On the weakly nonlinear seakeeping solution near the critical frequency

We study theoretically and numerically the nonlinear seakeeping problem of a submerged or floating body translating with constant forward speed $U$ parallel to the undisturbed free surface while at the same time undergoing a small oscillatory motion and/or encountering small-amplitude waves at frequency $\unicode[STIX]{x1D714}$. It is known that at the critical frequency corresponding to $\unicode[STIX]{x1D70F}\equiv \unicode[STIX]{x1D714}U/g=1/4$, where $g$ is the gravitational acceleration, the classical linear solution is unbounded for a single point source, and the inclusion of third-order free-surface nonlinearity due to cubic self-interactions of waves is necessary to remove the associated singularity. Although it has been shown that the linear solution is in fact bounded for a body with full geometry rather than a point source, the solution still varies sharply near the critical frequency. In this work, we show theoretically that for a submerged body, the nonlinear correction to the linear solution due to cubic self-interactions of resonant waves in the neighbourhood of $\unicode[STIX]{x1D70F}=1/4$ is of first order in the wave steepness (or body motion amplitude), which is the same order as the linear solution. With the inclusion of nonlinear effects in the dispersion relation, the wavenumbers of resonant waves become complex-valued and the resonant waves become evanescent, with their amplitudes vanishing with the distance away from the body. To assist in understanding the theory, we derive the analytic nonlinear solution for the case of a submerged two-dimensional circular cylinder in the neighbourhood of $\unicode[STIX]{x1D70F}=1/4$. Independent numerical simulations confirm the analytic solution for the submerged circular cylinder. Finally, we also demonstrate by numerical simulations that similar significant nonlinear effects for a surface-piercing body exist in the neighbourhood of $\unicode[STIX]{x1D70F}=1/4$.

High Sea State Linear Seakeeping Applicability Metric

Linear seakeeping predictions are attractive for design space exploration and preliminary or simple case motion estimates due to calculation speed and relatively simple input requirements. Linear seakeeping theory is an established prediction method with well-known assumptions. One of these assumptions is the assumption that the motions are small. The validity of this assumption is investigated by comparisons with a body exact nonlinear seakeeping code over a range of significant wave heights. A modern naval destroyer and a generic tumblehome ship are examined over a range of speeds, wave headings, and sea significant wave heights. A comparison between linear and nonlinear seakeeping results for the two hull forms show range of linear behavior for different geometries. A general metric based on relative motion is proposed to quantify the validity of the assumption and indicate up to what point linear seakeeping is appropriate including effect of hull form, speed, and relative wave heading.