**2. Problem Setup and Methods**

The problem under consideration is the turbulent annular flow of an incompressible Newtonian fluid, for which the governing equations are the equation of continuity and the Navier–Stokes equation, as described in the classical cylindrical coordinate system of (*x*,*r*, *θ*):

$$\nabla^\* \cdot \mathbf{u}^\* \;=\; \; 0,\tag{1}$$

$$\frac{\partial \mathbf{u}^\*}{\partial t^\*} + \left(\mathbf{u}^\* \cdot \nabla^\*\right) \mathbf{u}^\* = \left. -\nabla^\* p^\* + \frac{1}{Re\_w} \Delta^\* \mathbf{u}^\* - \frac{\mathbf{d} P^\*}{\mathbf{d} x^\*} \mathbf{e}\_\mathbf{x} \right. \tag{2}$$

Here, the velocity vector is represented by **u**, or (*ux*, *ur*, *u<sup>θ</sup>* ), which are the respective components in (*x*,*r*, *θ*); *p* is the pressure, and *t* is the time. These quantities are non-dimensionalized and are marked by a <sup>∗</sup> superscript: **u**<sup>∗</sup> = **u**/*uw*, *p*<sup>∗</sup> = *p*/*ρu*<sup>2</sup> *<sup>w</sup>* (*ρ*, density), *t* <sup>∗</sup> = *tuw*/*h*, *x*<sup>∗</sup> = *x*/*h*, and *r*<sup>∗</sup> = *r*/*h*, where *uw* and *h* are the inner-cylinder axial velocity and the gap between the two cylinder radii, respectively, as illustrated in Figure 1. The Reynolds number *Rew* is therefore based on *uw*, *h*, *ρ*, and the fluid viscosity *μ*, whereas another definition using one-quarter of *Rew* is more conventional for studies on the pCf [15,16,22]. In only the axial-direction component of Equation (2), a constant pressure gradient in *x* is added as an external force term, −d*P*/d*x*, with the axial unit vector **ex**. In addition to the imposed pressure gradient, the flow is driven by an axial translation of the inner rod with a constant velocity of *uw* > 0. The *x*-axis corresponds to the central axis common to both cylinders, and the radius ratio of *η* is an important geometrical parameter and is set to 0.1 for the main analysis in this study. Periodic boundary conditions are imposed in both the *x* and *θ* directions, and no-slip boundary conditions are enforced at the wall surfaces of the cylinders. In the following sections, the imposed pressure gradient is re-defined as the pressure gradient function *F*(*p*), which is normalized as

$$F(p) \equiv -\frac{\mathrm{d}P^\*}{\mathrm{d}x^\*} \cdot \mathrm{Re}\_{\mathrm{\mathcal{W}}} = \frac{-\mathrm{d}P/\mathrm{d}(x/h)}{\mu u\_{\mathrm{w}}/h} \tag{3}$$

and can be interpreted as the ratio of the imposed pressure gradient (i.e., the Poiseuille-like driving force) against the wall-bounded viscous shear stress (the Couette-like driving force).

As introduced above, there are two control parameters for the flow under consideration, i.e., *Rew* and *F*(*p*). Poiseuille-like flows are realized for a large *F*(*p*), whereas Couette-like flows are obtained for a small *F*(*p*), and a specifically pure Couette flow corresponds to *F*(*p*) = 0. As indicated in [44], the ratio of the shear stress at the two walls, which can be defined by *γ* = *τ*in/*τ*out in an aCPf, is another candidate of the control parameter relevant to a Couette–Poiseuille flow. Flows with *γ* ≈ 0, or a shear-less inner cylinder wall, are of special interest because they exhibit nearly zero mean shear at the moving rod, and can thus be a model for an understanding of the puff dynamics in a pure pipe. Under such conditions, the inner cylinder practically affects the core flow only as an impermeable

thin rod, and the coherent turbulent structures and turbulent production that dominantly occur near the static outer cylinder wall mimic those found in a canonical system of the CPF. Although the system chosen here is closer to a CPF than to an aPf or an aCf, it should be noted that the different boundary conditions regarding the inner rod preclude a mathematical homotopy continuation with the CPF. Except for a fully laminar flow state, *F*(*p*) providing *γ* = 0 is not explicitly obvious, and thus, a parametric survey must be conducted for each given *Rew*. In this study, we conducted a preliminary survey of the *F*(*p*) dependence of *τ*in for several *Rew* values using a DNS with a medium-scale computational domain, as reported in Section 3. Based on these results, we selected *F*(*p*), which will provide *τ*in ≈ 0 (*γ* ≈ 0) at each tested *Rew*, and accordingly applied the main DNS using a large-scale domain to reduce the spatial limitation on the laminar–turbulent coexistence.

**Figure 1.** Couette–Poiseuille flow in an annular channel between two concentric cylinders with a radius ratio of *η* ≡ *r*in/*r*out = 0.1, driven by a constant pressure gradient and an inner-cylinder axial movement. In this study, the pressure gradient is adjusted such that the mean velocity gradient on the inner-cylinder surface is approximately zero; that is, *τ*in = *μ* [*∂ux*(*r*)/*∂r*]*r*=*r*in ≈ 0.

The numerical conditions of the preliminary and main simulations for *η* = 0.1 are summarized in Tables 1 and 2, respectively. Long domain sizes of 51.2*h* and 409.6*h* were employed in the axial direction to capture a single turbulent puff and expected multiple puffs, whereas the radial and azimuthal domain lengths were of geometrically determined values of *h* and 2*π*, respectively. The grid resolutions have been confirmed to be fine such that fine-scale eddies in turbulent patches are well resolved, at least for the particularly interesting transitional regime of *Rew* ≤ 1600.

Equations (1) and (2) were discretized using a staggered central finite-difference method, where the fourth-order central difference scheme was used in both *x* and *θ*, along with the second-order scheme in *r* on a non-uniform radial grid. A time advancement was performed using a fractional-step second-order Adams–Bashforth scheme in combination with a Crank–Nicolson scheme for the radial viscous term. The Courant–Friedrichs–Lewy (CFL) condition was continuously monitored in all directions, and accordingly, the time-step Δ*t* constraint for the nonlinear terms was enforced to ensure stability. The details of the numerical method were reported in the literature [34,53]. The code validation carried out is discussed in the next section.
