3.1. Typical dynamics

The succession of snapshots shown in figure 2 represents a typical annealing between $Re_i=430$ and $Re_f=300$ at $\dot {Re}=0.3\ \textrm {s}^{-1}$. It is visualized from streamwise velocity fields. The dynamics is as follows. Laminar holes open up while the turbulent intensity in the turbulent zones remains almost at the same level. Indeed, the spatial density of turbulent kinetic energy, i.e. the kinetic energy within turbulent patches divided by their total area, is a slowly decreasing function of the Reynolds number (not shown), as already reported by Manneville (Reference Manneville2011) in his figure 9, for example. Later on, the holes get larger and eventually organize into oblique stripes typical of extended internal shear flows. When the turbulent fraction is too low to maintain a band, isolated turbulent spots remain and slowly disappear by fading away due to viscosity rather than by receding spatially.

3.2. Time traces

From now on, the turbulent fraction $F_t$ will be used as a single scalar parameter to follow the annealing dynamics. This is obviously restrictive but allows us, nevertheless, to gain a relevant insight into the studied system. Given our system, time can be made non-dimensional in different ways. For experiments or simulation performed at given $Re$, i.e. at given belt velocity $U$, one can built an advective time unit $h/U$ and a diffusive time unit $h^2/\nu$, the former being more relevant to the kind of dynamics we are interested in. Since the advective time scale depends on $U$, it is evolving with time and we have thus chosen to rescale time using $\dot {Re}$, which has the correct dimension to do so, which is constant through the whole experiment and, as we will see, provides a good scaling of the control parameter. For the same reasons, the values of $\dot {Re}$ are given in $\textrm {s}^{-1}$ rather than in dimensionless units.

Figures 3–5 present the time evolution of the turbulent fraction. To improve readability, only a subset of the studied $Re_f$ are displayed. Nevertheless, figure 6 and our comments gather all our observations. In figures 4 and 5, time traces are coloured according to the value of $\dot {Re}$ and gathered in the same panel by $Re_f$ values while in figure 3 it is the opposite. In these figures, time is given either in seconds or in units of $\dot {Re}$.

Figure 3.

Turbulent fraction versus time for all $Re_f$ for four different $\dot {Re}$ values. Final Reynolds numbers $Re_f$ are indicated in $(d)$. Traces with the same colour correspond to various experiments performed with the same parameters. In panel $(d)$, the bars labelled with values of $Re_f$ indicate the span of $F_t$ values reached asymptotically between different realizations with the same parameters. See figure 6 for a more quantitative picture. $(a) \ {\rm Quench}, \ (b) \ \dot{R}e = 5, \ (c)\ \dot{R}e = 1 \ {\rm and} \ (d) \ \dot{R}e = 0.3.$

Figure 4.

Turbulent fraction versus time for all $\dot {Re}$ for four different $Re_f$ values. Traces with the same colour correspond to various experiments performed with the same parameters. Vertical dash-dotted lines indicate $t_f$, the time at which the final Reynolds number $Re_f$ is reached. For quenches, this line is sometimes hidden by the ordinate axis. $(a) \ \textit{Re}_{f} = 300, \ (b) \ \textit{Re}_{f} = 320, \ (c) \ \textit{Re}_{f} = 340 \ {\rm and} \ (d) \ \textit{Re}_{f} = 360.$

Figure 5.

Turbulent fraction versus rescaled time $t\dot {Re}$ for all $\dot {Re}$ for four different $Re_f$ values. Traces with the same colour correspond to various experiments performed with the same parameters. Vertical dash-dotted lines indicate $t_f$, the time at which the final Reynolds number $Re_f$ is reached. Note that with this scaling they occur at the same $t\dot {Re}$ value for all experiments performed at the same $Re_f$, regardless of $\dot {Re}$ value. Quenches are not presented here because, due to our experimental protocol, the time rescaling used here is not relevant for quenches. $(a) \ \textit{Re}_{f} = 300, \ (b) \ \textit{Re}_{f} = 320, \ (c) \ \textit{Re}_{f} = 340 \ {\rm and} \ (d) \ \textit{Re}_{f} = 360.$

All together, these figures show that the turbulent fraction is globally decreasing from $F_t=1$ at $t=0$ toward lower values when $t$ increases. Regardless of $\dot {Re}$, when the final Reynolds number $Re_f$ is larger than 350 the dynamics is a trivial decay to asymptotic values, occurring with little fluctuation. For $Re_f=300$, the decay to $F_t=0$ is also fairly simple and with little fluctuation, but for decay rates $\dot {Re} \geq 1\ \textrm {s}^{-1}$, transient plateaus are often seen, as has already been observed by Prigent & Dauchot (Reference Prigent and Dauchot2005). The following sections provide a statistical analysis of these results.

The time at which the final Reynolds number $Re_f$ is reached is denoted in figures 4 and 5 with vertical dash-dotted lines. Interestingly, transient plateaus are seen when the system has already reached a Reynolds number value for which the asymptotic turbulent fraction is lower than that of the visited plateau. When rescaling time with $\dot {Re}$, as in figure 5, the final Reynolds number $Re_f$ is reached at $t\dot {Re}=Re_i-Re_f$, regardless of the value of $\dot {Re}$. The $F_t(t)$ curves associated with different $\dot {Re}$ get closer to each other when plotted against $t\dot {Re}$ but do not completely collapse. However, the order of the plots is reversed when comparing figure 4 with figure 5. When $F_t$ is plotted versus the non-dimensional time $t\dot {Re}$, it appears that, with this time scale, the dynamics is faster for lower $\dot {Re}$ or delayed for larger decay rate. This will be further discussed in the next section when commenting on the results from figure 8.

3.3. Asymptotic states

Figure 6(a) presents the $Re_f$ dependence of $\overline {F_t}$, the time average of the turbulent fraction calculated over the asymptotic plateaus described in the previous paragraph. Numerical simulation data from Duguet et al. (Reference Duguet, Schlatter and Henningson2010) were added for comparison; they have been obtained from the asymptotic state reached when a localized perturbation is added to the laminar flow. To quantify the level of fluctuations around the dynamics, we have calculated $\sigma _{F_t}$, the moving standard deviation of the turbulent fraction over time intervals of $T=15\ \textrm {s}$, as

(3.1)\begin{equation} \sigma_{F_t}(t) = \sqrt{\frac{1}{T}\int_{t-T/2}^{t+T/2}F_t^2(t')\,\textrm{d}t' - \left(\frac{1}{T}\int_{t-T/2}^{t+T/2}F_t(t')\,\textrm{d}t'\right)^2}. \end{equation}

Figure 6.

$(a)$ Asymptotic turbulent fraction $\overline {F_t}$ as a function of $Re_f$ for various $\dot {Re}$. Error bars represent the dispersion between experiments performed with the same control parameter values. Data from Duguet et al. (Reference Duguet, Schlatter and Henningson2010) are also presented. $(b)$ Time average of the moving standard deviation calculated over time intervals of 15 s. This quantity measures the level of fluctuations around the smoothed dynamics.

Figure 6(b) presents $\overline {\sigma _{F_t}}$, the time average of this moving standard deviation versus $Re_f$; $\overline {F_t}$ has an expected behaviour – it grows from zero at $Re_f\leq Re_g$ to one when $Re_f\geq Re_t$. The dispersion between similar experiments is very low, often less than the size of the symbol. For most values of $Re_f$ (when $Re_f$ is not in the range 310–330) we do not see any dependency of the results on $\dot {Re}$, indicating that, as far as annealing and quenches are concerned, the average asymptotic turbulent fraction does not depend on the specific history of the system. Numerical values obtained by Duguet et al. (Reference Duguet, Schlatter and Henningson2010) are systematically lower than ours which could be due to either the different protocol (transition from versus transition to turbulence) or the domain size, which is four times larger in the numerical simulations. In the range $310\leq Re_f\leq 330$, a large variability of the results is observed, as expected in the vicinity of a critical point. Figure 6(b), presenting the average level of fluctuations around the moving average state, shows that these inner fluctuations present a maximum for $Re_f \simeq 330\text {--}350$, which indeed corresponds to a range where large-scale flows have larger amplitudes, see Couliou & Monchaux (Reference Couliou and Monchaux2015). We thus interpret these low-frequency fluctuations as slow reorganizations of the turbulent regions involving strong secondary flows.

3.4. Turbulent fraction dynamics

Figure 7 presents some trajectories selected from figure 4 for all available $\dot {Re}$ values at $Re_f=300$ (continuous lines) and $Re_f=320$ (dash-dotted lines). We have chosen them to illustrate the turbulent fraction dynamics slowing down on the transient plateaus introduced in the previous section. Three well-defined plateaus can be identified: one around $F_t=0.5$, one around $F_t=0.3$ and one around $F_t=0.1$. We systematically observed the topology of the velocity fields associated with these plateaus and found that they correspond respectively to two bands, one band and a minimal spot, as shown in figure 7(c,d). Minimal spots will be clearly defined in § 3.5. Statistically, from more than 200 studied experiments, it appears that approximately $50\,\%$ of the trajectories that can visit these states, i.e. trajectories whose asymptotic state is below $F_t=0.5$, actually visit them. Surprisingly, the turbulent fraction dynamics is not monotonic, even if the controlling Reynolds number dynamics is. More surprisingly, most of the complex and non-monotonic dynamics takes place when the current Reynolds number is already equal to $Re_f$, i.e. to the right of the vertical dash-dotted lines of figures 4 and 5. Note that this remains true even when the final Reynolds number is below the global stability threshold $Re_g$. Consider, for example, the green trajectories in figure 7 that have been performed at $\dot {Re}=1$ and $Re_f=300$ or $Re_f=320$: in both cases, at $t=110\ \textrm {s}$, the final Reynolds number is reached and the expected state is $F_t=0$. Instead, we see that the system either stays on transient plateaus until much later times at approximately $t=200\ \textrm {s}$ or, for $Re_f=320$, develops long-term fluctuations between transient plateaus at times exceeding the experiment duration (400 s or even 600 s). These observations raise many questions about the nature of these states and in particular about their topology in the phase space. They also reveal the influence of the $\dot {Re}$ decay rate on the dynamics and highlight the great difference between quench and annealing. Indeed, transient plateaus are not observed for the lowest decay rates for which the dynamics almost follows the asymptotic $F_t(Re)$ curve, as can be seen in figure 8 where the asymptotic data of figure 6 are plotted together with the dynamic data of figure 5. This simultaneous plotting is possible because, when the time is scaled with $\dot {Re}$, $Re_i - t\dot {Re}$ is the current Reynolds number. We can see that the two curves collapse on top of each other, except perhaps for the larger $Re$ for which the dynamics is still delayed with respect to the evolution of the control parameter. A better collapse should be achieved by an even slower annealing at a lower $\dot {Re}$.

Figure 7.

(a) Some particular turbulent fraction dynamics picked up to illustrate the robustness of the plateaus. Traces are coloured according to $\dot {Re}$ as in figure 4, continuous lines are for $Re_f=300$, dash-dotted lines for $Re_f=320$. (b,c) Typical velocity fields associated with turbulent fraction plateau at $F_t\simeq 0.5$ (b) and $F_t\simeq 0.1$ (c). They respectively correspond to experiments performed at $\dot {Re}=0.3$ and $\dot {Re}=1$ for a common $Re_f=300$.

Figure 8.

Turbulent fraction versus Reynolds number. The continuous blue line is the ensemble average of data from figure 5 obtained for $Re_f=300$ and $\dot {Re}=0.1$, in this case $Re$ is the current Reynolds number. Black symbols with error bar are the ensemble average and standard deviation of data displayed in figure 6, in this case $Re$ is the final Reynolds number. See text for details.

3.5. Phase portrait

Trying to gain insight into the nature of the transient states and of the trajectories visiting them in the phase space, we have intended to draw phase portraits from first return maps to represent the transient trajectories discussed here. The most natural idea was to focus only on $F_t$ and we thus started by plotting $F_t(t)$ versus $F_t(t+\Delta t)$ for various time increments $\Delta t$. If this method is suitable for simple low-dimensional dynamical systems, it may be more difficult to use it to get relevant information in our spatially extended case. We have thus introduced other global physical quantities linked to spatial inhomogeneities of the flow. Due to the complex spatio-temporal nature and the high dimensionality of the studied system, no obvious reconstruction leading to insightful interpretation of the results could be obtained to better understand the nature of the transient states. Nevertheless, the outcomes of this work, that we will present as phase portraits, have allowed us to identify different turbulence retreat scenarios.

From our full resolution PIV system and 2-D power spectrum calculations, we have access to the power spectral density for wavelengths between 0.2 and 200 in dimensionless units. Examples of such instantaneous spectra obtained for an isolated band and a spot are presented in figure 9. From $1.6h$ to $10h$ the energy content is associated with streaks and corresponds to turbulent areas. Then, a spectral gap is observed around $\lambda _c\simeq 20h$. Above, from $30h$ to $100h$, the spectral content corresponds to large-scale flows. See Couliou & Monchaux (Reference Couliou and Monchaux2015) for a similar discussion from narrower PIV fields. By integrating the whole large-scale flow energy content, we have a measure of the large-scale flow intensity $A_{LSF}$ that is obtained as follows. We first calculate the instantaneous $u_z$ 2-D premultiplied power spectral density $S_{u_z}$ as

(3.2)\begin{equation} S_{u_z}(k_x,k_z,t) =|k|\times |\hat{u}_z(k_x,k_z,t)|^2, \end{equation}

where $\hat {u}_z(k_x,k_z)$ is the instantaneous two-dimensional Fourier transform of $u_z$ and $k=\sqrt {k_x^2+k_z^2}$. Then, we integrate the premultiplied spectrum along the azimuthal direction as

(3.3)\begin{equation} E_z (k,t)= \frac{1}{2{\rm \pi}}\int_0^{2{\rm \pi}}S_{u_z}(k_x,k_z,t)\,\textrm{d}\theta, \end{equation}

with $\theta =\arctan (k_z/k_x)$. Finally, integration over the large-scale wavelength yields an estimation of the large-scale flow intensity as

(3.4)\begin{equation} A_{LSF}(t) = \int_{\lambda/h=30}^{\lambda/h=100}kE_z(k,t)/U^2\,\textrm{d}k, \end{equation}

where $k=2{\rm \pi} /\lambda$.

Figure 9.

Typical power spectral density of $u_z$ averaged azimuthally for a flow containing bands (as in figure 7b) or a spot (as in figure 7c). They respectively correspond to experiments performed at $\dot {Re}=0.3$ and $\dot {Re}=1$ for a common $Re_f=300$. Turbulent streaks produce a rise in energy in a band of $\lambda /h\in [10^0\text {--}10^1]$ while large-scale flows produce energy for $\lambda /h\in [3\times 10^1\text {--}2\times 10^2]$.

Phase portraits for each experiment have been obtained by plotting $F_t(t)$ versus $A_{LSF}(t)$. The systematic observation of all the phase portraits has led to the identification of three types of trajectories exemplified in figure 10. Some trajectories go from $F_t=1$ toward $F_t=0$ with few fluctuations and without developing any large-scale flow while others quickly develop large-scale flows. These last much longer and have a much more complex dynamics. Finally, hybrid trajectories exist that start on the quick path but start to develop large-scale flow at some point and end on the complex and long lasting path. Figure 10 also presents snapshots of the streamwise velocity field taken along these three types of trajectories. Comparing them to each other, it appears that the same levels of turbulent fraction in both cases do not correspond to the same physical state at all. On the large-scale path, the turbulent part of the velocity field is actually made of intense, well defined and instable streaks while on the fast path we observe smooth spatial velocity fluctuations organized as streaks resembling what can be observed in the laminar wake of an object placed in a shear flow or around a localized turbulent spot, see, for example, Couliou & Monchaux (Reference Couliou and Monchaux2016) or Couliou & Monchaux (Reference Couliou and Monchaux2017). These two behaviours actually correspond to two ways in which turbulence declines. In the first case, when laminar holes appear, turbulence (i.e. streaks) intensity remains the same in the turbulent regions whose area progressively diminishes. Thus, turbulence recedes in area but not in intensity. In the second case the emergence of laminar holes goes along with a diminution of the turbulent intensity within turbulent regions and turbulence decreases more in intensity than in area. We thus call these trajectories fading ones. Interestingly, even on large-scale paths, when only a turbulent spot is left, it disappears as it fades, the turbulence losing intensity without decreasing in area. This shows that the imposed shear cannot maintain a turbulent spot which is smaller than a given size. We thus refer to these spots of minimal size as minimal spots. An example of such a minimal spot can be found in figure 7.

Figure 10.

Phase portrait presenting three representative trajectories in the plane ($F_t(t),A_{LSF}(t)$). Snapshots along these three paths are also shown to illustrate the differences between them. From top to bottom, turbulent fraction is 1, 0.9, 0.7, 0.5, 03 and 0.1. Panel frames are coloured according to the followed path.