Simultaneous velocity and concentration measurements over a rippled boundary subjected to oscillating fluid forcing

Abstract

We describe an oscillating boundary layer apparatus (OBLA) to investigate mass and momentum transfer in the wave bottom boundary layer. The facility is designed such that near-bed shallow water orbital velocities are physically modeled in full scale. A PIV/PLIF system allows for simultaneously resolving the intra-ripple velocity and dye concentration fields. We examine two cases by injecting dye at the trough and crest of the rippled boundary. The extent of the plume is the largest near the zero-crossing of the free-stream velocity and 40$^\circ$ later for the trough and crest case, respectively. Both cases showed periodic turbulent vortical structures influencing the phase-averaged concentration plumes. For normalized concentrations greater than 0.01, the plumes remained within the boundary layer and traveled half a ripple length for both cases. Dye spread vertically upward about 2 and 1.5 ripple heights from the crest and trough sources, respectively. Stronger advection was observed over the crests, along with a clear dependence on bedform asymmetry.

1 Introduction

Important biogeochemical and ecological processes occur near the wave-bottom boundary layer (WBBL) of the coastal ocean. For example, feeding and reproduction of organisms (Webster and Weissburg 2009), and exchanges of dissolved oxygen controlling the metabolism and nutrient cycling of biota (O’Connor and Hondzo 2008; Higashino and Stefan 2011). For the case of a rippled sediment bed, studies have found that mass sources, such as particulate carbon, dissolved oxygen, and algae, can preferentially organize below a particular ripple feature (i.e., trough, crest, or flank), resulting in significant across-ripple variability in porewater properties (Shum 1993; Precht et al. 2004; Huettel et al. 2007; Ramey et al. 2009). Hence, understanding mass and momentum transport near the WBBL is vital to asses, predict, and potentially mitigate these and other processes at the sediment-water interface (SWI). Resolving mass and momentum fluxes across the SWI requires detailed measurements of the unsteady velocity field, sediment movement, bed morphology, and scalar fields, all of which can vary significantly in both time and space (Rodríguez-Abudo and Foster 2014, 2017; Aponte-Cruz and Rodríguez-Abudo 2024).

Experimental measurements of mass and momentum fluxes are often conducted using acoustic techniques, which involve point or profile measurements of fluid velocity and scalar concentrations (Yorke and Oberg 2002; Kostaschuk et al. 2005); or acquisitions of one-point scalar concentration using fluorometers (Reidenbach et al. 2010; Elliott and Brooks 1997). Although these techniques provide valuable insights into scalar fluxes at the bottom boundary, they are unable to depict the 2D flow and concentration fields. For the case of uneven bottom boundaries, subjected to oscillating fluid forcing, resolving the 2D fields simultaneously is crucial, as the problem is highly space dependent and unsteady (Rodríguez-Abudo et al. 2013). Non-invasive techniques such as particle image velocimetry (PIV) and planar laser-induced fluorescence (PLIF) are a popular way to simultaneously resolve scalar and concentration fields in bottom boundary layers. Several efforts have been reported using these techniques in steady unidirectional turbulent flows. Crimaldi et al. (2002) obtained velocity and concentration fields in three orthogonal planes by injecting fluorescence dye from a flatbed; Connor et al. (2018) also used PLIF to resolve acetone concentration injected at three different heights in a wind tunnel, while Wagner et al. (2007) obtained scalar fluxes by injecting dye from a crest of a wavy bed.

Resolving mass fluxes over rippled beds subjected to oscillating forcing requires a more complex experimental setup, acquisition and analysis, given the unsteady and uneven nature of the flow. Comprehensive 2D observations can be performed in laboratory settings using wave makers or oscillating flow tunnels (OFTs). However, wave makers have limitations regarding size or capabilities, while OFTs represent a significant investment. An alternative approach involves oscillating a plate in an otherwise still fluid, building upon the pioneering work of Bagnold (1946). Hay et al. (2012) utilized this concept to resolve profiles of the turbulent oscillatory boundary layer above fixed sand and gravel beds. In this study, we further develop the concept into an oscillating boundary layer apparatus (OBLA) to study mass and momentum transport in the WBBL. The OBLA system can reproduce full-scale WBBL dynamics, including shallow water wave orbital velocities, velocity, and acceleration skewness, ripple formation and migration, among others. In addition, this effort presents unprecedented acquisition resolution of velocity and scalar fields over rippled walls subjected to oscillating fluid forcing. Near-bed hydrodynamics and concentration are reported herein, along with their variation in time and space.

2 Apparatus

The OBLA comprises three main structures (Fig. 1): a 7.4$\times$0.7$\times$0.7 m Plexiglass tank, an independent 3$\times$1$\times$2.5 m rigid aluminum frame with legs attached to the floor using 12.7-mm stainless steel screws, and a movable aluminum mechanism supporting a 2.5$\times$0.015$\times$0.53 m PVC tray. The moving mechanism is connected to the rigid frame through a heavy-duty linear roller system (Parker IPS) consisting of four LR-14HD bearings, each capable of supporting up to 1500 N. Water in the tank remains stationary while the suspended tray oscillates back and forth, simulating near-bed hydrodynamics under shallow water waves.

Fig. 1

a Front and b top views of the OBLA showing its three main structures: water tank (light blue), rigid frame (green), and moving mechanism (dark blue). All dimensions are given in m

The reference frame is located on the moving mechanism, which is driven by a high-torque (30 N$\cdot$m) stepper motor connected to a 3-m-long carbon-coated steel ball screw (25.4 mm diameter). The motor has 200 steps, providing displacement resolutions of 0.127 mm/step with the 25.4 mm screw lead. The apparatus can achieve maximum orbital displacements ($d_0$) of 1.5 m, velocity amplitudes ($U_0$) of 0.52 m s$^{-1}$, and accelerations of 0.4 m s$^{-2}$ (Fig. 2). Communication and power cables, as well as laser cooling hoses, are handled by a 150-mm Nylatrac$\circledR$ drag chain. A piston-like pump supplies a constant inflow of Rhodamine dye at any location along the longitudinal axis of the tray through injectors consisting of Luer-lock-type metallic needles.

Fig. 2

Horizontal velocity amplitudes ($U_0$, grayscale) and dimensionless wave height (H/h, solid lines) as a function of OBLA inputs ($d_0$ and T)

Near-bed velocity fields are obtained using a Dantec Particle Image Velocimetry (PIV) system, which includes a 200-mJ Nd:YAG laser, a 25-MP camera, and a DualScope optical splitting system. A Dantec planar laser-induced fluorescence (PLIF) system is used to measure relative concentrations of Rhodamine dye in a field-of-view (FOV) that is colocated with the PIV. The PIV and PLIF systems are integrated through an image acquisition system, allowing simultaneous measurement of velocity and concentration using a single camera connected to an AF Nikkor 50-mm f/1.4D lens, which is further attached to the DualScope. On the DualScope acquisition end, 532-nm and 527-nm filters enable PIV and PLIF detection, respectively. Background concentration is verified using a Cyclops-7F fluorometer, while a Nortek Vectrino Acoustic Doppler Profiling Velocimeter (ADPV) allows for PIV validation and high-frequency (100 Hz) velocity measurements in 30-mm profiles at 1-mm resolution. The OBLA is controlled by a set of two microcontrollers. An Arduino Nano 33BLE handles loops for the stepper motor, controlling the preconfigured wave period, orbital displacement, and number of cycles. An Arduino MKR1000 acts as a master controller, providing a human–machine interface (HMI) for operating the OBLA through a touchscreen. The MKR1000 initializes environmental sensors (barometric pressure, water and air temperature, and humidity), the HMI, the motor microcontroller, and emits the start pulse for the Dantec synchronizer, which initiates data acquisition.

User-defined inputs for the OBLA are orbital displacement ($d_0$) and wave period (T), which for regular monochromatic waves translate to horizontal velocity amplitudes $U_0=\pi \,d_0/T$ (Fig. 2). Assuming shallow water waves (flat wave orbits near the bed), the corresponding nondimensional wave heights (H/h, where H and h are wave height and water depth, respectively) for self-similar field conditions can be computed as:

$$\begin{aligned} \frac{H}{h} = 2\pi \frac{d_0}{g\,T^2}\frac{\lambda _\textrm{w}}{h}\cosh \left( 2\pi \frac{h}{\lambda _\textrm{w}}\right) , \end{aligned}$$

(1)

where $\lambda _\textrm{w}$ is the wavelength, and a threshold for wave breaking ($H/h=0.78$) has been imposed. Figures 2 and 3 depict the full range of $U_0$ and corresponding H/h values achievable by the OBLA for different $d_0$ and T combinations.

Fig. 3

Dimensionless wave height (H/h) in shallow water ($h/\lambda _\textrm{w} \le 0.05$) as a function of OBLA inputs ($d_0/g\,T^2$ and nondimensional water depth, $h/\lambda _\textrm{w}$)

The range of shear stresses achievable by the OBLA was assessed with the grain roughness Shields parameter (Fig. 4):

$$\begin{aligned} \theta _{2.5} = \frac{f_{2.5}}{2}\frac{{U_0}^2}{(s-1)g\,d_{50}}, \end{aligned}$$

(2)

where s is the sediment specific gravity, $d_{50}$ is the median grain diameter, g is the gravitational constant, and $f_{2.5}$ is the wave friction factor (Swart 1974) given by

$$\begin{aligned} f_{2.5} = \exp \left[ 5.213\left( \frac{2.5\,d_{50}}{A_0}\right) ^{0.194} - 5.977\right] , \end{aligned}$$

(3)

with $A_0=d_0/2$ as the wave excursion amplitude. For quartz sand ($s=2.65$), the OBLA is capable of achieving $\theta _{2.5}=[0.02-0.5]$ (Fig. 4), which includes shear stresses at the threshold for incipient motion all the way to the bedform regime. Sheet flow conditions ($\theta _{2.5}>0.8$) may also be achieved with lighter sediment grains (see for example Fig. 4 dashed lines for $s=1.2$).

Fig. 4

Grain roughness Shields parameter ($\theta _{2.5}$) as a function of OBLA inputs ($d_0/gT^2$) and dimensionless grain size ($d_{50}/d_0$) for two different grain specific gravities (s)

Similarly, the range of Reynolds numbers ($Re={U_0}^2T/2\pi \nu$, with $\nu$ as the kinematic viscosity) achievable by the OBLA ($Re<260,000$) include laminar and turbulent conditions (Fig. 5). For smooth walls, laminar conditions are found for $Re<10^4$, while turbulent regimes are expected for $Re>10^5$ (Davies and Thorne 2005). Flow over rough walls is considered fully turbulent with strong dependence on $d_0$, and ripple size (Jensen et al. 1989). Typical Reynolds number under field and large-scale laboratory conditions are in $O(10^5)$ (Davies and Thorne 2005).

Fig. 5

Reynolds number as a function of OBLA inputs ($d_0/gT^2$) and dimensionless grain size ($d_{50}/d_0$)

3 Observations

For this study, velocity and concentration measurements were taken for 10 consecutive cycles of regular monochromatic waves with $T=5$ s, $d_0=0.5$ m, and $U_0=0.31$ m s$^{-1}$ (Fig. 2). These wave characteristics correspond to a shallow water wave height (H) of 0.15 m in 0.54 m water depth (h) (Fig. 2). They also correspond to an intermediate water wave with $H=1.34$ m at $h=10$ m. According to Wiberg and Harris (1994), these wave characteristics are expected to produce ripples heights ($\eta _\textrm{r}$) of approximately 16 mm and lengths ($\lambda _\textrm{r}$) of 74 mm when interacting with fine sand (median diameter $d_{50}=0.14$ mm, and specific gravity $s=2.65$). To simulate this morphology and demonstrate the functionality of the OBLA, a corrugated plastic boundary with sinusoidal ripples ($\eta _\textrm{r}=16$ mm, $\lambda _\textrm{r}=74$ mm) was attached to the tray.

PIV images (at 8-bit depth) were acquired with a field of view (FOV) of 236$\times$137 mm at a rate of 15 Hz, resulting in a total of 750 image pairs and a 50-second acquisition period. The time interval between laser pulses was set to 1 ms. Velocity vectors were calculated using adaptive PIV algorithms with minimum and maximum interrogation areas (IA) of 32 and 64 pixels, respectively, and a 50% overlap. The resulting velocity field had a spatial resolution of 1 mm in both directions.

To validate the free-stream velocity vectors, measurements from the ADPV located 110 mm above a ripple trough were used. The uncertainty of horizontal velocities at the 95% confidence level was estimated by comparing them with the ADPV measurements. A bootstrap analysis with 1000 replicas (Efron and Tibshirani 1994) resulted in an uncertainty of ±0.034 m/s (11% of $U_0$). Potential velocity errors caused by laser reflections, dark areas, and other factors were assessed using the peak-to-peak ratio (PPR) of correlation peaks. On average, all interrogation areas satisfied the 1.2 PPR threshold proposed by Keane and Adrian (1992) (Fig. 6a). The spatial distribution of PPRs below 1.2 (Fig. 6b) indicated that less than 5% of the interrogation areas failed to meet this threshold, primarily in regions with high velocity gradients. An example of an instantaneous sample with less than 0.5% of interrogation areas below 1.2 PPR is shown in Fig. 6c. Outliers were reduced using a 5$\times$5 moving average validation filter, which rejected observations exceeding 10% of the average value within the window. The total number of spurious vectors detected was less than 3.1% for both horizontal (u) and vertical (w) velocities. After detection, outliers were replaced with the local average within the same moving window. Finally, a 9-point temporal smoothing was applied to all vectors.

Fig. 6

a Time-averaged PPR; b percentage of IAs that do not satisfy the PPR threshold; c instantaneous PPR; d time-averaged velocity field; e RMS velocity field; and f instantaneous PIV image corresponding to (c). The PPR-color scale is indicated at top in (a). Red dots indicate PPR values $<1.2$. Vectors in (d, e) are plotted every 7 positions in the horizontal direction and 2 in the vertical. The quiver scale for panels (d, e) is indicated at the top of panel (e). The dashed and solid lines in d enclose the areas presented in Figs. 9 and 10, respectively

Bulk statistics of the velocity field aligned well with the expected dynamics of the WBBL layer. The time-averaged velocity field showed values near zero, as expected for a regular sinusoidal wave (Fig. 6d). The root-mean-squared (RMS) velocities (Fig. 6e) were higher near the ripple crests and lower at the troughs, consistent with the anticipated distribution of flow acceleration, deceleration, and vortex formation. Vertical RMS velocities (wRMS) were analyzed to assess potential vibrations of the structure that might induce artificial flow patterns. In this dataset, the free-stream wRMS was 0.025 m s$^{-1}$, approximately 8% of $U_0$, and there was no evidence of artificial mean flows or recirculation in the tank.

For the PLIF protocol, a calibration was performed using 12 points with known concentrations ranging from 0 to 1000 $\mu$g/L. Each camera pixel was calibrated using a first-order polynomial to account for spatial differences in intensity. The resulting concentration resolution for the 8-bit camera was 0.39 $\mu$g/L. Crimaldi and Koseff (2001) identified several error sources for PLIF including background fluorescence, flume temperature, and attenuation, among others. To assess the possible presence of these optical errors, a black reference calibration was performed. Dye concentration was carefully chosen to avoid saturating the camera pixels.

Unlike PIV, which uses interrogation areas, PLIF operates on individual pixels, resulting in a much finer spatial resolution of 66$\times$66 $\mu$m per pixel. Rhodamine dye with an initial concentration of $C_0=1000$ $\mu$g L$^{-1}$ was injected at two locations, the crest and trough of the fixed-rippled bed, at a rate of 9.5 cm$^3\,$min$^{-1}$.

4 Results

Velocity and concentration measurements were phase-averaged ($\tilde{\;\;}$) such that

$$\begin{aligned} {\widetilde{Q}}({\tilde{t}}) = \frac{1}{M}\sum _{m=1}^M Q(t+mT), \end{aligned}$$

(4)

where Q refers to any flow or concentration quantity, ${\tilde{t}}$ is the phase, t is time, and M is the total number of realizations.

4.1 Hydrodynamics

During maximum acceleration phases, following the zero-crossings in velocity, streamlines closely adhere to the ripple geometry (Fig. 7a, g). From this moment, the vortex grows in diameter during deceleration stages, reaching its maximum size at about 20$^\circ$ before the next zero crossing (Fig. 7e, k). The vortex structures rise (Fig. 7f, l) to a height of about 1 $\eta _\textrm{r}$ measured from the crest, and eventually dissipates. This behavior is similar to what was observed in the three-dimensional mixture simulations conducted by Penko et al. (2013), and previous efforts such as those of Blondeaux and Vittori (1991) and Scandura (2007). The dissipation of the vortex is challenging to observe as it stretches horizontally along the flow direction.

Fig. 7

(Top) Free-stream horizontal velocity. a–l Velocity field (solid black arrows) depicted over flow streamlines (light gray lines). Vectors are plotted every 10 positions in the horizontal direction, and every 2 positions in the vertical

The boundary layer thickness ($\widetilde{\delta _{99}}$) was computed by finding the vertical location where the phase-averaged velocity profile reached 99% of the free-stream velocity. $\widetilde{\delta _{99}}$ is both phase and space-dependent due to the uneven bottom boundary. To evaluate the variation of $\widetilde{\delta _{99}}$ solely with respect to phase, a spatial average over one ripple length ($\lambda _\textrm{r}$) was performed such that

$$\begin{aligned} \langle \widetilde{\delta _{99}}\rangle = \frac{1}{\lambda _\textrm{r}} \int _0^{\lambda _\textrm{r}} \widetilde{\delta _{99}}\,\text {d}x , \end{aligned}$$

(5)

where the angle brackets denote space average (Fig. 8). $\langle \widetilde{\delta _{99}}\rangle$ shows a fairly linear growth from $\sim$5$^\circ$ after the zero crossing of horizontal free-stream velocity up to $\sim$175$^\circ$, when it quickly falls to zero. This drop occurs at phases when the coherent vortices rise and disappear from the viewing area. A bulk value of the boundary layer thickness was found by time averaging $\langle \widetilde{\delta _{99}}\rangle$ over the entire cycle, resulting in 22.7 mm. This value is slightly lower than the formulation by Kajiura (1968), where $\delta =f_\textrm{w}\,A_0/2$, with $f_\textrm{w}$ as the wave friction factor defined in eq. (3) using the large scale roughness ($Kn=27.7\,{\eta _\textrm{r}}^2/\lambda _\textrm{r}$, Grant and Madsen (1982)) instead of the grain roughness ($2.5\,d_{50}$).

Fig. 8

Space-averaged boundary layer thickness ($\langle \widetilde{\delta _{99}}\rangle$, solid line) and free-stream velocity ($U_\infty$, dashed line) as a function of wave phase

4.2 Dye concentrations

4.2.1 Trough-injection case

The plume observed in the phase-averaged concentration fields (${\widetilde{C}}$) normalized by the concentration at injection ($C_0$) is highly dependent on fluid velocity (Fig. 9). Overall ${\widetilde{C}}/C_0$ barely reaches 0.05, and values lower than 0.015 remain inside the two consecutive crests and do not overpass $\widetilde{\delta _{99}}$. The plume reaches its maximum longitudinal spread around the phases of zero crossings (Fig. 9e–g, l), whereas at high velocities the plume tends to take a rounder shape, likely due to the flow being trapped in the troughs which causes low local velocities and less advection (Fig. 9c, d, j). The coherent vortex lifts the plume over the flanks and crests, allowing for dye transport in those regions (Fig. 9e, k). The effect of the phase shift is notorious, see for example evidence of dye transport very close to the zero crossing of $U_\infty$ well beyond high velocities (Fig. 9e–g, l). Similarly, at high flows dye concentrations remain close to the injection point (Fig. 9c, d, i, j) indicating weak transport during these phases. A cross correlation analysis of ${\tilde{u}}$ against $U_\infty$ indicates phase shifts of $\sim$30$^\circ$. This is consistent with the well documented boundary layer leads (Jonsson 1980; Rodríguez-Abudo et al. 2013) (Jonsson 1980; Rodríguez-Abudo et al. 2013).

Fig. 9

(Top) Free-stream horizontal velocity. a–l Phase-averaged normalized concentration field (colormap) with corresponding streamlines (light-gray lines) and $\widetilde{\delta _{99}}$ (dashed-black line). The phases chosen are the same as in Fig. 7. Test case corresponds to dye injection at the trough

4.2.2 Crest-injection case

For the crest-injection case, the effect of the boundary layer phase shift is less notorious. Higher magnitudes of dye concentration with respect to the trough case are shown very close to the injection point for all phases, with ${\widetilde{C}}/C_0$ values above 0.05 as opposed to the ${\widetilde{C}}/C_0<0.02$ observed in the trough injection case. Locations with values higher than 0.01 are never above $\widetilde{\delta _{99}}$. The plume follows the bed during phases where coherent structures are not present (Fig. 10a, b, g, h, l). Near maximum velocities, the plume detaches from the bed (Fig. 10c) because of the coherent vortex located at the trough, which causes reverse flow in the vicinity of the bed decreasing the size of the plume at maximum velocities (Fig. 10d). When the vortex reaches its maximum size (Fig. 10e), the plume remains close to the crest and low magnitudes of dye rise and are advected into the next vortex on the right. At the zero crossing (Fig. 10f), the plume is very close to the bed and spreads to the right ahead of $U_\infty$.

Fig. 10

(Top) Free-stream horizontal velocity time series. a–f Phase-averaged velocity fields (first column), and normalized concentration fields (colormap) with streamlines (light-gray lines) and $\widetilde{\delta _{99}}$ (dashed-black line). Test case corresponds to dye injection at the crest

Asymmetries in bed morphology are evidenced in the plume of dye, as stronger plumes are observed in the second half cycle, when the dye is being advected toward the shallower trough (Fig. 10h, i, j). On the other hand, the first half cycle shows weaker plumes around maximum velocities (Fig. 10b–d), since deeper troughs form bigger vortices causing stronger flow reversals and recirculation (Lin and Yuan 2005).

4.2.3 Spread of dye

In order to assess the extent of dye concentrations, two measures of dye spread were computed. A radial spread ($S_\textrm{r}$) was found by quantifying the maximum distance between the injection point and any pixel containing ${\widetilde{C}}/C_0>0.005$ ($R_i$) and normalizing by the ripple height, such that $S_\textrm{r}({\tilde{t}})=R_i/\eta _\textrm{r}$. An area spread ($S_A$) was also computed for regions with ${\widetilde{C}}/C_0>0.005$, and normalized by $\eta _\textrm{r}\,\lambda _\textrm{r}$.

Ripple and source location asymmetry have considerable effects on the spread of dye. The injector in the trough case is located slightly closer to the left, causing a longer plume in that direction (Fig. 11b, black-dotted line); $S_\textrm{r}$ is a maximum at the first zero crossing, with a value of 1.5 at 0$^\circ$, $S_A$ is also higher during these phases (340 to 60$^\circ$). The minimum spread appears at maximum negative velocities (90 - 120$^\circ$), consistent with the rounded shape close to the source, shown in Fig. 9d. At maximum positive velocities, the plume also takes a rounded shape (Fig. 9i, j), but $S_\textrm{r}$ and $S_A$ are higher than the first half-cycle likely due to the injector location (slightly further to the left) and the previously discussed ripple asymmetry.

Fig. 11

a Free-stream horizontal velocity. b Radial spread ($S_\textrm{r}$). c Area spread ($S_A$)

For the case of the crest, ripple asymmetry promotes more spreading toward the shallower trough on the right, with the longest plume ($S_\textrm{r}=1.9$) observed at $\sim$230$^\circ$. $S_\textrm{r}$ decreases to 0.72 at 340$^\circ$, but $S_A$ remains high. This indicates a more symmetric plume around the source (Fig. 10k), where the plume is trapped between the coherent vortices at both sides, both of which have reached their maximum size. Similar behavior is observed in the first half-cycle but to a lesser extent.

The plume at the crest is generally stronger than the plume at the trough. This is evident in the averaged values of $S_\textrm{r}$, which are 1.05 and 0.72 for the crest and trough cases, respectively, and $S_A$ with maximum values of 0.08 and 0.04, respectively. This is expected given the increased higher velocities over the crests which promote higher advection. It is also consistent with the model of troughs acting as sinks in the mass transfer problem at the sediment-water interface, as observed by O’Connor and Hondzo (2008).

4.3 Bulk characteristics

The time-averaged concentration fields ($C_m$) (Fig. 12a, b) display relatively low values compared to the standard deviation fields (Fig. 12c, d), as expected for turbulent flows (Crimaldi et al. 2002; Wagner et al. 2007; Connor et al. 2018). $C_m$ values are present within one ripple length for the trough-injection case, while the crest case exhibits a higher advection toward the right due to ripple asymmetry. The standard deviation fields ($C_{\textrm{std}}$) reveal information about the presence of dye across the domain. For example, at trought injection it is clear that dye reaches almost the entire horizontal extent, and even surpasses the time-averaged boundary layer thickness ($\overline{\delta _{99}}$) (Fig. 12c). On the other hand, crest injection shows the presence of dye remaining very close to the ripple and barely exceeding $\overline{\delta _{99}}$ or consecutive ripple troughs. The standard deviation field allows for the observation of blobs and filaments that are not evident in the time average. For the crest-injection case, the dye blobs are more intense near the injection point but generally dissipate faster, causing $C_{\textrm{std}}$ to cover a lower area than in the trough case. The standard deviation field also reveals the presence of concentration filaments that are not persistent over time and/or fade in the phase averaging operation. The time-averaged concentration signature (for both cases) do not surpass $\overline{\delta _{99}}$ (see black-dotted lines in Fig. 12). $C_m$ shows, for both cases, persistence of dye within half a ripple length from the injection point. Vertically, the dye is persistent within one ripple height regardless of the injection case.

Fig. 12

Time-averaged of normalized concentration fields ($C_m$) for trough (a) and crest (b) case; and standard deviation fields ($C_{\textrm{std}}$) of normalized concentration for trough-injection (c), and crest-injection (d). The time-averaged boundary layer thickness ($\overline{\delta _{99}}$) is depicted as dotted-black lines in all panes

To further evaluate the concentration characteristics, intermittency fields were computed from the temporal probability of dye presence in each pixel (Crimaldi et al. 2002). We use Chatwin (1989) definition of dye presence, which refers to any non-zero concentration value. To mitigate experimental noise and device resolution, a threshold of $C_\textrm{T}=0.004\,C_0$ was chosen, which is one order of magnitude above the concentration resolution of the device and sufficiently low compared to $C_0$. Thus, the intermittency field is defined as:

$$\begin{aligned} \gamma (x,z) = \textrm{prob}\left( \frac{C(x, z, t)}{C_0} > C_\textrm{T}\right) , \end{aligned}$$

(6)

where x and z are the spatial coordinates, and t is time. $\gamma$ is related to the probability density function (PDF) of the spatial distribution of dye concentration (Chatwin 1989), although its magnitude is higher than that of the PDF. In both cases $\gamma$ is nearly equal to one very close to the source location, indicating persistent presence of dye at those regions (Fig. 13a, b).

Fig. 13

(Top panels): Intermittency ($\gamma$) fields for trough-injection case (a), and crest-injection case (b). (Bottom panels): Corresponding $\gamma$ profiles with top panels at several distances from the ripple boundary

Profiles of intermittency values collected at different vertical positions show a narrower peak for trough injections case (Fig. 13c). This suggests that dye injected at the crest is rapidly transported horizontally, causing $\gamma$ to decay quickly over the flanks. For the case of trough injection, there is a slight accumulation of dye above the flanks, followed by a rapid decay once it reaches the crest. Similarly, for the crest injection scenario (Fig. 13d), intermittency exhibits a steeper vertical decay above the source location, as seen by the local minima near $x=80$ mm for all profiles, except for the one closest to the bed (0.5 mm). Given that over the crest the mean flow remains predominantly horizontal, a local maxima appears over the flanks for profiles with a vertical offset greater than 0.5 mm.

Nonzero intermittency values (Fig. 13a, b) encompass a greater area than the nonzero time-averaged values (Fig. 12a, b) and a smaller area compared to the standard deviation (Fig. 12c, d). High concentration structures outside the colored area observed in the intermittency fields are possible but unlikely. However, their occurrence, no matter how rare, is significant for active tracking applications involving benthic organisms (Webster and Weissburg 2009).

The double-averaged concentration and velocity profiles (Fig. 14) were computed as in Rodríguez-Abudo and Foster (2014):

$$\begin{aligned} \langle Q \rangle = \frac{1}{L_\textrm{f}} \int _{L_\textrm{f}} Q\,\text {d}x , \end{aligned}$$

(7)

where Q represents the quantity to be double averaged, the angle brackets indicate the spatial averaging operation, and $L_\textrm{f}$ is the horizontal length occupied by the fluid in the limits defined by the vertical lines in Fig. 13a, b. Note that the domain is different for each case in order to keep the injection point at the center.

Fig. 14

Free-stream horizontal velocity (a); and double-averaged velocity profiles over double-averaged concentration profiles for trough (b), and crest-injection (c) cases. The lines A, B, and C indicate the vertical positions where the crest, shallower tough, and deeper trough, respectively, are located

The double-averaged velocity profiles exhibit the typical behavior of a regular (sinusoidal) wave. Velocity magnitudes below the ripple crests remain lower than those just above the crests (Fig. 14b, c). The maximum vertical ejection occurs before the zero crossing for both injection cases. The asymmetry observed in the time-dependent behavior (Fig. 11) is also evident in the double-averaged concentration profiles for the crest case, with a stronger presence of dye above the trough during the positive half-cycle. In these phases, the flow carries the dye toward the right trough, which promotes dye accumulation because of its shallower depth. The vertical extent of nonzero values does not exceed one ripple height over the crest and is limited at the bottom by the impermeable bed.

The double-averaged concentration profile for the trough case (Fig. 14b) shows a thicker average stain of dye, reaching approximately one ripple height. This suggests higher dissipation of dye above the crest level in the trough-injection scenario.

5 Conclusions

An oscillating boundary layer apparatus for full-scale physical modeling of wave bottom boundary layer dynamics has been presented. The system is able reproduce oscillations typical of shallow water waves, achieving maximum orbital velocities of 0.52 m s$^{-1}$, accelerations of 0.4 m s$^{-2}$ and orbital diameters of 1.5 m. The observational component is capable of simultaneously resolving velocity and concentration fields at 1-mm and 0.06-mm resolutions, respectively. The OBLA can recreate several grain-mobility regimes such as, near incipient motion, bedload transport with and without bedforms, and sheet flow. This last one can be reached using lighter sediments than the usual specific gravity value of 2.65.

The characterization presented herein shows very good agreement with the expected boundary layer dynamics in terms of formation of coherent structures, boundary layer thickness, and phase differences with respect to the free-stream velocity, which provides a hydrodynamic setting to resolve additional processes, such as mass transport across the sediment-water interface.

Unprecedented observations of dye transport in oscillatory flow have been presented. Significant advection and diffusion occurs over the crests for almost all phases regardless of the source location; whereas dissipation and transport over the troughs occurs mainly due to vortical coherent structures. The spread of dye is higher when the source is located at the crest likely due to higher velocities, and consequently more advection. This case also presents a plume consistently oriented with the flow that reaches its maximum before the maximum-velocity phase. For the trough-injection case, the plume almost disappears near the free-stream maximum velocity. The maximum transport of dye with ${\widetilde{C}}/C_0$ exceeding 0.01 barely extends beyond 0.5 $\lambda _\textrm{r}$ in horizontal distance and 1.5 $\eta _\textrm{r}$ in the vertical. Right after the zero crossing, the dye is transported following the bed morphology in a thin layer with filaments of approximately 1/3 $\lambda _\textrm{r}$. Throughout most phases, blobs with concentration peaks of around 0.015 are visible, resulting from broken filaments. $\widetilde{\delta _{99}}$ is vertically exceeded only around the inflection point of velocity at the source and close to the zero crossings over the crest. The intermittency profiles of the concentration fields do not match the typical Gaussian profile. Instead $\gamma$ profiles in this study show a narrower peak and a slight accumulation over the flanks.

The results presented in this paper provide a first look into the 2D mass exchange problem taking place at the SWI of a rippled bed. While percolation was not allowed in this set of experiments, important implications to the mass exchange problem can be drawn: 1) mass is more persistent near the ripple flanks; 2) mass emanating from the ripple crests will more likely reach further distances due to much higher advection; and 3) regardless of its source location across the rippled bed, mass is unlikely to reach elevations higher than the wave-induced boundary layer thickness, which is roughly O(1-10) cm for nearshore environments.