Experimental study of bore-driven swash–swash interactions on an impermeable rough slope

Coastal Engineering 108 (2016) 10–24

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Experimental study of bore-driven swash–swash interactions on an impermeable rough slope Bo-Tao Chen a, Gustaaf Adriaan Kikkert a,⁎, Dubravka Pokrajac b, Han-Jing Dai a a b

Department of Civil and Environmental Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China School of Engineering, King's College, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom

a r t i c l e

i n f o

Article history: Received 22 April 2015 Received in revised form 26 October 2015 Accepted 31 October 2015 Available online xxxx Keywords: Bores Experimental measurements Hydrodynamics Swash–swash interactions Turbulence

a b s t r a c t Experimental measurements are obtained to investigate the detailed hydrodynamics of wave–wave interactions in the swash-zone. Two bores are generated using a double dam-break mechanism and interact on a 1:10 impermeable rough slope. Measurements of the hydrodynamics are obtained via acoustic displacement sensors and a combined particle image velocimetry and laser induced fluorescent system. Two types of interactions are investigated: wave capture, with the second bore reaching the first one during the uprush, and weak wave–backwash interactions when interaction happens during the backwash of the first wave. The relative strength of the bores at the initial shoreline and time between their arrivals determines the initial type of interaction, however the type as well as the intensity of the interaction may vary throughout the swash-zone for the same swash event. During wave capture and weak wave–backwash interactions, the fluid of the first bore is advected upwards, then mixed with the fluid of the second bore by intense shearing and highly turbulent vortices generated at the front of the second bore because of the relative velocity differences between the two bores. The details of the hydrodynamics during interaction confirm the potential of swash–swash interactions to suspend sediment and transport it shoreward or interrupt seaward sediment transport. © 2015 Elsevier B.V. All rights reserved.

1. Introduction The flow in the swash zone is characterised by a high uprush velocity, a direction-reversal and a high backwash velocity while it is also highly turbulent, unsteady and non-uniform. On a mobile bed, this results in large sediment transport fluxes and hence the swash zone plays an important role in the dynamics of the foreshore and the constant evolution of the beach profiles. Many studies have investigated the detailed hydrodynamics of the flow in the swash zone (e.g. Barnes et al., 2009; Cowen et al., 2003; Kikkert et al., 2012, 2013; O'Donoghue et al., 2010; Petti and Longo, 2001), however these have focused on a single incoming bore climbing an initially dry beach. In comparison, there are relatively few investigations that focus on the interaction by successive swash events, despite field observations suggesting that the impact of the interactions on the sediment transport may be considerable (e.g. Hughes and Moseley, 2007; Masselink et al., 2009). Swash–swash interaction occurs when the period of the incident wave is smaller than the period of the swash event. The downshift in frequency between the incident wave and the swash is caused by uprush– backwash interactions when bores arriving at the initial shoreline are strong, and by standing long-waves in the swash-zone when bores ⁎ Corresponding author. E-mail addresses: [email protected] (B.-T. Chen), [email protected] (G.A. Kikkert), [email protected] (D. Pokrajac), [email protected] (H.-J. Dai).

http://dx.doi.org/10.1016/j.coastaleng.2015.10.010 0378-3839/© 2015 Elsevier B.V. All rights reserved.

arriving are weak (Mase, 1995). The degree of swash–swash interaction depends on beach slope, incident wave period and its height (Brocchini and Baldock, 2008). Swash–swash interactions do not occur along the full extent of the swash zone. Hughes and Moseley (2007) defined the region where interactions occur as the outer swash-zone. In the inner swash-zone, which is further landward, no interactions occur and the flow is a pure swash motion. Mase and Iwagaki (1985) defined two different types of interactions. The first occurs during the uprush of the first bore and hence the second bore overtakes the first bore. This is referred to as wave–capture interaction. The second type occurs during the backwash of the first bore, either before or after the backwash flow has become supercritical and formed a backwash bore (Hibberd and Peregrine, 1979). Propagation of the second bore is suppressed by the first bore, and this is referred to as wave–backwash interaction. A further distinction was made by Hughes and Moseley (2007) and Cáceres and Alsina (2012). For a weak wave–backwash interaction, the receding backwash has a limited amount of energy and the overrun of the incoming uprush results in an onshore flow. For a strong wave–backwash interaction, the backwash is stronger than the incoming uprush, therefore the resulting flow is offshore directed. The importance of swash–swash interactions was recognised by Kemp (1975) who qualitatively related the interactions to the sediment transport by suggesting that swash collisions may indicate whether a beach erodes or accretes. Swash–swash interactions also transfer wave momentum to longshore flows (Brocchini, 1997; Brocchini and

B.-T. Chen et al. / Coastal Engineering 108 (2016) 10–24

Peregrine, 1996; Ryrie, 1983). Mase and Iwagaki (1985) found that the ratio of the number of run-up events to the number of incoming waves decreases with decreasing slope and increasing deep-water wave steepness and hence the number of interactions increased. Field measurements by Weir et al. (2006) found good correlation between the boundary of erosion and accretion in the swash zone and the landward limit of swash interactions. Field measurements by Holland and Puleo (2001) indicated that if no interactions occur, because the swash duration is shorter than the period of the incoming wave, entrained sediments are deposited outside of the swash zone therefore flattening the beach. However, this also increases the swash duration until eventually the swash duration is equal to the period of the incoming waves and the rate of change of the profile becomes zero. If interactions do occur, the sediments are deposited in the swash zone, steepening the beach and thereby decreasing the duration of the swash event. This is in agreement with the results from Alsina et al. (2012), who compared experimental data from a steep beach with those from an artificially flattened beach. On the flattened beach, the swash period increased and hence the number of interactions increased. As a result the strength of the backwash was reduced and this reduced the offshore sediment transport. Alsina and Cáceres (2011) and Cáceres and Alsina (2012) found that all swash–swash interaction types induce high concentrations of suspended sediments. The strong wave–backwash interaction induces the highest concentration which is generally directed offshore. Sediments suspended by weak wave–backwash interaction, which occurs most frequently, and wave–capture interaction are generally directed onshore. Erikson et al. (2005) included the effects of swash– swash interactions into their model that predicts shoreline motions and found that the predictions matched laboratory measurements obtained on gentle beaches significantly better than when the interactions were not included in the model, but on steep beaches there was little improvement. This matched results by Hughes and Baldock (2004) whose model did not take into account the effects of interactions, but predictions on steep beaches, where swash interactions did occur, nevertheless provided a reasonable match with field measurements. Detailed measurements of the fundamental kinematics of swash– swash interactions, that cause the suspension of sediments, are rare. Barnes and Baldock (2007) carried out laboratory experiments to obtain measurements of flow depth, velocity and bed shear stress of wave– backwash interaction, but used a simplified set-up generating a quasisteady hydraulic jump. Field studies (e.g. Masselink et al., 2009) and large scale laboratory studies (e.g. Alsina and Cáceres, 2011) that included measurements of flow depth and velocity of interacting swash

Kanya Support Framework

Actuator 2

Actuator 1

Gate 2

Gate 1

events focused on the overall swash processes that generate beach erosion or accretion, not on the specific kinematic details of the interactions. Numerical models based on the non-linear shallow water (NLSW) equations have been used to simulate the behaviour of incoming wave groups in the nearshore zone, and therefore include interactions in the swash-zone (e.g. Orszaghova et al., 2014; Watson and Peregrine, 1992; Watson et al., 1995). However the focus of these studies was predominantly the surf-zone. In addition, experimental data used to validate these model predictions was also obtained predominantly in the surfzone and hence the accuracy of the predictions in the swash-zone is still unknown. This knowledge gap and lack of data have motivated the present investigation which aims to increase our fundamental understanding of swash–swash interactions and to create a data set suitable for testing of numerical models of swash–swash interactions in the swash zone. A new series of experiments has been carried out in the laboratory on a rough impermeable slope. The bore-driven swash is generated through the use of a dam-break mechanism (e.g. Kikkert et al., 2012; O'Donoghue et al., 2010). A second reservoir is added to enable a second bore to be generated with a pre-determined time lag. A range of swash– swash interactions is generated by varying the relative water levels in the two reservoirs and the time lag between bores. Flow depth measurements from acoustic displacement sensors are used to study the general behaviour of swash–swash interactions. For two specific cases, a wave–capture and weak wave–backwash interaction, simultaneous measurements of the flow depth (using laser induced fluorescence) and velocity (using particle image velocimetry) are obtained. Analysis of the ensemble averaged quantities and turbulence is used to study the hydrodynamics of the swash–swash interactions in detail. 2. Methodology 2.1. Experimental set-up The experiments were carried out in the 12.5 m long, 0.45 m high and 0.30 m wide glass-sided Armfield SII flume located in the Water Resources Laboratory of the Hong Kong University of the Science and Technology. Swash–swash interactions of similar scale as found in the field were generated in the laboratory by a double dam-break system positioned at one end of the flume (Fig. 1). The system consisted of a reservoir, two gate mechanisms and an aluminium support framework. The reservoir, made from Perspex, had an internal width of 0.279 m,

1:10 Impermeable sand rough bed

700mm ADS 1

Reservoir 2

1.0m

800mm

ADS 2

800mm

ADS 3

ADS 4

800mm ADS 5

z

Reservoir 1

1.0m

11

x

3.0m

PIV/LIF 1

PIV/LIF 2

845mm

PIV/LIF 3

800mm

Fig. 1. Experimental set-up of double dam-break system and impermeable rough bed with locations of ADS and PIV/LIF measurements.

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B.-T. Chen et al. / Coastal Engineering 108 (2016) 10–24

height of 0.630 m and length of 2.443 m. The bottom of the reservoir was 10 mm above the bottom of the flume. At the exit of the reservoir, Perspex wedges were inserted in the flume to create a smooth transition for the flow from the reservoir into the flume. The gates were made from 8 mm thick aluminium and the side and bottom edges of the gates were wrapped in elastic rubber. Each of the gates was attached to an electronic actuator system (CMS linear motion system from Bosch Rexroth AG). The actuators were computer controlled through software DriveTop 16V14 which raised the gates to 0.566 m in precisely 0.322 s. The software controlling the gates was triggered externally using a BNC 575 pulse generator. The first pulse which opened the first gate was generated manually, while the second pulse which opened the second gate was generated automatically a predetermined time later. The downward force of the actuator and the elastic rubber of the gates created a perfect seal for the water in the reservoir. Kanya aluminium profiles were used to build a framework over the top of the flume and the actuators were attached to the framework. To dissipate any vibrations caused by the rapid opening of the gates, the framework was bolted to the floor and ceiling of the laboratory. The impermeable 1:10 slope began at a distance of 3.006 m from the 1st gate and 4.015 m from the second gate (Fig. 1). The first 267 mm of the slope consisted of an aluminium wedge to create a smooth transition for the flow onto the slope and help make the slope watertight. The rough bed was created by gluing sediments with d50 = 2.0 mm to marine plywood planks using silicon glue. The coarse sand particles were chosen as they are commonly found on medium sloped beaches in the field. At the three locations where measurements of velocity and depth were obtained, 200 mm long and 20 mm wide glass slots were inserted into the plywood and again partially covered with sediments so that a 5 mm wide gap remained. An aluminium framework with a slope of 1:10 was inserted into the flume and the plywood planks were bolted onto the framework to form the impermeable rough bed. The gaps between the planks and glass wall of the flume were filled with silicone glue and covered in sediments to make the bed watertight. The time t = 0 s corresponds to the opening of the first gate. The origin of the spatial coordinates is the location where the impermeable rough bed intersected the initial water level in front of gate 1 when its height was 35 mm. This was the case for all experiments with a water level of 350 mm in the first reservoir. When the water level in the first reservoir was 400 mm, the initial water level in front of gate 1 was 40 mm, but the location of the origin was not altered. The x-axis is parallel to the bed and positive shoreward, the z-axis is normal to the bed and positive upward and u and w are the velocity components in the x and z-directions respectively.

2.2. Experimental conditions Two series of experiments were carried out. A set of exploratory experiments was carried out in which a large number of different swash– swash interactions were generated in order to investigate the overall features of the flows. Two different reservoir water level pairs were used. The first pair, 35–50, had a water level of 35 cm relative to the bottom of the flume in the first reservoir and 50 cm in the second reservoir. The second pair, 40–40, had a water level of 40 cm in both reservoirs. For these two water level pairs, the time delay between the opening of gate one and gate two, td, was varied from 1.5 s to 6.5 s in 0.5 s intervals. Analysis of the results of the exploratory experiments showed that water levels of 35–50 and a gate opening time delay of 1.5 s generated a typical wave capture swash interaction and water levels of 40–40 and a time delay of 3.5 s generated a typical weak wave–backwash interaction. These two cases were selected for the investigation of the detailed hydrodynamics of the flow. 2.3. Measurements During the exploratory experiments, the flow depth was measured at five locations along the bed at x = 48 mm, 748 mm, 1548 mm, 2348 mm and 3148 mm (Fig. 1). The flow depths were measured using Microsonic acoustic displacement sensors (ADS, mic + 35/IU/ TC) with an application range from 65 mm to 600 mm, a nominal accuracy of 0.2 mm and recording frequency of 2 kHz. The ADS were positioned along the centreline of the flume (Fig. 2) and calibrated in the flume under dry conditions. The signals of the sensors were recorded using a National Instruments cDAQ-9178 and two analog input modules NI 9239. The process was controlled via computer software LabVIEW. The recording of the depth measurements was triggered by the pulse generator that controlled the opening of the first gate, therefore the measurements started at t = 0 s. Measurements were obtained for 12 s and for each set of initial conditions the experiments were repeated five times to obtain the representative ensemble-averaged depths. Visual observations of the maximum positions of the wet–dry boundary on the beach and a steel ruler positioned along the slope yielded measurements of the maximum run-up. For the investigation of the detailed hydrodynamics of the flow, simultaneous measurements of the velocity and flow depth were obtained using a combined particle image velocimetry (PIV) and laser induced fluorescence (LIF) system at the three locations along the slope where the glass slots were inserted in the bed (x = 313 mm, 1158 mm and 1958 mm). Dantec Dynamics' silver coated hollow glass spheres with mean diameter of 10 μm, density of 1.4 g/cm3 and concentration of

ADS

LIF Camera

PIV Camera

Flume

0.7-1.3m Perspex Slots

0.6m Laser

Mirror

0.5-1.0m

Bed

1.5-1.9m 1.3-1.7m 1.2m

Fig. 2. Experimental equipment set-up for PIV/LIF and ADS measurements.

B.-T. Chen et al. / Coastal Engineering 108 (2016) 10–24

0.60 mg/l and Exciton Inc. Rhodamine 590 fluorescent dye with concentration of approximately 0.060 mg/l were added to the water which was illuminated by a 5 W, continuous wave, Nd YAG Laser (Dantec Dynamics RayPower 5000). The laser was positioned next to the flume, generating a horizontal light sheet (Fig. 2). BK7 right angle prism laser mirrors, positioned below the bed and fixed to the framework, directed the light sheet vertically upwards to illuminate the flow from below (O'Donoghue et al., 2010). The reflected light from the particles and emitted light by the dye were captured by two Speedsense 9040 digital cameras that were positioned on the other side of the flume from the laser and rotated to align with the slope of the bed (Fig. 2). The PIV camera was fitted with a 532 nm narrowband green filter and, to eliminate any interference by the free surface, was rotated slightly backward while the LIF camera was fitted with an orange filter and rotated slightly forward (Kikkert et al., 2012). The cameras recorded 2 megapixel (1632 × 1200 pixels), 8 bit, grey-scale images. The settings of the cameras and image acquisition were controlled through computer software Dynamic Studio v3.4. The time between images of a PIV image pair was 1 ms. The recording frequency of PIV image pairs and LIF images was 50 Hz. In total 12.0 s worth of data was recorded and at each measurement location experiments were repeated 50 times to obtain the appropriate ensemble-average and turbulence quantities (O'Donoghue et al., 2010). The system was triggered by the pulse generator that also controlled the opening of the first gate. Hence image recording began at t = 0 s. PIV image analysis began by dewarping of the images, to eliminate the effect of the backward rotation of the camera, followed by applying the adaptive cross-correlation algorithm which used three iterative steps, finishing with interrogation areas of 32 × 32 pixels, and 50% overlap. Outliers were removed through a range validation, a moving average validation and an ensemble average validation. The analysis yielded instantaneous u and w velocity vectors with a spatial resolution between 2.4 and 3.4 mm and random error between 15 and 21 mm/s. Entrained air was briefly present throughout the flow column after the arrival of the first bore and in the upper part of the flow column after the arrival of the second bore. The air bubbles blocked the camera view of the PIV seeding particles and hence the PIV algorithm obtained velocity vectors by using air bubbles instead, i.e. by correlating the locations of the shadows generated by the air bubbles. Entrained air is

13

affected by buoyancy, however the time for the bubbles to accelerate as a result of buoyancy was small. In addition, at the time when entrained air was present, the flow was highly turbulent, which decreased the rise velocity (e.g. Aliseda and Lasheras, 2011), and the bed-normal velocity was large. Therefore the percentage of the velocity attributable to the buoyancy effect was small and the accuracy of the flow velocity measurements based on the entrained air bubbles was still acceptable. Hence the velocity results that were based on the air bubbles have not been removed from the time-series shown. However, the random errors in the instantaneous velocity estimates may be greater, because the air bubbles are capable of changing shape and the shadows of the bubbles recorded by the camera are not necessarily in the plane illuminated by the laser. Larger errors in the instantaneous velocity estimates at the times of bore arrival result in overestimated magnitudes of the turbulent quantities, hence these estimates have been removed from the dataset. An edge detection algorithm was applied to the LIF images that determined the location of the interface between the areas of high intensity (water) and low intensity (air) which resulted in estimates for the instantaneous flow depth with a random error of approximately 0.2 mm. During the times of bore arrival, the three-dimensionality of the flow affected the ability of the algorithm to correctly determine the location of the free surface. These images were checked manually and the free surface location was adjusted if necessary.

3. Overall features of swash–swash interactions This section presents the experimental results from the exploratory experiments. Time-series of the ensemble-averaged depth, h, obtained by ADS are presented for reservoir levels 35–50 in Fig. 3 and 40–40 in Fig. 4. Before the arrival of the second bore at a particular location, the depth time-series are similar to those recorded by O'Donoghue et al. (2010) and Kikkert et al. (2012). Close to the initial shoreline, the flow depth increases rapidly after bore arrival and reaches maximum depth soon after. Decrease in flow depth is initially gradual, followed by a rapid decrease. Further up the slope, the flow depth increase after bore arrival is more gradual and decrease in flow depth after maximum depth is gradual but accelerates. Due to the interaction, the front of the

Fig. 3. Ensemble-averaged depth timeseries for initial conditions 35–50 and td = 6.0 s (a), 4.0 s (b) and 1.5 s (c), including the time of flow reversal of the first bore (○).

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B.-T. Chen et al. / Coastal Engineering 108 (2016) 10–24

second bore is highly turbulent and as a result very steep. Arrival of the second bore therefore results in an increase of the flow depth that is more abrupt than for the first bore irrespective of the location along the bed and similar to the arrival of the first bore at the initial shoreline. The maximum flow depth is reached soon after the arrival of the second bore. Once the second bore reaches a dry bed (e.g. td = 6.0 s) the depth time-series become similar to those observed for a single swash event further up the slope. The time of flow reversal, which is defined by the change in direction of the depth-averaged velocity, separates the uprush and backwash phases of the swash event (Kikkert et al., 2012; O'Donoghue et al., 2010). For the ADS experiments, the time of flow reversal is obtained from the NLSWE model by O'Donoghue et al. (2010) which accurately predicted the times of flow reversal obtained from laboratory experiments. The model was calibrated for the present experimental set-up using results from a single bore swash event with a reservoir water level of 500 mm (Dai and Kikkert, 2013) and then run for single swash events with reservoir water levels of 350 mm and 400 mm to obtain the times of flow reversal which are added to Figs. 3 and 4. A large opening delay of the second gate (td = 6.0 s, Figs. 3a and 4a) results in very weak backwash interaction or no interaction. For a medium delay of the second gate opening (td = 4.0 s, Figs. 3b and 4b), the second bore arrives when the first bore is in the backwash (i.e. after the time of flow reversal of the first bore) at all locations along the slope. After the interaction, the second bore travels shoreward and, if it is stronger than the first one (i.e. for reservoir levels 35–50), it reaches a larger maximum run-up (see also Table 1). According to the definitions by Hughes and Moseley (2007) and Cáceres and Alsina (2012), this swash–swash interaction is a weak wave–backwash interaction. A short delay (td = 1.5 s, Figs. 3c and 4c) results in a more complex behaviour. For reservoir levels 35–50 (Fig. 3c), the second bore arrives during the uprush of the first bore at the first four locations and therefore generates wave capture interaction. No interaction occurs at x = 3148 mm as the second wave overtakes the wave tip of the first bore shoreward of this location. For reservoir levels 40–40 (Fig. 4c), the second bore arrives at the initial shoreline location (x = 48 mm) during the uprush, and therefore also generates wave capture interaction. However, as it takes time for the second bore to travel up the slope, the second bore

Table 1 Maximum run-up of the first bore and second bore for reservoir levels 35–50 and 40–40 and gate opening delay times from 1.5 s to 6.5 s. td (s) 35–50 R1 (m) R2 (m) 40–40 R1 (m) R2 (m)

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

6.5

– 4.39

3.10 4.07

3.10 3.84

3.10 3.74

3.10 3.69

3.10 3.68

3.10 3.67

3.10 3.77

3.10 3.79

3.10 3.93

3.10 3.87

3.49 3.21

3.49 3.03

3.49 2.86

3.49 2.74

3.49 2.54

3.49 2.62

3.49 2.66

3.49 2.77

3.49 2.86

3.49 3.22

3.49 3.32

arrives at locations further up the slope at progressively later stages of the first swash event. The first bore has already changed its direction when the second bore arrives at x = 1548 mm (and all higher locations), hence creating a weak wave–backwash interaction. In summary, during the same swash event the type of interaction caused by the second bore may vary across the swash zone. The bore arrival delay of the two bores (tbad) is the time between the arrival of the first bore and arrival of the second bore at a measurement location. The relative bore arrival delay (trd) is the bore arrival delay minus the gate opening delay (td) and is plotted versus the gate opening delay in Fig. 5. The interaction between the first and second bores reduces the propagation of the second bore. For the 35–50 cases, trd is small because the second bore is stronger and hence faster than the first bore. Once it meets the second bore their interaction is relatively weak because the stronger second bore dominates. The relative bore arrival delay does increase slightly with gate opening delay, but is almost independent of location on the slope. This indicates that the majority of the delay occurs before the second bore reaches the slope which is the result of the greater horizontal distance from the second gate to the beach and the collision between the second bore and the wave that reflects back from the slope after the passage of the first bore. This collision is stronger for greater opening delay times. For the 40–40 cases (as well as 35–50 from x = 2348 mm shoreward), relative bore arrival delay does depend on location and gate opening delay. When the gate opening delay is small, the interaction occurs when the first bore is in the early backwash and hence velocities

Fig. 4. Ensemble-averaged depth timeseries for initial conditions 40–40 and td = 6.0 s (a), 4.0 s (b) and 1.5 s (c) including the time of flow reversal of the first bore (○).

B.-T. Chen et al. / Coastal Engineering 108 (2016) 10–24

15

Fig. 5. Relative bore arrival delay versus gate opening delay for reservoir levels (a) 35–50 and (b) 40–40.

throughout the swash zone are small. Interaction intensity, and thus the impedance of the first bore on the propagation of the second bore, is therefore relatively small. For medium gate opening delays, the backwash interaction occurs when the first bore is close to the maximum backwash velocity. Because the reservoir levels are the same, the momentum of the uprush of the second bore is similar to that of the backwash of the first bore at the time of interaction, causing significant collisions impeding the propagation of the second bore. For large gate opening delays, the first bore is close to the end of the backwash and velocities and flow depths are much smaller again. The interaction intensity and therefore the impedance is smaller as well, resulting once more in similar bore arrival delays along the slope. These results are mirrored in the maximum run-up measurements (Table 1). The greater initial momentum of the second bore in the 35–50 cases results in a greater maximum run-up for the second bore, R2, than the first bore, R1, and the difference is largest for both very small and very large gate opening delays. For the 40–40 cases, the interaction results in a smaller maximum run-up for the second bore. The differences are smallest for small and large gate opening delays.

4. Detailed hydrodynamics of wave capture and wave–backwash interaction The detailed investigation of the hydrodynamics swash interaction focuses on two different types of swash–swash interaction: (i) wave capture interaction, referred to in the remainder of the text as ‘capture interaction’ and (ii) weak wave–backwash interaction, referred to as ‘collision interaction’. The cases selected as capture and collision interaction cases are, respectively, the 35–50 case with td of 1.5 s, and the 40–40 case with td of 3.5 s. 4.1. Flow depth and depth-averaged velocity The ensemble-averaged flow depth, h , and ensemble-averaged depth-averaged bed parallel velocity, hui, time series are shown for the capture interaction case, i.e. reservoir levels 35–50 and td of 1.5 s, in Fig. 6. The features of the depth time-series have already been discussed using the ADS measurements. The features of the depth

Fig. 6. Time series of ensemble-averaged depth (top panel) and ensemble-averaged depth-averaged bed parallel velocities for capture interaction case.

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B.-T. Chen et al. / Coastal Engineering 108 (2016) 10–24

Table 2 Times of bore arrival and flow reversal for capture and collision interactions. x

Capture interaction

Collision interaction

1st bore arrival

2nd bore arrival

Flow reversal

1st bore arrival

1st flow reversal

2nd bore arrival

3rd flow reversal

(mm)

(s)

(s)

(s)

(s)

(s)

(s)

(s)

313 1158 1958

1.96 2.44 3.00

3.43 3.92 4.36

5.62 5.75 5.99

1.80 2.22 2.74

4.30 4.36 4.50

6.06 6.78 7.83

7.80 8.47 8.93

averaged velocity time-series include multiple saw-tooth shapes that were also observed in the field (Masselink et al., 2009) and in predictions from NLSW based models (e.g. Watson and Peregrine, 1992). Fig. 6 confirms that the interaction at the measurement locations is wave capture interaction, as it occurs before the time of flow reversal (Table 2). Until the time of arrival of the second bore, the features of the velocity time-series are those observed previously for the swash event without interaction (e.g. Kikkert et al., 2012). The velocity increases almost instantaneously to reach the maximum uprush velocity very soon after bore arrival, followed by an almost constant deceleration. The magnitude of the maximum uprush velocity decreases with distance up the slope. Upon arrival of the second bore, the velocity again increases almost instantaneously, hence for a brief time the flow has a positive (and very significant) local acceleration which is not present throughout the single bore swash event. The magnitude of this second velocity peak also decreases with distance up the slope. After the interaction, the flow combines the mass and momentum of the first and second bores before the interaction (minus the momentum lost during the interaction). The greater water level in the second reservoir results in a second bore with greater momentum. However, as a result of the mixing with the relatively low momentum fluid of the first bore, the peak velocity after the interaction is smaller than the peak velocity after the first bore arrival. The greater momentum of the second bore causes the peak velocity after the interaction to decrease more slowly with distance up the slope than the peak velocity after the arrival of the first bore. The difference in the magnitude of two velocity peaks has therefore all but disappeared at x = 1.958 m. In addition, the deceleration of the velocity once the front of the second bore has passed the measurement location is also reduced, but otherwise the velocity timeseries are again similar to that of the single bore swash event (e.g.

Kikkert et al., 2012). Deceleration of the flow is approximately constant for the remainder of the uprush resulting in times of flow reversal that increase with distance up the slope. At the time of flow reversal, the total mass of water in the swash-zone is the combined mass of the two bores and hence significantly larger than for a single bore swash event, resulting in a backwash of longer duration (Dai and Kikkert, 2013) but similar overall features. When the flow passes locations further down the slope, it has had more time to accelerate down the slope. This results in greater maximum backwash velocities, which are reached at later times in the swash event and for very small flow depths, than for the shoreward locations. Fig. 7 shows the depth and depth-averaged velocity time-series for the weak wave–backwash interaction case, i.e. reservoir levels 40–40 and td of 3.5 s. As before, until the arrival of the second bore the velocity time-series are the same as for the bore-driven swash event without interaction. The arrival of the second bore now occurs after flow reversal of the first bore (Table 2). Upon the arrival of the second bore the depthaveraged velocities become positive once again, generating a second flow reversal and confirming the interaction is weak wave–backwash interaction. At x = 313 mm, the second bore arrives before the first bore has reached the maximum backwash velocity and hence is still accelerating, at x = 1158 mm the first bore is approximately at the maximum backwash velocity, whereas at x = 1958 m the backwash velocity of the first bore is very small. At the time of interaction the respective depths of the first bore for the three locations (x = 313 mm, 1158 mm, 1958 mm) are rapidly decreasing at 79.7 mm, 21.6 mm and 2 mm respectively, whereas the velocities of the first bore are − 0.88 m/s, −1.22 m/s and approximately −0.2 m/s (velocity estimates just before the time of the second bore arrival at x = 1958 mm are affected significantly by the very shallow flow depth). Thus the corresponding

Fig. 7. Time series of ensemble-averaged depth (top panel) and ensemble-averaged depth-averaged bed parallel velocities for collision interaction case.

B.-T. Chen et al. / Coastal Engineering 108 (2016) 10–24

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Fig. 8. Ensemble-averaged bed-parallel velocity profiles for capture interaction case at x = 313 mm before (right panel) and after (left panel) interaction. The number above the profile is the time.

Fig. 9. Ensemble-averaged bed parallel velocity profiles for capture interaction case close to the time of the second bore arrival (3.43 s and 3.92 s at 313 mm and 1158 mm respectively). Δt is the time since the second bore arrival.

momentum (equal to depth times velocity) of the first bore at the time of interaction diminishes rapidly with distance up the slope. The momentum is greatest at x = 313 mm, resulting in the most energetic collision and thus the interaction with the greatest effect on the second bore. In contrast, the (positive) velocity of the combined bore immediately after the collision is greatest at the most shoreward location (0.68 m/s), probably because the (negative) velocity just before the collision is smallest, so it is easier to reverse and accelerate the flow than at the other two locations. After the arrival of the second bore, the velocity time-series are once again similar to that of the swash event without interactions, including a third flow reversal and second maximum backwash velocity.

4.2. Bed-parallel and bed-normal velocity profiles Example ensemble-averaged bed-parallel velocity profiles for the capture interaction case, i.e. reservoir levels 35–50 and td of 1.5 s, are presented in Figs. 8 (over the whole swash event) and 9 (close to the interaction, with a finer time resolution). Before the arrival of the second bore, the velocity profiles have the same features as the profiles of the single bore swash event (e.g. Kikkert et al., 2012). The profiles are relatively uniform, but a boundary layer develops near the bed (Fig. 8). As

the first bore is overtaken by the second bore, a step in the velocity profile is generated (Fig. 9). Up to the flow depth of the first bore at the time of interaction, the velocity is relatively small and uniform, while above this height the velocity increases very rapidly. This step is initially very pronounced. However, the steep velocity gradients generate intense shearing resulting in an increase in the mixing layer thickness, hml (see Table 3). Hence, the relatively high momentum fluid of the second bore is rapidly mixed with the relatively low momentum fluid of the

Table 3 Statistics of mixing layer present after the second bore arrival. Δū is the velocity difference between the local maximum and minimum velocities, hml is the mixing layer thickness defined as the difference between the locations of the maximum and minimum velocities and Δū/hml indicates the velocity gradient. Capture interaction x

Δt

Δū

hml

Collision interaction Δū/hml −1

(mm)

(s)

(m/s)

(mm)

(s

313 313 1158 1158

0.06 0.22 0.06 0.22

0.55 0.64 1.00 0.46

24 75 53 83

22 8.5 19 5.5

)

Δū

hml

Δū/hml

(m/s)

(mm)

(s−1)

1.96 1.88 1.38 0.70

34 106 14 25

57 17 100 28

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Fig. 10. Ensemble-averaged bed-parallel velocity profiles for collision interaction case at x = 313 mm before (right panel) and after (left panel) collision. The number above the profile is the time.

first bore. As a result, the velocity of the flow near the bed increases and the velocity of the flow near the free surface decreases, reducing the velocity difference between the local maximum and minimum velocities, Δū (see Table 3), and therefore also the step in the velocity profile. After approximately 1 s at x = 313 mm and 0.7 s at x = 1158 mm, the step is no longer present in the velocity profiles. The reason for the shorter time for locations further up the slope is probably that the shallower first bore loses momentum more rapidly than the second one (as can be seen in Fig. 6), so the second bore is relatively stronger i.e. the velocity difference is larger (Table 3) resulting in more intense mixing and faster development of the smooth velocity profile. The mixing during interaction destroys the developing boundary layer and hence the profiles after the interaction are again very uniform down to the bed (Fig. 8), thus also increasing bed-shear stress. The profiles are still uniform at the time of flow reversal which results in a less distinct maximum velocity near the bed, as the flow near the bed changes direction first, and quicker re-establishment of the standard velocity profile with the maximum velocity at the free surface than for the single bore swash event (Kikkert et al., 2012). For the remainder of the backwash, the velocity profiles are again very similar to those of the single bore swash event. The increasing velocity and decreasing flow depth result in very steep velocity profiles at the end of the backwash.

Figs. 10 and 11 present the bed-parallel velocity profiles for the collision interaction case, i.e. reservoir levels 40–40 and td of 3.5 s. Profiles are again similar to those of the single bore swash event until second bore arrival (Fig. 10 right). However, interaction occurs during the backwash of the first bore and therefore results in more complex development of the combined velocity profile (Fig. 11). Just before the collision the first bore has seaward velocities of approximately −1 m/s, while the second bore has shoreward velocities of similar but somewhat higher magnitude (judging by the final combined uniform velocity profile, which has shoreward direction and low magnitude). Upon collision the second bore minimises the impact of the collision by ‘climbing’ the first bore. This ascent is much more abrupt than for the capture interaction case hence causes a rapid increase in flow depth and a sharp change of velocity at the location of the first bore's free surface from seaward (beneath the free surface) to shoreward (above it). The magnitudes of the seaward and shoreward velocities just before the collision are similar, resulting in near-zero velocity at the level of the first bore's free surface. In the vicinity of this level the sharp velocity gradients, Δū, up to 2 m/s (Table 3) causes mixing across the free surface of the first bore which reduces the velocity magnitudes of both shoreward moving fluid above the free surface and seaward moving fluid below it. This in turn reduces the velocity gradients as well, however these remain larger than during the capture interaction case

Fig. 11. Ensemble-averaged bed parallel velocity profiles for collision interaction case close to the time of the second bore arrival (6.06 s and 6.78 s at 313 mm and 1158 mm respectively). Δt is the time since the second bore arrival.

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Fig. 12. Ensemble-averaged bed normal velocity profiles at x = 313 mm for capture interaction case close to the time of the second bore arrival (3.43 s). Δt is the time since the second bore arrival.

throughout the interaction (Table 3). In the lower part of the flow profile the efficiency of mixing decreases with the distance from the free surface hence leaving relatively strong seaward velocities in the middle of the first bore's depth. The efficiency of the flow reversal recovers again at the lowest end of the velocity profile, where the velocity of the first bore just before the collision was smallest. The collision of the two bores therefore results in a velocity profile with a complex shape: the velocity is close to zero near the bed and near the free surface of the first bore at the time of interaction, while the velocity between these two vertical locations remains negative with a maximum at approximately half the free surface height of the first bore. Above the height of the free surface of the first bore the velocity is positive, reaching a maximum close to the free surface of the second bore. This shape is short-lived, since mixing across the two regions of steep velocity gradients (above and

below the first bore's mid-depth) continues and results in a uniform profile seen about 1 s after the collision. After the interaction, the profiles are therefore similar to those after the interaction for the capture case, and the features of the profiles for the remainder of the swash event are therefore also similar. Ensemble-averaged bed normal velocity profiles for the capture and collision interaction cases are presented in Figs. 12 and 13. Away from the time of interaction, the bed-normal velocities are similar to those in the single bore swash event, i.e. they are approximately zero (e.g. Kikkert et al., 2012). After arrival of the second bore, the bed-normal velocity below the height of the first bore at the time of interaction for the capture case is positive, i.e. upwards, and approximately one order of magnitude smaller than the bed-parallel velocities (Fig. 12). This indicates that the incoming bore advects the slow moving fluid, primarily

Fig. 13. Ensemble-averaged bed normal velocity profiles at x = 313 mm for collision interaction case close to the time of the second bore arrival (6.06 s). Δt is the time since the second bore arrival.

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Fig. 14. Example images of capture interaction case at x = 313 mm with Δt = 0.045 s, the incoming second bore travels from left to right.

Fig. 15. Example images of collision interaction case at x = 313 mm with Δt = 0.200 s, the incoming second bore travels from left to right.

of the first bore, upwards into the flow column where it becomes part of the wave-tip of the second bore. As the second bore propagates up the slope it therefore accelerates the fluid of the first bore that it encounters rather than flowing over the top of the fluid of the first bore and overtaking it. This is facilitated by the previously mentioned fluid momentum transfer across the zone of intense shearing. After 0.29 s the advection of fluid is complete and the bed-normal velocity near the bed is again zero, however the mixing in the upper part of the flow column continues for approximately another 0.5 s, resulting in alternating positive and negative bed-normal velocities near the free surface. After 0.85 s, the profile is again approximately zero throughout which coincides with the re-establishment of the standard bed-parallel velocity profile (Fig. 9). For the collision case, the period when fluid from the first bore is advected upwards into the flow column is followed by the period when fluid from the second bore ascents the first bore and hence the time with significant bed-normal velocities is longer (Fig. 13). The magnitudes of the bed-normal velocities are of the same order as the bed-parallel velocities. Comparison of the bed-parallel and bednormal velocities indicates the existence of a strong clockwise vortex

(i.e. shoreward and downward flows at the upper part of the profile and seaward and upward flows in the middle) that enhances the already present turbulent mixing. Coherent flow structures found for both capture and collision interaction cases are further discussed in Section 4.3. After 0.80 s, the interaction no longer affects the bedparallel velocities (Fig. 11) and the bed-normal profiles are approximately zero again throughout the flow column. The bed-normal velocity profiles at locations further up the slope (not shown) have the same features as those at x = 313 mm. However, the magnitudes of the bednormal velocities decrease which again indicates that the effect of the interactions on the second bore reduces with distance up the slope. 4.3. Coherent flow structures The bed-parallel and bed-normal velocity profiles of both the capture and collision cases imply the existence of coherent vortices as the intense shearing due to the steep velocity gradients results in Kelvin– Helmholtz instabilities. Example images from high speed videos of the flow obtained just after the time of the second bore arrival are shown in Figs. 14 and 15 for the capture and collision interaction cases. Due

Fig. 16. Ensemble-averaged velocity vector and Okubo–Weiss parameter field for capture interaction case at x = 313 mm and Δt = 0.17 s (t = 3.60 s).

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Fig. 17. Ensemble-averaged velocity vector and Okubo–Weiss parameter field for collision interaction case at x = 313 mm and Δt = 0.34 s (t = 6.40 s).

to the presence of the air bubbles, the images reveal the presence of a large vortex at the interface of the two bores for both cases. The images also indicate that the mixing layer for the capture interaction is confined closer to the free surface whereas for the collision case it extends over the majority of the flow column. The presence of coherent vortices can be confirmed using the Okubo–Weiss parameter (Okubo, 1970; Weiss, 1991). The Okubo–Weiss parameter, OW, is calculated as: OW ¼ S2n þ S2s −ω2

ð1Þ

where Sn ¼ ∂u=∂x−∂w=∂z; Ss ¼ ∂w=∂x þ ∂u=∂z; ω ¼ ∂w=∂x−∂u=∂z

ð2Þ

are the normal component of strain, the shear component of strain and the vorticity respectively. Therefore positive values of OW indicate that strain dominates vorticity and negative values indicate that vorticity dominates strain. Example fields of the Okubo–Weiss parameters as

well as the ensemble-averaged velocity vectors for the capture and collision interaction cases are shown in Figs. 16 and 17, respectively. For the capture case, the bed-normal velocities during interaction are approximately an order of magnitude smaller than the bed-parallel velocities and hence the vortices are not directly apparent from the velocity vector fields. However, the Okubo–Weiss parameter confirms the impression from the images that a relatively large vortex is generated at the free surface depth of the first bore (Fig. 16). The intensity of the large vortex as well as the smaller ones near the free surface decreases quickly and 0.30 s after arrival of the second bore the vortices have all but disappeared and fluid is no longer advected across the flow column. During the collision interaction case, the bed-normal and bedparallel velocities have a similar magnitude and hence the complex nature of the velocity field is clearly visible (Fig. 17). The velocity vectors show the large clockwise rotating vortex at the interface depth of the first and second bores observed in the images, as well as a second, smaller anti-clockwise rotating vortex near the bed which is probably generated and driven by the first one and the ascending fluid of the

Fig. 18. Depth-averaged TKE time-series at x = 313 mm, 1158 mm and 1958 mm for capture interaction case (top panel) and collision interaction case (bottom panel).

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Fig. 19. Ensemble-averaged TKE profiles at x = 313 mm for capture interaction case close to the time of the second bore arrival (3.43 s). Δt is the time since the second bore arrival.

second bore. The presence of the two vortices is confirmed by the Okubo–Weiss parameter (Fig. 17). The magnitude of the Okubo– Weiss parameter indicates that the intensity of the large vortex is up to five times larger than for the capture case and is highest during the period Δt = 0.24–0.50 s. However, the intensity of the vortices decreases again rapidly and the vortices have disappeared by approximately Δt = 0.70 s which is also in agreement with the time when the bed-normal velocities are zero once more.

4.4. Turbulent quantities Based on the two components of velocity obtained from the PIV measurements, turbulent kinetic energy, TKE, is calculated as u0 u0 þ w0 w0 and Reynolds shear stress as −u0 w0 (Kikkert et al., 2012), where the prime denotes a fluctuating velocity component and the overbar denotes

ensemble-averaging. The depth-averaged turbulent kinetic energy, 〈TKE〉, is obtained by integrating the TKE profiles. The resulting timeseries for both the capture and collision interaction cases are presented in Fig. 18. For the single bore swash event peaks in TKE occur at bore arrival due to bore-generated turbulence, and near the end of the backwash due to bed-generated turbulence (Kikkert et al., 2012). These peaks are also observed in the present time-series, however the arrival of the second bore causes a third peak in TKE which is larger than the other two for the capture case and clearly dominates for collision. The reason is twofold: the wave-tip of the second bore transports bore-generated turbulence up the slope, but additional turbulence is generated locally due to the shearing and presence of turbulent vortices during interaction. For the capture interaction case, i.e. reservoir levels 35–50 and td = 1.5 s, the magnitude of TKE at x = 313 mm after the arrival of the second bore is therefore greater than that after the arrival of the first bore and its decrease with distance up the slope is more gradual. For the collision

Fig. 20. Ensemble-averaged TKE profiles at x = 313 mm for collision interaction case close to the time of the second bore arrival (6.06 s). Δt is the time since the second bore arrival.

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Fig. 21. Ensemble-averaged Reynolds stress profiles at x = 313 mm for collision interaction case close to the time of the second bore arrival (6.06 s). Δt is the time since the second bore arrival.

interaction case, i.e. reservoir levels 40–40 and td of 3.5 s, the greater velocity gradients (Table 3) result in more intense shearing and vortices of greater intensity than for the capture collision case. Therefore the magnitude of TKE after arrival of the second bore is much greater than for the wave capture case at x = 313 mm and 1158 mm. At x = 1958 mm, dissipation of previously generated turbulence and little production of turbulence due to the very small intensity of the interaction between the two bores result in a significant reduction in the magnitude of TKE after the second bore arrival, highlighting again the reduction in the intensity of the interaction along the slope. Profiles of TKE for the capture and collision cases are presented in Figs. 19 and 20, respectively which are generally very similar in shape although the magnitude of TKE in the profiles of the collision case is approximately a factor three larger. The advection of fluid from the first bore upwards into the flow column results initially in relatively small levels of TKE in the lower part of the flow column, while intense shearing and the presence of turbulent vortices in the upper part of the flow column result in relatively large levels of TKE. With time, TKE generated in the upper part of the flow column is diffused downwards to approximately the free surface depth of the first bore which coincides with the location of the centre of the large vortices seen in Figs. 16 and 17. The Reynolds stress profiles in Fig. 21 are obtained at the same location and times as the TKE profiles in Fig. 20 and yield similar conclusions in regard to the turbulent features of the flow during interaction. Initially the Reynolds stress in the upper part of the flow column is large due to the intense shearing and presence of the turbulent vortices and close to zero near the bed. With time, the magnitude of the Reynolds stress within the profile decreases and the location of the peak moves downward. During the same period the previously mentioned second (and weaker) vortex develops and generates negative Reynolds stress close to the bed. The Reynolds stress profiles are approximately zero again from 0.74 s after bore arrival onwards which agrees with the time of disappearance of the vortices. 5. Conclusions To investigate the detailed hydrodynamics of swash–swash interactions, experiments were carried out in the laboratory using a double dam-break mechanism to generate two bores that ran up an impermeable, 1:10 sloped, sand rough beach. Experimental measurements were

obtained via acoustic displacement sensors and a combined PIV/LIF system. Analysis of the experimental results showed that the type of interaction is dependent on the relative momentum of the two bores when they arrive at the initial shoreline location and the time delay between their arrivals. Swash interaction cannot necessarily be described by a single interaction type along the whole beach. During the time it takes for the second bore to run up the beach, the first bore may reach the time of flow reversal and hence wave capture interaction further down the slope is followed by weak wave–backwash interaction further up the slope. If the momentum of the second bore is significantly larger than that of the first bore, the intensity of the interaction is relatively weak irrespective of the delay time. If the momenta of the two bores are similar, then the intensity of the interaction is larger, especially if the interaction occurs when the first bore is in the midbackwash, close to the time of maximum backwash velocity. This also results in a smaller maximum run-up of the second bore. For wave–capture and weak wave–backwash interaction, the interaction processes are similar. The arrival of the second bore creates very steep gradients in the bed-parallel velocity profiles at the free surface depth of the first bore just before the interaction. This results in a mixing layer with intense shearing and generation of turbulent vortices. At the same time, fluid from the first bore is advected upwards into the flow column where it is rapidly mixed with the fluid from the second bore by the intense shearing and turbulent vortices. Development of the mixing layer causes the momentum in the lower part of the flow column to increase as well. As a result, after approximately 1 s since the arrival of the second bore, the steep gradients in the bed-parallel velocity profiles are no longer present. After the interaction, the velocity profiles are very uniform. During the weak wave–backwash interaction the bores travel in opposite directions and therefore the overall effects of interaction are stronger, i.e. velocity gradients are steeper, and the bednormal velocities and turbulence intensities are larger than during wave–capture interaction. The characteristics of the hydrodynamics during interaction confirm the potential of swash–swash interactions to suspend significant amounts of sediments high into the flow column and then transport the sediments shoreward with the second bore or interrupt the seaward transport of sediments by the backwash of the first bore. To quantify the sediment transport as a result of swash interactions requires more detailed measurements under controlled conditions to enable sediment

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transport models that take into account the effect of swash–swash interactions to be properly evaluated. The detailed data reported here are available on request from the authors. Acknowledgements The authors gratefully acknowledge financial support from the Hong Kong Research Grant Commission under grant number 613711 and the TUYF Charitable Trust under grant number TUYF12EG06. The authors also thank Tom Baldock and an anonymous reviewer for their constructive comments. References Aliseda, A., Lasheras, J.C., 2011. Preferential concentration and rise velocity reduction of bubbles immersed in a homogeneous and isotropic turbulent flow. Phys. Fluids 23. Alsina, J.M., Cáceres, I., 2011. Sediment suspension events in the inner surf and swash zone. Measurements in large-scale and high-energy wave conditions. Coast. Eng. 58, 657–670. Alsina, J.M., Cáceres, I., Brocchini, M., Baldock, T.E., 2012. An experimental study on sediment transport and bed evolution under different swash zone morphological conditions. Coast. Eng. 68, 31–43. Barnes, M.P., Baldock, T.E., 2007. Direct bed shear stress measurements in bore-driven Swash and Swash interactions. Coastal Sediments '07 — Proceedings of 6th International Symposium on Coastal Engineering and Science of Coastal Sediment Processes. Barnes, M.P., O'Donoghue, T., Alsina, J.M., Baldock, T.E., 2009. Direct bed shear stress measurements in bore-driven swash. Coast. Eng. 56, 853–867. Brocchini, M., 1997. Eulerian and Lagrangian aspects of the longshore drift in the surf and swash zones. J. Geophys. Res. C 102, 23155–23168. Brocchini, M., Baldock, T.E., 2008. Recent advances in modeling swash zone dynamics: influence of surf–swash interaction on nearshore hydrodynamics and morphodynamics. Rev. Geophys. 46, RG3003. Brocchini, M., Peregrine, D.H., 1996. Integral flow properties of the swash zone and averaging. J. Fluid Mech. 317, 241–273. Cáceres, I., Alsina, J.M., 2012. A detailed, event-by-event analysis of suspended sediment concentration in the swash zone. Cont. Shelf Res. 41, 61–76. Cowen, E.A., Sou, I.M., Liu, P.L., Raubenheimer, B., 2003. Particle image velocimetry measurements within a laboratory-generated swash zone. J. Eng. Mech. 129, 1119–1129. Dai, H.J., Kikkert, G.A., 2013. Characteristics of the entrained air bubble cloud in the swash-zone. Proceedings of 35th IAHR World Congress Chengu, China.

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