*3.1. Temporal Modulation of Statistically Steady Pu*ff *Dynamics*

Figure 2a shows the typical intermittent behaviour of statistically steady pipe flow at *Re* = 2400. Here, the vorticity is viewed from a reference frame co-moving at the constant bulk speed *us* = *ubt*. This case was run for 6000 convective time units ( *<sup>D</sup> us* ) or an equivalent to more than 100 periods beforehand in order to relax from its initial conditions and to let the flow develop its typical patchy and intermittent character: Turbulence is spatially localised and surrounded by laminar regions of relative calm. The time scale of laminar–turbulent interactions is on the order of 100 *<sup>D</sup> us* , as can be seen in Figure 5a, where we plot the temporal evolution of the turbulent (volume) fraction (*Ft* = *<sup>V</sup>*turb *<sup>V</sup>*pipe ) in the computational pipe domain (*V*pipe). It changes considerably every two or three hundred time units, reflecting the interactions visible in the corresponding space–time diagram. We computed *Ft* based on the streamwise vorticity plotted in Figure 2a and the threshold to separate laminar regions (deep blue in Figure 2) from turbulent ones was set to ω<sup>2</sup> *<sup>z</sup>* = <sup>4</sup> <sup>×</sup> <sup>10</sup>−<sup>2</sup> in order to match the average turbulent fraction reported by Avila and Hof [12] at *Re* = 2400 (approximately 50%, as in Figure 5b).

We started all pulsatile IC SSPF runs from the same initial flow field and set the initial time to *t <sup>T</sup>* = 0.25 to match the instantaneous bulk velocity of the pulsation (see Equation (1)) to the one of the steady flow. This ensured a smooth evolution from the initial condition and further allowed us to track the exact same realisations of localised flow structures in space and time as *A* was increased. The resulting spatiotemporal dynamics are shown in Figure 2b–d,f in a frame co-moving at the instantaneous bulk speed

$$\mathbf{x}^\*(t) = \int\_{t\_0}^t \boldsymbol{\mu}\_b \mathbf{d}t. \tag{9}$$

Already at *<sup>A</sup>* = 0.2, the time scale of the flow modulation (*<sup>T</sup>* <sup>≈</sup> <sup>60</sup> *<sup>D</sup> us* ) dominates the dynamics (Figure 2b). In general, as the amplitude of the pulsation increases, the turbulent fraction in the flow decreases, as seen in Figure 5b. Many structures in the initial flow field decay quickly and do not survive the first acceleration (AC) phase. At *A* = 0.2, only two puffs survive after *t* = 5*T*, and the dynamics appear to reach an equilibrium state that repeats cyclically. The two surviving puffs grow in intensity and in length during the early deceleration (DC) phase of the flow, and they split into two in the late stages of DC before the minimum flow rate is reached. Out of these two, only the upstream puff survives the entire AC phase and reaches the peak flow rate, where this cycle starts over. Indeed, it is well known that for SSPF, only the upstream puff survives in puff interactions [26]. Overall, it appears that for *A* = 0.2, the flow is clearly self-sustained (above the critical point) and that a successful splitting event may occur at later times. However, the length of the computational domain (100*D*) may not be sufficient to accommodate three puffs without strong interactions due to the periodic boundary conditions. Similar results were obtained for *A* = 0.4 and 0.5 and are shown in Figure 2c,d; the question of whether, in these cases, the puffs will ultimately decay or successfully split would require substantially longer runs than those performed here and is not further pursued. Figure 6 shows typical localised structures at four equispaced points of the cycle for *A* = 0.5, illustrating the cyclically occurring splitting attempts. In agreement with Xu et al. [15], Xu and Avila [16], these figures show that the surviving puffs (Figure 6d,c) are very similar to the puffs in the steady case (Figure 1c) even at this relatively large pulsation amplitude. For *A* = 0.6, no turbulent structure survived the first pulsation period, and the flow fully relaminarised. We checked amplitudes up to *A* = 1.4 (see Figure 5b). In general, with increasing amplitude, the downstream puff separates farther away from the upstream puff during AC before it dies at almost the end of AC.

**Figure 6.** Instantaneous representation of localised turbulent structures in a pulsatile pipe flow (*Re* = 2400, *Wo* = 8, *A* = 0.5). Grey surfaces represent low-speed streaks (*u <sup>z</sup>* = −0.4*us*) and blue/red surfaces represent positive/negative axial vorticity (ω*<sup>z</sup>* <sup>=</sup> <sup>±</sup><sup>8</sup> *us <sup>D</sup>* ). (**a**) Death of downstream puff. (**b**) Splitting event. (**c**) Growing puff. (**d**) Isolated puff. The exact location and time for each snapshot are as indicated in Figure 2d. The direction of the mean bulk flow (*us*) is always from left to right.
