$1/f$ noise and long-term memory of coherent structures in a turbulent shear flow

A shear flow of liquid metal (Galinstan) is driven in an annular channel by counter-rotating traveling magnetic fields imposed at the end caps. When the traveling velocities are large, the flow is turbulent and its azimuthal component displays random reversals. Power spectra of the velocity field exhibit a $1/f^{\alpha }$ power law on several decades and are related to power-law probability distributions $P\left(\tau \right)\sim {\tau }^{-\beta }$ of the waiting times between successive reversals. This $1/f$ type spectrum is observed only when the Reynolds number is large enough. In addition, the exponents $\alpha$ and $\beta$ are controlled by the symmetry of the system; a continuous transition between two different types of Flicker noise is observed as the equatorial symmetry of the flow is broken, in agreement with theoretical predictions.

Figure 1

Schematic view of the experiment. A flow of liquid metal (Galinstan) is driven in an annular channel by the Lorentz force due to a traveling magnetic field created by magnets placed on rotating disks.

Figure 2

Most probable velocities measured in the midplane as a function of $F$ for $\text{Re}=7.1\times {10}^3$. The two vertical dashed lines indicate the region of bistability between positive and negative flow velocity. Upper-left inset: time series of the velocity in the bistable regime. Lower-right inset: bimodality of the PDF related to the bistability of the flow.

Figure 3

Frequency power spectra $S\left(f\right)$ of the velocity $V\left(t\right)$ for different Reynolds numbers ($\text{Re}=4.7\times {10}^3$, $1.6\times {10}^4$, and $3.6\times {10}^4$ from bottom to top). For clarity, the spectra have been multiplied by 1, 10, and 1000. Note the $-\frac{5}{3}$ slope at high frequency, and the occurrence of $1/f^{\alpha }$ noise at low frequency. Inset: exponent $\alpha$ as a function of Re.

Figure 4

Power-law distribution of the waiting time $P\left(\tau \right)$ between two successive reversals of the flow for $\text{Re}=7.1\times {10}^4$. Inset: $\beta$ as a function of Re.

Figure 5

Probability density function of the velocity field for different values of the asymmetry $F$ of the forcing. Note the transition from a Gaussian distribution at large $\left|F\right|$ to bimodal behavior for $F\sim 0$.

Figure 6

Frequency power spectra $S\left(f\right)$ of the velocity $V\left(t\right)$ for different values of the asymmetry parameter ($F=0.21$, 0.14, 0.09, 0, $-0.14$, and $-0.21$ from bottom to top). For clarity, the spectra have been multiplied by 10 for each increment of $F$. The inset shows the exponent $\alpha$ as a function of $F$.

Figure 7

$\alpha$ as a function of $\beta$ for various values of $F$ and Re. The dashed line indicates the regime $\beta -\alpha =1$, while the solid line corresponds to $\beta +\alpha =3$. The skewness $\theta$ of the PDFs determines in which regime the system lies ($\theta <0.1$ for blue circles, $\theta >0.1$ for red squares).