*3.5. Intermittent Production and Dissipation*

In order to investigate the physical mechanisms by which puffs arise and survive in pulsatile pipe flow, we computed the production and dissipation of turbulent kinetic energy,

$$P\_{cl}(r) = -\langle \mu\_r' \mu\_z' \rangle\_a \frac{\langle \mu\_z \rangle\_a}{\delta r} \text{and} D\_{cl}(r) = -\frac{1}{\text{Re}} \langle \nabla \mathbf{u}' : \nabla \mathbf{u}' \rangle\_a. \tag{10}$$

Angled brackets denote averaging with respect to α and prime denotes the fluctuation around the respective average. Here, α can be any combination of averaging in the two homogeneous directions θ and *z*, as well as time *t*, or at a fixed phase φ. For the cases where puffs survive, *P*θ,*z*,<sup>φ</sup> and *D*θ,*z*,<sup>φ</sup> are strongly modulated by the pulsation of the flow, as exemplified in Figure 10 for *A* = 0.6. During AC, production and dissipation are low, whereas during DC, they are high. Peak production takes place during the early DC and is very similar to steady pipe flow in terms of magnitude and wall-normal distribution. However, at the phase of maximum production, the dissipation inside the Stokes layer is much more intense than in the steady case. Right after the peak in flow rate, the mean velocity profile develops an inflection point at the wall, which satisfies the Fjortoft criterion [6]. With ongoing deceleration, the inflection point moves away from the wall and catches up with the point of peak production. Both travel together further towards the pipe centre. Near to the minimum flow rate, the unstable inflection point loses the Fjortoft condition (because of new inflection points arising in the velocity profile,) and the production collapses. Hence, it appears that the puff is taking advantage of this inflection point during DC to survive the upcoming AC.

**Figure 9.** Instantaneous streamwise velocity profiles (*uz*) at five axial locations along the pipe for an IC SWOP simulation at *Re* = 2400, *Wo* = 8, and *A* = 0.5. To not interfere with one another, they are scaled in arbitrary physical units, since, in this representation, only the development in time and deviation from the SW profile are of interest. Thus, the velocity is scaled so its all-time maximum *uz*(*r*, θ = 0, *z*) is equal to 10*D*. Each profile is compared with the corresponding instantaneous SW profile (grey lines, also scaled) and its inflection point (grey circles) if they fulfil the Fjortoft criterion. The shaded grey area shows the instantaneous cross-sectional average of the streamwise vorticity ( ω2 *z r*,θ ) scaled so its all-time maximum is equal to 0.5*D*.

Figure 11 compares the production and dissipation profiles for the growth and decay of the localised helix during the first pulsation period for *A* = 1. Here, phase-logged time averaging is not possible, and averaging was performed only in the θ and *z* directions. During DC, the rate of production is negative in a small region inside the Stokes layer, meaning that turbulent kinetic energy is fed back to the mean flow and acts as an additional energy sink. This promotes relaminarisation and explains why the helix does not evolve into puff dynamics, as in the low-amplitude cases. Overall, the phenomenology is similar to that reported for oscillatory pipe flow, where negative production causes turbulence decay in cases initialised with fully developed turbulent flow fields of SSPF at high Reynolds numbers [29].
