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Table 3. Results obtained from the simulations carried out using the k-e turbulence model and relative errors with respect to the experimental values of the incident wave. The results obtained for the k-e model were close to the experimental ones, so that the same mesh was used for the simulations carried out with the SST turbulence model. The value for the turbu kinetic energy was the same as for the k-e model, as well as the value of the specific dissipation rate The similar behavior of both two-equation models reinforces the idea of the use of each model for ent (a). the problem that best suits each of them. The tendency in the relative errors is the same in both models, as it can be seen in Table 4. However, the SST model has a worst definition in the measurements of wavelength and celerity, while its definition of the wave height has marginal improvements w comparing it to the k-e model. the hen The wave parameters obtained from the simulations using the k-e turbulence model can be seen in Table 3. A limit of 10% for the wave height and 2% for the period were selected to validate the models. Although the errors are low, always below 2% in period, a better definition of the paddle movement is needed because errors become higher when higher waves are generated. It is important to highlight the satisfactory validation of the equation that the moving wall follows. Next, the wavelength was calculated with the phase-shift of the signals between probes and finally the wave propagation velocity, c. It is worth mentioning that the calculation of the wavelength through the phase-shift introduces a cumulative error from the signal fitting, so it will be further studied for its improvement.

Table 3 Results obtained from the simulations carried out using the k-e turbulence model and relative errors with respect to the experimental values of the incident wave. The results obtained for the k-e model were close to the experimental ones, so that the same mesh was used for the simulations carried out with the SST turbulence model. The value for the turbu kinetic energy was the same as for the k-e model, as well as the value of the specific dissipation rate The similar behavior of both two-equation models reinforces the idea of the use of each model for ent (a). the problem that best suits each of them. The tendency in the relative errors is the same in both models, as it can be seen in Table 4. However, the SST model has a worst definition in the measurements of wavelength and celerity, while its definition of the wave height has marginal improvements w comparing it to the k-e model. the hen The wave parameters obtained from the simulations using the k-e turbulence model can be seen in Table 3. A limit of 10% for the wave height and 2% for the period were selected to validate the models. Although the errors are low, always below 2% in period, a better definition of the paddle movement is needed because errors become higher when higher waves are generated. It is important to highlight the satisfactory validation of the equation that the moving wall follows. Next, the wavelength was calculated with the phase-shift of the signals between probes and finally the wave propagation velocity, c. It is worth mentioning that the calculation of the wavelength through the phase-shift introduces a cumulative error from the signal fitting, so it will be further studied for its improvement.

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All the simulations are replicas of the tests carried out in the EWF of the Energy Engineerir Jepartment at the School of Engineering in Bilbao (University of the Basque Country, UPV/EHL ‘he EWF is 12.5 m long, 0.6 m wide, and 0.7 m hi nd tempered glass and the tank base of stainless s gh. The walls of the flume are made of laminate teel. The commercial software ASDA-Soft (V5) [3. ontrols the piston-type wavemaker of the flume by controlling the servo drive and servo mot ‘he period and amplitude of the paddle are defined in the software, which converts the input data in inusoidal curve of progressive acceleration and deceleration of the motor. The motor is connecte o a linear actuator, which is attached to the pad ‘hen, the intended wave is generated according dle, creating the linear movement of the padd to the wavemaker theory [33,34] for a piston-tyt vavemaker. The wave energy extinction area consists of a self-designed parabolic extinction syster vhich can be adjusted in height and slope to situate it in the position of maximum wave absorptio ‘he scheme of the extinction system can be seen in Figure 1. Figure 1. (a) The general view of the EWF from the piston-type wavemaker (left) to the extinctior system (right). (b) Picture of the self-designed extinction system installed in the wave flume and th: highlights of the join (A) and the position of the cranks (B and C) to modify the angle Figure 1. (a) The general view of the EWF from the piston-type wavemaker (left) to the extinction
All the simulations are replicas of the tests carried out in the EWF of the Energy Engineerir Jepartment at the School of Engineering in Bilbao (University of the Basque Country, UPV/EHL ‘he EWF is 12.5 m long, 0.6 m wide, and 0.7 m hi nd tempered glass and the tank base of stainless s gh. The walls of the flume are made of laminate teel. The commercial software ASDA-Soft (V5) [3. ontrols the piston-type wavemaker of the flume by controlling the servo drive and servo mot ‘he period and amplitude of the paddle are defined in the software, which converts the input data in inusoidal curve of progressive acceleration and deceleration of the motor. The motor is connecte o a linear actuator, which is attached to the pad ‘hen, the intended wave is generated according dle, creating the linear movement of the padd to the wavemaker theory [33,34] for a piston-tyt vavemaker. The wave energy extinction area consists of a self-designed parabolic extinction syster vhich can be adjusted in height and slope to situate it in the position of maximum wave absorptio ‘he scheme of the extinction system can be seen in Figure 1. Figure 1. (a) The general view of the EWF from the piston-type wavemaker (left) to the extinctior system (right). (b) Picture of the self-designed extinction system installed in the wave flume and th: highlights of the join (A) and the position of the cranks (B and C) to modify the angle Figure 1. (a) The general view of the EWF from the piston-type wavemaker (left) to the extinction
Figure 2. Disposition of the resistive probes in the EWF. The arrows highlight the join with the structure that allows them to move along the flume. The free surface variation (7 (m)) is measured in three different points by using resistive-type probes, the position of which can be modified to adjust them according to the wavelengths of the waves generated. The free surface displacement is recorded by means of an ad hoc LABVIEW in time intervals of 3 ms. It is important to highlight that the probes must be separated in equal distances, as it can be observed in Figure 2, avoiding the half and quarter of the wavelength of study to avoid errors due to their positioning. The first resistive probe is situated at 5.5 m from the paddle to ensure, at least, a minimum distance of two wavelengths of the wave generated. In this study, the wavelengths of all the waves are smaller than 2.75 m, which allow, on the one hand, the waves to be fully developed when reaching the first probe and, on the other hand, to have a maximum flume length for each study 2.2. Numerical Wave Flume
Figure 2. Disposition of the resistive probes in the EWF. The arrows highlight the join with the structure that allows them to move along the flume. The free surface variation (7 (m)) is measured in three different points by using resistive-type probes, the position of which can be modified to adjust them according to the wavelengths of the waves generated. The free surface displacement is recorded by means of an ad hoc LABVIEW in time intervals of 3 ms. It is important to highlight that the probes must be separated in equal distances, as it can be observed in Figure 2, avoiding the half and quarter of the wavelength of study to avoid errors due to their positioning. The first resistive probe is situated at 5.5 m from the paddle to ensure, at least, a minimum distance of two wavelengths of the wave generated. In this study, the wavelengths of all the waves are smaller than 2.75 m, which allow, on the one hand, the waves to be fully developed when reaching the first probe and, on the other hand, to have a maximum flume length for each study 2.2. Numerical Wave Flume
Figure 3. (a) Mesh corresponding to a depth of 50 cm, (b) focus on the prism layers around the generation paddle. (c) The different areas of the mesh: (1) air, (2) below the paddle, (3) “deep-water”, (4) free surface, and (5) extinction system. All the boundaries in the 2-D meshes have a non-slip condition and the bottom and the top have a morphing boundary plane constraint to reproduce a realistic movement of the paddle. In the 3-D mesh the walls satisfy the non-slip condition except the lateral ones, with a slip condition to delete the friction influence of the walls in the wave generation and propagation. This allows a proper comparison with the experimental flume, in which this friction can be neglected due to its much larger width.
Figure 3. (a) Mesh corresponding to a depth of 50 cm, (b) focus on the prism layers around the generation paddle. (c) The different areas of the mesh: (1) air, (2) below the paddle, (3) “deep-water”, (4) free surface, and (5) extinction system. All the boundaries in the 2-D meshes have a non-slip condition and the bottom and the top have a morphing boundary plane constraint to reproduce a realistic movement of the paddle. In the 3-D mesh the walls satisfy the non-slip condition except the lateral ones, with a slip condition to delete the friction influence of the walls in the wave generation and propagation. This allows a proper comparison with the experimental flume, in which this friction can be neglected due to its much larger width.
Table 1. Theoretical values of the studied waves.
Table 1. Theoretical values of the studied waves.
Table 2. Experimental results and relative errors of the incident wave with respect to the theoretically intended values.
Table 2. Experimental results and relative errors of the incident wave with respect to the theoretically intended values.
Figure 4. Fitting of the free surface displacement for the wave 3: fitting (orange) and experimental (blue) measured by the first probe of the EWF. The fittings of the free surface displacement signals were done using the incident wave interval, which corresponds to all the data between the ts and tp time values. This fitting, shown in Figure 4, matches almost perfectly with the free surface variation data obtained from the experimental runs. This adjustment constantly achieves a quality of overlapping over 95% in the 3 probes for every wave.
Figure 4. Fitting of the free surface displacement for the wave 3: fitting (orange) and experimental (blue) measured by the first probe of the EWF. The fittings of the free surface displacement signals were done using the incident wave interval, which corresponds to all the data between the ts and tp time values. This fitting, shown in Figure 4, matches almost perfectly with the free surface variation data obtained from the experimental runs. This adjustment constantly achieves a quality of overlapping over 95% in the 3 probes for every wave.
Table 4. Results obtained from the simulations carried out using the SST turbulence model and relative errors with respect to the experimental values of the incident wave.
Table 4. Results obtained from the simulations carried out using the SST turbulence model and relative errors with respect to the experimental values of the incident wave.
Table 5. Results obtained from the simulations carried out using the LES turbulence model and relative A comparison of the free surface displacement was carried out between ex simulations incident and displacemen models used generating a reflected waves and observation of whether the extinction system affects in the first probe. The fitting is significantly good throughout all t similar energy dissipation by the extinction system, resulting in a simi displacemen periments and because of the low errors measured. This way allows a direct comparison between he free surface t. Figure 5 shows these comparisons between the experimental and the three turbulence he experiment, ar free surface at all three points. Figure 5 shows a good definition of the wave crests in all the models, but they fall short on the definition of the troughs, for which the LES models achieve a more constant definition of them.
Table 5. Results obtained from the simulations carried out using the LES turbulence model and relative A comparison of the free surface displacement was carried out between ex simulations incident and displacemen models used generating a reflected waves and observation of whether the extinction system affects in the first probe. The fitting is significantly good throughout all t similar energy dissipation by the extinction system, resulting in a simi displacemen periments and because of the low errors measured. This way allows a direct comparison between he free surface t. Figure 5 shows these comparisons between the experimental and the three turbulence he experiment, ar free surface at all three points. Figure 5 shows a good definition of the wave crests in all the models, but they fall short on the definition of the troughs, for which the LES models achieve a more constant definition of them.
Table 6. Comparison of the wave height and period of the resultant wave between the experimental and k-e model, and the error between them. Similar results were obtained with the SST model. While the definition of the wave periods is almost identical, a slight improvement is made in the wave height definition. The results can be seen in Table 7, where wave heights do not surpass a 7% error, while wave periods are always maintained at lower than 2%. This can be due to the improvements introduced by Menter, where the behavior of
Table 6. Comparison of the wave height and period of the resultant wave between the experimental and k-e model, and the error between them. Similar results were obtained with the SST model. While the definition of the wave periods is almost identical, a slight improvement is made in the wave height definition. The results can be seen in Table 7, where wave heights do not surpass a 7% error, while wave periods are always maintained at lower than 2%. This can be due to the improvements introduced by Menter, where the behavior of
Figure 5. Comparison of the free surface displacement for the turbulence model of (a) k-e, (b) SST, and (c) LES with the experimental data (blue) obtained in the EWF for the wave 3. the adverse pressure gradients is improved and thus, a better definition of the troughs is obtained, reaching, in some cases, a better definition.
Figure 5. Comparison of the free surface displacement for the turbulence model of (a) k-e, (b) SST, and (c) LES with the experimental data (blue) obtained in the EWF for the wave 3. the adverse pressure gradients is improved and thus, a better definition of the troughs is obtained, reaching, in some cases, a better definition.
Table 7. Comparison of the wave height and period of the resultant wave between the experimental and SST model, and the error between them. Finally, the LES results are shown in Table 8. Although a better definition of the waves is achieved with this model in deeper depths, it shows limitations when working at smaller depths. However, the tendency of the two other turbulence models when defining the wave period is constant and a better definition of the troughs is reached.
Table 7. Comparison of the wave height and period of the resultant wave between the experimental and SST model, and the error between them. Finally, the LES results are shown in Table 8. Although a better definition of the waves is achieved with this model in deeper depths, it shows limitations when working at smaller depths. However, the tendency of the two other turbulence models when defining the wave period is constant and a better definition of the troughs is reached.
Table 8. Comparison of the wave height and period of the resultant wave between the experimental and LES model, and the error between them. Finally, the reflection of the extinction system was studied to obtain more data for a proper comparison of the behavior of the turbulence models. The wave breaking, which is related to reflection, is the most turbulent process seen in the flume and therefore, it can result in higher differences between the turbulent models under investigation. Besides, the reflection coefficients allow comparison of the amplitudes of the incident and reflected waves, which confirms both fittings made to the free surface displacement signal.
Table 8. Comparison of the wave height and period of the resultant wave between the experimental and LES model, and the error between them. Finally, the reflection of the extinction system was studied to obtain more data for a proper comparison of the behavior of the turbulence models. The wave breaking, which is related to reflection, is the most turbulent process seen in the flume and therefore, it can result in higher differences between the turbulent models under investigation. Besides, the reflection coefficients allow comparison of the amplitudes of the incident and reflected waves, which confirms both fittings made to the free surface displacement signal.
Figure 6. Reflection coefficients measured for each wave in the experimental and numerical runs.
Figure 6. Reflection coefficients measured for each wave in the experimental and numerical runs.
Figure 7. Comparison of the incident and reflected amplitudes of the waves by the reflection method developed by Mansard and Funke for the experimental and numerical runs.
Figure 7. Comparison of the incident and reflected amplitudes of the waves by the reflection method developed by Mansard and Funke for the experimental and numerical runs.
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