In order to obtain turbulent velocities, the measured velocity time series were decomposed into mean, phase-averaged, and turbulent components by means of Reynolds decomposition: 1u=u‾+ũ+u′, where u is the measured velocity, u‾ is the time-averaged velocity, ũ is the phase-averaged velocity, and u′ is the turbulent (or fluctuating) velocity in the current direction. The same applies for v and w, respectively, in the wave and vertical upward direction. The time-averaged velocities u‾, v‾, and w‾ were obtained by time-averaging the instantaneous velocity time series measured by all the ADVs. Phase-averaged velocities in the current direction, ũ, and wave direction, ṽ, were computed as follows: 2ũ=1Nw∑i=1Nw(ui-u‾),3ṽ=1Nw∑i=1Nw(vi-v‾), where Nw is the number of waves used for the phase average. Waves were generated for 4 min for the wave period T=2.0 s and for 2 min for T=1.0 s in order to generate 120 waves per test, which is fairly larger than the minimum of 50 waves necessary for phase averaging .

Turbulent velocities u′, v′, and w′ in the x, y, and z directions, respectively, were then obtained by subtracting the time-averaged and phase-averaged velocities from the instantaneous velocity time series.

This type of decomposition is, however, subject to possible contamination due to fluctuations of the wave height. In other words, if the oscillatory flow is not perfectly regular, fluctuations of wave height will contaminate the turbulent velocity component. To account for the incidence of oscillatory flow contamination, turbulent velocities were also computed using the empirical mode decomposition method, or EMD . EMD is a promising approach to decompose the signal and remove oscillatory contamination in the turbulent component from a time series due to large wave height variability. The procedure of EMD used in the present work follows the one described by , which employed EMD in wave–current interaction experiments characterized by substantial wave height variability. Figure  shows turbulence intensity profiles Iu (= u′2‾/u‾), which are an indication of the turbulence activity along the water column, for Run 22 (WC, H=0.08 m, T=2.0 s) and Run 21 (WC, H=0.08 m, T=1.0 s), obtained with the Reynolds decomposition (black circles) and EMD (gray crosses).

Run 22 (Fig. a) and Run 21 (Fig. b) have a wave-height-normalized standard deviation σH/Hm of 0.05 and 0.16, respectively, which are respectively the smallest and the largest wave height variability of the entire dataset. The decomposition of turbulence velocities using both Reynolds decomposition and EMD reveals that, despite variations in wave height, both methods produce similar turbulence intensity profiles. However, the EMD method consistently shows slightly higher turbulence intensity values. If the turbulent velocity time series were contaminated by Reynolds decomposition, an increase in Reynolds turbulence intensity relative to EMD would be expected. Instead, the opposite occurs: Reynolds decomposition results in lower turbulence intensity, suggesting that turbulence contamination from wave motion is not occurring. This finding is consistent across all experiments.

Turbulent velocity data were processed in order to remove spikes in the time series. The presence of spikes is a common issue in acoustic velocimetry, and their removal (known as despiking) is considered an essential operation in velocity data processing . However, a problem arises in turbulent flows as distinguishing spikes between actual turbulent fluctuations is crucial. Several despiking methods have been developed over the last decades . The selected technique in this study is the one by . This method employs a bivariate kernel density function to generate a density map of the data, effectively isolating the turbulent data cluster from surrounding spike clusters. This technique outperforms previous methods that rely on universal noise thresholds, especially in the presence of turbulent flows, ensuring the preservation of the -5/3 slope of the turbulent velocity frequency spectrum. The maximum percentage of removed data is 15 %.

In rough flows, velocity flow fields are strictly related to the location of the point where they are measured. To investigate the spatially averaged characteristics of the flow, the velocity field close to a rough boundary can be made globally homogeneous by means of space averaging. The time-averaged velocities u‾, v‾, and w‾, obtained by time-averaging the instantaneous velocities measured by all the ADVs, were further averaged in order to obtain time- and space-averaged 〈u‾〉, 〈v‾〉, and 〈w‾〉 according to the following: 4〈u〉=1NADV∑i=1NADVu‾i, where NADV is the number of ADVs. The procedure of averaging by time and space is referred in the text as double averaging and allows filtering out the heterogeneous flow characteristics that depend to the specific position of the ADV.

Significant dimensional quantities and nondimensional parameters were computed from double-averaged velocities to characterize the flow field. Current freestream velocity Uc was computed by depth-averaging the double-averaged current velocities above the expected current boundary layer upper limit . Wave orbital velocity Uw was computed by considering the phase-averaged velocity maximums at the first measurement point above the wave boundary layer thickness. According to , the expected wave boundary layer thickness in rough flows depends on the relative wave orbital amplitude Abm/k, where Abm is the wave orbital amplitude (equal to Uw/ω) and k is the bottom roughness. Once the expected wave boundary layer thickness is computed, the wave orbital velocity is measured considering the lowest measurement point above the wave boundary layer thickness in the crest velocity profile. Then, the orbital velocities measured by each ADV were space-averaged in order to obtain double-averaged orbital velocity Uw. Once Uc and Uw were obtained, current and wave Reynolds numbers were computed with the following set of equations: 5Rec=Uchν,Rew=UwAbmν, where Abm is the wave orbital amplitude (Uw/ω, where ω=2π/T is the wave frequency). A nondimensional wave–current parameter Uw/Uc was computed as an indicator of the relative importance of the waves compared to the current. Moreover, the wave–current parameter is used to distinguish two wave–current regimes: the current-dominated regime (Uw/Uc<1) and the wave-dominated regime (Uw/Uc>1).

Current shear velocity u* and equivalent roughness ks were computed through a best-fitting technique .

The best-fit procedure is different depending whether on the flow is hydraulically smooth, i.e., when the viscous sublayer thickness is larger than the bed grain size, or hydraulically rough, when the viscous sublayer is destroyed as the grains are larger than the supposed thickness of the viscous layer. In hydraulically smooth flow, the velocity profile in the logarithmic region follows the law of the wall, 6uu*=1κln⁡zu*ν+5.0, where κ is the von Kármán constant (=0.4). In hydraulically rough flow, the near-bed velocity distribution follows a logarithmic law: 7uu*=1κln⁡zzo, where z0=ks/30, with ks being the equivalent roughness. In this case, a hypothesis on the position of the theoretical bottom needs to be made. The procedure follows the one suggested by by distinguishing between different hypotheses of theoretical bottom distance: the one that grants the larger logarithmic profile. Then, shear velocity was obtained from the slope of the linear fitting of u and log⁡(z), whereas ks was obtained through its intercept.

The computations to obtain the shear velocity and related quantities are subject to uncertainty. Especially ks may be very different depending on the measurement chosen to be part of the logarithmic profile linear fitting. A 95 % confidence interval for the slope was computed by means of a Student's t distribution to assess the uncertainty range of the slope. A similar procedure was applied to ks. From the shear velocity the Reynolds shear number was then computed, according to the relation 8Re*=u*d50ν.

Table  shows u* and ks alongside their confidence interval values and Re* for all CO and WC runs.

This specifically includes measured freestream current velocity Uc, measured orbital velocity Uw, wave–current regime parameter Uw/Uc, current Reynolds number Rec, and wave Reynolds number Rew. Their values are shown in Table . Target current velocities were chosen to have a relatively “weaker” (Fr=0.058) and a “stronger” (Fr=0.106) current, both in subcritical flow (Fr<1). The current Reynolds number Rec is greater than 4000 for all the experiments; therefore, the regime is fully turbulent, whereas the wave boundary layer is laminar as Rew<2×105 according to .

Current velocity profiles in the case of current only and waves plus current are compared in Fig.  to investigate the hydrodynamic effects of the waves superposed to the current.

Shear velocity u* and equivalent roughness ks are reported in the figure, alongside the fitting lines used for their computation . Subscripts CO and WC indicate current-only and wave-plus-current conditions, respectively.

Figure a, which reports a comparison between runs 1 (CO) and 8 (WC, H=0.08 m, T=2.0 s), both of them over a sand bed and with Fr=0.106, shows that the superposition of the waves determines an increase in resistance experienced by the current, revealed by the increase in shear velocity by 14 % and an increase in equivalent roughness by more than an order of magnitude.

Analogously, Fig. b shows the comparison of Run 28 (CO) and Run 33 (WC, H=0.08 m, T=2.0 s) over a gravel bed with Fr=0.106. With respect to the sand bed case in Fig. a, a similar behavior is observed over the gravel bed, although in this case the CO shear velocity is more than doubled in comparison with that of the sand bottom runs due to the presence of the rough bottom. Also in this case shear velocity and equivalent roughness increase as waves are superposed to the current by 17 % and 35 %, respectively.

In comparison with the experiments shown in Fig. a and b, the experiments shown in Fig. c and d were carried out with a larger water level (from 0.40 to 0.60 m) in order to generate a weaker current (Fr=0.058). Specifically, Fig. c shows a comparison between Run 10 (CO) and Run 11 (WC, H=0.08 m, T=2.0 s), both of them over a sand bed.

In this case, the superposition of waves determines a reduction of bottom shear, with a decrease in u* of 40 % and a decrease in ks of an order of magnitude.

Similarly, Fig. d illustrates a comparison between Run 19 (CO) and Run 22 (WC, H=0.08 m, T=2.0 s) over a gravel bed with Fr=0.058. In this case a slight decrease is observed for both uWC* (6 %) and ks,WC (57 %).

In order to provide an overall view of how the wave motion affects the current bottom flow, Fig.  shows the wave–current parameter Uw/Uc versus the shear Reynolds number ratio ReWC*/ReCO* for all the wave-plus-current tests.

The wave–current parameter indicates the relative strength of the wave motion compared to the current flow, separating the current-dominated regime (Uw/Uc<1) and the wave–dominated regime (Uw/Uc>1). The shear Reynolds number ratio is an expression of the shear experienced by the current relative to the current-only case. ReWC*/ReCO*<1 indicates a shear stress reduction compared to the current-only case, whereas ReWC*/ReCO*>1 indicates a shear stress enhancement.

In the presence of a stronger current (Fr=0.106, full markers), the superposition of the oscillatory flow always determines an increase in bed resistance, as shown by the ReWC*/ReCO* always being greater than 1, no matter the bed roughness. The superposition of the laminar wave boundary layer seems to determine a stress enhancement, proved by the increase in shear velocity and equivalent roughness. This is in accordance with most of experimental evidence in the literature . As the relative importance of the waves increases, i.e., as Uw/Uc increases, the shear seems to be enhanced in a nonmonotonous fashion, with a plateau around Uw/Uc≈1.

In the presence of a weaker current (Fr=0.058, empty markers), a different trend is observed, since the superposition of waves determines a decrease in flow resistance compared with the current-only case, as the ratio ReWC*/ReCO* is always below 1. Specifically, two behaviors are reported depending on the bed roughness. Over SB, the shear increases with a linear trend as Uw/Uc increases, whereas over GB, the shear experienced by the current remains fairly constant as Uw/Uc increases, with values of relative shear stress closer to 1.

An analysis of the turbulent velocity data was carried out and is presented in this section. Figure a shows turbulence intensities Iu =u′2‾/u‾ along the current direction for Run 1 (CO), Run 7 (WC, H=0.12 m, T=2.0 s), and Run 8 (WC, H=0.08 m, T=2.0 s). All runs are over a sand bed with Fr=0.106.

It can be observed that the presence of the waves always enhances turbulence intensity, both close to and far from the bottom. Nevertheless, even though the relative importance of the waves to the current increases, turbulence intensity profiles tends to collapse on top of each other. This is supported by the fact that the two cases have a very similar value of ReWC*/ReCO*.

Figure b shows turbulence intensities in the current direction Iu for Run 28 (CO), Run 33 (WC, H=0.08 m, T=2.0 s), and Run 34 (WC, H=0.12 m, T=2.0 s) over a gravel bed with the same current velocity Fr=0.106. The presence of the GB determines larger gradients of turbulence intensities in comparison with the corresponding SB case (Fig. a) with the same U. Notwithstanding the different wave–current regime, the two WC profiles show a very similar behavior. However, the increase in turbulence intensity in the larger-Uw/Uc case (Run 34) seems to extend to a larger part of the water column (approximately up to 0.03÷0.04 z/h).

Figure c shows turbulence intensities Iu in the current direction of Run 10 (CO), Run 11 (WC, H = 0.08 m, T=2.0 s), and Run 13 (WC, H=0.18 m, T=2.0 s) over a sand bed with Fr=0.058. As Uw/Uc increases, a turbulence intensity enhancement is observed in proximity to the bed, while there is a decrease in the upper part of the water column. The increase in the parameter Uw/Uc also seems to affect the profile gradient in close proximity to the bottom boundary, determining an increase in the bottom turbulence intensity gradient.

Figure d shows turbulence intensities Iu of Run 19 (CO), Run 21 (WC, H=0.08 m, T=2.0 s), and Run 22 (WC, H=0.12 m, T=2.0 s) over a gravel bed with Fr=0.058. The figure shows a larger gradient of Iu in comparison with the corresponding sand bed case with the same U in Fig. c, approximately up to z/h=0.10 m. The CO case shows a larger turbulent intensity very close to the bottom boundary compared with all the corresponding WC cases. This could confirm the results of Fig. d, which shows a slightly larger shear experienced by the current in the absence of waves. However, such a behavior was not observed in the SB case in Fig. c. Moreover, slightly larger gradients of the turbulence intensity profiles are observed in the CO case.

Turbulence at a wall boundary in a steady flow is mainly generated by the succession of two cyclic events: ejections and sweeps . These events are mainly responsible for turbulent vertical momentum transport and determine most of the generation of Reynolds shear stress . Quadrant analysis is a well-established technique to study the ejection–sweep cycle . In quadrant analysis the turbulent events, defined as the fluctuating velocities (u′(t),w′(t)) at an instant t, where u′ and w′ are the streamwise and vertical upward direction turbulent velocities, are subdivided into four quadrants (Q1–Q4) depending on their signs. The second (u′<0, w′>0) and the fourth (u′>0 and w′<0) quadrants are the ones associated with the ejections and sweeps, respectively.

Figure a and b show the quadrant analysis for a current-only case (Run 1 SB, Fr=0.106) and a wave-plus-current case (Run 8, SB, Fr=0.106, H = 0.08 m, T=2.0 s), respectively, both at the same relative depth z/h=0.040.

The blue number inside each quadrant indicates the percentage of turbulent events that fall into the quadrant, excluding a central region defined by a hyperbolic threshold , which excludes turbulent events not intense enough to be considered ejections or sweeps.

The dispersion of u′ and w′ around their mean value can be quantified by computing the determinant of the covariance matrix S: 9dS=det(S)=σu′2σw′2-Cov(u′,w′)2, where σu′ and σw′ are the standard deviations of u′ and w′, respectively, whereas Cov(u′,w′) indicates the covariance of u′ and w′. While larger variances indicate more dispersion, a large covariance (whether positive or negative) indicates a stronger linear relationship between u′ and w′, which reduces the overall dispersion. Thus, the determinant decreases as covariance increases, reflecting the reduced spread in the 2D space due to the linear dependency between the variables. The determinant dS was computed for all tests and is shown in Fig. A1 in Appendix A.

Comparison between Fig. a and b illustrates that the presence of waves determines an increase in the intensity of turbulent activity, shown by the turbulent events being more dispersed, with dS increasing from 5×10-8 to 5×10-7. However, the number of turbulent events above the hyperbolic threshold is almost halved in both ejection and sweep quadrants, with the number of events inside the hyperbolic hole reaching 62 %. Such a result indicates that the presence of the waves determines a decrease in the number of ejections and sweeps but at the same time an increase in their intensity.

In order to observe the occurrence of turbulent bursts during a wave phase, Fig. d illustrates the phase-averaged number of ejections and sweeps for Run 6 (WC, H=0.18 m, T=2.0 s) at z/h=0.04. Figure c shows the correspondent phase-averaged wave velocity.

As the wave phase progresses, a clear oscillation in the number of turbulent bursts is observed. Turbulent bursts progressively increase when the wave phase progresses from nodes to antinodes, i.e., towards crest and through phases, whereas as wave phase progresses from antinodes to nodes, a reduction of the number of ejections and sweeps is observed. The reported pattern seems to follows the nonlinearity of the wave, with the increase in the number of bursts during the crest stage being shorter and more intense than the ones in the trough stage. This behavior appears to be consistent for both ejections and sweeps (black and gray line, respectively). This occurrence was observed for all the runs with larger wave height cases (H=0.12 m) but not easily recognizable for the cases with lower H, which showed no recognizable oscillation in the number of ejections and sweeps along the wave phase.