.. _wave_excitation:
Wave excitation
===============
The wave excitation loads represent the combination of both the diffraction loads and the Froude-Krylov loads.
.. math::
\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:
.. math::
\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
- :math:`\omega_m` the wave frequency;
- :math:`\theta_n` the wave direction;
- :math:`\bar{x}_n = x \cos(\theta_n) + y \sin(\theta_n)`;
- :math:`\omega_m^e = \omega_m - k_mU` is the encounter circular frequency, calculated using the steady forward speed;
- :math:`A_{mn}` the wave amplitude for the wave frequency :math:`\omega_m` and the wave direction :math:`\theta_n`;
- :math:`\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 :math:`\omega` and the wave direction :math:`\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_FroudeKrylov:
Linear model
************
The Froude-Krylov force, given by the linear approximation, is:
.. math::
\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
- :math:`\omega_m` the wave frequency;
- :math:`\theta_n` the wave direction;
- :math:`\bar{x}_n = x \cos(\theta_n) + y \sin(\theta_n)`, with :math:`x` and :math:`y` the coordinates of the :any:`equilibrium frame `
- :math:`\omega_m^e = \omega_m - k_mU` is the encounter circular frequency, calculated using the steady forward speed;
- :math:`A_{mn}` the wave amplitude for the wave frequency :math:`\omega_m` and the wave direction :math:`\theta_n`;
- :math:`\mathbf{f}_{fk}(\omega_m,\theta_n)` is the frequency-domain Froude-Krylov force component for the wave frequency :math:`\omega` and the wave direction :math:`\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 :any:`weakly nonlinear hydrostatic model `,
the weakly nonlinear Froude-Krylov load is computed by integrating the incident pressure over the wetted body surface,
:math:`S_0`, defined by the exact position of the body mesh, clipped by the plane :math:`z = 0`.
..
Following the same method as for the weakly nonlinear hydrostatics, the computation of the weakly nonlinear Froude-Krylov
loads involves the incident pressure integration over the body mesh at its real position. The free surface is represented
by the plane :math:`z = 0`. Consequently, at every evaluation of the Froude-Krylov loads, the body mesh is clipped by the
plane :math:`z = 0`. The expression of the Froude-Krylov force is:
.. math::
\mathbf{f}_{FK}(t)= -\iint_{S_0} P_I \mathbf{n} dS
where
- :math:`P_I` is pressure of the incident wave field;
- :math:`\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:
.. math::
\mathbf{\Gamma}_{FK}(t)= -\iint_{S_0} P_I (\mathbf{OM}-\mathbf{OG})\times\mathbf{n} dS
with
- :math:`\mathbf{OG}` the position of the center of gravity of the body;
- :math:`\mathbf{OM}` the position of the centroid of the panel.
Fully nonlinear model
*********************
Following the same approximation made for the :any:`nonlinear hydrostatic model `, the nonlinear
Froude-Krylov load is computed by integrating the incident pressure over the exact wetted body surface, :math:`S_I`, defined
by the exact position of the body mesh, clipped by the undisturbed free surface : :math:`z = \eta_I`.
..
In the fully nonlinear model for evaluating the Froude-Krylov loads, the mesh used for the pressure integration is delimited by the incident wave field :math:`z = \eta_I` and not the plane :math:`z = 0` as in the weakly nonlinear model. Consequently, the expression of the Froude-Krylov force and torque become:
.. math::
\mathbf{f}_{FK}(t)= -\iint_{S_I} P_I \mathbf{n} dS
.. math::
\mathbf{\Gamma}_{FK}(t)= -\iint_{S_I} P_I (\mathbf{OM}-\mathbf{OG})\times\mathbf{n} dS
Diffraction force
-----------------
.. _linear_diffraction:
Linear Model
************
The diffraction force, given by the linear approximation, is:
.. math::
\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
- :math:`\omega_m` the wave frequency;
- :math:`\theta_n` the wave direction;
- :math:`\bar{x}_n = x \cos(\theta_n) + y \sin(\theta_n)`, with :math:`x` and :math:`y` the coordinates of the :any:`equilibrium frame `
- :math:`\omega_m^e = \omega_m - k_mU` is the encounter circular frequency, calculated using the steady forward speed;
- :math:`A_{mn}` the wave amplitude for the wave frequency :math:`\omega_m` and the wave direction :math:`\theta_n`;
- :math:`\mathbf{f}_D(\omega_m,\theta_n)` is the frequency-domain diffraction force component for the wave frequency :math:`\omega` and the wave direction :math:`\theta_n`.
.. note::
The frequency-domain diffraction forces are obtained from a linear potential flow-based solver such as *Helios*, *Nemoh* or *WAMIT*.