2.1 Eigenvalue analysis of linear stability to two-dimensional infinitesimal perturbation

Figure 1. (a) Different velocity profiles of the flows (with zero flux, upper and lower velocity equal to $1$ and $\unicode[STIX]{x1D70E}$ , respectively) for which the linear stability to two-dimensional infinitesimal perturbation is analysed: plane Couette–Poiseuille (green line), Couette (red line), Poiseuille (blue line). The magenta profile ( $\unicode[STIX]{x1D70E}=0.309$ ) represents the case at which linear stability disappears. (b) Dependence of the linear instability threshold on the nondimensionalized speed of the lower wall. The black crosses are the results of Balakumar (Reference Balakumar1997).

We parametrize the laminar Couette–Poiseuille flow family as:

where $y_{\ast }\in (-1,1)$ (see figure 1). This equation is derived by assuming a generic quadratic function with zero net flux and boundary conditions such that $U_{\ast }(1)=1$ , $U_{\ast }(-1)=\unicode[STIX]{x1D70E}$ , $\unicode[STIX]{x1D70E}\in (-1,1)$ . The plane Couette–Poiseuille flow analysed in this paper corresponds to $\unicode[STIX]{x1D70E}=0$ (green line in figure 1 a). The two limiting cases, pure plane Poiseuille ( $\unicode[STIX]{x1D70E}=1$ ) and Couette ( $\unicode[STIX]{x1D70E}=-1$ ) flows, are marked in figure 1(a) by blue and red curves, respectively. The linear stability was investigated by computing the least stable eigenvalue of the Orr–Sommerfeld operator. When $\unicode[STIX]{x1D70E}$ is decreased, the critical Reynolds number $Re_{L}$ , monotonically increases up to $\unicode[STIX]{x1D70E}=0.309$ , where it diverges to infinity ( $Re_{L}\rightarrow \infty$ , figure 1 b). Our plane Couette–Poiseuille flow ( $\unicode[STIX]{x1D70E}=0$ ) is thus linearly stable for any Reynolds number, similar to plane Couette and pipe flow.

If we substitute $\unicode[STIX]{x1D70E}=1-2\unicode[STIX]{x1D70E}_{2}$ and apply a Galilean transformation of $-1$ and reflection in $x$ and $y$ , we obtain the same formulation and results as in a previous study of Balakumar (Reference Balakumar1997).

2.2 Transient growth

Even if a shear flow is linearly stable, as in our case, a perturbation may grow transiently due to the non-normality of the linearized Navier–Stokes equations. In previous work, Bergström (Reference Bergström2005) calculated transient growth in plane Couette–Poiseuille flow for a single Reynolds number $Re=1000$ and for different combinations of Couette and Poiseuille components. Here, we calculate for a range of Reynolds numbers $Re\in (100,1000)$ the transient growth for plane Couette–Poiseuille flow with zero mass flux, given by $U_{\ast }(y_{\ast })=(3/4)(y_{\ast }^{2}-1)+(y_{\ast }+1)/2$ . For each streamwise ( $\unicode[STIX]{x1D6FC}_{\ast }$ ) and spanwise ( $\unicode[STIX]{x1D6FD}_{\ast }$ ) wavenumber combination we determine the maximal gain $G_{max}$ and the time $t_{\ast max}$ at which it occurs. We define:

where $\boldsymbol{q}=[\boldsymbol{u}_{\boldsymbol{\ast }}^{\boldsymbol{\prime }},\boldsymbol{v}_{\boldsymbol{\ast }}^{\boldsymbol{\prime }},\boldsymbol{w}_{\boldsymbol{\ast }}^{\boldsymbol{\prime }}]$ corresponds to velocity fluctuations, $\Vert \boldsymbol{q}_{\boldsymbol{o}\boldsymbol{u}\boldsymbol{t}}\Vert ^{2}=\Vert \boldsymbol{q}(t_{\ast max})\Vert ^{2}$ is the energy of the velocity fluctuations at $t_{\ast max}$ calculated in the entire domain and $\Vert \boldsymbol{q}_{\boldsymbol{o}\boldsymbol{p}\boldsymbol{t}}\Vert ^{2}=\Vert \boldsymbol{q}(t_{\ast }=0)\Vert ^{2}$ is the energy of the initial perturbation optimized for all $\boldsymbol{q}_{\mathbf{0}}$ that leads to the maximal energy gain. The details of the calculations are described in Schmid & Henningson (Reference Schmid and Henningson2001).

Figure 2. The dependence on streamwise ( $\unicode[STIX]{x1D6FC}_{\ast }$ ) and spanwise ( $\unicode[STIX]{x1D6FD}_{\ast }$ ) wavenumbers in plane Couette–Poiseuille flow for: (a) maximal amplification $G_{max}$ for $Re=500$ . The highest amplification occurs for ( $\unicode[STIX]{x1D6FC}_{\ast }=0,\unicode[STIX]{x1D6FD}_{\ast }=1.83$ ) and it is independent of Reynolds number. (b) Onset of unconditional stability $Re_{E}$ . The minimal Reynolds number, $Re_{E}=32.53$ , is reached for ( $\unicode[STIX]{x1D6FC}_{\ast }=0,\unicode[STIX]{x1D6FD}_{\ast }=1.728$ ).

In figure 2(a) we present the dependence of $G_{max}$ on $\unicode[STIX]{x1D6FC}_{\ast }$ and $\unicode[STIX]{x1D6FD}_{\ast }$ for $Re=500$ . There is a distinct peak at $\unicode[STIX]{x1D6FC}_{\ast opt}=0,\unicode[STIX]{x1D6FD}_{\ast opt}=1.83$ , with no streamwise dependence as is the case for plane Poiseuille flow (Schmid & Henningson Reference Schmid and Henningson2001). We verify that this wavenumber pair is optimal within $Re\in (100,1000)$ . We also determine that $G_{max}$ and $t_{\ast max}$ scale with $Re^{2}$ and $Re$ , respectively (table 1). Our present measurements with two-dimensional (2D) particle image velocimetry (PIV) are performed in one plane at $y_{\ast }=0.3$ . In contrast, the global quantity $G_{max}$ measures the perturbation energy over the entire gap (in $y_{\ast }$ ) and for this reason it cannot be used for quantitative comparison of our experimental and theoretical results. In figure 3 we present that the maximal amplitude of streaks occurs at $y_{\ast }=0.33$ , independently of Reynolds number, and so we define the local quantity $G_{max}^{\prime }$ :

We verify that both the global maximal gain $G_{max}$ and the local maximal gain of the streamwise velocity component $G_{max}^{\prime }$ occur almost at the same time $t_{\ast max}$ . In table 1 we show the scaling for $G_{max}^{\prime }$ .

Figure 3. Velocity profile of the streamwise velocity fluctuations $u_{\ast }^{\prime }(y_{\ast })$ at $t_{\ast max}$ for ( $\unicode[STIX]{x1D6FC}_{\ast }=0,\unicode[STIX]{x1D6FD}_{\ast }=1.83$ ) and for $Re=500$ . The profile is normalized with the maximal value $u_{\ast }^{\prime }(t_{\ast max})$ calculated over the entire gap in the wall-normal direction. Normalized velocity profiles for $Re\in (100,1000)$ collapse to a single curve. The magenta dashed line marks $y_{\ast }=0.33$ , at which the streaks reach maximal value.

2.3 Condition for no transient growth – unconditional stability

To complete the full characterization of Couette–Poiseuille flow, we also calculate the energy Reynolds number $Re_{E}$ (Joseph Reference Joseph1976) below which our flow is unconditionally stable ( $\text{d}\Vert \boldsymbol{q}\Vert ^{2}/\text{d}t<0$ for all $q_{0}$ ). In figure 2(b) we present the dependence of $Re_{E}$ on $\unicode[STIX]{x1D6FC}_{\ast }$ and $\unicode[STIX]{x1D6FD}_{\ast }$ , whose minimum is $Re_{E}(\unicode[STIX]{x1D6FC}_{\ast }=0,\unicode[STIX]{x1D6FD}_{\ast }=1.728)=32.53$ . The mode $\unicode[STIX]{x1D6FC}_{\ast }=\unicode[STIX]{x1D6FD}_{\ast }=0$ , which represents base flow modification, is unconditionally stable up to $Re=10^{8}$ . This suggests that mean flow modification cannot extract energy from the base flow by itself and can be sustained only by nonlinear energy transfer from other modes.

Figure 4. Experimental configuration: (a) perspective view; (b) cross-section in $xy$ plane showing the base flow in the gap.

3.1 Experimental set-up

The experimental set-up is presented in figure 4. It consists of a tank filled with water and with one closed-loop moving belt made of Mylar (of $0.175~\unicode[STIX]{x03BC}\text{m}$ thickness) near one bounding wall of the test section. The other wall, a glass plate, remains stationary. The moving wall and induced streamwise pressure gradient generate plane Couette–Poiseuille flow with nearly zero mean flux (for details see Klotz et al. (Reference Klotz, Lemoult, Frontczak, Tuckerman and Wesfreid2017)). The experiments reported here were performed with a gap between moving and stationary walls of $2h=10.8~\text{mm}$ and the aspect ratios of the test section in the streamwise/spanwise directions are $L_{x}/h=370.4$ and $L_{z}/h=96.3$ , respectively. A water jet is injected through a hole of $\unicode[STIX]{x1D719}=1.6~\text{mm}$ $=0.30h$ in the wall-normal direction at the centre of the test section and triggers the turbulent spot. The jet comes from a small high-pressure water container with a pressure controller (SMS ITV2010), as well as an electromagnetic valve, which controls its duration. We determine the average jet speed $\langle V_{jet}\rangle$ by repeating injections and measuring the total volume of the injected water with a measuring cylinder. A localized turbulent spot is triggered by a jet injection of short duration (approximately 1 advection unit, $\unicode[STIX]{x0394}T_{\ast }\simeq 1$ ) with very weak amplitude ( $A=\langle V_{jet}\rangle /U_{belt}\in (1.8,3.2)$ ) to minimize the nonlinear interactions. To test the limits of validity of linear transient growth, we also investigate higher perturbation amplitudes ( $A\in (5.7{-}34.9)$ ). Even if $\langle V_{jet}\rangle$ is greater than the typical velocity of the base flow, the duration of the injection is very short and the ratio of the injected volume $Q_{injected}$ to the total volume of the fluid volume in the channel $Q_{test\,section}$ is very low ( $Q_{injected}/Q_{test\,section}\simeq A\times \unicode[STIX]{x0394}T_{\ast }\times 10^{-6}$ ). A similar situation was described in Darbyshire & Mullin (Reference Darbyshire and Mullin1995).

Our experimental set-up has one stationary wall without a moving plastic belt. This grants us the advantage of free access to introduce a well-controlled perturbation without the necessity of synchronizing the phase of the belt motion with the moment of injection, as was necessary in classical plane Couette experiment, in which the water jet was introduced through the hole in the plastic belt (Bottin et al. Reference Bottin, Dauchot, Daviaud and Manneville1998), which may alter the direction of the injection.

We present the velocity fluctuations acquired with 2D PIV. The laser sheet was located parallel to the bounding walls at plane $y_{\ast }=0.33$ , where the base flow has nearly zero streamwise velocity and where linear theory predicts the highest amplitude of streaks for the optimal response. We use a Darvin Due laser (double-headed, maximum output 80 W, wavelength 527 nm) and Phantom Miro M120 camera ( $1920\times 1600$ pix, pixel pitch $0.28~\text{mm}~\text{pix}^{-1}$ ). The moment at which we start the acquisition was synchronized by a National Instrument NI PCI-6602 synchronization device. The sequence of acquired images was cross-correlated by Dantec Dynamic Studio 4.0 software using rectangular interrogation windows $64\times 8~\text{pix}$ with 50 % overlap. This unconventional choice is justified by the dominant streamwise component, which implies that the pixel displacement in the streamwise direction is an order of magnitude larger than in the spanwise direction. This aspect ratio also provides a high spatial resolution in the spanwise direction. For each $Re\in (330,380,480,520,580)$ , we acquire 15 different realizations with acquisition frequency $f=10~\text{Hz}$ . This frequency was sufficient to follow the dynamics of the streaks due to the nearly zero advection velocity of spots.

Our base flow is slightly affected by the belt phase motion due to the joining of two extremities of the belt (see Klotz et al. (Reference Klotz, Lemoult, Frontczak, Tuckerman and Wesfreid2017) for quantitative analysis), which introduces weak three-dimensionality. In addition, there is a small back-flow in the gap between the glass plate and the layer of the moving belt closest to it (see red curve in inset of figure 4). As a result, our base flow has a slightly non-zero mean flux (the time-averaged base flow has $U_{avg}<0.07U_{belt}$ ). In order to filter out the dependence of the base flow on the belt phase motion, we first measure the reference base flow (without triggering the turbulent spot) and then we subtract it for each actual realization (with a turbulent spot), keeping the same phase of belt motion as in the reference flow. In this way we calculate the velocity fluctuations: $u^{\prime }=U_{measured}-U_{base\,flow}$ and $w^{\prime }=W_{measured}-W_{base\,flow}$ , where $W_{base\,flow}\simeq 0$ .

3.2 Experimental evidence for transient growth

Figure 5. An example of the energy fluctuations for $Re=480$ , $A=2.3$ representing typical transient growth evolution. Each point corresponds to a single instant measured at 10 Hz; (a) mean energy of streamwise ( $\bar{E}_{u^{\prime }}$ ) and spanwise ( $\bar{E}_{w^{\prime }}$ ) velocity fluctuations (note that $\bar{E}_{w^{\prime }}$ is multiplied by a factor of 10); (b) $E_{u^{\prime }}$ for a single realization (blue solid curve), interpolation proposed by Kim & Moehlis (Reference Kim and Moehlis2006) (red dashed curve), linear interpolation (green dotted line) and exponential decay interpolation (cyan dotted curve). The black dashed horizontal line represents the noise level due to the variation of the base flow. The black dotted vertical line marks the reference time.

We denote mean and ensemble averaged energy by $\bar{E}$ and $\langle E\rangle$ , respectively: for the former we calculate the evolution of the energy fluctuations for each realization separately and then we average them, while for the latter we ensemble-average the sequence of velocity fields of 15 realizations and then calculate the energy.

From the experimental point of view, the most difficult task is to determine the very weak energy of the initial perturbation $E_{0}$ . To calculate it, we analyse the streamwise and spanwise velocity components in the initial frame in the temporal sequence of $\langle E\rangle$ for each ( $Re,A$ ) pair. Ensemble averaging filters out the variation of the base flow and enhances the signal that corresponds to the deterministic and repeatable perturbation. In addition, we consider only the region in the vicinity of the jet (by applying an appropriate mask covering $x_{\ast }\in (-6.6,8.2)$ , $z_{\ast }<|5.8|$ ). We further enhance the accuracy of $E_{0}$ by filtering out the signal close to the spatial homogeneous $(0,0)$ mode with a fast Fourier transform. We normalize both $\bar{E}$ and $\langle E\rangle$ with $E_{0}$ .

In figure 5(a) we show that the mean energy of spanwise velocity fluctuations ( $\bar{E}_{w^{\prime }}$ , in red) is more than one order of magnitude lower than that of the streamwise component ( $\bar{E}_{u^{\prime }}$ , in blue). For this reason in the following we consider only $\bar{E}_{u^{\prime }}$ . In figure 5(b) we present a typical evolution of $\bar{E}_{u^{\prime }}$ for a single realization (blue points), where the linear growth at initial stage (called also algebraic growth) is followed by the eventual exponential decay. It is further illustrated by a sequence of streamwise velocity fluctuation fields ( $u_{\ast }^{\prime }$ ) measured with 2D PIV (figure 6). The internal structure of the localized spot is dominated by streaks, with two dominant wavenumbers calculated using two-dimensional FFT transform: $\unicode[STIX]{x1D6FD}_{1\ast }=1.84$ (mostly at the left front and at the tips of the turbulent spot) and $\unicode[STIX]{x1D6FD}_{2\ast }=2.97$ (at the right front). These wavenumbers correspond to the wavenumbers $\unicode[STIX]{x1D706}_{z1\ast }=3.4$ and $\unicode[STIX]{x1D706}_{z2\ast }=2.1$ , respectively. The former value is in perfect agreement with our theoretical predictions (see figure 2 a). One possibility to explain the presence of the second wavenumber is the existence of two layers of asymmetric vortices that occupy different regions in $y$ . This problem is the subject of ongoing investigation. On each subplot in figure 7(a–e) we present all realizations acquired for a given ( $Re,A$ ) pair. Figure 7(a–d) corresponds to the weakest jet amplitude $A$ used in our experiment, which is appropriate for analysing linear transient growth (note that $\langle V_{jet}\rangle$ is kept constant and $A$ varies only due to $U_{belt}$ ). Up to $Re=480$ all realizations show the typical behaviour of linear transient growth: initial algebraic growth followed by exponential decay (see also interpolation in figure 5 b). This is also true for most realizations for $Re=520$ , with the exception of a single realization, in which the turbulent spot becomes self-sustained. We present this spatial and temporal evolution in figure 8, where the modulation of streaks can be observed. As we increase $Re$ further to $580$ , more spots behave in this way. However, we note that this process is random and some realizations still show transient growth and decaying dynamics. On each subfigure in figure 7 we mark one typical example of transient growth evolution and one example of a self-sustained spot (if any exists) by a thick black/blue curve, respectively. In figure 7(e–f) the same transition from transient (figure 7 e) to self-sustained dynamics (figure 7 f) is observed for higher $A$ , but it occurs for lower $Re$ .

Figure 6. Temporal evolution of the spot, represented by isovalues of the streamwise velocity fluctuations $u_{\ast }^{\prime }(x_{\ast },y_{\ast }=0.3,z_{\ast })$ at different times, measured with PIV for $Re=480$ , $A=2.3$ and for a single realization; $t_{\ast }=69.4$ corresponds to the spatial structure when the maximal energy gain is reached.

Figure 7. On each subfigure the evolution of $E_{u^{\prime }}$ for 15 different realizations for a given $Re$ and $A$ are shown. For each combination we mark with a thick line a single typical realization, for which a turbulent spot shows transient growth and decay (black solid line) and self-sustained dynamics (blue thick line). Also shown are the mean ( $\bar{E}_{u^{\prime }}$ ) and ensemble averaged ( $\langle E\rangle _{u^{\prime }}$ ) energy evolution (red and green dotted lines, respectively). The blue curve in (c) corresponds to figure 8.

Figure 8. The same as figure 6 but for $Re=520$ , $A=2.0$ . The evolution represents a realization without decay (self-sustained case).

Figure 9. Experimental and theoretical values for: (a) $t_{\ast max}$ (red square at ( $Re=580$ , $t_{\ast max}=133$ ) is determined from $\bar{E}_{u^{\prime }}$ evolution); (b) energy gain $G_{max}^{\prime }$ .

The behaviour of our measured localized turbulent spot can be compared with the dynamics of double-localized (in $x,z$ directions) exact coherent structure in plane Couette flow after being perturbed in its most unstable direction, as observed numerically by Brand & Gibson (Reference Brand and Gibson2014). For low enough Reynolds number, all cases led to monotonic relaminarization, preceded in most cases by a period of transient growth (compare with figure 7 a,b,e). Above a given Reynolds number some realizations led to relaminarization and the others produce long-lived turbulent spots with complex, long-term, and perturbation- and Reynolds-dependent behaviour (compare with figure 7 c,d,f). This sensitive dependence of the dynamics on the perturbation implies that their solution must lie on the laminar/turbulent boundary. One may also use the same argument for our results, keeping in mind that in the present case, the spatial structure can differ slightly for different experimental realizations, thus these do not represent a single point in a phase space. Nevertheless, this suggests that the turbulent spots that we observed may be related to the laminar/turbulent boundary.

Our measured structures also resemble the optimal wave packets localized in the both homogeneous directions calculated for the Blasius boundary layer using linearized Navier–Stokes equations (Cherubini et al. Reference Cherubini, Robinet, Bottaro and Palma2010b ). These optimal wave packets are dominated by streamwise-localized streaks, as in our case.

However, we recall that these numerical simulations were performed for different examples of wall-bounded shear flows (plane Couette flow for doubly-localized exact coherent structure and boundary layer flow for optimal wave packets). To our knowledge for the moment no results concerning optimal wave packets or exact coherent structures are available for plane Couette–Poiseuille flow, and for this reason more quantitative comparison with our experimental work is not possible.

We analyse separately every realization representing typical transient growth evolution and then calculate the mean values of $t_{\ast max}$ and $G_{max}^{\prime }$ . To improve accuracy of determination of these two quantities and to systematically determine the reference time from which we calculate $t_{\ast max}$ , we fit our experimental data to the formula proposed by Kim & Moehlis (Reference Kim and Moehlis2006) (red dashed curve in figure 5 b):

where $a$ stands for the amplitude of the streaks and $\unicode[STIX]{x1D6FE}_{1},\unicode[STIX]{x1D6FE}_{2}<0$ . In addition, $|A_{1}|\approx -|A_{2}|$ , $\unicode[STIX]{x1D6FE}_{1}\approx \unicode[STIX]{x1D6FE}_{2}$ , which provides non-normal behaviour. We add a supplementary term ( $B.E.$ ) representing the background experimental noise and possible variation of the base flow, whose amplitude is represented by the black dashed horizontal line in figure 5(b). We define the reference time as the moment at which interpolation reaches the level of the background noise and we measure $t_{\ast max}$ from this reference. One can see in figure 9 that for the lowest amplitude perturbation $A$ (black crosses) $t_{\ast max}$ is well predicted by linear transient growth theory and the value of $G_{max}^{\prime }$ seems to be slightly higher. As we increase the perturbation amplitude, $t_{\ast max}$ increases and deviates from the theoretical prediction. $G_{max}^{\prime }$ seems to slightly increase with both amplitude and $Re$ . We also note that a full comparison of the energy gain between the theory and experiment would require one to measure also the wall-normal component.

Nonlinear transient growth concepts described in (Duguet et al. Reference Duguet, Monokrousos, Brandt and Henningson2013; Kerswell, Pringle & Willis Reference Kerswell, Pringle and Willis2014; Farano et al. Reference Farano, Cherubini, Robinet and Palma2016) can give further insight into the dynamics of transient states. These theories are based on fully nonlinear Navier–Stokes equations and provide the nonlinear optimal perturbation (NLOP) (typically in the sense of the minimal energy difference from the laminar flow) that can lead to turbulence. Specifically, in contrast to the streamwise-extended optimal perturbations given by the linear approach, the NLOP is fully localized is space and has a higher energy gain than the linear theory prediction (Cherubini et al. Reference Cherubini, De Palma, Robinet and Bottaro2010a ; Pringle & Kerswell Reference Pringle and Kerswell2010; Kerswell et al. Reference Kerswell, Pringle and Willis2014). In our experiment we also use the spatially localized perturbation and for the highest Reynolds numbers the measured energy gain is larger than the value obtained with linear theory. Furthermore, using numerical simulations, Pringle, Willis & Kerswell (Reference Pringle, Willis and Kerswell2015) observed that before the lift-up mechanism (related to transient growth) takes over, causing the streaks to develop and elongate, the NLOP must initially unpack in space, which explains why we observed non-zero reference time in our experiments. Specifically, they reported that the initial unpacking of NLOP takes approximately 10 advective units, which is of the same order as typical values of the reference time in our measurements.