**3. Preliminary Simulations**

The reliability of the current simulation code may be demonstrated through a comparison with the existing pCPf DNS database at a comparable Reynolds number. Kasagi and coworkers [43,54] applied a DNS of several pCPfs and released their database obtained, from which a condition of (*Rew*,(*p*)) = (6000, 15.96) was chosen for the code validation during this test. At this Reynolds number and the mean pressure gradient, the pCPf is under a fully turbulent state throughout the channel, and no large-scale intermittency occurs. Its friction Reynolds number *Reτ*, normalized by the friction velocity on the fixed wall and the half width of the gap, is 154. Kuroda et al. [43] adopted a spectral method with a 128 × 128 Fourier series in the horizontal directions and Chebyshev polynomials up to order 96 in the wall-normal direction. Their domain size was 2.5*πh* × *h* × *πh*, whereas our counterpart simulation on an aCPf of *η* = 0.9 employed a nearly equal domain size of 8*h* × *h* × *π*/8 (≈ 3*h* at the gap center) in (*x*,*r*, *θ*). Figure 2 shows comparisons of the present mean and second-order statistics. An overbar, such as *ux*(*r*), denotes an ensemble-averaged quantity with respect to *t*, *x*, and *θ*, and subscript 'rms' indicates a root-mean-square value. The present control parameters of *Rew* and *F*(*p*) for our aCPf are 6000 and 16.0, respectively, and the resulting friction velocity and the friction Reynolds number on the fixed outer cylinder wall are 0.050*uw* and 151. The present results shown in Figure 2 are in reasonable agreement with the reference study, despite the wall curvature of the aCPf. A noticeable difference is detected only near the fixed wall (*y*/*h* ≈ 0.9), where the profile of *ux* exhibits a steep gradient, and thus, those of the streamwise turbulent intensity *u <sup>x</sup>*rms and Reynolds shear stress *u xu <sup>r</sup>* have peaks. The rather coarse grid resolution and the low-order spatial discretization (our finite difference code versus the previous spectral code) might affect the accuracy of the present simulation. In addition to the peak values of *u <sup>x</sup>*rms and *u xu <sup>r</sup>*, the second-order statistics from the present DNS and those of Kuroda et al. [43] agree well, particularly considering the differences in the flow geometry. The current Fortran code has been employed in different studies for several different boundary conditions [34,35,37,38,55], and thus, no further validation will be shown here.

**Figure 2.** Code validation by comparison with a previous direct numerical simulation (DNS) study on Couette–Poiseuille flow at *Rew* = 6000. (Left) Mean streamwise velocity profile; (middle three panels) root-mean-square values of velocity fluctuations in the streamwise, wall-normal (radial), and spanwise (azimuthal) directions; and (right) Reynolds shear stress. Lines and symbols represent the results obtained by this study for the annular Couette–Poiseuille flow (aCPf) with *η* = 0.9, and the result by Kuroda et al. [43] for plane Couette–Poiseuille flow (pCPf), respectively. Here, the wall-normal coordinate *y* represents the distance from the inner (bottom) wall, that is, *y* = *r* − *r*in.

In this study, we simulated a low-*η* annular flow that mimics a CPF by approximating the base flow, or the mean velocity profile, to that in the CPF. In the CPF, the velocity profile reaches its maximum at the pipe center; the velocity gradient becomes zero at the pipe center, and is at maximum on the surface of the pipe (i.e., the outer-cylinder surface). To match the base-flow characteristics of a CPF in an annular system, it is necessary to conduct a parametric investigation on the appropriate magnitude of the pressure gradient applied in the annular channel. As a preliminary analysis, we employed a medium-scale computational domain to reduce the computational cost of the parametric study. The computational domain size is smaller in the *x* direction than the present main analysis shown in Section 4. The streamwise length of the domain, *Lx*, was sufficient to capture one turbulent puff. The purpose of the preliminary analysis is to identify the value of the pressure gradient function

*F*(*p*) at each Reynolds number such that the friction coefficient on the inner cylindrical wall, *Cf* ,*in*, is practically zero, where *Cf* ,*in* is defined by the following:

$$C\_{f,in} = \frac{\tau\_{\rm in}}{\frac{1}{2}\rho l I^2},\tag{4}$$

where *U* is the bulk mean velocity obtained through a simulation. The positive/negative sign of *Cf* ,*in* corresponds to the positive/negative velocity gradient on the wall surface of the inner cylinder. Given d*u*/d*r* < 0, *Cf* ,*in* < 0, and vice versa. Table 1 shows the calculation conditions and the ranges of *Rew* and *F*(*p*) in the preliminary analysis. In the preliminary analysis, the calculation area in the mainstream direction was set to a smaller calculation area than that for the main analysis, but can capture one turbulent puff. Figure 3 shows the *F*(*p*) dependence of *Cf* ,*in* at several values of *Rew* near the global critical value. In each analysis plotted in the figure, a turbulent field with a high Reynolds number at equilibrium for each given *F*(*p*) was set as the initial flow field, and the ensemble-averaged *Cf* ,*in* value was acquired after reaching a statistically steady state. Note that laminarization did not occur in any of the cases shown here. In general, as *F*(*p*) increases, *Cf* ,*in* increases monotonically while changing from a negative to a positive value. This is consistent with the transition of the mean velocity profile from Couette-like to Poiseuille-like, and it can be confirmed that "the turning point" of *F*(*p*) indicating *Cf* ,*in* = 0 increases with *Rew*. According to this Reynolds-number dependence, an extrapolation predicts a value of *F*(*p*) that brings *Cf* ,*in* = 0 at a lower *Rew*, by which the main DNS analysis in the next section was executed.

**Table 1.** Numerical conditions for preliminary DNS with a moderate computational domain.

**Figure 3.** Friction coefficient on the inner cylinder surface as a function of the pressure function *F*(*p*) for different Reynolds numbers, obtained through the preliminary DNS study on an aCPf.
