnetic field (parameter settings can be found in the Methods and Supplementary Information).
• Figure 2: Probability density distribution of $\kappa$.a, In the subcritical regime ($\omega_H/\omega_a=0.1<1/2$), the PDFs of $\kappa = \sin^2{\alpha}$ for different levels of noise intensities ($\omega_\eta/\omega_a$). b, In the supercritical regime ($\omega_H/\omega_a=1.1>1/2$), the PDFs of $\sin^2\alpha$ for different levels of noise intensities ($\omega_\eta/\omega_a$). Symbols with the same color correspond to the same value of $\omega_\eta/\omega_a$ in a.
• Figure 3: Three distinct regimes of magnetic particle rotation dynamics in turbulence under a rotating magnetic field.a, Turbulence-dominated regime ($\omega_a \ll \omega_\eta$). b, The normalized particle angular velocity, $\omega_{p}/\omega_H$, exhibits turbulent fluctuations with a zero average and the phase lag derivative, $\dot{\beta}$ is randomized (c). d, The evolution of the tip of preferred magnetization direction of the particle, $\bm{n}$, is visualized in space, with color indicating time progression (Supplementary Movie 6). The random distribution of orientations confirms the turbulent-dominated nature. e, "Phase-locked" regime ($\omega_a \gg \omega_\eta$, $\omega_H<\omega_{cr}$): The angular velocity of the particle along the magnetic field rotation axis ($z$-axis) is locked to $\omega_H$ (f, green line), with zero net drift for other components (f, blue and orange lines). The phase lag derivative $\dot{\beta}$ is effectively zero (g) and the particle orientation follows a 2D trajectory (h, Supplementary Movie 7). i, "Back-and-forth" regime ($\omega_a \gg \omega_\eta$, $\omega_H>\omega_{cr}$): The angular velocity of the particle along the $z$-axis exhibits periodic acceleration and deceleration oscillations (j, inset), and $\dot{\beta}$ varies similarly (k, inset). The orientation vector oscillates periodically (back-and-forth looping in l, Supplementary Movie 8). m, Phase diagram of particle rotation regime, colored by the normalized variance $\sigma(\omega_{p,z})$ of the particle angular velocity along the $z$-direction, $\sigma(\omega_{p,z})=\langle \omega_{p,z}^2 - \langle \omega_{p,z}\rangle^2\rangle/\omega_\eta^2$. Black dot-dashed line: the critical frequency, $\omega_{cr} = \omega_a/2$. Green dashed line: the condition where the turbulence fluctuation intensity is balanced by the magnetic strength, i.e., $\omega_a = \omega_\eta$. The phase-locked regime is bounded with the blue thick line, predicted by the scaling analysis, i.e., $\omega_a = 2\omega_H + \omega_\eta$. Symbols (square, star, and circle): representative cases from the three distinct regimes shown in earlier panels. Triangle markers: experimental settings from Fig. \ref{['fig:fig2']}b-d.