*2.1. Geometry of aCf*

Annular Couette flow is the flow in the interstice between two coaxial cylinders of formally infinite length, driven by the motion at velocity *Uw* > 0 of the inner cylinder in the *x*-direction. The annular geometry of this flow is common to both Taylor–Couette flow and annular Pipe flow; however, the forcing is different and no spin of the walls is considered. A sketch of that geometry is displayed in Figure 1 with the usual notations for the cylindrical coordinates (*x*,*r*, *θ*). Assuming that the inner and outer cylinder have respective dimensional radii *rin* and *rout*, the main geometrical parameter of this study is the radius ratio *η* = *rin*/*rout*, which varies in the open interval (0, 1). We also introduce the gap *h* between the two cylinders *h* = *rout* − *rin*.

Computationally, the pipes require to have either finite length or to be spatially periodic. The use of a spectral Fourier-based method to solve the pressure Poisson equation requires axial and azimuthal periodicity. This introduces the two wavelengths *Lx* and *Lθ*, respectively, as the domain length and the angular periodicity. While *Lx* is a free parameter, the natural value for *L<sup>θ</sup>* is 2*π* because of the cylindrical geometry. However, there is no computational obstruction to choosing other values for *Lθ*, for instance *L<sup>θ</sup>* = 8*π* or 16*π* as in Ref. [36]. In what follows, we keep the generic notation *Lθ*.

**Figure 1.** Sketch of annular Couette flow in the cylindrical coordinate system.

Like in other wall-bounded shear flows, the main lengthscale ruling out the transitional dynamics at onset is the gap *h* between the two solid walls, which here depends directly on the value *η* via the relation *h* = *rout*(1 − *η*). The perimeter on the internal cylinder, at mid-gap or on the external cylinder, now expressed in units of *h*, is shown in Figure 2 when the original dimensional value of *L<sup>θ</sup>* is 2*π* (Figure 2a). The inner perimeter is also displayed when *L<sup>θ</sup>* is a multiple of 2*π* (Figure 2b), with *L<sup>θ</sup>* = 2*πn*. The theory developed in Refs. [33,36] shows that azimuthal large-scale flows cannot be accommodated by the geometry unless *L<sup>θ</sup> r*/*h* 1 everywhere in the domain. The data for the inner cylinder play the role of a lower bound. For *L<sup>θ</sup>* = 2*π*, it is clear from Figure 2a that, for the lowest values of *η*, no azimuthal large-scale flow is possible. However, increasing *n* leads to azimuthal large-scale flows being possible for smaller and smaller values of *η*. This leads to the possibility to artificially restore large-scale flows otherwise ruled out by geometrical confinement.
