*3.2. Data Binarization*

Velocity fluctuations with respect to the mean flow are defined as *u -* <sup>=</sup> *u* <sup>−</sup> *u*, where *u* is the space-averaged time-dependent velocity averaged along *x* and *θ*, as defined in Equation (3). Here, *y* denotes the (dimensional) distance from the inner cylinder to the outer cylinder as *y* = *r* − *rin*, instead of using *r*.

$$
\overline{u}(y,t) = \frac{1}{L\_x L\_\theta} \int\_0^{L\_x} \int\_0^{L\_\theta} \mathfrak{u}(x,y,\theta,t) \mathrm{d}x d\theta. \tag{3}
$$

The flow is separated into its laminar and turbulent components by postulating a threshold independently of the Reynolds number. The local criterion chosen is |*u <sup>r</sup>*/*Uw*| ≥ 0.01 for turbulence and |*u <sup>r</sup>*/*Uw*| < 0.01 for laminar flow, with *u <sup>r</sup>* the radial velocity component, which vanishes everywhere for strictly laminar flow. As in Figures 4 and 5, localized turbulent regions are visualized by contours of *u r* <sup>∗</sup> in steps of ±0.01. The turbulent fraction *Ft* is evaluated at mid-gap (*<sup>y</sup>* = *<sup>h</sup>*/2) by estimating the percentage of grid points for which the turbulent criterion above is fulfilled.

**Figure 4.** Contours of radial velocity fluctuations *u*∗ *<sup>r</sup>* at mid-gap for *η* = 0.3 around *Rew* = *Reg*. Typical snapshots of instantaneous flow fields obtained after reaching each equilibrium state are shown here. The main flow is from left to right. (**a**,**b**) original aCf with *L<sup>θ</sup>* = 2*π*, and (**c**,**d**) artificially extended with *L<sup>θ</sup>* = 96*π*.

**Figure 5.** The same as Figure 4, but for *η* = 0.1. (**a**,**b**) *L<sup>θ</sup>* = 2*π*; (**c**,**d**) *L<sup>θ</sup>* = 32*π*; and (**e**,**f**) *L<sup>θ</sup>* = 128*π*.

The dynamics of the proliferation process for *η* = 0.1 and 0.3 is illustrated in Figure 6 using space-time diagrams and compared one to another in the case *n* = 1. The spatial variable is *x* − *Uf t*, i.e., the streamwise coordinate in a frame moving with constant velocity *Uf* , which is close to the average velocity *um*. The space-time diagram is based on the binarized radial velocity *u <sup>r</sup>*. The absolute value of the radial velocity evaluated at mid-gap is first averaged azimuthally according to

$$
\langle u'\_{r\text{rms}} \rangle\_{\theta}(\mathbf{x}, t) = \sqrt{\frac{1}{2\pi} \int\_0^{2\pi} u'\_r^2(\mathbf{x}, h/2, \theta, t) \, d\theta}. \tag{4}
$$

and the binarization criterion is *u <sup>r</sup>*rms*θ*/*Uw* ≥ 0.01. The frame velocity *Uf* for *η* = 0.3 is chosen to be same with *um*, which is estimated in two steps. First, a spatially average velocity is evaluated at every time *t*

$$u\_m(t) = \frac{1}{L\_x(r\_{out}^2 - r\_{in}^2)L\_\theta} \int\_0^{L\_x} \int\_{r\_{in}}^{r\_{out}} \int\_0^{L\_\theta} u\_x(\mathbf{x}, \mathbf{y}, \theta, t) \, r \mathbf{dx} d\mathbf{r} d\theta,\tag{5}$$

then it is time-averaged using a classical moving average technique over a time interval Δ*T* (with Δ*T* > 104*h*/*Uw* after reaching equilibrium).

$$\overline{u\_{\rm ll}} = \frac{1}{\Delta T} \int\_{T}^{T + \Delta T} u\_{\rm m}(t) \mathbf{d}t. \tag{6}$$

We found that, for *η* = 0.1, an optimal value of *Uf* for the frame to move with puffs was slightly slower than *um*. For each value of *η*, three space-time diagrams are displayed, respectively below, close to and above the corresponding critical point *Reg*(*η*). The shorter aspect of the coherent structures for *η* = 0.1 is striking compared to *η* = 0.3. Many more splitting and decay events, qualitatively similar to the pipe flow case [40,42,43], occur for *η* = 0.1 despite equal pipe lengths. This suggests that the status of the present simulations for *η* = 0.1 is qualitatively much closer to the thermodynamic limit than it is for *η* = 0.3. As a by-product, the critical scaling is expected to converge at a lower price than at higher *η*. Given the cost obstacles induced by the diverging lengthscales/timescales in most critical phenomena, the above conclusion is positive news.

**Figure 6.** Space-time (*x* − *Uf t*) diagram of original aCf (*L<sup>θ</sup>* = 2*π*) for *η* = 0.1 (three leftmost columns) and 0.3 (three rightmost columns). Black: turbulence according to the criterion *u <sup>r</sup>*rms*θ*/*Uw* ≥ 0.01. The values of the frame velocity *Uf* for *η* = 0.1 are 0.288*Uw* at *Rew* = 407, 0.2875*Uw* at *Rew* = 407.5, 0.2815*Uw* at *Rew* = 415, and those for *η* = 0.3 are approximately equal to *um*.
