Probing the large scale velocity statistics in turbulence*

Turbulent flows in three dimensions of space involve a direct cascade of energy from a large scale $l_0$ to a much smaller Kolmogorov length scale $\eta$. In the limit of large Reynolds number, the statistical properties of the inertial range $1 \ll k l_0 \ll l_0/\eta$ have been well studied, starting from the analytical predictions given by Kolmogorov in 1941, and much refined since then. The dynamics of large scales ($k l_{0} \ll 1$) in the absence of bidimensionalization (rotation and/or stratification), however, have been much less studied experimentally. Two analytical predictions for the mean wavenumber energy spectrum ($E(k)$) have been proposed over the years, one by Saffman with $E(k) \propto k^{2}$ and one by Batchelor with $E(k) \propto k^{4}$. Up to date, both the Saffman and Batchelor spectra lack direct experimental evidence. We present a novel experiment that achieves a significant scale separation between the forcing scale and the experiment size to exhibit large scale statistics. Using index matching technique, we also obtain full optical access to measure in the bulk, the statistics of the large scales in a turbulent flow. As a result, we are able to test the theoretical predictions by Saffman and Batchelor.