Experimental characterization of extreme events of inertial dissipation in a turbulent swirling flow

The three-dimensional incompressible Navier–Stokes equations, which describe the motion of many fluids, are the cornerstones of many physical and engineering sciences. However, it is still unclear whether they are mathematically well posed, that is, whether their solutions remain regular over time or develop singularities. Even though it was shown that singularities, if exist, could only be rare events, they may induce additional energy dissipation by inertial means. Here, using measurements at the dissipative scale of an axisymmetric turbulent flow, we report estimates of such inertial energy dissipation and identify local events of extreme values. We characterize the topology of these extreme events and identify several main types. Most of them appear as fronts separating regions of distinct velocities, whereas events corresponding to focusing spirals, jets and cusps are also found. Our results highlight the non-triviality of turbulent flows at sub-Kolmogorov scales as possible footprints of singularities of the Navier–Stokes equation.

cameras of resolution 1600x1200 pixels, set at 45 • with respect to that plane, take successive snapshots of the flow. Then, the velocity field is reconstructed using peak correlation performed over 50% overlapping windows of size 16 to 32 pixels. As a result, we get instantaneous snapshots of the three components of the velocity field on a grid of approximate size 90 × 70 (see Fig. 3).
Typical maps of the instantaneous (top panels) and time averaged (middle panels) velocity fields for the global experiments are provided on Fig. 3, along with maps of the standard deviation (bottom panels) of the three components of the velocity at each grid points at Re ≈ 3 × 10 5 . The statistics for these maps have been obtained from 3 × 10 4 instantaneous snapshots. We observe that the instantaneous velocity fields are highly disordered contrary to the mean flow and the standard deviation which have well defined structures.
Along with local measurements, global diagnostics can be obtained. The torque applied to the top and bottom shafts are monitored using SCAIME technology, which allows us to measure the total power injected by the impellers into the flow (see Fig. 4). The calibration procedure, along with several other details on the experimental set-up may be found in 2, 3 and references therein.  Fig. 5 for an instantaneous velocity field at Re ≈ 3 × 10 5 . As the scale is decreased, the D 2D (u) does not vanish, but instead points towards localized points which we identify as strong inertial dissipation event with h ≤ 1/3.
For this computation, we have used a spherically symmetric function of x given by: where N is a normalization constant such that d 3 rG (r) = 1. According to 1 , the results should not depend on the choice of this function, in the limit → 0.
To estimate the scaling range of the extreme event, we have performed the computation of D (u) at different resolutions, using different averaging windows to reconstruct the velocity flow from the same image. An example is provided in Fig. 6. One sees that, as the resolution is increased, the region of elevated D (u) becomes sharper and sharper, but globally remains at the same location (emphasized by the white dot). On the other hand, the plot of D (u) at this location ( Fig. 6d) as a function of shows that there is a continuity between the measurements. For this event, D (u) is slowly varying at large scale, suggestive of a flow sturture with h ≈ 1/3 and then increases at the smallest scales. This is corroborated by a local plot of the in-plane velocity field around the event (Fig. 3). One clearly observes a front-like structure of the velocity field at this location.
This study is however only performed at scales larger than about 10 times the dissipative scale. Similar structures at the resolution scale is provided in the main part of this paper.
Supplementary Note 3: Simulation of dissipation around a cusp singularity We have simulated an artificial vorticity line with a cusp ω(x) on a 64 3 grid (Fig. 8-a). and computed the associated velocity field v BS using Biot-Savart law. To obtain a non-zero velocity along the vortex