# Determination of Submerged Breakwater Efficiency Using Computational Fluid Dynamics

^{*}

## Abstract

**:**

## 1. Introduction

- Section 2.1 presents a detailed explanation of the ICFD method used for breakwater analysis, including a description of the theoretical background and its advantages over another method for solving fluid–structure interaction (FSI) problems, namely the ALE method.
- Three possible topologies of breakwater systems with geometrical particulars are presented in Section 2.2.
- Section 3 details the numerical setup for assessing the most efficient topology, including comprehensive information about the mesh and boundary conditions of the breakwater system.
- Section 4 presents the results of both analyses, emphasising wave height reduction and drag estimation.
- The conclusion in Section 5 focuses on the findings obtained and points out the potential issues and/or limitations of the applied methods.

## 2. Material and Methods

#### 2.1. ICFD

_{i}

^{n}

^{+1}) is a convective term for non-linear advection, $\Upsilon $ is the time integration parameter, G is the gradient operator matrix, F is the external force vector, D is the divergence operator matrix, $\dot{\stackrel{~}{U}}$ is the source term vector, and u

_{i}and p

_{n}represent velocity and pressure.

_{i}

^{*}and p

^{(}

^{n}

^{+1)}are computed from Equations (4) and (5), respectively, and then reintroduced into the iterative loop to determine the next step velocity u

_{i}

^{(}

^{n}

^{+1)}. This approach enhances the accuracy of the final pressure and velocity values, although it is more time-consuming. LS-DYNA employs the Finite Element (FE) Method for spatial discretisation, adopting strategies similar to those in the ALE method but with a few notable differences. In particular, the element type used in the ICFD cases is tetrahedral, while the ALE method typically uses hexahedral elements. While the solver of the ALE method is designed for compressible fluids and uses an explicit time integration method, the ICFD solver is an implicit solver optimised for incompressible fluids and is able to perform double-precision calculations required for turbulence modelling and boundary layer definition. In addition to the previously mentioned advantages of ICFD over ALE for wave damping problems, the ICFD solver incorporates wave theory [21], offering options for Stokes’ 1st-, 2nd-, and 3rd-order theories. The choice of Stokes’ theory over linear theory is based on the ratio of wave height to wavelength. For 2nd-order Stokes’ theory, this ratio lies in the range 0.04 < H/L < 0.141 [22], which corresponds with the measured average wave height and length [23,24], making it the appropriate choice.

#### 2.2. Breakwater Geometry

## 3. Numerical Setup

#### 3.1. Mesh Particulars

#### 3.2. Numerical Setup for Assessing Most Efficient Topology

#### 3.3. Convergence Analysis

_{P}is the drag force due to pressure, with P being pressure and dA being the differential area:

_{v}is the viscous component of drag force, μ is viscosity, and $\frac{\partial u}{\partial y}$ is the shear velocity at the wall. The convergence diagram showed that results with less than 15 elements per amplitude diverge significantly from the results obtained with 15 or more elements as wave generation is irregular, Figure 5b.

#### 3.4. Numerical Setup for Submerged Breakwater System

## 4. Results

#### 4.1. Breakwater Efficiency

#### 4.2. Submerged Breakwater System Analysis

_{t}for different positions along the Y-axis is presented in Table 3. While the centreline passes above the wall of the breakwater, Cut 1 passes between two neighbouring breakwater elements, and Cut 2 directly above holes in the breakwater, Cut 3 passes further away from the submerged breakwater system. The transmission coefficient shows small variations between the centreline and Cuts 1 and 2, with Cut 2 having the lowest K

_{t}, meaning that breakwater holes reduce the wave the most but only for a small margin.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Towing tank at the University of Zagreb, Faculty of Mechanical Engineering; (

**b**) width between breakwater rows.

**Figure 2.**Breakwater topology: (

**a**) cylindrical 6-hole, (

**b**) Quadratic 16-hole, and (

**c**) Quadratic 36-hole topologies.

**Figure 6.**The numerical setup for a submerged breakwater system: (

**a**) the global domain, (

**b**) a breakwater system with 6 elements in a line (angle 3 case).

**Figure 9.**Comparison of different breakwater topologies based on (

**a**) drag, (

**b**) wave profile, and (

**c**) drag for different wave steepness ratios.

**Figure 10.**Comparison of different breakwater topologies based on (

**a**) wave velocity and (

**b**) wave profile.

**Figure 13.**Maximum turbulence creation at both wave crest and trough at (

**a**) first row of breakwater and (

**b**) last row of breakwater.

**Figure 15.**Force comparison for different wave amplitudes in (

**a**) drag force in X direction for δ = 1/20; (

**b**) relation between drag force in X direction and steepness ratio.

Model Name | Number of ICFD Elements | Number of Solid Elements | |
---|---|---|---|

Cylindrical holes | δ = 1/10, T = 1.75 s | 206 k | 500 |

δ = 1/15, T = 3.75 s | 206 k | 500 | |

δ = 1/20, T = 5 s | 206 k | 500 | |

δ = 1/33, T = 7.5 s | 206 k | 500 | |

Quadratic 16 holes | δ = 1/10, T = 1.75 s | 206 k | 1284 |

δ = 1/15, T = 3.75 s | 206 k | 1284 | |

δ = 1/20, T = 5 s | 206 k | 1284 | |

δ = 1/33, T = 7.5 s | 206 k | 1284 | |

Quadratic 36 holes | δ = 1/10, T = 1.75 s | 206 k | 2148 |

δ = 1/15, T = 3.75 s | 206 k | 2148 | |

δ = 1/20, T = 5 s | 206 k | 2148 | |

δ = 1/33, T = 7.5 s | 206 k | 2148 |

Model Name | Number of ICFD Elements | Number of Solid Elements | Real-World Time | |
---|---|---|---|---|

7.5 cm amplitude, 1.5 m wavelength, δ = 1/20, T = 3.75 s | Angle 1 | 435 k | 22 k | 12 h, 29 min |

Angle 2 | 435 k | 33 k | 12 h, 43 min | |

Angle 3 | 435 k | 45 k | 13 h, 13 min | |

Angle 4 | 435 k | 60 k | 13 h, 50 min | |

10 cm amplitude, 2 m wavelength, δ = 1/20, T = 5 s | Angle 1 | 284 k | 22 k | 3 h, 50 min |

Angle 2 | 284 k | 33 k | 3 h, 55 min | |

Angle 3 | 284 k | 45 k | 4 h, 6 min | |

Angle 4 | 284 k | 60 k | 4 h, 15 min | |

12.5 cm amplitude, 2.5 m wavelength, δ = 1/20, T = 6.25 s | Angle 1 | 197 k | 22 k | 2 h, 58 min |

Angle 2 | 197 k | 33 k | 3 h | |

Angle 3 | 197 k | 45 k | 3 h, 2 min | |

Angle 4 | 197 k | 60 k | 3 h, 12 min | |

12.5 cm amplitude, 1.25 m wavelength, δ = 1/10, T = 3.1 s | Angle 1 | 197 k | 22 k | 4 h, 3 min |

Angle 2 | 197 k | 33 k | 4 h, 15 min | |

Angle 3 | 197 k | 45 k | 4 h, 54 min | |

Angle 4 | 197 k | 60 k | 8 h, 21 min | |

12.5 cm amplitude, 1.875 m wavelength, δ = 1/15, T = 6.25 s | Angle 1 | 197 k | 22 k | 4 h, 45 min |

Angle 2 | 197 k | 33 k | 6 h, 40 min | |

Angle 3 | 197 k | 45 k | 7 h, 3 min | |

Angle 4 | 197 k | 60 k | 7 h, 14 min | |

12.5 cm amplitude, 3.6 m wavelength, δ = 1/33, T = 9 s | Angle 1 | 197 k | 22 k | 5 h, 46 min |

Angle 2 | 197 k | 33 k | 6 h, 36 min | |

Angle 3 | 197 k | 45 k | 6 h, 42 min | |

Angle 4 | 197 k | 60 k | 7 h, 24 min |

δ = 1/20 | Model Name | Centreline K_{t} [-] | Cut 1 K_{t} [-] | Cut 2 K_{t} [-] | Cut 3 K_{t} [-] |
---|---|---|---|---|---|

7.5 cm amplitude | Angle 1 | 0.34 | 0.36 | 0.33 | 0.4 |

Angle 2 | 0.3 | 0.32 | 0.29 | 0.33 | |

Angle 3 | 0.07 | 0.09 | 0.067 | 0.16 | |

Angle 4 | 0.04 | 0.05 | 0.032 | 0.12 | |

10 cm amplitude | Angle 1 | 0.46 | 0.48 | 0.45 | 0.53 |

Angle 2 | 0.33 | 0.35 | 0.32 | 0.42 | |

Angle 3 | 0.22 | 0.24 | 0.21 | 0.33 | |

Angle 4 | 0.15 | 0.17 | 0.14 | 0.28 | |

12.5 cm amplitude | Angle 1 | 0.53 | 0.55 | 0.53 | 0.6 |

Angle 2 | 0.28 | 0.29 | 0.27 | 0.46 | |

Angle 3 | 0.2 | 0.22 | 0.19 | 0.37 | |

Angle 4 | 0.13 | 0.15 | 0.12 | 0.18 |

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**MDPI and ACS Style**

Rudan, S.; Sviličić, Š.
Determination of Submerged Breakwater Efficiency Using Computational Fluid Dynamics. *Oceans* **2024**, *5*, 742-757.
https://doi.org/10.3390/oceans5040042

**AMA Style**

Rudan S, Sviličić Š.
Determination of Submerged Breakwater Efficiency Using Computational Fluid Dynamics. *Oceans*. 2024; 5(4):742-757.
https://doi.org/10.3390/oceans5040042

**Chicago/Turabian Style**

Rudan, Smiljko, and Šimun Sviličić.
2024. "Determination of Submerged Breakwater Efficiency Using Computational Fluid Dynamics" *Oceans* 5, no. 4: 742-757.
https://doi.org/10.3390/oceans5040042