Wave excitation

The wave excitation loads represent the combination of both the diffraction loads and the Froude-Krylov loads.

\[\mathbf{f}_E(t) = \mathbf{f}_D(t) + \mathbf{f}_{FK}(t)\]

In the case of a linear wave excitation model, the diffraction and Froude-Krylov loads are computed together:

\[\mathbf{f}_E(t) = \sum_m \sum_n \Im\left(A_{mn} \mathbf{f}_e(\omega_m,\theta_n) e^{j(k_m\bar{x}_n - \omega_m^e t + \Phi_{mn})}\right)\]

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 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:

\[\mathbf{f}_{FK}(t) = \sum_m \sum_n \Im\left(A_{mn} \mathbf{f}_{fk}(\omega_m,\theta_n) e^{j(k_m\bar{x}_n - \omega_m^e t + \Phi_{mn})}\right)\]

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 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 \(z = 0\).

\[\mathbf{f}_{FK}(t)= -\iint_{S_0} P_I \mathbf{n} dS\]

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:

\[\mathbf{\Gamma}_{FK}(t)= -\iint_{S_0} P_I (\mathbf{OM}-\mathbf{OG})\times\mathbf{n} dS\]

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 : \(z = \eta_I\).

\[\mathbf{f}_{FK}(t)= -\iint_{S_I} P_I \mathbf{n} dS\]
\[\mathbf{\Gamma}_{FK}(t)= -\iint_{S_I} P_I (\mathbf{OM}-\mathbf{OG})\times\mathbf{n} dS\]

Diffraction force

Linear Model

The diffraction force, given by the linear approximation, is:

\[\mathbf{f}_D(t) = \sum_m \sum_n \Im\left(A_{mn} \mathbf{f}_d(\omega_m,\theta_n) e^{j(k_m\bar{x}_n - \omega_m^e t + \Phi_{mn})}\right)\]

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 Nemoh or WAMIT.