*4.2. Space-Time Diagrams*

As for the puff turbulence in CPF, it is well known that a turbulent puff can split into two puffs over time, the turbulence between puffs should attenuate and become a laminar pocket, and one or both puff(s) should decay quasi-stochastically because of their finite lifetime [3,6,8,9]. Avila et al. [10] observed the puff-turbulence sustainment only due to puff-splitting events that have time scale shorter than the puff-decay time scale. These features may be identified from the temporal development of the puff spatial distribution. The STDs of the present aCPf are shown in Figures 6–8, where the horizontal axis is the streamwise coordinate in a frame of reference moving at a certain velocity, and the vertical axis represents the dimensionless time at each Reynolds number. The frame-moving velocity is nearly the mean gap-center velocity, which also corresponds to the propagation velocity of an observed single puff. The color contour shows the azimuthal average of the radial velocity at mid-gap, *ur<sup>θ</sup>* <sup>=</sup> <sup>2</sup>*<sup>π</sup>* <sup>0</sup> *ur*(*x*, *h*/2, *θ*, *t*)d*θ*/2*π*, such that the laminar and turbulent regions can be clearly distinguished. Although the apparent length of each turbulent puff depends on the criterion used to discriminate it from the surrounding laminar flow, a different choice does not change the qualitative conclusions obtained.

**Figure 6.** Space-time diagram for (**a**) Case 1 at *Rew* = 1600 and *F*(*p*) = 6.5, (**b**) Case 2 at *Rew* = 1600 and *F*(*p*) = 6.5 with a different initial condition from that in Case 1, and (**c**) a typical discrete expansion process of turbulence in a subcritical transitional pipe flow at *Re* = 2300, cited from Avila et al. [10]. In (**c**), the contour color indicates the cross-sectional average of the streamwise vorticity squared, where red and blue correspond to turbulent puff and laminar regions, respectively, and the Reynolds number *Re* is based on the mean velocity *U* and the pipe diameter *D*. In (**a**,**b**), the contour shows the azimuthally averaged radial velocity *ur<sup>θ</sup>* at the gap center. The axial distribution is monitored from a moving frame of reference with a speed close to the puff propagation. The temporal development is monitored from *t* = 0, that is, the beginning of each DNS with a higher-*Rew* field with more puffs; therefore, some initial puffs decayed immediately after the start of the simulation. Not to scale (aspect ratio *x*:*y* = 10:1).

We first present the results for *Rew* = 1600 and *F*(*p*) = 6.5, as discussed in Section 4.1. The flow field visualized in Figure 4 was first achieved through an adiabatic decrease in *Rew* (with a change in *F*(*p*), accordingly) from a fully turbulent regime, and was then used as the initial condition for the following simulation to trace the behavior of the puff in the phase diagram of the *Lx*-space and time for as long as possible. The STD obtained is shown in Figure 6b, which monitors the pattern starting from an initial state with several puffs—the isolated turbulent patch featured as a red and blue segment at given time *t*. The overall puff pattern remains intrinsically spatiotemporally intermittent and exhibits both puff decay and splitting very frequently. These individual puffs have statistically well-defined lengths, similar to those in a CPF [8]. The number of puffs captured in the present domain is roughly constant between 5 and 7, and it is again confirmed that the puff intervals tend to be constant even if splitting or attenuation occurs in each individual puff. For this reason, Figure 5a reveals regular oscillations in the correlation function *Rxx*(Δ*x*), while a snapshot visualized in Figure 4 happens to have no periodicity in the puffs when considering the complete pipe length. Figure 6b may invoke an STD obtained from experimental and numerical observations of a DP-like feature in other flow systems [39,42]. Another DNS labeled as Case 1 was repeated for the same parameter set of (*Rew*, *F*(*p*)), but with a different initial condition with a single puff, which was prepared from a lower-*Rew* DNS

(see Figure 6a). The initial single puff is sustained for a long period of >5000*h*/*uw*, during which it splits irregularly, the first time at *tuw*/*h* ≈ 3000 and the second time at *tuw*/*h* ≈ 4000, but both newborn puffs decay after they are separated from their parents. A newly emitted daughter puff by the third splitting *tuw*/*h* ≈ 5500 grows and successively produces grandchild puffs. In addition, there are many signs of puff splitting. The puff turbulence eventually covers the entire domain, yet is intrinsically patchy, as in Case 2. It can be concluded that, in an aCPf similar to a CPF, the turbulent puff can split, regardless of the initial field, in qualitative agreement with a typical STD sample of a CPF [10], as displayed in Figure 6c.

**Figure 7.** Space-time diagram for (**a**) *Rew* = 1600 (*ReD* ≈ 2190) and *F*(*p*) = 6.5, as well as (**b**) *Rew* = 1500 and *F*(*p*) = 5.3 (*ReD* ≈ 2045). The contour shows *ur<sup>θ</sup>* at the gap center. The axial distribution was monitored from a moving frame of reference. The same initial condition was applied for all cases presented here. Not to scale (aspect ratio *x*:*y* = 10:1).

Figure 7a shows the STD of *Rew* = 1600 and *F*(*p*) = 6.5 (Case 2), but the speed of the moving frame of reference is modified such that puffs appear to be stationary with respect to space. With this adjustment, the propagation speed of the puff can be estimated as approximately 0.625*uw*. According to Figure 6a, when tracking a single puff in Case 1, the propagation speed is slightly faster and ≈0.65*uw*. The result is reasonable because the bulk velocity generally decreases with the expanding turbulent region. Figure 7b is an STD at *Rew* = 1500 with the same horizontal coordinate of (*x* − 0.625*uw*)/*h*, showing the eventual return to laminar flow. Once a puff starts to decay, its turbulent patch seems to accelerate slightly and takes approximately 300*uw*/*h* to attenuate completely. Before that, it took more than 4200*h*/*uw* before the system settled to the fully laminar state. While the flow at *ReD* = 2190 of Figure 7a exhibits frequent puff splitting or those signs during a period of *tuw*/*h* ≈ 5000, the flow at *ReD* = 2045 in Figure 7b undergoes only the puff decay with no puff splitting, and the flow field simply reached a laminar flow.

**Figure 8.** Space-time diagram for (**a**) *Rew* = 1550, (**b**) *Rew* = 1540, (**c**) *Rew* = 1530, and (**d**) *Rew* = 1525: see Table 1 for each given *F*(*p*) value. The contour shows *ur<sup>θ</sup>* at the gap center. The axial distribution is monitored from a moving frame of reference. The same initial condition was applied for all cases presented here. Not to scale (aspect ratio *x*:*y* = 10:1).

We further investigated the intermediate range between the two above-discussed cases (2045 < *ReD* < 2190) to elucidate the trends in the frequency or time of the puff-splitting events. Figure 8 presents an STD at each control-parameter set. In all DNSs presented in the figure, the initial conditions are exactly the same. In the figure, six puffs can be seen initially, but two or three of them decay immediately, particularly in the lower-Reynolds-number cases. At the lowest *Rew* shown in Figure 8d, puffs disappear one after another on a time scale of *O*(1000*h*/*uw*), and finally, one puff remains. There is no sign of decay in the surviving puff even after 13,000*h*/*uw*, but it is likely that the puffs will stochastically disappear and laminarize if a much longer simulation is available. This might also be true for the other cases presented here. At *Rew* < 1600, no puff splitting was observed, resulting in only puff damping. In only Figure 8b, a sign of puff splitting is detected at *tuw*/*h* ≈ 7500, although the "daughter puff" is not perfectly formed, and is finally attenuated before leaving the parent puff. Note here that a further DNS indicates no qualitative change in the flow pattern at least until *tuw*/*h* = 11,500 also for *Rew* = 1540, although not shown in the figure. According to a similar type of study [10], the puff splitting in the CPF was observed both numerically and experimentally for *ReD* > 2200, whereas clear splitting was measured in their experiments down to *ReD* = 2025 < *Reg* (=2040). If our observations were continued as long as 10<sup>7</sup> outer time units, as Avila et al. [10] experimentally did, the current system of the aCPf could exhibit a puff-splitting event even below the true *Reg*, which is not exactly determined as of now. At least, it can be said that the puff decay and splitting rates at this stage differ strongly from those observed at *Rew* = 1600 (*ReD* = 2190). As for this regime, a conclusion similar to an experimental study on a CPF [3] can be drawn, i.e., the cluster of puffs in a wave-like fashion results in fewer puff-splitting events in the STD, whose visual appearance differs from the STD for a DP universality class. Such well-organized distances between active sites (corresponding to the puffs) and the absence of splitting events are different features from those of the DP.

Figure 9 shows the temporal change in the turbulent fraction, *Ft*(*t*), which is the spatial ratio of the turbulent region to the entire calculated region, including both the turbulent and laminar regions. Here, *Ft*(*t*) ≈ 1 indicates a fully turbulent state, and *Ft*(*t*) = 0 is a fully laminar state. We set a threshold *vth* to distinguish between laminar and turbulent regions such that *Ft*(*t*) ≈ 0.5 in the Reynolds number region where turbulent puffs densely appear in an axial extent, as in the case of *Rew* = 1600 and *F*(*p*) = 6.5 visualized in Figure 7a. Figure 9a shows the temporal change of *Ft*(*t*) at *Rew* = 1500 and *F*(*p*) = 5.3, that is, the case diagnosed as a laminar regime by a visualization in Figure 7b, employing three different threshold values (*vth*, *vth*2, and *vth*3). It can be confirmed that the time change of *Ft*(*t*), particularly the gradient of the curve, does not depend on the threshold value. When *vth* = 0.005, the temporal changes in *Ft*(*t*) at several Reynolds numbers below *Rew* = 1600 are plotted in Figure 9b. In the vicinity of the critical point, a (1+1)-D DP universality class should obey a power law of *Ft*(*t*) ∝ *t* <sup>−</sup>0.159 over time. From the figure, the current data at *Rew* = 1575–1550 seem to be consistent with (1+1)-DP, although more data and more exponents will be needed to properly confirm this trend. However, it should be noted that, for *Rew* ≤ 1550, none of the puffs split and turbulent puffs were only attenuated, as shown in Figure 8a. This result suggests that a value close to the critical exponent of DP can be obtained even under a non-DP phenomenon of a simple decaying process without splitting. We should regard this result as a 'spurious' DP feature because the puff splitting (or an active site that creates offspring) is a requisite for the critical point and, hence, DP behavior. In other words, this reminds us to take caution regarding the judgment of a DP within the laminar–turbulent intermittency.

**Figure 9.** Time series of turbulent fractions: (**a**) for *Re* = 1500 with different threshold values and (**b**) for *Re* = 1525–1600 with a threshold value of *vth*.
