3.1. Reynolds stresses

Figure 6 presents the wall normal profiles of Reynolds stresses, $\overline{u'u'}$ , $\overline{v'v'}$ and $-\overline{u'v'}$ , and compares them with DNS data obtained for smooth-wall rigid wall channel flow at Re _τ = 1000 (Graham et al. Reference Graham2016) and Re _τ = 5200 (Lee & Moser Reference Lee and Moser2015), both of them obtained from the Johns Hopkins Turbulence Database (Li et al. Reference Li, Perlman, Wan, Yang, Meneveau, Burns, Chen, Szalay and Eyink2008). Here, the prime indicates the fluctuation from mean value and the overbar refers to ensemble averaging. The present Reynolds stresses are calculated based on spatial ‘binning’ of the particle tracking data, using tools available in the Davis 10.2 software, with 8 × 8 × 8 voxel windows (0.24 × 0.24 × 0.24 mm³). The corresponding window size in terms of wall units are presented in table 1. This method first calculates the mean velocity using all the tracks in the interrogation volume, and then subtract the mean from each track, followed by ensemble averaging to obtain the Reynolds stresses. The contribution of each track is also weighted by its distance from the interrogation volume centre using a Gaussian weighting function. Consequently, the stresses are under-resolved, especially close to the wall. In the outer part of the boundary layer, the present distribution of $\overline{u'u'}^{+}$ and $\overline{v'v'}^{+}$ increase with Reynolds number in the outer layer, as expected, and mostly fall between the two DNS datasets. The profiles of both components collapse as the wall is approached, also consistent with the smooth wall data. However, the present increase in $\overline{u'u'}^{+}$ , and the decrease in $\overline{v'v'}^{+}$ with decreasing y ⁺ appear to be faster than those of the smooth wall. In figure 6(c), the three profiles of $-\overline{u'v'}^{+}$ peak in the log layer, but their magnitudes are considerably lower than those of the rigid wall DNS. As discussed in Wang et al. (Reference Wang, Koley and Katz2020) based on comparisons of 2-D PIV data obtained for rigid and complaint walls, the spatial extent of u^′u^′ and ${v}^{\prime}{v}^{\prime}$ , two-point correlations over the compliant wall are substantially lower than those of the smooth wall. This finding implies that the interaction with the compliant material ‘scrambles’ the turbulence, reducing the correlation between the two velocity components, hence the Reynolds shear stress. Yet, figure 6(c) shows that the locations of shear stress peaks are consistent with the empirical relationship obtained for smooth wall boundary layers, y _m ⁺ = 2.17 $\sqrt{\delta ^{+}}$ , by Morill-Winter, Philip & Klewicki (Reference Morrill-Winter, Philip and Klewicki2017). Interestingly, y _m ⁺ is quite close to the critical height, y _c ⁺ , at Re _τ = 6700 and 8900, but is nearly twice of y _c ⁺ at Re _τ = 3300.

Figure 7 demonstrates that trends are different when the same results are plotted using mixed scaling, following Schultz & Flack (Reference Schultz and Flack2013), i.e. the Reynolds stresses are scaled with u _τ ², and y with δ. Replotting the data with y scaled with y _c (not shown) is not significantly different since y _c /δ remains nearly a constant in our measurements (table 1, Lu et al. Reference Lu, Xiang, Zaki and Katz2024). To assess the effect of the limited spatial resolution, these plots also compare the results obtained for binning volumes of 8 × 8 × 8 voxels with those obtained using 6 × 6 × 6 voxels. As is evident, trends with elevations remain similar, and that increasing resolution increases $\overline{u^{\prime}u^{\prime}}^{+}$ and $\overline{v^{\prime}v^{\prime}}^{+}$ by 10 %– 20 % at low elevations and overlap in the outer layer, while $-\overline{u^{\prime}v^{\prime}}^{+}$ is affected by less than 4 %. Both normal stresses nearly collapse in the outer layer, and deviate below the log layer, where $\overline{u^{\prime}u^{\prime}}^{+}$ increases and $\overline{v^{\prime}v^{\prime}}^{+}$ decreases with decreasing Re _τ . The mixed scaling seems to nearly collapse the $-\overline{u^{\prime}v^{\prime}}^{+}$ profiles (figure 7 c). These trends, but not the magnitudes, are consistent with those of smooth wall data (DeGraaf & Eaton Reference De Graaff and Eaton2000; Hultmark et al. Reference Hultmark, Vallikivi, Bailey and Smits2012; Schultz & Flack Reference Schultz and Flack2013).

Figure 7. Wall normal profiles of Reynolds stresses plotted using mixed scaling: (a) $\overline{u'u'}^{+}$ , (b) $\overline{v^{\prime}v^{\prime}}^{+}$ , (c) $-\overline{u^{\prime}v^{\prime}}^{+}$ . Circles, 8 × 8 × 8 voxels binning; lines, 6 × 6 × 6 voxel binning.

We have also tried to compare the present profiles with those presented in Wang et al. (Reference Wang, Koley and Katz2020) and Greidanus et al. (Reference Greidanus, Delfos, Picken and Westerweel2022) for compliant wall boundary layers. In both cases, results do not agree (not shown here). The Greidanus et al. (Reference Greidanus, Delfos, Picken and Westerweel2022) data corresponds to at least an order of magnitude lower E/ρU ₀ ², but Re _τ on the 5800–8000 range, and their spatial resolution is lower than the present levels. The Wang et al. (Reference Wang, Koley and Katz2020) stresses are calculated based on standard 2-D PIV measurements at a similar spatial resolution, speeds and wall properties. In mixed scale plots (not shown), trends of $\overline{u'u'}^{+}$ are similar to the present data, but the present magnitudes are slightly higher. The trends of $\overline{v'v'}^{+}$ and $-\overline{u'v'}^{+}$ with increasing Re _τ do not agree, especially their observed sharp increase in stresses near the wall at high Re _τ .

3.2. Decomposition of the wave-coherent motion and stochastic turbulence

The velocity and pressure fluctuations in a boundary layer over travelling waves originate either from the shear-driven turbulence or from wave-correlated motions. The wavenumber-frequency spectral analyses in Lu et al. (Reference Lu, Xiang, Zaki and Katz2024) have shown that turbulence is advected with the deformation wave below the critical layer, but follow the mean flow at higher elevations. Furthermore, at all three Reynolds numbers, the flow–deformation coherence peaks at the critical height. To understand the flow–deformation interactions, we would like to examine the characteristic turbulent flow phenomena that propagate with the deformation. Hence, the unsteady flow is decomposed into a wave-coherent part that presumably travels with the wave, and a ‘stochastic’ turbulent part that is not correlated with the wave. To this end, we adopt the ‘triple decomposition’ approach introduced by Hussain & Reynolds (Reference Hussain and Reynolds1970) and the Hilbert projection method introduced by Hristov et al. (Reference Hristov, Friehe and Miller1998), as described briefly here. Each flow variable $f$ is decomposed into its mean $\overline{\!f}$ (ensemble averaged value), and fluctuating/unsteady part, $f'$ . The latter is further decomposed into a wave-coherent, $\tilde{f}$ , and stochastic (incoherent), $f''$ , parts, i.e.

(3.1) \begin{equation} f=\overline{\!f}+f'=\overline{\!f}+\tilde{f}+f''. \end{equation}

Figure 8. (a) Power spectral densities of the wall deformation, and (b) sample time segments of the instantaneous wall deformation at the indicated times, z ⁺ = 0, and Re _τ = 3300.

The Hilbert projection method used for determining $\tilde{f}$ , is particularly suitable for data that does not have a dominant periodic frequency or wavelength. A detailed summary of the procedure is provided in the Appendix. The reason that one cannot rely on simple phase averaging is discussed next. Figure 8(a) presents the power spectral densities of the wall deformation, E _dd , at the present three Reynolds numbers. They are calculated using fast Fourier transform with a Hanning window available in MATLAB for every point on the surface, and then spatially averaged. The angular frequency ω in rad⁻¹ is normalized by U ₀ /l ₀, and E _dd by l ₀ d _rms ² /U ₀, where d _rms is the r.m.s. of the wall deformation, presented in table 1. Clearly, the three deformation waves have broad ranges of frequencies, all peaking at a frequency corresponding to a wavelength of 3l ₀ advected at 0.53U ₀, the present wave speed. Results for the two higher Reynolds numbers nearly collapse, but the peak at the lowest Reynolds number is lower, and has a larger low frequency content. Furthermore, figure 8(b) provides sample instantaneous one-dimensional deformation waves at three instants, t ₁, t ₂ and t ₃, that are separated by a time gap of 50ν/u _τ ², all at Re _τ = 3300. As is evident, the wave amplitude varies spatially and in time, making phase averaging extremely challenging. In contrast, Hristov & Plancarte (Reference Hristov and Ruiz-Plancarte2014) and Hristov et al. (Reference Hristov, Friehe and Miller1998) show for field ocean waves that a Hilbert projection method can be used for estimating the wave-coherent component of flow variables in the atmospheric boundary layer. Further details about this procedure, including a demonstration that it is effective in extracting the wave-coherent part of a flow variable, are provided in the Appendix. The procedure involves the following steps. First, the wall deformation d(x,z,t) is divided into multiple narrowband signals, d _k (x,z,t), each with a frequency bandwidth of Δω = 0.15U ₀ /l ₀. The division is performed in the Fourier domain, and each bandpass filtered signal is converted to the time domain. Second, the unsteady flow variable $f^{\prime}$ is projected onto each d _k using

(3.2) \begin{equation} \tilde{f}=\sum _{k}\left\{\frac{\langle f',{d}_{k}\rangle }{\left\| {d}_{k}\right\| ^{2}}{d}_{k}+\frac{\langle f',\mathcal{D}_{k}\rangle }{\left\| \mathcal{D}_{k}\right\| ^{2}}\mathcal{D}_{k}\right\},\end{equation}

where $\mathcal{D}_{k}$ is the Hilbert transform of d _k , $\langle$ · $\rangle$ indicates an inner product and ||·||² is the squared norm. The Appendix shows that $\tilde{d}$ has a 99.9 % correlation with d, i.e. they are nearly identical, and that the maximum d– $\tilde{p}$ and d– $\tilde{v}$ correlations exceed 96 % and 75 %, respectively. The remaining incoherent part of the signal, $f''$ , is obtained by subtracting the $\tilde{f}$ from $f'$ .

To verify that the wave coherent and stochastic turbulence are orthogonal, we have calculated the vertical profiles of the cross term, $\overline{\tilde{f}f^{\prime\prime}}$ , and compared them with the corresponding coherent values for the streamwise and vertical velocity components as well as for the pressure. In all cases, the mean cross terms normalized by the corresponding coherent terms (not shown) remain below 10^–7, indicating that the wave-coherent motion and the stochastic turbulence are essentially orthogonal. It appears that the Hilbert projection method is effective even for the present non-stationary wave condition. Considering the small magnitudes of cross terms, they are neglected in subsequent discussions.

The next discussion compares the two-point correlation between the wall deformation and the wave-coherent flow variables with that between the deformation and the stochastic turbulence. The conditional correlation is conditioned on (i) a deformation magnitude exceeding d _rms , the local temporal r.m.s. deformation averaged over the entire area (table 1), and (ii) a surface peak (bump) or a minimum (dimple) at $\Delta$ x = 0. The $\tilde{f}$ –d correlation for a bump is defined as

(3.3) \begin{align} & \left.C_{\tilde{f}-d}\left({\unicode[Arial]{x0394}} x,y,{\unicode[Arial]{x0394}} z\right)\right| _{d\left(x_{0},z_{0},t\right)\gt {d_{\textit{rms}}}} \nonumber \\ & =\dfrac{\left.\langle \tilde{f}\left(x_{0}+{\unicode[Arial]{x0394}} x,y,z_{0}+{\unicode[Arial]{x0394}} z,t\right)d\left(x_{0},z_{0},t\right)\rangle \right| _{d\left(x_{0},z_{0},t\right)\gt {d_{\textit{rms}}}}}{\left.\left(\langle \tilde{f}\left(x_{0}+{\unicode[Arial]{x0394}} x,y,z_{0}+{\unicode[Arial]{x0394}} z,t\right)^{2}\rangle \left\langle d\left(x_{0},z_{0},t\right)^{2}\right\rangle \right)^{\frac{1}{2}}\right| _{d\left(x_{0},z_{0},t\right)\gt {d_{\textit{rms}}}}}.\end{align}

Figure 9. Conditional correlations of the deformation with the indicated flow variables based on (a) bumps, and (b) dimples, both at Re _τ = 3300. In each set, subpanels (i), (iii), (v), (vii) and (ix) show the correlations with wave-coherent components ( $C_{\tilde{f}-d}$ ), subpanels (ii), (iv), (vi), (viii) and (x) the correlation with the ‘stochastic’ turbulence ( $C_{f''-d}$ ). The subpanels from (i) to (vi) show the distributions of $C_{u-d}$ , $C_{v-d}$ and $C_{p-d}$ . Subpanels (vii) and (viii) present the conditionally averaged wall shape, and (ix) and (x), the variations of peak magnitudes of conditional correlations with Reynolds number.

A corresponding expression is used for a dimple. For the stochastic signal, $\tilde{f}$ is replaced by $f''$ . The results for both a bump and a dimple are shown in figure 9(a,b). In each set, (i,iii,v,vii,ix) and (ii,iv,vi,viii,x) correspond to the $\tilde{f}$ –d and $f^{\prime\prime}$ –d correlations, respectively. Only results for Re _τ = 3300 are presented here since those of the higher Reynolds numbers look very similar. The conditional wall shapes are plotted in (vii,viii), where the bump is bounded by dimples on both sides, and the dimple by bumps. The distance between dimples (or bumps), representing the characteristic wavelength of the deformation is 2.8l ₀, very close to the predicted theoretical value based on a linear analysis of the compliant wall response to harmonic excitations. The critical height is marked by a yellow dashed line. The conditional correlations of the coherent velocity and pressure based on bumps and dimples are essentially identical (figure 9 a). Both are highly correlated with the deformation, with $C_{\tilde{u}-d}$ and $C_{\tilde{v}-d}$ having peaks of around ±0.7 on both sides of the bump/dimple. The domains of high correlation extend well above the critical layer, with the high $C_{\tilde{v}-d}$ peaks extending to further than those of $C_{\tilde{u}-d}$ . Near the wall, the transitions between negative and positive $C_{\tilde{u}-d}$ are offset slightly downstream of the bump. Away from the wall, at y ⁺ > 400, the patterns switch phase for reasons discussed in the next section. In contrast, the transitions between negative and positive $C_{\tilde{v}-d}$ occur above the bump, and there are no phase shifts in the correlations at higher elevations. As expected, the $\tilde{p}$ –d correlations are negative above the bump, with values lower than –0.9 extending deep into the log layer and remaining below –0.5 at y ⁺ = 900. The minimum correlations, –0.98, are measured at y ⁺ = 90, close to the critical height. The negative correlation peaks are bounded on both sides by positive peaks that also extend to y ⁺ > 800, with peak magnitudes of 0.6 aligned with the deformation troughs in the data conditioned on bumps, and bumps in the analysis conditioned on dimples. The implications and flow phenomena associated with these patterns are discussed in the next section. In contrast, the incoherent velocities and pressures are poorly correlated with the deformation, confirming that the $f^{\prime\prime}$ variables are incoherent with the deformation. The correlations with the incoherent velocity components (u^′′ and $v^{\prime\prime}$ ), conditioned on bumps and dimples, have opposite signs with peak magnitudes remaining below 0.2. They indicate a broad sweeping flow above the bump, and a diminishing sweeping flow above the dimple, with $C_{v''-d}$ > 0 to the left of the dimple and nearly zero to the right of it. There is also non-zero but very low correlation with the pressure, indicating a weak minimum above the bump, and a weak maximum above the dimple. It should be noted, as discussion follows, that except for the pressure near the wall, for most of the flow field, the magnitudes of u^′′ , $v^{\prime\prime}$ and p^′′ are substantially higher than the corresponding coherent variables. Furthermore, as shown in subpanels (ix) and (x), the peak correlation magnitudes involving coherent variables, namely the highest values of $C_{\tilde{u}-d}$ , $C_{\tilde{v}-d}$ and $C_{\tilde{p}-d}$ , decrease with increasing Reynolds number. This decrease occurs in spite of the increase in the scaled deformation height (table 1). Finally, the maxima of $C_{u''-d}$ , $C_{v''-d}$ and $C_{p''-d}$ magnitudes, which are also presented in subpanels (ix) and (x), remain low and decrease with increasing Reynolds number.

In addition, while the magnitudes of the u^′′–d correlations for a bump and a dimple are low, their inclination and elongation resemble those of u^′–u^′ correlations in a typical rigid wall boundary layer (e.g. Ganapathisubramani et al. Reference Ganapathisubramani, Hutchins, Hambleton, Longmire and Marusic2005; Sillero, Jiménez & Moser Reference Sillero, Jiménez and Moser2014). This trend likely arises from the fact that conditional correlations are restricted to deformations exceeding the r.m.s. value. Such a restriction might cause a weak bias towards sweeping motions, where the near wall velocity is elevated, and the flow is subjected to adverse pressure gradients. Furthermore, the inclination angle of the u^′′–d correlation is ∼15°, consistent with that of the u^′–u^′ correlation in a solid wall boundary layer, a trend that has been associated with coherent structures (Ganapathisubramani et al. Reference Ganapathisubramani, Hutchins, Hambleton, Longmire and Marusic2005; Adrian Reference Adrian2007; Jiménez Reference Jiménez2018). This trend suggests that high wall deformations might be weakly correlated with ‘naturally occurring’ coherent structures in the boundary layer.

Figure 10 presents the distributions of phase-averaged coherent flow field, $\widehat{\tilde{f}}$ , as a function of deformation phase, at the three Reynolds numbers. The phase averaging is performed by dividing a wave cycle into 20 phase bins, i.e. each with a width of π/10, and averaging all the data (> 8.8 × 10⁵ samples) inside each bin. The wall-normal axis is scaled both as y/y _c and y/δ, with the results for Re _τ = 3300, 6700 and 8900 plotted in figures 10(a,b), 10(c,d) and 10(e,f), respectively. The phase-averaged wall deformation is presented in figure 10(g) and repeated three times for convenience. Note that the phase-averaged dimples are slightly deeper than the bumps, consistent with the PDFs of deformation in figure 3, e.g. at Re _τ = 8900, d _max ≈ −0.93d _min . For all the three cases, when the vertical axis is scaled with outer variables, they display similar distributions of flow quantities penetrating deep into the log layer in spite of substantial differences in the heights of the deformation. The velocity contours have alternate signs, which are not in phase with the windward (–π < φ< 0) or leeward (0 < φ <π) sides. The $\widehat{\tilde{u}}$ contours (figure 10 a,c,e) are inclined upstream at y/δ≤ 0.15, and shift significantly at y/δ ∼ 0.19. Near the wall, the velocity vectors indicate Q2 events (ejections, $\widehat{\tilde{u}}$ < 0, $\widehat{\tilde{v}}$ > 0) generally on the windward side and near the crest, and Q4 events (sweeps, $\widehat{\tilde{u}}$ > 0, $\widehat{\tilde{v}}$ < 0) on the leeward side and the trough. The distribution of $\widehat{\tilde{v}}$ appear to be periodic at all elevation, with positive values above the windward side and negative on the leeward side, without the phase shift observed in the horizontal velocity.

Figure 10. Variations of the wave-coherent, spatially and temporally phase-averaged: (a,c,e) $\widehat{\tilde{u}}$ ⁺, (b,d,f) $\widehat{\tilde{v}}$ ⁺ and (g) $\hat{d}/\delta$ with deformation phase; (a,b) Re _τ = 3300, (c,d) Re _τ = 6700 and (e,f) Re _τ = 8900. The arrows in (a,c,e) show the velocity vectors, and the yellow dashed lines indicate the critical heights. Panels (h) and (i) display the variations of peak (h) $\widehat{\tilde{u}}$ ⁺ and (i) $\widehat{\tilde{v}}$ ⁺ with Reynolds number.

Due to the Q2 and Q4 events, the horizontal velocity maxima are located slightly above the critical layer, whereas the coherent vertical velocity maxima extend to around three times the critical height. Overall, the coherent velocity peaks decrease with increasing Reynolds numbers, in contrast with the increase in deformation height. Further understanding of the flow structure can be obtained by examining the corresponding distributions of $\widehat{\widetilde{\omega _{z}}}$ and $\widehat{\tilde{p}}$ presented in figure 11. Like the velocity, the vorticity and pressure magnitudes decrease with increasing Reynolds number. Below the critical height, $\widehat{\widetilde{\omega _{z}}}$ (figure 11 a,c,e) is negative on the leeward side and positive on the windward side. This pattern shifts at higher elevations, where the $\widehat{\widetilde{\omega _{z}}}$ < 0 region is aligned with the top of a bump, and $\widehat{\widetilde{\omega _{z}}}$ > 0 with the trough. At y > y _c , the transitions between positive and negative areas remain largely vertical. The negative regions in the distributions of pressure (figure 11 b,d,f) appear to be aligned with the wave crests, with the peaks around π/10 ahead of the summits. The positive peaks are aligned with the troughs, with their maxima also located approximately π/10 ahead of the valleys. The ∼π/10 phase shift is consistent with the trends of p^′–d correlations reported in Zhang et al. (Reference Zhang, Wang, Blake and Katz2017) for an order of magnitude stiffer compliant wall, as inferred from the offset of the peak correlation and the deformation wavelength. They attribute this delay primarily to the flow structure in the boundary layer, and to a lesser extent, to the material response. Esteghamatian et al. (Reference Esteghamatian, Katz and Zaki2022) also observe a similar phase lag, but in their case, the delay decreases with the deformation magnitude, e.g. from ∼π/4 at d _rms ⁺ = 5.6 to ∼π/8 at d _rms ⁺ = 0.55. Finally, figures 10(h,i) and 11(h,i) show that for all the flow variables presented in figures 10 and 11, the magnitudes of the coherent parts decrease with increasing Reynolds number in spite of the significant increase in d _rms ⁺ . A plausible cause for these contradicting trends is discussed later in the paper.

Figure 11. Variations of the wave-coherent, spatially and temporally phase-averaged: (a,c,e) $\widehat{\widetilde{\omega _{z}}}$ ⁺, (b,d,f) $\widehat{\tilde{p}}$ ⁺,and (g) $\hat{d}/\delta$ with deformation phase; (a,b) Re _τ = 3300, (c,d) Re _τ = 6700 and (e,f) Re _τ = 8900. The yellow dashed lines indicate the critical heights. Panels (h) and (i) display the variations of peak (h) $\widehat{\widetilde{\omega _{z}}}$ ⁺ and (i) $\widehat{\tilde{p}}$ ⁺ with Reynolds number.

The locations of the negative and positive vorticity peaks above the crest and trough, respectively, are consistent with results of conditional averaging of the flow structures above bumps and dimples, without decomposition to coherent and stochastic turbulence, as mentioned briefly (but not shown) in Lu et al. (Reference Lu, Xiang, Zaki and Katz2024). Here, the conditional averaging also uses d > d _rms for a surface bump and d < d _rms for a dimple. The results of this conditional sampling, which is summarized in figure 12, show that bumps preferentially form under a negative spanwise vortex, owing to the pressure minimum that this vortex generates (figure 12 a,c). A sweeping flow induced downstream of this vortex impinges on the surface, generating a pressure maximum and a dimple at the sweep to ejection transition (figure 12 b,d). The present coherent part of the flow contains a negative vorticity peak above the bump, where the pressure is minimum, and a sweeping-ejection transition above the dimple, where the pressure is maximum. In attempts to explain the curious shift in the vorticity distribution below the critical height, one possibility is generation of counter rotating vorticity as the vortical structure above interacts with the boundary, generating shear in the opposite direction. Another possibility might involve viscous vorticity flux from the wall associated either with the pressure gradients (Lighthill Reference Lighthill and Rosenhead1963) or the surface-parallel material acceleration (Morton Reference Morton1984). These two contributors are compared based on DNS data by Esteghamatian et al. (Reference Esteghamatian, Katz and Zaki2022). Results show these two contributors have opposite effects, with a net impact that varies with the compliant material stiffness and amplitude of the deformation. Unfortunately, the present data does not have sufficient spatial resolution to determine the near wall vorticity or the surface acceleration, so we cannot comment on the viscous vorticity diffusion. Esteghamatian et al. (Reference Esteghamatian, Katz and Zaki2022) also show for relatively large deformation (d _rms ⁺ = 5.6), thickening of the boundary layer and flow separation close to the wave crest inject negative vorticity to the flow above the trough. However, the present deformations are significantly smaller, and the vorticity distribution shift occurs also for d _rms ⁺ << 1 (figure 11 a,c,e). Hence flow separation is unlikely to be a significant contributor. This discussion suggests that the first option, namely formation of counter-rotating vorticity as the flow induced by a vortex interacts with the wall, seems to be the most viable option.

Figure 12. Conditionally averaged flow variables and deformation at Δz ⁺ = 0 for (a,c) a surface bump, and (b,d) a dimple, both at Re _τ = 3300: (a,b) pressure contours and in-plane velocity vectors, and (c,d) compliant wall shape.

Before proceeding, it should be noted that the present distributions of $\widehat{\tilde{u}}$ and $\widehat{\tilde{v}}$ are not consistent with results obtained for wind-wave interactions (e.g. Buckley & Veron Reference Buckley and Veron2019; Cao & Shen Reference Cao and Shen2021; Do, Wang & Chang Reference Do, Wang and Chang2024), and the simulated flow over a very soft compliant material (Esteghamatian et al. Reference Esteghamatian, Katz and Zaki2022). Discrepancies exist also among results obtained for ocean waves owing to differences in wave amplitude, slope and ‘age’ (U $_{\textit{s}w}$ /u _τ ). There is better consistency among the distributions of pressure, where the minima are persistently centred in the vicinity of the wave crest, and the maxima around the trough. The differences from Esteghamatian et al. (Reference Esteghamatian, Katz and Zaki2022) might be related to the more than an order of magnitude higher wave amplitude owing to the much softer material in their simulations. Consequently, their wave crest is more prone to flow separation. Another possibility is related to the relationship between wave speed and flow velocity, which might affect the compliant wall–flow interactions. In the simulations, the magnitude of U $_{\textit{s}w}$ is close to the shear wave speed, corresponding to the advection speed of Rayleigh waves in elastic material (∼0.95C _t , Freund Reference Freund1998). In contrast, in the present experiments as well as the data presented in Carpenter et al. (Reference Carpenter, Davies and Lucey2000), Zhang et al. (Reference Zhang, Wang, Blake and Katz2017), Wang et al. (Reference Wang, Koley and Katz2020), Greidanus et al. (Reference Greidanus, Delfos, Picken and Westerweel2022) and Lu et al. (Reference Lu, Xiang, Zaki and Katz2024), U $_{\textit{s}w}$ varies between 40 %–80 % of U ₀ and does not scale with C _t .

Questions remain whether it is possible to relate the unsteady wave-coherent motion to the flow induced by a steady wall roughness. Nakato et al. (Reference Nakato, Onogi, Himeno, Tanaka and Suzuki1985) suggest that a rough wall boundary layer should be considered as flow over a wavy surface when the roughness slope is less than 6°. In this case, the momentum deficit in the log layer is strongly affected by the slope of the roughness element, increasing from ∼1 to ∼9.5 when the slope increases from 1° to 6° (e.g. Napoli, Armenio & Marchis Reference Napoli, Armenio and De Marchis2008; Schultz & Flack Reference Schultz and Flack2009). In the present study, the slope changes, 0.01° to 0.07°, are two orders of magnitude smaller, yet the downward shifts in velocity profiles increase from 1.1 to 2.6, i.e. it is of the same order as that of the rough wall. However, in the present measurements, the deformation amplitude also increases with the Reynolds number, which is also expected to affect the momentum deficit. Furthermore, the flow over stationary wavy surfaces with low slopes has also been modelled theoretically by Hunt, Leibovich & Richards (Reference Hunt, Leibovich and Richards1988). This model assumes that the wave amplitude and the roughness length are much smaller than wavelength, conditions that are satisfied in the present experiments. They divide the flow field into an inner region, where surface-induced shear stress is significant, and an outer region, where the flow perturbations are inviscid. The present measurements do not resolve the inner region. In the outer region, the present coherent vertical velocity component (figure 10 b,d,f) peaks at the elevation as the theoretical prediction. However, the present coherent horizontal velocity peaks in the outer layer, in contrast to the model prediction that places this peak in the inner layer. Therefore, some of the trends of the compliant wall boundary layer appear to be consistent with those of the flow over a stationary surface undulation, while others do not.

Figure 13. Variations of the wave-coherent, spatially and temporally phase-averaged: (a,d,g) $\widehat{\tilde{u}\tilde{u}}$ ⁺, (b,e,h) $\widehat{\tilde{v}\tilde{v}}$ ⁺, (c,f,i) $-\widehat{\tilde{u}\tilde{v}}$ ⁺ and (j) $\hat{d}/\delta$ with deformation phase: (a–c) Re _τ = 3300, (d–f) Re _τ = 6700 and (g–i) Re _τ = 8900. Yellow dashed lines indicate the critical heights. Panels (k–m) display the variations of peak (k) $\widehat{\tilde{u}\tilde{u}}$ ⁺, (l) $\widehat{\tilde{v}\tilde{v}}$ ⁺ and (m) $-\widehat{\tilde{u}\tilde{v}}$ ⁺ with the Reynolds number.

The next discussion compares the magnitudes of $\widetilde{u_{{i}}}\widetilde{u_{{j}}}$ with those of $u_{i}^{\prime\prime}u_{j}^{\prime\prime}$ , and $\tilde{p}_{\textit{rms}}$ with $p^{\prime\prime} _{\textit{rms}}$ . As figure 13 demonstrates, all the coherent stresses are concentrated near the wall, decaying rapidly with elevation. The values of $\widehat{\tilde{u}\tilde{u}}$ peak at or just above the critical layer, and those of the other stresses, at higher elevations, all consistent with the peak locations of their mean values. All the coherent stresses have maxima on the windward and leeward sides, which are asymmetric both in phase and in magnitude. The phase lags are consistent with the delay between the coherent velocity and the deformation, as depicted in figure 10. All the stresses are stronger on the leeward side, and decrease with increasing Reynolds number, trends that are evident in figure 13(k–m). The difference between the windward and leeward values appears to be associated with infrequent, high amplitude sweep-ejection events above the troughs. The formation of stress maxima on the windward and leeward sides is consistent with the DNS data in Esteghamatian et al. (Reference Esteghamatian, Katz and Zaki2022), with some phase differences. The simulations also show strong negative shear stress peaks between the positive maxima very near the wall (y ⁺ ≤ 4), which the experimental data cannot resolve.

Figure 14. Variations of the stochastic, spatially and temporally phase-averaged: (a,d,g) $\widehat{\textit{u}^{\prime\prime}\textit{u}^{\prime\prime}}$ ⁺, (b,e,h) $\widehat{v^{\prime\prime}v^{\prime\prime}}$ ⁺, (c,f,i) $-\widehat{\textit{u}^{\prime\prime}v^{\prime\prime}}$ ⁺ and (j) $\hat{d}/\delta$ with deformation phase; (a–c) Re _τ = 3300, (d–f) Re _τ = 6700 and (g–i) Re _τ = 8900. Panels (k–m) display the variations of peak (k) $\widehat{\textit{u}^{\prime\prime}\textit{u}^{\prime\prime}}$ ⁺, (l) $\widehat{v^{\prime\prime}v^{\prime\prime}}$ ⁺ and (m) $-\widehat{\textit{u}^{\prime\prime}v^{\prime\prime}}$ ⁺ with the Reynolds number.

Figure 14 presents the phase-averaged distributions of $u_{i}^{\prime\prime}u_{j}^{\prime\prime}$ for the three Reynolds numbers. As is evident, the magnitudes of the incoherent stresses are higher than the coherent ones by nearly an order of magnitude everywhere, indicating that most of the unsteady motion is incoherent, and that the trends with elevation are consistent with those of the total Reynolds stresses. In particular the maximum in $-\widehat{\textit{u}^{\prime\prime}{v}^{\prime\prime}}$ at the two higher Reynolds number occurs at the critical height. While the variations of $\widehat{\textit{u}^{\prime\prime}\textit{u}^{\prime\prime}}$ and $-\widehat{\textit{u}^{\prime\prime}v^{\prime\prime}}$ with deformation phase are milder than those of the coherent motions, the ‘stochastic’ stresses are still not distributed uniformly. Both $\widehat{\textit{u}^{\prime\prime}\textit{u}^{\prime\prime}}$ and $-\widehat{\textit{u}^{\prime\prime}v^{\prime\prime}}$ peak on the downwind sides of the bump, with the maxima in $\widehat{\textit{u}^{\prime\prime}\textit{u}^{\prime\prime}}$ occurring upstream of those of $-\widehat{\textit{u}^{\prime\prime}v^{\prime\prime}}$ . Their variations with phase seem to decrease with increasing Reynold number. Since the leeward side experience an adverse pressure gradient, the turbulence level is expected to increase. Interestingly, the peaks in $\widehat{\tilde{u}\tilde{u}}$ do not occur in the same phase as those of $\widehat{\textit{u}^{\prime\prime}\textit{u}^{\prime\prime}}$ , but the maxima in - $\widehat{\tilde{u}\tilde{v}}$ and $-\widehat{\textit{u}^{\prime\prime}v^{\prime\prime}}$ do coincide. The distributions of $\widehat{v^{\prime\prime}v^{\prime\prime}}$ are more uniform, and do not display consistent or clear trends with Reynolds number. As discussed later, these phase variations will affect the energy exchange between the coherent and incoherent parts of the kinetic energy.

Figure 15. Variations of the spatially and temporally phase-averaged: (a,c,e) $\widehat{\tilde{p}}_{\textit{rms}}$ ⁺, (b,d,f) $\widehat{p''}_{\textit{rms}}$ ⁺ and (g) $\hat{d}/\delta$ with deformation phase: (a,b) Re _τ = 3300, (c,d) Re _τ = 6700 and (e,f) Re _τ = 8900. Panels (h,i) display the variations of peak (h) $\tilde{p}_{\textit{rms}}$ ⁺, and (i) $p''_{\textit{rms}}$ ⁺ with Reynolds number.

The phase-averaged distributions of coherent and stochastic r.m.s. pressure are compared in figure 15. The coherent r.m.s. pressure peaks are concentrated near the wall, peaking slightly beyond the crest and the trough of the deformation (figure 15 a,c,e), and their magnitudes decrease with increasing Reynolds number (figure 15 h). The stochastic r.m.s. pressure also peaks at the wall but does not vary significantly with phase (figure 15 b,d,f). In contrast to the coherent values, the stochastic r.m.s. pressure increases with Reynolds number (figure 15 i). Owing to these opposite trends, while the coherent and stochastic maxima have similar magnitudes at Re _τ = 3300, the stochastic r.m.s. is more than twice higher than that at Re _τ = 8900.

3.3. Kinetic energy budgets for wave-coherent flow and stochastic turbulence

Following the established framework, the total turbulent kinetic energy can also be separated into a coherent ‘wave kinetic energy’ (WKE), and an incoherent ‘stochastic kinetic energy’ (SKE),

(3.4) \begin{equation} \begin{array}{c} \frac{1}{2}\overline{u_{{i}}'u_{{i}}'}=\frac{1}{2}\overline{\widetilde{u_{{i}}}\widetilde{u_{{i}}}}+\frac{1}{2}\overline{u_{{i}}^{\prime\prime}u_{{i}}^{\prime\prime}}.\end{array} \end{equation}

As verified before, the cross terms are negligible since $\widetilde{u_{{i}}}$ and $u_{{i}}^{\prime\prime}$ are uncorrelated. Following Reynolds & Hussain (Reference Reynolds and Hussain1972), the turbulent kinetic energy budget equation is also decomposed. The WKE budget is given by

(3.5) \begin{equation} \begin{array}{c} 0=\left(-\overline{\widetilde{u_{{i}}}\widetilde{u_{{j}}}}\right)\partial _{j}\overline{u_{{i}}}+\overline{\left(\widetilde{u_{{i}}^{\prime\prime}u_{{j}}^{\prime\prime}}\right)\partial _{{j}}\widetilde{u_{{i}}}}-\partial _{j}\left(\overline{\widetilde{u_{{j}}}\tilde{p}}\right)-\overline{u_{{j}}}\partial _{j}\left(0.5\overline{\widetilde{u_{{i}}}\widetilde{u_{{i}}}}\right)-\partial _{j}\left[\overline{\widetilde{u_{{j}}}\left(0.5\widetilde{u_{{i}}}\widetilde{u_{{i}}}\right)}\right]\\\quad -\partial _{j}\left[\overline{\widetilde{u_{{i}}} \left(\widetilde{u_{{i}}^{\prime\prime}u_{{j}}^{\prime\prime}}\right)}\right] +\nu \partial _{j}\left[\overline{\widetilde{u_{{i}}}\left(\partial _{{j}}\widetilde{u_{{i}}}+\partial _{{i}}\widetilde{u_{{j}}}\right)}\right]-2\nu \overline{\left(\partial _{{j}}\widetilde{u_{{i}}}+\partial _{{i}}\widetilde{u_{{j}}}\right)\left(\partial _{{j}}\widetilde{u_{{i}}}+\partial _{{i}}\widetilde{u_{{j}}}\right)}, \end{array} \end{equation}

where $(-\overline{\widetilde{u_{{i}}}\widetilde{u_{{j}}}})\partial _{j}\overline{u_{{i}}}$ is the WKE production by mean flow; $\overline{(\widetilde{u_{{i}}^{\prime\prime}u_{{j}}^{\prime\prime}})\partial _{{j}}\widetilde{u_{{i}}}}$ is the WKE production by the stochastic turbulence; $-\partial _{j}(\overline{\widetilde{u_{{j}}}\tilde{p}})$ is the coherent pressure diffusion; $-\overline{u_{{j}}}\partial _{j}(0.5\overline{\widetilde{u_{{i}}}\widetilde{u_{{i}}}})$ is the advection by mean flow; and the fifth to seventh terms represent the transport of WKE by the wave-coherent stresses, stochastic turbulent stresses and viscous stresses, respectively. The last term is the viscous dissipation. Accordingly, the budget equation for the SKE is

(3.6) \begin{equation} \begin{array}{c} 0=\left(\!-\overline{u_{{i}}^{\prime\prime}u_{{j}}^{\prime\prime}}\right)\partial _{j}\overline{u_{{i}}}-\overline{\left(\widetilde{u_{{i}}^{\prime\prime}u_{{j}}^{\prime\prime}}\right)\partial _{{j}}\widetilde{u_{{i}}}}-\partial _{j}\left(\overline{u_{{j}}^{\prime\prime}\textit{p}^{\prime\prime}}\right)-\overline{u_{{j}}}\partial _{j}\left(0.5\overline{u_{{i}}^{\prime\prime}u_{{i}}^{\prime\prime}}\right)-\partial _{j}\!\left[\overline{u_{{j}}^{\prime\prime}\left(0.5u_{{i}}^{\prime\prime}u_{{i}}^{\prime\prime}\right)}\right]\\ -\overline{\widetilde{u_{{j}}}\partial _{{j}}\left(0.5\widetilde{u_{{i}}^{\prime\prime}u_{{i}}^{\prime\prime}}\right)}+\nu \partial _{j}\left[\overline{u_{{i}}^{\prime\prime}\left(\partial _{{j}}u_{{i}}^{\prime\prime}+\partial _{{i}}u_{{j}}^{\prime\prime}\right)}\right]-2\nu \overline{\left(\partial _{{j}}u_{{i}}^{\prime\prime}+\partial _{{i}}u_{{j}}^{\prime\prime}\right)\left(\partial _{{j}}u_{{i}}^{\prime\prime}+\partial _{{i}}u_{{j}}^{\prime\prime}\right)},\end{array} \end{equation}

where $(-\overline{u_{{i}}^{\prime\prime}u_{{j}}^{\prime\prime}})\partial _{j}\overline{u_{{i}}}$ and $-\overline{(\widetilde{u_{{i}}^{\prime\prime}u_{{j}}^{\prime\prime}})\partial _{{j}}\widetilde{u_{{i}}}}$ are production of SKE by the mean flow and the wave-coherent motions, respectively. The other terms follow the same order as those associated with WKE. One term, $\overline{(\widetilde{u_{{i}}^{\prime\prime}u_{{j}}^{\prime\prime}})\partial _{{j}}\widetilde{u_{{i}}}}$ , appears in both equations with the opposite sign, representing transfer of energy from the stochastic turbulence to the wave-coherent motions when positive.

The profiles of WKE and SKE budget terms at Re _τ = 3300 are compared in figure 16. As is evident, near the wall, the production by mean flow is larger than the rest of the terms by more than an order of magnitude, with shear productions, namely $(-\overline{\tilde{u}\tilde{v}})\partial \overline{u}/\partial y$ and $(-\overline{u''v''})\partial \overline{u}/\partial y$ , being the dominant contributors. Near the wall, the WKE production is only 6 %–10 % of the SKE counterpart, decreasing to 3 % in the log layer. Figure 17(a,b) shows the variations of shear production terms with Reynolds number. When plotted using inner variables, the profiles of both production terms collapse far away from the interface, but they deviate near the wall, consistent with the reduction of the stochastic and coherent shear stresses with increasing Reynolds number. Trends do not collapse when plotted using mixed variables either (insert in figure 17 a). While the stresses nearly collapse, the mean velocity gradients do not. Both production terms still increase with decreasing elevation at the lowest point that can be resolved by the present 3-D measurements, implying that both terms peak below the critical height. For the resolved range, the profiles of SKE production (figure 17 b) are consistent with those obtained in DNS of a smooth rigid wall channel flow at Re _τ = 5200 (Lee & Moser Reference Lee and Moser2015), especially in the outer layer. However, compliant wall boundary layer DNS (Esteghamatian et al. Reference Esteghamatian, Katz and Zaki2022) shows that the Reynolds stress becomes negative at y ⁺ < 4, implying that reverse energy transfer, from the wave to the mean flow, occurs very near the wall. Finally, figures 17(c) and 17(d) compare the variation of shear production terms with wave phase. In accordance with the distributions of shear stresses, the WKE production peaks are located slightly upstream of the crest and the trough. The variation in SKE production with phase is milder, with higher values on the leeward side, in the same area as the phase-averaged transition from sweep to ejection.

Figure 16. Wall-normal profiles of the ensemble-averaged kinetic energy budget terms for: (a) WKE and (b) SKE, both at Re _τ = 3300.

Figure 17. Shear production rates: (a,b) ensemble-averaged profiles of (a) WKE, and (b) SKE at the indicated Reynolds numbers; (c,d) distributions of (c) $(-\widehat{\tilde{u}\tilde{v}}\partial \overline{u}/\partial y)^{+}$ and (d) $(-\widehat{u''v''}\partial \overline{u}/\partial y)^{+}$ at Re _τ = 3300.

Figure 18. The axial contributor to wave–turbulence energy exchange: (a,b,c) temporally and spatially phase-averaged distributions at (a) Re _τ = 3300, (b) Re _τ = 6700 and (c) Re _τ = 8900. (d) Variations of the peak values with Reynolds number.

The next discussion shifts to the energy exchange between WKE and SKEs. Consistent with trends of wind–wave interaction (Zhang, Wang & Liu Reference Zhang, Wang and Liu2024), the axial extension/contraction term, i.e. $\overline{(\widetilde{u''u''})\partial \tilde{u}/\partial x}$ , is the dominant contributor. Plots of the phase-averaged axial contraction (figure 18 a–c) indicate that this term is negative on the windward side, i.e. energy is transferred from WKE to SKE owing to the wave-induced streamwise contraction, and positive on the leeward side, owing to axial extension. The net energy flux is therefore the difference between the contributions of contraction and extension. At all three Reynolds numbers, the axial contraction and extension peaks are aligned along the critical height but decrease in magnitude with increasing Re _τ (figure 18 d). Profiles of the net energy fluxes (figure 19) are presented scaled using inner variables (figure 19 a), and mixed variables, u _τ ³/λ (figure 19 b), where the wall unit is replaced by the surface wavelength (λ = 3l ₀). Both have been used for normalizing data in studies of wind–wave interaction (e.g. Yousefi et al. Reference Yousefi, Veron and Buckley2021; Zhang et al. Reference Zhang, Wang and Liu2024). With either scaling, the magnitudes of energy flux decrease with increasing Reynolds number, but the differences between them are significantly smaller under mixed scaling. Finally, the energy fluxes at all the three Reynolds numbers change sign across the critical layer. At y> y _c , kinetic energy is transferred from the stochastic turbulence to the wave-coherent field. Conversely, at y< y _c , kinetic energy flows from the coherent to the stochastic turbulence. A sign change across the critical layer is also observed in the coherent pressure diffusion, in which the wall-normal gradient of the $\tilde{p}$ – $\tilde{v}$ correlation is the main contributor. Profiles of the ensemble averaged values (figure 20 a,b) indicate that mixed scaling leads to better collapse of results than the inner variables. In contrast to the wave–turbulence exchange, pressure diffusion adds energy to the wave-coherent field at y< y _c and depletes it at y> y _c . A comparison with the profiles of WKE, shown in figure 21, indicates that the peak negative pressure diffusion is located at nearly the same elevation as the maximum WKE, and the sign change indicates transport towards the wall, across the critical layer.

Figure 19. Wall-normal profiles of the ensemble-averaged axial wave–turbulence energy exchange term for the three Reynolds numbers, normalized using (a) inner and (b) mixed parameters.

Figure 20. Wall-normal profiles of the ensemble-averaged coherent pressure diffusion term for the three Reynolds numbers, normalized using (a) inner and (b) mixed parameters.

Figure 21. Wall-normal profiles of the WKE at the indicated Reynolds numbers.