Steep Wave Effects on Wave Induced Vertical Bending Moment Using a 3D Numerical Wave Tank

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This paper presents a detail comparative study on wave induced vertical bending moment (VBM) between linear and approximate nonlinear calculations using a 3D numerical wave tank (NWT) method. The developed numerical approach is in time domain where the ambient incident waves can be defined by any suitable wave theory. Certain justifying approximations employed in the solution of the interaction hydrodynamics (diffraction and radiation) enabling the NWT to generate stable long duration time histories of all parameters of interest. This is an extension of our earlier works towards the development of a practical NWT based solution for wave-structure interactions [1]. After a brief outline of the implemented numerical details, a comprehensive validation and verification of vertical shear force (VSF) and bending moment RAOs computed using the linearized version of the NWT against the usual linear results of strip theory and 3D panel codes are presented. Next we undertake the comparative study between the fully linear and approximate nonlinear versions of the present code for different incident wave steepness. In the approximate nonlinear formulation, the ambient incident wave is defined by the full nonlinear numerical wave model based on Fourier approximation method which can generate very steep steady periodic nonlinear waves up to the near wave breaking limit. The nonlinearities associated with the incident Froude Krylov and hydrostatic restoring forces/moments are exact up to the instantaneous wetted surface at the displaced location, but the hydrodynamic diffraction and radiation effects are linearized around the mean wetted surface. The standard S175 container hull is considered for the comparative studies because of its geometric nonlinearities. Numerical simulations are performed for four different wave lengths with increasing wave steepness. It is observed that the computed wave induced VBM amidships from the approximate nonlinear results can be almost 30% higher compared to the results from a purely linear solution, which can be a critical issue from the safety point. Significant higher harmonics are also observed in the approximate nonlinear results which at some times may be responsible for exciting the undesirable whipping/springing responses.