# Free-Decay Heave Motion of a Spherical Buoy

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Measurements

## 3. Numerical Simulations

#### 3.1. High-Fidelity Model Based on Solution of Navier–Stokes Equations

**u**and p, respectively) are expressed as

**g**is the gravitational acceleration vector,

**f**

_{γ}is the surface tension force, and $\sigma =-p\mathit{I}+\mu [\mathsf{\nabla}\mathit{u}+{\left(\mathsf{\nabla},\mathit{u}\right)}^{T}]$ is the average stress tensor, with

**I**and $\mu $ being the identity tensor and mixture viscosity, respectively. It is worth mentioning that, as shown in a number of previous studies (see [29] for a recent example), the dynamics of the buoy and the flow around it can be accurately captured without needing to include a turbulent model in the simulations. We assumed that the two-phase fluid is initially at rest (i.e.,

**u**= 0 at t = 0) and considered the boundary conditions

**n**is the unit normal vector pointing towards the fluid, S

_{b}represents the surface of the buoy, and S

_{t}denotes the top boundary of the computational domain that confines the air column from the above. The instantaneous linear and angular velocities of the buoy (

**U**and $\mathsf{\Omega}$, respectively) were determined via

**U**and $\mathsf{\Omega}$ were set to zero at t = 0, and the initial position of the buoy was set to its corresponding value in the experiments. Note that the contribution of surface tension to the load exerted on the buoy was ignored while still accounting for its contribution to Equation (1).

_{γ}was calculated following the continuum surface force (CSF) model [33] from

#### 3.2. Reduced-Order Models Based on Linear Potential Flow Theory

#### 3.2.1. Cummins Model

_{z}= dz

_{b}/dt is the velocity of the buoy in the z direction, R = 0.142 m is the buoy’s radius, and d = 0.090 m and ${z}_{b}^{eq}=H+R-d=1.052\mathrm{m}$ denote the draft and the center of mass position of the buoy in the z direction at static equilibrium, respectively. The infinite frequency added mass coefficient and radiation impulse response function (IRF) in the above equation can be expressed (see, e.g., [49]), respectively, as

^{2}, and $\mathrm{\Phi}$ is the dimensionless, complex-valued velocity potential field that satisfies the following boundary-value problem:

_{∞}and k(t), we iteratively solved integro-differential Equation (8) for the position of the buoy, where at each iteration, the convolution integral was treated as a forcing function whose values were known from the velocity U

_{z}calculated in the previous iteration. The forcing function was set to zero in the first iteration.

#### 3.2.2. Mass-Spring-Damper Model

## 4. Results and Discussion

^{☆}= 9.82 s

^{−1}and δ = 1.86. Comparing these values with the ones reported in Equation (16) suggests that the reduced-order models underpredict the effective added mass and damping coefficients corresponding to the buoy’s heave motion by about 9% and 24%, respectively. This may not be surprising, since these models are based on the linear potential flow theory. However, Kramer et al. [29] reported closer agreements between the predictions of their reduced-order models and their measurements for the case where the normalized drop height, defined as $({z}_{b}{|}_{t=0}-{z}_{b}^{eq})/R$, is $0.6$, which is very close to its magnitude in our work that is $0.61$. This discrepancy can be due to two differences between our study and the free-decay tests performed by Kramer et al. [29], namely, disparities in the draft lengths of the buoys at static equilibrium and the position of the sidewalls. The buoy is half submerged at equilibrium (i.e., d = R), and the distance between the center of the buoy to the sidewalls was roughly 28R and 43R in the experiments of Kramer et al. [29], whereas, in our case, d = 63R, and the buoy-to-wall distances were about 10R and 35R. It is worth noting that the variations in both the added mass and damping coefficients with λ are greatly affected by the mean position of the buoy’s draft line. For instance, it is known that, for a half-submerged sphere, b/ρR

^{3}ω decays to zero very rapidly as 1/λ

^{4}when λ → ∞ (see, e.g., [13]). Yet, we found that, for our buoy, the dimensionless damping coefficient decays more slowly to zero with a rate close to 1/λ

^{2}. Overall, our findings imply that, while reduced-order models are generally accurate, the extent to which their predictions deviate from the measurements or the results of high-fidelity models depends on the specific configuration of the problem under consideration.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**(

**a**) Diagram of experimental setup (top–down view). (

**b**) Annotated image of the experimental setup with assigned wave gauge numbers.

**Figure 3.**(

**a**) Representative snapshot from numerical simulations illustrating the wave created as a result of buoy oscillations. (

**b**) Discretization of the entire computational domain and (

**c**) a close-up view of the overset mesh around the buoy.

**Figure 4.**Plots of the dimensionless (

**a**) added mass and (

**b**) damping coefficients (see Equation (10) for the definitions) versus the dimensionless parameter $\mathsf{\lambda}$. (

**c**) Plot of impulse response function (IRF) as a function of time.

**Figure 5.**(

**a**) Comparison between experimental measurement and three types of numerical calculations for the time evolution of the vertical position of the buoy. (

**b**) Comparison between experimental measurements and VOF calculations for the time history of the vertical position of the air–water interface at eight locations near the buoy. Experimental data represent averages over ten realizations, where the coefficient of variation (the ratio of the standard deviation to the mean) for the vast majority of the collected data was below a few percentage points. Subfigures (

**b**–

**i**) correspond to wave gauges 1–8 (see Figure 2a,b). The mean absolute differences between the plots presented here are reported in Table 3.

Item | Purpose |
---|---|

Edinburgh Designs wave tank ($10\times 3\times 1\mathrm{m}$) | Pool for experiments |

Eight resistive wave gauges | Water height measurement |

Eleven-camera Qualisys motion tracking system | Buoy motion measurement |

Spherical buoy | Test article |

Twelve 19 mm reflective markers | Motion tracking |

Wave Gauge Index | X Position (m) | Y Position (m) | $\sqrt{{\mathit{X}}^{2}+{\mathit{Y}}^{2}}$ |
---|---|---|---|

1 | 1.393 | 0.004 | 1.393 |

2 | 1.292 | 0.004 | 1.292 |

3 | 0.027 | 1.159 | 1.159 |

4 | 0.025 | 0.950 | 0.950 |

5 | 0.023 | 0.705 | 0.705 |

6 | 0.017 | 0.458 | 0.458 |

7 | −1.293 | 0.019 | 1.293 |

8 | −1.394 | 0.021 | 1.394 |

**Table 3.**Mean absolute differences between the curves plotted in Figure 5.

a_{exp-vof} | a_{exp-Cumm} | a_{exp-msd} | b | c | d | e | f | g | h | i |
---|---|---|---|---|---|---|---|---|---|---|

0.0028 | 0.0041 | 0.0045 | 0.0008 | 0.0010 | 0.0012 | 0.0015 | 0.0018 | 0.0023 | 0.0007 | 0.0006 |

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**MDPI and ACS Style**

Colling, J.K.; Jafari Kang, S.; Dehdashti, E.; Husain, S.; Masoud, H.; Parker, G.G.
Free-Decay Heave Motion of a Spherical Buoy. *Fluids* **2022**, *7*, 188.
https://doi.org/10.3390/fluids7060188

**AMA Style**

Colling JK, Jafari Kang S, Dehdashti E, Husain S, Masoud H, Parker GG.
Free-Decay Heave Motion of a Spherical Buoy. *Fluids*. 2022; 7(6):188.
https://doi.org/10.3390/fluids7060188

**Chicago/Turabian Style**

Colling, Jacob K., Saeed Jafari Kang, Esmaeil Dehdashti, Salman Husain, Hassan Masoud, and Gordon G. Parker.
2022. "Free-Decay Heave Motion of a Spherical Buoy" *Fluids* 7, no. 6: 188.
https://doi.org/10.3390/fluids7060188