*3.2. Optimal Infinitesimal Perturbations of Pulsatile Pipe Flow*

We performed a linear non-modal stability analysis of Sexl–Womersley flow at (*Re* = 2400, *Wo* = 8) and amplitudes up to *A* = 1.6, as described in Section 2.3. For *A* ≤ 0.4, the optimal perturbation is the same as for statistically steady pipe flow: an axial two-roll configuration (not shown here). For *A* ≥ 0.5, the optimal perturbation is a streamwise helix and the optimal initial time of perturbation is *<sup>t</sup>*<sup>0</sup> *<sup>T</sup>* ∈ [0.2, 0.3]. In Figure 3b,c, we show the optimal perturbation for *A* = 1 at the optimal time of perturbation (*t*0) and at the point of maximum energy amplification (*tf*), respectively. Initially, the optimal helical perturbation is localised very close to the pipe wall at the border of the Stokes layer

(δ = <sup>1</sup> <sup>√</sup> <sup>2</sup>*Wo* ), and it is tilted towards it. Within the rest of the DC phase and the first stages of AC, the perturbation rapidly grows by four orders of magnitude in energy within only 40% of the period. By the time of maximum energy amplification (early AC phase at *<sup>t</sup> <sup>T</sup>* = 0.6), the optimal helix has separated from the Stokes layer and moved completely to the outer bulk region, where the stabilising effect of acceleration arrives later. This wall-normal phase lag increases with *Wo* [13] and can be nicely seen for the profiles close around the peak flow rate ( *<sup>t</sup> <sup>T</sup>* = 0.5 and 1.0) in Figure 3a. At the end of the process, the helix has been tilted opposite to its original configuration in a process reminiscent of the Orr mechanism. See Xu et al. [20] for more details and for a comprehensive parametric exploration.
