3.1 Single-jet disturbance

In this experimental set-up at $Re\gtrsim 1200$ , disordered flow is initiated naturally by imperfections, roughness, fluid impurities, etc. For $400\lesssim Re\lesssim 1200$ , the flow remains laminar and only disturbances of sufficiently large amplitude can initiate localised turbulence. For uniform pipe flow disturbed by cross-flow jets, the dependencies of the flux, the duration, the orientation and the jet diameter were already investigated (Hof et al. Reference Hof, Juel and Mullin2003; Peixinho & Mullin Reference Peixinho and Mullin2007). Specifically, the critical flux for turbulence is independent of the injection time when the dimensionless disturbed length is larger than 6 pipe diameters. It is also known that smaller jet diameters lead to lower flux thresholds (Peixinho & Mullin Reference Peixinho and Mullin2007; Tasaka et al. Reference Tasaka, Schneider and Mullin2010).

The effect of the cross-flow jet on the sudden expansion pipe flow is shown in figure 3. At small $V_{R}$ , the jet modifies the incoming velocity profile, but the flow recovers its laminar state downstream. The flow is considered transient when the laminar state is disturbed for a small fraction of the injection duration (e.g. see figure 4 a) and it is considered above the threshold for localised turbulence when the flow is disturbed for most of the duration of the applied disturbance (e.g. see figure 4 b). In figure 3, the critical velocity ratio, $V_{R}$ , decreases with increasing $Re$ indicating that the pipe is increasingly sensitive to the jet disturbance. A least-squares fit to the single-jet data gives a good description through a power law: $V_{R}\propto Re^{-2.3\pm 0.1}$ . The scaling for subcritical turbulence is fairly abrupt and could be related to the amplification behaviour of the expansion flow. Clearly, the range of $Re$ is limited. Furthermore, the scaling cannot hold for smaller $Re$ . For $Re\lesssim 400$ , disturbances cannot be sustained downstream.

Figure 3.

Critical velocity ratio, $V_{R}$ , for the onset of localised turbulent patch for single jet and suction as a function of $Re$ . The continuous line is a power law fit: $Re^{-2.3}$ and the dashed line is a guide to the eye. The error bars indicate the range of parameters where hairpin vortices are observed and open symbols represent visual laminar transient disturbances for the single jet. The inset shows the same results in log–log scale.

Once $V_{R}$ exceeds the threshold, localised turbulence is formed downstream. Typical examples of transient, intermittent and localised turbulent patches are show in the top image of figures 6(a), 6(b) and 6(c), respectively. Figure 4 shows examples of space–time diagrams of the brightness along the centreline axis of the pipe for $Re=700$ and 900. The vertical black lines mark the junction between pipe sections. As the single jet is started at $t=0$ , the disturbance moves downstream and develops before the localised turbulent patch forms downstream of the expansion at $x/d\simeq 3$ . The oblique lines on the diagram represent particles that move downstream and the slope can be related to the mean advection speed of these particles. At $t/t_{inj}=1$ , $t_{inj}$ being the injection total duration, the injection is stopped and the turbulent localised patch downstream of the expansion decays as it is carried downstream. Below the threshold, $V_{R}$ is sufficient to produce disturbed flow that evolves as transient localised turbulence. It is supposed that the injection induces a counter-rotating vortex, typical of a cross-flow jet (Camussi, Guj & Stella Reference Camussi, Guj and Stella2002; Vétel et al. Reference Vétel, Garon, Pelletier and Farinas2008; Ilak et al. Reference Ilak, Schlatter, Bagheri and Henningson2012; Karagozian Reference Karagozian2014), which breaks the azimuthal symmetry of the recirculation region downstream of the expansion and induces non-axisymmetric modes as proposed by Cantwell et al. (Reference Cantwell, Barkley and Blackburn2010). Yet, the quantitative identification of such azimuthal modes from the experiments would require three-dimensional PIV and is out of scope of the present study. An example of transient localised turbulence is presented in figure 4(a) for $Re=700$ and $V_{R}=0.32$ . Supplementary movie 1 (available at https://doi.org/10.1017/jfm.2018.421) shows an example of localised turbulence for $Re=700$ and $V_{R}=0.35$ .

Figure 4.

Space–time diagram of localised turbulence patches (a) below threshold (transient) at $Re=700$ and $V_{R}=0.32$ , (b) above threshold at $Re=900$ and $V_{R}=0.54$ .

The dependence of the dynamics of the disturbances on the various $V_{R}$ and $Re$ was studied through space–time diagrams. These diagrams can be characterised using $t_{c}$ , the onset time for the turbulent patch and the patch duration, $t_{patch}$ , as sketched by the arrows in figure 4(b). The results of a systematic investigation are presented in figure 5 and show a decrease of $t_{c}$ and a rapid increase and saturation of $t_{patch}$ as $V_{R}$ increases. When scaled by the advection time, $U/d$ , the data for different $Re$ and different $V_{R}$ seem to collapse. Altogether, the data suggest that the subcritical behaviour for single jets depends only on $V_{R}$ , the intensity of the jet compared to the mean pipe velocity.

Figure 5.

(a) Onset time, $t_{c}$ , of the turbulent localised patch as a function of $Re$ and (b) variations of duration time of a turbulent patch, $t_{patch}$ , with respect to $V_{R}$ at different $Re$ . The dotted line indicates the end of the injection.

For low $V_{R}$ , transient growth and decay of the turbulent patches are observed. In order to describe the spatio-temporal dynamics, figure 6 shows spatio-temporal diagrams for different axial locations along the diameter for $Re=700$ . In figure 6(a), $V_{R}=0.26$ , the disturbance breaks first at $20d$ from the expansion, grows slightly up to $x/d=30$ and then decays. In figure 6(b), for $V_{R}=0.34$ , an intermittent regime is observed. Finally, in figure 6(c) for $V_{R}=0.75$ , the growth remains localised between $x/d=10$ and 40.

Figure 6.

Flow visualisation photographs and space–time diagrams of transient growth of disturbances at $Re=700$ for (a) $V_{R}=0.26$ , (b) 0.34 and (c) 0.75. The arrow on the photographs indicates the position of the single-jet disturbance. The dashed-dot lines on the diagrams indicate the end of the jet injection.

3.2 Hairpin vortices

Hairpin vortices have been observed before in uniform pipes (Peixinho & Mullin Reference Peixinho and Mullin2007; Åsén et al. Reference Åsén, Kreiss and Rempfer2010; Tasaka et al. Reference Tasaka, Schneider and Mullin2010) or expansion pipe flow (Vétel et al. Reference Vétel, Garon, Pelletier and Farinas2008). Here, these have been observed specifically for the single-jet disturbance and their domain of existence is indicated by the error bars in figure 3 above the threshold for localised turbulent patches. It is interesting to note that the domain of existence for hairpin vortices appears at $0.5\lesssim V_{R}\lesssim 1.5$ . This range of $V_{R}$ is in agreement with the range reported in previous studies of jet in cross-flow in large or unbounded domains (Camussi et al. Reference Camussi, Guj and Stella2002; Ilak et al. Reference Ilak, Schlatter, Bagheri and Henningson2012; Karagozian Reference Karagozian2014). This suggests that counter-rotating vortex pairs are present in the experiments and may induce instabilities.

Figure 7 shows these hairpin vortices emerging from a single-jet disturbance at the inlet, propagating through the expansion and breaking down into a turbulent patch. On side view images in figure 7(b), the inclination of the hairpin vortices is enhanced due to the stretching effect when passing through the expansion.

Figure 7.

Photographs of a development of hairpin vortex array visualised through fluorescein dye taken for $Re=700$ and $V_{R}=0.7$ ; (a) top view and (b) side view.

3.3 Suction disturbance

In the case of the suction disturbance, the effect of the aspiration is to distort the inlet velocity profile that induces a strong asymmetry of the flow. A snapshot of the base flow without any disturbance is shown in figure 8(a), and the distorted flows for different $V_{R}$ in 8(b) and 8(d), respectively below and above the critical value of $V_{R}$ . The vertical white bands correspond to the connections between the pipe sections and the black shadow corresponds to the separation between the two lamps used during acquisition. Figure 8(c) is a sketch illustrating the flow deformation and the position of shear-unstable regions in grey. Specifically, the recirculation area on the opposite side of the aspiration gets several times bigger.

Figure 8.

Photographs for the suction disturbance for $Re=600$ . (a) Laminar flow: $V_{R}=0$ , (b) low amplitude suction at $V_{R}=2.2$ , (c) schematic of the streamlines of case (b), where the grey area represents the shear zone, and (d) suction at $V_{R}=2.5$ .

Below the critical value of $V_{R}$ , transient wavy disturbances appear at the boundary between the main flow and the recirculation area. These are relaminarised without initiating localised turbulence. Supplementary movie 2 corresponds to the case $Re=600$ and $V_{R}=2.2$ . On the contrary, in figure 8(d), $V_{R}$ is above the critical value. The amplification of the instability between the main flow and the recirculation area is observed, the previously mentioned wavy disturbances grow and lead to the formation of a turbulent patch fixed in space. Small disturbances are formed and amplified during the aspiration, sustaining the turbulent patch. In addition, a region of high shear appears at $x/d\approx 15$ from the expansion and the analysis of the brightness intensity on the centreline indicates the existence of small amplitude oscillations with a frequency of approximately 0.43 Hz. Such oscillations and frequency have been reported before in numerical simulations (Boujo, Ehrenstein & Gallaire Reference Boujo, Ehrenstein and Gallaire2013; Selvam et al. Reference Selvam, Peixinho and Willis2015) and their authors have suggested Kelvin–Helmholtz instabilities between the main flow and the recirculation region behind an expansion or a bump.

The finite amplitude thresholds of the suction disturbance for the onset of the localised turbulent patch are also presented in figure 3. Distinct scalings are observed for the jet and the suction. It is also clear that larger $V_{R}$ values are required to cause transition. Indeed, the suction preserves the recirculation region and somehow selects axisymmetric modes that require a higher threshold to induce instability. This is consistent with the numerical study of Cantwell et al. (Reference Cantwell, Barkley and Blackburn2010) which showed that the energy growth of azimuthal modes is larger than the axisymmetric ones.

Note that the injection and suction modify the flux in the inlet pipe. Moreover, the scaling for the critical $V_{R}$ has a different behaviour as the suction seems to reach a plateau corresponding to a shear consistent with the constant critical velocity required to cause Kelvin–Helmholtz instability. When the suction is stopped, the flow recovers the axisymmetric recirculation pattern after a short transient time.

3.4 Synthetic jet disturbance

In the previous subsections, the cross-flow jet and suction have been tested in a systematic way and indicate clear thresholds for localised turbulence. Now, these disturbances are combined to achieve a periodic zero-net-mass-flux disturbance, since an equal quantity of fluid was pushed then pulled in and out of the pipe through the 3 mm diameter hole. Although periodic excitation is very different from constant-flux disturbances, the periodic injection and suction should lead to essentially the same effect at a very low frequency of oscillation.

Hence, the key parameter is the frequency of the oscillations, $f$ , which is made dimensionless through the Strouhal number: $St=fd/U$ . The synthetic disturbance was systematically tested for different $Re$ and $St$ values. Figure 9(a) shows the threshold for localised turbulence as a function of $Re$ for two initial oscillation amplitudes: $A=50$ and $100~\unicode[STIX]{x03BC}\text{m}$ . Note that the values of $A$ are relatively small in this study in order to agree with transient growth theory. The general shape of the threshold data is a sharp cutoff at low $Re$ and a decreasing curve as $Re$ increases, indicating the expansion pipe flow is more sensitive to disturbances of low frequency as $Re$ increases. The threshold away from the cutoff can be fitted with a power law: $Re^{-2.4\pm 0.2}$ with an exponent similar to the exponent for the single jet. This is because the maximum $V_{R}$ can transiently reach values typical of the threshold for a single jet. A systematic shift is observed toward low $Re$ and low $St$ as the eccentricity of the cam, $A$ , is increased. For high $St$ , the behaviour of the synthetic jet is basically the periodic formation of stretched vortex rings. As there is a sharp cutoff, the method to estimate the threshold is to keep the oscillation frequency $f$ constant and vary $Re$ . This leads to horizontal error bars with the left-hand error bar indicating rapid decay of the disturbance and the right-hand a persistence of the disordered motion.

Figure 9.

Stability of the synthetic jet disturbance: (a) $St$ versus $Re$ . The lines are least-square fits: $St\propto Re^{-2.2\pm 0.2}$ . (b) $St$ versus turbulent patch position, $x_{patch}/d$ , for different $Re$ and $A=50~\unicode[STIX]{x03BC}\text{m}$ . The schematic diagram indicates the position of the disturbance, the expansion and the turbulent patch.

The disturbance involves and forms a localised turbulent patch, which is approximately 25 $d$ long with an active core and a decaying waves downstream. The location of the active core, $x_{patch}$ , remains at a steady axial position from the expansion as long as the disturbance is applied. $x_{patch}$ varies sensitively upon $St$ and is almost independent of $Re$ as shown in figure 9(b). High $St$ disturbances caused disturbed flow at $x\simeq 7d$ . At $St\approx 0.1$ , oscillatory disturbances induced unsteady flow further downstream at $x\simeq 25d$ from the expansion.

Transient growth of infinitesimal perturbations has been discussed in various flows (Westin et al. Reference Westin, Bakchinov, Kozlov and Alfredsson1998; Marais et al. Reference Marais, Godoy-Diana, Barkley and Wesfreid2011; Boujo et al. Reference Boujo, Ehrenstein and Gallaire2013; Klotz & Wesfreid Reference Klotz and Wesfreid2017) where the position of the centroid of perturbations is reported to move upstream depending on $Re$ and the perturbation frequency. For an expansion pipe, the transient growth analysis from Cantwell et al. (Reference Cantwell, Barkley and Blackburn2010) indicates that stochastic noise-driven flow at $Re=600$ and 900 has non-axisymmetric azimuthal modes of disturbances with a centroid of perturbation energy at $x/d=18$ and 32, respectively. This is very close to the measured $x_{patch}$ reported in figure 9(b). However, the experiment did not show clear periodic optimal flow structures. The simulations of Sanmiguel-Rojas et al. (Reference Sanmiguel-Rojas, Del Pino and Gutiérrez-Montes2010) have predicted the first unstable mode at $x\approx 150d$ from the expansion for $Re\simeq 3273$ and $St\simeq 0.13$ . It suggests that smaller amplitude disturbance could initiate disturbed flow further downstream. In the present experiment, when the disturbance is stopped, the patch is no longer sustained.

In figure 10, a photograph and space–time diagrams along the pipe diameter at fixed axial positions, where the turbulent patch starts to develop, are presented. Using fluorescein solutions, with a mass concentration of $0.001~\text{mg}~\text{l}^{-1}$ , injected at the inlet of the pipe with a flow rate of $1~\text{ml}~\text{min}^{-1}$ , the growth of the wavy disturbance can be monitored. The dye travels downstream, passes in front of the disturbance mechanism and then passes through the expansion. Downstream of the expansion, the periodic patterns are captured using the cameras.

Space–time diagrams along the diameter are produced, which show the amplification of the oscillations. Figures 10(b)–10(d) present space–time diagrams for $St=0.1$ , 0.17 and 0.42, respectively. Note the $x$ -position has to be adjusted and the maximum amplification growth changes with $St$ . For $St=0.1$ , the amplification occurs at $x\simeq 26$ and the wavy disturbance is merely disturbed. For $St=0.17$ , the maximum amplification occurs at $x/d\simeq 13$ and the space–time pattern contains the main wave and presumably one harmonic. For $St=0.42$ , the maximum amplification occurs at $x/d\simeq 5$ and seems to contain several harmonics. Yet, no frequency shift is observed upon the main frequency according to fast Fourier transform analysis of the light intensity (not shown). Hence, the synthetic jet disturbance in the range of parameters studied behaves as an amplifier. Further, the space–time diagrams suggest that the instabilities are convective as the flow contains weak or almost no swirl around the axes (Miranda-Barea et al. Reference Miranda-Barea, Fabrellas-García, Parras and del Pino2016).

The results seem to be consistent with previous experiments from Chun & Sung (Reference Chun and Sung1996), where the turbulent separation flow downstream of a backward-facing step was controlled using a synthetic jet from a thin slit at the corner. The corner location has also been shown to be highly sensitive to disturbances in expansion pipe flow (Sreenivasan & Strykowski Reference Sreenivasan and Strykowski1983; Cantwell et al. Reference Cantwell, Barkley and Blackburn2010). It is anticipated that disturbances introduced further upstream from the expansion may need to be compensated with larger amplitudes to observe transition. Yet, the scalings exponents for the thresholds could still hold and may be used to control the drag or the mixing in the expansion.

Figure 10.

Effect of the synthetic jet. (a) A snapshot of the propagating oscillating disturbance from fluorescein visualisation images of the synthetic jet disturbance at $Re=600$ and $St=0.1$ . See supplementary movie 3. Space–time diagrams at constant $x/d$ along the diameter for (b) $St=0.1$ at $x/d\simeq 26$ , (c) $St=0.17$ at $x/d\simeq 13$ and (d) $St=0.42$ at $x/d\simeq 5$ . The location $x/d$ corresponds to the trailing edge (upstream) of the disturbed flow.