*3.3. Intermittency Statistics*

The statistical post-processing protocol for STI is vastly similar to that used by other authors: the first step is to monitor the decay in the time of the turbulence fraction *Ft*(*t*) when the system is initiated with turbulence everywhere. By dichotomy, this yields a good approximation of *Reg* and allows one to define the reduced control parameter *ε* = (*Rew* − *Reg*)/*Reg*. This decay is expected to be algebraic exactly at onset, i.e., of the form *Ft*(*t*) = *O*(*t* <sup>−</sup>*α*). This yields as well the so-called dynamic exponent *α*. In a second phase, the equilibrium turbulent fraction (i.e., its time average) is monitored as a function of *ε*. For *ε* > 0, the data *versus* the expected scaling *Ft*(*t*) = *O*(*εβ*) yield the exponent *β*. Eventually, the mean correlation length *ξ*(*Rew*) (either *ξ<sup>x</sup>* in the streamwise direction or *ξθ* in the azimuthal one) can be estimated at equilibrium by monitoring the cumulative distribution function (CDF) of the laminar gaps *Plam*(*lx* > *L*), where *lx* stands for the length of a laminar trough and *L* is a dummy variable. A critical exponent *μ*<sup>⊥</sup> can be evaluated from fits as the algebraic decay exponent of the CDF.

We begin by describing the results from the critical quench experiments of Figure 7 for *η* = 0.1 and *n* = 64. The initial condition corresponds to a turbulent velocity field from a long simulation well above *Reg*, here taken as *Rew* = 280. The same initial condition is used for new simulations at another target value of *Rew*, in principle such that *Rew* is "close" to *Reg*. As expected, the flow relaminarizes (attested by the monotonic decrease of *Ft*(*t*)) for sufficiently low values of *Rew*, whereas it stays turbulent for the higher values. In the latter case, the turbulent fraction reaches a non-zero mean value *Ft*, which will be reported in the next figure. The set of colored curves in Figure 7a straddle the decay curve corresponding to the critical value *Rew* = *Reg*, whose best approximation in the figure is the red curve associated with *Rew* = 262.5. For continuous phase transitions, the corresponding decay is expected to be of power-law type, i.e., *Ft* = *O*(*t* <sup>−</sup>*α*). This fact of 260 < *Reg* < 262.5 yields an approximation of *Reg* = 261.7, which allows for defining *ε* as before. The present approach rests on the hypothesis of a critical scaling in the vicinity of the critical point. If that hypothesis is correct then, by rescaling time and turbulent fraction, the curves of Figure 7a should collapse onto two master curves, one for the relaminarization process and the other for the saturation process. This is tested in Figure 7b by plotting *t <sup>α</sup>Ft*(*t*) as a function of the rescaled time *<sup>t</sup>*|*ε*<sup>|</sup> *<sup>ν</sup>* . As for *<sup>α</sup>* and *<sup>ν</sup>*, the approximate values from (1 + 1)-D DP theory, respectively 0.451 and 1.733, have been used for the rescaling. The match is satisfying, which confirms that a critical range has been identified in this system.

As a by-product of Figure 7, the values of the mean turbulent fraction *Ft*, obtained after reaching equilibrium, are reported in Figure 8 as functions of *Rew*. Critical theories all predict a scaling *Ft* = *O*(*εβ*) close enough to the critical point. The algebraic scaling revealed in the previous plots of critical quench suggests that, for instance, *Re* = 262.5 belongs to the range where algebraic fits apply for *η* = 0.1 and *L<sup>θ</sup>* = 128*π*. Consequently, if, for these parameters, *ε* is defined using the approximated *Reg* = 261.7, the dependence of *Ft versus ε* is also expected to be algebraic in the same range of values of *Re*. In that case, the power-law exponents can be classically estimated using log-log plots and compared to those from DP theories. Algebraic fits of *Ft* are shown in Figure 8 both for *η* = 0.1 (left) and 0.3 (right). For each case, the main plot of *Ft versus Rew* is displayed in linear coordinates, while the inset displays *Ft versus ε* in log-log coordinates, in order to highlight the quality of the estimation of the power-law exponent.

**Figure 7.** Critical quenches from *Rew* = 280 to each Reynolds number. Temporal variation of turbulent fraction *Ft* for *η* = 0.1 and *L<sup>θ</sup>* = 128*π* (log-log scale). In (**a**), the black dashed-dotted line and dashed line each indicate possible algebraic fits with the dynamic exponent *α* from (2 + 1)-D and (1 + 1)-D directed percolation (respectively *α* = 0.451 and 0.159). See also Supplementary; (**b**) test of the 1D scaling hypothesis by plotting *t <sup>α</sup>Ft* vs. *tε <sup>ν</sup>*|| (log-log scale), with *<sup>ν</sup>*||=1.733 for (1 + 1)-D DP.

**Figure 8.** Reynolds-number dependence of the time-averaged turbulent fraction *Ft* vs *Rew* for the different radius ratios in the original domain (*L<sup>θ</sup>* = 2*π*) and in artificially extended domains (*L<sup>θ</sup>* 2*π*). Vertical error bars: standard deviations of *Ft* during the averaging period. Dashed/dashed-dotted line: algebraic fits *Ft* = *O*(*εβ*), with exponent *β* obtained either as best fit *β*fit or from the (1 + 1)-D DP universality class *β*1D = 0.276. In each figure, the insets are plotted in log-log coordinates *versus ε* that is determined with *Reg* presented in Table 2.

The details of the fitting procedure for the various parameters used are given in Table 2. It includes the values of the best fitted exponents as well as the approximate fitting range. As could already be deduced graphically from the insets in Figure 8a, for *η* = 0.1, the compatibility of the exponent *β* with the theoretical value of *β*1D = 0.276 from (1 + 1)-D DP is good (to the second digit). This is confirmed for both *η* = 32*π* and *η* = 128*π*, which suggests that the thermodynamic limit is already reached, at least as far as the determination of the exponent *β* is concerned. For *L<sup>θ</sup>* = 2*π* the approximated exponent is 0.31 which constitutes a less accurate, but still consistent approximation of the theoretical exponent. For *L<sup>θ</sup>* = 2*π*, the range of validity of the algebraic fits extends up to ≈5%, whereas it exceeds 10% for *L<sup>θ</sup>* ≥ 32*π*. For *η* = 0.3, the situation is slightly different: for a large azimuthal extent *L<sup>θ</sup>* = 96*π*, there is a very good match with the 1D theoretical exponent all the way up to *ε* ≈ 20%. For *L<sup>θ</sup>* = 2*π*, however, although an algebraic fit seems consistent with the data below *ε* < 1% the measured exponent is closer to 0.12 than to 0.276: none of these values matches any of the percolation theories.

**Table 2.** Critical Reynolds number *Reg* and critical exponent *β* depending on geometrical parameters *η* (radius ratio) and *L<sup>θ</sup>* (azimuthal extension). In addition, shown is the fitting range to estimate *Reg* and *β*. † : not measured.


The interpretation is delicate. On one hand, algebraic fits seem always verified as soon as *ε* is small enough; on the other hand, (1 + 1)-D percolation exponents are well approximated only for sufficient azimuthal extension of the order of 100*π* or more. The original system with *L<sup>θ</sup>* = 2*π* hence needs to be interpreted as a system with the DP property that experiences a *geometrical frustration* due to lateral confinement. The present data support the hypothesis that the frustration effect is stronger for *η* = 0.3 than for *η* = 0.1, and thus that the quality of the DP fit will be correspondingly worse. Conversely, the convergence towards the thermodynamic limit seems slower for larger *η*.

Importantly, we emphasize the main difference between the present conclusion and that by Kunii et al. [36], where the azimuthal extension for *η* = 0.1 was limited to *L<sup>θ</sup>* = 16*π* (to be compared to the present values of 32*π* and 128*π*). The fits reported in Figure 16 of that article suggested a fit compatible with the (2 + 1)-D exponent *β*2D = 0.583. This former result, in the light of the present computations, is re-interpreted now as a finite-size effect.

A power-law dependence of *Ft* alone does not warrant the proximity to the critical point, as pointed out by Shimizu and Manneville [23] for pPf. Although the critical quenches reported earlier also suggest power-law statistics near the picked up values for *Reg*, the classical determination relies on, at least, three independent algebraic exponents. In order to lift this ambiguity, we chose to report in Figure 9 statistics of laminar gap size for different values of *Rew* near the suspected critical point. Expecting possible anisotropy when the domain is artificially extended in *θ*, two kinds of statistics have been monitored, similarly to the study of Chantry et al. [20]. The axial extent of the gaps for *η* = 0.1 and *L<sup>θ</sup>* = 128*π* is shown in Figure 9a in log-log coordinates (and Figure 9c in lin-log representation). The azimuthal extent of the laminar gaps is shown in Figure 9b in log-log coordinates (and Figure 9d in lin-log representation). All four figures support a cross-over from exponential to power-law statistics as *Rew* approaches the value of 262.5, with a decay exponent graphically compatible with the decay exponent *μ*<sup>⊥</sup> of (1 + 1)-D DP. The cross-over appears, however, more clearly in the azimuthal where the match with the theoretical value of *μ*<sup>⊥</sup> is valid over a full decade. In the streamwise direction, the trend is not clear enough to extract a critical exponent with full accuracy. This confirms, however, that the present statistics are indeed gathered in a relevant neighborhood

of the critical point and that, for these parameters, *Rew* = 262.5 is a decent working approximation of *Reg*.

**Figure 9.** Time-averaged distributions of laminar gap in (**a**,**c**) the streamwise direction and (**b**,**d**) the azimuthal direction, evaluated at mid-gap. (**a**,**b**) log-log plots vs. (**c**,**d**) lin-log plots. *L<sup>θ</sup>* = 128*π*, *η* = 0.1 as in Figure 5e,f. In both figures, black dashed-dotted line (- · -) and dashed lien (- - -) indicate theoretical distributions *<sup>P</sup>*(Δ*L*∗) <sup>∼</sup> <sup>Δ</sup>*L*∗−*μ*<sup>⊥</sup> with exponents *<sup>μ</sup>*<sup>⊥</sup> from the universality classes of (2 + 1)-D DP and (1 + 1)-D DP, respectively, i.e., *<sup>μ</sup>*⊥2D = 1.84, and *<sup>μ</sup>*⊥1D = 1.748.
