Airflow measurements at a wavy air–water interface using PIV and LIF

The experiments presented here were conducted in the large wind-wave-current tank at the Air Sea Interaction Laboratory of the University of Delaware, in Lewes, Delaware. The facility is specifically designed and equipped for studies in air–sea interactions. The tank is 42 m long, 1 m wide and 1.25 m high. During the experiments described in this study, the water depth was kept at 0.70 m. An artificial permeable wave-absorbing beach was placed at the downwind end of the tank to dissipate wave energy and eliminate wave reflections. The tank is equipped with a programmable, computer-controlled, recirculating wind tunnel, with a honeycomb flow straightener used to condition the airflow before it reaches the test section of the tank. The main test section where the waves and airflow were measured was placed at a fetch of 22.7 m, where fetch is defined as the distance from the point where the wind tunnel reaches the water, to the center of the PIV imaging area.

A complex imaging system, using a combination of PIV and LIF techniques, was specifically developed for this study. As a result, we were able to measure velocities in the air above waves, on average as close as 0.1 mm above the air–water interface. In addition to wind velocities, temporal and spatial wave properties were measured by LIF, simultaneously with the velocity measurements. A three-dimensional sketch of the instrument setup is presented in Fig. 1. The specifications of all imaging devices are listed in Table 1 and additional details are provided in the following sections.

We developed a PIV system, capable of measuring along-channel 2D velocity fields in the air above wind waves. For this, the airflow was seeded with 8–12 \(\upmu\)m water droplets generated by a commercial freshwater fog generator (Microcool Inc.) equipped with 28 fog nozzles, affixed to the airflow straightener at the location of zero fetch. Freshwater seeding particles were employed to avoid any influence on the surface tension of the (freshwater) waves. In the PIV technique, small particles seed the flow and act as near Lagrangian tracers that are then illuminated with a flashed light source and imaged with a digital camera. The motion of the particles between two flashes then yields a local velocity estimate. Here, the Stokes time scale of the particles is \(\tau _p = 0.3\) ms (e.g., Raffel et al. 2013). At the lowest wind speed, \(\tau _p\) is 130 times smaller than the Kolmogorov time scale \(\tau _k\) (estimated in the bulk flow at the height of a wavelength, and using the measured friction velocity—Table 2). At the highest wind speed, \(\tau _p\) is at least five times smaller than \(\tau _k\). Given these Stokes numbers \(S_t = \tau _p/\tau _k\), the seeding particles are anticipated to accurately follow the turbulent fluid motions in all conditions presented, with an accuracy of order 1% or better (Tropea et al. 2007). Furthermore, using the transfer function of tracing particles in a turbulent flow (i.e. the particle response) from Mei (1996), and a representative turbulent energy spectrum of the form \(E(\omega ) \propto \omega ^{-5/3}\) (e.g., Kaimal and Finnigan 1994; Li et al. 2015), we estimate that more than 99% of the fluid’s velocity variance is resolved from the tracer particle motion, for all wind speeds studied.

The particles were illuminated by a high intensity green laser sheet generated by a pulsed dual-head Nd–Yag laser system (New Wave Research, 200 mJ/pulse, 3–5 ns pulse duration, 532 nm wavelength), fitted with a set of cylindrical and spherical lenses. It should be noted that each Nd–Yag laser system is actually composed of two pulsed lasers. The output beams of both internal lasers are internally routed and aligned such that their optical paths are identical, once outside the laser head. Each laser can flash at a frequency of up to 15 Hz. The two lasers can be flashed simultaneously, or with a time delay \(\Delta t\). For PIV, \(\Delta t\) is generally adjusted such that particle displacements are optimal on the PIV images. This time delay depends on the flow velocity and the size of the image detected (Raffel et al. 2013). In this study, the adjustable laser sheet thickness was set to approximately 0.3 cm, to provide crisp, in focus, high resolution images of the PIV particles (see the example image in panel a of Fig. 2 below).

The particles illuminated by the green flashed laser were imaged by two side-by-side CCD cameras (Jai TM4200CL, 2048 \(\times\) 2048 pixels) (see Fig. 1). Each of these PIV cameras was fitted with a 105 mm telephoto lens (Nikon). The adjacent PIV frames were collated to obtain a single high resolution (47 \(\upmu\)m/ pixel) 18.7 \(\times\) 9.6 cm PIV image. The PIV cameras operated at 14.4 frames/second yielding velocity fields at a 7.2 Hz rate. To minimize direct high intensity laser light reflections from the wavy water surface into the PIV cameras, the PIV laser sheet was slightly tilted, with an angle ≤ 3° with respect to the vertical. The laser sheet tilt was kept minimal so as to assure that the measurements were all taken in the same along-wind plane. To avoid the obstruction of the imaged near-surface region by the waves or the water meniscus at the glass wall of the tank, all cameras were also slightly tilted downward by ≤ 4°. The angle of camera tilt was chosen sufficiently small such that the entire imaged plane (laser sheet plane) was still fully in focus on the images. This was possible because the depth of the cameras’ field of view was made large enough (O(1) cm), by properly adjusting the camera lens apertures.¹ The wall meniscus problem was encountered, for example, by Belden and Techet (2011), who also solved it by tilting their PIV cameras. Finally, it should be noted that the acquired images were corrected for both lens distortions and variations in spatial resolution due to the slight tilt.

The PIV images were processed using an algorithm based on the adaptive PIV algorithm described in Thomas et al. (2005), which relies on a pyramid cascade of increasingly smaller and shifting interrogation windows to achieve large dynamical range in the detected velocity. A central difference scheme was used to compute displacements within the PIV images. The PIV images were processed with final interrogation windows of 8 \(\times\) 8 pixel, with 50% window overlap, yielding 1 velocity vector measurement on a 2D grid with 188 \(\upmu\)m spacing in each (horizontal and vertical) direction.

While we expect the seeding particles to accurately follow the turbulent motions in the fluid, errors in detecting the position of the particles can nonetheless lead to errors in the measured fluid velocity. Here, subpixel accuracy was obtained by means of a three-point Gaussian interpolation on the cross-correlation field obtained at the last level of the iterative PIV routine. Given the size of the particles (\(\approx\) 2–3 pixels), we estimate a subpixel accuracy on the order of 0.05 pixels (Raffel et al. 2013). This yields an error on the velocity of up to 4.7 cm s\(^{-1}\) (at the highest wind speed) which corresponds to 0.3% of the bulk flow velocity. The velocity error is as low as 7.8 mm s\(^{-1}\) at the lowest wind speed, corresponding to about 0.9% of the bulk flow speed. Finally, correlation is known to decrease in the presence of shear (Adrian and Westerweel 2011), particularly if the flow is not homogeneous within an interrogation window. We estimate, as we did for the Stokes time scales above, that the Kolmogorov length scale is on the order of the resolution of the PIV measurements. Thus, turbulent particle motions within a PIV interrogation window should be minimal. Nonetheless, we expect high shear near the interface (see below) and as such, only displacement estimates with a cross-correlation of more than 50% were kept. Remaining outliers, if any, were subsequently discarded.

To obtain spatial gradients of the velocities, first cubic smooth spline shells were fitted to the instantaneous velocity fields. Spatial gradients of the velocities were computed by performing an analytical derivative of the spline fits (Spedding and Rignot 1993; Fouras and Soria 1998; Cohn and Koochesfahani 2000; Kimmoun and Branger 2007). Given the errors on the velocity estimates mentioned above, we estimate that the minimum detectable shear ranges from 20 s\(^{-1}\) at the lowest wind speed to 125 s\(^{-1}\) at the highest wind speed (Adrian 1997). Practically, this means that there is a minimum measurable stress; in these experiments this threshold is at most 6% of the total turbulent stress \(\rho u*^2\). At the highest wind speed, the shear near the surface is anticipated to be substantial. In these extreme conditions, we estimate that the averaging over an interrogation window inherent to the PIV technique would lead to an underestimate of the surface shear of up to 15% at the surface, within the air-side viscous sublayer.

To avoid contaminating airflow velocity calculations with the water-side portion of the image, the PIV algorithm was run only on the portion of the image that is in the air above the instantaneous water surface (André and Bardet 2014). Here, the subsurface portion of the image was entirely removed. However, locating the air–water interface (automatically) in the PIV images is difficult because the interface scatters the laser light only weakly. To address this issue, we employed a LIF technique described below.

To precisely and programmatically (using Matlab, Mathworks) detect the position of the interface within the thousands of acquired PIV images (see NOF values in Table 1), we acquired high resolution (100 \(\upmu\)m/pixel) LIF images (PIV surface detection images, noted “PIVSD” hereafter) of the wave surface profiles. Previous PIV investigations of the flow below or above waves were generally challenged by the lack of high quality water surface elevation data, rendering surface detection on the PIV images difficult. The PIVSD images encompass the PIV field of view and were acquired simultaneously with the PIV images. They were acquired using another CCD camera (Jai TM4200CL, 2048 \(\times\) 2048 pixels), and using the illumination provided by the PIV laser (Fig. 1). The PIVSD camera lens (Nikon, 50 mm) was fitted with an amber acrylic bandpass optical filter (Kentek ACRX-BB2 optical filter, with 6.1 optical density (OD) at 532 nm, and OD < 0.6 at 566 nm). Rhodamine 6G dye (excitation at 532 nm, and maximum emission at 566 nm) was added to the water at a concentration of 8 \(\times\) 10\(^{-6}\) g L\(^{-1}\). Thus, the pulsed PIV laser sheet not only illuminated the air-side PIV tracer fog particles to be imaged by the PIV cameras, but also excited the fluorescence of Rhodamine at the air–water interface, to be imaged by the LIF camera (PIVSD). Note that the amber optical filter on the PIVSD camera lens allowed the resulting PIVSD images to only show the fluorescing water (amber color), without being contaminated by light scattered from the (green) PIV fog particles (see PIVSD image in Fig. 2b). A sample stitched PIV-PIVSD image is also plotted in Fig. 2, in panel a.

In addition, automotive window wipers were placed in front of the cameras inside the flume to clean the accumulated fog off the tank windows. Special attention was given to the placement of the wipers so that they would not interfere with the imaged airflow. Finally, a rectangular piece of rubberized hog hair (generally used as filtering material in industrial HVAC systems) was placed at the bottom of the tank, under the imaging area, to scatter refracted laser light and avoid intense reflections back into the imaging area.

To locate the interface in the LIF images (PIVSD), we developed a surface (or edge) detection algorithm based on local variations of image intensity gradients, computed by kernel convolution. Using this method, the uncertainty on the location of the water surface on the PIV images was estimated to be ± 0.1 mm. The algorithm was then applied to the PIVSD images to locate the surface. High resolution surface detection on the PIV images is necessary to accurately detect the height of our airflow PIV measurements with respect to the water surface. Since one airflow velocity vector was measured by PIV approximately every 0.2 mm, we estimate that velocity measurements closest to the water surface were achieved on average 0.1 mm above the water surface. This proximity to the surface provided velocity measurements within the viscous layer in low wind conditions (e.g., \(U_{10}\) = 2.19 m s\(^{-1}\)), as well as the ability to detect a critical height (where the phase-averaged airflow velocity matches the phase speed of the average wave, see the phase-averaging method below) as close as 0.4 mm to the air–water interface (Buckley and Veron 2016). In contrast, the seminal PIV airflow measurements of Reul et al. (1999) above mechanically generated waves were achieved no less than 1.5 mm above the water surface.

While the surface detection images mentioned above are useful to locate the surface within the PIV images, in most cases, they did not have a sufficiently large footprint to capture entire wavelengths of the surface waves and/or did not acquire with sufficient temporal resolution to satisfactorily measure the wave frequency spectrum. This experimental setup, is, to the best of our knowledge, the first of it’s kind to combine PIV measurements with a large spatial LIF snapshots of the water surface elevation. Indeed, since the footprint of PIV images is generally limited by a requirement of high resolution, past studies were only able to resolve small fractions of the sampled waves. Since the wave phase of a single point along the water surface depends upon the wavenumber spectral analysis of the water surface elevation, it is important to “look” as far as possible up and downwind of the PIV image footprint. To quantify the wave field, two types of wave data were collected during these experiments: along-wind spatial surface profiles with high spatial resolution \(\eta (x)\), and single point high frequency wave height measurements \(\eta _i(t)\). Large along-wind spatial profiles of the wavy surface were obtained by LIF, using a CCD camera (Jai TM4200CL, 2048 \(\times\) 2048 pixels, noted “LFV” hereafter, as in “large field of view”—Fig. 1), focused on the intersection with the surface of a large green laser sheet, generated by another pulsed dual-head Nd-Yag laser (New Wave Research, 120 mJ/pulse, 3–5 ns pulse duration). The LFV laser sheet was slightly tilted, by an angle of ≤ 5°, to clear the underlying PIV mirror (see Fig. 1, bottom right panel). This laser was pulsed 20 \(\upmu\)s before the first flash of the PIV laser. The LFV camera was fitted with a wide angle (115°) lens (Rokinon, 14 mm) and an amber acrylic bandpass optical filter (Kentek, 566 nm). The resulting LFV images provided measurements of the along-wind surface elevation in the center line of the channel over a length of 51.2 cm (250 \(\upmu\)m/pixel resolution), and at a rate of 7.2 Hz. The LFV field of view was positioned in the same along-channel plane as the PIV images, and extended 16.7 cm upwind and 15.8 cm downwind of the PIV field of view (Fig. 2b). Using the same edge detection algorithm as for the PIVSD images, the uncertainty on the location of the water surface on the LFV images was estimated to be ± 0.5 mm. Duncan et al. (1999) for example, used a similar LIF technique to study the surface profiles of mechanically generated breaking waves.

In addition, a single point optical wave gauge system, noted “WG”, provided time series of the water height, respectively 1.4 and 2.8 cm upwind and 2.7 and 4.2 cm downwind of the PIV airflow velocity measurements. The system consisted of two CCD cameras (Jai CV-M2CL— 300 \(\times\) 1600 pixel), each of which was fitted with a telephoto lens (Nikon, 60 mm) and an amber acrylic bandpass filter (Kentek, 566 nm). The cameras are shown in Fig. 1. Each WG camera imaged the intersection with the surface of two 200 mW continuous green laser beams (see Fig. 2b). The resulting LIF images provided measurements of the water height with a resolution of 66 \(\upmu\)m/pixel, and at a frequency of 93.6 Hz.

The entire imaging system was automated and computer-controlled, using National Instruments software and hardware. The triggers to all 6 cameras, the 2 pulsed Nd-Yag lasers, the 4 WG lasers, and the window wipers, were timed by PCI 6602 timing boards (National Instruments), and generated through BNC-2121 (National Instruments) connector blocks (see Fig. 3 below).

The acquisition sequence is as follows: first, the LFV laser was flashed at t = 0 s. The corresponding image was acquired with the LFV camera. Since total darkness was achieved in the laboratory, the LFV camera CCD only detected light during the LFV laser flash, that lasted 3–5 ns. This provided crisp sharp snapshots of the air–water interface (see Fig. 2). The LFV LIF imaging system only acquired one LFV image per PIV image pair, every 1/7.2 s. Next, the PIV (and PIVSD) images were acquired. The first PIV laser flash occurred, 20 \(\upmu\)s after the LFV images at t = 20 \(\upmu\)s, yielding the first images of a PIV pair. Note here that the wavy water surface did not display any detectable displacements within this 20 \(\upmu\)s time interval. (A “PIV pair” here, represents actually a total of 6 images, since 2 PIV pairs and 1 PIVSD pair are acquired). The second PIV laser flash took place after a time interval \(\Delta t\). Once again, since there was no other light source than the 3–5 ns laser flash, the PIV particle images were extremely sharp and in focus (see again Fig. 2a), even though the full exposure of the second PIV frame was constrained by camera electronics and forced to 67 ms. The PIV time interval \(\Delta t\) was adjusted for each experimental wind speed, to achieve optimal particle displacement on the PIV images for velocity vector calculation and ranged from \(\Delta t=300\) \(\upmu\)s down to \(\Delta t=50\) \(\upmu\)s at the highest wind speed.

In the meantime, the four continuous WG laser beams were on continuously. These did not contaminate the PIV and PIVSD images, because the WG beams were outside of the PIV and PIVSD fields of view. They were equally undetectable on the LFV fields of view, because the LFV camera exposure time was too short to detect these low power laser beams on such a large field of view. The two WG cameras were simultaneously triggered at a frequency of 93.6 Hz. This particular frequency was chosen because it is an integer multiple (\(\times\)13) of 7.2 Hz, the PIV pair frequency. This allowed us to acquire 13 WG samples between each LFV/PIV/PIVSD measurements, with no drift in the time intervals between LFV/PIV/PIVSD and WG samples. In fact the WG exposure time was kept short (8 \(\upmu\)s), to optimize the WG signal resolution, and also to avoid contamination of every 13th WG image by the LFV/PIV/PIVSD laser flashes. The entire LFV/PIV/PIVSD laser flash and camera exposure sequence took place between two consecutive WG image acquisitions.

Each wind-wave experiment followed a fully automated, repeatable procedure. A data flow chart is provided in Fig. 3. At first, the wind was slowly increased to its target steady value. The wind blower was controlled by analog signals, generated and sampled by a PCIe 6353 data acquisition board (National Instruments) coupled with a BNC-2090A connector block (National Instruments). After the wave field had sufficiently developed and reached a fetch-limited equilibrium state, the fog generator was digitally triggered (see Fig. 3) and the system acquired simultaneously PIV data, LIF PIV surface detection data, LIF large field of view data, and LIF single point wave height data. During each experiment, the inside of the tank windows was dried using the window wipers every 30 s, and for a period of 3 s. The images altered by the presence of the wiper were later systematically removed from the dataset. During the acquisition, images were transferred to hard drive striped sets by IO Industries frame grabbers. The images were then accessed through Streams 5 software (IO Industries), and stored on a large data storage striped set after acquisition had ended. The data were later processed for image distortion correction, surface detection, and PIV velocity calculation using routines executed in Matlab.

For this study, wind speeds were generated ranging from 0.86 to 16.63 m s\(^{-1}\) (10-m extrapolated values, see Table 2). The peak wave phase celerity \(C_p\) and wave length \(\lambda _p\) were estimated using the linear deep-water dispersion relationship. The wave amplitudes \(a_{p}\) were computed as \(a_{p}=\sqrt{2}a_{rms}\), where the rms surface elevation \(a_{rms}\) was obtained from the single point optical WG measurements. Overall, we studied 6 different wind speeds. Note that the lowest wind speed of 0.86 m s\(^{-1}\) did not generate any detectable surface waves.

Each of the experiments were performed for durations varying from 4.5 to 14 minutes, depending on the wind-wave conditions. Those durations were chosen based on the estimated dominant wavelength and wave speed for each experiment, with the objective of sampling the same number of waves (approximately 2000) for each condition. This resulted in a total number of images acquired by the imaging system ranging from 85,000 to 200,000, depending on the wind-wave conditions. The results presented in this study were produced from the acquisition and processing of a total of over 1.9 Million images.

Direct outputs of the PIV processing described above are shown in Fig. 4. Here, we show u, the horizontal component of the airflow velocity vector \({\mathbf {u}}\) collected over wind waves for wind speeds of \(U_{10}\) ranging from 2.19 to 16.63 m s\(^{-1}\) (\(U_{10}\) is the wind speed extrapolated to a height of 10-m above the mean water level, a common practice in ocean science). These instantaneous horizontal velocity fields shown are not “best picks”; they were chosen because they give a representative picture of the dynamics upwind and downwind of crests, above waves at the peak of the spectrum. At the \(U_{10}=0.86\) m s\(^{-1}\) wind speed (not shown here), wind waves are not detected and the overall airflow resembles that over a solid flat plate. At higher wind speeds, the presence of waves significantly alters the airflow. In particular, for winds \(U_{10}\geqslant 5.00\) m s\(^{-1}\), large wakes in the lee of the waves are clearly visible and suggest airflow separation. However, airflow separation from the surface cannot be assessed from velocity measurements alone which are not Galilean invariant. A more robust way to evaluate the occurrence of airflow separation events is through the visualization of kinematic fields obtained from the gradient velocity tensor \({\mathbf {\nabla }} {\mathbf {u}}\) (Simpson 1989).

In Fig. 5, we show instantaneous kinematic fields in the airflow for a wind speed of \(U_{10}=5.00\) m s\(^{-1}\). The wave shown has a slope of \(ak=0.29\), where a and k are respectively the amplitude and dominant wavenumber of the wave, measured directly on the LFV image (see Fig. 6). This relatively steep wave has a sharp crest and a flat trough, as well as what appear to be parasitic capillary waves just downwind of the crest (e.g., Longuet-Higgins 1963; Phillips 1981; Ebuchi et al. 1987; Perlin et al. 1993). In Fig. 5, the axes have been normalized with the peak wavenumber \(k_p=2\pi / \lambda _p\), velocities are normalized using the 10-m equivalent velocity \(U_{10}\), and kinematic fields involving components of \({\mathbf {\nabla }} {\mathbf {u}}\) are normalized using \(u_*^2/\nu\), i.e. the ratio of the friction velocity squared to the kinematic viscosity of air.² Above the crest of this steep wave, the airflow moves very fast throughout the entire height of the sampled air column, and the streamwise velocity decreases only very near the surface (Fig. 5a). This suggests that, over the crest, the viscous boundary layer is very thin, and the near surface shear is large. In aerodynamics, boundary layer thinning is known to occur in a similar fashion in favorable pressure gradient conditions, on the leading edge of an airplane wing or the upwind side of a hill for example (Baskaran et al. 1987; Simpson 1989). Just past the wave crest, the shear layer appears to detach from the water surface, and the near surface streamwise velocity drops dramatically. We observe a region of negative velocity near \(k_px=0.8\), which indicates that the crest of the wave is completely sheltering this region from the wind, and that the airflow reverses within the sheltered region. The vertical velocity field \(w/U_{10}\) (Fig. 5b) shows that, upwind of the crest, the airflow is significantly forced upward near the surface. Downwind of the crest, near \(k_px=0.8\), the airflow moves upward near the surface, then back downward near \(k_px=1\) and continues to vary significantly above the wave trough (Fig. 5b). These variations are coherent with the separated sheltering hypothesis, which suggests the presence of recirculations past the point of separation (e.g., Simpson 1989). The spanwise vorticity field, plotted in Fig. 5c, confirms quite unequivocally the conjecture that the airflow is separating past the crest of the wave. The layer of high positive spanwise vorticity (mostly due to shear, see below), attached to the surface on the windward side of the wave, completely detaches from the surface past the crest of the wave. This (now free) high vorticity layer then appears to oscillate (near \(k_px=0.8 - 1\)), and break up into small centers of high (now essentially vortical, see below) vorticity. The oscillations of the unstable free shear layer past the wave crest are reminiscent of shear instabilities. The vortices that are shed past the crest show signs of vortex pairing or “roll-up” (for example just before \(k_px=2\)), similarly to those observed in perturbed free shear layers (e.g., Winant and Browand 1974; Ho and Huang 1982; Ho and Huerre 1984). We may also note here that there is no positive vorticity below the flow-reversed sheltered region, but rather slightly negative spanwise vorticity, which suggests flow reversal in that region.

To distinguish between eddies in the flow and vorticity that is shear-induced, we show estimates of the “swirling strength” \(\lambda _{ci}\) in Fig. 5d. The “swirling strength” is shown by Zhou et al. (1999) to be a measure of the strength of the local swirling motion. It is defined as the imaginary part of the complex eigenvalue of the velocity gradient tensor (Zhou et al. 1999). Here, in two dimensions, \(\lambda _{ci}\) is estimated asWe note that a number of other vortex identification criteria exists such as the Q-criterion (Hunt et al. 1988), the \(\Delta\)-criterion (Chong et al. 1990), or the \(\lambda _2\)-criterion (Jeong and Hussain 1995). These can be utilized as well but for the purpose of this paper, the details of the vortex identification method are not critical. Here, in Fig. 5d, we observe centers of intense positive spanwise swirling strength predominantly past \(k_px=0.8\), which is where the detached high vorticity layer begins to lose coherence. This confirms that the observed centers of high vorticity (Fig. 5c) are due to eddies in the flow.

Conversely, to estimate the importance of shearing motions in the airflow, we decompose the velocity gradient tensor into a symmetrical partand an anti-symmetrical partThe resulting tensors \({\mathsf {S}}\) and \({\mathsf {\Omega }}\) are the strain rate tensor and rotation rate tensor respectively. The latter, also called vorticity tensor, contains only 3 distinct elements \(\omega _i=-\epsilon _{ijk} {\mathsf {\Omega }}_{jk}\), corresponding to the components of the vorticity vector \(\mathbf { {\Omega }}=\mathbf {\nabla } \times \mathbf {u}\). Thus \({\mathsf {S}}\) contains deformation of fluid elements other than solid body rotations. The deviatoric part of \({\mathsf {S}}\), defined as:further contains the so-called “pure shear”, or deformations that are not associated with divergence in the flow since \(tr\left( {\mathsf {S}}\right) ={\mathbf {\nabla \cdot u}}\). In two dimensions, \({\mathsf {\Gamma }}\) contains two independent components and can, therefore, be represented by a vector \(\mathbf { {\Gamma }}\). The “strength of anisotropic stretching”, or “shearing strength”, defined by Zhang et al. (2009) as the magnitude of \(\mathbf { {\Gamma }}\) can then be expressed as:In Fig. 5e, we have plotted \(\gamma\) corresponding to the velocity field of Fig. 5a, b. A thin layer of high \(\gamma\), is present along the windward face of the wave, detaches from the crest, and extends up to \(k_px=0.8\). The existence of this high \(\gamma\) layer, combined with the absence of significant swirling strength along that layer, suggests that the vorticity is essentially shear related, at least until the vortical layer becomes unstable and dislocates. Past \(k_px=1\), downwind of the now disintegrated free shear layer, a number of vortices and small high shear structures occupy the turbulent boundary layer. Then, just before \(k_px=3\), a high shear layer is gradually regenerated at the surface, indicating a progressive restoration of the surface-attached airflow.

In panel Fig. 5f we show along-wave surface viscous stress measurements, taken within the airflow’s viscous sublayer (averaged from 100 and to 500 \(\upmu\)m above the air–water interface). It can be noted that viscous stress is also commonly referred to as skin friction drag, in aerodynamics, (e.g. Schlichting and Gersten 2000). The viscous stress is computed from Eq. 9 below. Over the steeper, airflow separating wave, the viscous stress peaks at the crest, where it matches the total stress. It dramatically drops just past the crest, at the point of separation of the airflow. The surface viscous stress then becomes negative (below the flow reversal region), and remains close to zero up to \(k_px=3\), and then slowly increases back up to \(\approx\) 50% of the total stress, on the windward side of the next downwind wave. These stress measurements compare very well with the measurements of Banner and Peirson (1998) (made on the waterside) and the measurements of Grare et al. (2013b)

To further the analysis of instantaneous data such as those presented above and to extract the turbulence, temporal or spatial averaging is generally required. Below, we present a triple decomposition (Phillips 1977) which splits kinematic fields into the sum of spatial mean, wave coherent, and turbulent components. This triple decomposition relies on proper phase detection and the use of a surface following coordinate system. In the following section, we show how our experimental design makes this analysis possible.

From here, we can compute phase or ensemble averages of a number of data products derived from wave-coherent or turbulent fields. For example, Fig. 8 shows normalized phase averaged velocities, plotted over the phase averaged water surface elevation for wind speeds \(U_{10}\) of 2.19, 5.00, and 16.63 m s\(^{-1}\). The panels on the left show the horizontal velocity component \(\langle u \rangle / U_{10}\), overlaid with phase averaged horizontal velocity profiles at wave phases \(\phi =-\pi /2\), 0, \(\pi /2\), and \(\pi\). For these slow and short wind waves, the airflow is faster on the upwind face of the waves than on the downwind face. This effect corresponds to a sheltering effect past the crest of the waves (Belcher and Hunt 1998) and leads to a thinning of the boundary layer as the airflow approaches the crest, and a boundary layer thickening past the crest. This effect is particularly clear in the velocity profiles for \(U_{10}=16.63\) m s\(^{-1}\). In fact, the normalized wave-coherent horizontal velocity \(\widetilde{u}/ (a_p k_p U_{10})\), shown in the middle panels (d, e, f) of Fig. 8, indeed indicate that the effect of waves is to accelerate the horizontal flow upwind of the surface waves, and decelerate the flow downwind of the waves. In addition, this effect is confined near the interface and decays exponentially away from the interface, as intuition would suggest, at a rate proportional to the surface wave peak wavenumber. The normalized wave-coherent vertical velocities \(\widetilde{w}/ (a_p k_p U_{10})\) show common phase locked features: the air moves upward above positive wave slopes, and downward above negative slopes, a behavior that is expected and consistent with that of \(\widetilde{u}/ (a_p k_p U_{10})\). At the lowest wind speed, however, within a thin layer near the surface, the behavior of \(\widetilde{w} / (a_p k_p U_{10})\) is reversed. The height of this reversal corresponds to the height of the so-called critical layer. At this height, the local wind speed matches that of the (dominant) surface wave. Below the critical layer the wind is slower than the surface wave and \(\widetilde{w} / (a_p k_p U_{10})\) is forced by wave orbital motion; above the critical layer, the wind is faster than the dominant wave which then acts in a similar way as a fixed obstacle would (see Buckley and Veron (2016) and references therein for details).

Additionally, Fig. 9 shows phase and ensemble averages of higher order data products such as wave and turbulent stresses for wind speeds \(U_{10}\) of 2.19, 5.00, and 16.63 m s\(^{-1}\). The normalized phase averaged wave stress \(\widetilde{u}\widetilde{w} /u_*^2\) (Fig. 9 panels a, b, and c) shows positive regions of intense wave stress along most of the upwind face of waves, as well as along most of the downwind face, where the positive stress is slightly less intense. These regions are interlaced with regions of negative wave stress just downwind of crests (intense negative stress) and of troughs (less intense negative stress). The negative stress contours are pinched thin near the surface, such that the negative stress regions are barely connected to the surface, and they widen away from the surface. At the lowest wind speed, while the alternating positive-negative stress pattern exists in a broadly similar form away from the surface, near the surface the mean wave stress pattern is considerably different: the upwind and downwind faces of the wind waves are exposed to regions of negative wave stress, more intense downwind of crests, and slightly less intense upwind. In this case, the situation is almost exactly the mirror image of the higher wind speed cases. As in Fig. 8, this effect is confined below the critical layer height. Belcher and Hunt (1998) predicted this pattern and dubbed it “negative asymmetry” effect. On the right hand side of panels Fig. 9a–c, we show profiles of the normalized ensemble averaged wave stresses \(\overline{ \widetilde{u}\widetilde{w} }/u_*^2\). As expected, the wave effects are confined near the interface. Note that these profiles are plotted with respect to the surface-following vertical coordinate \(\zeta\). Figures 9d–f show the normalized phase averaged turbulent stress \(-\langle u'w' \rangle /u_*^2\). Regions of strong Reynolds stress are found on average downwind of the waves, which is where the boundary layer thickens. This phase-locked “jet” of increased turbulent stress can be related to the sheltering effect (separated or not) that occurs downwind of waves. The separation of the airflow past wave crests is the obvious candidate to explain increased turbulence production past crests since detached free shear layers and airflow separation are expected to produce substantial turbulence and vorticity (see Fig. 5c, d). The lowest wind speed here also differs in that the regions of high Reynolds stresses (downwind of crests) are only marginally more intense that the surrounding stresses at other phases. It is likely that these smaller slope waves do not cause as much separation as the others (Reul et al. 2008). We also note a near-surface layer of reduced turbulent stress at all phases. This suggests that within the critical layer, the work of turbulent forces on the waves is reduced.

Finally, to identify dominant motions in the airflow above waves, POD (proper orthogonal decomposition) can be applied to a set of velocity snapshots measured by PIV. POD analysis has been typically used to decompose flows in the vicinity of fixed boundaries (e.g., Palmer et al. 2012). In this paper, we describe a method of adapting the POD method to the airflow above moving surface waves, and we present some preliminary results for one low wind speed experiment (\(U_{10}\) = 2.19 m s\(^{-1}\), see results below). Briefly, the proper orthogonal decomposition of turbulent velocity fields \({\mathbf {u}}'\), consists in finding a basis of orthonormal spatial modes \({\mathbf { {\Psi }}_k}({\mathbf {x}})\), and temporal coefficients \(a_k(t)\), such that:To obtain the kth mode, the following eigenvalue problem needs to be solved:where the two-point autocorrelation tensor is given by:and \({\mathcal {T}}\) and \({\mathbf {\mathcal {X}}}\) represent time and physical spaces³ respectively. Orthonormality of the spatial modes requires thatwhich in turn implies that the temporal coefficients are orthogonal and scaled such that:The decomposition outlined above (Eq. 12) is particularly interesting because it leads to a convenient physical interpretation. Indeed, since the Reynolds stress tensor is a particular case of the autocorrelation tensor \({\mathsf {R}}\), each eigenvalue \(\lambda _k\) is proportional to the TKE content of the corresponding mode \(\mathbf { {\Psi }_k}\), and the sum of the eigenvalues \(\sum _{k} \lambda _k\) is equivalent to twice the turbulent kinetic energy contained in the domain \({\mathbf {\mathcal {X}}}\) (see Luchtenburg et al. (2009) for details). In other words, the total TKE is:Furthermore, modes with large eigenvalues represent the spatial patterns that contain a significant amount of TKE.

For the dataset presented here, as suggested by the instantaneous observations as well as the phase-averaged results presented thus far, the kinematics of the airflow above the waves strongly depend on the phase of the waves, as well as on the height above the water surface. Thus, we choose to perform a wave-phase dependent POD analysis. To do so, similarly to how we obtained phase averages, we construct an ensemble of instantaneous velocity profiles \({\mathbf {u}}'_s(\zeta )\) measured by PIV above the water surface at a wave phase \(\phi _0\). Thus, the time domain \({\mathcal {T}}\) consists of snapshots in time, and \({\mathbf {\mathcal {X}}}\) reduces to a single dimension along \(\zeta\). In that sense, as described above in Sect. 3.2.3, \({\mathbf {u}}'_s(\zeta )\) form a sub-ensemble of all available measurements \({\mathbf {u}}'_i(\xi ,\zeta )\), at a location \(\xi\) along the interface where the wave phase is \(\phi _0\).

Performing the snapshot POD analysis outlined above yields the spatial modes \(\mathbf { {\Psi }_k}(\mathbf {x})=(\sigma _k,\theta _k)\), corresponding to the turbulent velocity \({\mathbf {u}}'(\phi , \zeta )\). \(\sigma _k\) and \(\theta _k\) are the horizontal and vertical components. \(\sigma _k\) and \(\theta _k\) are the horizontal and vertical components of vector \(\mathbf { {\Psi }_k}(\mathbf {x})\) respectively. It can be noted that, following Eq. 12, both \(\sigma _k\) and \(\theta _k\) represent energy resulting from both components (\(u'\) and \(w'\)) of the velocity vector field. “Snapshot POD” was applied for example by Palmer et al. (2012) to PIV airflow measurements solid barchan dunes. Additional details on this technique were given by Sirovich (1987) and Cazemier et al. (1998). Here, we’ve adapted this method to the airflow over moving water surface waves, by defining a wave-dependent version of snapshot POD.

Figure 10 shows the 3 POD modes with the largest fractional TKE content, for the experiment with \(U_{10} = 2.19\) m s\(^{-1}\) (see panels a through f). For reference, the phase-averaged turbulent velocity variances and TKE for the same experiment are plotted on the right (panels g, h, i). Above \(k_p\zeta \sim 0.2\), the first mode shows patterns consistent with a turbulent boundary layer over a flat wall: intense streamwise and downward turbulent motions. This mode also shows a slight intensification of streamwise motions above wave crests (see panel a). These effects are signs of the modulation of this boundary layer by the waves, otherwise known as non-separated sheltering. Interestingly, over these very small waves in low winds, it is mostly at the higher order modes (panels b and f for example), that we can see closed phase-locked contours downwind of crests, reminiscent of those in the turbulent stress (Fig. 9, panel d) and in the TKE components \(\langle u'^2 \rangle / u_*^2\) and \(\langle u'^2 + w'^2 \rangle / u_*^2\) (Fig. 10, panels g and i). These features are probably due to intermittent airflow separation events, which have been found to take place over a small percentage of the waves in these low wind conditions. Here, POD analysis allows to classify flow patterns in order of importance for the total TKE: In this low wind-speed experiment, turbulent flow patterns due to airflow separation events have a lower contribution to the total TKE than simple non-separated sheltering downwind of wave crests.