**1. Introduction**

The dynamics and intermittency of transitional turbulence in statistically steady pipe flow have been extensively studied for over a century [1–4], and the underlying mechanisms are reasonably well understood [5]. The only control parameter is the Reynolds number (*Re* = *ub<sup>t</sup> D* <sup>ν</sup> ), which quantifies the relative magnitude of inertia and viscous forces in the system. Here, *ub*, *D*, and ν denote the bulk velocity, the pipe diameter, and the fluid's kinematic viscosity, respectively. Angled brackets indicate an averaging operation with respect to time (*t*). Although statistically steady pipe flow is linearly stable [6,7], turbulence can be triggered with finite-amplitude perturbations [1,8]. Independently of their type [9], if successful, these perturbations result in spatially localised turbulent puffs, provided that the Reynolds number is not too high [10]. More specifically, for *Re* < 2250, puffs can remain in equilibrium for long times until they either proliferate or decay. Both processes are stochastic (memoryless) and, beyond the critical point (*Re* > 2040), ultimately lead to patterns consisting of several puffs separated by quiescent flow regions [5,11]. For *Re* > 2250, the spatiotemporal dynamics become much richer. Here, puffs may grow and split into two, as for lower Reynolds numbers, or expand continuously to become slugs (see Figure 1). In addition, laminar holes may appear inside the slugs and eventually close, leading to a merger of structures [12]. Figure 2a provides a representation of the resulting spatiotemporally intermittent behaviour of localised turbulent structures (red) and laminar islands (blue) at *Re* = 2400.

**Figure 1.** Instantaneous representation of localised turbulent structures in a statistically steady pipe flow (*Re* = 2400, *A* = 0.0). Grey surfaces represent low-speed streaks (*u <sup>z</sup>* = −0.4*ub*) and blue/red surfaces represent positive/negative axial vorticity (ω*<sup>z</sup>* <sup>=</sup> <sup>±</sup><sup>6</sup> *ub <sup>D</sup>* ). (**a**) Puff splitting. (**b**) Single puff. (**c**) Weak slug. The exact location and time for each snapshot are indicated in Figure 2a. The direction of the mean bulk flow (*us*) is always from left to right.

In many systems, internal fluid transport is statistically unsteady. Pumps never run perfectly uniformly, blood flow in arteries is pulsatile (due to the systolic contractions of the heart), and air oscillates in and out of the lungs while breathing. A simple mathematical model for these examples is pipe flow driven at a harmonically varying rate

$$
\mu\_b(t) = \langle \mu\_b \rangle\_l \Big( 1 + A \cdot \cos \left( 2\pi \frac{t}{T} \right) \Big). \tag{1}
$$

In this case, two more control parameters come in to play in addition to the Reynolds number. The Womersley number (*Wo* = *<sup>D</sup>* 2 2<sup>π</sup> *<sup>T</sup>*<sup>ν</sup> ) quantifies the relative magnitude of the viscous time scale with respect to the time scale of the imposed flow pulsation, i.e., the oscillation period *T*. The amplitude (*A* = *uo us* ) is the relative strength of the oscillating component of the flow (*uo*) with respect to the steady component of the flow (*us* = *ubt*). For *A* = 0, the statistically steady case is recovered, whereas for large *A*, the purely oscillatory flow is approached (as the steady part becomes negligible). According to Sexl [13] and Womersley [14], there is an analytical solution to the Navier–Stokes equations for laminar flow through a smooth pipe and single harmonic driving. The Sexl–Womersley (SW) velocity profile (*uSW*(*r*, *t*)) can be added to the (parabolic) Hagen–Poiseuille profile to obtain an analytical (laminar) solution for any combination of *Wo* and *A*. As an example, we show in Figure 3a the temporal evolution of *uSW* for a pulsatile pipe flow at *Wo* = 8 and *A* = 1.

Understanding the transition to turbulence in statistically unsteady pipe flows remains incomplete, although progress has recently been made [15–17]. The puff dynamics for relatively small amplitudes (*A* ≤ 0.5) are well understood. For *Wo* ≤ 5, the flow stays for a long time in the low Reynolds number regime. A low instantaneous *Re* enhances the decay of puffs, and hence, puffs only survive if the mean Reynolds number is substantially increased with respect to the steady case [15,16]. For *Wo* ≥ 12, the minimum Reynolds number necessary for puffs to survive tends asymptotically to the one for statistically steady pipe flow [15,16,18,19]. For intermediate Womersley numbers, the threshold decreases smoothly from the low to high *Wo* regime [15,16]. This can be seen, for example, in Figure 8 of Xu et al. [15].

Puffs, however, are not the only mechanism through which pulsatile pipe flow may become turbulent. A new instability was discovered recently in laboratory experiments by Xu et al. [17]. In their experiments, curvature, misalignment of pipe segments, small contractions, and, in general, finite-size geometric imperfections led to the cyclic development of sudden bursts of turbulence. At each period, helical-like structures grew and triggered turbulence during the deceleration phase of the pulsation before the flow relaminarised again during the acceleration phase. This behaviour was observed for relatively high amplitudes (*A* ≥ 0.5), intermediate Womersley numbers (5 ≤ *Wo* ≤ 8), and mean Reynolds numbers as low as *Re* = 800. Motivated by this finding, Xu et al. [20] carried out a comprehensive non-modal stability analysis of pulsatile pipe flow. They showed that certain helical perturbations exploit an Orr-like mechanism to grow by several orders of magnitude in energy. They linked this mechanism to the inflection points of the SW velocity profile that emerge during the deceleration phase (see Figure 3a–c). Inflectional SW velocity profiles are indeed known to be linearly

unstable in the quasi-steady limit [21], as long as they satisfy the Fjortoft criteria. The smaller the Womersley number, the longer the velocity profile is unstable, thus effectively providing a more fertile ground for instabilities to grow. However, as the Womersley number is reduced, the velocity profile becomes increasingly parabolic and, hence, loses its inflection points. The amplification of helical disturbances is most efficient for *Wo* ≈ 7 [20], exactly in the regime where the helical instability was observed experimentally [17].

**Figure 2.** Spatiotemporal representation of the turbulence activity in the computational 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. Steady (*A* = 0, **a**) and pulsatile (**b**–**f**) pipe flow at *Re* = 2400, *Wo* = 8, and different amplitudes *A*. Initial conditions for all *A* - 0 were either taken from the steady case at time *<sup>t</sup> <sup>T</sup>* = 0.25 (**b**–**d**,**f**) or composed of a localised helical perturbation on top of the laminar Sexl–Womersley velocity profile (**e**).

**Figure 3.** Sexl–Womersley (SW) flow and its optimal perturbation for (*Re* = 2400, *Wo* = 8, *A* = 1.0). (**a**) Time-dependent velocity profile (*u*SW) for 20 equispaced points within one pulsation period (*T*). Circles denote the maximum and minimum peak flow (PF), whereas upward- and downward-facing triangles denote phases of acceleration (AC) and deceleration (DC), respectively. (**b**) Optimal helical perturbation during DC ( *<sup>t</sup> <sup>T</sup>* = 0.2) according to our transient growth analysis based on the linearised Navier–Stokes equations. To be used as initial condition in our direct numerical simulation (DNS) (Section 3.3), the helix is scaled to an amplitude of 4 <sup>×</sup> <sup>10</sup>−<sup>2</sup> *us*. (**c**) Evolution of the optimal perturbation under the constraints of the linearised Navier–Stokes equations at the later time of maximal energy amplification. Note that, in the framework of transient growth analysis, the absolute amplitude of the initial helix is not important; only the relative growth rate is of interest. The dashed lines correspond to the Stoke layer thickness (δ).

The purpose of this paper is to investigate the spatiotemporal intermittency of turbulence in pulsatile pipe flow. More specifically, we aim to characterise the intermediate regime in which helical structures and puffs are expected to compete. To that end, we perform transient growth analysis and direct numerical simulations of pulsatile pipe flow at fixed *Re* = 2400 and *Wo* = 8, as well as different pulsation amplitudes *A*.
