*4.1. Puffs in Annular-Pipe Flow*

Figure 4 presents a three-dimensional visualization of localized turbulence in the form of puffs, which is observed as an equilibrium state reached after a lengthy simulation under the condition of *Rew* = 1600 and *F*(*p*) = 6.5. The turbulent region can be clearly detected by showing the radial velocity fluctuations or the wall-normal velocity component. The threshold value of ±0.03*uw* for the iso-surface was arbitrarily chosen to extract its typical arrowhead shape similar to that of a puff. A slight change in this threshold value does not significantly affect the interpretation of the present results. In the snapshot, multiple turbulent patches, called 'puffs' hereafter, can be confirmed to be distributed intermittently with respect to the streamwise direction. The blank regions between neighboring puffs can be regarded as being in a laminar flow because of an insignificant fluctuating velocity, implying the well-established coexistence of laminar and turbulent regions in the aCPf. As is clear from the enlarged figure, the puff has an arrowhead shape, and the puff extends downstream in the center of the outer pipe. Although the average velocity gradient on the inner cylindrical wall surface is almost zero, this situation is considered to be due to the similar driving mechanism of the puff of the CPF. For the CPF, Shimizu et al. [11] reported that turbulence in the puff originates from low-speed streaks, as well as from streamwise vortices along the (outer-)pipe wall and across the trailing edge of the puff through the Kelvin–Helmholtz instability, which induces velocity fluctuations that propagate downstream faster than the puff itself in the core region. Such a driving mechanism of the puff is also common to the present aCPf with nearly zero *Cf* ,*in*. The streamwise size of each puff is approximately 30 times

the gap width *h*, which corresponds to 15 times the hydraulic diameter, and is consistent with that of the puff observed in the CPF [1,7,8,12]. The array of puffs seems variable in intervals, but is likely not less than 30*h*. The wavelength and periodicity of the puffs are examined using two-point correlation functions of the turbulence quantities. In the *x* and *θ* directions, the auto-correlation coefficients are defined as follows:

$$R\_{\rm ii}(\Delta x) = \frac{\overline{u\_i'(\mathbf{x}, r\_{\rm ref}, \theta) u\_i'(\mathbf{x} + \Delta \mathbf{x}, r\_{\rm ref}, \theta)}}{u\_{i \rm rms}'(r\_{\rm ref}) \cdot u\_{i \rm rms}'(r\_{\rm ref})} \tag{6}$$

and

$$R\_{\vec{u}}(\Delta\theta) = \frac{\overline{u\_i'(\mathbf{x}, r\_{\text{ref}}, \theta) u\_i'(\mathbf{x}, r\_{\text{ref}}, \theta + \Delta\theta)}}{u\_{i\text{rms}}'(r\_{\text{ref}}) \cdot u\_{i\text{rms}}'(r\_{\text{ref}})},\tag{7}$$

where *i* ∈ (*x*,*r*, *θ*). Figure 5 shows the two-point correlation coefficients of each velocity component for the case visualized in Figure 4. The statistical dataset was accumulated over the time of 5000*h*/*uw* after achieving a pseudo-equilibrium state of multiple puffs.

**Figure 4.** Instantaneous flow field for *Rew* = 1600 and *F*(*p*) = 6.5. Iso-surfaces of radial velocity fluctuation are shown: red, *u <sup>r</sup>* = 0.03*uw*; blue, *u <sup>r</sup>* = −0.03*uw*. The left-to-right direction corresponds to the direction of the main flow, by which the observed puffs propagate. Not to scale.

From Figure 5a, the axial periodicity and interval of the puff can be estimated. First, we note that the three curves at different *y*∗ exhibit consistency, implying that flow state and patterning are only weakly dependent on *y* or *r*. As also plotted in (b) and (c) for the other directional components, fine-scale turbulent structures inside a puff should have a rather short streamwise extent, and indeed, the profiles of *Rrr* and *Rθθ* fall to almost zero at Δ*x* < 5*h*. The profile of *Rxx* also decreases drastically for a small Δ*x*, although its significant oscillation for a long axial extent suggests a spatial coexistence of laminar and turbulent regions rather than turbulent structures, since these two flow states have different mean velocity profiles, particularly near the walls. The oscillations observed in Figure 5a are somewhat strong at both the inner and outer walls, relative to the gap center. The profile of *Rxx* takes the first negative local minimum at Δ ≈ 30*h* and shows regular spikes at intervals of approximately 60*h*. The correlation is not zero even at half the computational domain length (*Lx*/2 = 204.8*h*). Peaks at 60*h*, 120*h*, and 180*h* manifest the presence of seven distinct puffs in *Lx* on average. This suggests that the puffs at this Reynolds number tend to be arranged regularly throughout the axial extent. If the puff spacing is irregular, the correlation coefficient distribution should not show periodic fluctuations and should asymptotically approach zero. This regularity of the puff arrangement may differ from the characteristic of the DP universal class, which should exhibit a wide-scale invariant pattern close

to the critical point [39,41]. Mukund and Hof [3] reported a similar aspect on multiple puffs in a CPF, where they referred to the wave-like fashion as 'puff clustering'; that is, the resultant pattern of clustering puffs was observed to propagate like waves. They also pointed out that interactions between puffs were responsible for the approach to the statistical steady state and strongly affected the percolation threshold. This may predict a difference in the global stability between a single puff (i.e., isolated puffs) and multiple puffs (puff clustering), as discussed in Section 4.2.

**Figure 5.** Two-point correlation coefficient of velocity fluctuation for *Rew* = 1600 and *F*(*p*) = 6.5. (**a**–**c**) Streamwise spatial correlation as a function of Δ*x*, and (**d**–**f**) azimuthal correlation as a function of Δ*θ*. (**a**,**d**) Auto-correlation of streamwise velocity component *u <sup>x</sup>*, (**b**,**e**) that of *u <sup>r</sup>*, and (**b**,**e**) that of *u θ* . Here, the reference radial position *r*ref is translated as the inner-wall-normal height *y*<sup>∗</sup> = *r*ref − *r*/*h*.

The azimuthal two-point correlation functions shown in Figure 5d–f indicate the azimuthal intervals between fine-scale turbulent structures, such as low-speed streaks inside the puff. There exists no large-scale pattern in the azimuthal direction, unlike those of the helically shaped turbulent patches in high-*η* aPf [35] and aCf [37]. The blue curve in Figure 5d, measured near the outer cylinder wall, only has a peak at Δ*θ* = *π*/2. The cross-sectional flow pattern observed here consists of four low-speed streaks close to the outer wall spaced at *π*/2. This azimuthal configuration regarding turbulence inside the puff is in agreement with those found in the CPF [6].

The presence of turbulent equilibrium puffs was observed even at *Rew* < 1600, and the flow field finally reached the fully laminar state at *Rew* = 1500. Although the space-time diagram (STD) and the turbulent fraction, *Ft*(*t*) (the plot of which is shown later), reveal a tendency toward laminarization at *Rew* = 1525, one turbulent puff was maintained in the present computational domain at least during the present observation time of >1.3 × 104, and the laminarization was not completed. If normalized by the hydraulic equivalent diameter 2*h* and bulk velocity *U*, the Reynolds numbers of *Rew* = 1600 and 1500 correspond to *ReD* = 2190 and 2045, respectively. This range of *ReD* = 2045–2190 is close to or slightly narrower than that for the counterpart of the CPF (*ReD* = 2000–2700 [1], 2040–2400 [10], 2300–3000 [2], and 2000–2200 [4]). In particular, a discrepancy in the lower bound value of the subcritical transition regime, that is, the global critical Reynolds number, is of interest, although the similarity with the results by Avila et al. [10] is rather surprising. A cause of this discrepancy remains unclear: One of the main causes may be the presence of the inner cylinder, which suppresses turbulent motions across the central axis in the case of an aCPf. Another cause may be the non-slip inner-cylinder surface, which prevents a puff from splitting into two puffs in the case of an aCPf. Shimizu et al. [8] proposed a model process of puff splitting in the CPF, which starts with an azimuthally isolated streak propagating downstream through the laminar–turbulent interface of the puff. An emitted streaky disturbance can be a seed of a "daughter puff," which spreads again in the azimuthal direction and grows into a turbulent puff after leaving the parent puff sufficiently far away. As a system even closer to the CPF, an ideal aPf with a stress-free boundary condition at the inner wall can be analyzed, although such an unpractical situation will be considered as a future task. In terms of the conjecture that puff splitting is unlikely in the aCPf relative to the CPF, we traced puffs with lengthy simulations, and their STDs are shown in Section 4.2.
