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Numerical setup

This simulation reproduces a two-dimensional wave flume with a constant-depth region and a steep slope (1 on 3). The two-dimensional setup, shown in Figure 10, is similar to the one used in Hi guera et al. (2018). The numerical domain is 2 m long in the wave propagation direction and 0.187 m in the vertical direction. The solitary wave height is H = 2.8 cm and the water depth is h = 8 cm. The wave is generated with the wave generation boundary conditions in olaFlow (Higuera, 2015) on the left boundary, applying Grimshaw’s third order theory (Grimshaw, 1971). The wave propagates over a 1.5 m-long smooth and flat bottom before arriving at the toe of the slope. The slope is made from sand grains, which are contained in a sandbox that extends 5 mm vertically.

The mesh has been created to minimise the computational cost, allowing lower resolution in areas where mesh quality is not critical (e.g., away from the water, near the top boundary). Cell size variations have been performed by geometric progression to ensure smooth transitions. The smallest cell size is 1 x 1 mm, over the slope, in the vicinity of the beach. The horizontal cell size increases gradually from 1 mm at the toe of the slope to 5 mm at the wave generation boundary. The vertical cell size also increases from 1 mm at Z = h + H to 5 mm at the top boundary. The mesh which is unstructured but hexahedral-cell dominant, totals 83,500 cells. Unlike the mesh used in Higuera et al. (2018), no boundary layer refinement has been performed near the bottom walls in order to accommodate the sediment particles.

Sediment grains are represented by 32,000 individual spherical particles. Each sphere has a diameter Dr,o = 0.25 mm, a density of 2600 kg/m3, a Young’s modulus of E = 75 GPa and

Sketch of the beach flume. Not to scale

Figure 10: Sketch of the beach flume. Not to scale.

a Poisson’s ratio of v = 0.17, corresponding to typical values for fine sand (Daphalapurkar et al., 2011: Roman-Sierra et al., 2014). The initial positions of the sediment particles are initialized randomly, thus preventing an artificial perfect packing. The cell porosity has been limited numerically to a minimum value of 40%, also to maintain typical values.

Turbulence has been modelled with the volume-averaged к — oj SST turbulence model, modified according to Larsen and Fuhrman (2018) and Devolder et al. (2017). No turbulence enhancement due to the sediment particles has been considered.

The simulation time is 4 seconds and takes 20 days in a single Xeon core (2.50 GHz), which points out the needs for optimisation and parallelization of DEM techniques to achieve reasonable computational times.

Numerical results

For consistency with Higuera et al. (2018), the zero time reference is chosen when the crest of the solitary wave is located directly above the toe of the slope (x = 0 m). During the simulation five phases can be distinguished. Initially, during the propagation phase, the wave travels over the horizontal bottom with a constant form, subjected to frictional effects induced by the bottom. The second stage, the shoaling phase, starts as the wave propagates over the slope. The shoaling wave changes its shape, losing its symmetry and developing a steeper front, but it does not break for the selected nonlinearity and steep slope values. It. can be noted that up to this stage, there are no evident hydrodynamic changes when compared to the simulation in Higuera et al. (2018).

The next phase is runup, in which the wave-driven uprush flow over the slope is decelerating and thinning due to gravity, until the maximum runup takes place and the flow tip stops before moving down. The starting point of the runup phase is defined when the flow tip (i.e., shoreline) becomes the highest point in the water domain. This happens at t = 0.31 s, slightly behind t = 0.27 s in the original experiments and t = 0.29 s in the numerical simulation in Higuera et al. (2018). This difference is probably caused by the additional friction imposed by the sand grains.

As the water tongue flows up the slope, the flow tip advances over the beach, slowly percolating and trapping pockets of air between the solid bottom and the beach surface. As presented in Figure 11, the number of particles dragged initially by the uprush (left panel) is limited. The right panel in Figure 11 shows a latter stage, before maximum runup. As it can be observed, the water tongue mobilises a larger number of particles from the top layers of the beach, producing sheet-flow-like sediment transport, which was also observed in similar experiments by Sumer et al. (2011). In both panels, the velocities inside the porous beach are significantly lower than in the free-flow region.

The comparison of free surface elevation (FSE) profiles during the runup phase is shown in the top panel of Figure 12. Generally, the shape of the free surface over the sediment is very similar to that of the glass-bottom experiments, especially away from the flow tip. In that area, results are comparable with those reported in Higuera et al. (2018) too, except for t = —0.14 s and t = 0.06 s, in which the profile elevation is slightly above the smooth bottom simulation results. The major FSE differences between the present simulation and the experimental measurements take place during the latter runup phase (t = 0.45 — 0.65 s) at the flow tip, where significant decrease in the runup extent can be observed, despite the good agreement in the offshore region. For example, maximum runup takes place at t = 0.66 s, almost at the same instant (t = 0.65 s) in Higuera et al. (2018). However, the present runup height is noticeably lower: ггц = 7.1 cm, as compared to ггц = 8.4 cm and 8.0 cm of the experiment and numerical simulation in Higuera et al. (2018), respectively. This difference can be explained by the additional friction and the percolation that the flow

Flow and sediment velocities near the shoreline during the initial

Figure 11: Flow and sediment velocities near the shoreline during the initial (left panel) and mid (right panel) runup phase. The horizontal component of velocity is represented for the flow, whereas the velocity magnitude is presented in coloured arrows for the sand grains.

Comparison of experimental and numerical free surface elevation profiles during the runup (top panel) and rundown phases (bottom panel)

Figure 12: Comparison of experimental and numerical free surface elevation profiles during the runup (top panel) and rundown phases (bottom panel).

experiences during runup because of the porous beach, which produces additional energy dissipation.

At the start of the rundown phase the hydrodynamics have already transitioned from being wave-driven to gravity-driven. The potential energy of the flow is progressively converted into kinetic energy as the flow accelerates due to gravity. The increasing momentum produces a fast and shallow flow (supercritical flow regime) which moves into the quiescent deeper waters (subcritical regime), eventually producing a hydraulic jump that overturns.

Some of the pockets of air that were trapped during the runup phase start playing an important role at this stage. The air tries to flow up towards the free surface but the friction of the sand grains prevents it from ascending as fast as a free bubble. This creates a “fluidised bed” effect, which induces lift forces on the particles, thus easing the transport by the retreating swash flow. Furthermore, the flow velocities during the rundown phase are significantly larger than during the runup phase and the sediment transport in the offshore direction is favoured by gravity, therefore, this induces a sheet-like sediment flow. The amount of particles transported is massive, as shown in Figure 13.

The comparison of free surface elevation (FSE) profiles during the rundown phase is presented in the bottom panel of Figure 12. Since the starting point of this phase, the maximum runup, was lower than in the experiments, the retreating flow tip is at all times below the experimental mark. Furthermore, undulations in the FSE start to develop at the lower part of the slope. This effect is caused by the sediment as it does not appear in the numerical simulation in Higuera et al. (2018). Apart from these discrepancies, the overall agreement of the initial FSE profiles follows the same trend.

Before the hydraulic jump occurs, the difference in water depth between the supercritical and subcritical regions induces a large pressure gradient, which overcomes the relatively low momentum in the boundary layer near the beach interface, reversing the flow in the area of transition, as can be seen in the top left panel in Figure 13. Flow reversal starts taking place at t = 0.93 s in the present simulation, whereas in the experiments and numerical simulation for the smooth bottom in Higuera et al. (2018), flow reversal occurs at t = 0.91 s and t = 0.81 s, respectively. As an immediate result of flow reversal, an anti-clockwise vortex, which is perfectly visible in the LIC pattern, is created and evolves together with the hydraulic jump, which starts overturning at t = 0.98 s. This eddy picks sediment up from the beach surface and puts it in suspension. This phenomenon was already observed in Matsunaga and Honji (1980), in which they reported that the sediment was lifted up from the bed by the backwash vortex, and not by the plunging wave. Similar observations were made in Sumer et al. (2013). The vortex also acts like a trampoline, projecting the incoming sediment upwards, while it is being advected in the offshore direction by the downrush flow, from which the eddy gains energy to grow. At this stage (t = 0.98 s) there is a smaller second anti-clockwise vortex up the slope from the main vortex, which is more noticeable in the following panels. By t = 1.03 s the hydraulic jump is plunging onshore. The second vortex becomes more visible, and between the two vortices mentioned, a new vortex is created, which rotates in the clockwise direction for compatibility reasons. Although it is partially covered by the sediment, half of it is above the beach interface. Another difference of this simulation when compared with Higuera et al. (2018) is the shape of the overturning wave (see t = 1.06 s in the bottom panel of Figure 12), which initially traps less air when breaking. The system of three vortices continues to be advected downstream, growing and deforming, as it can be appreciated perfectly at t = 1.11 s, when the hydraulic jump has overturned and broken, projecting the wave lip shoreward. Up to this stage, prior to wave breaking, the evolution of the system of vortices is extremely similar to that under smooth bed conditions (Higuera et ah, 2018).

Wave breaking creates very complex flow patterns (t = 1.23 s), trapping significant amounts of air and transporting plumes of sediment towards the surface. Nevertheless, it must be noted that this is a 2D simulation, therefore, breaking cannot be characterised completely, as it is a 3D process predominantly. For example, when air is trapped in a 2D simulation, it will disturb and “cut” the flow as it moves up to the surface since 2D air entrainment occurs in infinitely wide pressurised rollers, whereas in 3D simulations the disturbance might be localised in certain areas only, as air and water can mobilise sideways and create the low pressure tubes and bubbles typical in air entrainment processes during

Flow velocities and sediment grains over the slope during the rundown phase and the development of the hydraulic jump

Figure 13: Flow velocities and sediment grains over the slope during the rundown phase and the development of the hydraulic jump. The horizontal component of velocity is represented for the flow with Line Integral Convolution technique (LIC, i.e., showing instantaneous streamlines). Sand grains are depicted as black dots.

Beach profiles during the simulation

Figure 14: Beach profiles during the simulation. The differences in the vertical direction with respect to the initial profile (dotted line) have been amplified by a factor of 5. The straight solid line represents the bottom of the sand pit.

wave breaking. The situation towards the end of the simulation (t = 1.92 s), when the system is almost at rest, shows some particles that are still in suspension due to the remaining vortices, but most of the sediment has settled down already.

The time evolution of the beach profile is presented in Figure 14. The reader should note that the differences in the elevation of the sediment bed with respect to the original profile have been amplified by a factor of 5 to ease comparisons.

The initial stage depicts an almost perfect straight line (blue dotted line) following the 1:3 slope, parallel to the physical bottom of the tank (black continuous line). At the maximum runup instant most of the profile (red dashed line) shows a lower level, up to a maximum of 2.9 mm below the original line. This is caused because the initial packing is random and, as can be seen in Figure 11, some gaps exist within the internal structure, therefore, the sediment packing compacts due to the wave-induced velocities and pressures during the runup phase. Moreover, at this instant the flow has already started moving down the slope except at the tip. The sediment is also dragged down and some bumps appear and possibly induce the free surface oscillations that were previously observed in Figure 13. The beach profile when wave breaking has ended (green dash-dot line) shows that the sediment has piled up. The largest peak appears at a location between the downrush flow reversal and the hydraulic jump, 7.5 cm offshore from the shoreline, and is 8.6 mm above the original beach profile, which is consistent with the observations in Sumer et al. (2011) and Li et al. (2019). Nevertheless, the sediment on the bottom is still in movement and avalanches, which are captured by the DEM method, are ongoing at this instant. The final beach profile, after the sediment has come to a stop (grey solid line) shows the results of the avalanches, reducing the height of the peak by extending the mound down the slope and producing smoother slopes. The sediment uphill and downhill from the new berm remains almost unchanged in the last 2 snapshots. Finally, it can be noted that two areas of the slope remain unaltered for longer time than others. There is a small area within 2.3 cm from the toe of the slope that presents no erosion or accretion during the whole simulation. Additionally, the area above x = 0.478 m (maximum runup reaches x = 0.453 m) shows no movement after the initial settling since maximum runup occurs at this stage.

Concluding remarks

This case has shown the potential of RANS modelling for simulating systems in which sediment is represented particle by particle. This simulation must be viewed as a proof of concept, pointing out the benefits of this method, and can in principle be applied in diverse applications such as to estimate scour around coastal or offshore structures.

There are two main challenges that must be overcome in order to use this technique for real applications (i.e., consultancy cases). The first factor is performing three-dimensional simulations, which are required to represent important physical processes such as wave breaking. For example, 3D effects were found negligible prior to wave breaking in Higuera et al. (2018), whereas in the present simulation the air pockets trapped in the sediment bed flow up towards the free surface, eventually cutting through the downwards flow. This is an obvious side effect of the 2D simulation, as in a 3D simulation air could also escape sideways. However, running a 3D simulation with particles would increase the simulation time exponentially. In view of this, the second factor is computational cost, which in this case is extremely large even for the limited number of particles in 2D, due to the particular implementation of DEM in OpenFOAM®. Fortunately, more efficient DEM solvers exist (Klossr and Goniva, 2010) and can be coupled with OpenFOAM®.

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