Heaving sphere in irregular waves

This comparison corresponds to the simulation of a sphere in heave motion submitted to irregular waves. The model is presented by the International Energy Agency (IEA) Offshore Energy System (OES) Task 10 [OES10] as a benchmark case for model validation and verification regrouping 25 organizations. The goal of this section is to compare the use of a linear and nonlinear approach for the computation of the hydrostatic and Froude-Krylov loads.

Description of the test case

The sphere considered in this simulation has a radius of \(5\) \(m\) and a total mass of \(2,618 .10^5\) \(kg\). At equilibrium, the center of the sphere is located on the mean water level and its center of gravity is located \(2\) \(m\) below the water line. Main properties of the sphere are presented in the next table.

Parameters Values
Radius \(5\) \(m\)
Initial sphere location (\(0\), \(0\), \(0\))
Center of gravity (\(0\), \(0\), \(-2\))
Mass \(261.8\times10^3\) \(kg\)
Ixx \(1.690\times10^6\) \(kg.m^2\)
Iyy \(1.690\times10^6\) \(kg.m^2\)
Izz \(2.606\times10^6\) \(kg.m^2\)
Water detph Inf
Water density \(1000\) \(kg/m^3\)
K33 \(7.695\times10^5\) \(N/m\)
K44 \(5.126\times10^6\) \(N.m\)
K55 \(5.126\times10^6\) \(N.m\)

The sphere is submitted to an irregular wave field propagating positive along the x-axis. A Jonswap wave spectrum is considered with a significant wave height (Hs) of \(0.5\) \(m\), a spectral peak period (Tp) of \(4.4\) \(s\) and a gamma factor (\(\gamma\)) of \(1\).

Effects of a nonlinear hydrostatic and Froude-Krylov approach

The time series of the floating heaving sphere in irregular waves are now compared. Two models are considered:

  • a fully linear model;
  • a weakly nonlinear model: the hydrostatic and Froude-Krylov loads are computed with a fully nonlinear approach.

The two time series are plotted in Fig. 23. Due to the small steepness of the waves (\(0.26\) %), the two models match perfectly, which validates their mutual implementation in irregular waves.

../../_images/Comparison_Sphere_IW_Lin_Nonlin_hs_fk.png

Fig. 23 Comparison of the time series of a floating heaving sphere in an irregular wave field using a linear (blue) and fully nonlinear (orange) hydrostatic and Froude-Krylov model

References

[OES10]
  1. Wendt, Y-H Yu, K. Ruehl, T. Bunnik, I. Touzon, B. W. Nam, J. S. Kim, K-H Kim, C. E. Janson, K-R. Jakobsen, S. Crowley, L. Vega, K. Rajagopalan, T. Mathai, D. Greaves, E. Ransley, P. Lamont-Kane, W. Sheng, R. Costello, B. Kennedy, S. Thomas, P. Heras, H. Bingham, A. Kurniawan, M. M. Kramer, D. Ogden, S. Girardin, A. Babarit, P.-Y. Wuillaume, D. Steinke, A. Roy, S. Betty, P. Shofield, J. Jansson and J. Hoffman, “International Energy Agency Ocean Energy Systems Task 10 Wave Energy Converter Modeleing Verification and Validation”, European Wave and Tidal Energy Conference, Cork, Ireland, 2017
[Nemoh]
  1. Babarit and G. Delhommeau, “Theoretical and numerical aspects of the open source BEM solver NEMOH”, in Proc. of the 11th European Wave and Tidal Energy Conference”, Nantes, France, 2015.