3.1. Global flow modifications

Starting from the Eulerian–Eulerian formulation of the momentum equation (2.2), the mean stress in the streamwise direction can be expressed in terms of the unified field variables:

(3.1)\begin{equation} \overbrace{\frac{1}{Re} \frac{\boldsymbol{\text{d}}{\langle u \rangle}}{\text{d}{y}}}^{{\tau_{\mu}}} ~ \overbrace{-\langle u' v' \rangle}^{{\tau_{R}}} + \overbrace{G\langle {(1-\varGamma)}\mathsf{B}_{xy}\rangle }^{{\tau_{e}}} = \left(1 - \frac{y}{2}\right) \tau_{\text{w}} + \frac{y}{2} \tau_{{w,t}}.\end{equation}

From left to right, the total stress is comprised of the viscous contribution $\tau _\mu$, the turbulent Reynolds stress $\tau _R$ and the elastic term $\tau _e$. On the right-hand side, $\tau _{{w}}$ is the mean shear stress at the nominal height of the compliant surface $y=0$ and $\tau _{{w,t}}$ is the mean stress at the top wall $y=2$. We multiply both sides of (3.1) by $2/h_0$, where $h_0$ is the height at which the total stress changes sign, $h_0=1 + (\tau _{{w}}+\tau _{{w,t}})/(\tau _{{w}}-\tau _{{w,t}})$, and integrate over $0< y< h_0$:

(3.2)\begin{equation} \frac{2}{h_0} \int_0^{h_0} \left( \tau_\mu + \tau_R + \tau_e \right)\text{d}y = \tau_{{w}}. \end{equation}

The right-hand side of (3.2) expresses the wall shear stress at the mean location of the compliant surface and the left-hand side shows the contribution of different stress constituents. By normalizing (3.2) with mean wall shear stress in rigid simulation, $\tau _{{w}}^{\{R_{180},R_{590}\}}$, we can directly compare the contributors to the stress in flow over a compliant surface with that of a rigid wall (figure 4a). Similarly to the recent experimental (Wang et al. Reference Wang, Koley and Katz2020) and numerical (Rosti & Brandt Reference Rosti and Brandt2017) studies, wall compliance increases the drag. The drag increase is associated with an increase in the Reynolds shear stress $\tau _R$, and is largest in the main case $C$. The differences between the stress budgets for cases $R_{180}$ and $C_G$ are within the statistical and discretization uncertainty, which is an indication that the stiffer wall has a minimal impact on the turbulence. The average stresses in (3.1) are evaluated in Cartesian coordinates, and therefore differ from averages performed at locations that are equidistant to the surface. In order to resolve this issue, in § 3.4 we adopt wave-fitted coordinates which also allow us to compute the stress due to the pressure acting on the deformed interface.

Figure 4. (a) Contribution of different stress components to the wall drag (3.1), (3.2). (b) Profiles of the stresses for case $C$: viscous stress $\tau _\mu$ ($\cdot \cdot \cdot \cdot \cdot$, black); turbulent Reynolds stress $\tau _{R}$ (- - -, blue); elastic stress $\tau _e$ (——, black); sum of the three components (——, grey). The stresses are plotted in (left axis) outer and (right axis) wall units. (c) Mean streamwise velocity profiles for cases $C$ (——, black), $C_L$ (-$\cdot$-$\cdot$, black), $C_G$ ($\cdot \cdot \cdot \cdot \cdot$, black), $C_H$ (- - -, black), $R_{180}$ (——, grey) and $R_{590}$ (- - -, grey), compared with experimental data of Wang et al. (Reference Wang, Koley and Katz2020) at $Re_\tau = 5179$ and $G^\star /(\rho _f^\star {U_0^\star }^2)=0.797$ (blue circle). The $y^+$ coordinate is shifted vertically in order to account for the effect of roughness (Jackson Reference Jackson1981).

Note that the drag increase in case $C_H$ relative to the rigid wall $R_{590}$ is only $5\,\%$, compared with the $46\,\%$ drag increase in case $C$ relative to $R_{180}$. This difference is despite channels $C_H$ and $C$ being designed to have the same amplitude and wavenumber of surface displacement in viscous units. From a roughness perspective, both cases $C_H$ and $C$ belong to a ‘transitionally rough’ regime with $d^+<28$. Therefore, the Reynolds number is expected to influence the normalized drag (Nikuradse Reference Nikuradse1950). A similar trend was observed in the experiments of Wang et al. (Reference Wang, Koley and Katz2020). Those authors reported that the drag increase due to wall compliance relative to a rigid wall reduced from $10.7\,\%$ to $5.0\,\%$ when the Reynolds number was increased by $84\,\%$.

Figure 4(b) shows the stress profiles in case $C$ where the drag increase is most substantial. Except in case $C_G$ which is almost in the one-way coupling regime, the trends are similar in other compliant cases and therefore are omitted for brevity. The total stress profile, plotted in outer (left axis) and wall (right axis) units, shows the sum of left-hand-side terms in (3.1). The total stress varies linearly with $y$, and its magnitude is larger by approximately $33\,\%$ at $y=0$ than at $y=2$. An important observation is that the Reynolds shear stress changes sign near the surface. This effect is consistently observed in all compliant cases, and is discussed in detail in § 3.4.

Figure 4(c) shows the mean velocity profiles in a semi-logarithmic coordinate in wall units (for completeness, the profiles of the Reynolds normal and shear stresses are provided in Appendix C). For the sake of consistency with the previous literature, the data are first presented in a standard Cartesian coordinate and without any fluid/solid conditional sampling. Since the effective location where the mean drag is exerted on the surface may not coincide with the nominal surface height, in figure 4(c) we use a vertical displacement $d_h$ proposed by Jackson (Reference Jackson1981) to shift the coordinate. This model is widely used in the study of turbulent flows over rough walls (Leonardi & Castro Reference Leonardi and Castro2010; Ismail, Zaki & Durbin Reference Ismail, Zaki and Durbin2018), permeable walls (Breugem et al. Reference Breugem, Boersma and Uittenbogaard2006) and, more recently, compliant surfaces (Rosti & Brandt Reference Rosti and Brandt2017). The value of $d_h$ is chosen in a way to attain a constant slope in the inertial range, i.e. $(y+d_h)^+ ({{\rm d}\langle u^+ \rangle }/{{{\rm d}y}^+})$ remains approximately constant over the logarithmic layer. The enhanced drag is accompanied by a downward shift in the logarithmic region, even if $d_h=0$. The reduced momentum over compliant material is evident in the shown experimental data of Wang et al. (Reference Wang, Koley and Katz2020) at $\mbox { {Re}}_\tau = 5179$ and $G^\star /(\rho _f^\star {U_0^\star }^2)=0.797$, where $U_0^\star$ is the free-stream velocity. These data were sampled in the flow only and, therefore, are not contaminated by samples within the material; they are also plotted with $d_h=0$.

Unlike the experiments, the computational results are not conditioned on the fluid phase, and hence include samples from the compliant material. In addition, both the experimental and numerical results are plotted in Cartesian coordinates with reference to the nominal interface height, as opposed to the instantaneous interface position. Therefore, averages at a fixed $y$ location include samples from a range of distances to the surface, which most significantly affects the statistics near the interface. In order to better capture the mean flow in the viscous sublayer, we adopt a surface-fitted coordinate which follows the interface near the compliant surface and smoothly transitions to a Cartesian coordinate with distance from the interface (see Appendix B for details of the surface-fitted coordinates).

Figure 5 shows the mean-velocity profiles compared with the smooth-wall simulations. For cases $\{C, C_L, C_H\}$, the mean momentum deficit in the logarithmic layer is still observed in the surface-fitted coordinates. The slope of the logarithmic layer, however, does not change significantly, similar to the experimental observations of Wang et al. (Reference Wang, Koley and Katz2020). The viscous sublayer is still retained for the most part, and a decrease in momentum in the buffer layer is observed only in cases $C$ and $C_H$ which, as will be discussed, experience large surface displacements $d^+ \approx 20$. For the stiff material $C_G$, the maximum difference in the profiles from the reference $R_{180}$ case is less than $2\,\%$, which is comparable to the statistical uncertainty (Oliver et al. Reference Oliver, Malaya, Ulerich and Moser2014) and is therefore immaterial – similar to the trends reported by Wang et al. (Reference Wang, Koley and Katz2020) and Rosti & Brandt (Reference Rosti and Brandt2017) for stiff materials; little further attention will be directed to this case.

Figure 5. Mean streamwise velocity profiles in a surface-fitted coordinate: (a) $C$ (——, black), $C_L$ (-$\cdot$-$\cdot$, black), $C_G$ ($\cdot \cdot \cdot \cdot \cdot$, black), $R_{180}$ (——, grey); (b) $C_H$ (- - -, black), $R_{590}$ (- - -, grey). The logarithmic law for a smooth wall, $u^+=(1/0.41)\log \eta ^+ + 5.5$, and the viscous sublayer velocity profile, $u^+ = \eta ^+$, are also plotted for reference ($\cdot \cdot \cdot \cdot \cdot$, blue).

3.2. Deformation and pressure spectra

Visualizations of the instantaneous surface deformation from the compliant wall simulations are shown figure 6. The amplitudes of the displacements are relatively large in cases $C$ and $C_H$, while the material properties were selected to strongly couple with the turbulence in the channel. Case $C_G$, on the other hand, was designed to have a high material shear-wave speed and, as a result, decouple from the turbulence; this case has the smallest displacements which are of the order of one wall unit. The most salient feature in all cases is the formation of spanwise-oriented surface-displacement patterns that propagate in the streamwise direction. The visualizations also capture a streamwise-oriented pattern with relatively low spanwise wavenumber. The co-presence of the spanwise and streamwise undulations gives rise to a complex topography which is reminiscent of ripples on a water surface. Spanwise-oriented deformation patterns were observed in the experiments of Wang et al. (Reference Wang, Koley and Katz2020), although in their case the surface displacements were more chaotic. Similar to the experiments, the length and width of surface displacements do not vary appreciably with $G$, which is in agreement with the presumption that the peak-wavenumber material response is controlled by the thickness of the layer rather than its modulus of elasticity (Chase Reference Chase1991; Benschop et al. Reference Benschop, Greidanus, Delfos, Westerweel and Breugem2019).

Figure 6. Instantaneous visualization of compliant wall surface, coloured by displacement in wall units, for cases (a) $C$, (b) $C_L$, (c) $C_G$ and (d) $C_H$.

In order to demonstrate the wave propagation at the material–fluid interface, we examine both the streamwise and spanwise wavenumber–frequency spectra of the surface displacement (figures 7 and 8). In figure 7, streamwise travelling modes are observed in all cases with speeds marginally slower than those of the shear waves in the compliant materials, $\sqrt {G^\star /\rho _s^\star } = \{0.7, 0.7, 1.0, 0.59\}$ in cases $\{C, C_L, C_G, C_H\}$. As is discussed in § 3.3, the wave motion is very similar to the Rayleigh wave propagating in an elastic material (Rayleigh Reference Rayleigh1885), whose advection speed is similarly slightly smaller than that of the shear wave, i.e. $0.954\sqrt {G^\star /\rho _s^\star }$.

Figure 7. Streamwise wavenumber–frequency spectra of surface deformation for cases (a) $C$, (b) $C_L$, (c) $C_G$ and (d) $C_H$. Vertical lines indicate the wavenumber corresponding to $3L_e$ and inclined dashed lines indicate different phase speeds.

Figure 8. Spanwise wavenumber–frequency spectra of surface deformation for cases (a) $C$, (b) $C_L$, (c) $C_G$ and (d) $C_H$. Vertical lines indicate the wavenumber corresponding to $3L_e$ and inclined dashed lines indicate different phase speeds.

In case $C_G$ with stiff material, the phase speed is relatively high and the resonance is weak, since pressure fluctuations near the wall have lower phase speeds and hence weakly couple to the material deformation. In all cases, the range of dominant wavenumbers are clearly controlled by the material thickness $L_e$, and the peak response shifts to higher wavenumbers in case $C_L$ with the thinner compliant layer (compare figures 7a and 7b). This trend is in qualitative agreement with the predictions by linear models (Chase Reference Chase1991; Benschop et al. Reference Benschop, Greidanus, Delfos, Westerweel and Breugem2019) and the experiments of Wang et al. (Reference Wang, Koley and Katz2020). Relative to the main case $C$, the peak frequencies are higher in cases $C_L$ and $C_G$, the former due to higher range of triggered wavenumbers and the latter due to the larger $u_{w}$.

In the high-Reynolds-number case $C_H$, the shear-wave speed and layer thickness match with those of case $C$ in wall units. The amplitude and wavenumber–frequency range of excited modes are similar in the two cases, which confirms that selecting the material properties ($G,L_e$) to be matched in inner scaling was appropriate.

The spanwise wavenumber–frequency spectra (figure 8) do not show clear travelling modes with constant speed. Instead, stationary modes are observed in the spanwise direction, since the pressure fluctuations can trigger spanwise-travelling waves with equal probability of positive and negative velocities. This description is consistent with the time evolution of the surface deformation from the simulations. The frequencies where high energy is observed match the frequencies of the streamwise-travelling waves (compare figures 7 and 8). The interpretation in physical space is one where surface deformations have a nearly standing-wave appearance in the span while they propagate downstream at approximately the shear-wave speed.

The general picture of the spanwise wavenumber–frequency spectra is qualitatively similar to the experimental observations of (Wang et al. Reference Wang, Koley and Katz2020). However, those experiments had reflective lateral boundaries which are inherently different from the periodic ones employed in our simulations. As such, there are some differences, in particular when comparing with their low-fluid-velocity case (equivalent to large non-dimensional $G$) in which the excited modes were distributed across both low and high frequencies. The authors attributed the energy in the low-frequency range of the spectra to the spanwise-travelling modes, which are only dominant in cases with higher values of $G$. In our simulations, the spanwise waves are mostly stationary, although some weak traces of travelling waves at the shear-wave speed can still be detected, particularly in case $C_H$ (figure 8d). Similar to the experiments, the energy is spread across a range of wavenumbers, and the peak wavelength is smaller than $3L_e$. The peak modes are also at much lower wavenumbers compared with the $k_x$–$\omega _t$ spectra, implying that the surface structures are primarily spanwise oriented.

In order to probe the impact of the travelling surface waves on the flow, streamwise wavenumber–frequency spectra of pressure for case $C$ are shown in figure 9. The $y^+$ location is selected to be near the interface and beyond the wave crest, in order to avoid sampling the pressure inside the material. For comparison, figure 9(b) shows the spectra for the rigid-wall case at the same Reynolds number and $y^+$ location. The spectra show elevated energy in case $C$ compared with case $R_{180}$. The amplification of pressure spectra is particularly noticeable at advection velocity equal to the Rayleigh wave speed, which is approximately $0.68$ in case $C$; a similar trend was observed for the other compliant cases (not shown). Hence, the modulus of elasticity in wall units can affect both the magnitude and the advection speed of pressure fluctuations near the wall. The amplification of the spectra does not, however, appear to be confined by the wavenumber corresponding to $3L_e$, i.e. the energy is elevated across a wider range of wavenumbers travelling with the same advection speed.

Figure 9. Wavenumber–frequency power spectra of pressure for cases (a) $C$ and (b) $R_{180}$. The $y^+$ location is above the the crest of surface waves in the compliant case. Vertical lines indicate the wavenumber corresponding to $3L_e$ and inclined dashed lines indicate the bulk velocity $u=1$ and the phase speed of the Rayleigh wave in elastic material, $u_R=0.954\sqrt {G^\star /\rho _s^\star }$.

We interpret the surface deformation through the lens of a response primarily to pressure fluctuations, due to a strong correlation between $p'$ and $d$ that we report in § 3.3. Note, however, that wave propagation in compliant walls can also be triggered in response to shear-stress fluctuations at the interface (Chase Reference Chase1991; Gad-El-Hak Reference Gad-El-Hak2003). A hint of this effect was recorded in our simulations, where the spanwise shear-stress fluctuations at the surface have a weak correlation with the spanwise undulation of the interface (not shown). In other words, streamwise-aligned near-wall turbulence structures can contribute to the observed spanwise surface undulations in figure 6. Ultimately, since the most prominent effect in our simulations is the streamwise wave propagation, we place our focus on the dominant pressure–deformation interactions.

To identify the pressure disturbances that correlate with the surface deformation, cross-spectra were evaluated in the $k_x$–$\omega _t$ plane and are plotted in figure 10. The cross-spectra are shown at two different $y^+$ locations for case $C$. In the near-wall region, $y^+=28$, a clear advection band is observed with maximum magnitude coinciding with the peak mode in the surface spectra (compare figure 10a with figure 7a). Below the wavenumber corresponding to $3L_e$, the amplitudes of the cross-spectra reduce and shift to phase speeds closer to the bulk velocity, which correspond to wall signature of larger structures that travel at higher phase speeds. In the logarithmic layer (figure 10b), the overall amplitudes of the cross-spectra diminish, and the peak values are shifted to lower wavenumbers. This trend is expected since relatively larger eddies in the logarithmic layer have a wall signature that can cause deformation of the compliant material.

Figure 10. Pressure–deformation wavenumber–frequency cross-spectra for case $C$ at (a) $y^+=28$ and (b) $y^+ = 90$. Vertical solid lines indicate the wavenumber corresponding to $3L_e$ and inclined dashed lines indicate the advection velocity of Rayleigh wave in elastic material, $0.954\sqrt {G^\star /\rho _s^\star }$.

3.3. Wave-correlated motions and flow instabilities

So far we have described the wave propagation at the surface of the compliant wall, and its impact on integral flow properties, e.g. drag, mean-flow profiles and pressure spectra. In this section, we closely examine the wave-correlated motions and important features of the pressure and velocity fields in the vicinity of the interface. In doing so, it is helpful to know the velocity field associated with the wave inside the compliant material. Figure 11 shows an instantaneous $x$–$y$ plane with contours of spanwise vorticity near the interface, and the in-plane velocity vectors inside the compliant material. The observed wave motion consists of counter-rotating spanwise rolls, with positive and negative spanwise vorticity below the crest and the trough, respectively. This pattern resembles Rayleigh waves which were first identified at the surface of an isotropic elastic material (Rayleigh Reference Rayleigh1885) and can also be sustained in viscoelastic media (Carcione Reference Carcione1992); they are comprised of vertical and tangential motions that decrease exponentially in amplitude with depth from the surface.

Figure 11. Instantaneous contour plot of spanwise vorticity near the compliant surface and the in-plane velocity vectors inside the compliant wall. Counter-rotating spanwise rolls inside the compliant wall are induced by the Rayleigh wave propagating in the streamwise direction.

We start by evaluating the conditional two-point correlation between the surface displacement and the pressure:

(3.3)\begin{equation} R_{dp}({\rm \Delta} x, y, {\rm \Delta} z) = \frac{\langle d(x_0,z_0,t) p(x_0 + {\rm \Delta} x, y, z_0 + {\rm \Delta} z, t) \rangle}{(\langle d(x_0,z_0,t)^2 \rangle\langle p(x_0 + {\rm \Delta} x, y, z_0 + {\rm \Delta} z, t)^2 \rangle)^{1/2}}. \end{equation}

The adopted condition is strong positive displacement, specifically $d(x_0,z_0,t)>d_{rms}$, and only points in the fluid are sampled. Figure 12 shows $R_{dp}({\rm \Delta} x,y,0)$ for cases $C$ and $C_G$, which feature the largest and smallest surface deformation in wall units. A positive pressure at the surface induces a depression in the compliant material, while a pressure deficit gives rise to a protrusion. Therefore, the displacement is expectedly anti-correlated with the pressure at the surface in both cases $C$ and $C_G$. A phase shift between the displacement and pressure at higher locations is observed, which increases with $y$ and saturates in the logarithmic layer. Zhang et al. (Reference Zhang, Wang, Blake and Katz2017) also reported a phase shift between the near-wall pressure and surface displacement for much stiffer compliant material with surface displacements smaller than one wall unit. In our simulations, the increase in this phase shift with $y$ is more gradual and smooth in case $C_G$ where $d^+_{rms}= 0.55$, while a sharp increase is observed near $y^+\approx 35$ in case $C$ where $d^+_{rms}= 5.6$. Also, while the pressure is minimum at the reference position $(y^+, {\rm \Delta} x)=(d_{rms},0)$ in both cases, a local minimum is observed only in case $C$ at $(y^+,{\rm \Delta} x^+)=(48,41)$. Therefore, it is possible that in case $C$ there are additional unsteady effects due to large surface displacements and strong two-way coupling that lead to a pressure minimum away from the surface. We therefore focus the analysis on this case.

Figure 12. (Top) Spatial correlation between surface displacement and pressure, $R_{dp}({\rm \Delta} x, y, 0)$, defined in (3.3). Dashed line contours mark negative values, and the positions of minimum value at each $y$ location are indicated by green dots. (Bottom) Phase-averaged surface displacement. All plots are conditioned on strong positive displacement, $d>d_{rms}$. Cases (a) $C$ and (b) $C_G$.

Phase-averaged flow quantities were evaluated for case $C$ in the surface-fitted coordinate, and are plotted in figure 13. The contravariant wave-correlated velocities, $\tilde {u}_\xi$ and $\tilde {v}_\eta$, are significant near the surface, and penetrate up to $\eta ^+$ locations in the logarithmic layer. Near the surface, the velocity contours are tilted upstream when viewed in the laboratory frame, and are slightly adjusted further away from the wall. In a frame travelling with the wave speed, $\langle u\rangle - u_{w}$ is negative, i.e. the mean flow is in the opposite direction to the wave propagation near the surface; far from the surface the relative flow is positive. The $\eta$ location where the mean velocity matches the wave speed, $\langle u \rangle = u_{w}$, is the critical layer which is marked by a green dashed line in figure 13. Below the critical layer, surface-induced velocity perturbations are advected upstream relative to the wave due to the negative sign of $\langle u\rangle - u_{w}$, which gives rise to velocity contours that are tilted upstream. In contrast, the tilt in the pressure contours is in the forward direction at all heights because it is due to a different mechanism. In congruence with the pressure–deformation correlations (figure 12a), pressure contours away from the surface exhibit a phase lead compared to the pressure at the surface. It is interesting to note that the $\eta ^+$ location of the pressure minimum coincides with the height of the critical layer. One explanation is that unsteady effects in the lee side of the wave, which are further discussed below, give rise to additional turbulent motions and a pressure drop. The positive pressure gradient on the lee side, $0 \le \tilde {x}^+ \le 100$, increases the chance of flow destabilization and shear-layer detachment.

Figure 13. Line and colour contours in surface-fitted coordinates of wave-correlated (a) streamwise and (b) wall-normal contravariant velocities, (c) pressure and (d) surface displacement. In (a–c), negative contour values are plotted with black dashed lines and the green dashed line shows the mean critical-layer height. In (a), the line plot in the right-hand panel shows the mean streamwise velocity in the frame of the wave and averaged over all phases.

Instantaneous flow visualizations in the frame of the wave clarify the unsteady flow features that are not visible in phase-averaged plots. Figure 14 shows a series of snapshots of the vorticity field over a wave crest, which capture the shear-layer detachment near the surface. The vorticity contours are overlaid with line contours of pressure (figure 14a) and velocity vectors in the wave frame (figure 14b). Below the critical layer, where the flow is reversed in this wave frame, the negative vorticity layer on the lee side is lifted up resulting in a significant amount of low-speed fluid being ejected from the wall region. The vertical component of the surface velocity is positive on the lee side (cf. figure 13), which contributes to this lifting process. Once the negative vorticity crosses the critical layer, it is exposed to velocities that are faster than the wave speed, and is detached and transported downstream. The detachment process is reminiscent of a two-dimensional Kelvin–Helmholtz instability and roll-up. The pressure line contours show a drop at the core of the detached vortex, which explains the existence of a pressure minimum in phase-averaged fields (figure 13) at the critical layer downstream of the crest.

Figure 14. Instantaneous visualizations of spanwise vorticity contours in the $x$–$y$ plane over a sample wave crest from case $C$. Colour contour plots are overlaid with (a) line contours of pressure with dashed lines for negative values and (b) velocity vectors in the frame of the wave $(u-u_{w},v)$. Green dashed and solid lines are the instantaneous critical-layer height and the fluid–solid interface, respectively.

Figure 15 shows a three-dimensional view of another instance of this unsteady phenomenon. The isosurface of vorticity coloured by the pressure shows a lifted shear layer which is about to detach. The shape of the layer is locally two-dimensional, which is consistent with the spanwise-aligned rolls formed at the compliant surface. Once the layer is completely detached from the surface, more complex and three-dimensional structures form. This unsteady phenomenon is frequently repeated on the lee side of the wave crest in cases with strong two-way coupling, i.e. $C$, $C_L$ and $C_H$. Although the instantaneous visualizations show that the lift-up takes place in the lee side, the exact location at which the instability is triggered is not evident from these snapshots. We will investigate the origin of these events, describe the role of surface accelerations and estimate the locations where the velocity profile is most prone to instability.

Figure 15. Instantaneous isosurface of spanwise vorticity, $\omega _z^+ = 0.275$, coloured by pressure; visualization is from case $C$ and shows a subregion of the domain. The fluid–solid interface is also shown and is displaced vertically for clarity.

The near-interface velocity profile is directly related to the flux of vorticity at the surface. For a solid surface, the ‘vorticity flux density’ is directly related to the tangential pressure gradients (Lighthill Reference Lighthill1963). Morton (Reference Morton1984) extended Lighthill's theory to include the effect of wall acceleration. Since then, active flow control strategies have exploited the relation between vorticity flux, pressure gradient and surface acceleration with the aim of reducing drag (Koumoutsakos Reference Koumoutsakos1999; Zhao, Wu & Luo Reference Zhao, Wu and Luo2004). Since the spanwise vorticity is primarily due to the wall-normal gradient of the streamwise velocity, the vorticity flux at a moving boundary can be approximated by

(3.4)\begin{equation} \left.\frac{1}{\mbox{{Re}}}\frac{\partial \omega_z}{\partial \eta}\right\vert_{\eta = 0} \approx \underbrace{\left.-\frac{\partial p }{\partial \xi}\right|_{\eta = 0}}_{S_p} \quad \underbrace{-\frac{\boldsymbol{\text{d}}{u_{s,\xi}}}{\text{d}{t}}}_{S_u}. \end{equation}

The two source terms on the right-hand side are the contributions from the pressure gradient and surface acceleration, where $\text {d} u_{s,\xi }/\text {d}t$ is the material derivative of the tangential surface velocity. When these terms lead to ${\partial \omega _z}/{\partial \eta }|_{\eta = 0}>0$, the surface is a source of negative vorticity and the velocity profile is stabilized. In contrast, when ${\partial \omega _z}/{\partial \eta }|_{\eta = 0}<0$, the surface is a sink of negative vorticity. As a result, a peak negative vorticity is established near the wall, and the sign of the vorticity gradient changes across it; the location of the peak is therefore an inflection point in the velocity profile.

The sources of vorticity flux were phase-averaged and are plotted in figure 16(a) for case $C$. The pressure gradient contribution $\overline {S}_p$ is positive (stabilizing) on the windward side and negative (destabilizing) on the lee side of the wave. This picture is consistent with intuition. Interestingly, the surface acceleration source term $\overline {S}_u$ has nearly the same amplitude as $\overline {S}_p$, and is almost out of phase. Due to asymmetries, however, the two contributions do not completely negate each other. The net source term is positive (stabilizing) over the crest and negative (destabilizing) near the troughs. In figure 16(c–f), the profiles of $\overline {\omega _z}^+$ and $\overline {\partial {\omega _z}/\partial {\eta }}^+$ are shown for the phase locations that experience positive (blue) and negative (red) source terms. The blue curves show the enhanced vorticity in the stabilized phases. The red curves show that $\overline {\partial {\omega _z}/\partial {\eta }}^+$ changes sign for a number of phases that are associated with net negative source term, $\overline {S}_p+\overline {S}_u < 0$. While the instabilities are initiated near the troughs where $\overline {S}_p+\overline {S}_u$ is minimum, the lifted shear layer is typically transported backward and towards the lee side of the wave because the local near-interface velocities are negative in the wave frame (cf. figure 14). Also, despite the mean negative values of $\overline {\omega _z}^+$ at the interface, brief instances of $\overline {\omega _z}^+>0$ are common when an instability is triggered, although a complete separation event with flow reversal is extremely rare.

Figure 16. (a) Phase-averaged source terms of vorticity flux (3.4): surface acceleration term $\overline {S}_u$ (-$\cdot$-$\cdot$, black); pressure gradient term $\overline {S}_p$ (- - -, black); $\overline {S}_u+\overline {S}_p$ (——, black). (b) Phase-averaged surface deformation coloured blue and red for $\overline {S}_u+\overline {S}_p$ positive and negative, respectively. (c–f) Profiles of $\overline {\omega _z}^+$ and $\overline {\partial {\omega _z}/\partial {\eta }}^+$ are conditioned on (c,d) $\overline {S}_u+\overline {S}_p < 0$ and (e, f) $\overline {S}_u+\overline {S}_p > 0$. Dark to light correspond to (c,d) $-20<\tilde {x}^+<40$ and (e, f) $-84<\tilde {x}^+<-20$ and $40<\tilde {x}^+<84$.

We now direct our focus to the impact of wall properties and Reynolds number on the competing effects of pressure gradient and surface acceleration (figure 17). In both cases $C_L$ and $C_G$, the two sources of vorticity flux remain out of phase. However, the pressure source is larger, and the net term $\overline {S}_p+\overline {S}_u$ is mostly negative (stabilizing) in the windward side of the wave and vice versa. Note that in these two compliant cases the surface displacements, and in turn the surface accelerations, are small in wall units relative to the main case $C$. Therefore, the compliant walls for cases $C_L$ and $C_G$ approach the behaviour of rigid roughness where the pressure gradient contribution is the only source of vorticity flux. In the case with higher Reynolds number $C_H$, the net source term and individual contributions are similar to those of case $C$. The wave amplitude in wall units, which is almost equal in cases $C$ and $C_H$, is therefore a controlling parameter in the balance between $\overline {S}_p$ and $\overline {S}_u$.

Figure 17. Similar to figure 16(a,b), except for cases (a,b) $C_L$, (c,d) $C_G$ and (e, f) $C_H$.

3.4. Form drag, pressure work and near-wall stresses

In this section we examine the impact of the compliant wall on the flux of streamwise momentum in the surface-normal direction and the energy exchange with the flow. Due to the surface deformation, form drag becomes relevant. While the impact of form drag is unambiguous in rough walls (Jiménez Reference Jiménez2004), studies of air flow over surface gravity waves suggest a more nuanced role, e.g. form drag can reverse sign and drive the wind for fast waves (Gent Reference Gent1977; Sullivan et al. Reference Sullivan, McWilliams and Moeng2000).

The phase-averaged pressure $\overline {p}^+$ and form drag $\overline {p \partial d /\partial x}^+$ for case $C$ are shown in figure 18(a). As discussed in § 3.3, the pressure is out of phase with the surface displacements. There is an asymmetry in the pressure relative to the wave crest, and the minimum pressure is slightly shifted towards the lee side of the wave. The pressure drag onto the surface $\overline {p \partial d /\partial x}^+$ (figure 18b) is also asymmetric relative to the crest. The net effect is a positive drag on the interface, which is most pronounced in case $C$ (see also table 3). In cases with smaller surface displacements (not shown), such as $C_G$, not only is the wave-correlated pressure smaller, but also the pressure is more symmetric with respect to the crest. Both effects result in a reduced form drag.

Figure 18. Phase-averaged surface quantities in case $C$. (a) Pressure $\overline {p}^+$ (——, black) and surface-normal velocity $\overline {v_{s,\eta }}^+$ (- - -, green), (b) form drag $\overline {p\partial {d}/\partial {x}}^+$ and (c) pressure work exerted by the fluid onto the surface $\overline {-p v_{s,\eta }}^+$. Surface displacement is schematically marked in the bottom of each panel.

Table 3. Space- and time-averaged pressure work exerted by the fluid onto the surface $\langle -p v_{s,\eta } \rangle ^+$, form drag $\langle p \partial {d}/\partial {x}\rangle ^+$ and surface shear stress $-\langle u'_s v'_{s,\eta } \rangle ^+$.

Unlike for a rigid wall, the fluid can exert pressure work onto the compliant material, $\langle -p v_{s,\eta } \rangle ^+$, where $v_{s,\eta }$ denotes the interface-normal velocity. The phase-averaged $\overline {v}^+_{s,\eta }$ is included in figure 18(a) along with the pressure. The surface motion is downward on the windward side and upward on the lee side of the wave, and lags the pressure by a phase shift of nearly ${\rm \pi} /2$. The correlation is plotted in figure 18(c), and inherits the asymmetry of the pressure ($\overline {v}_{s,\eta }^+$ will be shown to closely match the symmetry of Rayleigh waves; cf. figure19). Again, the net effect integrated over the entire surface is positive in all cases (table 3). Therefore, the fluid exerts work on the compliant material and this energy exchange is primarily due to the vertical motion of the wave.

Figure 19. Phase-averaged surface quantities in case $C$. (a) Surface velocities in streamwise $\tilde {u}^+_s$ (——, blue) and surface-normal $\tilde {v}^+_{s,\eta }$ (——, red) directions. (b) Phase-averaged wave-induced shear stress $-\tilde {u}^+_s\tilde {v}^+_{s,\eta }$ (——, black). Velocities and stresses are also plotted from solution of the Rayleigh wave equations: horizontal velocity $u_R^+$ (- - -, blue), vertical velocity (- - -, red) and shear stress $-u_R^+v_R^+$ (- - -, black). Surface displacement is schematically marked in the bottom of each panel.

In § 3.1 we showed that the Reynolds shear stress, evaluated in Cartesian coordinates, changes sign near the surface and is negative at $y=0$. This effect is revisited here in more detail, using wave-fitted coordinates and phase averaging. We recall that Rayleigh waves at the free surface of an elastic material have zero shear stress because its sinusoidal horizontal and vertical velocities are ${\rm \pi} /2$ out of phase. Therefore, in our configuration the turbulent flow alters the surface wave motion in a manner that gives rise to finite net shear stress. We plot our phase-averaged interface velocities and the surface motion of classical Rayleigh waves in figure 19; the comparison is qualitative and only intended to contrast the relative phases of the two velocity components. It is clear that $\tilde {v}^+_{s,\eta }$ is essentially sinusoidal, while $\tilde {u}^+_s$ deviates from the Rayleigh wave motion in particular on the lee side. This deviation is due to the reaction force of the fluid onto the surface, in response to the total drag. As is discussed below, turbulent and pressure drag are particularly large in the lee side of the wave near $0<\tilde {x}^+<50$. The reaction force on the material is, therefore, also large and in the positive $x$ direction, which decreases the magnitude of the negative tangential velocity of the surface in that $\tilde {x}^+$ range. Ultimately, $-\tilde {u}^+_s\tilde {v}^+_{s,\eta }$ also deviates from the Rayleigh sinusoidal pattern, with the largest difference on the lee side. Consistent with our earlier observations in Cartesian coordinates (§ 3.1), $-\langle u'_s v'_{s,\eta } \rangle ^+$ averaged over the entire surface is negative in all cases (table 3).

The extent to which the negative shear stress at the surface persists into the flow is examined in figure 20. We report the phase-averaged shear and pressure stresses in surface-fitted coordinates. We only present case $C$ because the results are qualitatively similar for cases $C_L$ and $C_H$, and the wave-correlated stresses are negligible for case $C_G$. The contours of the wave-correlated component of the Reynolds shear stress $- \tilde {u}\tilde {v}_\eta$ reflect the importance of the critical layer and the wave boundary-layer height (figure 20a). The latter is defined as $h_\text {w} \equiv \sqrt {\nu \lambda _x /u_{w}}$, where $\lambda _x$ is the dominant streamwise wavelength and is approximately $h_{w}^+ \approx 4$. Below this height the motion of the fluid is appreciably influenced by the wave, and hence the patterns of $- \tilde {u}^+\tilde {v}_\eta ^+$ are very similar to those reported at the surface (compare with 19b); the net contribution $\langle - \tilde {u}\tilde {v}_\eta \rangle ^+$ is therefore negative. Between $h_{w}^+\approx 4$ and the critical-layer height, the negative stress decays and the positive stress increases substantially, resulting in a change of sign of the net contribution, $\langle - \tilde {u}\tilde {v}_\eta \rangle ^+$. The flow below the critical layer, which spans the viscous sublayer and part of the buffer layer, is simultaneously influenced by the wave motion and the turbulence. Above the critical layer, the magnitude of $- \tilde {u}\tilde {v}_\eta$ decays quickly and is practically negligible. As such, the critical-layer height demarcates the region of impact of the wave-correlated stresses above the surface, similar to turbulent air flow above gravity waves (Sullivan et al. Reference Sullivan, McWilliams and Moeng2000; Yousefi et al. Reference Yousefi, Veron and Buckley2020).

Figure 20. Phase-averaged stresses in surface-fitted coordinates for case $C$. (a) Wave-correlated $- (\tilde {u}\tilde {v}_\eta )^+$ stress, (b) stochastic Reynolds shear stress $- \overline {u'' v''_\eta }^+$ and (c) pressure stress $-\overline {(p/J) \partial \eta /\partial x}^+$ and phase-averaged surface displacement $\overline {d}^+$. The line plots in the right-hand panels show the quantities averaged over one wavelength. The horizontal lines show the height of the wave boundary layer ($\cdot \cdot \cdot \cdot \cdot$, black) and the height of the critical layer (- - -, black).

The stochastic turbulent stresses $- \overline {u'' v''_\eta }^+$ are shown in figure 20(b). Except for the region below $h_{w}$ on the windward side of the wave, $- \overline {u'' v''_\eta }^+$ is positive everywhere. The stresses peak between the trough and the critical layer, where the unsteady shear-layer detachment events described in § 3.3 are most probable. Above the critical layer, the patterns of strong positive $-\overline {u'' v''_\eta }^+$ are tilted forward due the higher local velocities.

The wave-correlated and stochastic components of the Reynolds stresses are interdependent. For example, the production of the latter includes $P_{ij} = -\langle \overline {u_i'' u_k''} \partial \tilde {u}_j/\partial {x_k} + \overline {u_j'' u_k''} \partial \tilde {u}_i/\partial {x_k} \rangle$, which involves the wave-correlated velocity gradient. This term represents the production of small-scale turbulence energy in phase with the wave (Barbano et al. Reference Barbano, Brogno, Tampieri and Di Sabatino2022), and the same term appears with an opposite sign in the equation of the wave-correlated stresses. An in-depth analysis of the energy exchange between the mean flow, wave-correlated velocities and stochastic fluctuations is left to future research.

The pressure stress $-\overline {(p/J) \partial \eta /\partial x}^+$, where $J$ is the Jacobian of the coordinate transformation, is shown in figure 20(c). Below the wave boundary layer, the pressure drag is equal to the form drag at the surface, and alternates between negative and positive stress on the windward and lee sides of the wave. Above the wave boundary layer, however, only positive drag is observed. This positive pressure stress has a phase lead above $h_{w}$, and is due to the unsteady shear-layer detachment events that take place on the trough. Note that the pressure stress by definition depends on the curvature of the isolevels of $\eta$, and therefore it gradually decays to zero as the impact of surface undulations diminishes away from the surface. The pressure itself, however, remains correlated with the wave at much higher $\eta$ locations, as previously seen in figure 13(c).

3.5. Three-dimensional effects

Since the propagating waves in the compliant layer are primarily spanwise-oriented rolls, we have so far focused on a two-dimensional analysis of the wave-correlated velocities. However, as shown previously in spanwise wavenumber–frequency spectra (figure 8), finite $k_z$ deformations are present although at lower wavenumbers compared with the streamwise ones. In this section we investigate the impact of three-dimensionality of the surface on the wave-correlated velocities and Reynolds shear stress.

We start by analysing the surface velocities associated with the wave propagation (figure 21). Despite the interface undulation in the span, the surface-velocity vectors are mostly two-dimensional and dominated by their $x$ and $y$ components. These two components are associated with the streamwise-propagating rolls, which dominate the surface motion. The spanwise interface velocity (colour contours) is smaller in magnitude, and has a wave-correlated pattern: the sign of $\overline {w}_s$ matches that of $\overline {v}_{s,\eta }\overline {n}_z$, where $\overline {n}_z$ is the spanwise component of the surface-normal vector. The observed pattern is better understood considering the combined effects of streamwise wave propagation and spanwise surface deformation. At $\tilde {z}=0$, the wave propagation is sustained by upward velocities on the lee side giving way to downward velocity on the windward side of the crest (vector plots in figure 21). In the region ($0<|\tilde {z}|<|\lambda _z/2|$), where $\lambda _z$ is the dominant spanwise wavelength, the interface-normal velocity has a component in the spanwise direction. The product of $\overline {v}_{s,\eta }$ and $\overline {n}_z$ determines the sign and magnitude of $\overline {w}_s$. Due to the phase relation between $\overline {v}_{s,\eta }$ and $\overline {n}_z$, the maximum amplitude of $\overline {w}_s$ occurs at $\tilde {z} = \pm \lambda _z/4$. Physically, the interface on the lee side is stretched outward in the span as it is pulled away from the wall, and on the windward side it is squeezed in the span towards the middle as it retracts towards the wall. While these three-dimensional features of the surface are more clearly observed in cases $C_L$ and $C_H$, similar patterns are also observed in cases $C$ and $C_G$ (not shown). In case $C$, the spanwise-oriented rolls are dominant and hence the structures are less three-dimensional. In case $C_G$, the wave-correlated velocities are in general smaller due to the weaker coupling between surface and flow.

Figure 21. Phase-averaged surface displacement, velocity vectors and contours of the spanwise component of surface velocity near the crest. Cases (a) $C_L$ and (b) $C_H$. The surface displacement is multiplied by a factor of two for clarity.

Figure 22 shows phase-averaged quantities below and above the surface for case $C_L$. The phase-averaged pressure isosurfaces are shown in figure 22(b). They are predominantly two-dimensional, echo the wave motion in the streamwise direction and their magnitude is largest in the span above the crest. The vertical velocity component inside the compliant layer and in the fluid is clearly dominated by the wave motion (figures 22c and 22d). The spanwise fluid velocity $\bar {w}$ in figure 22(e) matches the above description of figure 21; the velocity structures are, however, asymmetric in the direction of wave propagation, specifically stronger on the windward side of the wave. This asymmetry is due to the presence of a secondary flow which we discuss below. The streamwise velocity fluctuations with respect to the Cartesian average $\overline {u}^+-\langle u\rangle ^+$ are shown in figure 22( f). While influenced by the interface wave motion in the streamwise direction, there is also a clear spanwise dependence due to the three-dimensionality of the interface. Specifically, the streamwise velocity is slower above the spanwise crests and faster above the spanwise troughs. Momentum transport across this spanwise gradient, $-\bar {w}{\partial \bar {u}}/{\partial z}$, can contribute to drag (Jelly, Jung & Zaki Reference Jelly, Jung and Zaki2014). In particular, on the windward side of the travelling wave where the magnitude of $\bar {w}$ is larger, the term $-\bar {w}{\partial \bar {u}}/{\partial z}$ is positive and, therefore, a drag penalty.

Figure 22. Phase-averaged isosurfaces from case $C_L$. (a) Fluid–solid interface, (b) pressure $\bar {p}$, (c) wall-normal velocity inside the compliant wall $\overline {v_s}^+$, (d) wall-normal velocity in the fluid $\overline {v}^+$, (e) spanwise velocity $\bar {w}$ and ( f) streamwise velocity fluctuations $\overline {u}^+-\langle u\rangle ^+$.

The three-dimensionality of the flow is further examined by plotting phase-averaged fields in cross-flow planes on the windward (figure 23a,b) and lee (figure 23c–f) sides of the wave. Of particular interest is the phase-averaged streamwise vorticity which can capture secondary flows that may arise due to inhomogeneity in near-wall turbulence (Perkins Reference Perkins1970). For compliant walls, an additional source of streamwise vorticity is the spatial variation in surface velocity, specifically $\partial \overline {v}_{s,\eta } / \partial {\zeta }$, where $\zeta$ is the unit-tangent vector to the surface in the cross-flow plane. In the figure, vectors show the in-plane components of the phase-averaged flow velocity. Since the range of magnitudes differs appreciably near to and away from the interface, the vector fields are plotted twice, with vector lengths proportional to the velocity magnitudes in figures 23(a) and 23(c), and using uniform vector lengths in figures 23(b) and 23(d). On the windward side, strong patterns of $\overline {\omega _x}$ with opposite signs are observed at the surface, which are primarily generated due to the gradient of wall-normal surface velocities, $\partial \overline {v}_{s,\eta } / \partial {\zeta }$ (figure 23a). The sign of this near-surface vorticity is reversed on the lee side, again due to the surface motion. We focus our attention on the weaker pattern of $\overline {\omega _x}$ that is observed away from the surface, which does not reverse direction from the windward to the lee side. The uniform vector fields show an associated pair of counter-rotating streamwise vortices whose cores coincide with the local extrema of $\overline {\omega _x}$, and whose core-to-core spacing is roughly half the dominant wavelength of the spanwise surface undulations.

Figure 23. Phase-averaged contours of (a–d) streamwise vorticity $\overline {\omega _x}^+$, (e) streamwise velocity $\overline {u}^+$ and ( f) $\overline {u''w''}^+$. Vectors are in-plane phase-averaged velocity $(\overline {w}^+,\overline {v}^+)$. Vector lengths in (a,c) are scaled by the magnitude of the in-plane velocity and in (b,d) are uniform. The streamwise locations are (a,b) in the windward side, $\tilde {x}^+ = -17$, and (c–f) in the lee side, $\tilde {x}^+=17$, of the streamwise wave.

The outer vortex motion is best characterized as Prandtl's secondary flows of the second kind, and is comparable to those observed above streamwise-aligned riblets (Goldstein & Tuan Reference Goldstein and Tuan1998) and superhydrophobic textures (Jelly et al. Reference Jelly, Jung and Zaki2014). It is generated due to the inhomogeneity in Reynolds stresses in the span. In particular, $\partial ^2 \overline {w''w''} /\partial {y} \partial {z}$ is the dominant source term in the streamwise vorticity equation, and changes sign across the spanwise crest. The pair of counter-rotating vorticies are therefore generated by the turbulence above the spanwise surface undulation, and are largely independent of the streamwise phase: the secondary-flow vorticies away from the surface are similar in figures 23(b) and 23(d).

With the above description in mind, it is helpful to recall the pattern of the spanwise surface velocities (figure 22e). The stronger $\bar {w}$ on the windward side arises because the spanwise motion of the surface (figure 21a) has the same sign as the spanwise velocity of the outer secondary vortex near $\tilde {y}^+ \approx 25$. In contrast, on the lee side of the wave, there is a reversal in the spanwise wall velocity alone while the outer secondary flow remains unchanged. As a result, their superposition leads to a weaker $\bar {w}$.

Another implication of the spanwise surface undulations is the generation of a stochastic Reynolds stress $\overline {u''w''}$ (figure 23f). Patterns of positive and negative $\overline {u''w''}$ on both sides of the crest are visible, and arise due to turbulence production against the spanwise shear, $-\overline {w''w''}\partial \bar {u}/\partial {z}$ (figures 23f and 23e). A fluid parcel which is transported by a stochastic perturbation from the crest towards the positive $z$ direction ($w''>0$) carries low-momentum fluid towards the high-momentum zone ($u''<0$). This and the opposite motion results in $\overline {u''w''}<0$. Conversely, $\overline {u''w''}>0$ is generated on the other side of the crest. The spanwise gradient of $\overline {u''w''}$ is therefore negative near $\tilde {z} = 0$, implying that the lateral turbulent transport above the crest results in a drag penalty. This effect, however, is expected to be cancelled by the negative drag contribution over the spanwise troughs where ${\partial \overline {u''w''}}/{\partial z}>0$.

These results highlight the central role of the propagation of quasi-two-dimensional surface waves in studying the interaction of turbulence with a compliant surface. In addition to the roughness effect and the incurred form drag, the surface velocities modify the flow structures near the interface and up to the logarithmic layer, contribute to the flux of vorticity at the interface and give rise to energy exchange between the flow and the surface. While the surface waves are spanwise-elongated, they also express low-wavenumber spanwise undulations. The three-dimensionality of the wave motion generates streamwise vorticity near the interface and a secondary outer flow with an associated spanwise inhomogeneity.