**1. Introduction**

The discontinuous reverse transition of wall-bounded turbulence into a laminar flow is a fundamental problem that has been studied for many years, while the laminar-to-turbulent transition is rather smooth, or its critical point is often well predicted by linear stability theory. Subcritical flows in the reverse transition are known to feature two regimes in competition, namely, laminar and turbulent, in which there occurs large-scale intermittency that coexists spatially with a laminar flow. The large-scale nature of localized turbulence often forms a regular pattern once established. The intermittent structure or formation pattern of localized turbulence varies depending on the flow system, and a number of studies have been conducted on canonical flows, such as a circular pipe flow (CPF) and planar flows. In the CPF, a so-called equilibrium turbulent puff, or simply a "puff," is localized in the streamwise direction, resulting in uni-directional intermittency. The puff turbulence is sustained within a Reynolds-number range based on the bulk velocity *U* and the pipe diameter *D* of *ReD* = 2000–2700 [1]: Although there are some differences depending on the experimental conditions, such as the disturbance introduction method and the pipe length [2–4], studies have indicated that the puff's nature is deeply related to the determination of the lower-limit Reynolds number (the global critical Reynolds number, *Reg*), above which turbulent motions can survive globally. Streamwise-localized solutions underlying the puff have been found, and Hopf bifurcations to new branches including unstable periodic orbits are expected to cover the turbulent attractor [5]. It is also known that puffs can split (or proliferate) more frequently than their decay and have a finite lifetime even at *Re* > *Reg* [6–9]. Avila et al. [10] identified *Reg* = 2040 ± 10 for the CPF by monitoring both the puff-splitting time and the decay time. Recent attempts have been made to elucidate the puff-driving

mechanism [11] and one-dimensional modeling [12], and an understanding of the puff's nature is progressing compared to other intermittent structures.

In planar flows, intermittent structures with bi-directionality were discovered during the last two decades (excluding the spiral turbulence in a Taylor–Couette flow [13]), which are called oblique turbulent stripes/bands with a certain inclination with respect to the streamwise direction, and were found in a plane Couette flow (pCf) [14–18] and a plane Poiseuille flow (pPf) [19–21]. A wall turbulence that is stably stratified by body forces, such as the Coriolis force and buoyancy, also undergoes the stripe regime [22,23]. The stripe pattern has attracted recent interest, and some studies have found families of relevant localized solutions [24,25]. As the Reynolds number approaches the relevant *Reg*, the stripe pattern becomes isolated oblique bands, which fall into a non-equilibrium state accompanied by band growth, a break (not the same as the splitting of the puff), and a mutual collision [26–28]. Because the laminar gap surrounding the isolated bands is large at near criticality, a large-scale channel setup or computational domain is required for precise tracking of the process toward a fully laminar state and for estimating *Reg*. For this reason, research is still ongoing, such as elucidating the mechanism of an isolated oblique band [29] and the statistical characteristics [30]. For details, also see recent review papers [31–33].

The two kinds of intermittent structures mentioned above were observed in different canonical flows, that is, the CPF and the planar flow, and the direct relationship between the turbulent puff and stripe is unknown. Our research group therefore focused on an annular flow between concentric cylinders. Depending on the radius ratio *η* ≡ *r*in/*r*out (where *r*in and *r*out are the inner and outer cylinder radii, respectively), the curvature and the circumferential length (relative to the gap width) should change and may affect the large-scale nature of the intermittency. With *η* ≈ 1 or 0, the flow system can be regarded as a planar flow or a CPF system, respectively. Ishida et al. [34–36] conducted direct numerical simulations (DNSs) to study the subcritical transition process of the annular Poiseuille flow (aPf) using *η* as a parameter in addition to the Reynolds number. The authors observed both the turbulent puff and the stripe according to *η*, i.e., a helical turbulence (i.e., a turbulent stripe in the annular flow) at *η* ≥ 0.5, puff turbulence similar to the transitional CPF at *η* < 0.2, and an intermediate state at 0.2 ≤ *η* ≤ 0.4. At *η* = 0.1, the observed puff split and decayed over time. A similar tendency was also uncovered in an annular Couette flow (aCf); it was reported that puffs occur at *η* = 0.1, and they split and attenuate over time [37,38]. The authors found a speckled irregular intermittent structure that differs from turbulent stripes and puffs, which was shown to have characteristics of the (1+1)-dimensional directed-percolation (DP) universal class. Recent studies have focused on the relationship between the subcritical transition phenomenon and DP [39–42].

In this study, by employing an annular system as a platform, we aim to unify uni- and bi-directional intermittent structures observed in the CPF and planar flows, respectively. The key to achieving this aim is bridging between the two different systems in terms of the base flow. The base flows of the studied aPf and aCf are qualitatively different from that of the CPF. This mismatch motivated us to simulate the annular Couette–Poiseuille flow (aCPf) at a low *η*, which should be more similar to the CPF. However, the presence of the inner cylinder may affect both the onset and splitting of the puff. The main purpose of this study is to answer whether puff splitting would occur in a low *η*. Moreover, *Reg* and the Reynolds-number dependence of puff splitting are investigated, and the DP feature is discussed.

Previous DNS studies on Couette–Poiseuille flow mainly focused on the planar turbulence. Kuroda et al. [43] compared the mean velocity profiles and various turbulence statistics for three patterns of imposed mean pressure gradients in the flow path. In particular, among the three patterns, the authors analyzed the shear stress near the moving wall surface in a turbulent field such that it approaches zero. A similar attempt was also conducted by other researchers [44–46]. As an experimental study, Nakabayashi et al. [47] also measured the turbulence statistics of a plane Couette–Poiseuille flow at high Reynolds numbers, and classified the flow field into a Couetteor Poiseuille-type depending on the base flow. In addition, Klotz et al. [48,49] eliminated the net

flow in a plane Couette–Poiseuille flow (pCPf), allowing localized turbulence to be tracked for long periods of time while stationary in the observation window. A recent quench experiment on the decay of Couette–Poiseuille turbulence is likely to approach the crossover of the decay rate, that is, the quantitative identification of *Reg* [50]. However, to the best of our knowledge, except for in limited studies [51,52], there is no DNS available for the subcritical transition process of the aCPf. The present DNS is the first to explore laminar–turbulent intermittency in a low-*η* aCPf.

The remainder of this paper is organized as follows. Section 2 presents the flow configuration, dimensionless parameters, and equations used in our simulations. In Section 3, which is dedicated to the preliminary results, we validated the current code and illustrated the parameter dependence of the base flow in terms of the mean friction on the inner cylinder. Section 4 begins with a puff characterization of the observed turbulent patches. Space-time diagrams of a turbulent quantity revealing the puff splitting and decay are then presented. All results are summarized and discussed in Section 5.
