4 Discussion

The often conflicting results reported in previous studies of the onset of turbulence in pulsatile flows reflect the difficulty in quantifying the transition process in linearly stable flows, in particular if there are several control parameters. Puffs are the first turbulent structures to arise in the subcritical regime of pulsatile pipe flow, and lifetimes of turbulent puffs provide a natural and very accurate measure of the transition threshold, because small changes in the governing parameters can change mean lifetimes by several orders of magnitude. Our detailed studies of puff survival probabilities show that the transition threshold overall decreases with $\unicode[STIX]{x1D6FC}$ despite the fact that most earlier studies (Nerem et al. Reference Nerem, Seed and Wood1972; Nerem & Seed Reference Nerem and Seed1972; Falsetti et al. Reference Falsetti, Carroll, Swope and Chen1983; Peacock et al. Reference Peacock, Jones, Tock and Lutz1998) reported the opposite trend. A main difference is that while in the present study turbulence is triggered actively for each measurement, in their case the flow was left to its own devices without any knowledge or control of the perturbation levels. Turbulence in their studies may have hence existed at considerably lower parameters but simply may have not been triggered, lacking sufficiently strong disturbances. If perturbation levels to trigger turbulence depend on $\unicode[STIX]{x1D6FC}$ or if the noise levels present in the experiments change with $\unicode[STIX]{x1D6FC}$ , experiments lacking an active triggering mechanism become very difficult to interpret.

Our observations that the transition threshold remains unchanged in the large frequency limit ( $\unicode[STIX]{x1D6FC}>12$ ) agrees very well with the earlier studies by Stettler & Hussain (Reference Stettler and Hussain1986), Trip et al. (Reference Trip, Kuik, Westerweel and Poelma2012). Note that in these two studies turbulence was actively triggered (just like in our study). Once the Womersley number decreases below $10$ however, puff lifetimes decrease notably and transition is delayed to larger $Re_{s}$ . While this regime was not covered by Trip et al. (Reference Trip, Kuik, Westerweel and Poelma2012), Stettler & Hussain (Reference Stettler and Hussain1986) had also reported transition delay for $5<\unicode[STIX]{x1D6FC}<10$ although threshold values between their and our study differ by ${\sim}20\,\%$ . In contrast to our study they found however that for $\unicode[STIX]{x1D6FC}<5$ the transition threshold decreases. On the other hand it should be noted that the pipe length used in their study is insufficient to investigate such low Womersley numbers: turbulent structures created at the inlet can only experience part of the oscillation cycle as they were flushed out of the pipe before a full cycle finished. The average $Re$ that turbulence experiences over such a partial cycle then depends on the phase at which turbulence is created, which will necessarily cause deviations from the true transition threshold. In particular turbulent structures that traverse the pipe during the faster half of the cycle ( $Re(t)>Re_{s}$ ) experience a larger effective Reynolds number and will reach the end of the pipe, although in a longer pipe they would have decayed during the slower half of the cycle ( $Re(t)<Re_{s}$ ).

When varying pulsation amplitudes, we observed that the transition delay increases monotonically. This delay with increasing amplitude is observed for all frequencies studied here, but it is much more pronounced at low $\unicode[STIX]{x1D6FC}$ . Earlier studies had reported a transition delay in this regime (Sarpkaya Reference Sarpkaya1966; Stettler & Hussain Reference Stettler and Hussain1986), but they reported that at larger amplitudes the transition moved back to lower Reynolds number, e.g. Sarpkaya (Reference Sarpkaya1966) reported that for fixed $\unicode[STIX]{x1D6FC}(=5.8)$ as the pulsation amplitude increases from $A=0.5$ to $A=0.6$ $Re_{s}=3800$ decreases to $Re_{s}=3000$ , which is not confirmed in our study. It should however be noted that for larger amplitudes other instabilities may arise and the present study only considers the subcritical transition to turbulence via puffs.

The surprising observation that in the small Womersley number limit the upward shift of the transition point persists can be understood when looking at the growth and decay processes of turbulence. This argument is most easily illustrated for the simpler case of a square-wave pulsation, i.e. where the Reynolds number jumps between two levels. We have carried out a direct numerical simulation setting the amplitude to $40\,\%$ and the mean Reynolds number to $2040$ (corresponding to the critical point of steady pipe flow, Avila et al. (Reference Avila, Moxey, de Lozar, Avila, Barkley and Hof2011)). The simulation was carried out in a $100D$ long pipe with periodic boundary condition imposed in the axial direction using the hybrid finite difference Fourier spectral method described in Willis & Kerswell (Reference Willis and Kerswell2009). The spatial resolution is $64\times 96\times 2560$ grid points in radial, azimuthal and axial directions. At $Re=2856$ , i.e. the faster half, we perturbed the flow with a puff that we simulated at $Re=2040$ in the same pipe. Then the puff starts to grow turning into a slug. At the jump to the lower cycle at $Re=1224$ , we abruptly changed the mass flux (mean flow speed) while keep the turbulent fluctuations unchanged. This mimics the situation in experiment in which the mass flux is changed quickly by, e.g. abruptly adjusting the speed of the driving piston. An oscillation period of $120$ advective time units (corresponding to a Womersley number of $5$ ) was simulated.

As shown in figure 6 during the faster half of the square-wave cycle ( $Re=2856$ , bottom half of figure 6) turbulence rapidly expands. After half of the oscillation period ( $60$ time units) $Re$ changes to $1224$ , and turbulence abruptly decays (top half of figure 6). A key difference is that in the faster part of the cycle the turbulent patch expands outwards from its interfaces, whereas at low $Re$ turbulence collapses throughout (rather than receding from its interfaces). The decay is much faster and clearly dominates over the growth process when viewed over one full pulsation period. This imbalance between the decay and growth implies that in the low Womersley number regime turbulence can only become sustained at significantly larger $Re$ than in steady pipe flow, in qualitative agreement with the experimental observations.

Figure 6.

Direct numerical simulation of a puff in a square-wave motion at $Re_{s}=2040$ with $A=0.4$ . From bottom to top, each snapshot shows the contour of the streamwise vorticity normalised by mean velocity of $Re_{s}$ , from $t=0$ to $t=T$ with an equal time interval of $T/8$ .

For sinusoidal pulsation the collapse of turbulence will be less abrupt than in above example due to the smoother Reynolds number decrease, nevertheless it will equally dominate the prior growth if Womersley number are small. In particular we would expect that here, for sufficiently slow Reynolds number variations, expanding turbulent structures (i.e. slugs) collapse to puffs when $Re$ decreases to the puff regime. To investigate if at sufficiently low Womersley numbers this proposed structural adjustment of turbulence indeed occurs we compare flows at $\unicode[STIX]{x1D6FC}=$ 8, 5 and 3. In all three runs the mean Reynolds number is equal to $3000$ while the pulsation amplitude is $0.4$ . In each case the laminar flow is perturbed at $t=T/4$ where the flow speed is at its maximum ( $Re=4200$ ). We then follow the evolving turbulent flow downstream and visualise the flow field at $t=3T/4$ , i.e. at the minimum flow speed ( $Re=1800$ ). The main difference among the three runs is the rate at which the flow speed sinusoidally adjusts between the maximum and minimum Reynolds number: for $\unicode[STIX]{x1D6FC}=8$ the time that elapses corresponds to $(T/2=)37.5D/U_{s}$ , while for $\unicode[STIX]{x1D6FC}=3$ the Reynolds number adjustment is spread out over a much longer time of $250D/U_{s}$ . At $\unicode[STIX]{x1D6FC}=8$ turbulence takes the form of a slug-like structure (figure 7 a), whereas in a steady flow at the same Reynolds number ( $1800$ ) slugs are unstable and only much shorter puffs can be found (see figure 7 d). The fact that in this case (figure 7 a) an elongated turbulent structure is found despite the low instantaneous Reynolds number must be attributed to the relatively fast drop from a much higher Reynolds number. The Reynolds number change appears to be too fast for turbulence to adapt to the structure characteristic for the instantaneous speed. While puffs are prone to decay, slugs are stable and hence turbulence at $Wo=8$ , $Re_{s}=3000$ is sustained and well above the transition threshold. At the intermediate Womersley number (figure 7 b) the slug observed at the minimum of the cycle (i.e.  $Re=1800$ ) is shorter than at $\unicode[STIX]{x1D6FC}=8$ but still larger than the structures would be under steady flow conditions at this instantaneous $Re$ . Finally for $\unicode[STIX]{x1D6FC}=3$ at the cycle minimum turbulence has reduced to a puff (figure 7 c), structurally equivalent to puffs in steady pipe flow at this Reynolds number (figure 7 d). Although during the faster part of the cycle turbulence had spread considerably (not shown) once $Re$ drops sufficiently (here shown for $Re=1800$ ), the extended slug (in this example ${>}100D$ at its maximum length) collapses and in this case only a single puff (at the position of the slug’s trailing edge) remained. It hence appears that at low Womersley numbers where Reynolds number changes are much slower, turbulent structures can more readily adapt to the instantaneous Reynolds numbers. As Womersley numbers further decrease one may expect that eventually turbulent structures follow the very slow changes of Reynolds number in a quasi-steady fashion. In this case the survival of turbulence would be decided over the low Reynolds number phase where turbulence consists of distinct puffs with finite lifetimes. In this limit it should then be possible to estimate the transition threshold based on puff lifetimes by integrating over the corresponding steady flow values for the corresponding Reynolds numbers. For instance, the survival probability $P$ is obtained by integration over the instantaneous decay rates which are $\unicode[STIX]{x1D705}(t)\unicode[STIX]{x1D6FF}t$ at time $t$ , where $\unicode[STIX]{x1D705}$ is the escape rate (Avila et al. Reference Avila, Willis and Hof2010):

Figure 7.

Flow structures recorded at an instantaneous Reynolds number $Re(t)=1800$ for various $\unicode[STIX]{x1D6FC}$ : (a) $\unicode[STIX]{x1D6FC}=8$ , (b) $\unicode[STIX]{x1D6FC}=5$ , (c) $\unicode[STIX]{x1D6FC}=3$ ; (d) steady pipe flow at $Re=1800$ . For the pulsating flows (a–c) all other parameters are identical $Re_{s}=3000$ and $A=0.4$ and the images were recorded at the minimum of the cycle (i.e. at $Re=1800$ ). At $\unicode[STIX]{x1D6FC}=8$ turbulence forms an elongated slug, despite the low instantaneous Reynolds number ( $1800$ ). For $\unicode[STIX]{x1D6FC}=3$ on the other hand the observed flow structure is a puff in close correspondence to puffs in steady flow of the same $Re$ (d).

Figure 8.

Estimates of the transition threshold, obtained by determining the Reynolds number value where puffs have a $50\,\%$ survival probability (i.e.  $Re_{s}(\unicode[STIX]{x1D6FC})|P=0.5$ ) over the respective pipe length $L$ of the three different pipe set-ups. For comparison the (purple) dashed curve shows the $50\,\%$ survival probability over a single period obtained from the quasi-steady assumption. For $\unicode[STIX]{x1D6FC}\rightarrow 0$ the curve asymptotes towards an upper bound ( $Re_{s}=3400$ for $A=0.4$ ) (see inset). The black dotted line marks the $50\,\%$ survival probability for $L=350D$ (same as red circles) for steady pipe flow.

Here $[t_{1},t_{2}]$ is the time interval of the pulsation cycle for which $Re\leqslant 2040$ and the escape rate $\unicode[STIX]{x1D705}$ is equal to the inverse of the median lifetime $\unicode[STIX]{x1D70F}(Re)$ which for steady pipe flow is $\unicode[STIX]{x1D70F}=\text{exp}(\text{exp}(0.00556Re-8.5))$ (Avila et al. Reference Avila, Moxey, de Lozar, Avila, Barkley and Hof2011). For fixed values of $\unicode[STIX]{x1D6FC}$ we then determine from (4.1) the mean Reynolds number ( $Re_{s}$ ) at which $50\,\%$ of the puffs survive over one period, i.e.  $Re_{s}|P=0.5$ . When decreasing the Womersley number while keeping $Re_{s}$ fixed, the length of the time interval $[t_{1},t_{2}]$ increases and consequently the survival probability (equation (4.1)) decreases. In order to retain the same survival probability (i.e.  $P=0.5$ ) at lower $\unicode[STIX]{x1D6FC}$ the mean Reynolds number has to go up. Therefore the quasi-steady assumption predicts monotonically increasing threshold values $Re_{s}|P=0.5$ for decreasing $\unicode[STIX]{x1D6FC}$ . This at least qualitatively agrees with the increasing trend observed in experiments. To test the quantitative agreement the results of the quasi-steady analysis are shown for comparison in figure 8 (dashed curve). For $\unicode[STIX]{x1D6FC}<2.5$ they indeed closely capture the values measured in the pulsating flow experiment. It would appear that for the long period times (and hence the slow Reynolds number variations at these low Womersley numbers) flows indeed can be considered as quasi steady. The threshold increase predicted by the quasi-steady assumption is bounded from above because decay rates become zero at $Re=2040$ (as suggested by Avila et al. (Reference Avila, Moxey, de Lozar, Avila, Barkley and Hof2011)). If $Re$ remains above $2040$ for the entire cycle turbulence ceases to decay. Determining the $P=0.5$ threshold from (4.1) for very low Womersley number values $Re_{s}|P=0.5$ indeed levels off at a value of $3400$ , as shown in the inset of figure 8. Here the minimum Reynolds number of the cycle is $3400(1-A)=2040$ , the point where decay rates become zero. Unfortunately due to the very long periods at these low Womersley numbers an experimental verification of this predicted asymptotic value is not possible.