Susceptibility divergence, phase transition and multistability of a highly turbulent closed flow

Using time-series of stereoscopic particle image velocimetry data, we study the response of a turbulent von K\'{a}rm\'{a}n swirling flow to a continuous breaking of its forcing symmetry. Experiments are carried over a wide Reynolds number range, from laminar regime at $Re = 10^{2}$ to highly turbulent regime near $Re = 10^{6}$. We show that the flow symmetry can be quantitatively characterized by two scalars, the global angular momentum $I$ and the mixing layer altitude $z_s$, which are shown to be statistically equivalent. Furthermore, we report that the flow response to small forcing dissymetry is linear, with a slope depending on the Reynolds number: this response coefficient increases non monotonically from small to large Reynolds number and presents a divergence at a critical Reynolds number $Re_c = 40\,000 \pm 5\,000$. This divergence coincides with a change in the statistical properties of the instantaneous flow symmetry $I(t)$: its pdf changes from Gaussian to non-Gaussian with multiple maxima, revealing metastable non-symmetrical states. For symmetric forcing, a peak of fluctuations of $I(t)$ is also observed at $Re_c$: these fluctuations correspond to time-intermittencies between metastable states of the flow which, contrary to the very-long-time-averaged mean flow, spontaneously and dynamically break the system symmetry. We show that these observations can be interpreted in terms of divergence of the susceptibility to symmetry breaking, revealing the existence of a phase transition. An analogy with the ferromagnetic-paramagnetic transition in solid-state physics is presented and discussed.