# Determination of Formulae for the Hydrodynamic Performance of a Fixed Box-Type Free Surface Breakwater in the Intermediate Water

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

_{t}, reflection coefficient C

_{r}, and dissipation coefficient C

_{d}(hereinafter referred to as RTD coefficients). The other is the wave force, which concerns the safety and stability of the breakwater, including the horizontal wave force and vertical wave force.

_{t}, and dissipation coefficient C

_{d}are also an indispensable part of the wave attenuation performance of F-BW. The horizontal and vertical wave forces are related to the security of the F-BW. However, the prediction formulas based on tests or numerical simulations for horizontal and vertical wave forces of the F-BW in the above studies were rare. Therefore, an attempt is necessary to present a proposed formula for the prediction of RTD coefficients and wave forces, which will provide design judgments for the relevant practitioners in intermediate waters.

## 2. Theoretical Introduction

#### 2.1. Governing Equations

_{i}represents the Cartesian coordinate, and u

_{i}is the fluid velocity along the x- and z-axes. A

_{x}and A

_{z}are the area fractions open to flow in the x and z directions, respectively, ρ is the fluid density, p is the pressure, v is the dynamic viscosity, and g is the gravity force. The Reynolds stresses term, $\rho \overline{{{u}_{i}}^{\prime}{{u}_{j}}^{\prime}}$, is modeled by the renormalized-group (RNG) turbulence model.

#### 2.2. RNG Turbulence Model

_{T}and dissipation rate ε

_{T}in this model are as follows:

#### 2.3. Principle of Mass Source Wavemaker

_{i}is the incident wave height, k is the wavenumber, σ is the wave frequency, and h is the still water depth.

_{fr}into the mass source model. The expression of the volume flow rate V

_{fr}is as follows:

_{fr}is multiplied by an increasing envelope function to make the wave increase gradually in the first three wave periods. The equation of the increasing function is as follows:

#### 2.4. Principle of Numerical Solution

_{i}and the reflected wave heights H

_{r}by using this method. To prove that the horizontal wave force of the F-BW is related to the free surface onshore and offshore of the breakwater, probe WG3 is placed 0.02 m in front of the F-BW, while probe WG4 is placed 0.02 m behind the F-BW to measure the wave profile at the front (η3) and back (η4) of the F-BW. The wave gauge (WG5) is mounted on the onshore side of the F-BW to obtain the surface elevation of the transmitted wave heights H

_{t}. The wave transmission, reflection, and wave energy dissipation coefficients are defined by solving Equation (8a)–(8c).

_{t}is the transmission coefficient; C

_{r}is the reflection coefficient; and C

_{d}is the wave energy dissipation coefficient.

_{x}and F

_{z}, respectively. The horizontal wave force is consistent with the direction of wave propagation, and the vertical wave force is vertically upward. To facilitate the research, obtaining the extreme value of the steady part of the wave force time series, we define the average value of the horizontal wave force positive and negative peak as the horizontal positive maximum wave force F

_{x}

^{+max}and horizontal negative maximum wave force F

_{x}

^{−max}, the vertical wave force positive and negative peak as the vertical positive maximum wave force F

_{z}

^{+max}and vertical negative maximum wave force F

_{z}

^{−max}. The representative time series of the dimensionless wave elevation, horizontal, and vertical wave forces are shown in Figure 4. The numerical results of H

_{i}, H

_{r}, H

_{t}, F

_{x}

^{±max}, F

_{z}

^{±max}were acquired based on the stable elevations in this figure. To facilitate discussion, we define F

_{x}

^{±max}/0.005 ρgh

^{2}and F

_{z}

^{±max}/0.005 ρgh

^{2}as the dimensionless horizontal and vertical maximum wave forces on the F-BW, respectively. The crest and trough values of the time series of the wave forces are studied because the extreme values of the horizontal and vertical wave forces on the F-BW under the Stokes second-order wave have a slightly sharper crest and flatter trough.

## 3. Model Setup and Validation

#### 3.1. Numerical Wave Tank Setup

_{S}and high H

_{S}was added to the numerical flume. The symmetry boundaries were used overspreading with the mass source form, and the y-direction width of the mass source block was consistent with the width of the NWT. Pledging each edge of the mass block to coincide with the grid line of the NWT is shown in Figure 3b,c.

#### 3.2. Numerical Model Validation

#### 3.2.1. Grid Independent Verification

_{i}= 0.06 m, T = 1.2 s, and h = 1.2 m by Ren et al. are close to the target cases in this paper [23]. This wave condition is applied to complete the grid independence verification. Different grid arrangements can be seen in Table 2, and the time series of the wave profiles under the three grid sizes are compared with the theoretical results by solving Equation (5), as shown in Figure 5. The error of the numerical simulation results was calculated according to Equation (10). The wave profile deviations among the coarse mesh, medium mesh, and fine mesh are compared. The wave profiles under the medium mesh and the fine mesh are closer, and the deviation from the theoretical value is less than 5%, which meets the requirements of Det Norske Veritas (DNV) [39]. It can be judged that only medium meshes and refined meshes meet the requirements of numerical simulation. Considering the balance between calculation accuracy and calculation efficiency, the following numerical simulations always chose a medium mesh.

_{theoretical}is the wave height of the theoretical result and H

_{numerical}is the wave height of the numerical result.

#### 3.2.2. Validation of Wave Forces

_{i}= 0.06 m, T = 1.2 s, h = 1.2 m, and draft dr = 0.2 m, a rectangular box of width B = 0.8 m and wave height H

_{i}= 0.4 m is fixed and semi-immersed, as proposed by Ren et al. [23]. The horizontal and vertical wave forces of the F-BW were verified by comparison with the theoretical results of Mei and Black [40] and the numerical simulation results of Ren et al. [23]. The time series of the wave forces are compared in Figure 6. The total simulation time of this case is 16 wave cycles. Since it takes some time for the progressive wave to arrive at the F-BW from the source, the horizontal and vertical wave forces begin to reach the stable state at t = 7 T seconds in Figure 6. By comparison, the simulated time series of horizontal and vertical wave forces are almost consistent with those presented by Ren et al. [23] and Mei and Black [40]. This result indicated that the present NWT could meet the calculation accuracy.

## 4. Results and Discussion

#### 4.1. Influence Analysis of Four Factors on the Hydrodynamic Performance of F-BW

_{i}/h). For the mechanism analysis of the interaction between waves and breakwater, the mechanism study of the horizontal wave force is rather complicated. Since the breakwater is in a semisubmerged state, the Morison formula is no longer applicable to the guidance of the calculation of the horizontal wave force. The horizontal wave force is studied separately from the water particle velocity; see the free surface difference (η3–η4) in the front and back sides of the F-BW and the water particle streamline in Figure 7 and Figure 8 for details. Among them, five representative cases are selected from all cases in this article for comparative analysis corresponding to Figure 7a–e. Note that case (a) T1.2dr0.14B0.5Hi0.07 represents a wave period of 1.2 s, draft of 0.14 m, breakwater width of 0.5 m and incident wave height of 0.07 m. Due to the effect of water blockage, flow separation is generated at the bottom corner of the offshore side of the breakwater, and the generated clockwise vortex destroys the original motion path of the wave water particles without structure in Figure 8a and allows the free surface difference in the front and back of the F-BW to gradually reach a maximum. At time instant t

_{0}in Figure 7, the horizontal wave force also reaches a maximum. It can be seen in Figure 8b that the vertical wave force is easier to analyze. When the vertical wave force is at its maximum, the streamline realizes complete diffraction, and no vortex is generated. Furthermore, to understand the mechanism and contribution of each influencing factor on the hydrodynamic performance of the F-BW in detail, the statistical results are shown in Figure 9, Figure 10, Figure 11 and Figure 12.

#### 4.1.1. Effect of Draft

_{0}= 11.48 s, the maximum free surface difference is 0.068 m, and the maximum horizontal wave force is 7.98 N. In the second column of Figure 7b, when time t

_{0}= 11.52 s, the maximum free surface difference is 0.083 m, and the maximum horizontal wave force is 15.91 N. Obviously, the increase in the draft enhances the water blockage action in front of the F-BW, weakens the diffraction effect of the wave, and delays the time for the horizontal wave force to reach its maximum. Figure 9a shows that F

_{x}

^{+max}increases with increasing draft under wave heights of H

_{i}= 0.05 m and H

_{i}= 0.07 m. Similarly, the absolute values of F

_{x}

^{−max}exhibit a similar law. The absolute values of F

_{z}

^{−max}and F

_{z}

^{+max}decrease with increasing draft under wave heights of H

_{i}= 0.05 m and H

_{i}= 0.07 m in Figure 9b, which is related to the exponential decay of the wave kinetic energy along the water depth. It is not difficult to see in Figure 7a,b that the wave hydrodynamic pressure on the lower surface of the F-BW decreases with decreasing wave kinetic energy as the water depth increases. The effective action area increases as the draft reduces the penetration of waves. Figure 9c shows that the transmission coefficient decreases with increasing draft under wave heights of H

_{i}= 0.05 m and H

_{i}= 0.07 m. Due to the increase in the interaction area between waves and F-BW, the reflected wave energy increases in Figure 7, and Figure 7b is more obvious than Figure 7a. The wave energy dissipation coefficient increases with decreasing draft in Figure 9c. Since the wave energy is mainly concentrated on the still water level, the fluid particle velocity maximum at the lower corner of F-BW is 0.30 m/s in Figure 7a is more than the 0.17 m/s in Figure 7b, more wave energy is dissipated when the fluid particle with higher velocity collides with F-BW due to decreasing draft.

#### 4.1.2. Effect of Breakwater Width

_{x}

^{−max}and F

_{x}

^{+max}is not obvious. When the vertical wave force is at its maximum, the streamline realizes complete diffraction, and no vortex is generated in Figure 8b. Therefore, the vertical wave force is related to the acting area of the F-BW lower surface. Figure 10b shows that the absolute values of F

_{z}

^{−max}and F

_{z}

^{+max}increase with increasing breakwater width. In the second column of Figure 7c, when time t

_{0}= 11.42 s, the free surface difference and the horizontal wave force reach a maximum faster than in case (a). Obviously, the increase in the breakwater width increases the wave diffraction difficulty. Figure 10c shows that the transmission coefficient decreases with increasing breakwater width, and the reflection coefficient increases with increasing breakwater width. Due to fluid particle velocity maximum is similar between Figure 7a,c. The increase in breakwater width has little influence on wave energy dissipation.

#### 4.1.3. Effect of Wave Period

_{0}= 11.13 s, the maximum free surface difference is 0.051 m, and the maximum horizontal wave force is 6.90 N. According to Equation (9), because the wave energy is more abundant on the two sides of the breakwater in case (4), the horizontal wave force is comparable even if the free surface difference is smaller than that in case (1). Figure 11a shows that F

_{x}

^{−max}and F

_{x}

^{+max}are weakly related to the wave period under wave heights of H

_{i}= 0.05 m and H

_{i}= 0.07 m. Because the long-period waves possess a large wave energy in Figure 7d, they increase the wave pressure on the lower surface of the F-BW. Therefore, the absolute values of F

_{z}

^{−max}and F

_{z}

^{+max}increase linearly with the wave period in Figure 11b. Figure 11c shows that the transmission coefficient increases with increasing wave period under wave heights of H

_{i}= 0.05 m and H

_{i}= 0.07 m. Long-period waves have a better diffraction ability at the same depth, and more wave energy passes through the F-BW. The decreasing ratio of the breakwater width to wavelength weakens the ability to block progressive waves, and the reflection coefficient decreases accordingly. The wave energy dissipation coefficient shows an alphabetic symbol “M” distribution with the wave period. This indicates that the wave energy dissipation is more complex and requires further study. When the dimensionless wave period is 5.06, both the transmission and reflection coefficients are close to 0.71, the dissipation coefficient is at the minimum by applying Equation (8c).

#### 4.1.4. Effect of Wave Height

_{0}= 11.44 s, the maximum free surface difference is 0.031 m, and the maximum horizontal wave force is 3.43 N. Obviously, the increase in wave height increases the diffraction difficulty of the wave and delays the time when the horizontal wave force reaches its maximum. The higher the wave height, the more abundant the wave energy in Figure 7a,e. The water particle velocity maximum is 0.11 m/s in Figure 7e, which is much less than the water particle velocity maximum in Figure 7a. The larger wave height causes a larger wave elevation difference, and the larger horizontal wave force under other variable conditions is consistent by comparing Figure 7a,e. Therefore, F

_{x}

^{−max}and F

_{x}

^{+max}increase linearly with increasing wave height under drafts dr = 0.14 m and dr = 0.28 m in Figure 12a. The increase in wave height leads to increasing dynamic wave pressure, which in turn leads to increasing wave pressure on the F-BW lower surface and an increase in vertical wave force. Therefore, F

_{z}

^{−max}and F

_{z}

^{+max}increase linearly with increasing wave height under drafts dr = 0.14 m and dr = 0.28 m in Figure 12b. Figure 12c shows that the increasing wave height results in more wave reflection and less transmission due to the increasing blockage effect. The reflection ability weakens with decreasing interaction area (the ratio of the wetted surface height of the front wall of the F-BW to the wave height). The water particle velocity maximum of 0.11 m/s in Figure 7e is less than the water particle velocity maximum of 0.3 m/s in Figure 7a. The increasing water particle velocity with increasing wave height results in better vortex dissipation near the F-BW. Hence, the wave energy dissipation coefficient increases.

#### 4.2. Prediction Equations of F-BW Hydrodynamic Performance Parameters

_{i}, draft d

_{r}, breakwater width B, and still water depth h. In Equation (11), the RTD coefficients C

_{t,r,d}and wave force extremum F

_{x,z}

^{±max}are expressed as follows:

_{i}/h ≤ 0.12.

#### 4.3. Deviation Analysis of the Prediction Equations

## 5. Conclusions

_{i}) on the hydrodynamic performance (RTD coefficients and wave forces) are highlighted. The vital conclusions are as follows:

- (1)
- The performance of two-dimensional viscous numerical wave tanks (NWTs) with a mass source wave maker and small length scale (1:40) are analyzed. By comparison, the wave model employed in this paper is competent for the numerical simulation of the F-BW.
- (2)
- The results show that the increase in the four influence factors, except the wave period, benefits the decrease in the wave transmission. The increase in draft and breakwater width is beneficial to the increase in the wave reflection, and the wave period and wave height are opposite. The increase in draft benefits the decrease in wave energy dissipation, and the wave height is opposite.
- (3)
- The increase in the draft and wave height benefits the increase in the horizontal positive and negative maximum wave forces. In addition to the draft, the increase in the other three influence factors benefits the increase in the vertical positive and negative maximum wave forces.
- (4)
- Applying multiple linear regression presents the prediction equations of RTD coefficients and the extreme wave force. The prediction equations are verified by comparing them with literature observation datasets.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

_{i}, draft d

_{r}, breakwater width B, and water depth h are the main factors that affect the wave dissipation performance and wave force of an F-BW in the intermediate waters. Therefore, the wave force of an F-BW can be expressed as a function of the above factors as follows:

^{0}L

^{1}T

^{0}], [g] = [M

^{0}L

^{1}T

^{−2}], [ρ] = [M

^{1}L

^{−3}T

^{0}], where wave force per breakwater length in the vertical wave direction $F$, expressed as [F = ρgh

^{2}], Equation (A1) can be written as follows:

_{1}, x

_{2}, x

_{3}, and x

_{4}are the unknown coefficients.

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**Figure 2.**Wave parameter conditions analyzed in this study and their relations in the Le Méhauté diagram.

**Figure 4.**Time histories of wave elevation η measured by WG1, WG2, and WG5 and horizontal and vertical wave forces of F-BW at H

_{i}= 0.07 m, T = 1.4 s, B = 0.5 m, dr = 0.14 m, and h = 0.75 m.

**Figure 7.**Comparative analysis of five different cases under the interaction between waves and breakwater: First column: numerically obtained snapshots of free surface profile and velocity field; Second column: time history of free surface and horizontal wave force.

**Figure 8.**Snapshots of the velocity streamline field: (

**a**) Time instant of the horizontal positive maximum wave force; (

**b**) Time instant of the vertical positive maximum wave force.

**Figure 9.**Effect of the draft dr on the hydrodynamic performance of the F-BW at wave heights H

_{i}= 0.05 m and H

_{i}= 0.07 m. (

**a**) Horizontal positive and negative maximum wave forces F

_{x}

^{+max}and F

_{x}

^{−max}; (

**b**) Vertical positive and negative maximum wave forces F

_{z}

^{+max}and F

_{z}

^{−max}; (

**c**) Transmission coefficient C

_{t}, reflection coefficient C

_{r}, and dissipation coefficient C

_{d}.

**Figure 10.**Influence of the breakwater width B on the hydrodynamic performance of the F-BW at wave heights H

_{i}= 0.05 m and H

_{i}= 0.07 m. (

**a**) Horizontal positive and negative maximum wave forces F

_{x}

^{+max}and F

_{x}

^{−max}; (

**b**) Vertical positive and negative maximum wave forces F

_{z}

^{+max}and F

_{z}

^{−max}; (

**c**) Transmission coefficient C

_{t}, reflection coefficient C

_{r}, and dissipation coefficient C

_{d}.

**Figure 11.**Influence of the wave period T on the hydrodynamic performance of the F-BW at wave heights H

_{i}= 0.05 m and H

_{i}= 0.07 m. (

**a**) Horizontal positive and negative maximum wave forces F

_{x}

^{+max}and F

_{x}

^{−max}; (

**b**) Vertical positive and negative maximum wave forces F

_{z}

^{+max}and F

_{z}

^{−max}; (

**c**) Transmission coefficient C

_{t}, reflection coefficient C

_{r}, and dissipation coefficient C

_{d}.

**Figure 12.**Influence of the wave height H

_{i}on the hydrodynamic performance of the F-BW at draft dr = 0.14 m and dr = 0.28 m. (

**a**) Horizontal positive and negative maximum wave forces F

_{x}

^{+max}and F

_{x}

^{−max}; (

**b**) Vertical positive and negative maximum wave forces F

_{z}

^{+max}and F

_{z}

^{−max}; (

**c**) Transmission coefficient C

_{t}, reflection coefficient C

_{r}, and dissipation coefficient C

_{d}.

**Figure 13.**Comparison of the results between previous studies and the numerical results of this study. (

**a**) Transmission coefficient C

_{t}, reflection coefficient C

_{r}, and dissipation coefficient C

_{d}(Koutandos [13], Liang et al. [14]) and (

**b**) Maximum wave force (Mei and Black [40]; Ren et al. [23]).

Variable 1 | Variable 2 | Variable 3 | Variable 4 | |
---|---|---|---|---|

dr | B | T | H_{i} | |

[m] | [m] | [s] | [m] | |

Case 1 | 0.07 | 0.05 | 1.2 | 0.05 |

Case 2 | 0.07 | |||

Case 3 | 0.14 | 0.05 | ||

Case 4 | 0.07 | |||

Case 5 | 0.21 | 0.05 | ||

Case 6 | 0.07 | |||

Case 7 | 0.28 | 0.05 | ||

Case 8 | 0.07 | |||

Case 9 | 0.35 | 0.05 | ||

Case 10 | 0.07 | |||

Case 11 | 0.14 | 0.2 | 1.2 | 0.05 |

Case 12 | 0.07 | |||

Case 13 | 0.3 | 0.05 | ||

Case 14 | 0.07 | |||

Case 15 | 0.4 | 0.05 | ||

Case 16 | 0.07 | |||

Case 17 | 0.6 | 0.05 | ||

Case 18 | 0.07 | |||

Case 19 | 0.14 | 0.5 | 1 | 0.05 |

Case 20 | 0.07 | |||

Case 21 | 1.4 | 0.05 | ||

Case 22 | 0.07 | |||

Case 23 | 1.6 | 0.05 | ||

Case 24 | 0.07 | |||

Case 25 | 1.8 | 0.05 | ||

Case 26 | 0.07 | |||

Case 27 | 0.14 | 0.5 | 1.2 | 0.03 |

Case 28 | 0.28 | |||

Case 29 | 0.14 | 0.09 | ||

Case 30 | 0.28 |

Mesh Type | Computation Domain Grid Size (cm) | Nested Domain Grid Size (cm) | Cell Number | Elapsed Time (×10^{4} s) | Wave Height (cm) | Error % |
---|---|---|---|---|---|---|

Coarse | 2 | 1 | 701460 | 0.6496 | 5.642 | 5.96 |

Middle | 1 | 0.5 | 3411180 | 7.6832 | 5.768 | 3.87 |

Fine | 0.5 | 0.25 | 13350960 | 48.1437 | 5.769 | 3.85 |

Theoretical | - | - | - | - | 6.000 | - |

Equation Number | Equations | R^{2} |
---|---|---|

(12a) | ${C}_{t}={0.003\left(\frac{dr}{h}\right)}^{-0.935}{\left(\frac{B}{h}\right)}^{-0.519}{\left(\frac{{gT}^{2}}{h}\right)}^{1.039}{\left(\frac{{H}_{i}}{h}\right)}^{-0.064}$ | 0.948 |

(12b) | ${C}_{r}={3.650\left(\frac{dr}{h}\right)}^{0.213}{\left(\frac{B}{h}\right)}^{0.187}{\left(\frac{{gT}^{2}}{h}\right)}^{-0.436}{\left(\frac{{H}_{i}}{h}\right)}^{-0.074}$ | 0.958 |

(12c) | ${C}_{d}={3.100\left(\frac{dr}{h}\right)}^{-0.235}{\left(\frac{B}{h}\right)}^{-0.113}{\left(\frac{{gT}^{2}}{h}\right)}^{-0.493}{\left(\frac{{H}_{i}}{h}\right)}^{0.426}$ | 0.695 |

(12d) | $\frac{{{F}_{x}}^{+max}}{0.005\rho g{h}^{2}}={21.398\left(\frac{dr}{h}\right)}^{0.866}{\left(\frac{B}{h}\right)}^{0.139}{\left(\frac{{gT}^{2}}{h}\right)}^{-0.127}{\left(\frac{{H}_{i}}{h}\right)}^{1.027}$ | 0.992 |

(12e) | $\frac{{{F}_{x}}^{-max}}{0.005\rho g{h}^{2}}={-14.199\left(\frac{dr}{h}\right)}^{1.062}{\left(\frac{B}{h}\right)}^{0.057}{\left(\frac{{gT}^{2}}{h}\right)}^{-0.042}{\left(\frac{{H}_{i}}{h}\right)}^{0.912}$ | 0.988 |

(12f) | $\frac{{{F}_{z}}^{+max}}{0.005\rho g{h}^{2}}={0.079\left(\frac{dr}{h}\right)}^{-0.519}{\left(\frac{B}{h}\right)}^{0.852}{\left(\frac{{gT}^{2}}{h}\right)}^{0.829}{\left(\frac{{H}_{i}}{h}\right)}^{0.798}$ | 0.988 |

(12g) | $\frac{{{F}_{z}}^{-max}}{0.005\rho g{h}^{2}}={-2.314\left(\frac{dr}{h}\right)}^{-0.254}{\left(\frac{B}{h}\right)}^{0.890}{\left(\frac{{gT}^{2}}{h}\right)}^{0.282}{\left(\frac{{H}_{i}}{h}\right)}^{1.145}$ | 0.989 |

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## Share and Cite

**MDPI and ACS Style**

Niu, G.; Chen, Y.; Lv, J.; Zhang, J.; Fan, N.
Determination of Formulae for the Hydrodynamic Performance of a Fixed Box-Type Free Surface Breakwater in the Intermediate Water. *J. Mar. Sci. Eng.* **2023**, *11*, 1812.
https://doi.org/10.3390/jmse11091812

**AMA Style**

Niu G, Chen Y, Lv J, Zhang J, Fan N.
Determination of Formulae for the Hydrodynamic Performance of a Fixed Box-Type Free Surface Breakwater in the Intermediate Water. *Journal of Marine Science and Engineering*. 2023; 11(9):1812.
https://doi.org/10.3390/jmse11091812

**Chicago/Turabian Style**

Niu, Guoxu, Yaoyong Chen, Jiao Lv, Jing Zhang, and Ning Fan.
2023. "Determination of Formulae for the Hydrodynamic Performance of a Fixed Box-Type Free Surface Breakwater in the Intermediate Water" *Journal of Marine Science and Engineering* 11, no. 9: 1812.
https://doi.org/10.3390/jmse11091812