*3.6. E*ff*ect of Local Geometric Imperfections*

We performed a third set of DNS using the laminar SW velocity profile as the initial condition and the volume force described in Section 2.4. For this third set of simulations, we considered only the four amplitudes *A* ∈ {0, 0.5, 1.0, 1.4}. In line with the experiments of Xu et al. [17], we found no transition to turbulence at all for the axisymmetric contraction in all cases considered. By contrast, when the force is localised in all three dimensions, the response of the flow depends strongly on the amplitude of the pulsation. For *A* = 0, i.e., statistically steady pipe flow, there is no surge of turbulence or localised transition arising from the bump. This confirms that our force represents a small perturbation to the flow. As we increase the amplitude to *A* = 0.5, some vorticity is generated

during peak flow rate, but no turbulent dynamics develop (see Figure 12a). The picture changes for amplitude *A* = 1. As shown in Figure 12b, during early DC, the presence of the bump is able to trigger turbulence in every period. Turbulence grows until late DC, and then laminarises. Occasionally, puffs emerge and are able to survive for more than one period if they interact (again) with the local bump due to the periodic boundary conditions used in our DNS. However, if the upstream puff interacts with a new turbulent spot arising from the bump, both die. For *A* = 1.4, on the other hand, no puffs develop, and the dynamics are solely characterised by bursts of turbulence arising at the bump, which proceed downstream as they decay (see Figure 12c). In all cases, the time at which the perturbation is triggered and grows is in agreement with our non-modal stability analysis and with the experiments of Xu et al. [17].

**Figure 10.** Production (**a**) and dissipation (**b**) of turbulent kinetic energy compared for different phases of the pulsation period for *A* = 0.6 using the SWOP initial conditions. Averages are taken over spaceand phase-logged time instants (α = θ, *z*,φ) over four periods of puff dynamics, excluding the initial period without puffs. Circles denote the existence and wall-normal location of the inflection points of the corresponding mean profile <sup>∂</sup>2*uz*φ,θ,*z*/∂2*<sup>r</sup>* = 0 that satisfy the Fjortoft criterion. The vertical dashed line denotes the Stokes layer.

**Figure 11.** Production (**a**) and dissipation (**b**) of turbulent kinetic energy compared for different phases of the initial pulsation period for *A* = 1 using the SWOP initial conditions.

**Figure 12.** Spatiotemporal representation of the turbulence activity in the pipe domain based on the cross-sectional average of the streamwise vorticity (ω*z*) plotted on a logarithmic scale and in a stationary reference frame. Pulsatile pipe flow at *Re* = 2400, *Wo* = 8, and different amplitudes *A*. Initial conditions are based on the Sexl–Womersley velocity profile, and there is a permanent body force. (**a**–**c**) Local bump. (**d**) Tilted bump.

Interestingly, the structures that the local bump triggers are mirror symmetric and not helical (see Figure 13a–d). They resemble structures resulting from the optimal non-modal disturbances in pulsatile pipe flow past a constriction [25]. They grow in axial length and magnitude during the late stages of DC while retaining their mirror symmetry. This is only lost in the last stages of DC, as low-velocity streaks form in the centre of the pipe. Finally, either a puff emerges from these streaks, or the flow laminarises during AC.

We also performed simulations with a tilted bump. In this case, the emerging structures exhibit not only mirror-symmetric, but also helical-like features (see Figure 13e–h). They also grow during the late stages of DC and either decay or trigger puffs depending on the pulsation amplitude. From the point of view of spatiotemporal intermittency, their evolution is quite similar to the evolution of the structures triggered by the local bump for all the amplitudes considered, as exemplarily shown in Figure 12c,d.

The fact that different geometric disturbances can trigger different structures is consistent with the non-modal stability analysis of Xu et al. [20]. The analysis showed that the instantaneous Sexl–Womersley profile is linearly unstable (in the quasi-steady limit) during most of the DC phase. Out of all the perturbations that could grow on top of this unstable profile, helical modes have the highest potential to do so. This holds for helical modes spiralling in positive and negative axial

directions. That means that, if we were to disturb a flow in a way such that helical modes are excited, unless there is a preferred direction, helical modes and their swirling counterparts can grow simultaneously on top of the laminar flow profile. Our local bump represents a highly symmetric perturbation that allows this to happen, which explains why we observe mirror-symmetric structures. If we introduce some non-symmetric perturbation instead, then we see a preferred direction for the structures to swirl, as in the simulations with the tilted bump.

**Figure 13.** Instantaneous representation of localised turbulent structures in a pulsatile pipe flow DNS at (*Re* = 2400, *Wo* = 8, *A* = 1.4). The DNS was initialised at *<sup>t</sup> <sup>T</sup>* = 0.25 using the corresponding SW profile and by introducing a local bump like body force, as described by Equation (3) and Table 1. Grey surfaces represent low-speed streaks (*u <sup>z</sup>* = −0.2 *us*) and blue/red surfaces represent positive/negative axial vorticity (ω*<sup>z</sup>* <sup>=</sup> <sup>±</sup><sup>2</sup> *us <sup>D</sup>* for all panels except (**d**) and (**h**). There, it is <sup>±</sup>0.8 *us <sup>D</sup>* . (**a**–**d**) Local bump. (**e**–**h**) Tilted bump. The exact instants in time are given in Figure 12c,d. The direction of the mean bulk flow (*us*) is always from left to right.
