Investigation of inner-outer interactions in a turbulent boundary layer using high-speed particle image velocimetry

The structure of inner-outer interactions in a smooth-wall turbulent boundary layer is investigated using high-frame-rate particle image velocimetry (PIV) with two overlapping fields of view. A refractive-index-matched facility is used to enable the resolution of turbulent structures very close to the wall where these modulation effects are dominant. Amplitude and frequency modulation correlation coefficients were first investigated by computing single-, two-, and multiple-point correlation analyses now routinely reported in the literature. The wall-normal trends in amplitude modulation for the streamwise and wall-normal velocities and in frequency modulation for the streamwise velocity are consistent with those previously reported in the literature, validating the efficacy of the PIV approach used herein. The frequency modulation behavior for wall-normal velocity, however whose behavior in this regard has received little attention in the literature, differs from its amplitude modulation counterpart in the near-wall region. It is not clear whether this difference is physical or a by-product of enhanced sensitivity to measurement noise and/or finite spatial resolution of the measurements. Leveraging the spatial and temporal nature of the PIV data, a conditional average-based method is extended to the PIV data to directly capture the spatiotemporal signature of these interactions. This spatiotemporal picture reveals that the small-scale variance in streamwise and wall-normal velocity fluctuations not only correlates with the slow variations of the large scales in the logarithmic region, but also leads the local large scales (measured using a single hot wire, for example). These observations clearly indicate a structure akin to that proposed by Baars et al. [W. J. Baars et al., Exp. Fluids 56, 188 (2015)], including the relative positions of the modulated and modulating scales. The independence of these observations from filter choice is also shown, with equivalent correlation coefficients developed based on the conditional averaging analysis.

Figure 1

(a) Schematic of the two-FOV PIV camera arrangement (the floor is indicated in green). (b) Photo highlighting this experimental arrangement in the RIM flow facility.

Figure 2

Example instantaneous snapshot of streamwise velocity contours from the two cameras. The small camera (sFOV) is embedded inside the big camera (bFOV) near the wall. Also shown in inset is the velocity difference between the two cameras within the sFOV [contours ranging from $-0.04$ m/s (red) to 0.04 m/s (blue)].

Figure 3

Wall-normal profiles of (a) mean streamwise velocity, (b) in-plane Reynolds stresses, and (c) streamwise turbulent skewness in inner units. Profiles from the bFOV and sFOV are shown, as are LES and hot-wire results at comparable ${\text{Re}}_{\tau }$ from [65, 66].

Figure 4

One-point and two-point amplitude modulation correlation coefficients based on the (a) streamwise and (b) wall-normal velocity components, respectively, using the envelope-based method from the sFOV, along with similar coefficients from the ${\text{Re}}_{\tau }=2000$ channel-flow DNS of Hoyas and Jiménez [69] computed by Wu et al. [42]. Also shown are multipoint AM correlation coefficients for (c) streamwise and (d) wall-normal velocity fluctuations. The blue dashed line indicates two-point correlation coefficient values and the red dash-dotted line indicates the one-point counterparts reported in (a) and (b).

Figure 5

One-point and two-point frequency modulation correlation coefficients based on the (a) streamwise and (b) wall-normal velocity components, respectively, using wavelet-energy-based method from the sFOV. Also shown are the multipoint FM correlation coefficients for (c) streamwise and (d) wall-normal velocity components. The blue dashed line indicates two-point correlation coefficient values and the red dash-dotted line indicates the counterparts reported in (a) and (b).

Figure 6

(a) Example outer large-scale time series $u_{0L}\left(t^{+}\right)$, with all positive zero-crossing points ${\tau }_{0^{+}}$ marked in blue circles and two example neighborhoods of width $200t^{*}$ in red. (b) Ensemble average of all such positive [${\langle u_{0L}\left(\tau \right)\rangle }_{{\tau }_{0^{+}}}$ (red)] and negative [${\langle u_{0L}\left(\tau \right)\rangle }_{{\tau }_{0^{-}}}$ (blue), not marked in (a)] zero-crossing neighborhoods. Also shown as dashed lines are ensemble averages of $u$ (red) and $v$ (green) spatial variations corresponding to ${\tau }_0^{+}$ corresponding to Taylor hypothesis equivalent times (i.e., $\mathrm{\Delta }{x}^{+}=-{\tau }^{+}{U}_0$), along with $-{\langle u_{0L}\left(\tau \right)\rangle }_{{\tau }_{0^{-}}}$ (blue dotted line) and ${\langle u_{0L}(-\tau )\rangle }_{{\tau }_{0^{-}}}$ (blue dashed line).

Figure 7

Plot of (a), (c), and (e) ${\langle u\rangle }_{0c}$ and (b), (d), and (f) ${\langle v\rangle }_{0c}$, the spatial signature of the large scale, corresponding to (a) and (b) $\tau ={\tau }_{\downarrow }$, (c) and (d) $\tau =0$, and (e) and (f) $\tau ={\tau }_{\uparrow }$.

Figure 8

Shown as filled contours are (a) small-scale energy $\langle u_s^2\rangle$ and small-scale energy discrepancy in (b) streamwise ($\mathrm{\Delta }\langle u_s^2\rangle$) and (c) wall-normal ($\mathrm{\Delta }\langle v_s^2\rangle$) turbulence at three times (i) $\tau ={\tau }_{\downarrow }$, (ii) $\tau =0$, and (iii) $\tau ={\tau }_{\uparrow }$. Also shown as line plots are the corresponding large-scale structures $u_L$ with the same levels as shown in Fig. 7. Negative levels are shown as dashed lines. See the video in [74] which illustrates the full time evolution of the results.

Figure 9

(a) Plot of $\mathrm{\Delta }\langle u_s^2\rangle (x^{+}=0,y^{+}=18,\tau )$ (solid line), ${\langle u\rangle }_{0c}(0,18,\tau )$ (dash-dotted line), and ${\langle u\rangle }_{0c}(0,y_o,\tau )$ (dashed line). (b) One-probe and two-probe correlation coefficients similar to those in Figs. 4 and 4.

Figure 10

Same as Fig. 8, but for large and small scales defined by a sharp spectral filter. See the video in [74] which illustrates the full time evolution of the results.