Stochastic Chaos in a Turbulent Swirling Flow

We report the experimental evidence of the existence of a random attractor in a fully developed turbulent swirling flow. By defining a global observable which tracks the asymmetry in the flux of angular momentum imparted to the flow, we can first reconstruct the associated turbulent attractor and then follow its route towards chaos. We further show that the experimental attractor can be modeled by stochastic Duffing equations, that match the quantitative properties of the experimental flow, namely, the number of quasistationary states and transition rates among them, the effective dimensions, and the continuity of the first Lyapunov exponents. Such properties can be recovered neither using deterministic models nor using stochastic differential equations based on effective potentials obtained by inverting the probability distributions of the experimental global observables. Our findings open the way to low-dimensional modeling of systems featuring a large number of degrees of freedom and multiple quasistationary states.

Figure 1

(a),(b) Time series of the global observable $\theta$ reflecting the symmetries of the von Kármán turbulent flow experiment with torque forcing at $\mathrm{Re}=3\times 10^5$. The dynamics of $\theta (t)$ for $\gamma \simeq 0$ consists of fluctuations around an average value. Instead, for higher values of $|\gamma |$, an irregular switching between different quasistationary states appears. (c) PSD of $\theta (t)$ for different values of $\gamma$. The black and gray arrows highlight the two spectral peaks $f_{s1}\simeq 4.5\mathrm{Hz}$ and $f_{s2}\simeq 7\mathrm{Hz}$ corresponding to the typical mean propeller frequency of the two quasistationary states.

Figure 2

The von Kármán turbulent flow attractor reconstructed for the experiment performed at $\gamma =0.067$. The attractor is obtained by embedding the peaks ${\theta }_m$ extracted from the time series $\theta (t)$ in a three-dimensional space. One can identify two quasistationary states (labeled as $s_1$ and $s_2$) corresponding to the average velocity fields (dark red insets, arrows, and color scale show, respectively, the in-plane and the orthogonal average velocity and three cycles (highlighted by colored arrows). The direction of the arrows indicates the only possible paths to switch from one state to the other. See also Supplemental video [19].

Figure 3

(a)–(e) The von Kármán turbulent flow attractors for five different $\gamma$ values. From (c), the sequence shows how the attractor bifurcates symmetrically into a noisy periodic motion (b),(d) and into a noisy chaotic attractor (a),(e). (f)–(j) The same bifurcation sequence reproduced in the stochastic Duffing attractors for five different $\mu$ values.

Figure 4

(a) First Lyapunov exponents ${\lambda }_1$ measured for different $\gamma$ in the von Kármán turbulent flow (blue circles). For $\gamma \le 0.02$, ${\lambda }_1\simeq 0$ corresponds to the experiments whose attractor is a random point attractor. For larger $|\gamma |$, ${\lambda }_1>0$ and noisy attractors are found. (b) First, second, and third Lyapunov exponents for different $\mu$ measured in the stochastic Duffing equation with $\sigma =0.2$. The behavior of ${\lambda }_1$ is qualitatively similar to that of the experiments. See also Figs. 13 and 14 in Ref. [19] for further robustness tests.