# Highly Accurate Experimental Heave Decay Tests with a Floating Sphere: A Public Benchmark Dataset for Model Validation of Fluid–Structure Interaction

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## Abstract

**:**

## 1. Introduction

#### 1.1. The Test Case

_{w}, while the density of air is disregarded. A fixed global Cartesian coordinate system is defined from the still water level; the xy-plane coincides with the plane of the free water surface, and the z-axis is vertical oriented upwards towards the air phase; see Figure 2. The sphere is half-submerged when at rest, and with the CoG on the z-axis (underneath the center of buoyancy), the local and global coordinate system axes will coincide when the sphere is at rest; see Figure 2. The seabed is horizontal with a depth of d = 3D.

_{1}, sway x

_{2}, and heave x

_{3}, respectively. Rotations relative to the rest condition around the local x, y, and z-axes are defined as roll x

_{4}, pitch x

_{5}, and yaw x

_{6}, respectively. Three initial test setups are investigated with displacements of the sphere in positive heave given by the drop height H

_{0}= {0.1D, 0.3D, 0.5D}; see Figure 3.

#### 1.2. Numerical Modelling Blind Tests of the Test Case

#### 1.3. RANS Models

#### 1.4. FNPF Models

#### 1.5. LPF Models

## 2. Materials and Experimental Setup

#### 2.1. The Sphere Model

#### 2.2. Experimental Setup and Equipment

^{®}832 line with 8 braided fibers and 32 weaves per inch (thickness 0.30 mm, weight 0.18 g/m).

_{0}, as given in Table 1. The sphere model was kept at a given drop height, until the model and the free water surface were at rest; see Figure 12. The initial calmness of the sphere model (measured drop heights, velocities, and accelerations) and the free water surface are quantified in Section 3.

## 3. Results

#### 3.1. Decay Measurements and Expanded Uncertainty

_{Cv}following the Student’s t distribution [30]. C refers to the confidence level and v is the number of degrees of freedom (not to be confused with the previously introduced rigid body motions, but rather the independent variables in the calculation of ${u}_{\overline{X}}$) given by v = N − 1 with N being the number of repetitions.

_{0.95,3}= 3.182 [30].

^{2}for H

_{0}= 0.5D), which is in the same order of magnitude as$g$, allowing the deflection to be assessed by including the weight of an additional reflective marker. Conservatively, the systematic standard uncertainties introduced from deflections in the global$z$-direction of the support rods of the reflective markers were included as 0.1 mm (ISO Type B) for H

_{0}= 0.5D. The systematic standard uncertainties for the lower drop heights were linearly scaled down.

_{e}

_{0}< 8 multiplied with ${\overline{H}}_{0,m}$ are 0.44, 0.24, and 0.09 mm for the target drop heights of $0.5D$, $0.3D$, and $0.1D$, respectively, which correspond to about 0.3% of the drop height for all cases.

#### 3.2. Six DoF Motions

_{0}= 0.5D. The measured six-DoF motions for${H}_{0}=\left\{0.1D,0.3D\right\}$are presented in Appendix D. The influences on the heave measurements from roll and pitch of the sphere model were included in the uncertainty analysis; see Table 3.

#### 3.3. Initial Calmness of the Sphere Model

_{e}

_{0}< 0), were assessed; see Figure 17. The position time series were subtracted with the respective measured drop heights to get zero as reference value. A moving average filter with a size of 21 samples was utilized to filter the acceleration time series.

_{e}

_{0}< 0 were calculated. The mean and standard uncertainty of the position time series are both 0.0000 m (0.0 mm). The mean and standard uncertainty of the velocity time series are 0.0000 m/s and 0.0004 m/s, respectively. The mean and standard uncertainty of the acceleration time series are −0.0002 m/s

^{2}and 0.0097 m/s

^{2}, respectively.

#### 3.4. Frequency Content

#### 3.5. Reflections and Initial Calmness of the Water Phase

_{r}

_{0}= 8.44/c, where$c$is the celerity of a linear wave with period T

_{e}

_{0}, is included in Figure 19. Reflected waves propagated past the locations of wave gauges 1, 2, and 3 for around 2.0, 1.3, and 0.7 periods before t

_{r}

_{0}, respectively. Decay time series presented up to t/T

_{e}

_{0}= 8 are not under the influence of reflections from waves with the period T

_{e}

_{0}; see Figure 19. This can be considered a conservative estimate, as the main wave front of radiated waves would have propagated with the group velocity rather than the phase velocity. The measured surface elevations from the other drop heights are included in Appendix D.

_{e}

_{0}< 0 are both 0.0000 m.

#### 3.6. Uncertainties of Physical Parameters

#### 3.7. Comparison of Decay Measurements to Numerical Modelling Blind Tests

_{0}= 0.5D is shown. The first trough and crest of the decay time series are shown in Figure 23 and Figure 24, respectively. In Figure 25, the comparison of decay time series is shown merely for the numerical models of higher fidelity, i.e., FNPF and RANS models.

## 4. Discussion

_{e}

_{0}); see Figure 17b. This broadly correlates with the time-variation of the expanded uncertainty in Figure 15, for which the time-variation is governed by the random uncertainty (over the systematic). Over the three drop heights, the random uncertainty decreases with the drop height, where obviously the sphere model will oscillate with lower speeds for lower drop heights; see Figure 15 and Figure 16b. The observations of dependence between the random uncertainties and the speed of the sphere model are ascribed to marker-image-shape-distortions increased by higher relative speeds between the optical motion capture system and the test specimen, as reported in [32].

#### Comparison to Numerical Modelling Blind Tests

_{0}), occur for the LPF0 and LPF1 models at H

_{0}= 0.5D. Not considering the phase shifts, but merely the magnitudes of troughs and crests, the LPF0 and LPF1 models, respectively, deviate with around 12–13 and 1–5 mm (i.e., 9% and 1–3% of H

_{0}) at the first trough and crest; see Figure 23 and Figure 24. The LPF0 model oscillates with the damped natural frequency of a one-DoF spring-mass-damper system with constant hydrodynamic coefficients, and thus is not capable of including broader frequency contents, which may explain the larger phase shifts for larger drop heights; see Figure 21. The linearization of the hydrostatic force in the LPF1 model spuriously increases the acceleration, as discussed in Appendix C. As the drop height is decreased, the heave decay will oscillate with${T}_{e0}$and the assumption of linear hydrostatics will become more accurate. Consequently, the LPF0 and LPF1 models become increasingly accurate in both amplitude and phase for lower drop heights; see Figure 21. The inclusion of nonlinear hydrostatics in the LPF2 and 3 models significantly reduces the phase shifts; see, e.g., Figure 23. The constant${a}_{33}^{\infty}$term in the LPF2 model, however, spuriously delays the decay at initiation—see Figure 22—and in general increases the deviation from the physical tests when the sphere is displaced from its rest condition at which the constant${a}_{33}^{\infty}$term is evaluated; see Figure 21 and Figure 24. Only including the draft-dependency of the${a}_{33}^{\infty}$term in the radiation force as in the LPF3 model (see Appendix C) introduces large deviations at the first trough at H

_{0}= 0.5D; see Figure 23. The inclusion of draft-dependency of the convolution part of the radiation force, as done in the LPF4 model (refer to Appendix C for further information), does not yield more accurate results. Despite the large deviations at the first trough, the LPF3 model captures all subsequent crests and troughs in the H

_{0}= 0.5D case with an accuracy close to those of the RANS models, and is thus significantly more accurate than the LPF2 model with constant ${a}_{33}^{\infty}$. At H

_{0}= 0.1D, the LPF2 and 3 models perform with maximum deviations of around 1 mm, which are comparable to the deviations of the models of higher fidelity.

_{0}= 0.1D, corresponding to 3% of H

_{0}. At the first trough, the models FNPF1, RANS1 and RANS5 lie within the 95% CI of the physical measurements, while the RANS2 and RANS4 models deviate with less than 0.3 mm (i.e., less than 1% of H

_{0}). Deviations at the first trough have the same order of magnitude for H

_{0}= 0.3D, whereas at H

_{0}= 0.5D, the deviations increase to around 1–3 mm (i.e., 1–2% of H

_{0}), with the exception of the RANS2 and RANS3 models, which are actually within the (narrow) 95% CI. The kinematics, and thus velocity gradients, are largest within the first natural period, leading to high demands on the near-wall meshing and treatment (mesh morphing, wall functions, etc.) in the RANS models. However, from Figure 25, there is a general tendency of the largest deviations to occur at 1 < t/T

_{e}

_{0}< 4 (even when taking into account the decrease of the CI width; see Figure 15). Assuming the time-error of the motion capture system to be negligible, the reasoning behind the tendency of largest deviations to not occur during the first natural period is two-fold: (i) in a RANS model, errors from the numerical discretization and iterations accumulate, and (ii) turbulence increases over the first periods and when the sphere changes direction. The former includes numerical errors of turbulence parameters if calculated in a turbulence model, while the latter refers to the increase of the complexity of the water phase over time (emergence of high-frequency perturbations of the free surface and sub-grid vortices) and how model errors of either not including a turbulence model (laminar simulations) or the inaccuracies associated with a given model thus become more pronounced with time. The deviations tend to reduce for 4 < t/T

_{e}

_{0}which is ascribed to the low amplitudes themselves rather than an increase in the accuracy, as the continued increase in the phase shifts (up to around 0.04 s, i.e., 0.05T

_{e0}) also suggests. An increased accuracy from inclusion of a turbulence model (k-omega-SST) can be seen by comparing the RANS2 and RANS3 models in Figure 25.

_{0}= 0.5D are up to 8 mm or 5% of H

_{0}. For H

_{0}= 0.5D, the maximum of deviations at troughs and crests are an order of magnitude higher for the LPF models than the RANS models, which indicates the potential pitfalls of LPF models for large-amplitude motions.

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

_{3}[m]; see Figure A2. The three columns WG1, WG2, and WG3 [m] contain the surface elevation time series at three wave gauges locations, introduced in Section 2.2, and are included for the experimental results and for certain numerical results. Four repetitions were performed of the physical heave decay tests, all of which are included in the result files under Experimental results. The heave decay time series are presented in a raw and in a normalized format, as explained in Section 3. The normalized results are also represented in a file containing the sample mean and the upper and lower bounds of the 95% CI around the sample mean; see Section 3.

**Figure A1.**Directory structure of the Supplementary Materials.

**Figure A2.**Structure of result files. Example with first part of raw measurements (

**a**) and mean of normalized data with 95% CI (

**b**).

## Appendix B

Name | Institution and Authors | Framework | Description | Comp. Effort [CH] * |
---|---|---|---|---|

RANS1 | Aalborg University; C.E., J.A. | OpenFOAM-v1912 | 3D URANS model. Incompressible, isothermal. Volume of fluid method. Two vertical symmetry planes. Reflective side walls. Mesh morphing using SLERP method. Cell count of 6-9 M cells. No turbulence model. Second-order accurate in time and space. CFL criterion of 0.5 | ~3000–6500 |

RANS2 | University of Plymouth; E.R., S.B. | OpenFOAM 5.0 | 3D URANS model. Incompressible, isothermal. Volume of fluid. Two vertical symmetry planes. Reflective side walls. Mesh morphing using SLERP method. Cell count of ~12 M cells. No turbulence model. CFL criterion of 0.5. | ~1000–4200 |

RANS3 | University of Plymouth; E.R., S.B. | OpenFOAM 5.0 | Same as RANS2 except k-Omega SST turbulence model. Only conducted for H_{0} = 0.5D. | ~1800 |

RANS4 | National Renewable Energy Lab.; Y.-H.Y., T.T.T. | STAR-CCM+ 13.06 | 3D URANS model. Incompressible, isothermal. Volume of fluid. Two vertical symmetry planes. Cell count of 6 M cells. Mesh morphing with one DOF. k-Omega SST turbulence model. second-order accurate in time and space. CFL criterion of 0.5. Max. time step of 0.1 ms. | ~1000–2600 |

RANS5 | Budapest University of Technology and Economics; J.D., C.H. | OpenFOAM 7 | 2D URANS model. Incompressible, isothermal. Volume of fluid method. Axisymmetric wedge geometry. Cell count of approx. 20 K cells. No turbulence model. second-order accurate in time and space. CFL criterion of 0.25. Water depth changed to 1.8 m to allow mesh morphing. | ~0.5–2.5 |

FNPF1 | Chalmers University of Technology; C.-E.J. | SHIPFLOW-Motions 6 | Fully nonlinear potential flow BEM. 1600 panels were used on the sphere and 4600 panels were used on the free surface. The time step was 0.005 s. | ~6 |

LPF0 | Aalborg University; M.B.K., J.A. | WAMIT and MatLab | Analytical solution to one-DoF mass-spring-damper system with hydrodynamic coefficients from BEM (for ω = ω_{e}_{0}) | - ** |

LPF1 | Floating Power Plant; M.B.K. | WAMIT and MatLab/Simulink | Model with linear hydrostatics and linear coefficients from BEM. Time-step: 1 ms, solver: ode4 (Runge-Kutta). | - ** |

LPF2 | Floating Power Plant; M.B.K. | WAMIT and MatLab/Simulink | Model with nonlinear hydrostatics and linear coefficients from BEM. Time-step: 1 ms, solver: ode4 (Runge-Kutta). | - ** |

LPF3 | Floating Power Plant; M.B.K. | WAMIT and MatLab/Simulink | Model with nonlinear hydrostatics, linear radiation function from linear BEM but position dependent infinity added mass. Time-step: 1 ms, solver: ode4 (Runge-Kutta). | - ** |

LPF4 | Floating Power Plant; M.B.K. | WAMIT and MatLab/Simulink | Model with nonlinear hydrostatics and position dependent radiation functions (based on linear coefficients from BEM). Time-step: 1 ms, solver: ode4 (Runge-Kutta). | - ** |

## Appendix C

#### Appendix C.1. Linearization of Hydrostatics

#### Appendix C.2. The LPF0 Model

T_{e}_{0} | ω_{e}_{0} | δ | A_{33}(ω_{e}_{0}) | B_{33}(ω_{e}_{0}) | C_{33} | C_{1} | C_{2} |
---|---|---|---|---|---|---|---|

[s] | [rad/s] | [rad/s] | [kg] | [Ns/m] | [N/m] | [m] | [m] |

0.76 | 8.30 | 0.695 | 2.97 | 13.95 | 692.89 | H_{0} | 0.0839H_{0} |

#### Appendix C.3. The LPF1–4 Models

Model | Hydrostatics C_{3} | $\mathbf{Added}\mathbf{Mass}{\mathit{a}}_{33}^{\mathit{\infty}}$ | Radiation Convolution Function K_{33} |
---|---|---|---|

LPF1 | Constant | Constant | Constant function |

LPF2 | Draft-dependent | Constant | Constant function |

LPF3 | Draft-dependent | Draft-dependent | Constant function |

LPF4 | Draft-dependent | Draft-dependent | Draft-dependent functions |

#### Appendix C.4. Measured Hydrostatics

**Figure A3.**Measured hydrostatic forces as function of the draft. Nonlinear (Equation (A1)) and linear (Equation (A3)) analytical expressions of the hydrostatic force are included.

#### Appendix C.5. Added Mass at Infinite Frequency

**Figure A4.**Draft-dependent normalized added mass at infinite frequency with a fifth order polynomial fit.

#### Appendix C.6. Radiation IRF

_{33}, see Equation (A9), was calculated using WAMIT hydrodynamic damping coefficients for different drafts of the sphere. The curves in Figure A5 show the spread in the functions when going from zero draft (flat curve) to full submergence with draft equal to the diameter D (largest curve). A resolution in draft of 1 mm was used (a total of 300 functions). The radiation impulse function to be used at a particular time step during the simulation was thus pieced together of the radiation impulse functions corresponding to the drafts of previous time history. Linear interpolation in the functions was used to get the values corresponding to the actual drafts.

**Figure A5.**Normalized radiation impulse response functions for different drafts. Steps of 15 mm draft are shown for better visualization.

#### Appendix C.7. Comparison of the LPF1–4 Models

## Appendix D

_{0}= 0.1D and H

_{0}= 0.3D are presented in Figure A9. The locations of wave gauges can be seen in Figure 8. The measured motions in all six DoF for the drop heights H

_{0}= 0.1D and H

_{0}= 0.3D are presented in Figure A10 and Figure A11, respectively.

**Figure A7.**Normalized decay time series for the three investigated drop heights (enlarged version). The 95% CI was scaled up by a factor of 30 to be able to visualize the time-dependency.

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**Figure 5.**Unballasted hemisphere shell with a diameter of 300 mm (

**a**). Ballasted hemisphere shell with rubber gasket (

**b**).

**Figure 9.**Test setup in the wave basin: center—half-submerged sphere model; left—camera; front—wave gauges; right—motion capture cameras; above—release system fixed to the bridge.

**Figure 10.**Test setup in the wave basin: center—half-submerged sphere model; background—motion capture cameras (strobes in purple); right—wave gauge (number 3).

**Figure 12.**Photos of investigated drop heights: H

_{0}= 0.1D (

**a**), H

_{0}= 0.3D (

**b**), and H

_{0}= 0.5D (

**c**).

**Figure 17.**Position (

**a**), velocity (

**b**), and acceleration (

**c**) time series with zooms of the limits −0.3 < t/T

_{e}

_{0}< 0.

**Figure 20.**Surface elevation time series with a zoom of the limits −1 < t/T

_{e}

_{0}< 0 for H

_{0}= 0.5D.

**Figure 25.**Comparison of physical and high-fidelity numerical test results with a zoom of 0 < t/T

_{e}

_{0}< 2.

Parameter | D | m | CoG | g | H_{0} | ρ_{w} | d |
---|---|---|---|---|---|---|---|

Unit | mm | kg | mm | m/s^{2} | mm | kg/m^{3} | mm |

Value | 300 | 7.056 | (0, 0, −34.8) | 9.82 | {30 90 150} | 998.2 | 900 |

Parameter | M | CoG | I_{xx} | I_{yy} | I_{zz} | I_{xz} | I_{xy}, I_{yz} |
---|---|---|---|---|---|---|---|

Unit | g | mm | gmm^{2} | gmm^{2} | gmm^{2} | gmm^{2} | gmm^{2} |

Value | 7056 | (0, 0, −34.8) | 98251·10^{3} | 98254·10^{3} | 73052·10^{3} | 0·10^{3} | 10·10^{3} |

Systematic Error Source k | $\mathbf{Elemental}\mathbf{Systematic}\mathbf{Standard}\mathbf{Uncertainty}{\mathit{b}}_{\overline{\mathit{X}},\mathit{k}}{\overline{\mathit{H}}}_{0,\mathit{m}}\left[\mathbf{mm}\right]$ | ISO Types |
---|---|---|

Calibration of motion capture system (Oqus7+) | 0.01 | A |

Vibrations of bridge (reference frame) | 0.01 | B |

Vibration of support rods for reflective markers (for ascending H _{0}) | 0.02, 0.06, 0.10 | B |

Influence on heave measurements from roll and pitch | Time-dependent, <0.02 | A |

Parameter | Value | Standard Uncertainty | Unit | ISO Type | |
---|---|---|---|---|---|

Test case values | Diameter of sphere | 300 | 0.1 | mm | B |

Mass of sphere | 7056 | 1 | g | B | |

Centre of gravity | (0.0, 0.0, −34.8) | (0.1, 0.1, 0.1) | mm | B | |

Acceleration due to gravity | 9.82 | 0.003 | m/s^{2} | B | |

Drop heights (mean); H _{0} = {0.1D,0.3D,0.5D} | {29.16, 89.18, 150.06} | {0.8, 0.5, 0.3} | mm | A | |

Density of water [31] | 998.2 | 0.4 | kg/m^{3} | B | |

Water depth | 900 | 1 | mm | B | |

Initial velocity in heave | 0.0000 | 0.0004 | m/s | A | |

Initial acceleration in heave | −0.0002 | 0.0097 | m/s^{2} | A | |

Additional values (Recommended for high-fidelity models) | Temperature of air and water | 20 | 2 | °C | B |

Kinematic viscosity of water [31] | 1.0·10^{−6} | 0.1·10^{−6} | m^{2}/s | B | |

Density of air [31] | 1.20 | 0.012 | kg/m^{3} | B | |

Kinematic viscosity of air [31] | 15.1·10^{−6} | 0.2·10^{−6} | m^{2}/s | B | |

Surface tension water-air [31] | 0.07 | 0.004 | N/m | B | |

Moments of inertia of the sphere model; I = {I _{xx},I_{yy},I_{zz},I_{xy},I_{xz},I_{yz}} | {98251, 98254, 73052, 0, 10, 0}·10^{3} | {37, 37, 1, 0, −77, 96}·10^{3} | gmm^{2} | B | |

Initial surface elevation | 0.0 | 0.01 | mm | A |

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**MDPI and ACS Style**

Kramer, M.B.; Andersen, J.; Thomas, S.; Bendixen, F.B.; Bingham, H.; Read, R.; Holk, N.; Ransley, E.; Brown, S.; Yu, Y.-H.;
et al. Highly Accurate Experimental Heave Decay Tests with a Floating Sphere: A Public Benchmark Dataset for Model Validation of Fluid–Structure Interaction. *Energies* **2021**, *14*, 269.
https://doi.org/10.3390/en14020269

**AMA Style**

Kramer MB, Andersen J, Thomas S, Bendixen FB, Bingham H, Read R, Holk N, Ransley E, Brown S, Yu Y-H,
et al. Highly Accurate Experimental Heave Decay Tests with a Floating Sphere: A Public Benchmark Dataset for Model Validation of Fluid–Structure Interaction. *Energies*. 2021; 14(2):269.
https://doi.org/10.3390/en14020269

**Chicago/Turabian Style**

Kramer, Morten Bech, Jacob Andersen, Sarah Thomas, Flemming Buus Bendixen, Harry Bingham, Robert Read, Nikolaj Holk, Edward Ransley, Scott Brown, Yi-Hsiang Yu,
and et al. 2021. "Highly Accurate Experimental Heave Decay Tests with a Floating Sphere: A Public Benchmark Dataset for Model Validation of Fluid–Structure Interaction" *Energies* 14, no. 2: 269.
https://doi.org/10.3390/en14020269