Accurately assessing the response of sediments to oscillatory flows requires high-resolution fluid velocity and sediment transport measurements at the fluid–sediment interface. Fluid and sediment grain velocities were measured simultaneously with combined particle image and tracking velocimetry under oscillatory flows over movable sand ripples. Three high-speed cameras equipped with varying optical filters were used to distinguish between fluorescent fluid tracers and the grains, from which the fluid and grain velocities were determined, respectively. Individual grains were tracked during transport to determine velocities and trajectories. Sediment grains were first mobilized by a vortex impacting the bed during flow reversal and suspended into the water column just prior to vortex ejection from the ripple crest, similar to previous observations. During phases of maximum flow velocity, additional grains were mobilized by the shear stress and were subsequently suspended. The flow reversed and similar observations were made in the opposite direction. Consequently, four peaks in suspended sediment concentration were observed throughout the flow cycle, consistent with previous observations. However, some previous researchers attributed peaks in suspended sediment concentration occurring during phases of maximum flow velocity to sediment-laden vortices that were shed from adjacent ripples. The measured sediment grain velocities were of similar magnitude and phase to the near-bed fluid velocities when the grains were being advected with the flow. Measurements of suspended sediment concentration agreed well with semiempirical formulations having an average root-mean-square deviation of approximately 4 × 10−5 m3 m−3. Predictions of settling velocity also compared well with the laboratory estimates, agreeing to within 90%.
Sand ripples enhance suspended sediment transport and wave energy dissipation in the near shore. Hence, much research effort has focused on understanding the hydrodynamics and evolution of these ripples in coastal regions through field, laboratory, and numerical studies (e.g., Bagnold and Taylor 1946; Blondeaux et al. 2004; Faraci and Foti 2002; Doucette and O’Donoghue 2006; Charru et al. 2013). Typically, small-scale ripples have wavelengths of 10–100 cm and heights of 1–10 cm (Traykovski et al. 1999; Hurther and Thorne 2011). Laboratory and field efforts have shown that the hydrodynamics over small-scale ripples are dominated by the generation and ejection of turbulent coherent structures (i.e., vortices) (e.g., Blondeaux et al. 2004; Davies and Thorne 2005). The velocities near the ripple crest may be up to 50% larger than the free-stream velocity with a phase lead of up to 108° ahead of the free streamflow due to vortex formation (e.g., Stachurska and Staroszczyk 2016). The sediments used in these experiments range from fine (Nakato et al. 1977; Nichols and Foster 2007; Hurther and Thorne 2011; Stachurska and Staroszczyk 2016) to medium (van der Werf et al. 2007) to coarse sand (García et al. 2002). Suspended sediment is predominantly located very near the bed, typically within a few ripple heights of the ripple crest (van der Werf et al. 2007). The average suspended sediment concentration profile above the ripple crest may be approximated with exponential (Sleath 1984; van der Werf 2006) or power-law formulations (Nielsen 1992). The concentration decay length may also be calculated as a function of the ripple height (Nielsen 1986, 1992; van der Werf 2006).
Three to five peaks in suspended sediment concentration have been observed throughout the flow cycle. The strongest peaks occur just after flow reversal associated with vortex ejection, with smaller peaks occurring later in the flow cycle, which some attribute to sediment grains being advected by vortices shed from neighboring ripples (Nakato et al. 1977; Nielsen 1992; van der Werf et al. 2007). Although sediment transport in oscillatory flows has previously been correlated with the Shields parameter (Shields 1936; Grant and Madsen 1979; van der Werf 2006), laboratory experiments demonstrate that ripples may persist even when the critical value of the Shields parameter for sheet flow is exceeded (Nichols and Foster 2009). In these experiments, the Sleath parameter was found to have a significant effect on ripple morphology. The ripple profile flattened only after the Sleath parameter exceeded the critical value for plug flow of 0.1.
Numerical models have also been used to assess the hydrodynamics and sediment transport over ripples under oscillatory flows. The range of models employed include direct numerical simulations (Scandura et al. 2000; Shimizu et al. 2001; Bhaganagar and Hsu 2009; Finn et al. 2016), large-eddy simulations (Chang and Scotti 2004; Zedler and Street 2006), turbulence closure models solving the Reynolds-averaged Navier–Stokes equations (Andersen 1999; Fredsøe et al. 1999; Eidsvik 2004), and discrete vortex models (Hansen et al. 1994; Malarkey and Davies 2002). A mixture theory model has also been developed that treats the fluid and sediment as a continuum of fluid having varying density and viscosity (Penko et al. 2013). The numerical results have also shown that wave energy dissipation is enhanced over ripples (Barr et al. 2004) and that the hydrodynamics over ripples are influenced by vortex formation and ejection (Zedler and Street 2006; Penko et al. 2013). While these models provide high-resolution information on small-scale bottom boundary layer processes, more detailed experiments are needed to validate these numerical models to improve predictive capability of sediment transport over ripples.
Particle image velocimetry (PIV) has provided detailed measurements of hydrodynamics in the laboratory for the past several decades (Adrian 1991). PIV has been used to measure fluid velocities and fluid–sediment interactions within the bottom boundary layer over flat beds (Nezu and Azuma 2004; Campbell et al. 2005; Frank et al. 2015a). PIV has also been used to measure fluid velocities and bedform morphology over rippled beds (García et al. 2002; O’Donoghue et al. 2006; van der Werf 2006; Nichols and Foster 2007; Rodríguez-Abudo et al. 2013). However, there are some limitations to standard PIV techniques that may inhibit measurements at the fluid–sediment interface. Intense laser light reflections from the high sediment concentrations at the bed reduce the contrast of the near-bed fluid tracers from the background, making the tracers indistinguishable. The intense reflections also prohibit the calculation of fluid velocities near the fluid–sediment interface, specifically during sediment mobilization and entrainment into the water column, which are important stages for sediment transport analyses (Nichols and Foster 2007). As a possible solution, the laser may be operated in low-power mode to reduce the strong reflections at the bed and improve velocity resolution (Rodríguez-Abudo et al. 2013). However, while reducing the intensity of the laser light may improve the quality of the image at the fluid–sediment interface, it potentially decreases the image quality in the water column and therefore could increase flow measurement uncertainty. Additionally, when suspended sediment grains are used as the fluid seeding agent, the observed fluid velocities may have an additional vertical component because of the settling of the sediment grains that must be accounted for (van der Werf 2006; van der Werf et al. 2008). Low sampling frequency also may limit the ability for PIV to fully resolve turbulence parameters or to perform particle tracking. Last, phase separation is challenging when using standard PIV techniques. Separating the fluid from the sediment phase enables Lagrangian tracking of individual sediment grains from the onset of motion to entrainment to better relate mobility with near-bed hydrodynamics (Radice et al. 2010; Frank et al. 2015a).
Phase separation from PIV measurements of two-phase flows has been attempted using several methods such as observed differences in intensity and size of scattered light by each phase (Khalitov and Longmire 2002; Zhang et al. 2008), differences in the color of the scattered light from suspended particles and fluorescent seeding tracers (Hassan et al. 1993; Deen et al. 2000; García et al. 2002; Yang et al. 2011), differences in the size of particles (Gui and Merzkirch 1996; Nezu and Azuma 2004; Noguchi et al. 2008), using a two-dimensional median filter (Kiger and Pan 2000), using spatial frequency and correlation peak properties (Delnoij et al. 2000), and through filtering and subtraction between consecutive frames (Radice et al. 2010). These efforts included the measurement of bubbles through a liquid, solid particles in air, and solid particles in liquids.
Within the field of sediment transport, phase separation of sediment grains from fluid tracers has been performed in unidirectional flows based on the difference in size between the grains and tracers (Nezu and Azuma 2004; Muste et al. 2005; Noguchi et al. 2008), the intensity and size of scattered light by each phase (Tapia et al. 2006), and through filtering and subtraction between consecutive frames (Radice et al. 2010). In oscillatory flows, mobile sediment grains near the bed have been distinguished from fluid tracers through filtering and subtraction between consecutive frames (Frank et al. 2015a) and by seeding the fluid with fluorescent tracers (García et al. 2002; Admiraal et al. 2006). Filter techniques may have high uncertainty particularly for images with low contrast and are limited by the sampling frequency. Seeding the fluid with fluorescent tracers enables a clearer distinction between the fluid tracers and sediment grains, significantly reducing the effects of strong reflections near the bed, while improving data quality throughout the water column (Pedocchi et al. 2008).
Particle tracking velocimetry (PTV) determines the location of identified dispersed particles in subsequent time steps in order to estimate the particle velocities and trajectories. The velocities and trajectories of individual sediment grains within the bottom boundary layer could provide additional information regarding the fluid forces responsible for initiating sediment motion, as well as how and where the sediment grains are transported, leading to more accurate predictions of larger-scale sediment transport phenomena such as beach erosion. Several particle tracking algorithms have been proposed to track dispersed particles in a variety of multiphase flows including the nearest-neighbor approach (Papantoniou and Dracos 1989; Malik et al. 1993; Crocker and Grier 1996), two-frame match probability (Baek and Lee 1996), three-frame minimum acceleration (Ouellette et al. 2006), four-frame minimum change in acceleration (Malik et al. 1993; Dracos 1996), and the modified four-frame minimum acceleration predictive tracking algorithm (Ouellette et al. 2006). Additional particle tracking methods have been developed based on discrete relaxation (Ohmi and Li 2000), Kalman filters (Takehara et al. 2000), velocity gradient tensors (Ishikawa et al. 2000), correlations (Saga et al. 2001), hybrid adaptive schemes (Kim and Lee 2002), polynomial regressions (Biwole et al. 2009), probability hypothesis density filters (Wood et al. 2012), and more recent shake-the-box routines (Schanz et al. 2016). Though not the most sophisticated algorithm, the nearest-neighbor technique is applicable to a large range of flows including oscillatory and large-scale flows and is computationally inexpensive (Li et al. 2013). However, connecting particle locations to long trajectories may not be possible if the particles are identical, if particle movements cannot be predicted with a model, or if the particle displacement between images is large relative to the interparticle spacing (Papantoniou and Dracos 1989; Crocker and Grier 1996). A detailed description of several PTV algorithms was provided by Dracos (1996) and a brief updated overview reported by Cierpka et al. (2013).
Many of these existing particle tracking algorithms have been successfully applied to unidirectional flows and diffusive flows with no imposed free stream. Tracking sediment grains in oscillatory flow presents an additional challenge as sediment grains change direction with flow reversal. Moreover, the phase lead of near-bed velocities ahead of the free stream causes the flow to reverse at different times throughout the water column, further complicating the assessment. Individual sand grains have been tracked over a fixed rippled bed (Chu et al. 2012), and sand grains have been distinguished from fluorescent tracers over a moveable rippled bed (García et al. 2002); however, Lagrangian tracking of individual sediment grains over a moveable rippled bed has not been previously performed.
High-resolution observations of fluid–sediment interactions in oscillatory flows are needed to further develop theoretical formulations and validate numerical models to better predict the morphodynamics within the bottom boundary layer. Presented here are simultaneous laboratory measurements of fluid and sediment grain velocities in oscillatory flows with PIV and PTV, respectively. Fluorescent seeding tracers were used to distinguish the fluid tracers from the sediment grains. Independent measurements were made of the fluid and sediment grain velocities during grain mobilization and transport over moveable rippled sediment beds. PTV was used to determine the velocities and trajectories of sediment grains. The sediment in the water column detected by the PTV algorithm was then used to estimate the suspended sediment concentration and bulk settling velocity. The estimated suspended sediment concentration and settling velocities were then compared with several semiempirical formulations.
2. Experimental setup
The experiment was conducted in the small-oscillatory flow tunnel located at the U.S. Naval Research Laboratory, Stennis Space Center, Mississippi. The flow tunnel is described in detail by Calantoni et al. (2013). The test section is 2 m long with a flow cross section of 0.25 m × 0.25 m above a 0.30-m-deep sediment well. Oscillatory flow is generated with a piston and flywheel assembly. The facility is capable of generating symmetric or asymmetric oscillatory flows with periods T ranging from 2.3 to 10 s and maximum orbital velocity amplitudes ranging from 0.05 to 0.6 m s−1. The sediment well in the test section of the flow tunnel was filled with well-sorted, coarse quartz sand with a median grain diameter of and density of . The bed was initially manually planed. Oscillatory flow with a period of 2.35 s and a maximum orbital velocity amplitude of 0.26 m s−1 was then driven over the planed bed to generate ripples. The ripples reached equilibrium after approximately 4 h when they were observed to be two-dimensional with equal ripple wavelengths and heights in the test section. At this stage, measurements of fluid and sediment grain velocities were made. Subsequently, the flow was reduced to a maximum orbital velocity amplitude of 0.19 m s−1 and period of 3.02 s. The reduced flow resulted in smaller sediment grain displacements between time steps, thereby facilitating more accurate determination of the sediment grain trajectories throughout the flow cycle. The flow was reduced for approximately 1 min to collect one dataset for particle tracking analysis and was increased immediately afterward. Therefore, the ripples did not have time to adjust to the slower flow. Only the faster flow that was in equilibrium with the ripples was used to estimate the suspended sediment concentration and compared with the theoretical formulations. The water temperature in the flow tunnel was observed to be between 20° and 21°C.
The PIV system includes three high-speed, 1-megapixel cameras and a 100 mJ dual-cavity neodymium-doped yttrium–aluminum–garnet (Nd:YAG) laser. The flow was seeded with neutrally buoyant hollow glass spheres coated with Rhodamine B having a density of 1050 kg m−3 and median diameter of 10 µm. The fluorescent tracers absorbed the 532-nm laser light and reemitted at a higher wavelength of 584 nm. Optical long-pass filters with a cut-on wavelength of 550 nm were installed on cameras A and C, set up to measure stereo PIV of the two-dimensional, three-component fluid velocities (Fig. 1). The third high-speed camera (B in Fig. 1) was equipped with a neutral-density filter to reduce the intensity of the laser light observed at the sensor. The third camera was dominated by the higher-intensity light scattered by the sediment, not the fluid tracers, allowing for sediment grain tracking. Overlapping, simultaneously recorded images from the three cameras provided independent sediment grain and fluid velocities in the same field of view at high temporal (100 Hz) and spatial (~0.8 mm) resolution. Image pairs were collected at 100 Hz with a laser pulse separation of 2000 μs between the paired images over an 11 cm × 11 cm field of view. The laser light sheet was approximately 2 mm thick in the cross-tunnel direction. Datasets were collected in bursts of 27.28 s limited by the internal storage capacity of the cameras.
The LaVision flow processing software DaVis was used to calibrate the images, improve image contrast, normalize the light intensity, and calculate the three components of fluid velocity from the images captured by cameras A and C (Fig. 1). The PIV images were calibrated with a two-level 10 cm × 10 cm black calibration plate with white dots spaced 1 cm apart. The lid of the tunnel was removed, and the calibration plate was hung in the water in the intended plane of the laser sheet. The three high-speed cameras captured images that were reconstructed based on the preprogrammed dimensions of the calibration plate in DaVis. The images were preprocessed to improve image contrast between the fluorescent tracer particles and the background by subtracting a sliding minimum intensity from each image. The image intensity was then normalized with a sliding minimum–maximum filter to homogenize the intensities of the fluorescent tracer particles. First, a sliding minimum was subtracted from the image to remove the local background intensity level. Next, the local maximum intensity was determined over the specified window, as well as a global maximum intensity over 10 times the window size. The image was then multiplied by the ratio of the global maximum to local maximum to normalize the intensity of the light reemitted by the fluorescent tracers (LaVision 2014). Fluid velocities were determined by stereo cross correlation of the PIV image pairs using a round interrogation window with an initial diameter of 64 pixels having 50% overlap by diameter. Multiple passes were performed with a decreasing interrogation window, resulting in a final window diameter of 16 pixels. Spurious velocity vectors were removed with a median filter and replaced with the average of the nearest neighbors.
PTV was performed by first dynamically masking the rippled bed. Next, the images were binarized to distinguish between the light scattered by the sediment grains and the surrounding background. Following binarization of the images, a particle tracking algorithm (Crocker and Grier 1996) was implemented (Blair and Dufresne 2016) to bandpass filter the images, detect individual sediment grains by locating contiguous regions of high intensity, and analyze grain locations between image pairs of subsequent time steps to determine the new location. The time-resolved grain locations were then linked to determine the trajectories of the identified grains. The sediment grain velocities were calculated by dividing the change in position by the time step between the paired images. The PTV algorithm of Crocker and Grier (1996) was chosen because it was easily adaptable to oscillatory flows, could handle tracking in any direction, and did not require a constant time step between images.
The sediment bed elevation and ripple profiles at the centimeter–meter scale were measured with the Bed Laser Surface Tracking (BLAST) system. The BLAST system consists of two 520 nm, continuous wave (CW), fan beam lasers that project a ~1-m laser line on the sediment bed in the along-tunnel direction. Two digital single-lens reflex (DSLR) cameras captured images and video of each of the laser lines projected on the bed. One CW laser and camera with a 10-mm fish-eye lens were translated across the width of the small-oscillatory flow tunnel (S-OFT) with a high-precision stepping motor in 0.5-mm increments, while the camera captured an image of the laser line at every step. Each image size is 5184 × 3456 pixels, resulting in a 2–6 pixels per millimeter resolution. The sediment bed elevation was scanned before and after each experiment. The second DSLR camera captured high-definition video (1080 pixels) at 30 frames per second of sediment bed elevation profiles in the along-tunnel direction at the center of the test section over several flow cycles. The BLAST measurements were used to determine the ripple wavelength and ripple height because the ripple wavelength was approximately 0.02 m longer than the PIV field of view. Additionally, the BLAST measurements were used to determine the ripple profile and assess bedform evolution as they were less sensitive to suspended sediment and provided measurements over longer time segments than the PIV.
Ripples were generated from an initially flat sediment bed with regular oscillatory flow having period and maximum orbital velocity amplitude . The flow was slightly skewed and asymmetric in the negative direction, having values of skewness and asymmetry of −0.07 and −0.02, respectively (Elgar et al. 1988). The flow statistics were calculated from the free-stream velocity time series measured at and above the bed, more than three ripple heights away from the ripple trough to avoid wake effects of the bedform. The measured period was calculated as the average time between negative and positive zero crossings, and the maximum orbital velocity amplitude was calculated as the standard deviation of the velocity time series multiplied by . The resulting ripples had wavelengths of 0.13 m and heights of 0.025 m as measured by the BLAST system.
a. PIV and PTV
The near-bed velocity field above the ripple is very important for sediment transport, as this is the region where fluid forces directly impact the sediment bed to initiate sediment motion. Time series of ensemble-averaged free-stream horizontal velocity and snapshots of the velocity field across the ripple are shown for flow having and (Fig. 2). The horizontal free-stream velocity was ensemble averaged over 11 flow cycles, during which the ripple migrated approximately 1.2 mm. The migration rate was deemed too small to affect the ensemble average. The PIV images of the ripple displayed (Figs. 2b–e) correspond to the phases indicated (Fig. 2a). The sediment grain and fluid velocities are indicated by cyan and white vectors, respectively. The maximum orbital velocity amplitude is indicated as a scale vector in magenta in the top left of each image in the direction of the free streamflow. The fluid velocity field was estimated with a spatial resolution of ~0.8 mm, for a total of 147 profiles across the field of view. However, for better visualization only every third vector is displayed, resulting in 39 profiles displayed across the ripple. The vectors were also down sampled in the vertical to display only every third vector.
Shortly after the negative–positive flow reversal, the flow was accelerating to the right (Fig. 2b). The near-bed flow velocity above the ripple crest was approximately 0.10 m s−1 larger than the free-stream velocity because of flow constriction over the ripple crest. Sediment grains mobilized during flow reversal were advected over the ripple crest, suspended into the water column, and deposited on the ripple slope (Fig. 2b). Subsequently, additional sediment grains were suspended during maximum positive flow (Fig. 2c) and advected across the ripple crest. Flow separation was observed along the ripple slope and in the trough. Velocity vectors indicated the formation of a vortex in the flow separation region (Figs. 2c,e). In Fig. 2d, the flow had just reversed from positive to negative (right to left). Sediment grains mobilized during flow reversal were being advected to the left over the ripple crest and deposited on the opposite slope. In this trial, more sediment grains were suspended during flow reversal than during peak velocity (note the greater number of suspended sediment grains in Figs. 2b and 2d than in Figs. 2c and 2e).
The phase lead between near-bed and free-stream horizontal velocities was determined by assessing the relative timing between flow reversals (van der Werf et al. 2007). The near-bed velocities were taken at a height of 0.028 m. A maximum phase lead of approximately 50° was observed along the ripple slope where the vortex was generated during flow reversal (Fig. 3). A phase lead of 50° is significantly larger than the 10°–21° expected for fully turbulent flow over a rough flat bed (O’Donoghue and Wright 2004; Foster et al. 2006; Terrile et al. 2006; Frank et al. 2015b). Although the observed phase lead of 50° is smaller than the phase lead of 108° (van der Werf et al. 2007) observed for more energetic flows with larger periods and velocities over ripples with finer sediments, the location of the maximum phase lead along the ripple slope was in good agreement with previous observations. The maximum phase lead was located on the left side of the ripple for a wave phase of 0° (Fig. 3b) and the right side of the ripple for a wave phase of 180° (Fig. 3c), suggesting that the large phase lead of the near-bed fluid velocities ahead of the free stream was predominantly due to vortex generation, which develops on opposite sides of the ripple depending on the flow direction, consistent with previous observations (van der Werf et al. 2007).
To further analyze the suspended sediment transport, the percentage of suspended sediment grains moving with a particular grain velocity was assessed (Fig. 4). The sediment grain velocities were calculated with the grain locations identified by the PTV algorithm between individual images of each pair having a time step of 2000 μs. The horizontal and vertical sediment grain velocities were then ensemble averaged over 11 wave cycles. The ensemble-averaged horizontal sediment grain velocities had a maximum value of 0.35 m s−1 and a phase lead of between 40° and 50° over the free stream at flow reversals, similar to the near-bed horizontal fluid velocities. During the accelerating phase of the positive flow (0°–40°), more than 80% of the suspended grains had positive horizontal and vertical velocities, indicating that they had been picked up and were being advected with the flow upward over the ripple to the right. Starting at a phase of 45°, negative vertical grain velocities suggest that the grains were beginning to settle to the bed. The flow then reversed and analogous observations were made in the opposite direction.
b. Suspended sediment concentration
The spatial and temporal variability of the volumetric suspended sediment concentration were analyzed over the ripple throughout the flow cycle. Additionally, the Shields parameter, viscous shear stress, and vorticity were calculated to determine the fluid mechanism suspending the sediment grains at flow reversal and peak flow. The quantity of suspended sediment grains detected with PTV was used to estimate the instantaneous volumetric suspended sediment concentration C above the ripple. The volumetric suspended sediment concentration was calculated in volume bins with areas of 3.3 mm × 3.3 mm over the two-dimensional PIV field of view and a depth of 2 mm in the cross-tunnel direction y based on the laser sheet thickness. The ensemble-averaged volumetric suspended sediment concentration in each bin and at each time step was defined as the volume of sediment per unit volume of water,
where is the number of sediment grains detected in each bin and at each time step, is the volume of one sediment grain, and is the bin volume (21.78 mm3). The calculations were made with bins that overlapped by 50% to increase spatial resolution. The bin volume was chosen to minimize errors in the estimated volumetric suspended sediment concentration (Simeonov et al. 2015).
The ensemble-averaged free-stream horizontal velocity and the ensemble-averaged volumetric suspended sediment concentration fields corresponding to eight phases of the flow are illustrated in Fig. 5. The free-stream velocities and suspended sediment concentrations were ensemble averaged over 11 flow cycles. All suspended sediment was located within one ripple height of the crest. In Fig. 5b, the volumetric suspended sediment concentration over the ripple was minimized when the flow reversed from negative to positive. The volumetric suspended sediment concentration was at a maximum in Fig. 5c due to grains suspended shortly after flow reversal, illustrating a phase lag between the maximum volumetric suspended sediment concentration and free-streamflow reversal. During the maximum positive free streamflow (Fig. 5d), additional sediment grains were suspended and subsequently deposited on the ripple slope during the decelerating positive flow (Fig. 5e). The flow reversed (Fig. 5f) and similar observations were made in the opposite direction. The time-averaged volumetric suspended sediment concentration field over all 11 flow cycles was fairly symmetric about the ripple crest (Fig. 6). Additionally, the suspended sediment was very localized near the ripple crest with very few grains detected above the ripple trough.
To better understand the temporal variability of the sediment dynamics, the horizontal free-stream velocity, Shields parameter, volumetric suspended sediment transport rate, and volumetric suspended sediment concentration ensemble averaged over 11 flow cycles are plotted in Fig. 7. The Shields parameter was calculated using the ensemble-averaged horizontal free-stream velocity (Ribberink 1998),
where is the wave friction factor, is the horizontal free-stream velocity at time t, s is the specific gravity of quartz sand, g is the gravitational acceleration, and is the median grain size diameter. The wave friction factor was calculated as (Swart 1974),
where is the Nikuradse roughness , and A is the orbital excursion amplitude .
The suspended sediment transport rate was defined as the volume flux of sediment per unit width of bed per unit time (e.g., Drake and Calantoni 2001). The total horizontal and vertical volumetric suspended sediment transport rate throughout the flow cycle were calculated by
respectively, where Abed is the laser illuminated, planar projection of the ripple (=0.0002 m2); i and j are counting variables; Nx is the total number of bins in the horizontal dimension across the field of view; Nz is the total number of bins in the vertical dimension above the bed surface; u is the ensemble-averaged horizontal fluid velocity; and w is the ensemble-averaged vertical fluid velocity. The factor of 1/2 compensates for the 50% overlap so that sediment grains were not added twice. The volumetric suspended sediment concentration was calculated from the number of sediment grains in the same volume bins as previously described. The vertical sediment transport was very small relative to the horizontal sediment transport (note the dotted line in Fig. 7c). Pick up of the sediment grains was not accounted for because the near-bed region (within one grain diameter) was masked out in the determination of the suspended sediment concentration as the particle tracking algorithm had difficulty identifying and tracking individual grains in this near-bed region. The total suspended sediment concentration throughout the flow cycle (Fig. 7d) was calculated in a similar manner with
Four peaks in the horizontal suspended sediment transport and volumetric suspended sediment concentration were observed throughout the flow cycle (Fig. 7c and Fig. 7d, respectively). The two largest peaks occurred at flow phases of approximately 34° and 220° due to grains mobilized during flow reversal subsequently suspended into the water column and advected over the ripple crest. The two largest peaks in suspended sediment transport were not symmetric for each half of the flow cycle. The peak in the negative direction was larger, possibly because of the slight skewness and asymmetry of the flow in the negative direction. The two smaller peaks, observed at 120° and 282°, occurred during maximum free-streamflow velocities and were similar in magnitude for each half of the flow cycle.
As would be expected for the vortex ripple regime, the Shields parameter exceeded the critical threshold for sediment motion during maximum flow velocities, indicating that the shear stress was strong enough to mobilize the sediment during maximum flows. The two smaller peaks in suspended sediment concentration occurring at 120° and 282° were likely mobilized by the maximum shear stresses in each direction. However, the two largest peaks in suspended sediment transport and concentration indicate that more sediment grains were mobilized during flow reversals, when the Shields parameter was well below the critical threshold, which suggests that the shear stress was not responsible for mobilizing the sediment at flow reversal. The near-bed viscous shear stress (, where μ is the dynamic viscosity of water) across the ripple also confirmed that the critical threshold of 0.05 was only exceeded at phases of 120° and 282° during peak flow (Figs. 8c,e) and not during the other two peak-concentration events (Figs. 8b,d).
The vorticity was calculated with the method of Nichols and Foster (2007). Regions of strong vorticity were observed along the ripple slope at phases corresponding to the two largest peaks in suspended sediment concentration (Figs. 9b,d). Sediment grains were first mobilized as bedload during vortex generation along the ripple slope just prior to flow reversal. The sediment grains were then transported as suspended load over the ripple crest ahead of the vortex, which was advected when the flow velocities increased. The volumetric suspended sediment transport peaks at phases 34° and 220° indicate that more sediment grains were suspended by vortex shedding following flow reversal than by the shear stress during maximum flow velocities. Similar observations have been made in large-scale wave flume experiments (Nichols and Foster 2009).
c. Sediment grain trajectories
The trajectories of individual sediment grains were determined through Lagrangian tracking of identified sediment grains throughout the flow cycle. Tracking grains from the onset of motion to entrainment into the water column better relates sediment mobility to hydrodynamics within the bottom boundary layer such that the fluid forces mobilizing the grains can be determined and the dominant mode of sediment transport assessed. The sampling frequency of 100 Hz was not fast enough to resolve long trajectories throughout the cycle for the flow with T = 2.35 s and Uo = 0.26 m s−1. When grain displacement between time steps was more than half the intergranular spacing of the suspended grains, grain positions became confused between images and the algorithm could not accurately track the grains (Crocker and Grier 1996). A higher sampling frequency may have improved the tracking efficiency at high velocities. The oscillatory flow presented additional problems when the flow accelerated, since more grains were suspended and their velocities increased; therefore, neither the grain displacement nor the intergranular spacing of the suspended grains was constant throughout the flow cycle.
A slower oscillatory flow was generated with T = 3.02 s and Uo = 0.19 m s−1 in order to resolve the grain trajectories with PTV. Time series of ensemble-averaged free-stream horizontal velocity and snapshots of the velocity field from the ripple crest to trough are shown in Fig. 10. Sediment grain trajectories are shown during short phase segments (Fig. 10a) with each grain trajectory represented by a different line color. The fluid velocities are indicated by white vectors. The maximum orbital velocity amplitude is indicated as a scale vector in magenta in the top left of each image. The fluid velocity field was measured with a spatial resolution of ~0.8 mm, for a total of 147 profiles across the field of view. However, for better visualization only 25 profiles are displayed from ripple crest to trough.
The flow reversed from negative to positive and was accelerating to the right (Fig. 10b). The flow above the crest was greater than that above the trough due to the flow constriction by the ripple crest. The suspended sediment grains that were mobilized during flow reversal were advected across the ripple crest and deposited on the slope of the ripple. During maximum positive flow (Fig. 10c), a second set of sediment grains were mobilized and advected across the ripple crest as well. Flow separation from the bed was observed along the lower part of the slope and in the trough. The flow reversed from positive to negative and similar observations were made in the opposite direction (Figs. 10d,e). Shorter trajectories were observed within a couple grain diameters of the bed surface, possibly representing sediment grains that were being transported in short bursts through saltation. Longer sediment grain trajectories were observed farther up in the water column, representing grains that were being advected as suspended load.
a. Sediment settling velocity
The settling velocity is an important parameter for many sediment transport calculations, including estimating the suspended sediment concentration and solving the advection–diffusion equation. One commonly used settling-velocity formulation is that of Gibbs et al. (1971),
where is the terminal settling velocity, and ν is the kinematic viscosity of water. All dimensional quantities must be in the centimeter–gram–second (cgs) system in order for the constants to be valid. In addition, note that the equation has been recast in terms of the sediment grain diameter in accordance with Nielsen (1992). Another settling-velocity formulation was developed by Soulsby (1997),
where is the nondimensionalized grain diameter given by
While both formulations are dependent on the kinematic viscosity of the fluid, the median grain diameter, and sediment density, the formulation in (7) is slightly more sensitive to the kinematic viscosity of the fluid.
In fluids with high concentrations of suspended sediment grains, the settling velocity may be reduced or hindered as a result of grain–grain interactions. A correction factor to the terminal settling velocity to account for hindered settling was calculated with
where is the correction factor defined as the ratio of the terminal settling velocity to the settling velocity of the sediment grain in suspension, is given by (6), and the exponent was calculated based on the grain Reynolds number for these experimental conditions (Richardson and Zaki 1954; Penko et al. 2013). A correction factor of was obtained for the experimental data presented in this article, indicating that the suspended sediment concentrations were several orders of magnitude too small to significantly hinder the settling of the coarse sand grains.
The bulk settling velocity was estimated from the suspended sediment concentration and compared with the commonly used formulations in (7) and (8). The ensemble-averaged free-stream horizontal velocity and vertical profiles of ensemble-averaged suspended sediment concentration spatially averaged across the ripple are illustrated in Fig. 11. The height of the ripple crest is indicated by the black dashed line, and best-fit trend lines minimizing the residuals through the maximum suspended sediment concentration are indicated by white solid lines. Using the average slope of the two best-fit trend lines, the settling velocity of the suspended sediment grains was estimated as 0.099 m s−1. The settling velocity calculated with (7) (Gibbs et al. 1971) and (8) (Soulsby 1997) were 0.109 and 0.095 m s−1, representing an agreement of 90% and 95%, respectively. Therefore, the formulations agreed well with the settling velocity estimated from the detected suspended sediment grains, suggesting high confidence in using these formulations in model predictions of sediment transport under oscillatory flows.
b. Suspended sediment concentration
Suspended sediment transport is a critical mechanism for shaping the bathymetry in nearshore regions. Four peaks in suspended sediment concentration were observed throughout the flow cycle, with the two largest peaks occurring shortly after flow reversal because of grains mobilized by the vortex, in agreement with previous observations (Nakato et al. 1977; Nielsen 1992; van der Werf et al. 2007). Two smaller peaks in suspended sediment concentration were observed during maximum free-streamflow velocities because of grains mobilized by the shear stress. Previous observations with finer sand grains also had peaks in suspended sediment concentration during phases of maximum free-streamflow velocities, but some researchers attributed these peaks to sediment-laden vortices being advected from adjacent ripples (Nakato et al. 1977; Nielsen 1992; van der Werf et al. 2007). The median grain diameter of the suspended sediment used in the previous studies were smaller than those in this experiment, ranging from 0.14 to 0.33 mm (Nakato et al. 1977; van der Werf et al. 2007). The corresponding terminal settling velocities ranged from 0.014 to 0.047 m s−1, as calculated with the formulation of Gibbs et al. (1971). The sediment in this experiment was larger, with a median grain diameter of 0.7 mm and settled much faster with a settling velocity of approximately 0.10 m s−1. For the flows presented here, no coarse sediment grains were advected from the neighboring ripples because they typically settled on the ripple slopes and trough. Therefore, the smaller peaks in suspended sediment concentration during maximum flows were attributed to sediment grains that were mobilized by the shear stresses.
Typically, the suspended sediment transport rate is estimated from the vertical profile of the suspended sediment concentration and the fluid velocity. Therefore, many formulations have been proposed for the vertical profile of the suspended sediment concentration. The following formulation for the suspended sediment concentration profile of coarse sand grains has been proposed (Nielsen 1986, 1992):
where is the time-averaged vertical profile of the volumetric suspended sediment concentration. The reference concentration and the length scale of the suspended sediment distribution were calculated as
respectively, where is the modified effective Shields parameter over ripples, and η is the ripple height. The modified effective Shields parameter over ripples was calculated as follows:
The formulation of Nielsen (1986, 1992) only estimates the suspended sediment concentration profile above the ripple crest. However, Sleath (1984) proposed the following formulation, which accounts for the suspended sediment along the slope of the ripple near the crest:
where δ is a constant (here assumed =0) and σ is a constant (). The reference concentration was determined at the ripple crest and was calculated as (Nielsen 1979),
The critical Shields parameter was assumed to be 0.05 for quartz sand.
The laboratory observations were compared with the semiempirical formulations of suspended sediment concentration proposed by Nielsen (1992) and Sleath (1984) (Fig. 12). The suspended sediment concentration at the crest was calculated as the average over x ranging from −0.0127 to −0.0062 m, and the values at the slope were calculated to the left of the crest for x ranging from −0.029 to −0.0144 m. Previous research efforts have shown that the shape of the suspended sediment concentration profile above the ripple crest is concave for coarse sand grains but convex for finer sand grains (Nielsen 1992; Absi 2010). The general concave shape of the vertical profiles of suspended sediment concentration given by the semiempirical formulations agreed well with the laboratory observations of coarse sand grains. The root-mean-square deviations of the laboratory data from the model predictions by Nielsen (1992) and Sleath (1984) were 4.8 × 10−5 and 3.2 × 10−5 m3 m−3, respectively. The Sleath (1984) predictions of mean suspended sediment concentration above the ripple crest height were very similar to those below the ripple crest height. However, the laboratory measurements of mean suspended sediment concentration above and below the ripple crest were not very similar. The shape and magnitude of the profile predicted by Sleath (1984) agreed very well with the observations below the ripple crest height, having a root-mean-square deviation of 8.6 × 10−6 m3 m−3 in this region. Above the ripple crest, the laboratory measurements were larger than Sleath (1984) predicted, indicating that some physical process, possibly vortex entrainment, may not be accounted for in the Sleath (1984) formulation. Nielsen (1992) made no prediction of the profile below the ripple crest. The data were evenly scattered about both predictions just above the ripple crest. However, the formulation of Sleath (1984) was in better agreement with the data in the upper water column above z = 0.035 m, where it converged to zero more quickly than the formulation of Nielsen (1992). Davies and Thorne (2016) evaluated several decay-length-scale formulations for a range of sediment grain diameters and showed that convective effects may be more significant for larger sediment sizes. Perhaps this may account for some of the scatter in the present results.
The first set of independent simultaneous measurements of fluid and individual sediment grain velocities were made in the laboratory with the implementation of combined particle image and tracking velocimetry for sediment transport under oscillatory flows over movable rippled beds. Measured sediment velocities were of similar magnitude and phase to the near-bed fluid velocities when the grains were being advected with the flow. Sediment grain trajectories were also determined throughout the flow cycle. For the hydrodynamic conditions presented over the ripples, vortex generation near flow reversal mobilized the sediment grains in the form of bedload along the ripple slopes. Vortex ejection from the ripple crest during the accelerating phase of the flow cycle produced significantly larger quantities of suspended sediment transport than the shear stresses during phases of maximum velocities. Some previous studies with finer sediment grains attributed suspended sediment during peak flow to sediment-laden vortices shed by adjacent ripples.
Calculations of the sediment settling velocity compared very well with the semiempirical formulations. The comparisons of the time-averaged vertical profiles of suspended sediment concentration revealed that the formulations of Nielsen (1992) and Sleath (1984) captured the general concave shape of the profile as expected for coarse sand grains. However, only the Sleath (1984) model accounted for the suspended sediment concentration below the ripple crest where it demonstrated good agreement with the laboratory data in this region.
Donya Frank-Gilchrist was supported as a postdoctoral fellow through the National Research Council Research Associateship Program at the U.S. Naval Research Laboratory. Allison Penko and Joseph Calantoni were supported under base funding to the U.S. Naval Research Laboratory from the Office of Naval Research. The authors would like to acknowledge NRL staff members Edward Braithwaite and Abigail Hode for their assistance with instrument set up and data collection during the experiment.