# 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.

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] 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] 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.