# The Hydrodynamic Moment of a Floating Structure in Finite Flowing Water

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Design and Configuration

#### 2.1. Experimental Configuration and Dynamic Moment Measurement

_{1}and F

_{2}, and the distances from the center of the structure to the sensors are denoted D

_{1}and D

_{2}, respectively. The hydrodynamic moment was calculated with clockwise being negative and counterclockwise being positive as follows:

#### 2.2. Dimensional Analysis and Parameter Design of the Floating Structure

_{Floating}or M

_{F}for short. The influencing factors of M

_{F}[MLT

^{−2}] mainly include the width of the structure B ([L]), the height of the structure a ([L]), a draft of the structure h ([L]), downstream water depth H′ ([L]), gravitational constant of acceleration g ([LT

^{−2}]), water density ρ ([ML

^{−3}]), upstream and downstream water level difference ΔH ([L]), and average velocity of the upstream section v ([LT

^{−1}]). Figure 3 shows a schematic diagram of the main parameters affecting the floating structure. According to dimensional analysis, the expression of the parameters influencing the hydrodynamic moment is as follows:

^{2}] in the kinematic dimension, and water density ρ in the dynamic dimension ([ML

^{−3}]) are selected as the three basic physical quantities. According to the π theorem:

_{f}. We define π

_{2/}π

_{3}= B/a as the aspect ratio of the floating structure; h/H′ is the relative draft; ΔH/H′ is the relative water level difference, and π

_{6}= Fr = v/$\sqrt{gH}$ is the Froude number of the upstream section. The length of the section is five times the water depth of the flume at the upstream end of the floating structure, where the flow is stable and is not affected by the structure. The simplified result is:

^{2}/s, 0.06 m

^{2}/s, 0.07 m

^{2}/s, and 0.08 m

^{2}/s; the draft depths of the structure were 2.00 cm, 4.00 cm, 6.00 cm, and 8.00 cm; and the downstream water levels were 18.00 cm, 20.00 cm, 22.00 cm, and 24.00 cm, respectively (see in Table 1). To ensure the completeness of the test, orthogonal tests were also conducted, resulting in a total of 121 test groups.

## 3. Results and Discussion

#### 3.1. Shape of the Floating Structure

_{f}to the structure shape parameter B/a under the same hydraulic conditions. With increasing B/a, M

_{f}increases. During the experiment, with increasing B/a, the flow velocity decreases significantly near the center of the structure’s lower surface. The small recirculation zone near the lower center of the structure expands to the back half of the structure. When the shape parameter of the floating structure increases, the surface pressure difference caused by the uneven distribution of the dynamic pressure increases, and the distance between the action point and the center of the structure also increases, resulting in a rise in the hydrodynamic moment. For a larger structure, the pressure drop from the entrance of the bottom gap to the downstream part increases, causing an increased pressure difference and giving rise to the hydrodynamic moment.

_{f}was obtained through a fitting analysis:

_{f}= k

_{1}Bh/aH′ + c

_{1}

_{1}and c

_{1}are the coefficient and constant terms, respectively, of the independent variable Bh/aH′.

#### 3.2. Draft Depth

_{f}and h/ΔH, indicating that M

_{f}has an obvious logarithmic relationship with h/ΔH, and its value increases with increasing B/a. A non-uniform distribution of the flow velocity causes a change in the hydrodynamic moment at different draft depths. The hydrodynamic pressure acting on the surface of the structure decreases obviously if the draft depth is small. When the water level difference increases, the velocity around the floating structure rises, and the pressure difference between the front and back surfaces of the structure increases, resulting in an increase in the hydrodynamic moment.

_{2}. Hence, the correlation between M

_{f}and the draft depth can be expressed as follows:

_{2}and c

_{2}are the coefficient and constant terms, respectively, of the independent variable ln($\frac{h}{\Delta H}$)

#### 3.3. Relative Water Level Difference

_{f}and the relative water level difference ΔH/H′. M

_{f}increases with increasing ΔH/H′, and when the structure is large, the growth rate of the hydrodynamic moment shows an increasing trend. This is similar to the conclusion that the moment increases with increasing B/a. When ΔH/H′ increases, the pressure difference between the front and back surfaces of the structure increases. With an increasing water level difference, the hydrodynamic pressure acting on the upstream face of the floating structure is obviously higher than that acting on the downstream face; therefore, the structure tilts downstream, and the moment increases obviously. The hydrodynamic pressure acting on the downstream face was found to be relatively low due to the recirculation zone existing near the downstream face of the structure. As a consequence, the structure becomes increasingly unstable as a result of the increasing water level difference and thus can be easily unbalanced.

_{f}and ΔH/H′ was obtained through a fitting analysis:

_{f}= k

_{3}B/a∙ΔH/H′ + c

_{3}

_{3}and c

_{3}are the coefficient and constant terms, respectively, of the independent variable BΔH/aH′.

#### 3.4. Froude Number

_{f}and Fr

^{2}. The M

_{f}value of the floating structure increases with increasing Fr

^{2}. Moreover, with increasing B/a, the growth rate of M

_{f}tends to increase. With increasing velocity, the recirculation zone in the downstream area increases, thereby increasing the velocity difference between the upstream and downstream faces of the structure. The kinetic energy of the flow is converted into potential energy due to the water acting on the upstream face, and the dynamic pressure increases, thereby increasing the pressure difference between the front and back surfaces of the structure, which increases the hydrodynamic moment and decreases its stability.

^{2}/a on M

_{f}, the following formula can be obtained:

_{f}= k

_{4}BFr

^{2}/a + c

_{4}

_{4}and c

_{4}are the coefficient and constant terms, respectively, of the independent variable $B$Fr

^{2}/a.

## 4. Stepwise Regression Analyses and Validation of M_{f}

#### 4.1. Regression Analysis of M_{f}

_{f}was obtained by excluding the repetitive variable BΔH/aH′. The results of significance analysis of the regression equation coefficients are shown in Table 2.

_{f}of a floating structure of unit length is as follows:

#### 4.2. Verification and Error Analysis

_{f}; thus, the AMCC was used to verify the correctness of the prediction and to ascertain whether the regression model is satisfactory. The SEE value evaluates the reliability of the data, with smaller values indicating higher reliability, which indicates that the observations are closer to the fitted line. The AMCC and SEE are defined as follows:

^{2}is the coefficient of determination, K is the size of the dataset, J is the number of dimensionless independent variables, and y

_{k}is the measured value of M

_{f}. Here, $\tilde{{y}_{k}}$ is the value of M

_{f}estimated by Equation (9), and $\overline{y}$ is the mean value of y

_{k}.

_{f}. The F-test, DW test and VIF test formulas are given as follows:

_{t}is the error term at time t, and R

_{i}

^{2}is the multiple coefficient of determination of the independent variable.

_{f}, a comparison between the measured and calculated results is shown in Figure 10, suggesting good agreement. Thus, the development of a calculation formula for the overturning moment was deemed successful.

## 5. Conclusions

_{f}of a floating structure were obtained based on dimensional analysis. The characteristics of the moment and the correlations between the influencing factors and the moment were analyzed, and an expression of the moment was derived. The main conclusions are as follows.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Notations

a | Height of the structure |

B | Width of the structure |

Fr^{2} | Froude number: Fr^{2} = v^{2}/gH |

g | Gravitational constant of acceleration |

H | Upstream water depth |

H′ | Downstream water depth |

h | Draft of the structure |

L | Length of the structure |

q | Discharge per width of flow |

R^{2} | Coefficient of determination |

v | Average flow velocity |

ΔH | Water level difference |

ρ | Water density |

AMCC | Adjusted multiple correlation coefficient |

SEE | Standard error of estimation |

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**Figure 1.**Diagram of the test model: (

**a**) Panoramic view of the experimental configuration, (

**b**) The diagram of the floating structure.

**Figure 2.**Schematic of moment measurement: (

**a**) Layout of the sensor, (

**b**) Pressure profile of the floating structure.

a/m | B/m | h/m | q/(m^{2} s^{−1}) | H′/m | |||
---|---|---|---|---|---|---|---|

0.10 | 0.10 0.20 0.30 0.40 | 0.02 | 0.05 | 0.18 | 0.20 | 0.22 | 0.24 |

0.06 | 0.18 | 0.20 | 0.22 | 0.24 | |||

0.07 | 0.18 | 0.20 | 0.22 | 0.24 | |||

0.08 | 0.18 | 0.20 | 0.22 | 0.24 | |||

0.04 | 0.05 | 0.18 | 0.20 | 0.22 | 0.24 | ||

0.06 | 0.18 | 0.20 | 0.22 | 0.24 | |||

0.07 | 0.18 | 0.20 | 0.22 | 0.24 | |||

0.08 | 0.18 | 0.20 | 0.22 | 0.24 | |||

0.06 | 0.05 | 0.18 | 0.20 | 0.22 | 0.24 | ||

0.06 | 0.18 | 0.20 | 0.22 | 0.24 | |||

0.07 | 0.18 | 0.20 | 0.22 | 0.24 | |||

0.08 | 0.18 | 0.20 | 0.22 | 0.24 | |||

0.08 | 0.05 | 0.18 | 0.20 | 0.22 | 0.24 | ||

0.06 | 0.18 | 0.20 | 0.22 | 0.24 | |||

0.07 | 0.18 | 0.20 | 0.22 | 0.24 | |||

0.08 | 0.18 | 0.20 | 0.22 | 0.24 |

Independent Variable | Nonstandardized Coefficients | T | Sig. | Coefficient 95% Confidence Interval | Collinear Statistics | |||
---|---|---|---|---|---|---|---|---|

Coefficient | Standard Error | Lower Limit | Upper Limit | Tolerance | VIF | |||

Constant | −0.001 | 0.000 | −1.73 | 0.08 | −0.001 | 0.000 | ||

BΔH/aH′ | 0.045 | 0.002 | 19.99 | 0.00 | 0.040 | 0.049 | 0.66 | 1.51 |

BFr^{2}/a | 0.036 | 0.002 | 15.73 | 0.00 | 0.031 | 0.040 | 0.51 | 1.97 |

Δh | −0.001 | 0.000 | −2.21 | 0.00 | 0.000 | 0.000 | 0.73 | 1.38 |

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**MDPI and ACS Style**

Cui, Z.; Pan, S.-Y.; Chen, Y.-J.
The Hydrodynamic Moment of a Floating Structure in Finite Flowing Water. *Fluids* **2021**, *6*, 307.
https://doi.org/10.3390/fluids6090307

**AMA Style**

Cui Z, Pan S-Y, Chen Y-J.
The Hydrodynamic Moment of a Floating Structure in Finite Flowing Water. *Fluids*. 2021; 6(9):307.
https://doi.org/10.3390/fluids6090307

**Chicago/Turabian Style**

Cui, Zhen, Shi-Yang Pan, and Yue-Jun Chen.
2021. "The Hydrodynamic Moment of a Floating Structure in Finite Flowing Water" *Fluids* 6, no. 9: 307.
https://doi.org/10.3390/fluids6090307