Eulerian vs Lagrangian Irreversibility in an Experimental Turbulent Swirling Flow

In a turbulent fluid, the time-reversal symmetry is explicitly broken by viscosity, and spontaneously broken in the inviscid limit. Recently, Drivas [J. Nonlinear Sci. 29, 65 (2019).] proved the equivalence of two different local indicators of time irreversibility: (i) an Eulerian one, based on regularity properties of the velocity field [Duchon and Robert, Nonlinearity 13, 249 (2000).]; (ii) a Lagrangian one, based on symmetry properties of the trajectories under time reversal [Jucha et al., Phys. Rev. Lett. 113, 054501 (2014).]. We test this equivalence in a turbulent Von Kármán experiment at a resolution of the order of the Kolmogorov scale using a high resolution 4D-PTV technique. We use the equivalence to perform the first joined Eulerian-Lagrangian exploration of the dynamics leading to time irreversibility, and find that it is linked with vortex interaction, suggesting a link between irreversibility and singularity.

Figure 1

Instantaneous snapshots of Eulerian (a) and interpolated-Lagrangian (b) irreversible criteria normalized by the mean energy dissipation $\epsilon$, computed in the middle plane of the measurement volume of our SVK dataset. (c) PDFs of Lagrangian and Eulerian irreversible criteria for SVK and GVK.

Figure 2

Joint PDF of Lagrangian and Eulerian irreversibility indicators for SVK (a) and GVK (d). The dotted line has equation $y=x$. Joint PDF of Lagrangian and Eulerian maxima found at each snapshot for SVK (b) and GVK (e). Joint PDF of snapshot minima of the Lagrangian criterion and the corresponding value of the Eulerian criterion at the same position for SVK (c) and GVK (f).

Figure 3

3D visualization of Eulerian irreversibility isosurfaces (in red) at $T_{*}\approx 6\epsilon$ and highly irreversible trajectories selected from the Lagrangian criterion with $T_{*}\approx 10\epsilon$. This event was taken from the SVK dataset. There are 19 time steps plotted for each trajectory. Positive Lagrangian indicators are showed in black as negative values are coded in blue-green. For representation purposes, the Lagrangian indicator was fitted on trajectory segments of $5\mathrm{\Delta }t$. The Lagrangian scale is ${\ell }^{\mathcal{L}}=10\eta$ and the Eulerian scale is ${\ell }^{\mathcal{E}}=3.8\eta$.