Inner–outer interactions in a turbulent boundary layer overlying complex roughness

Hot-wire measurements were performed in a zero-pressure-gradient turbulent boundary layer overlying both a smooth and a rough wall for the purpose of investigating the details of inner–outer flow interactions. The roughness considered embodies a broad range of topographical scales arranged in an irregular manner and reflects the topographical complexity often encountered in practical flow systems. Single-probe point-wise measurements with a traversing probe were made at two different regions of the rough-wall flow, which was previously shown to be heterogeneous in the spanwise direction, to investigate the distribution of streamwise turbulent kinetic energy and large scale–small scale interactions. In addition, two-probe simultaneous measurements were conducted enabling investigation of inner–outer interactions, wherein the large scales were independently sampled in the outer layer. Roughness-induced changes to the near-wall behavior were investigated, particularly by contrasting the amplitude and frequency modulation effects of inner–outer interactions in the rough-wall flow with well-established smooth-wall flow phenomena. It was observed that the rough-wall flow exhibits both amplitude and frequency modulation features close to the wall in a manner very similar to smooth-wall flow, though the correlated nature of these effects was found to be more intense in the rough-wall flow. In particular, frequency modulation was found to illuminate these enhanced modulation effects in the rough-wall flow. The two-probe measurements helped in evaluating the suitability of the interaction-schematic recently proposed by Baars et al., Exp. Fluids 56, 1 (2015) for rough-wall flows. This model was found to be suitable for the rough-wall flow considered herein, and it was found that frequency modulation is a “cleaner” measure of the inner–outer modulation interactions for this rough-wall flow.

Figure 1

A true-scale schematic of the hot-wire measurement locations relative to the complex roughness shown in perspective view. The mean streamwise velocity is presented in a cross-flow plane (measured using sPIV), highlighting the spanwise positions of the current hot-wire measurements (denoted with “X”) at an LMP (left) and HMP (right), as previously identified by Barros and Christensen [43]. The current measurement points are 150 mm upstream of the PIV plane indicated and distributed on the solid line.

Figure 2

(a) Profiles of mean velocity normalized in inner units (squares), and in defect form with $y$-normalized in outer units (triangles). Also shown are logarithmic profiles with $\kappa =0.384$. (b) Corresponding mean streamwise turbulent kinetic energy profiles. Smooth; rough (LMP); rough (HMP).

Figure 3

(a) Premultiplied streamwise TKE spectrogram (dashed horizontal line corresponds to ${\lambda }_x^{+}=7000$—the cutoff wavelength) and (b) single-point AM correlation coefficient, $R_{1P}^a$ (regions I, II, and III marked in the correlation map) for smooth-wall flow.

Figure 4

(a, b) Premultiplied streamwise TKE spectrograms (dashed horizontal line corresponds to ${\lambda }_x^{+}=7000$—the cutoff wavelength) and (c, d) single-point AM correlation coefficients, $R_{1P}^a$, for rough-wall flow at an LMP and a HMP, respectively.

Figure 5

The zero-time-delay AM correlation coefficient, $R_{1P}^a{|}_{\tau =0}$, as defined by Mathis et al. [15], as a function of the wall-normal position. $◯$: smooth; $\square$: Rough (LMP); $△$: Rough (HMP).

Figure 6

(a–c) Premultiplied streamwise TKE spectrograms (dashed horizontal line corresponds to ${\lambda }_x^{+}=7000$—the cutoff wavelength) and (d–f) two-point AM correlation coefficients, $R_{2P}^a$, for smooth- and rough-wall flow at an LMP and a HMP, respectively. The blue lines in (d–f) indicate the location of the outer probe for each case.

Figure 7

Time delay for maximum AM correlation coefficient. $\square$: one probe; $△$: two probe. Smooth; rough (LMP); rough (HMP).

Figure 8

Premultiplied wavelet power spectrum (WPS) ($f\tilde{E}$, right) of a velocity time series measured over smooth-wall flow at $y^{+}=9$. The time series on the plot mark the corresponding instantaneous frequencies, $f_s$ and $f_{sL}$, and the cutoff frequency, $f_c$ (dashed). The time average of the WPS ($f{\langle \tilde{E}\rangle }_t$) is compared with corresponding smoothed premultiplied power spectrum ($k{\varphi }_{uu}$) on the left—both of which are measures of average distribution of energy among various scales.

Figure 9

One-probe FM correlation maps, $R_{1P}^f$, for (a) smooth and (b, c) rough-wall flow at an LMP and a HMP, respectively.

Figure 10

The zero-time-delay FM correlation coefficient, $R_{1P}^f{|}_{\tau =0}$, as a function of the wall-normal position. $◯$, smooth; $\square$, rough (LMP); $△$, rough (HMP).

Figure 11

Two-probe FM correlation maps, $R_{2P}^f$, for (a) smooth and (b, c) rough-wall flow at an LMP and a HMP, respectively. The blue lines indicate the location of the outer probe for each case.

Figure 12

A schematic representation of inner–outer interactions that was originally presented by Baars et al. [29], but modified to represent the current experiments and observations.