Heaving sphere in regular waves

This benchmark corresponds to the simulation of a sphere in heave motion submitted to regular waves. This simulation 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. Description of this test case and results obtained by FRyDoM are summarized in the following.

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 depth 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 a regular wave field propagating positive along the x-axis. The wave periods considered in this test case varies from \(3\) \(s\) to \(11\) \(s\) with a steepness of \(0.2\) %.

Results in RAO

The response amplitude operator (RAO) in heave motion obtained from FRyDoM is presented in Fig. 38. Very good agreement with the results obtained by Nemoh [Nemoh] can be observed.

Heave RAO

Fig. 38 Response Amplitude Operator (RAO) in heave motion. Comparison of the numerical results from FRyDoM to those from Nemoh

Effects of a nonlinear hydrostatic and Froude-Krylov approach

The time series of the same floating heaving sphere in 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.

A single regular wave of period \(3\) \(s\) and amplitude \(0.022\) \(m\) is present. The two time series are plotted in Fig. 39. Due to the small steepness of the waves (\(0.05\) %), the two models match perfectly, which validates their mutual implementation in regular waves.


Fig. 39 Comparison of the time series of a floating heaving sphere in a regular wave field using a linear (blue) and fully nonlinear (orange) hydrostatic and Froude-Krylov model


  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 Modeling Verification and Validation”, European Wave and Tidal Energy Conference, Cork, Ireland, 2017
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