Stochastic dynamics of particles trapped in turbulent flows

The long-time dynamics of large particles trapped in two nonhomogeneous turbulent shear flows is studied experimentally. Both flows present a common feature, a shear region that separates two colliding circulations, but with different spatial symmetries and temporal behaviors. Because large particles are less and less sensitive to flow fluctuations as their size increases, we observe the emergence of a slow dynamics corresponding to back-and-forth motions between two attractors, and a super-slow regime synchronized with flow reversals when they exist. Such dynamics is substantially reproduced by a one-dimensional stochastic model of an overdamped particle trapped in a two-well potential, forced by a colored noise. An extended model is also proposed that reproduces observed dynamics and trapping without potential barrier: the key ingredient is the ratio between the time scales of the noise correlation and the particle dynamics. A total agreement with experiments requires the introduction of spatially nonhomogeneous fluctuations and a suited confinement strength.

Figure 1

Sketches of the von Kármán setups and mean flows. Top: S-vK (square-von Kármán) configuration. Bottom: C-vK (cylindrical-von Kármán) configuration.

Figure 2

Time series of the axial position of a particle on short time (first column) and long time (second column) with the probability density functions (third column). The first and second lines stand for 6 and 18 mm particles freely advected in the S-vK configuration with a rotating frequency $f_{\text{prop}}=3\mathrm{Hz}$, while the last line is a 30 mm particle moving in the C-vK with $f_{\text{prop}}=3.16\mathrm{Hz}$. For the C-vK, the signal has been labeled according to the position of the shear layer corresponding to ${\mathrm{C}}^{+}$-vK (black) and ${\mathrm{C}}^{-}$-vK (gray) states. The PDF are conditioned on the two states ${\mathrm{C}}^{\pm }$-vK.

Figure 3

(a) Power spectrum densities (noted PSD) of the axial position $z$ for 6 mm (thin solid line) and 18 mm (thick solid line) particles in the S-vK with $f_{\text{prop}}=3\mathrm{Hz}$. (b) PSD of $z$ for 10 mm (thin solid line) and 30 mm (thick solid line) particles in the C-vK with $f_{\text{prop}}=3.16\mathrm{Hz}$.

Figure 4

Normalized spectra $f_{\text{prop}}\text{PSD}\left(z\right)$ of the axial position $z$ for 6 mm particles, plotted as a function of the reduced frequency $f/f_{\text{prop}}$, measured for three rotating frequencies $f_{\text{prop}}=2,3,4\mathrm{Hz}$ in the S-vK setup. $f_{\text{prop}}=4\mathrm{Hz}$ (solid line), $f_{\text{prop}}=3\mathrm{Hz}$ (dashed line), and $f_{\text{prop}}=2\mathrm{Hz}$ (dashed-dotted line).

Figure 5

(a) Position PSD for the double-well model. Thick solid: reference case $(\delta ,\lambda )=(0,0)$. Thin solid: symmetric case $(\delta ,\lambda )=(2,0)$. Dashed line: symmetric case, $(\delta ,\lambda )=(4,0)$. Open circles: asymmetric case $(\delta ,\lambda )=(2,0.35)$. The dotted line indicates a power of exponent $-2$. (b) Corresponding position PDF for the double-well model with conventions of figure (a). (c), (d) Position PSD and PDF for the nonhomogeneous model. Thick solid: reference case $({\tau }_p,\lambda )=(1,0)$. Thin solid: symmetric case $({\tau }_p,\lambda )=(10,0)$. Open circles: asymmetric case $({\tau }_p,\lambda )=(10,5)$.