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# Numerical results

The numerical simulations produce a large amount of data to analyse, considering the number of alternatives simulated and the fields available (VOF, velocity, pressure, turbulent variables...) Therefore, we have selected 5 variables to rationalize the analysis of the alternatives. Fx, F~ are the maximum wave-induced horizontal and vertical forces acting on the crown wall, measured in Newtons (N) and calculated by integrating the pressure field around the structure, filtering the time signal to eliminate impulsive loading. qm is the mean overtopping rate over the event, measured in m3/s; and ^ Q is the total overtopping volume, measured in m3. Consequently, the overtopping event duration can be calculated as J2Q/

5.2.1 Reference case

The base case is taken as a reference to analyse the performance of the new alternatives and the values are included in Table 2. The numerical experiments have been simulated at laboratory scale, therefore, some variables are scaled up to prototype scale for direct comparison with data available in literature. The model has been designed at a 1/10 length scale, which implies that the prototype structure is at 8 m of water depth. Froude number (defined as Fr = uj[gl, where и is velocity and l is a length scale) similarity scaling is applied to preserve the ratio between the inertial and gravitational forces. Therefore, assuming no scale effects, the laboratory-scale variables can be transformed into prototype-scale values by multiplying them by a factor. The factors for the different variables are as follows: 3.16 for time and velocity, 10 for length, 316.23 for flow rate and 1000 for volume and force.

The weight of the crown wall at laboratory scale is Fy = 4112 N, considering the emerged and submerged weights accordingly for the present still water level, and a dry concrete density of 2400 kg/m3. The maximum horizontal force induced by the solitary wave is 28% of F?, while the maximum uplift force totals 52% of Fw.

The overtopping results can be compared with the EurOtop manual (van der Meer et ah, 2018). As referenced in Table 2, the equivalent prototype-scale mean discharge rate is qm = 60 m3/s per unit metre and the total overtopping volume is 250 m3 per unit metre. These values are one to three orders of magnitude larger than the maximum values reported in the EurOtop manual. For example, a qm = 0.05 m3/s and ^ Q between 5 and 50 m3 would already pose a major threat, causing “significant damage or sinking of larger yachts”. This

Table 2: Reference variables and values obtained for the base case. LS stands for laboratory scale. PS stands for prototype scale.

 Variable Fx (kN) Fz (kN) qm (m3/s) Y.Q (m3) SCs Value (LS) 1.149 2.123 0.19 0.25 0.87 Value (PS) 1149 2123 60 250 0.87

makes clear that the solitary wave conditions chosen represent an extreme event such as a tsunami wave. Moreover, since the lowest value of SF.v during the simulation is 0.87, the structure is well below the stability limit for the wave conditions tested, and is at risk of failing due to the caisson sliding. This is the main justification that drives the need to find new alternatives with which to achieve structural stability.

5.2.2 Global analysis of alternative A cases

Given the numerous cases and variables to analyse, the results for alternatives A, and later B, are represented as an array of radar plots in Figures 4 and 5. Each of the variables except SF.v are represented in terms of the percentage of variation with respect to the reference case values. The central line indicates no variation (0%), while the upper and lower limits are indicated in the top left panel. Variable SF.v is scaled differently: the central line corresponds to the limit state, SF.v = 1. Any values below 1 (outer region) are considered potentially unsafe, as the crown wall will be likely to slide. Therefore, for all variables, a point in the inner region indicates a safer condition with respect to the original geometry, while the outer region signals a more dangerous condition.

The results for alternative A (new armour layer) are displayed in Figure 4. The variation of all the indicators is quite limited, below 5%. Changes are low, around 1%, especially for those alternatives with large negative freeboard and narrow berm width (upper and left sections). At the lowest freeboard values (-0.8 and -0.6, top two rows), increasing the berm width does not produce significant changes, as the crest of the new layer is too deep to alter the incoming wave significantly. As the freeboard increases (-0.4 and -0.2), the effect of the berm width starts being noticeable, reducing both qm and Y Q up to 5% and 2.5%, respectively. However, in all cases the SF.v is still below 0.9, thus, the stability of the crown wall continues to be compromised. The cases with the highest freeboard (/ = 0, bottom row) show a consistent and noticeable decrease in the hydrodynamic forces and overtopping up to a 6% as the berm width increases, bringing SFs closer to 1 progressively. The only case in which SF.v is just above 1 is (/ = 0, b = 1). This is the most expensive alternative in terms of building cost, and will be analysed and compared with the original structure later in the text.

5.2.3 Global analysis of alternative В cases

The results for alternative В (detached breakwater) are shown in Figure 5. In this plot the range of variation of each indicator variable is different, as indicated in the top left panel. Similarly to what occurred for alternative A, the cases with the lowest freeboard (-0.8, -0.6) do not show significant differences with respect to the original alternative. Also, separation “s" does not have a major impact either, as the four lines represented are almost collapsed into one. For example, a detached breakwater with / = —0.8 will only reduce the variables studied by 2% at most, independent of its berm width and separation from the main breakwater. Nevertheless, for / = —0.6, the reduction in the total overtopping volume starts becoming noticeable (5%) for the largest berm widths. Also, the separation of the detached breakwater (s) starts having a noticeable impact in qm for the longest berm widths (0.6-1) at / = -0.4.

As the freeboard continues to increase, the overtopping variables decrease significantly faster than forces. For example, at f = —0.2 the force reduction reaches 3-5% while Y Q decreases up to 29%. The main reason behind this is that the detached breakwater produces local wave transformations, but does not induce wave breaking for / = —0.2, thus, the amount of energy reflected and dissipated is limited. Moreover, since qm decreases much less than Y Q- if means that the time of the overtopping event (^ Q/qm) must be reduced too.

Figure 4: Percentage of variation for the horizontal and uplift forces, mean overtopping discharge rate and total overtopping volume and safety factor against sliding for the cases in alternative A. From top to bottom: / =(-0.8, -0.6, -0.4, -0.2, 0). From left to right: b =(0.2, 0.4, 0.6, 0.8, 1). The range of variation per variable is indicated in the top left panel.

It can be noted that there is no linear correlation between s and the value of qm. While the maximum reduction of qm corresponds to cases with s = 0, qm magnitude can also increase with respect to the original case for a certain range of parameters. For example, the largest reduction of q,„ is 9%, for s = 0 (square marker). Cases with s = 1 and s = 3 also show a reduction, although less significant. However, qm increases slightly in cases with s = 2 (triangle marker). This irregular evolution can only be explained due to local effects on wave transformation.

Even though overtopping reductions can be significant for / < —0.2 in alternative B, no single case fulfils that SFs > 1, hence, the structure would still be at risk of failing by sliding. The bottom row in Figure 5 (/ = 0) includes all the cases in which SF.v > 1. The effect of the berm width plays an important role in reducing the overtopping volume, whereas the separation s does not produce significant differences for this variable. The separation (s)

Figure 5: Percentage of variation for the horizontal and uplift forces, mean overtopping discharge rate and total overtopping volume and safety factor against sliding for the cases in alternative B. From top to bottom: / =(-0.8, -0.6, -0.4, -0.2, 0). From left to right: b =(0.2, 0.4, 0.6, 0.8, 1). The range of variation per variable is indicated in the top left panel. The separation of the structure (s) is: 0 for continuous line and square marker, 1 for dashed line and circumference marker, 2 for dotted line and triangle marker and 3 for dash-dot line and plus sign marker.

is, however, important when analysing SFs, especially for the narrowest berms. In case of / = 0, b = 0.2, the two smallest separations (s = 0 and 1) fail, while cases with s = 2 and 3 are above SFs = 1. For cases with b = 0.4,0.6, only the case with s = 0 would produce a SFs < 1. Above b = 0.6 all the cases, independent of s, would fulfil SFs > 1. The largest SFs obtained is 1.08, for case (/ = 0, b = 1, s = 3). Therefore, this case will be studied in depth below.

It must be remarked that there is no direct relation between the maximum reduction of horizontal and uplift forces and the increase in SF.v, because the maximum forces in both directions do not occur at the same instant .

5.2.4 Detailed analysis of the alternative A and В selected cases

In this subsection we perform a detailed comparison of the selective cases, alternative A (/ = 0, b = 1), alternative В (/ = 0, b = 1, s = 3) and the original reference case.

The evolution of the wave and velocity field before and during the overtopping phase are represented in Figures 6 and 7. At t = 7.30 s the additional layer in alternative A has not produced any significant effects yet, as the wave is still several water depths away from the toe of the breakwater, whereas the detached breakwater has induced wave breaking, thus changing the shape and kinematics of the incident wave. The most noticeable effects in alternative В include a larger particle velocity between the free surface and the top of the breakwater and a lower velocity inside the breakwater due to the porous medium drag force. Also, the maximum free surface elevation decreases as compared with the reference case, showing a flatter wave crest.

At t = 8.65 s the wave has just impacted the crown wall. Both alternatives show noticeable differences with respect to the original case. The most obvious difference is that overtopping is delayed, since it is incipient for the base case (bottom subpanel) but no water has surpassed the crown wall yet in either of the other two alternatives. In solution A, the deviation with respect to the initial case is limited, and localized near the crown wall because the new armour layer attached to the breakwater produces energy dissipation and a wake effect above it, effectively delaying the impact and overtopping phases. The water level at the front of the crown wall is significantly lower for alternative В at this stage, because of wave breaking occurrence. The detached breakwater also produces noticeable energy dissipation, inducing a wake after the structure that is intense and creating a clockwise vortex that remains in place for a long time.

Figure 7 shows two time steps during the overtopping phase. At t = 9.15 s, the overtopping jet in alternative A looks very similar to that in the original case, although in a previous stage of development, i.e., while the jet has recently impacted the esplanade of the crown wall and is flowing to the leeside, it has not yet been projected back towards the vertical wall,

Figure 6: Evolution of the solitary wave before overtopping. For each instant, the top subpanel is case A (/ = 0, b = 1), the middle subpanel is case В (/ = 0, b = 1, s = 3) and the bottom subpanel is the reference case.

Figure 7: Evolution of the solitary wave during the overtopping phase. For each instant, the top subpanel is case A (/ = 0, b = 1), the middle subpanel is case В (/ = 0, b = 1, s = 3) and the bottom subpanel is the reference case.

as in the base case. This effect will eventually occur for all three cases, the only differences being the magnitude of the back-flow and the time of occurrence. The alternative В presents a significantly different jet shape, caused by a deficit in horizontal momentum as a result of wave breaking. Nevertheless, the depth of the overtopping jet is thicker at this instant.

At the final panel, t = 9.50 s, all three overtopping jets show similar features as recirculation behind the vertical wall and impact on the quiescent water body behind the breakwater. The discharge volume is so large that the disturbance created on the leeside of the breakwater is of the order of magnitude of the crown wall height. This observation is aligned with the reports in EurOtop, by which even much smaller discharge volumes will cause safety threats or even sink large vessels. Another feature to note is the vortex located on the leeside of the detached breakwater in alternative B. As mentioned before, this coherent structure persists a long time in that position. Although it has not appeared yet, another vortex is created on the seaside of the new berm in alternative A, which also persists for long time, although with a lower absolute velocity.

The reflection coefficient R is calculated at a wave gauge near the wave generation boundary, as the ratio between the maximum elevation of the reflected wave and the height of the incident wave. The range of variation of R among all the cases simulated is 28.0% to 40.1%, with R = 32.9% for the original alternative. The selected alternative A yields the lowest global reflection coefficient, 28.0%, whereas the selected alternative В produces a reflection coefficient of 31.6%. This 3.6% difference is not very significant, as it corresponds to less than 1.5 cells, but indicates that since overtopping has also been reduced in both cases, the new structural elements produce additional energy dissipation.

The time evolution of the overtopping variables is represented in Figure 8. The instantaneous overtopping rate (q) shows that the alternative A causes a delay of 0.11 s in the start of overtopping, and the slope of the curve is also steeper initially. The rest of features, such as the maximum value or ending time, remain practically unchanged, thus the total overtopping time is reduced. An almost identical evolution can be observed for the installtaneous overtopping water depth (central panel), except that the alternative A produces a 15% reduction with respect to the highest value. This translates into a small reduction of the total overtopping volume (3%, right panel) and of the mean instantaneous overtopping rate (5%).

Alternative В produces significant changes because the solitary wave breaks before reaching the structure. First, the overtopping event starts 0.25 s after and finishes almost at the same instant as in the reference case, therefore, the duration of the overtopping event is reduced by almost 20%. The initial slope of q is almost identical to that of the reference case. However, the shape of the overtopping curve is very different, with a marked peak instead of a smooth crest. The variable hQ also follows the same trend, with a maximum at 30 cm water depth. The maximum value of q is larger than in the reference case by 9%, yet the area under the curve is significantly smaller, as can be seen in the right panel, in which the total overtopping volume is reduced substantially (32%). In spite of slightly increasing the maximum q value, alternative В reduces its mean value by 13%. Nevertheless, this reduction is not enough for the structure to be within the values reported in the EurOtop manual, as the original wave condition is extreme compared to the freeboard of the structure.

The wave-induced forces acting on the structure are presented in Figure 9. Some of the previous observations for overtopping are also applicable to the forces time series. The main difference is that forces increase very slowly at first, and start growing faster prior to the overtopping event. This is because the wave progressively impacts the vertical wall, increasing the water level steadily and building up the pressure prior to surpassing it. After overtopping starts, the growth rate decreases, and forces reach their maximum value. At

Figure 8: Overtopping analysis of the selected alternatives. The left panel represents the instantaneous overtopping rate (q). The central panel represents the instantaneous overtopping water depth (ha). The right panel represents the cumulative overtopping volume Q). The continuous line denotes the reference case, the dashed line is for the selected alternative A (/ = 0, b = 1) and the dotted line represents the selected alternative В (/ = 0, b = 1, s = 3).

Figure 9: Wave-induced force analysis. The left panel represents the horizontal force and the right panel represents the uplift force. The continuous line denotes the reference case, the dashed line is for the selected alternative A and the dotted line represents the selected alternative B.

this stage the total maximum force reduction is 5% for case A and 10% for case B. both for Fx and Fz. In view of the time series, it can be concluded that the wave breaking event in alternative В only has a significant effect on the initial part of the Fx curve. The evolution of the trailing edge is quite similar in all cases, with forces that start to decrease progressively, following the trend of the overtopping water depth. However, small differences in Fz, in terms of a slight delay in the end of the uplift force time series can also be observed. In any case, not only the magnitude of the maximum forces is reduced, but also the total duration of the impact event is shortened for case B.

As mentioned before, these cases were selected because the new design prevents the structural failure in terms of a quasi-static analysis. The reduction in the force magnitudes translates into a SF.y that is just above 1 for alternative A and a SFs equals to 1.08 in

case B.

# Concluding remarks

This example portraits the capabilities of VARANS modelling to evaluate different structural alternatives for a given scenario. The numerical model has proven to be a useful tool to calculate the overtopping and forces acting on a structure, therefore, to establish its functionality and stability regimes. However, numerical modelling is complementary to experimental modelling, as validation is a necessary step to check whether the model (in combination with the numerical schemes and mesh used) is able to simulate accurately all the physical processes involved in a specific case.

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