3.1.1 Downstream region

First, we give some explanations regarding the spatial stability analysis in RHPF. The reader must note that the determination of local absolute instabilities by adopting the Briggs–Bers criterion (Briggs Reference Briggs1964; Bers Reference Bers1975) requires the so-called parallel-flow approximation (see figure 9 below where we present the neutral curves in solid and dashed lines given by Fernandez-Feria & del Pino Reference Fernandez-Feria and del Pino2002). Actually, we set a pair of values $Re$ , $L_{0}$ in the ( $Re$ , $L$ )-plane to reach a given initial state before applying the spin-down. If RHPF is stable (S.), it will reach a steady base flow solution. However, if this flow has convective (C.I.) or absolute instabilities (A.I.), it will reach a limit cycle around the base flow solution, since a periodic unstable spiral ( $n=-1$ or $n=-2$ ) appears in the downstream region. In other words, we use the expression limit cycle to define the situation where RHPF becomes time-dependent because travelling waves appear in the downstream region while applying a critical rotation to the pipe flow $L_{0}$ greater than $L_{c}$ or $L_{a}$ , thus becoming convectively or absolutely unstable, respectively (see again figure 9). Shrestha et al. (Reference Shrestha, Parras, del Pino, Sanmiguel-Rojas and Fernandez-Feria2013) have reported that flow visualisations in the inlet region ( $2d\leqslant x^{\ast }\leqslant 11d$ ) have already provided a method to distinguish between stable and unstable solutions, and between C.I. and A.I., showing that the values of the measured frequencies and wavelengths in the downstream region are in good agreement with those obtained by a spatial, linear stability analysis with the parallel-flow approximation. This analysis works well when constant values of $Re$ and $L_{0}$ were tested experimentally, though it was impossible to determine the transition between C.I. and A.I. by direct observation in the downstream region.

In this new study we present a novel experimental methodology based on the spin-down process through direct observations in the upstream region (see supplementary material available online at http://dx.doi.org/10.1017/jfm.2016.122). However, we must ensure that the base flow is RHPF in the downstream region even after applying the spin-down, that is to say, when the flow changes from the pair $Re$ , $L_{0}$ to $Re$ , $L_{f}$ . To that end, we set a pair $Re$ , $L_{0}$ inside the absolutely unstable region and we will reduce the swirl parameter up to a certain value $L_{f}$ . The spatial and temporal axes are non-dimensionalised with the characteristic length $L_{c}=d/2=9.5~\text{mm}$ and characteristic time $t_{c}=L_{c}/W_{0}$ , $W_{0}=2U$ being the maximum axial velocity which depends on the Reynolds number tested. The non-dimensional variables $r=r^{\ast }/L_{c}$ , $x=x^{\ast }/L_{c}$ and ${\it\tau}=t/t_{c}$ stem from these characteristic magnitudes defined above. Figure 2 shows snapshots in a ( $r$ , $x$ ) plane at different dimensionless times in the spin-down process, (a–f) corresponding to ${\it\tau}=0$ –50 with time steps equal to 10, for a constant Reynolds number, $Re=250$ . As the reader will note in all of the snapshots the flow moves from left to right (positive $x$ indicates positive flow rate, see figure 2).

Figure 2.

Flow visualisations for $Re=250$ and $L_{0}=2$ and $L_{f}=1.0$ , at certain times ${\it\tau}=0$ (a), 10 (b), 20 (c), 30 (d), 40 (e) and 50 (f) in the spatial domain $164\leqslant x\leqslant 179$ .

Figure 3 represents the spatiotemporal diagram which results from the flow visualisations presented in figure 2 in a cut of the symmetry axis ( $r=0$ ) in the downstream region. The flow starts with an initial and final swirl parameters $L_{0}=2$ and $L_{f}=1$ , respectively. Patterns appear periodically up to ${\it\tau}=0$ , when the spin-down has been applied, and they reappear at ${\it\tau}={\rm\Delta}{\it\tau}_{1}$ , being ${\rm\Delta}{\it\tau}_{1}\approx 45$ , approximately. Obviously, the slope in the spatiotemporal diagram changes from $L_{0}$ to $L_{f}$ as a consequence of the variation in the swirl parameter, and because the flow reaches a new oscillatory state too. Recall that the flow is convectively unstable in figure 3 for $L=L_{f}$ .

Figure 3.

Spatiotemporal diagram for $Re=250$ and $L_{0}=2$ and $L_{f}=1.0$ .

Figure 4.

Dimensionless frequencies for $Re=250$ and $L_{0}=2$ ( ${\it\omega}_{0}$ , blue circles) with different $L_{f}$ ( ${\it\omega}_{f}$ , red squares). Only the highest frequencies of ${\it\omega}$ correspond to the predominant modes resulting from 1D FFT (marked with subscript 1). The star indicates the theoretical frequency for the transition from convectively to absolutely unstable flow given in Fernandez-Feria & del Pino (Reference Fernandez-Feria and del Pino2002).

On the other hand, we are able to compute dimensionless real frequencies, ${\it\omega}$ , from the spatiotemporal diagram presented in figure 3, being ${\it\omega}=\hat{{\it\omega}}R/W_{0}$ , where $\hat{{\it\omega}}$ is the dimensional frequency obtained by using one-dimensional (1D) fast Fourier transform (FFT). The results are shown in figure 4 for different reductions of the rotation speed, while keeping constant the Reynolds number, $Re=250$ , and the initial value of the swirl parameter, $L_{0}=2$ . We define ${\it\omega}_{0}$ (blue circles) as the frequencies which correspond to the initial swirl parameter $L_{0}$ (negative ${\it\tau}$ in figure 3), while ${\it\omega}_{f}$ (red squares) belong to the periodic patterns after applying the spin-down process ( ${\it\tau}$  greater than ${\rm\Delta}{\it\tau}_{1}$ in figure 3). One explores a set of experiments under the same initial conditions ( $Re$ , $L_{0}$ ), so the frequencies at which the flow is oscillating are roughly constant, ${\it\omega}_{01}\approx 3.15$ , thus confirming the reliability of the results. The subscript 1 in ${\it\omega}_{01}$ and ${\it\omega}_{f1}$ defines the predominant frequencies at $L_{0}$ and  $L_{f}$ , respectively. One must note that the experimental points ${\it\omega}_{01}$ and ${\it\omega}_{f1}$ tend to be equal as the reduction jump, defined as ${\rm\Delta}L=L_{0}-L_{f}$ , approaches zero ( ${\rm\Delta}L\rightarrow 0$ ), confirming again a good experimental procedure. In addition, the critical swirl parameter $L_{f}=L_{a}=1.3$ corresponds to the threshold between convective and absolute instabilities using the local, spatial stability analysis with a parallel-flow approximation for azimuthal wavenumber $n=-1$ (see again figure 9 below Fernandez-Feria & del Pino Reference Fernandez-Feria and del Pino2002). This stability analysis also provides the frequency for the onset of absolute instabilities. Therefore, the pair $(L_{a},{\it\omega}_{a})=(1.3,1.14)$ is also depicted in figure 4 with one star symbol. The theoretical prediction ${\it\omega}_{a}$ is in excellent agreement with the trend of the experimental points ${\it\omega}_{f1}$ . This fact confirms that the flow is RHPF in the downstream region after applying the spin-down process. In addition, other experimental peaks of (secondary) frequencies also appear while performing 1D FFT, and they are shown with subscript 2 ( ${\it\omega}_{02}$ and ${\it\omega}_{f2}$ ) in figure 4. It is worth mentioning that the values of ${\it\omega}_{02}$ are half the predominant frequencies ${\it\omega}_{02}={\it\omega}_{01}/2\approx 1.57$ . We observe the same behaviour after applying the spin-down process from $L_{0}$ to $L_{f}$ , so ${\it\omega}_{f2}\approx {\it\omega}_{f1}/2$ . Curiously, only peaks of (secondary) frequencies ${\it\omega}_{f2}$ are present for swirl parameters $L_{f}$ greater than $L_{a}$ .

Three main features stem from the analysis of figure 4: first, the experimental procedure is reliable because only small errors appear in the computation of the frequencies ( ${\it\omega}_{0}$ , ${\it\omega}_{f}$ ) and, second, the linear theory with a parallel approximation accounts quite well for the RHPF stability after applying the spin-down process. As mentioned in the introduction, this last characteristic is important as it was also highlighted in other flows such as two-dimensional wakes, see Delbende & Chomaz (Reference Delbende and Chomaz1998b ). Thirdly, we do believe that the appearance of second real frequency peaks, with a magnitude half the predominant ones, can only be related to the presence of other mode with $n=-2$ that interacts with the predominant one $n=-1$ within the absolutely unstable region. This mode shift from $n=-1$ to $n=-2$ will be presented in the last section.

Figure 5.

Images of convectively (a) and absolutely unstable flows (b) for $Re=500$ and $L=0.8$ and 1.5, respectively. The spatial domain corresponds to $2\leqslant x\leqslant 14$ .

Figure 6.

Spatiotemporal diagrams for $Re=250$ and originally unstable states: convectively $L_{0}=1$ (a–d) and absolutely $L_{0}=2$ (e–f). The reduction of the angular velocity is of the size: 30 % (a–e), 50 % (b–f), 80 % (c–g) and 100 % (d–h). We depict the time ${\rm\Delta}{\it\tau}_{2}$ and the negative and positive slopes ( $x^{-}$ and $x^{+}$ , respectively) with dashed black lines.

3.1.2 Upstream region

In this section we describe the different states that were found experimentally in RHPF in the pipe inlet next to the conditioning chamber. The onset of absolute instabilities was experimentally characterised in Shrestha et al. (Reference Shrestha, Parras, del Pino, Sanmiguel-Rojas and Fernandez-Feria2013) by means of the appearance of a spiral structure in the pipe inlet region, such as observed in figure 5 in the spatial domain $2\leqslant x\leqslant 14$ . These snapshots for $Re=500$ allow a precise distinction between convectively (a) and absolutely (b) unstable flows for $L=0.8$ and $L=1.5$ , respectively, in RHPF following the criterion given in Shrestha et al. (Reference Shrestha, Parras, del Pino, Sanmiguel-Rojas and Fernandez-Feria2013): the flow upstream of the pipe is considered stable, so the propagation of travelling waves downstream with negative group velocity interact with the flow in the inlet region forming the characteristic spiral pattern. However, direct visualisations of the absolutely unstable flow and the subsequent appearance of spirals in the pipe inlet region are far from being sharp just in the transition area (see figure 2d in Shrestha et al. (Reference Shrestha, Parras, del Pino, Sanmiguel-Rojas and Fernandez-Feria2013)). For this reason, it was necessary to generate a spatiotemporal diagram in the inlet region so that the spiral structures there would become much clearer. This is shown in figure 6 for $Re=250$ and $L_{0}=1$ (a–d) or $L_{0}=2$ (e–h) in the spatial domain $2\leqslant x\leqslant 30$ . We now pay special attention on the values ${\it\tau}$ lower than zero which correspond to a constant initial value of the swirl parameter $L_{0}$ , while the features regarding the application of the spin-down process (positive ${\it\tau}$ ) will be discussed below. Thereby, the onset of absolute instabilities was experimentally observed as a sequence of spirals in the pipe inlet region which, consequently, show up as different colour bands parallel to the temporal axis of the spatiotemporal diagram, the spatial domain being the symmetry axis, $r=0$ (see also figure 4c in Shrestha et al. (Reference Shrestha, Parras, del Pino, Sanmiguel-Rojas and Fernandez-Feria2013)). Thus, these parallel bands in figure 6(e–h) for negative values of ${\it\tau}$ reveal the presence of spiral patterns which correspond to the onset of absolute instabilities so the pair $L_{0}=2$ and $Re=250$ belongs to the absolutely unstable region. The same patterns are observed in these figures as a consequence of the same initial conditions. However, with swirl parameters close to the end of the convectively unstable flow, see figure 6(a–d) for $L_{0}=1$ when the critical value of the swirl parameter is $L_{a}\approx 1.3$ , we still observe a sequence of bands which might be taken as absolutely unstable cases. A plausible cause may be that the transition from convectively to absolutely unstable flows is gradual in RHPF in the sense that flow patterns do not change abruptly but rather slightly with increasing $Re$ or $L_{0}$ . In addition, the problem of visualisation-based direct observations in the transition area becomes harder and harder as $Re$ increases, so the spiral structure shown in figure 5(b) is not too clear. This shortcoming justifies this experimental visualisation-based alternative by spinning down the rotating pipe or spinning it down to rest has been developed in order to establish the transition from convective to absolute instabilities experimentally. Another inconvenience derived from direct visualisation is that there is no possibility to distinguish between $n=-1$ and $n=-2$ modes in a ( $r,x$ ) plane. We will be back to figure 6 after explaining in detail the video given as supplementary material.

For the characterisation of the flow states in the pipe inlet, namely, convectively and absolutely unstable flows, we resort to a new experimental method so as to distinguish between them. This new experimental procedure is based on the nature of the hydrodynamic instabilities (Huerre & Monkewitz Reference Huerre and Monkewitz1998): whereas the convective instabilities tend to disturb the flow downstream, the absolute ones affect all of the spatial domain, even the pipe inlet region where the flow is developing, and is supposedly stable. This is due to the fact that convectively unstable travelling waves tend to propagate more intensely downstream, i.e. the group velocity is positive ( $c_{g}>0$ ). However, absolute instabilities tend to destabilise the flow in the region where RHPF is developing, i.e. in the pipe inlet, thereby one could observe the change from the initial typical conical structure (figure 5 a) into the already-mentioned spiral structure (figure 5 b). The reason for this change in the flow structure is that the absolute-instabilities-produced wave packets tend to propagate more intensely both downstream and upstream, i.e. the group velocity is negative ( $c_{g}<0$ ). The transition takes place, in mathematical form, for the null group velocity and this is the experimental critical point when it comes to determining where the transition takes place.

On grounds of what has been previously explained, for the experimental analysis we set a given $Re$ and $L=L_{0}$ to establish the RHPF. Once the initial state has been established in the pipe, we reduce the angular velocity, its new value ranging from 1 % (infinitesimal variation) to 100 % (spin-down to rest or sudden stop). In fact, for a given reduction of the angular velocity, which depends on $Re$ and $L_{0}$ , we have been able to clearly identify the transition from convective to absolute instability. Further in the text we discuss the computation of the relative value of the angular velocity reduction required for an instability to take place but, first, it is essential to describe the temporal evolution of RHPF in the pipe inlet. The experimental methodology is based on the variation of the swirl parameter from an initial state, which can be convective or absolute, to a final state with one randomly-chosen swirl parameter, $L_{f}$ , lower than that initially adopted, $L_{0}$ . By following this experimental procedure, the video attached to this manuscript illustrates one temporal sequence in the ( $r$ , $x$ ) plane for $Re=250$ and initial convective state ( $L_{0}=0.8$ ) and absolute state ( $L_{0}=2$ ) with a 50 % reduction of the rotation speed for different times. The sequence of images show the temporal evolution of the perturbation-produced wave packet fronts in the pipe inlet region. Therefore, the criterion adopted to distinguish whether or not the wave packet fronts propagate towards to the inlet is the drastic variation observed in the perturbations which were originally convective or absolute. In convectively unstable flows, perturbations do not alter the structure of the inlet whatever reduction of the rotation speed is applied during the spin-down process, so the wave packet fronts only move downstream in the video for $L_{0}=0.8$ , confirming the nature of the perturbations given in Huerre & Monkewitz (Reference Huerre and Monkewitz1998). In addition, it is also clear in the video how the instability grows upstream towards to the pipe inlet region when $L_{0}=2$ . This change results as a consequence of the wave packet propagation upstream, thus confirming again the absolute nature of the instability. In some cases, the wave packet fronts disappear in the inlet region of the pipe where the convergent nozzle is placed, as its opacity does not lend itself to visualisation. As far as we know, this is the first experimental evidence of a temporal evolution of absolute instabilities in RHPF. Recall again that the flow moves to the right in the video, and in figures 5 and 6, as indicated above, providing the reference point for the distinction between upstream and downstream.

The advantage of the experimental method presented here with respect to that given in Shrestha et al. (Reference Shrestha, Parras, del Pino, Sanmiguel-Rojas and Fernandez-Feria2013) is that no spatiotemporal diagram is needed as one can distinguish whether the flow is convectively or absolutely unstable simply with a direct observation. However, the next step is to quantify the experimental measurements performed in order to validate the success of the experimental procedure reported in this paper.

As we indicated above, we repeat the same procedure to build spatiotemporal diagrams given in figure 3, but for the upstream region instead of the downstream region. Figure 6 displays the results for different reductions in the swirl parameter for $Re=250$ : convectively unstable states (a–d) and absolutely unstable states (e–h). The spatiotemporal diagrams in figures (a–d) correspond to an initial swirl parameter $L_{0}=1$ and reductions of 30 % (a), 50 % (b), 80 % (c) and 100 % (d), whereas the ones in figures (e–h) correspond to an initial swirl parameter $L_{0}=2$ , and the same values of the reduction given above. We will pay our attention on positive values of ${\it\tau}$ . The origin of the time axis corresponds to the beginning of the partial/total reduction of the swirl parameter within the timing errors of the mechanical response of the system.

In this work we go one step further studying the temporal evolution while varying the value of the swirl parameter as it may become an efficient experimental method to detect the onset of absolute instabilities. This process can be described as follows. While the transient process takes place as a result of a partial or total reduction of the swirl parameter applied to initial convectively unstable flows, a temporal evolution of the wave packet is observed, its transient front moving with positive velocity. However, when the reduction is applied to initial absolutely unstable flows, the wave packet front appears moving counterflow-wise (with negative velocity). Actually, we can confirm the presence of a wave packet travelling upstream until it is blocked by the inlet in our confined flow, see figure 6(e–f) for ${\it\tau}$ approximately greater than 5. The experimental results presented in figure 6(e–h) are also similar to the data reported by Chomaz (Reference Chomaz2003) in his numerical study for wakes in open flows while studying the fully nonlinear dynamics of this type of flows.

The experimental evidence presented in figure 6 depends on the level of reduction of the angular velocity which is also function of $Re$ and $L_{0}$ , ${\rm\Delta}L/L(\%)=f(Re,L_{0})$ . Thus, the characteristic velocity of the wave packet front ( $c^{+/-}$ ) can be computed with the quotient $x^{+/-}/t$ which corresponds to light bands in figure 6. One can observe that positive and negative upper scripts denote light bands from figure 6 with slopes greater or lower than zero. In other words, rays with positive slope show patterns flowing with positive flow rate (from left to right).

Figure 7.

Evolution of the velocity of the wave packet front in terms of the reduction of the swirl velocity for cases which are originally convectively unstable (blue diamonds, $L_{0}=1$ ) and absolutely unstable (grey circles, $L_{0}=2$ ) together with the theoretical threshold from convective to absolute instabilities (white squares, $L_{0}=1.3$ ). The labels (a–h) correspond to figure 6.

To quantify the velocity $c^{+/-}$ , different reference points have been chosen (between three or five rays, depending on the spatiotemporal diagram) and their mean values and standard deviation have been computed. In figure 7 the values of the velocity $c$ are plotted in terms of the decreasing rate of swirl for a constant Reynolds number, $Re=250$ , and for the same cases of swirl velocity reduction shown in figure 6, to which some extra cases have been added. It is observed that for the initial convectively unstable cases, depicted with diamonds ( $L_{0}=1$ ), the velocities are always positive ( $c=c^{+}$ ). However, in the initial absolutely unstable cases, which correspond to circles ( $L_{0}=2$ ), there exists a critical value of the swirl reduction, from which the flow visualisations start to give negative values ( $c^{-}$ ). In fact, the critical reduction of the swirl parameter, for which $c=0$ , is found to be near 65 %. There are values in the $x$ axis which are not plotted in figure 7, as for example those where the reduction is less than 20 % (convectively unstable, $L_{0}=1$ ) or less than 10 % (absolutely unstable, $L_{0}=2$ ). This lack of information stems from the fact that no characteristic $c^{+}$ or $c^{-}$ have been observed in the spatiotemporal diagram through direct flow visualisations. In other words, the flow does not present variations which can be identified visually for (infinitesimal) low reductions in the angular velocity, and then there is no experimental evidence of wave front movements. Therefore, in these particular cases, it is observed that the critical value ${\rm\Delta}L/L(\%)$ decreases almost to the region which cannot be observed experimentally. For values $L_{0}$ lower than the critical swirl parameter, $L_{0}=L_{a}=1.3$ , all the velocity values computed are positive ( $c=c^{+}$ , i.e.  $L_{0}=1$ ), confirming the experimental methodology used in this work. We also depict one solid and one dashed line in figure 7 to highlight the jump from $c^{-}$ to $c^{+}$ for $L_{0}=2$ and 1.3, respectively.

It remains an open question why spin-down produces a change in the flow behaviour allowing us to detect by direct observations the change from convective or absolute instabilities. We do believe that a plausible answer could be related to the linear impulse response of the RHPF. On the one hand, a small a variation in the rotation causes the inlet flow to have a negative ray (see, e.g., figure 6 e or f), while on the other hand a significant reduction of the swirl parameter makes the flow behave as it were convective; compare, e.g., figure 6(d) and (h). In our view the aspect ratio in the reduction of the rotation speed to visualise the change in the flow during the spin-down process (from negative to positive as we increase the reduction ratio of the swirl parameter) is conditioned by the initial values of the two parameters that govern this problem: $Re$ and $L_{0}$ . The further this pair of values inside the absolutely unstable region is from the neutral curve of the convectively unstable region in the ( $L,Re$ ) plane, the greater the amount of reduction of the swirl parameter is required. The flow behaves in a different manner starting with ( $L_{0},Re$ ) from the A.I. region and finishing in the A.I. region with ( $L_{f}$ , $Re$ ) or starting in the A.I. region and finishing on the C.I. region. If we apply a linear response while decreasing the swirl parameter, we need at least a 20 % reduction to observe the effect of the spin-down through flow visualisations. This amount of reduction of the angular velocity should be lower in case of a numerical study. Thus, once we force a reduction of the angular velocity, this jump is convected from the wall side to the axis in the radial direction. To attain the new final state the flow needs a certain period of time, so at the axis of the pipe it remains absolutely unstable with the same characteristics as the starting swirl number $L_{0}$ . We do know that one vortex responds to any perturbation by generating waves to readapt the flow. In this case, the waves travelling upwards through the axis of the pipe are quicker than those generated in the wall through a sudden reduction of the angular velocity. Depending on how strong the reduction rate is, those waves propagating from the wall in radial direction are stronger or weaker than the absolute instabilities. These unstable waves travel upwards finding a stable flow which reduces their energy in the entrance of the pipe. That is the reason why the flow needs a given strength of the reduction rate for the perturbation to be clearly distinguishable through flow visualisations in the entrance of the rotating pipe, and this value cannot be predicted theoretically by easy means: only the competition between absolute instabilities in RHPF and centrifugal modes due to the spin-down process could clarify this point. Finally, it is also worth mentioning that a small reduction of the swirl parameter during the spin-down process could be seen as an impulsive response of RHPF for large time. This is also discussed using the Batchelor vortex in Delbende & Chomaz (Reference Delbende and Chomaz1998a ), where they describe the numerical procedure to determine the change from convective to absolute instabilities by using the complex streamwise wavenumber and the complex frequency observed along each spatiotemporal ray $v_{g}=x/t$ for large time. Unfortunately, there is no possibility to compute the resulting growth rate given in Delbende & Chomaz (Reference Delbende and Chomaz1998a ) through flow visualisations.

Figure 8.

Dimensionless time ${\rm\Delta}{\it\tau}_{1}$ for the spin-down process in the downstream region (grey circles, see figure 3) and ${\rm\Delta}{\it\tau}_{2}$ corresponding to the upstream region (red diamonds, see figure 6) for different values of the final values of the swirl parameter $L_{f}$ , the initial state being $Re=250$ and $L_{0}=2$ . In dashed lines, it is depicted the critical values $L_{c}=0.11$ and $L_{a}=1.3$ given in Fernandez-Feria & del Pino (Reference Fernandez-Feria and del Pino2002).