# Numerical Modelling of an Innovative Conical Pile Head Breakwater

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## Abstract

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## 1. Introduction

_{t}), reflection coefficient (K

_{r}) and dissipation coefficient (K

_{d}) of non-perforated and perforated CPHBs are examined using the computational fluid dynamics (CFD) model REEF3D with both monochromatic and irregular waves. Before carrying out the numerical modelling of the CPHB, a sensitivity analysis is carried out to determine the optimal grid size and CFL number for accurate representation of waves. First, simulations on non-perforated and perforated CPHBs are performed using monochromatic waves. The numerical results are validated with experimental data [27,28], and the most efficient structural CPHB configurations are chosen. Further, the performance of the efficient CPHB models is examined with irregular waves. Finally, the performance characteristics with both types of waves are compared.

## 2. Modelling of CPHB Structure

#### 2.1. Details of Physical Modelling

_{12}= L/3 and X

_{13}= 2L/3. The position of the gauges was adjusted in relation to the wavelength (L) of each generated wave. Using the three-probe approach [32], the composite wave data recorded in three gauges were separated into incident and reflected wave components (H

_{i}and H

_{r}, respectively). The transmitted wave height (H

_{t}) was measured using WG4, which was located at a distance of ‘L’ from the structure. Further, an additional wave gauge (WG5) was employed to collect the wave data before placing the structure. These data were used for wave reconstruction in the numerical wave tank.

_{max}) and height of the pile heads (Y/H

_{max}) with different wave climates. The study found that b/D = 0.1, D/H

_{max}= 0.4 and Y/H

_{max}= 1.5 is the optimum configuration in terms of wave attenuation. Further, Sathyanarayana et al. [28] introduced perforations on the optimum CPHB and investigated the influence of perforation distribution (Pa), percentage of perforations (P) and relative size of perforations (S/D). The typical tabulation of Pa is illustrated in Figure 3. The percentage of perforations is the ratio of the total area of perforations to the corresponding surface area of the CPH. According to the study, the optimum configuration of perforated CPHBs is Pa = 50%, P = 19.2% and S/D = 0.25.

#### 2.2. Numerical Modelling

#### 2.2.1. REEF3D

_{i}is the averaged velocity over time t, $\rho $ is the density of water, $\nu $ is the kinematic viscosity, $\nu $

_{t}is the eddy viscosity, p is the pressure, and g is the acceleration due to gravity. The pressure terms in the RANS equation are solved by the projection method proposed by Chorin [43]. The BiCGStab algorithm [44] is applied to solve the Poisson equation for pressure. The fifth-order weighted essentially non-oscillatory (WENO) scheme developed by Jiang and Shu [45] is employed to discretise the convection terms of the RANS equation. Time discretisation is achieved through the third-order TVD Runge–Kutta scheme [46]. According to Brackbill et al.’s [47] continuum surface force (CSF) model, the material characteristics of the two phases are calculated for the numerical domain. REEF3D uses the Courant–Friedrichs–Lewy (CFL) criterion, which determines the optimal time steps to maintain numerical stability throughout the simulation. MPI (Message Passing Interface) is used for parallel computation between multiple cores to maximise the efficiency of the numerical model. The k-ω model presented by Wilcox [48] is applied for turbulence modelling, in which k and $\omega $ denote the turbulent kinetic energy and the specific turbulence dissipation rate, respectively. The two-equation k–$\omega $ model is defined by the following equations.

_{k}denotes the rate of turbulent production, and the values of the closure coefficients are σ

_{k}= 2, $\sigma $

_{$\omega $}= 2, $\alpha $= 5/9, β

_{k}= 9/100 and $\beta $ = 3/40. To limit the overproduction of eddy viscosity outside the boundary layer, the eddy viscosity is regulated by the eddy viscosity limiters presented by [48] as:

#### 2.2.2. Numerical Model Setup

_{t}and K

_{r}as per Isaacson [32] and Mansard and Funke [51]. The width of the tank is truncated by half (0.71 m to 0.355 m) using the symmetric plane boundary condition applied on one side of the tank. The other side of the tank has a no-slip wall boundary condition. Similar boundary conditions are also applied at the bottom of the tank. The waves are generated at one end using the Dirichlet inlet boundary condition. The active absorption method is adopted at the opposite end to absorb the transmitted waves, requiring no additional tank length. At the top of the NWT, a symmetric plane boundary condition is applied to represent the tank being open to the atmosphere. The potential effects of re-reflection in the NWT may be at the very minimum and not cognisable such that they may not have a noticeable effect on the results [52]. The details of the boundary conditions of NWT are presented in Figure 4. The same scale as that of the physical model (1:30) is adopted in the numerical model. In the case of monochromatic waves, the waves are reconstructed using the free surface elevation data measured by WG5 (refer to Figure 2).

_{r}for monochromatic waves is calculated using the three-probe approach in order to ensure consistency between physical and numerical modelling. The positioning of wave gauges is in accordance with the physical modelling (X

_{12}= L/3 and X

_{13}= 2L/3) as shown in Figure 4. For monochromatic waves, the K

_{t}, K

_{r}and K

_{d}are calculated as per Equations (7)–(9), respectively.

_{i}represents the incident wave height, and H

_{t}and H

_{r}represent the transmitted and reflected wave heights, respectively. The dissipation coefficient K

_{d}is computed using the wave energy conservation formula.

_{i}and H

_{r}, respectively). WG3 is positioned at a distance of L towards the seaside of the structure. WG1 and WG2 are separated by X

_{12}= L/10. The distance between WG1 and WG3 is constrained to lie within the ranges of L/6 < X

_{13}< L/3, X

_{13}≠ L/5 and X

_{13}≠ 3L/10. Hence, X

_{13}= L/4 is selected to adhere to these limitations. The transmitted wave height is measured using WG4, which is positioned at a distance of L, as shown in Figure 2.

_{p}is the peak angular wave frequency. The wave surface elevation data are generated theoretically for the required H

_{is}and T

_{p}by employing the equation of the Scott–Wiegel spectrum. Using the theoretical time-domain data, the waves are reconstructed in the NWT. The waves are simulated for a duration of 120 s. The wave transmission coefficient is calculated as K

_{t}= H

_{ts}/H

_{is}. The significant incident wave height (H

_{is}) and significant transmitted wave height (H

_{ts}) are obtained using a frequency domain analysis. The significant incident wave height is estimated as H

_{is}= 4.0$\sqrt{{m}_{0i}}$, where m

_{0i}is the zeroth moment of the incident wave spectrum obtained using the numerical probe located at x = 0.02 m. Similarly, the significant transmitted wave height is calculated using H

_{ts}= 4.0$\sqrt{{m}_{0t}}$, where m

_{0t}is the zeroth moment of the transmitted wave spectrum obtained using Wave Probe 4. The reflection coefficient (K

_{r}) is estimated using the procedure proposed by Mansard and Funke [51]. The dissipation coefficient (K

_{d}) is calculated using Equation (9), which is derived based on the law of conservation of wave energy.

## 3. Results and Discussion

#### 3.1. Validation of Wave Generation

#### 3.1.1. Monochromatic Waves

_{i}= 0.16 m, T = 1.8 s) and gentler (H

_{i}= 0.06 m, T = 2.0 s) wave heights for various grid sizes, as shown in Figure 5. For grid size optimisation, uniform grid sizes dx = dy = dz = 0.08 m, 0.04 m, 0.02 m or 0.01 m are considered while keeping the CFL number constant at 0.1. Table 2 presents the root-mean-square error (RMSE) values obtained by comparing the experimental data with the numerically reconstructed wave surface elevation. The grid size analysis clearly shows that lowering the grid size from 0.08 m to 0.04 m results in a reduction in the RMSE values. The free surface elevation is found to agree well with the measured data for a grid size of 0.02 m. Further reducing the grid size from 0.02 to 0.01 m shows a negligible improvement with higher computational time. It can be concluded from the grid refinement study that a grid size of 0.02 m is sufficient for accurate wave generation with a maximum RMSE of 0.0053 m. Therefore, dx= 0.02 m is used for further investigation of the influence of CFL number on wave generation. The CFL numbers considered for the sensitivity study are 0.4, 0.2, 0.1 and 0.05, as shown in Figure 5, and the errors associated are listed in Table 2.

_{i}= 0.16 m and T = 1.8 s case by employing the optimum grid size and CFL number (dx = 0.02 m and CFL = 0.1). It is found that the free surface elevations calculated in the 2D and the 3D NWT are in agreement.

#### 3.1.2. Irregular Waves

_{is}= 0.12 m and T

_{p}= 1.4 s). Similar to monochromatic waves, the simulations are performed in a 2D NWT by considering uniform grid sizes of 0.08 m, 0.04 m, 0.02 m or 0.01 m while maintaining CFL= 0.1. The free surface elevation is measured in the NWT using the numerical wave probe at x = 0.02 m. Figure 6a represents the spectral wave density obtained for different grid sizes. The numerical peak spectral wave density is higher than the experimental peak spectral wave density by 15.82% and 10.76% for grid sizes of 0.08 m and 0.04 m, respectively. The difference between the numerical and experimental peak spectral wave density decreases to 9.34% when the grid size is reduced to 0.02 m. Further reduction in grid size from 0.02 m to 0.01 m resulted in a reduction in error of only 0.38% (9.34% to 8.95%). Since the improvement of results is negligible between grid sizes of 0.02 m and 0.01 m, a grid size of 0.02 m is fixed for determining the optimum CFL number.

#### 3.2. Performance of CPHB with Monochromatic Waves

_{max}= 0.4 and 0.5). Perforations with the optimum size and arrangement (Pa = 50%, P = 19.2%, and S/D = 0.25) indicated by Sathyanarayana et al. [28] are used for the perforated structure. The results of the non-perforated and perforated CPHBs are validated with the experimental data and analysed to arrive at the best-performing configuration of CPHBs. The study is carried out in various wave energy regions with intermediate water depth conditions. The different combinations of wave heights and periods considered for the simulation of monochromatic waves are listed in Table 3. The cases are selected such that the wave steepnesses (H

_{i}/gT

^{2}) are of the same range as those in the experiments. Finally, The CPHBs with the best-performing configuration with and without perforations are subjected to irregular wave conditions.

#### 3.2.1. Validation of Numerical Results with Experimental Data

_{t}, K

_{r}and K

_{d}of two non-perforated pile heads (D/H

_{max}= 0.4 and 0.5) are compared to experimental data [27] in Figure 8. Figure 9 presents the validation of numerical results with the experimental data for the case of the perforated CPHB. Best-fit lines are drawn to gain a better understanding and interpretation of the results. The trend lines plotted for the numerical results of both non-perforated and perforated CPHBs match with those of the experimental results to a reasonable extent. In the case of non-perforated CPHBs, the numerical results slightly under-predict for K

_{t}(less than 4%) and over-predict for K

_{r}and K

_{d}(less than 9%) for both cases of D/H

_{max}. For the perforated CPHB, the variation is slightly higher (up to 12%) compared to the non-perforated structure. The RMSE calculated by comparing the experimental and numerical results is summarised in Table 4. The comparison of results shows that the numerically determined performance characteristics of both non-perforated and perforated CPHBs are in relatively good agreement with the experimental data.

#### 3.2.2. Effect of Relative Pile Head Diameter

_{max}= 0.4 and 0.5) of non-perforated CPHBs are performed to determine the influence of the pile head diameter and to arrive at the optimum-performing configuration. Figure 10 presents the simulated images of the wave crest interaction with non-perforated CPHBs for various D/H

_{max}at the same time step (t = 9.10 s). Due to the larger numbers and smaller spacing of pile heads, the horizontal velocity of waves (u

_{x}) is obstructed to a significant amount in the case of D/H

_{max}= 0.4 compared to that of D/H

_{max}= 0.5 (refer to Figure 10). To overcome the obstruction, a relatively considerable number of waves may enter into the hollow portion of the pile head for D/H

_{max}= 0.4 compared to D/H

_{max}= 0.5. The water that enters the perforated pile head flushes out and results in additional energy loss, as demonstrated in Figure 10a. When D/H

_{max}= 0.5, a relatively higher quantity of waves are transmitted between the pile heads with an intensified velocity, as seen in Figure 10b. The CPHB with D/H

_{max}= 0.4 configuration has a higher number of piles and about 9.8% higher blockage area compared to D/H

_{max}= 0.5. The higher blockage area increases the effectiveness of the obstruction of wave energy, leading to wave breaking over the structure along with higher wave reflection.

_{max}= 0.4, the horizontal propagation of the incident wave at the free surface is obstructed by the structure to a substantial extent, whereas, for D/H

_{max}= 0.5, the wave easily propagates through the larger gaps without a significant reduction in velocity. In addition, the formation of vortices is clearly noticed on the lee side of the structure for D/H

_{max}= 0.4 (Figure 11a), which contributes to energy losses. When D/H

_{max}= 0.5, the energy dissipation through vortex formation is reduced, possibly due to lower blockage resulting from a long distance between the pile heads.

_{max}). For higher wave steepness, D/H

_{max}= 0.4 exhibits about 8% lower K

_{t}, 18.2% higher K

_{r}and 6% higher K

_{d}compared to D/H

_{max}= 0.5. At lower wave steepness, it is noticed that the K

_{t}and K

_{d}are comparable. The lowest K

_{t}of 0.64 is obtained for D/H

_{max}= 0.4 at a higher wave steepness along with K

_{r}of 0.22 and K

_{d}of 0.73.

#### 3.2.3. Effect of Perforations

_{x}) as the wave interacts with the non-perforated CPHBs at different time instances (t). The increased area of the piles (CPHBs) contributes to a comparatively higher obstruction than conventional pile breakwaters, due to which the horizontal velocity of the waves is obstructed (refer to Figure 13b). Due to this obstruction, a part of the wave may propagate through the gaps between pile heads with an intensified velocity. Another part may flow over the pile head and enter the hollow portion of CPH (refer to Figure 13c). This results in turbulence and energy dissipation along with partial reflection of waves, as presented in Figure 13c.

_{t}by about 5% to 16.5%. The values of K

_{r}and K

_{d}are increased by about 27.25% and 10.28% on average, respectively. A minimum K

_{t}of 0.54 is calculated for the perforated CPHB at higher wave steepness, associated with a K

_{r}and K

_{d}of 0.28 and 0.80, respectively. The observed performances of the non-perforated and perforated CPHBs are in agreement with the experimental data [28] and other similar studies on pile breakwaters [21,22,24,26].

#### 3.2.4. Effect of Wave Steepness

_{i}/gT

^{2}) accounts for both the effects of wave height and period. Steeper waves tend to be unstable in nature, and the slightest obstruction to their propagation triggers wave breaking and energy loss, while gentler waves are comparatively stable. Figure 16 presents the wave interaction with the steeper and gentler waves, in which the steeper wave and gentler wave correspond to cases M1 and M8, respectively (refer to Table 3). As noticed in Figure 16, turbulence generation is higher for M1 than M8, which results in higher energy losses and higher wave attenuation. In the case of M8, the wave energy transmits smoothly around the pile heads without losing much energy, leading to higher K

_{t}values. In addition, higher reflection is calculated for the M1 case than for M8, as illustrated in Figure 16. In general, it can be inferred from Figure 8 and Figure 9 that K

_{t}is indirectly proportional to the wave steepness, while K

_{r}and K

_{d}are directly proportional. The wave attenuation capability of the CPHB is more pronounced for steeper incident waves than for gentler waves for both non-perforated and perforated CPHBs. For the non-perforated CPHB (D/H

_{max}= 0.4), the value of K

_{t}obtained against steeper waves is about 22% smaller than that of gentler waves. Similarly, for the same CPHB configuration with perforations, about a 23.6% smaller K

_{t}is obtained.

_{max}= 0.4 performs better than D/H

_{max}= 0.5. Further, perforations on the CPH surface proved to be advantageous in enhancing CPHB wave attenuation characteristics. Therefore, the best-performing non-perforated CPHB (D/H

_{max}= 0.4, Y/H

_{max}= 1.5 and b/D = 0.1) and perforated CPHB (Pa = 50%, P = 19.2% and S/D = 0.25) are considered for further investigation with irregular waves.

#### 3.2.5. Performance Comparison with Other Pile Breakwater Structures

#### 3.3. Comparison of CPHB Performance with Monochromatic and Irregular Waves

_{max}= 0.4, Y/H

_{max}= 1.5 and b/D = 0.1) and perforated CPHBs Pa = 50%, (P = 19.2% and S/D = 0.25) obtained against monochromatic waves. The combination of significant incident wave height (H

_{is}) and peak wave period (T

_{p}) is selected to match the experimental wave steepness range with uniform distribution. The T

_{p}and H

_{is}values considered in the study are listed in Table 6. Figure 17 presents the comparison of performance characteristics between monochromatic and irregular waves for both non-perforated and perforated CPHBs. The trend lines are drawn for the discrete data in order to clearly comprehend CPHB performance, and the results are then evaluated using the trend lines. A close examination of Figure 17 reveals that the trend of K

_{t}, K

_{r}and K

_{d}with respect to wave steepness is similar to that seen for monochromatic waves.

_{t}values obtained for the non-perforated CPHBs with monochromatic wave test conditions range from 0.83 to 0.64, whereas for irregular waves, the range is between 0.72 and 0.36. Similarly, for the perforated CPHBs, the K

_{t}varied from 0.8 to 0.54 with monochromatic waves and 0.68 to 0.33 in the case of irregular waves. It is observed that the K

_{r}obtained for the irregular waves is lower than that of the monochromatic waves for both the non-perforated and perforated CPHBs. Additionally, the dissipation characteristics are higher for both cases (non-perforated and perforated) of CPHBs with irregular waves. Therefore, it can be stated that the performance characteristics calculated using the monochromatic wave conditions are conservative. Similar deviations in the performance characteristics between monochromatic and irregular waves have been reported in the literature for other pile structures, including partially immersed twin vertical barriers [60], T-type breakwaters [61] and ⊥-type breakwaters [62]. Overall, it is evident that the non-perforated CPHB with the structural configuration of $D/H$

_{max}= 0.4, $Y/H$

_{max}= 1.5 and $\mathit{b}/\mathit{D}$ = 0.1 can attenuate waves up to 67% with irregular wave conditions. Incorporating the perforations enhances the wave attenuation capability of the structure by about 5% to 10% with irregular wave climates.

## 4. Conclusions

- (1)
- In general, K
_{t}is found to be indirectly proportional to the wave steepness, whereas K_{r}and K_{d}exhibit the opposite pattern. - (2)
- Validation of the numerical results with the experimental data shows that REEF3D produces reliable results with acceptable RMSE values.
- (3)
- The hydraulic performance of the CPHB structure is found to be more conservative with monochromatic waves than with irregular waves.
- (4)
- In the case of irregular waves, K
_{t}ranges from 0.72 to 0.36 for the non-perforated CPHB with an optimum configuration of D/${H}_{max}$ = 0.4, $Y/{H}_{max}$ = 1.5 and $\mathit{b}/\mathit{D}$ = 0.1. For the same configuration, K_{t}ranges between 0.83 and 0.64 with monochromatic waves. - (5)
- Introducing perforations with the optimum configuration ($Pa$ = 50%, $\mathit{S}/\mathit{D}$ = 0.25 and P = 19.2%) on the CPHs enhanced the transmission capability of the CPHB by about 5% to 16.5% with monochromatic waves and 5% to 10% with irregular waves.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 5.**Influence of grid size (dx) and CFL number on reconstruction of monochromatic wave surface. (

**a**) H

_{i}= 0.06 m, T = 2 s and CFL = 0.1, (

**b**) H

_{i}= 0.16 m, T = 1.8 s and CFL = 0.1, (

**c**) H

_{i}= 0.06 m, T = 2 s and dx = 0.02 m, (

**d**) H

_{i}= 0.16 m, T = 1.8 s and dx = 0.02 m.

**Figure 6.**Influence of grid size (dx) and CFL number on the reconstruction of irregular waves for the case of T

_{p}= 1.4 s and H

_{is}= 0.12 m. (

**a**) CFL = 0.1 and varying dx, (

**b**) dx = 0.02 m and varying CFL numbers.

**Figure 8.**Comparison of numerical and experimental results for various D/H

_{max}of non-perforated CPHBs. (

**a**) K

_{t}for D/H

_{max}= 0.4 and Y/H

_{max}= 1.5, (

**b**) K

_{r}for D/H

_{max}= 0.4 and Y/H

_{max}= 1.5, (

**c**) K

_{d}for D/H

_{max}= 0.4 and Y/H

_{max}= 1.5, (

**d**) K

_{t}for D/H

_{max}= 0.5 and Y/H

_{max}= 1.5, (

**e**) K

_{r}for D/H

_{max}= 0.5 and Y/H

_{max}= 1.5, (

**f**) K

_{d}for D/H

_{max}= 0.5 and Y/H

_{max}= 1.5.

**Figure 10.**Simulated free surfaces with velocity magnitude (m/s) during the wave–structure interaction for different D/H

_{max}of non-perforated CPHBs.

**Figure 11.**Plan-view of particle path lines during the interaction of the wave crest with the non-perforated CPHBs for different D/H

_{max}at t = 9.10 s.

**Figure 13.**Wave interaction with the non-perforated CPHB (D/H

_{max}= 0.4) at different time instances (t).

**Figure 14.**Simulated free surfaces of non-perforated and perforated CPHB cases with velocity magnitude (m/s).

**Figure 16.**Wave interaction with the CPHB structure (D/H

_{max}= 0.4) for gentler and steeper waves.

**Figure 17.**Comparison of performance characteristics between monochromatic and irregular waves for non-perforated and perforated CPHBs.

Governing Parameters | Expression | Test Range |
---|---|---|

Maximum wave height (m) | H_{max} | 0.16 |

Top diameter of conical pile head (m) | D | 0.064, 0.080 |

Diameter of supporting pile (m) | d | 0.04 |

Height of conical pile head (m) | Y | 0.24 |

Draft or submergence of pile head (m) | y | 0.12 |

Size of perforation (m) | S | 0.016 |

Water depth (m) | h | 0.40 |

Wave period (s) | T | 1.4, 1.6, 1.8, 2.0 |

Incident wave height (m) | H_{i} | 0.06, 0.08, 0.10, 0.12, 0.14, 0.16 |

Angle of wave attack (degrees) | $\theta $ | 90 |

Non-Dimensional Parameters | ||

Relative pile head diameter | D/H_{max} | 0.4, 0.5 |

Relative pile head height | Y/H_{max} | 1.5 |

Clear spacing between pile heads | b/D | 0.1 |

Clear spacing between the supporting piles | b_{0}/d | 0.76 |

Distribution of perforations (%) | Pa | 50 |

Percentage of perforation (%) | P | 19.2 |

Relative size of perforations | S/D | 0.25 |

Incident wave steepness | H_{i}/gT^{2} | 0.00152 to 0.0062 |

T (s) | H (m) | Grid Study (with CFL = 0.1) | CFL Study (with dx = 0.02 m) | ||
---|---|---|---|---|---|

dx (m) | RMSE | CFL No. | RMSE | ||

2.0 | 0.06 | 0.08 | 0.0033 | 0.40 | 0.0025 |

0.04 | 0.0021 | 0.20 | 0.0025 | ||

0.02 | 0.0020 | 0.10 | 0.0023 | ||

0.01 | 0.0018 | 0.05 | 0.0023 | ||

1.8 | 0.16 | 0.08 | 0.0085 | 0.40 | 0.0068 |

0.04 | 0.0053 | 0.20 | 0.0063 | ||

0.02 | 0.0053 | 0.10 | 0.0055 | ||

0.01 | 0.0036 | 0.05 | 0.0045 |

Cases | T (s) | H_{i} (m) | L (m) | H_{i}/gT^{2} | Wave Theory |
---|---|---|---|---|---|

M1 | 1.4 | 0.12 | 2.39 | 0.00624 | Stokes 3rd order |

M2 | 1.6 | 0.14 | 2.84 | 0.00557 | Stokes 3rd order |

M3 | 1.8 | 0.16 | 3.27 | 0.00503 | Cnoidal |

M4 | 1.8 | 0.10 | 3.27 | 0.00315 | Stokes 3rd order |

M5 | 2.0 | 0.16 | 3.70 | 0.00408 | Cnoidal |

M6 | 2.0 | 0.10 | 3.70 | 0.00255 | Stokes 3rd order |

M7 | 2.0 | 0.08 | 3.70 | 0.00204 | Stokes 2nd order |

M8 | 2.0 | 0.06 | 3.70 | 0.00153 | Stokes 2nd order |

CPHB | D/H_{max} | RMSE | ||
---|---|---|---|---|

K_{t} | K_{r} | K_{d} | ||

Non-perforated | 0.4 | 0.0313 | 0.0142 | 0.0242 |

0.5 | 0.0355 | 0.0090 | 0.0476 | |

Perforated | 0.4 | 0.048 | 0.017 | 0.0422 |

Type of Breakwater | Structural Details | No. of | K_{t} | K_{r} | K_{d} | ||
---|---|---|---|---|---|---|---|

d (m) | b_{0}/d | P (%) | Pile Units | ||||

(per m) | |||||||

Non-perforated hollow piles [24,26] | 0.034 | 0.15 | NA | 25.96 | 0.71 to 0.78 | 0.28 to 0.29 | 0.56 to 0.64 |

Perforated hollow piles [24,26] | 0.034 | 0.15 | 25 | 25.96 | 0.66 to 0.73 | 0.22 to 0.30 | 0.64 to 0.69 |

Non-perforated suspended pipes [23,25] | 0.034 | 0.15 | NA | 25.96 | 0.73 to 0.82 | 0.19 to 0.25 | 0.55 to 0.64 |

Perforated suspended pipes [23,25] | 0.034 | 0.15 | 25 | 25.96 | 0.67 to 0.79 | 0.16 to 0.22 | 0.59 to 0.71 |

Rectangular piles [58] | 0.006 | 1.77 | 21 | 56.41 | 0.73 to 0.88 | 0.09 to 0.28 | 0.49 to 0.64 |

Zigzag porous screens [59] | 0.040 | 0.22 | 40 | 20.49 | 0.67 to 0.83 | 0.16 to 0.18 | 0.57 to 0.73 |

Non-perforated CPHB | 0.040 | 0.76 | NA | 14.20 | 0.66 to 0.83 | 0.13 to 0.23 | 0.55 to 0.73 |

Perforated CPHB | 0.040 | 0.76 | 19.2 | 14.20 | 0.54 to 0.80 | 0.17 to 0.28 | 0.59 to 0.80 |

Cases | T (s) | H_{i} (m) | H_{i}/gT^{2} |
---|---|---|---|

M1 | 1.4 | 0.12 | 0.00624 |

M2 | 1.8 | 0.10 | 0.00315 |

M3 | 1.8 | 0.16 | 0.00503 |

M4 | 2.0 | 0.06 | 0.00153 |

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**MDPI and ACS Style**

Sathyanarayana, A.H.; Suvarna, P.S.; Umesh, P.; Shirlal, K.G.; Bihs, H.; Kamath, A.
Numerical Modelling of an Innovative Conical Pile Head Breakwater. *Water* **2022**, *14*, 4087.
https://doi.org/10.3390/w14244087

**AMA Style**

Sathyanarayana AH, Suvarna PS, Umesh P, Shirlal KG, Bihs H, Kamath A.
Numerical Modelling of an Innovative Conical Pile Head Breakwater. *Water*. 2022; 14(24):4087.
https://doi.org/10.3390/w14244087

**Chicago/Turabian Style**

Sathyanarayana, Arunakumar Hunasanahally, Praveen S. Suvarna, Pruthviraj Umesh, Kiran G. Shirlal, Hans Bihs, and Arun Kamath.
2022. "Numerical Modelling of an Innovative Conical Pile Head Breakwater" *Water* 14, no. 24: 4087.
https://doi.org/10.3390/w14244087