Lift-up and streak waviness drive the self-sustained process in wall-bounded transition to turbulence

Flow field measurements from a Couette-Poiseuille experiment are used to examine quantitatively certain steps of the self-sustained process (SSP) of wall-bounded transition to turbulence. Although the different parts of the SSP have been discussed at large in the literature, direct measurements from experiment are scarce and the present results show, using a local analysis of the turbulent patterns, that (1) the amplitude of streamwise rolls is related to streak waviness, bringing a quantitative picture to one of the main physical mechanisms of Waleffe's model of SSP, and (2), at low waviness, direct measurements of the correlation between the streak and roll amplitudes, respectively probed by the streamwise and wall-normal velocity perturbations, quantify the lift-up effect. This analysis method of the SSP does not rely on the specificity of Couette-Poiseuille flows and can be used to investigate this mechanism in other flows.

Figure 1

(a) Schematic diagram of the experimental Couette-Poiseuille setup and stereo-PIV configuration. (b) Position of the cylindrical vortex generators on the vertical $xz$ plane. (c) Side view of the vortex generator in the $xy$ plane. The velocity profile displayed is the laminar Couette-Poiseuille.

Figure 2

Stereo-PIV velocity fields on the $xz$ plane at $y=0.33h$: (a) streamwise velocity $u_x$, (b) spanwise velocity $u_z$, and (c) wall-normal velocity $u_y$. The velocities are nondimensionalized by the belt velocity $U_{\mathrm{belt}}$. Horizontal black and green dashed lines in (a) and (b) and vertical black and green dashed line in (c): axes along which the corresponding velocity profiles are displayed in (d), (e), and (f). These lines are located close to a wavy streak.

Figure 3

Decomposition, at $\text{Re}=550$, of the small-scale streamwise velocity into straight and wavy components $u_x=u_{x,\mathrm{stra}}+u_{x,\mathrm{wavy}}$, and the corresponding premultiplied energy spectra. (a) Small-scale flow $u_x$; (b) straight streaks $u_{x,\mathrm{stra}}$; (c) wavy streaks $u_{x,\mathrm{wavy}}$; (d) premultiplied energy spectrum $k{E}_{u_x}$ of $u_x$; (e) premultiplied energy spectrum $k{E}_{u_{x,\mathrm{stra}}}$ of $u_{x,\mathrm{stra}}$; (f) premultiplied energy spectrum $k{E}_{u_{x,\mathrm{wavy}}}$ of $u_{x,\mathrm{wavy}}$. The spectrum (d) is the sum of the filtered spectra (e) and (f).

Figure 4

Small-scale flow snapshots at $\text{Re}=550$; (a) $u_x$ field; (b) $u_z$ field; (c) $u_y$ field; (d) ${\omega }_{y,\mathrm{wavy}}$. On snapshots of $u_z, u_y, {\omega }_{y,\mathrm{wavy}}$ the contour of the selected streaks are shown in solid lines ($u_x=0.1$) and dashed line ($u_x=-0.1$). See the Supplemental Material [41] for a video corresponding to the time evolution of this figure in a full experimental run.

Figure 5

(a) Streak-averaged wall-normal velocity $\langle \left|u_y\right|\rangle$ as a function of the wavy vorticity $\langle \left|{\omega }_{y,\mathrm{wavy}}\right|\rangle$ for positive $u_x$ streaks at $\text{Re}=550$. (b) Streaks $u_x$ from one frame with numbers corresponding to the data points indicated in (a). The area outside the streaks is set to zero for clarity.

Figure 6

Streak-averaged wall-normal $\langle \left|u_y\right|\rangle$ velocity as a function of the streak-averaged wavy vorticity $\langle \left|{\omega }_{y,\mathrm{wavy}}\right|\rangle$ for (a) $\text{Re}=450$, (b) $\text{Re}=550$, (c) $\text{Re}=600$, (d) $\text{Re}=700$, and (e) $\text{Re}=1000$. The histograms on the top and right sides of each plot indicate the density of points at the corresponding position. (f) Zoom of the low-waviness region indicated in (b) showing that for the points with $\langle \left|{\omega }_{y,\mathrm{wavy}}\right|\rangle <0.04$ the increasing trend of $\langle \left|u_y\right|\rangle$ as a function of $\langle \left|{\omega }_{y,\mathrm{wavy}}\right|\rangle$ cannot be established. On each curve a running average over a window of 50 points is shown as a solid green line; the line is broken when the number of points is scarce and the running average is no longer meaningful.

Figure 7

$〈|u_y|〉$ vs $〈|u_x|〉$ for all five experiments at $\text{Re}=550$ (the color map indicates values of $\langle \left|{\omega }_{y,\mathrm{wavy}}\right|\rangle$). The red circles correspond to cases' values of $\langle \left|{\omega }_{y,\mathrm{wavy}}\right|\rangle <0.04$ (a linear fit of these values is shown in dashed line). The black dots are the filtered fields $\langle \left|u_{y,\mathrm{stra}}\right|\rangle$ vs $\langle \left|u_{x,\mathrm{stra}}\right|\rangle$. A linear fit of the black points is indistinguishable within the accuracy of the experimental measurements from the dashed line.

Figure 8

$\langle \left|u_y\right|\rangle$ vs $\langle \left|u_x\right|\rangle$ for all Reynolds numbers: (a) $\text{Re}=450$, (b) $\text{Re}=550$, (c) $\text{Re}=600$, (d) $\text{Re}=700$, and (e) $\text{Re}=1000$. The histograms on the top and right sides of each plot indicate the density of points at the corresponding position. The mean values are shown as a blue triangle on each plot. The points where $\langle \left|{\omega }_{y,\mathrm{wavy}}\right|\rangle <0.04$ are marked in red and the black dots are the filtered fields $\langle \left|u_{y,\mathrm{stra}}\right|\rangle$ vs $\langle \left|u_{x,\mathrm{stra}}\right|\rangle$, as in Fig. 7. The fit for the red points at $\text{Re}=450$ is recalled as a dashed line in all plots, showing that it serves as a lower bound and that in the turbulent regimes all points are above it. (f) Box plots representing the histogram of $\langle \left|u_y\right|\rangle$ for each Reynolds number. On each box, the central red line indicates the median and the bottom and top edges of the box indicate the 25th and 75th percentiles, respectively. The whiskers extend to the most extreme data points not considered outliers and the outliers are the red markers.