  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.
