*2.2. Governing Equations and Computational Methods*

Whereas *η* is a geometrical parameter only, we also introduce the Reynolds number *Rew* = *Uwh*/4*ν*, based on the half velocity of the cylinder sliding *Uw*/2, the half gap width *h*/2, and the kinematic viscosity *ν* of the fluid. The reason why half-gap and half-velocities are considered to non-dimensionalize the equations is a simple way to reconnect with the standard conventions for pCf as *η* goes towards unity. By choosing this convention for all values of *η*, the non-dimensional incompressible equations ruling the flow dynamics without any turbulence model read

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

$$\frac{\partial \boldsymbol{u}^\*}{\partial t^\*} + \left(\boldsymbol{u}^\* \cdot \nabla^\*\right) \boldsymbol{u}^\* = -\nabla^\* p^\* + \frac{1}{4\mathcal{R}c\_w} \Delta^\* \boldsymbol{u}^\*,\tag{2}$$

where superscripts <sup>∗</sup> indicate quantities non-dimensionalized with *Uw* and *<sup>h</sup>*, and where *<sup>u</sup>* <sup>=</sup> (*ux*, *ur*, *u<sup>θ</sup>* ) and *p* represent the velocity field and the pressure field, respectively.

Equation (2) is discretized in space using finite differences and with fine enough grid resolutions according to the standard criteria of direct numerical simulation (DNS) [26]. The time discretization is carried out using a second-order Crank–Nicolson scheme, and an Adams–Bashforth scheme for the wall-normal viscous term and the other terms, respectively. Further details about the numerical methods used here can be found in Ref. [38]. Table 1 lists the parameters used in this computational study.

**Figure 2.** (**a**) circumference of original annular pipe system at the outer cylinder, at mid-gap, and at the inner cylinder; (**b**) circumference at the inner cylinder for *L<sup>θ</sup>* ≥ 2*π*.

**Table 1.** Computational conditions. *L*∗ *<sup>i</sup>* : length of the computational domain in the direction *i*, non-dimensionalized by the gap width *h* = (*rout* − *rin*); *L*<sup>∗</sup> *out* (resp. *L*<sup>∗</sup> *in*) the circumference of the outer (resp. inner) cylinder surfaces, normalized by *h*; *Ni* : the number of grids.

