*3.3. Nonlinear Dynamics of Helical Perturbations*

In our second set of DNS (IC SWOP), we superimposed the optimal helical perturbation scaled to a small amplitude (4 <sup>×</sup> <sup>10</sup>−<sup>2</sup> *us*) on top of the SW profile. All simulations were started at the optimal initial time of perturbation (*t*0). We used a global, as well as an axially localised, helix as initial perturbation and we varied the pulsation amplitude *A* whilst keeping *Re* = 2400 and *Wo* = 8 fixed. In all runs, the global helix exhibited rapid growth, followed by a breakdown into turbulence and immediate decay within the first period, in good agreement with the DNS of Xu et al. [17], for *A* = 0.85, *Wo* = 5.6, and a shorter pipe domain. Our results hence extend their findings to larger *A* and *Wo*, and are not explicitly shown here.

Using a localised helix as initial condition instead also led to a very similar fate for the helix, but only for *A* ≥ 0.8 (see Figure 5b). The amplification of the local helix and its subsequent death is shown in Figures 7c–e and 8a–d. For smaller amplitudes (*A* ≤ 0.8), intermittent puff turbulence emerged after the growth and decay of the initial helix and then was sustained for many periods (see Figure 5). The dynamics of the generated localised puffs are the same as described in Section 3.1 and are exemplarily shown in Figure 8e–h. The puffs that survive AC grow during early DC and attempt to split into two puffs during late DC. In the subsequent AC, the splitting downstream puff decays and leaves only the upstream puff behind to start the cycle over. Figures 2e and 7a,b compare this cycle and its initialisation phase for different amplitudes. For *A* = 0.5, a self-sustaining puff develops only from the downstream end of the amplified helix. For *A* = 0.6, puffs develop from both ends of the localised helix. Shortly thereafter, both puffs interact, which leads to the death of the downstream puff (similar to what happens, for example, in Figure 2b). For *A* = 0.8, a puff develops only from the upstream end of the amplified helix. For this case, the puff is able to survive for four periods before the flow completely relaminarises.

For both large and small amplitudes, the initial optimum perturbation energy is amplified by about two orders of magnitude (Figure 8), which is much less than in the linear case. It is worth noting that perturbations obtained with a non-linear non-modal stability analysis should yield a more effective growth [27]. These methods would help to avoid the discrepancy between linear and non-linear behaviour of perturbations at least before their complete saturation, and should be considered in future works. Our linear optimum perturbation, once introduced into the DNS, also moves towards the bulk region of the pipe; however, before it can complete the growth predicted in the linear analysis, it breaks up into turbulent spots arising at its upstream and downstream ends. For the larger amplitudes, the helix further narrows and develops a turbulent puff with a central low-speed streak before decaying. For the lower amplitudes, on the other hand, the helix opens up again and develops a turbulent spot with several low-speed streaks closer to the wall.

We used a hyperbolic tangent, as in Equation (7), to localise the radial and axial velocities of the helix perturbation in the *z* direction with the parameters *Mz* = <sup>20</sup> *<sup>D</sup>* and *Lz* = 5*D*. The azimuthal velocity was calculated to preserve the divergence-free condition. For a perturbation magnitude of 4 <sup>×</sup> <sup>10</sup>−<sup>2</sup> *us*, this procedure leaves a remainder of the helix in the rest of the domain, which is everywhere < 10−7*us*. This remainder grows dramatically and results in the white bands visible in Figures 2e and 7a–e.

**Figure 7.** 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 co-moving reference frame (*z*∗ ). For pulsatile pipe flow at (*Re* = 2400, *Wo* = 8). (**a**–**e**) For different pulsation amplitudes *A*, always using the SWOP initial condition. Note that the optimal time of perturbation slightly changes with *A*. The horizontal straight lines mark regions for which three-dimensional representations of the localised flow structures are shown in Figure 8. (**f**) For a permanent body force and the unperturbed SW velocity profile as initial condition. The curved black line represents the fixed location of the highly localised body force viewed from the co-moving reference frame. The direction of the mean bulk flow (*us*) is always from left to right.


**Figure 8.** Instantaneous representation of localised turbulent structures in a pulsatile pipe flow DNS at *Re* = 2400, *Wo* = 8, and two different amplitudes. (**a**–**d**) Growth and decay of an initial helix at *A* = 1.0. (**e**–**h**) Development of a puff at *A* = 0.5. Both DNS were initialised at *<sup>t</sup> <sup>T</sup>* = 0.2 using the SWOP initial condition. 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>* ). The exact location for each snapshot is as indicated in Figures 2e and 7c, respectively. (**a**) Decay. (**b**) Breakdown into turbulence. (**c**) Amplification of helix. (**d**) Localised optimal helix perturbation. (**e**,**f**) Birth of a downstream puff. (**g**) Amplification of helix. (**h**) Localised optimal helix perturbation. Note that the initial perturbation is two orders of magnitude smaller. The direction of the mean bulk flow (*us*) is always from left to right.
