# Laboratory Study on Flow Characteristics during Solitary Waves Interacting with a Suspended Horizontal Plate

^{*}

## Abstract

**:**

_{0}and 0.8 C

_{0}, respectively, where C

_{0}is the maximum flow speed of the incident wave. Ritter’s analytical solution for the dam–break flow problem was examined and compared with the measured data. The accuracy of this solution for the present subject is significant in the period of T ∈ (0.6, 0.9). The adequate experimental data are valuable as a benchmark problem for further numerical model refinement and the improvement of fluid theory.

## 1. Introduction

## 2. Experimental Set–Up

#### 2.1. Experimental Apparatus and Model Structure

#### 2.2. Velocity Measurements by PIV

^{3}and keep well suspended and fluid–following in water. A high–speed complementary metal–oxide–semiconductor (CMOS) camera (PCO.dmaxS4) with a resolution of 2016 × 2016 pixels, 12–bit dynamic range, and 1000–Hz framing rate was used to capture the particle–laden images.

#### 2.3. Experimental Conditions

## 3. Results and Discussion

#### 3.1. Evolution of Flow Structures

_{0}, which is higher than the vertical component with a value of around 0.30 C

_{0}. In this paper, C

_{0}is the maximum flow velocity of the incident wave, as 0.68 m/s in this case. Figure 3b shows the velocity of splash–up along the front wall of the deck structure. Although the velocity is still horizontally dominated, there is a significant increase in vertical velocity. More specifically, the maximum horizontal and vertical velocity at this moment increases to the value of 1.28 C

_{0}and 0.63 C

_{0}, respectively. After that, as the water tongue extends further forward, the horizontal velocity increases to 1.46 C

_{0}. Meanwhile, the maximum vertical velocity is 0.36 C

_{0}, significantly decreased compared to the previous period. This is mainly since the splashing tongue of water has finished its falling process. Right after the wave crests through the front of the plate, the dominating momentum changes immediately from upward speed to downward direction due to the phase change of the solitary wave. As shown in Figure 3d, an obvious flow separation phenomenon is presented at the front of the plate while a significant increase in vertical velocity occurs. The maximum horizontal and vertical velocity is around 1.57 C

_{0}and 0.60 C

_{0}, respectively. At T = 0.74, the flow separation becomes more pronounced. As shown in Figure 3e, the maximum vertical velocity is around 0.81 C

_{0}, while the maximum horizontal velocity is around 1.47 C

_{0}. After T = 0.95, the green water is dominated by the separation phase from the plate, the measured velocity magnitude tends to reduce. More specifically, in Figure 3f–h, the maximum horizontal velocity decreases from 1.13 C

_{0}to 0.43 C

_{0}, while the maximum vertical velocity decreases from 0.75 C

_{0}to 0.32 C

_{0}.

_{mx}, maximum vertical velocity U

_{mz}, and the maximum velocity magnitude U

_{mc}is calculated. The time history of the maximum velocity is shown in Figure 6, where the vertical error bars represent the standard deviation of experimental repeated measurements. The velocity is normalized with C

_{0}, as C

_{0}is defined in Equation (5) mentioned before as 0.68 m/s in this case. Generally, the green water process is presented to be dominated by the horizontal velocity. After the wavefront impact with the plate front side, the horizontal velocity increases significantly. At T = 0.61, the horizontal velocity increases to a maximum of 1.59 C

_{0,}and then the horizontal velocity starts to decrease. Different from the horizontal velocity, the trend of vertical velocity is more complex. With the wave tongue runup along the front side, the vertical velocity shows a rapid increase (0 < T < 0.18). Then, with the splashed tongue falling and attached to the top side of the plate, the maximum vertical velocity declines. After T = 0.42, flow separation starts to occur, causing the vertical velocity restarts to increase. With the incident wave passing forward, the vertical velocity decreases gradually. During this whole process, the maximum vertical velocity presents as 0.83 C

_{0}.

#### 3.2. Effect of Wave Height and Structural Suspended Height

_{0}(incident–wave maximum velocity). Generally, the flow field is dominated by the horizontal velocity, as U

_{mc}presents little discrepancy with U

_{mx}. Note that the U

_{mc}is not the root mean square (RMS) of the U

_{mx}and U

_{mz}, as the maximum horizontal velocity and vertical velocity may not occur in the same position nor at the same time. Although the velocities are normalized by C

_{0}which is dependent on the wave height, the U

_{mx}/C

_{0}and U

_{mc}/C

_{0}present incremental over the relative wave height, see Figure 10a. The Maximum U

_{mx}/C

_{0}and U

_{mc}/C

_{0}is 1.90 and 1.98, which are measured in the case of H/h = 10/30, D/h = 2/30. For the vertical velocity, U

_{mz}/C

_{0}, the maximum velocity is almost invariable with a constant around 0.8, only slight changes can be seen. For the cases of fixed incident–wave height (H/h = 10/30, Figure 10b), the ratio of maximum velocity magnitude to incident–wave velocity decreases over D/h, while the variation of U

_{mz}/C

_{0}is still slight.

#### 3.3. Comparing with the Green Water Model

_{d}is the flow depth, $\overline{U}$ is the horizontal velocity, g is the gravitational acceleration, S

_{0}is the bottom slope, and S

_{f}is the friction slope.

_{0}is the initial water depth of the dam model.

_{0}and increases linearly concerning x. A diagrammatic sketch of Ritter’s solution is shown in Figure 11.

_{deck}is the plate upper surface elevation from the still water level.

_{deck}= 3 cm).

## 4. Conclusions

- (1)
- The flow evolution of green water can be categorized into the following three phases: (A) Green water tongue generation and run–up, (B) Green water overtopping along the plate, and (C) Flow separation from the plate. In Phase A, the wavefront contacts the plate at T = 0. At the front side of the plate, a green water tongue was generated and run up rapidly. Subsequently, the water tongue overturns and slaps the upper surface of the plate. Phase B includes the effects of the start of green water overtopping, the burst of entrapped bubbles, wave overtopping at the end of the plate, water collision behind the plate, and vortex generation under the plate front end. In Phase C, the flow separation starts at the front side of the plate and ends at the rear side of the plate;
- (2)
- Structural suspended height and incident wave size have different influences on the flow properties in each stage. Their evolutions present obvious similarities in general but several differences in detail. The increase in plate height will lead to a decrease in global velocities, less air entrapped in Phase B but a larger aerated area in Phase C. Besides, the increase in incident wave height can cause a significant increase in global velocities and more air involved in the water;
- (3)
- The maximum velocity around the plate was counted and normalized with C
_{0}(the maximum flow speed of the incident wave). There is little difference between the velocity magnitude and the horizontal component, indicating the flow kinematics is dominated by the horizontal flow. The normalized maximum horizontal velocity increase with larger incident waves and decrease with higher suspended height, while the ratio of maximum vertical velocity to C_{0}is almost invariable with a constant around 0.8; - (4)
- The Ritter’s dam–break flow solution for the prediction of green water flow in solitary wave conditions was validated. This solution agrees with the vertical–average horizontal velocities above the plate in the period of T ∈ (0.6, 0.9) when the flow field is in the transition stage from Phase B to Phase C. The rest of the time, the distribution of velocity does not conform to the description of Ritter’s solution. The minimum deviation coefficient throughout the present experiments appears in the case with the smallest submerged depth (under the wave crest, 3 cm).

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**General setup of present experiments: (

**a**) sketch of the experimental setup; (

**b**) definition of the temporal origin and the spatial coordinates; (

**c**) measured surface profiles in the absence of the model; and (

**d**) sketch map of the definition of the characteristic wave–passing period.

**Figure 2.**Snapshots of aerated flow evolution around the plate of Case 1, D/h = 6/30, H/h = 10/30. T = (

**a**) 0, (

**b**) 0.17, (

**c**) 0.42, (

**d**) 0.62, (

**e**) 0.74, (

**f**) 0.95, (

**g**) 1.45, (

**h**) 1.92.

**Figure 3.**Measured velocity fields around the plate on the side view of Case 1, D/h = 6/30, H/h = 10/30. T = (

**a**) 0; (

**b**) 0.17; (

**c**) 0.42; (

**d**) 0.62; (

**e**) 0.74; (

**f**) 0.95; (

**g**) 1.45; and (

**h**) 1.92.

**Figure 4.**Detailed weather–side velocity evolution around the plate of Case 1, D/h = 6/30, H/h = 10/30. T = (

**a**) 0; (

**b**) 0.17; (

**c**) 0.31; (

**d**) 0.42; (

**e**) 0.62; and (

**f**) 0.74.

**Figure 6.**Time history of maximum fluid velocities normalized by the wave velocity of Case 1, D/h = 6/30, H/h = 10/30, C

_{0}= 0.68 m/s.

**Figure 7.**Measured velocity fields around the plate of different cases at T = 0.17 (Phase A): (

**a**) case 1, D/h = 6/30, H/h = 10/30; (

**b**) case 2, D/h = 4/30, H/h = 10/30; (

**c**) case 3, D/h = 2/30, H/h = 10/30; (

**d**) case 4, D/h = 2/30, H/h = 8/30; and (

**e**) case 5, D/h = 2/30, H/h = 6/30.

**Figure 8.**Same with Figure 7 for T = 0.69 (Phase B). (

**a**) Case 1, D/h = 6/30, H/h = 10/30; (

**b**) Case 2, D/h = 4/30, H/h = 10/30; (

**c**) Case 3, D/h = 2/30, H/h = 10/30; (

**d**) Case 4, D/h = 2/30, H/h = 8/30; (

**e**) Case 5, D/h = 2/30, H/h = 6/30.

**Figure 9.**Same with Figure 7 for T = 1.03 (Phase C): (

**a**) case 1, D/h = 6/30, H/h = 10/30; (

**b**) case 2, D/h = 4/30, H/h = 10/30; (

**c**) case 3, D/h = 2/30, H/h = 10/30; (

**d**) case 4, D/h = 2/30, H/h = 8/30; and (

**e**) case 5, D/h = 2/30, H/h = 6/30.

**Figure 10.**Maximum horizontal velocity, vertical velocity, and velocity magnitude of different cases: (

**a**) case 3; case 4, and case 5, D/h = 2/30; and (

**b**) case 1, case 2, and case 3, H/h = 10/30.

**Figure 12.**Comparisons of measured velocities and Ritter’s solution above the plate for case 1. Red dot: measured depth–averaged velocity, blue line: Ritter’s solution: T = (

**a**) 0.35; (

**b**) 0.43; (

**c**) 0.52; (

**d**) 0.61; (

**e**) 0.69; (

**f**) 0.78; (

**g**) 0.87; and (

**h**) 0.95.

**Figure 13.**Comparisons of measured velocities and Ritter’s solution above the plate for T = 0.45. Red dot: measured depth–averaged velocity, blue line: Ritter’s solution: (

**a**) case 1, D/h = 6/30, H/h = 10/30; (

**b**) case 2, D/h = 4/30, H/h = 10/30; (

**c**) case 3, D/h = 2/30, H/h = 10/30; (

**d**) case 4, D/h = 2/30, H/h = 8/30; and (

**e**) Case 5, D/h = 2/30, H/h = 6/30.

**Figure 14.**Deviation coefficients between measured velocities and Ritter’s solutions: (

**a**) case 3, case 4, and case 5, D/h = 2/30; and (

**b**) case 1, case 2, and case 3, H/h = 10/30.

Case | Water Depth h (cm) | Plate Thickness (cm) | Plate Length L (cm) | Suspended Height D (cm) | Wave Height H (cm) | Max. Free Stream Velocity C_{0} (m/s) | Characteristic Period (s) |
---|---|---|---|---|---|---|---|

1 | 30 | 1 | 25 | 6 | 10 | 0.683 | 0.578 |

2 | 4 | 10 | 0.683 | 0.752 | |||

3 | 2 | 10 | 0.683 | 1.001 | |||

4 | 2 | 8 | 0.513 | 1.045 | |||

5 | 2 | 6 | 0.366 | 1.079 |

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**MDPI and ACS Style**

Niu, X.; Ma, Y.; Dong, G.
Laboratory Study on Flow Characteristics during Solitary Waves Interacting with a Suspended Horizontal Plate. *Water* **2022**, *14*, 2386.
https://doi.org/10.3390/w14152386

**AMA Style**

Niu X, Ma Y, Dong G.
Laboratory Study on Flow Characteristics during Solitary Waves Interacting with a Suspended Horizontal Plate. *Water*. 2022; 14(15):2386.
https://doi.org/10.3390/w14152386

**Chicago/Turabian Style**

Niu, Xuyang, Yuxiang Ma, and Guohai Dong.
2022. "Laboratory Study on Flow Characteristics during Solitary Waves Interacting with a Suspended Horizontal Plate" *Water* 14, no. 15: 2386.
https://doi.org/10.3390/w14152386