3 Spatio-temporal dynamics of pulsatile turbulence

Instantaneous snapshots of a simulation at $Re=2040$ , $A=0.4$ and $Wo=7.2$ are shown in figure 2(b–e) in a frame co-moving with the mean flow speed. Throughout the cycle, the turbulent region grows and shrinks with a certain delay with respect to the instantaneous Reynolds number $\widetilde{Re}(t)$ , but preserves the characteristic shape of a puff. The spatio-temporal dynamics of the flow is visualized in figure 4(c) and exhibits incipient stages of puff splitting and decay. Turbulence expands and retreats from the downstream interface, and the propagation speed of the puff oscillates about the mean flow speed. This is not surprising because in steady pipe flow puffs propagate at nearly the mean flow speed at $Re=2040$ (Avila et al. Reference Avila, Moxey, de Lozar, Avila, Barkley and Hof2011). As $Wo$ is reduced, while keeping $Re=2040$ and $A=0.4$ constant, puffs have more time to grow and shrink during the cycle. This leads to splitting and relaminarization, as exemplified in figure 4(d) for $Wo=5.6$ . Figure 4(e,f) shows that, by further reducing $Wo$ , the flow responds quasi-statically. Turbulence expands continuously as a slug while $\widetilde{Re}(t)$ increases, but collapses irreversibly during the lower half-cycle. By contrast, at high $Wo$ , the turbulent dynamics has little time to react to the rapid pulsation of the flow rate, and the behaviour of steady pipe flow is gradually recovered (see figure 4 a,b).

Figure 4.

Space–time diagrams of localized turbulence at $Re=2040$ and $A=0.4$ in a frame co-moving at the mean speed. Colour maps of $\bar{q}(z,t)$ are shown with the same colour scale as in figure 2. Note that the pipe length (horizontal axis) was varied with $Wo$ to avoid interaction of the upstream and downstream interfaces through the imposed periodic boundary conditions.

Figure 5.

Space–time diagrams of localized turbulence at $Re=2040$ and $A=0.2$ in a frame co-moving at the mean speed. The colour maps are as in figure 4.

Figure 5 shows analogous space–time diagrams also at $Re=2040$ , but for lower pulsation amplitude $A=0.2$ . Here the instantaneous Reynolds number $\widetilde{Re}(t)$ varies across a narrower range and this permits smaller variations in the size of turbulence. This hinders the occurrence of both splitting and relaminarization, enabling turbulence to survive for longer times. Together, the visualizations of figures 4 and 5 suggest that pulsation shifts the transition threshold to larger $Re$ . Reducing $Wo$ or increasing $A$ considerably enhances the stability of the laminar pulsatile flow.

These visualizations also highlight the need to use long pipes in experiments in order to capture the asymptotic dynamics of the system at low pulsation frequency. For instance, let us consider the case $(Re,A,Wo)=(2040,0.4,2.0)$ shown in figure 4(f). In the experiments of Stettler & Hussain (Reference Stettler and Hussain1986), localized turbulence would reach the end of the pipe at $t/{\mathcal{T}}\approx 0.41$ (space was converted into time via the mean flow speed). In their experiments, a long slug would be observed to reach the end of the pipe and $Re=2040$ would be taken as turbulent. However, turbulence begins decaying at the downstream interface at $t/{\mathcal{T}}\approx 0.5$ and has collapsed completely by $t/{\mathcal{T}}\approx 0.7$ . Hence $Re=2040$ is in fact below the transition threshold if a full cycle is considered.

Figure 6.

Probability density functions (p.d.f.) of turbulent length $L_{T}$ at $Re=2040$ , $A=0.4$ and several $Wo$ . The data with $L_{T}<5$ are shown as dotted lines because turbulence starts to decay irreversibly. Reference data for a puff in steady pipe flow are shown as a grey thick line.

The effect of pulsation frequency on the length of the turbulent region $L_{T}$ was systematically studied by performing 20 runs at each $Wo$ shown in figure 4. Here a threshold on the turbulence intensity ( $\bar{q}_{c}=2\times 10^{-3}$ ) was set to distinguish the beginning and end of turbulent flow regions. In the presence of split puffs, $\bar{q}$ may fall below this threshold in the gap between puffs. In such cases, $L_{T}$ was taken as the length encompassing all puffs. We note that larger (smaller) values of $\bar{q}_{c}$ result in shorter (longer) turbulent length, but such difference is trivial for the following analysis. Figure 6 shows the probability density function (p.d.f.) of $L_{T}$ . At $Wo=17.7$ , $L_{T}$ is narrowly distributed about $9.3$ (in steady pipe flow, shown in grey, $L_{T}\approx 9$ ), and no effect of the pulsation is observed because $\widetilde{Re}(t)$ varies too rapidly for the puff to adapt. At $Wo=10.2$ the length begins to adjust to the flow pulsation and varies between $16$ and $5$ , centred around a much broader peak at $10$ in the p.d.f. By further reducing to $Wo=7.2$ , the p.d.f. turns bimodal with a peak at $L_{T}=8$ corresponding to one puff, and another peak at $14$ corresponding to the incipient stages of splitting (see figure 4 c). At $Wo=5.6$ splitting events occur often and two clear peaks at $L_{T}=7$ and $20$ , corresponding to one and two puffs, can be clearly discerned. As $Wo$ is further reduced, the distribution becomes progressively flat because of the continuous expansion of the turbulent slug (see e.g. figure 4 e). Note that when $L_{T}\lesssim 5$ puffs begin to decay irreversibly and the flow fully relaminarizes.

Figure 7.

(a) Space–time diagram of localized turbulence at $(Re,A,Wo)=(2500,0.4,3.5)$ in a $300D$ pipe in a frame co-moving at the mean flow speed. The dashed lines indicate fixed downstream locations in the pipe. (b–d) Time series of the cross-sectionally averaged turbulence intensity $\bar{q}$ at the downstream locations marked in (a).

Figure 7(a) shows a space–time diagram obtained at $Re=2500$ , $A=0.4$ and $Wo=3.5$ , where the dashed lines denote the position of three (virtual) measurement points at fixed downstream locations. The measured time series of $\overline{q}$ are shown in figure 7(b–d) and are analogous to time traces measured in experiments. Initially, a puff forms (figure 7 b) and quickly begins to grow as a slug. As $\widetilde{Re}(t)>2900$ the downstream interface becomes increasingly sharp and the slug appears nearly symmetric (figure 7 c). As $\widetilde{Re}(t)$ decreases, the structure begins to shrink and finally collapses entirely. A stationary observer at $z=300$ (figure 7 d) sees first the passage of the sharp downstream interface, while at the time that the rear part of the structure reaches the measurement point, turbulence is monotonically decaying. As a result, a diffuse upstream interface is seen. The signal thus resembles an inverted puff, thereby explaining the observation of Stettler & Hussain (Reference Stettler and Hussain1986), but corresponds in fact to the decay of a slug in pulsatile pipe flow. Note also that if a continuous disturbance as the orifice of Stettler & Hussain (Reference Stettler and Hussain1986) were used, then such turbulent slugs would be observed periodically at the measurement point (phase-locked turbulence).

4 Lifetimes of localized turbulence in pulsatile pipe flow

Xu et al. (Reference Xu, Warnecke, Song, Ma and Hof2017) introduced two novelties with respect to previous studies of transition in pulsatile pipe flow. First, they disturbed the flow with an injection, which allowed triggering turbulence at lower $Re$ than the orifice of Stettler & Hussain (Reference Stettler and Hussain1986) and Trip et al. (Reference Trip, Kuik, Westerweel and Poelma2012). Their injection generated a single turbulent puff and is similar to the localized pair of rolls used here to disturb the flow. Second, at a fixed $Wo$ , they measured the survival probabilities of turbulent puffs $P_{exp}(L)$ , consisting of the number of puffs detected at distance $L$ downstream of the injection, divided by the total number of realizations. This was inspired by previous studies of steady Couette and pipe flows, where the underlying relaminarization process is memoryless (Bottin & Chaté Reference Bottin and Chaté1998; Faisst & Eckhardt Reference Faisst and Eckhardt2004). More specifically, the probability of a puff surviving beyond time $t$ in steady pipe flow is given by the survivor function $S(t)=\text{exp}[-(t-t_{0})/\unicode[STIX]{x1D70F}]$ , where $\unicode[STIX]{x1D70F}$ is the mean lifetime and $t_{0}$ is the initial formation time of the puff after the laminar flow is disturbed (Hof et al. Reference Hof, Westerweel, Schneider and Eckhardt2006). The key point here is that the dependence of initial condition (disturbance method) is fully contained in $t_{0}$ , whereas $\unicode[STIX]{x1D70F}$ is uniquely determined by $Re$  (see de Lozar & Hof Reference de Lozar and Hof2009). Hence stability thresholds determined with survival probabilities are intrinsic to the flow (disturbance independent so long as a single turbulent puff is generated). Xu et al. (Reference Xu, Warnecke, Song, Ma and Hof2017) defined the transition threshold $Re_{T}(Wo)$ such that $P_{exp}(L)=0.5$ . They performed measurement with $L=350$ , $1300$ and $2250$ , as indicated by the three different triangle styles shown in figure 1 (in their experiments, $L$ was increased, as $Wo$ was decreased).

Figure 8.

Survival probability $S(t)$ of localized turbulence at $(Re,A,Wo)=(1820,0.4,17.7)$ , $(2040,0.4,7.2)$ and $(2320,0.4,5.6)$ , respectively. In (a), grey triangles (circles) show the survival probability at $Re=1820$ ( $Re=1860$ ) in steady pipe flow (Avila, Willis & Hof Reference Avila, Willis and Hof2010). In (c), grey circles show the survival probability at $(Re,A,Wo)=(2040,0.2,5.6)$ .

We investigated the effect of pulsation on turbulence relaminarization at three selected points in $(Re,Wo)$ parameter space to allow a direct comparison to Xu et al. (Reference Xu, Warnecke, Song, Ma and Hof2017) for $A=0.4$ . In each case 150 runs were performed in a pipe of length $100$ to sufficiently resolve the survivor functions. At high frequency, $Wo=17.7$ and $Re=1820$ , the relaminarization process is memoryless as shown by the corresponding survivor function (red squares) in figure 8(a). In comparison to steady pipe flow (grey triangles), the survival probability slightly increases but remains below that of steady flow at $Re=1860$ (grey circles). Regarding lifetimes, the effect of pulsation at $Wo=17.7$ is equivalent to a small shift (less than $2\,\%$ ) in Reynolds number.

At $Wo=7.2$ the relaminarization process remains approximately memoryless (figure 8 b), but this character is lost as the pulsation frequency is further reduced to $Wo=5.6$ (figure 8 c). The survivor function is characterized by constant steps over a half-cycle (during which no relaminarization occurs), followed by exponential decay during the subsequent half-cycle. Table 2 summarizes the results of our simulations and includes a direct comparison to the experiments of Xu et al. (Reference Xu, Warnecke, Song, Ma and Hof2017). Despite the uncertainty in $t_{0}^{\ast }$ and the limited sample sizes, there is excellent quantitative agreement between the two datasets. The effect of pulsation amplitude was tested by computing lifetime statistics at $(Re,A,Wo)=(2040,0.2,5.6)$ . The corresponding survivor function is shown in grey circles in figure 8(c) and is very similar to that of $(Re,A,Wo)=(2320,0.4,5.6)$ , shown as blue triangles. Thus, doubling the pulsation amplitude from $A=0.2$ to $0.4$ requires an increase of the Reynolds number by $14\,\%$ to preserve similar relaminarization dynamics and probability. This clearly demonstrates the stabilizing effect of increasing pulsation amplitude.

5 Assessment of experimentally measured transition thresholds

Xu et al. (Reference Xu, Warnecke, Song, Ma and Hof2017) proposed that, in pulsatile pipe flow at low amplitude $A\leqslant 0.4$ , transition can be divided into three regimes according to the pulsation frequency. These regimes are summarized and analysed here in view of the new insights gained from our numerical simulations.

Figure 9.

Experimentally measured transition thresholds from experiments rescaled by the respective steady thresholds. The solid and dashed lines show the threshold given by (5.2) and (5.6), respectively. In a sufficiently long pipe, $Re_{T}/Re_{T,s}$ approaches the upper limit $1/(1-A)$ as $Wo$ approaches zero, whereas for insufficient length $1/(1+A)$ is approached. Both limits are shown as thin grey dotted lines for $A=0.4$ .

Table 2.

Lifetime statistics of localized turbulence in pulsatile pipe flow. Here $\unicode[STIX]{x1D70F}^{\ast }$ is the median lifetime and $t_{0}^{\ast }$ is the time of the first relaminarization. The survival probability at $L=350$ from our simulations is compared to the survival probability measured experimentally at $L=350$ by Xu et al. (Reference Xu, Warnecke, Song, Ma and Hof2017).

5.3 Low pulsation frequencies: experiments in long pipes

In the experiments of Xu et al. (Reference Xu, Warnecke, Song, Ma and Hof2017), the transition threshold continues to shift to larger Reynolds number as $Wo$ is reduced. This effect gradually saturates, and they explained this by considering the quasi-steady limit ( $Wo\rightarrow 0$ ) in a sufficiently long pipe. They argued that the flow should remain turbulent provided that the instantaneous Reynolds number remained above the transition threshold for steady flow throughout the cycle, i.e.  $\min \widetilde{Re}(t)\geqslant Re_{T,s}$ . This requirement yields the threshold

(5.1) $$\begin{eqnarray}Re_{T}(Wo=0)={\displaystyle \frac{Re_{T,s}}{1-A}}.\end{eqnarray}$$

Xu et al. (Reference Xu, Warnecke, Song, Ma and Hof2017) showed not only that their data approach this limit, but also that, for small $Wo\lesssim 3$ , survival probabilities can be approximated from the corresponding steady-state values. The thresholds measured by Xu et al. (Reference Xu, Warnecke, Song, Ma and Hof2017) are well approximated by the empirical correlation

(5.2) $$\begin{eqnarray}Re_{T}(Wo)=Re_{T,s}\left[1+{\displaystyle \frac{A}{2(1-A)}}\left(1+{\displaystyle \frac{2}{\unicode[STIX]{x03C0}}}\arctan (\unicode[STIX]{x1D6FC}^{2}Wo^{-1}-\unicode[STIX]{x1D6FD}^{2}Wo^{3})\right)\right],\end{eqnarray}$$

which guarantees by construction that the correct transition thresholds are recovered in both the low- and high-frequency limits, independently of the value of the fitting parameters (here $\unicode[STIX]{x1D6FC}\approx 2.08$ and $\unicode[STIX]{x1D6FD}\approx 0.07$ obtained with a nonlinear least-squares fit to their data). The same value of the fitting parameters renders good approximations of the data of Stettler & Hussain (Reference Stettler and Hussain1986) for $A=0.4$ and $0.2$ .

5.4 Low pulsation frequencies: prediction of the transition threshold in short pipes

The quasi-steady limit proposed by Xu et al. (Reference Xu, Warnecke, Song, Ma and Hof2017) relies on the assumption of a sufficiently long pipe subject to a single impulsive disturbance. However, the pulsation period diverges as $1/Wo^{2}$ , which ultimately prevents the observation of turbulence over a full cycle within the available measurement length. Even in their very long pipe with $L=2250$ , the minimum feasible Womersley number is $Wo^{\ast }\approx 1.5$ , whereas the pipe of Stettler & Hussain (Reference Stettler and Hussain1986) with $L=330$ has $Wo^{\ast }\approx 4$ . In many practical situations, lengths are even shorter and disturbances are present all the time. Here we propose a prediction of the transition threshold in continuously disturbed finite-length pipes.

Stettler & Hussain (Reference Stettler and Hussain1986) observed that, at low $Wo\lesssim 5$ , turbulence appeared in a phase-locked manner, when the instantaneous Reynolds number exceeded the transition Reynolds number of the steady case, i.e.  $\max \widetilde{Re}(t)\geqslant Re_{T,s}$ . From this they argued that, in the quasi-steady limit, the transition threshold should be given by

(5.3) $$\begin{eqnarray}Re_{T}(Wo=0)={\displaystyle \frac{Re_{T,s}}{1+A}},\end{eqnarray}$$

and this prediction was found to be in good agreement with their experiments. According to this argument, turbulence is first triggered at time $t_{tr}$ , with $\widetilde{Re}(t_{tr})=Re_{T,s}$ . By plugging this in (2.1), we obtain

(5.4) $$\begin{eqnarray}t_{tr}={\displaystyle \frac{{\mathcal{T}}}{2\unicode[STIX]{x03C0}}}\sin ^{-1}\left[{\displaystyle \frac{1}{A}}\left({\displaystyle \frac{Re_{T,s}}{Re}}-1\right)\right].\end{eqnarray}$$

Subsequently, turbulence travels downstream until it either decays or is flushed at the end of the pipe. For simplicity, we assume that turbulence decays when $\widetilde{Re}(t_{d})=Re_{D,s}\approx 1600$ , at which point the lifetime of a puff in steady pipe flow becomes negligible (Hof et al. Reference Hof, de Lozar, Kuik and Westerweel2008). This yields the following decay time:

(5.5) $$\begin{eqnarray}t_{d}={\displaystyle \frac{{\mathcal{T}}}{2\unicode[STIX]{x03C0}}}\sin ^{-1}\left[{\displaystyle \frac{1}{A}}\left({\displaystyle \frac{Re_{D,s}}{Re}}-1\right)\right].\end{eqnarray}$$

For $(Re,A,Wo)=(2040,0.4,2)$ , we obtain $t_{d}\approx 473$  ( $t_{d}/{\mathcal{T}}\approx 0.6$ ), which is in good agreement with our simulation of figure 4(f).

At a measurement point downstream of the disturbance, turbulence will be detected if the time lapsed between the generation and decay of turbulence is longer than its travel time. Otherwise, turbulence will decay between disturbance and measurement point and laminar flows will be detected. Hence the measured transition threshold should be given by the nonlinear equation

(5.6) $$\begin{eqnarray}{\displaystyle \frac{L}{\,\overline{c}\,}}=t_{d}-t_{tr},\end{eqnarray}$$

where $L$ is the distance between disturbance and measurement point and $\overline{c}$ is the average propagation speed of turbulence during the time interval $t\in [t_{tr},t_{d}]$ . By assuming that the instantaneous propagation speed of turbulence is equal to the instantaneous bulk speed, direct integration yields

(5.7) $$\begin{eqnarray}\overline{c}={\displaystyle \frac{1}{t_{d}-t_{tr}}}\int _{t_{tr}}^{t_{d}}\left[1+A\sin \left({\displaystyle \frac{2\unicode[STIX]{x03C0}t}{{\mathcal{T}}}}\right)\right]\text{d}t=1+{\displaystyle \frac{{\mathcal{T}}A}{2\unicode[STIX]{x03C0}(t_{d}-t_{tr})}}\left[\cos \left({\displaystyle \frac{2\unicode[STIX]{x03C0}t_{tr}}{{\mathcal{T}}}}\right)-\cos \left({\displaystyle \frac{2\unicode[STIX]{x03C0}t_{d}}{{\mathcal{T}}}}\right)\right].\end{eqnarray}$$

The dashed lines in figure 9 show the solution of (5.6) for $A=0.2$ and $A=0.4$ . Both curves capture qualitatively the trend of the transition threshold measured by Stettler & Hussain (Reference Stettler and Hussain1986). The agreement could be made more quantitative by computing $\overline{c}$ from the actual $Re$ -dependent speed of downstream laminar–turbulent interfaces in steady pipe flow (Barkley et al. Reference Barkley, Song, Mukund, Lemoult, Avila and Hof2015). Finally, we note that the choice $Re_{D,s}\approx 1600$ does not significantly affect the prediction. In fact, replacing $1600$ by $1700$ or $1800$ cannot be noticed in the scale used in figure 9.