**5. Conclusions**

We performed direct numerical simulations (DNSs) of the concentric annular Couette–Poiseuille flow (aCPf) and investigated the laminar–turbulent intermittent field of the so-called puff turbulence, particularly during its subcritical transition. From previous studies, the laminar–turbulent intermittency in annular flows (a pure Couette flow [37] or Poiseuille flow [34]) exhibits the helically shaped turbulent pattern with bi-directional spatial intermittency and puff turbulence with uni-directional intermittency, depending on the radius ratio. This fact leads to a unified understanding of the formation of localized turbulence patterns of different systems, including planar and circular pipe flows (CPFs); however, these analyses were conducted under conditions in which the basic velocity profiles do not qualitatively match those of the CPF. In this study, the radius ratio (of the inner/outer radii) was as low as 0.1, and the mean pressure gradient was imposed such that the inner-cylinder surface had a zero velocity gradient on average, so that the CPF was alternatively simulated by an annular system. Multiple puffs were demonstrated using a long computational domain in the axial direction, and the presence or absence of the puff-splitting event and its onset Reynolds number were investigated using a long-term DNS. The Reynolds number was reduced adiabatically from the fully turbulent field, and the following results were obtained.

• At *Rew* = 1600, puff-splitting events occur along with stochastic puff decay, resulting in wave-like fashion of multiple puffs with constant intervals.


The question considered in this study was whether puff splitting can occur in an aCPf, which essentially has a non-slip inner cylinder. In fact, puff splitting was clearly observed at *Rew* = 1600, and a sign of splitting was detected at *Rew* = 1540, which may be close to the global critical point, *Reg*. This result guarantees that the planar system and the in-pipe system can be linked via the annular system. Near the criticality, oblique turbulent stripes grow or split in the longitudinal direction of the band, but the mainstream directional splitting, as seen in the CPF, is less pronounced in the planar flows. Our results suggest that the localized structures seen in both the planar and pipe flows can cause mainstream directional splitting. However, we should note that no completed puff splitting was detected near *Reg*. The puff splitting could be observed for *Rew* < 1600 and even below *Reg* by increasing both the observation time and domain by orders of magnitude. Such a task to explore the exact *Reg* value as well as the Reynolds-number dependence of the puff-splitting time scale near *Reg* is a challenging one that is almost impossible at present. Another possible approach is to study lifetimes of single puffs [56] and time scales of splitting [10] at conditions away from *Reg*, as in earlier studies on the CPF. This may allow us to discuss whether the current system behaves more like quasi-1D Couette flow or like pipe flow, as done by Shi et al. [57] for a pCf. We would like to report on this issue in another paper. Moreover, the characterization of a DP universal class remains skeptical. Similarly to the critical phenomena of the DP universal class, the region of the absorbing state (laminar-flow gap among puffs) should increase as the criticality approaches. From these facts, it is important to verify the DP feature after further expanding the axial computational domain. In addition, since the transition process of an aCPf has a dependence on the radius ratio and *F*(*p*), a parametric study will also be addressed in the future.

**Author Contributions:** Conceptualization, T.T.; methodology, H.M. and T.T.; simulation and formal analysis, H.M.; visualization, H.M.; writing—original draft preparation, H.M. and T.T.; project administration, T.T.; funding acquisition, T.T.; all authors discussed the results and commented on the manuscript. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was funded by Grant-in-Aid for JSPS (Japan Society for the Promotion of Science) Fellowship 16H06066 and 19H02071.

**Acknowledgments:** Numerical simulations were performed on SX-ACE supercomputers at the Cybermedia Centre of Osaka University and the Cyberscience Centre of Tohoku University.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

## **Abbreviations**

The following abbreviations are used in this manuscript:

