*3.2. Second-Order Phase Transition*

In this section we present measurements of the turbulent fraction above the critical point. The measurement procedure was as in the previous section, with the exception that the recording started after a few minutes in order to ensure that the flow reached steady state conditions at the *Rei* of interest. Since turbulence was sustained in these measurements, the recording time was set from 90 s at the largest *Rei* to 15 min at *Rei* = 525 (corresponding to 1.4 <sup>×</sup> 106 advective units), which was the lowest *Rei* at which turbulence was sustained. In general, the observation time was increased, as the critical point was approached (in order to account for the expected critical slowing down). Note that the lids were stationary in these experiments (as for the results shown in Figure 1), which slightly stabilized turbulence when compared to the lifetime measurements with rotating lids discussed in the previous section; with stationary lids turbulence was sustained for *Rei* ≥ 525, whereas with rotating lids transient turbulence was found up to *Rei* = 532.

As shown in Figure 6a, the spatio-temporal dynamics of turbulent patterns at *Rei* = 525 is very rich. Oftentimes a single turbulent stripe spanning the whole system in the axial direction was observed. This then receded and eventually split into two or more arms, one of which would survive and extend to fill the system axially again. Only a slight increase of *Rei* to 532 was sufficient to almost triple the turbulent fraction, which is reflected by the persistence of more than two turbulent spiral arms (in average) as shown in Figure 6b.

The retrieved turbulent fractions from all measurements are plotted in Figure 7. The minimum measured turbulent fraction is about five times smaller than in previous plane Couette experiments [8], and the maximum observation time in advective units is about 30% longer. The turbulent fraction increases continuously with increasing *Rei* from its minimum value of about 7% (*Rei* = 525) to more than 50%, suggesting a second-order phase transition. The scaling of the turbulent fraction in the vicinity of the critical point is consistent with that expected from directed percolation in two dimensions, *Tf* = *a*(*Rei* − *Rei*,*c*) β , where β = 0.583, *Rei*,*<sup>c</sup>* is the critical Reynolds number and *a* is a proportionality constant. A least-square fit of this function to the data close to the critical point (524 < *Rei* < 540) yields *a* = 0.0667 and *Rei*,*<sup>c</sup>* = 524.1 and approximates very well the data (see the black line in Figure 7). However, measurements closer to the critical point (including a direct determination of the critical point itself) would be necessary to test the robustness and accuracy of this fit. For example, if the function above is fitted with a free exponent, then *a* = 0.0493, *Rei*,*<sup>c</sup>* = 523.5 and β = 0.703 is obtained. Finally, we stress that our system is too small to accurately determine critical exponents. Studies of quasi-one-dimensional Couette flow [26] and of quasi-two-dimensional Waleffe flow [28] show that determining the critical exponents requires a considerably larger system size. Indeed the observed interactions of the stripes with the axially bounding lids demonstrate that the the axial aspect ratio may be insufficient to probe the question of whether transition to turbulence in quasi-two-dimensional Couette flow falls into the directed percolation universality class.
