Structure of high and low shear-stress events in a turbulent boundary layer

Simultaneous particle image velocimetry (PIV) and wall-shear-stress sensor measurements were performed to study structures associated with shear-stress events in a flat plate turbulent boundary layer at a Reynolds number ${\text{Re}}_{\tau }\approx 4000$. The PIV field of view covers $8\delta$ (where $\delta$ is the boundary layer thickness) along the streamwise direction and captures the entire boundary layer in the wall-normal direction. Simultaneously, wall-shear-stress measurements that capture the large-scale fluctuations were taken using a spanwise array of hot-film skin-friction sensors (spanning $2\delta )$. Based on this combination of measurements, the organization of the conditional wall-normal and streamwise velocity fluctuations $(u$ and $v)$ and of the Reynolds shear stress $\left(-uv\right)$ can be extracted. Conditional averages of the velocity field are computed by dividing the histogram of the large-scale wall-shear-stress fluctuations into four quartiles, each containing 25% of the occurrences. The conditional events corresponding to the extreme quartiles of the histogram (positive and negative) predominantly contribute to a change of velocity profile associated with the large structures and in the modulation of the small scales. A detailed examination of the Reynolds shear-stress contribution related to each of the four quartiles shows that the flow above a low wall-shear-stress event carries a larger amount of Reynolds shear stress than the other quartiles. The contribution of the small and large scales to this observation is discussed based on a scale decomposition of the velocity field.

Figure 1

Experimental setup: (a) photograph of the flat plate in the wind tunnel, (b) schematic of the global experimental setup (flow left to right), (c) schematic of the PIV setup (flow right to left), (d) location of the wall-shear-stress sensors (flow left to right).

Figure 2

Statistical profiles of the turbulent boundary layer compared to data from Refs. [19] (hot-wire data, ${\text{Re}}_{\tau }=3910)$, [11] (LES data, ${\text{Re}}_{\theta }=8300)$, [21] (stereo-PIV, ${\text{Re}}_{\theta }=8100)$, and [25] (Tomo-PIV, ${\text{Re}}_{\theta }=4850)$. (a) Mean velocity profile $U^{+}$: right triangle (measured), solid line (LES data), dashed line (HW data), dashed gray line (Tomo-PIV), gray line with crosses (tereo-PIV); ${\overline{u^2}}^{+}$: square (measured), dotted line (LES data), and line with cross (HW data); (b) ${\overline{v^2}}^{+}$: diamond (measured) and solid line (LES data); Reynolds shear stress, $-{\overline{uv}}^{+}$: left triangle (measured) and dashed line (LES data).

Figure 3

Histogram of the normalized fluctuations of the friction velocity.

Figure 4

Two-point correlation map $C_{u_{\tau }{u}_{\tau }}$. The contour lines correspond to level 0.1 to 0.9 with 0.2 increment. The dashed lines are negative contour, and solid contours are positive $C_{u_{\tau }{u}_{\tau }}$.

Figure 5

Instantaneous fluctuations of the friction velocity, $u_{\tau }$, at the wall and of the streamwise velocity, $u$, in the laser sheet plane associated with a low shear-stress event (top) and high shear-stress event (bottom).

Figure 6

Decomposition of the streamwise velocity fluctuation: (top) total streamwise fluctuation, $u$; (middle) large-scale fluctuation, $u_l$, computed with moving average (box-filter) that is $1\delta$ long; (bottom) small-scale fluctuation computed by subtracting the field (top) and (bottom).

Figure 7

Extreme wall-shear-stress events: (a) decomposition of the histogram of the friction velocity fluctuations in four bins, each containing 25% of the occurrences. (b) Profiles of the conditional streamwise velocity fluctuation in the centerline $\left(z/\delta =0\right)$ at $x/\delta =1$ The color line corresponds to the part of the histogram of the same color. (c) Profiles of the change of the small-scale variance of the streamwise velocity component. (d) Profiles of the change of the small-scale variance of the wall-normal velocity component. Note that the conditional profiles were running-averaged from $0.5\delta$ to $1.5\delta$ to improve statistical convergence.

Figure 8

Change of the Reynolds shear-stress activity for the two extreme parts of the histogram in the plane $z/\delta =0$: (a) field associated with a very low wall-shear-stress event; (b) field associated with a very high wall-shear-stress event.

Figure 9

Profiles of the conditional Reynolds shear stress in the middle plan $z=0$ and averaged between $0.5\delta$ and $1.5\delta$. The blue, green, orange, and red correspond, respectively, to the very low shear, slightly low, slightly high, and very high stress event at the wall given the histogram decomposition.

Figure 10

Reynolds shear-stress profile based on the large- and small-scale decomposition: the solid line corresponds to $\overline{-u_l{v}_l}$, the dash-dotted line to $\overline{-u_s{v}_s}$, the dashed line to $\overline{-u_l{v}_s}$, and the triangles to $\overline{-u_s{v}_l}$.

Figure 11

Change of the Reynolds shear-stress activity given the large- and small-scale decomposition for the four bins of the histogram. Profile is located in the middle plan $z=0$ and averaged between $0.5\delta$ and $1.5\delta$: $(\mathrm{a})〈\widetilde{-u_s{v}_s}〉$, (b) $〈\widetilde{-u_l{v}_l}〉$.