Wave excitation¶
The wave excitation loads represent the combination of both the diffraction loads and the Froude-Krylov loads.
In the case of a linear wave excitation model, the diffraction and Froude-Krylov loads are computed together:
with
- \(\omega_m\) the wave frequency;
- \(\theta_n\) the wave direction;
- \(\bar{x}_n = x \cos(\theta_n) + y \sin(\theta_n)\);
- \(\omega_m^e = \omega_m - k_mU\) is the encounter circular frequency, calculated using the steady forward speed;
- \(A_{mn}\) the wave amplitude for the wave frequency \(\omega_m\) and the wave direction \(\theta_n\);
- \(\mathbf{f}_e(\omega_m,\theta_n)\) is the frequency-domain excitation force component (with both the diffraction and Froude-Krylov components) for the wave frequency \(\omega\) and the wave direction \(\theta_n\).
The frequency-domain excitation forces are obtained from a linear potential flow-based solver such as Helios, Nemoh or WAMIT.
Froude-Krylov force¶
The Froude-Krylov loads are due to the integration of the pressure of the incident wave field over the wetted body surface. Several approximations can be defined, as for the hydrostatic loads:
- a linear model;
- a weakly nonlinear model;
- a fully nonlinear model.
Linear model¶
The Froude-Krylov force, given by the linear approximation, is:
with
- \(\omega_m\) the wave frequency;
- \(\theta_n\) the wave direction;
- \(\bar{x}_n = x \cos(\theta_n) + y \sin(\theta_n)\), with \(x\) and \(y\) the coordinates of the equilibrium frame
- \(\omega_m^e = \omega_m - k_mU\) is the encounter circular frequency, calculated using the steady forward speed;
- \(A_{mn}\) the wave amplitude for the wave frequency \(\omega_m\) and the wave direction \(\theta_n\);
- \(\mathbf{f}_{fk}(\omega_m,\theta_n)\) is the frequency-domain Froude-Krylov force component for the wave frequency \(\omega\) and the wave direction \(\theta_n\).
Note
The frequency-domain Froude-Krylov forces are obtained from a linear potential flow-based solver such as Helios, Nemoh or WAMIT.
Weakly nonlinear model¶
Following the same approximations made for the weakly nonlinear hydrostatic model, the weakly nonlinear Froude-Krylov load is computed by integrating the incident pressure over the wetted body surface, \(S_0\), defined by the exact position of the body mesh, clipped by the plane \(z = 0\).
where
- \(P_I\) is pressure of the incident wave field;
- \(\mathbf{n}\) is normal vector, pointing outward the body surface;
The Froude-Krylov torque at the center of gravity of the body is expressed by:
with
- \(\mathbf{OG}\) the position of the center of gravity of the body;
- \(\mathbf{OM}\) the position of the centroid of the panel.
Fully nonlinear model¶
Following the same approximation made for the nonlinear hydrostatic model, the nonlinear Froude-Krylov load is computed by integrating the incident pressure over the exact wetted body surface, \(S_I\), defined by the exact position of the body mesh, clipped by the undisturbed free surface : \(z = \eta_I\).
Diffraction force¶
Linear Model¶
The diffraction force, given by the linear approximation, is:
with
- \(\omega_m\) the wave frequency;
- \(\theta_n\) the wave direction;
- \(\bar{x}_n = x \cos(\theta_n) + y \sin(\theta_n)\), with \(x\) and \(y\) the coordinates of the equilibrium frame
- \(\omega_m^e = \omega_m - k_mU\) is the encounter circular frequency, calculated using the steady forward speed;
- \(A_{mn}\) the wave amplitude for the wave frequency \(\omega_m\) and the wave direction \(\theta_n\);
- \(\mathbf{f}_D(\omega_m,\theta_n)\) is the frequency-domain diffraction force component for the wave frequency \(\omega\) and the wave direction \(\theta_n\).
Note
The frequency-domain diffraction forces are obtained from a linear potential flow-based solver such as Helios, Nemoh or WAMIT.