# Experimental Study on Submerged Horizontal Perforated Plates under Irregular Wave Conditions

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## Abstract

**:**

## 1. Introduction

## 2. Testing Conditions

#### 2.1. Experimental Arrangement

_{1}, W

_{2}, W

_{3}, W

_{4}, W

_{5}, W

_{6}) were positioned before and after the horizontal board to measure the wave heights before and after the horizontal board. Four Nortek Vectrino Profiler meters (V

_{1}, V

_{2}, V

_{3}, V

_{4}) were also arranged at the bottom of the horizontal plate to measure the velocity distribution at the bottom of the plate. The layout of the model and the relative positions of the wave height meters and velocity meters in the water tank are shown in Figure 2.

#### 2.2. Test Parameters and Groups

_{p}), which is the vertical distance between the static water surface and the upper surface of the plate; the Significant wave height of the irregular wave (H

_{s}), the significant wave period (T

_{s}), the significant wavelength (λ

_{s}), the reflection coefficient (C

_{R}), which equals the reflction wave height/incident wave height, the transmission coefficient (C

_{T}), which is the transmission wave height/incident wave height, and the spectral function (S (f)) (as shown in Figure 1).

_{s}” correspond to each “H

_{s}”) and Table 3 respectively. The test groups are presented in Table 4 (Nine “T

_{s}” correspond to “H

_{s}” of 0.05 and 0.1, five “T

_{s}” correspond to “H

_{s}” of 0.15).

_{max}and L

_{min}are the longest and shortest wavelengths among all wave elements in the experiment, and it takes Δl = 0.3 m in the paper.

## 3. Effect on Transmitted Wave Period

_{t}/T

_{p}), is used as the vertical coordinate. The relative plate length, expressed as the ratio of the plate length to the effective wavelength (L/λ

_{s}), is used as the horizontal coordinate. These two coordinates are used to plot the variation curve.

#### 3.1. Effect of Relative Dive Depth on the Cycle

_{p}/d) on the wave period. The results are compared with the same opening rate and significant wave height, with different relative dive depths by varying the relative plate lengths. The data from three wave height meters: W

_{4}(before), W

_{5}(middle) and W

_{6}(after), were extracted for comparison after each wave condition. For clarity, the applicable period of irregular waves is referred to as “period”, the significant wave height of irregular waves as “wave height”, and the effective wavelength of irregular waves as “wavelength”. It is worth noting that the relative submergence depth (inundation depth/water depth) and the fixed water depth of this test only differ by a coefficient. For ease of understanding, the more intuitive concept of submerged water depth is used for explanation in this section instead of relative submergence depth.

_{s}= 0.1 m and H

_{s}= 0.15 m, are presented in Figure 3. Here, the transmission period Tt is obtained using the average results of the three wave height meters located after the dike. In order to illustrate the differences in the observation results of the three wave altimeters, the relative average error amplitudes ($\frac{\sum \left|{\mathrm{T}}_{\mathrm{i}}-\overline{\mathrm{T}}\right|}{3}$/$\overline{\mathrm{T}}$) of the periodic results of the three observation points behind the embankment under different working conditions were calculated. Figure 3a shows that the relative average error amplitudes under different wave conditions are between 0.42% and 7.78%, while Figure 3b shows that the relative average error amplitudes under different wave conditions are between 0.46% and 6.16%. It was found that the transmission period decreases as the relative submergence decreases. The attenuation of the transmittance period, especially for shorter waves (with larger relative plate lengths), was significant. This indicates that the closer the perforated plate is to the water surface, the more pronounced the attenuation of the transmittance period becomes. The reason is that the closer the horizontal plate is to the water surface, the greater the blocking effect on the upper layer wave energy, thus having a significant impact on the period.

#### 3.2. Effect of Opening Rate on Cycle Time

_{t}/T

_{p}(transmitted wave effective period/incident wave effective period) as the vertical coordinate and the relative plate length L/λ

_{s}(plate length/effective wavelength) as the horizontal coordinate. The results are compared for each relative dive depth, with the same significant wave height but different opening ratios.

_{p}/d = 0.25, the period ratio of the long-period perforated plate is greater than 1.0, and that of the short-period plate is close to 1.0. Comparing the effect of the opening rate, it can be noted that the curves of the three perforated plates are almost identical. In addition, the period ratio of the solid plate in the long-wave (small relative plate length) case is significantly smaller than that of the perforated plate. while it is significantly higher in the short-wave case. The period ratios of the solid plate and perforated plate are also more consistent in the short-wave case. This indicates that the horizontal plate without an open hole can significantly reduce the transmitted wave period. But once the hole is activated, the wave period decay is not much related to the opening rate.

_{p}/d = 0.15, the period ratio is lower than 1.0 for most wave conditions (short wave case with larger relative length). On opening the hole, the curves became roughly similar under short wave conditions, and the solid plate made the period decrease significantly under long wave conditions.

_{p}/d = 0.05, the period ratio reaches its minimum. The attenuation of the transmission wave period is most significant, and the cycle ratio of some short waves (relative to the plate length is larger) is even smaller than that of the solid plate. By analyzing the curves for different opening ratios, it becomes evident that as the opening ratio decreases, the attenuation of the transmitted period initially increases and then decreases, with the inflection point occurring at an opening ratio of 15% under test conditions. In this paper, the perforation rate falls within the moderate range of 15%, and the most significant attenuation of the transmitted wave period is when the submergence depth is small. This phenomenon suggests that a judicious choice of opening ratio (neither too large nor too small) can optimize the period attenuation.

## 4. Effect on Transmitted Wave Height

_{T}(transmitted wave height/incident wave height) and reflection coefficient C

_{R}(reflected wave height/incident wave height) to study the variation of wave height around the structure. Since wave energy is proportional to the square of the wave height, this section mainly examines the transmission coefficient of the submerged horizontal plate. The effects of wave height, relative submergence and opening rate on wave height and wave energy are also discussed.

#### 4.1. Effect of Wave Height on Transmission Coefficient

#### 4.2. Effect of Relative Dive Depth on Transmission Coefficient

_{s}= 0.15 m is set, which is considered a relatively large value. It corresponds to five groups of periods (see Table 4 for details). Figure 6 shows the variation curves of transmission coefficient C

_{T}, reflection coefficient C

_{R}and energy dissipation with the relative plate length at three relative diving depths (submergence depth) of the horizontal plate. Since the water depth d was fixed as a constant in the tests, the corresponding inundation depths for the three relative dive depths (d

_{p}/d = 0.25, 0.15, 0.05) are 0.1 m, 0.06 m and 0.02 m, respectively. After calculation, the relative average error amplitude of each wave condition in Figure 6 is between 1.41% and 8.52%.

#### 4.3. Effect of Open Ratio on Transmittance Coefficient

_{p}/d = 0.05.

_{s}= 0.05 m. After calculation, the relative average error amplitude of each wave condition in Figure 7 is between 0.61% and 12.95%.

## 5. Effect on Flow Rate

#### 5.1. Measurement and Uniformity Analysis of Plate Bottom Velocity Field

_{1}, V

_{2}, V

_{3}, and V

_{4}) at each time point was measured. Based on this, the flow velocity time history curve in the time domain was plotted, and the time domain curve was transformed into the frequency domain velocity spectrum curve through Fourier transform. By comparing the velocity time history curves and velocity spectra of the four measurement points in the X-direction, it was found that the shapes of the velocity time history curves and velocity spectra of the four measurement points were basically the same, and the positions of the four spectral peaks and peaks were basically the same. This can prove that the X-direction velocity of each point at the bottom of the horizontal plate is close. In the experiment, it was found that the Z-direction velocity is generally in the order of 0.01 m/s, which is relatively small compared to the X-direction velocity. Therefore, it can be said that the distribution of the flow field at the bottom of the horizontal plate is relatively uniform. One point velocity can be used as a representative for research. The V

_{2}measurement point results will be used for analysis in the following text.

_{i}represents the instantaneous flow velocity at each time, and V

_{rms}represents the root mean square flow velocity.

_{1}–V

_{4}measurement points at the bottom of the solid plate.

_{1}–V

_{4}measurement points at the bottom of the solid plate are very close under different working conditions, further confirming that the flow velocity at the bottom of the horizontal plate is basically uniformly distributed. In the following discussion, only the root mean square velocity results at the V

_{2}measurement point will be taken for discussion.

#### 5.2. Effect of Relative Dive Depth on the Flow Rate of Perforated Plate

#### 5.3. Effect of Opening Ratio on Flow Rate

_{p}/d = 0.25 and significant wave height Hs = 0.05 m, the flow velocities in X and Z directions exhibit a general trend where maximum flow velocity decreases with an increase in relative plate length, while the Z-direction flow velocity is smaller in value and less varied. Comparing the four sets of curves with different opening ratios, it is evident that the opening ratio can affect the maximum flow velocity. In Figure 8a, comparing the two curves with K = 0 and K = 0.15 shows that the flow velocity of the solid plate (K = 0) is lower than that of the open-hole plate (K = 0.1) in all eight conditions except one. This finding indicates that perforation increases the circulation performance of the upper and lower water bodies, thus increasing the flow velocity in the lower part of the horizontal plate. Comparing these two types of perforated plates (K = 0.1 and K = 0.15), it is noted that the maximum flow velocity increased when the relative plate length was large (smaller wavelength) after the perforated rate was increased. This indicates that increasing the perforated area in the short-wave case can amplify the flow velocity at the bottom of the plate, whereas this effect is not significant in the long-wave cases. Comparing the two types of open-aperture plates, K = 0.1 and K = 0.2, it was found that the two curves were relatively close to each other, indicating that the perturbation effect of open-aperture on the bottom flow velocity becomes less noticeable after increasing the open-aperture ratio to a certain degree.

_{p}/d = 0.25 and H

_{s}= 0.15 m, it can be seen that the RMS flow velocity in the X-direction and Z-direction shows a gradual decrease with an increase in relative plate lengths, but this change is not significant at a lower velocity in the Z-direction. It is also observed from Figure 11a that the RMS flow velocity of a solid plate is the smallest, especially when the relative plate length (long wave) is small, and the flow velocity increases significantly after the hole is opened. For the open-perforated plate, the three curves almost overlapped, indicating that the effect of the open-perforation rate on the root-mean-square flow velocity is not prominent.

#### 5.4. Transmittance Coefficient and Flow Rate

_{p}/d = 0.35. Comparing the results for the three wave heights, the variation patterns of the transmission coefficient curve and the maximum flow velocity curve with large wave height (H

_{s}= 0.15 m) in Figure 12c shows a general decrease when the relative plate lengths decrease. The discussion mainly focused on the X-direction flow velocity due to insignificant Z-direction flow velocity. When wave height is small (H

_{s}= 0.05 m, H

_{s}= 0.1 m), the transmission coefficient decreases and then increases, while the flow velocity monotonically decreases. This phenomenon indicates that the correlation between the transmission coefficient and the maximum flow velocity at the bottom of the solid plate is not particularly strong, especially in cases with small wave heights.

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**The comparisons of wave period ratio of 20% opening perforated plate (H

_{s}= 0.1 m, H

_{s}= 0.15 m). (

**a**) Significant wave height H

_{s}= 0.1 m; (

**b**) Significant wave height H

_{s}= 0.15 m.

**Figure 4.**The comparisons of wave period ratio of significant wave height H

_{s}= 0.05 m. (

**a**) Relative submerged depth d

_{p}/d = 0.25; (

**b**) Relative submerged depth d

_{p}/d = 0.15; (

**c**) Relative submerged depth d

_{p}/d = 0.05.

**Figure 5.**Effect of significant wave height on transmission coefficient at different relativesubmerged depth. (

**a**) Relative submerged depth d

_{p}/d = 0.35; (

**b**) Relative submerged depth d

_{p}/d = 0.25; (

**c**) Relative submerged depth d

_{p}/d = 0.15; (

**d**) Relative submerged depth d

_{p}/d = 0.05.

**Figure 6.**Transmission coefficient, reflection coefficient and energy dissipation of perforated plate (K = 0.1, H

_{s}= 0.15 m).

**Figure 7.**Effect of porosity on transmission coefficient, reflection coefficient and energy dissipation (d

_{p}/d = 0.05, H

_{s}= 0.05 m).

**Figure 8.**Influence of relative plate length on maximum velocity under different relative submerged depths (k = 0.15, H

_{s}= 0.1 m). (

**a**) Positive maximum velocity in X direction; (

**b**) Negative maximum velocity in X direction; (

**c**) Positive maximum velocity in Z direction; (

**d**) Negative maximum velocity in Z direction.

**Figure 9.**The influence of relative plate length on the maximum flow rate under different opening ratios (dp/d = 0.25, H

_{s}= 0.05 m). (

**a**) Positive maximum velocity in X direction; (

**b**) Negative maximum velocity in X direction; (

**c**) Positive maximum velocity in Z direction; (

**d**) Negative maximum velocity in Z direction.

**Figure 11.**Effect of relative plate length on RMS velocity with different opening ratios (d

_{p}/d = 0.25, H

_{s}= 0.15 m). (

**a**) RMS velocity in the X direction; (

**b**) RMS velocity in the Z direction.

**Figure 12.**Biaxial diagram of perforated plate Velocity-Transmission coefficient (K = 0.2, d

_{p}/d = 0.25). (

**a**) Significant wave height H

_{s}= 0.05 m; (

**b**) Significant wave height H

_{s}= 0.1 m; (

**c**) Significant wave height H

_{s}= 0.15 m.

Main Parameters | Indication Symbols | Units | Main Parameters | Indication Symbols | Units |
---|---|---|---|---|---|

Testing water depth | d | m | Opening ratio | K | / |

Horizontal plate submergence depth | d_{p} | m | Effective wavelength | λ_{s} | m |

Length of horizontal plate | L | m | Wave frequency | f | Hz |

Significant wave period | T_{s} | s | Reflection coefficient | C_{R} | / |

Significant wave height | H_{s} | m | Relative dive depth | d_{p}/d | / |

Relative plate length | L/λ_{s} | / |

Prototypes | Models | ||||
---|---|---|---|---|---|

h/m | Ts/s | Hs/m | h/m | Ts/s | Hs/m |

6.4 | 10.68, 7.32, 5.8, 4.92, 4.36, 3.96, 3.64, 3.4, 3.24 | 0.8, 1.6, 2.4 | 0.4 | 2.67, 1.83, 1.45, 1.23, 1.09, 0.99, 0.91, 0.85, 0.81 | 0.05, 0.1, 0.15 |

Length/m | Broad/m | Thick/m | Submergence Depth/m | |
---|---|---|---|---|

Prototype | 16 | 16 | 1.6 | 2.24, 1.6, 0.96, 0.32 |

Model | 1 | 1 | 0.1 | 0.14, 0.1, 0.06, 0.02 |

Depth d/m | Submergence Depth dp/m | Opening Ratio | Irregular Wave Effective Wave | |
---|---|---|---|---|

Effective Period Ts/s | Height Hs/m | |||

0.4 | 0.14, 0.1, 0.06, 0.02 | 0 | 2.67, 1.83, 1.45, 1.23, 1.09, 0.99, 0.91, 0.85, 0.81 | 0.05, 0.1 |

2.67, 1.83, 1.45, 1.23, 1.09 | 0.15 | |||

0.4 | 0.1, 0.06, 0.02 | 0.1, 0.15, 0.2 | 2.67, 1.83, 1.45, 1.23, 1.09, 0.99, 0.91, 0.85, 0.81 | 0.05, 0.1 |

2.67, 1.83, 1.45, 1.23, 1.09 | 0.15 |

Wave Height m | Period s | Submergence Depth m | Root Mean Square Velocity in X-Direction m/s | Root Mean Square Velocity in Z-Direction m/s | ||||||
---|---|---|---|---|---|---|---|---|---|---|

V_{1} | V_{2} | V_{3} | V_{4} | V_{1} | V_{2} | V_{3} | V_{4} | |||

0.05 | 0.85 | 0.14 | 0.012 | 0.011 | 0.015 | 0.018 | 0.005 | 0.004 | 0.004 | 0.005 |

0.05 | 0.99 | 0.1 | 0.022 | 0.024 | 0.018 | 0.023 | 0.010 | 0.009 | 0.011 | 0.010 |

0.1 | 1.23 | 0.06 | 0.072 | 0.065 | 0.078 | 0.063 | 0.029 | 0.022 | 0.025 | 0.024 |

0.15 | 2.67 | 0.02 | 0.112 | 0.102 | 0.126 | 0.118 | 0.032 | 0.028 | 0.036 | 0.034 |

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## Share and Cite

**MDPI and ACS Style**

Zheng, Y.; Zhou, Y.; Jin, R.; Mu, Y.; He, M.; Zhao, L.
Experimental Study on Submerged Horizontal Perforated Plates under Irregular Wave Conditions. *Water* **2023**, *15*, 3015.
https://doi.org/10.3390/w15163015

**AMA Style**

Zheng Y, Zhou Y, Jin R, Mu Y, He M, Zhao L.
Experimental Study on Submerged Horizontal Perforated Plates under Irregular Wave Conditions. *Water*. 2023; 15(16):3015.
https://doi.org/10.3390/w15163015

**Chicago/Turabian Style**

Zheng, Yanna, Yifan Zhou, Ruijia Jin, Yingna Mu, Ming He, and Lingxiao Zhao.
2023. "Experimental Study on Submerged Horizontal Perforated Plates under Irregular Wave Conditions" *Water* 15, no. 16: 3015.
https://doi.org/10.3390/w15163015