Hydrodynamic loads on bodies (Part II)
======================================

First order loads
~~~~~~~~~~~~~~~~~

Hydrodynamic forces and moments are obtained from direct pressure integration over the body surface. The body gravity center will be used as a reference for body movements and moments acting on it.

.. math::
   \begin{aligned}
      & \boldsymbol{F}^1 = \int_{S_B^0}P_p^1 \boldsymbol{n}_p^0 ds - \int_{S_{TS}^0} P_p^1 \boldsymbol{n}_p^0 ds = \boldsymbol{F}_H^0 + \boldsymbol{F}_H^1 + \boldsymbol{F}_D^1 + \boldsymbol{F}_{TS}^1 
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      & \boldsymbol{M}^1 = \int_{S_B^0} P_p^1 \left( \boldsymbol{x}_G - \boldsymbol{x}_p \right) \times \boldsymbol{n}_p^0 ds - \int_{S_{TS}^0} P_p^1 \left( \boldsymbol{x}_G - \boldsymbol{x}_p \right) \times \boldsymbol{n}_p^0 ds = \\
      & = \boldsymbol{M}_H^0 + \boldsymbol{M}_H^1 + \boldsymbol{M}_D^1 + \boldsymbol{M}_{TS}^1 
   \end{aligned}
   :label:

where sub-index :math:`H` stands for hydrostatic loads, :math:`D` stands for dynamic loads, and :math:`TS` for transom stern.

Hydrostatic loads
~~~~~~~~~~~~~~~~~

The hydrostatic loads are split as follows:

.. math::
   \begin{aligned}
      & \boldsymbol{F}_H^0 = - \int_{S_B^0} \rho g z \boldsymbol{n}_p^0  ds = \rho g ∀ 
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      & \boldsymbol{M}_H^0 = - \int_{S_B^0} \rho g z \overrightarrow{\boldsymbol{G}^0 \boldsymbol{P}^0} \times \boldsymbol{n}_p^0  ds 
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      & \boldsymbol{F}_H^1 = - \int_{S_B^0} \rho g r_pz^1 \boldsymbol{n}_p^0  ds = \overline{\overline{\boldsymbol{K}_H}} \Delta ^1 
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      & \boldsymbol{M}_H^1 = - \int_{S_B^0} \rho g r_pz^1 \overrightarrow{\boldsymbol{G}^0 \boldsymbol{P}^0} \times \boldsymbol{n}_p^0  ds = \overline{\overline{\boldsymbol{K}_H}} \theta^1 
   \end{aligned}
   :label:

where :math:`∀` is the body displacement, and  :math:`\overline{\overline{K_H}}` is the hydrostatic restoring matrix (defined in section 3.2).

Dynamic loads
~~~~~~~~~~~~~

The dynamic loads are computed as:

.. math::
   \begin{aligned}
      & \boldsymbol{F}_D^1 = -\int_{S_B^0} \rho \left( \frac{\partial φ}{\partial t} + \boldsymbol{U} \cdot  \nabla_h φ + Q \right) \boldsymbol{n}_p^0  ds 
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      & \boldsymbol{M}_D^1 = -\int_{S_B^0} \rho \left( \frac{\partial φ}{\partial t} + \boldsymbol{U} \cdot  \nabla_h φ + Q \right) \overrightarrow{\boldsymbol{G}^0 \boldsymbol{P}^0} \times \boldsymbol{n}_p^0 ds 
   \end{aligned}
   :label:

where :math:`\boldsymbol{U}` and :math:`\boldsymbol{Q}` depends on the flow approximation used as follows: 

Kelvin flow type:

.. math::
   \begin{aligned}
      & \boldsymbol{U} = \boldsymbol{U}_b
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      & \boldsymbol{Q} = 0  
   \end{aligned}
   :label:

Double body flow type:

.. math::
   \begin{aligned}
      & \boldsymbol{U} = \boldsymbol{U}_b + \nabla_h ϕ^{DB}
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      & \boldsymbol{Q} = - \frac{1}{2} \nabla_h ϕ^{DB} \nabla_h ϕ^{DB}
   \end{aligned}
   :label:

Non-linear flow type:

.. math::
   \begin{aligned}
      & \boldsymbol{U} = \boldsymbol{U}_b + \nabla_h ϕ
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      & \boldsymbol{Q} = - \frac{1}{2} \nabla_h ϕ \cdot  \nabla_h ϕ
   \end{aligned}
   :label:

Transom stern added resistance
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

If a transom stern is defined, no hydrodynamic pressure will exist on the transom surface. However, since hydrostatic and dynamic loads are calculated integrating on the initial wet surface of the body, :math:`S_B^0` and the transom stern surface is included :math:`S_{TS}^0 ⊂ S_B^0`, imposing a zero pressure on the transom stern is equivalent to adding the corresponding negative pressure. Then, integrating over :math:`S_{TS}^0`, the loads obtained can be seen as added resistance due to the lack of hydrostatic and dynamic pressure.

.. math::
   \begin{aligned}
      & \boldsymbol{F}_{TS}^1 = -\int_{S_{TS}^0} P_p^1 \boldsymbol{n}_p^0 ds 
   \end{aligned}
   :label:

.. math::
   \begin{aligned}
      & \boldsymbol{M}_{TS}^1 = -\int_{S_{TS}^0} P_p^1 \left( \boldsymbol{x}_G - \boldsymbol{x}_p \right) \times \boldsymbol{n}_p^0 ds 
   \end{aligned}
   :label: