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# Energy balance

A strong point of the Л'-SPH method is that its energy equation can be cast in a conservative format, thus implying an accurate account for the energy exchanges between the fluid and solid phase.

The global mechanical energy equation is obtained by multiplying the momentum equation for ut scalarly and by summing all over the fluid particles. Using the symmetry properties of the arguments of the summations and assuming that the body force is conservative, i.e., / = V(f> with ф = ф(г), we obtain:

where the starred summations indicate the summations over the fluid particles and the global mechanical energy is given by:

Here, £k is the global kinetic energy and £p is the global potential energy. Note that £m, £f, and £p only refer to the fluid phase. To guarantee the conservation of the total energy of the fluid-solid system, we introduce the global internal energy £, and require:

By definition, the total energy of the fluid-solid system remains constant during the evolution while energy exchanges may occur between mechanical and internal energy. As a consequence of (34), the equation for the internal energy reads:

The expression on the right hand-side may be regarded as a source/sink term dragging energy from/to the mechanical energy to/from the internal energy. To show this, it is convenient to focus on the right-hand side of Equation (32). Using the continuity equation and the symmetry properties of arguments in the double summations, it is possible to rearrange it in the following form (see Antuono et ah, 2015 for details):

where

and the barred summations denote the summation over the ghost particles in the solid domain. Here, Vs represents the power associated with the diffusive term, Va is the power dissipated by the artificial viscosity and Vs indicates the power due to the interaction between the fluid with the solid. Finally, the symbol £c indicates the energy due to compressibility, namely

For a linear state equation like that adopted in (12), we obtain:

Then, in comparison to the standard SPH scheme, the (5-variants predicts a further term in the global equations of mechanical and internal energy. Specifically, the global mechanical energy equation can be rearranged in the following way:

The power term Va is always negative from the second principle of thermodynamics while Vs is always negative by construction (see Antuono et ah, 2015 for details). As a consequence, these contributions represent purely dissipative terms. Conversely, V3 has not a defined sign, since it takes into account the energy exchange between the solid and fluid phase. As a consequence of Equation (34), we can also write:

which represents a rearrangement of Equation (35). Regarding the energy' exchanges between the solid and fluid phase, a detailed study on the consistency of the mirroring techniques described in Equations (17), (18) and (19) has been tackled, for example, in (Cercos-Pita et ah, 2017).

# Applications

In this section we show some applications of the A-SPH scheme to hydrodynamic problems of interest in the coastal and marine engineering fields. The first example provides simulations of violent water impacts against solid walls in both two and three dimensions. These problems highlight the ability' of the <5-SPH in the prediction of complex ffee-surface dynamics, as well as in the evaluation of local loads on solid structures. Then, we consider an extreme water-impact event against a flap-type wave energy converter. This application encloses all the main topics/features discussed in the previous test cases, i.e., wave propagation, body motions and violent water impacts, in a real engineering application.

## Prediction of water impacts

Here the robustness of this scheme is tested by simulating violent free-surface flows, characterized by water impact events where the air-water interface is subjected to rapid dynamics inducing high pressure peaks.

The first problem consists in a dam-break flow impacting against a vertical wall. This benchmark test cases has been studied through the (i'-SPlI model in the paper (Marrone et al., 2011). Figure 4 shows the dam-break flow generated by the gravity collapse of a water column of height H and width 2H. The fluid is confined in a rectangular tank and, after the dam break, it evolves rightwards, impacts against the tank wall and generates a plunging breaking wave.

In (Marrone et al., 2011) different resolutions were adopted with a maximum of H/Ax = 320 (corresponding to about 200,000 particles), showing a good match with the experimental and reference data. In fact, the pressure signals recorded at the probes Pi and Pj (see Figure 4) did not show sensible variations for spatial resolutions finer than H/Ax = 320. In

Figure 4: Dam-break flow of a water column of height H and width 2H. The colors are representative of the pressure field (made dimensionless with respect to the initial hydrostatic value). During the simulation the pressure time histories at the probes Pi (y = 0.01 H) and Pi (y = 0.267H) are recorded.

Figure 5: Dam-break flow of a water column of height II and width 2H. Time histories of the kinetic and potential energy of the flow (made dimensionless with respect to the reference energy pgH3).

the present section the ratio is increased up to H/Ax = 1600 (about 5.2 million particles) in order to get results with an higher accuracy in the whole domain.

Figure 5 depicts the time histories of kinetic and potential energy. During the early stages of the evolution, namely for t{g/H)D2 g [0,3], the potential energy is converted in kinetic energy. The left plot of Figure 6 displays the flow configuration at t(gfH)1'2 = 3, when the maximum of the kinetic energy is reached. At this time instant the water is generating a run-up along the right wall of the tank while a high pressure region develops at the bottom with a maximum pressure value of about 1.3 pgH.

Figure 7 shows the time histories recorded at the probe Pi. The maximum pressure along the vertical wall occurs at time t(gfH)1'2 = 2.385 and corresponds to approximately ~ 2.9 pgH. This is in good agreement with the prediction obtained by the potential theory, namely 2.8pgH (see for details Dobrovol'Skava, 1969 and Marrone et al., 2011). In the same figure, the numerical output is also compared to the experimental data by Lobovskv et al. (2013). The maximum pressure peak of the experiments is close to the S—SPH prediction even if the whole pressure peak appears more localized in time. This is likely due to the fact that the numerical simulation is two-dimensional while in the experiments 3D effects were not completely negligible. Further, the advance in time of the (5-SPH signal with respect to the measurements is due to the fact that friction along the bottom is not modelled (e.g., a free slip condition is imposed along the bottom).

After t(g/H)ll2 = 3 the flow starts to decelerate until the time t{g/H)lI2 = 5.5 when the potential energy reaches a local maximum. This instant corresponds to the formation of a plunging jet (see the right plot of Figure 6). As shown in Figure 8 the latter hits the underlying layer of water at about t(g/H)1I2 = 6.10 and generates acoustic pressure

Figure 6: Dam-break flow of a water column of height H and width 2H impacting on a vertical wall. Enlarged views of the pressure field during two time instants: left) the water evolves upward along the wall, right) the water reaches its maximum vertical height and collapses under the gravity action.

Figure 7: Dam-break flow of a water column of height II and width 2H. Time histories of the pressure recorded on the probe Pi.

waves. This is a consequence of the weakly-compressible model adopted in the Л'-SPI I scheme which, during impacts, converts mechanical energy in acoustic wave energy (see, for example, Marrone et al., 2015).

On the top-left panel of Figure 9 the pressure field at time t(g/H)1^2 = 6.65 is shown. This time instant corresponds to the second local maximum of the kinetic energy (see Figure 5) and is related to the splash-up stage caused by the ricochet of the plunging jet. Since the simulation is performed with a single-phase model (and, consequently, no aircushioning effects are accounted for), the cavity entrapped by the plunging jet is subjected to large volume variations. In turn, these lead to large changes in the pressure field which, for example, rises up to 1.3 pgH at about t(g/Р/)1 /2 = 6.65 and almost decreases to zero at t{g/H)ll2 = 7.85 (see the top-right plot of Figure 9). Figure 7 shows that the pressure variations measured in the experiments are generally smaller during this stage. This is due to the presence of the air in the cavity (i.e., air-cushioning effects) and to three-dimensional effects.

Figure 10 shows the pressure time histories recorded at the probe P, placed at у = 0.267H. As stressed in (Lobovskv et al., 2014), it was not possible to obtain a good repeatability for the measures at P4, since the pressure signals showed large fluctuations

Figure 8: Dam-break flow of a water column of height H and width 2H impacting on a vertical wall. Enlarged view of the pressure field during the vertical drop of the water.

Figure 9: Dam-break flow of a water column of height H and width 2H impacting on a vertical wall. Enlarged views of the pressure field during four time instants.

Figure 10: Dam-break flow of a water column of height II and width 2H. Time histories of the pressure recorded on the probe P4.

between the different runs. For this reason, in Figure 10 we also reported the curves of the averaged value and of the percentiles 2.5 and 97.5 from Lobovsky et al., 2014. The numerical solution predicted by the rt-SPII scheme lays between these curves. In the past, repeatability issues with the experiments created several problems for the validation of the SPH model on this benchmark test-case. For example, in (Buchner et ah, 2011) the same experiments were performed by using pressure probes characterized by a large diameter, namely 9 cm.

As a consequence, those probes in such a critical region (characterized by very local loads and large oscillations) provided measures that were difficult to use for a reliable validation procedure (see, e.g., Colicchio et al., 2002; Marrone et ah, 2011).

At t(g/H)112 = 7.85 the potential energy reaches its second peak which corresponds to the high water column fed during the splash-up stage (see the top-right plot of Figure 9). At this time instant the cavity reaches is maximum extension and the pressure field becomes noisy, because of the impacts of falling water drops and because of the closure of many small cavities generated during the run-down of the flow on the vertical wall.

The bottom-left plot of Figure 9 displays the collapse of the cavity at time t(g///)1 ^2 = 8.50. This causes a large pressure peak of intensity ~ 2.bpgH (see, for example, Figure 13). The cavity closes with a velocity of about v = 0.25JgH, and the collapse induces a hammer pressure close to pvcq.

After this stage the flow evolves in a complex way, moving from right to left, with the fragmentation of the free-surface and the generation of several small jets and drops (see the bottom-right plot of Figure 9).

Finally, Figure 11 sketches the time histories of the forces evaluated on the left and right vertical wall and along the bottom. The forces are made dimensionless through the water weight W = pglll1 acting on the bottom before the dam release. Note that the water weight vector, namely —Wj, has been subtracted to the vertical force Fy, and, consequently, the signal for Fy oscillates around the zero level. Figure 11 shows that the force along the bottom is subjected to the largest variations even in comparison to the force along the right wall (where the first water impact occurs). In particular, the cavity collapse induces an overload of ~ 1.811' on the bottom while its effect on the right wall is more modest.

In references Colagrossi and Landrini, 2003 and Marrone et al., 2016 the same simulation has been performed including the presence of air. In fact, the latter substantially modifies the dynamics of the flow after the cavity closure, i.e., for t(g/H)^2 > 6. In (Colagrossi and Landrini, 2003) different values of the reference pressure of the air were considered and their influence on the cavity evolution and on the pressure recorded on the right wall was discussed. Conversely, in (Marrone et al., 2016) the reference pressure was set to Punr = 17-2 pgH, that corresponds to Poair = 1 atm in the length scale experiment of Buchner et al., 2011. In this latter case, the dynamic of the post-breaking stage is reported in Figure 12. Compared to the single-phase simulation, the presence of the air prevents the reduction of the cavity (top-right plot of Figure 12) which at t(g/H)C2 = 8.43 split in two parts (bottom-left plot of Figure 12).

Figure 13 depicts the time histories of the pressure recorded at the probe P3 positioned at у = 0.17H for both the single- and two-phase model. The pressure signal provided by the single-phase simulation has been filtered in order to remove the acoustic components (see, for example, Meringolo et al., 2017). No filtering procedure has been applied to the signal

Figure 11: Dam-break flow of a water column of height II and width 2H. Time histories of the forces predicted by the d—SPH acting on the left and right vertical walls and along the bottom.

Figure 12: Dam-break flow of a water column of height H and width 2H impacting on a vertical wall. Snapshots of the evolution using an air-water S—SPH model.

Figure 13: Dam-break flow of a water column of height H and width 2H. Time histories of the pressure recorded at the probe P3 (y = 0.17//) evaluated through a single- and a two-phase S—SPH model.

of the two-phase model whose oscillations are mainly due to the air cushion effect caused by the gas entrapped in the cavity. Incidentally, we observe that this behaviour is not visible in the experimental pressure records of (Lobovsky et al., 2014). A possible reason is that 3D effects lead to a complex cavity shape which is fragmented much faster than the cavity predicted by the 2D models.

Finally to close this section a more realistic case is considered moving to a three- dimensional framework. As in (Marrone et al., 2011) the dam-break flow against a prismatic column is here discussed. Figure 14 shows some snapshots of the free-surface evolution after the water impact against the column. At t = 0.36 s (top-left plot of Figure 14) part of the water hits the column and generates a run-up while the the remaining bulk of fluid moves forward. At t = 0.51 s (top-right plot) the run-up stage ends and the water along the column starts its descent, while the remaining part of the flow hits against the rear vertical wall. At t = 0.83 s (bottom-left plot) a plunging jet is formed just in front of the column with a dynamic similarly to the one discussed for the 2D case. The water impact on the vertical wall is more complicated because of the large 3D effects and the free-surface is characterized by a complex shape because of the fragmentation. The bottom-right plot, at t = 1.05, shows the formation of a splash-up in front of the column and a violent flow generated behind it by the flow reflected at the rear vertical wall.

Here, for the sake of brevity, the description is just limited to a qualitative point of view, while in (Marrone et al., 2011) a validation against experimental data is also provided. In (Marrone et al., 2011) about 1.1 millions particles were used (i.e., H/Ax = 75 where H is the

Figure 14: Dam-break flow against a vertical column. Snapshots of the free-surface evolution.

Figure 15: Dam-break flow against a vertical column. Contour plots of the pressure field on a vertical and a horizontal plane.

water column height at the beginning of the simulation). This resolution was fine enough to obtain a good match with the experimental data in terms of the forces acting on the column and local measures of the velocity in front of the column. Here, the simulation is repeated increasing the number of particles to 40 millions, i.e., H/Ax = 245. The time steps has been set equal to 0.2 milliseconds and the code marched for 10,000 steps. The simulation ran on 300 cores for 24 hours. Figure 15 reports a vertical and a horizontal slice of the flow field containing pressure contour plots.

## Extreme loads on a Wave Energy Converter (WEC)

As a final application the 3D simulation of an extreme wave impacting a flap-type wave energy converter is considered. Specifically, the survivability of a bottom-mounted pitching device, consisting of a partly submerged flap placed in front of a breakwater, is studied numerically. To this aim, a realistic site is considered: a flap-type WEC positioned ahead the north dike in the city of Bayonne, South west Atlantic coast of Fiance (further details can be found in Baudry et al., 2015). Wave measurements from the closest buoy are considered in order to have realistic characteristics of the extreme events occurring in the site. For the generation of the wave, the adopted numerical set-up is composed by a piston wave-maker inside a tank limited by a sloping dike (see the sketch in the top plot of Figure 16). In order to produce the desired wave characteristics in terms of period and height, a sinusoidal motion law has been prescribed to the piston following the formulae described in (Dean and Dairymple, 1991). In the bottom plot of Figure 16 the adopted geometrical configuration is shown. For the sake of completeness, a view of the full numerical domain adopted for the simulation is provided in Figure 17. The tank width has been set equal to 60 m in order to avoid possible reflections from the lateral walls.

In Figures 18 and 19 the free-surface evolution is depicted from two different perspectives before and after the impact. The wave breaking starts just before the impact and, then, the

Figure 16: Top: Sketch of the numerical domain and definition of the main parameters. Bottom: Geometrical configuration adopted for simulation.

Figure 17: Numerical set-up for the 3D simulations.

Figure 18: Free-surface evolution of the wave impacting a bottom-hinged wave energy converter from two different views, just before the impact.

Figure 19: Free-surface evolution of the wave impacting a bottom-hinged wave energy converter from two different views, just after the impact.

wave impacts the flap with a front almost parallel to the body surface, causing a complex 3D splashing process with several jet ejections and fragmentations. During the impact stage the flap rotation remains quite small. As a consequence, the loading process evolves as a sort of flip-through phenomenon Lugni et al., 2006, similar to what documented by Peregrine, 2003 for waves impacting vertical walls. For this kind of dynamics, the impact is characterized by very small time and space scales.

This behaviour can be better observed performing the same simulation within a 2D framework. The wave steepens as it approaches the flap and its profile becomes almost parallel to that of the flap (see Figure 20). As a consequence, the contact point between liquid and solid settles very rapidly along the body and the momentum transfer of the wave focuses in a very small region. This behaviour can be better observed in Figure 21 where the pressure measured along the flap is plotted on the time-space plane. A very narrow pressure peak is observed around t = 9.25 s. This pressure increase expires in few tenths of seconds and involves a space scale of about 0.1 m. Apart from the large force and torque experienced, this kind of loading is potentially harmful for the device, as it can imply local permanent deformations of the structure.

The time histories of the horizontal force and torque around the hinge are reported in Figure 22 for both 3D and 2D simulations (the latter multiplied by the flap width). In the 2D model the torque peak has a narrower shape and is overestimated by a factor close to 1.5. This can be justified by the fact that the flap shape is close to a square and, therefore, 3D effects can be non-negligible.

Figure 20: Pressure field evolution before and after the impact, in the 2D simulation.

Figure 21: Evolution of the measured pressure on a section of the flap wall in the time-space plane.

Figure 22: Comparison of the torque around the hinge evaluated by the 2D and 3D models (2D values are multiplied by the flap width for consistency with the 3D model).

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