2.3.6. Hydrodynamic loads on bodies (Part II)

2.3.6.1. 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.

(2.565)\[\begin{aligned} & \boldsymbol{F}^1 = \int_{S_B^0}P_p^1 \boldsymbol{n}_p^0 ds - \int_{S_{TS}^0} P_p^1 \boldsymbol{n}_p^0 ds = \boldsymbol{F}_H^0 + \boldsymbol{F}_H^1 + \boldsymbol{F}_D^1 + \boldsymbol{F}_{TS}^1 \end{aligned}\]

(2.566)\[\begin{split}\begin{aligned} & \boldsymbol{M}^1 = \int_{S_B^0} P_p^1 \left( \boldsymbol{x}_G - \boldsymbol{x}_p \right) \times \boldsymbol{n}_p^0 ds - \int_{S_{TS}^0} P_p^1 \left( \boldsymbol{x}_G - \boldsymbol{x}_p \right) \times \boldsymbol{n}_p^0 ds = \\ & = \boldsymbol{M}_H^0 + \boldsymbol{M}_H^1 + \boldsymbol{M}_D^1 + \boldsymbol{M}_{TS}^1 \end{aligned}\end{split}\]

where sub-index \(H\) stands for hydrostatic loads, \(D\) stands for dynamic loads, and \(TS\) for transom stern.

2.3.6.2. Hydrostatic loads

The hydrostatic loads are split as follows:

(2.567)\[\begin{aligned} & \boldsymbol{F}_H^0 = - \int_{S_B^0} \rho g z \boldsymbol{n}_p^0 ds = \rho g ∀ \end{aligned}\]

(2.568)\[\begin{aligned} & \boldsymbol{M}_H^0 = - \int_{S_B^0} \rho g z \overrightarrow{\boldsymbol{G}^0 \boldsymbol{P}^0} \times \boldsymbol{n}_p^0 ds \end{aligned}\]

(2.569)\[\begin{aligned} & \boldsymbol{F}_H^1 = - \int_{S_B^0} \rho g r_pz^1 \boldsymbol{n}_p^0 ds = \overline{\overline{\boldsymbol{K}_H}} \Delta ^1 \end{aligned}\]

(2.570)\[\begin{aligned} & \boldsymbol{M}_H^1 = - \int_{S_B^0} \rho g r_pz^1 \overrightarrow{\boldsymbol{G}^0 \boldsymbol{P}^0} \times \boldsymbol{n}_p^0 ds = \overline{\overline{\boldsymbol{K}_H}} \theta^1 \end{aligned}\]

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

2.3.6.3. Dynamic loads

The dynamic loads are computed as:

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

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

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

Kelvin flow type:

(2.573)\[\begin{aligned} & \boldsymbol{U} = \boldsymbol{U}_b \end{aligned}\]

(2.574)\[\begin{aligned} & \boldsymbol{Q} = 0 \end{aligned}\]

Double body flow type:

(2.575)\[\begin{aligned} & \boldsymbol{U} = \boldsymbol{U}_b + \nabla_h ϕ^{DB} \end{aligned}\]

(2.576)\[\begin{aligned} & \boldsymbol{Q} = - \frac{1}{2} \nabla_h ϕ^{DB} \nabla_h ϕ^{DB} \end{aligned}\]

Non-linear flow type:

(2.577)\[\begin{aligned} & \boldsymbol{U} = \boldsymbol{U}_b + \nabla_h ϕ \end{aligned}\]

(2.578)\[\begin{aligned} & \boldsymbol{Q} = - \frac{1}{2} \nabla_h ϕ \cdot \nabla_h ϕ \end{aligned}\]

2.3.6.4. 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, \(S_B^0\) and the transom stern surface is included \(S_{TS}^0 ⊂ S_B^0\), imposing a zero pressure on the transom stern is equivalent to adding the corresponding negative pressure. Then, integrating over \(S_{TS}^0\), the loads obtained can be seen as added resistance due to the lack of hydrostatic and dynamic pressure.

(2.579)\[\begin{aligned} & \boldsymbol{F}_{TS}^1 = -\int_{S_{TS}^0} P_p^1 \boldsymbol{n}_p^0 ds \end{aligned}\]

(2.580)\[\begin{aligned} & \boldsymbol{M}_{TS}^1 = -\int_{S_{TS}^0} P_p^1 \left( \boldsymbol{x}_G - \boldsymbol{x}_p \right) \times \boldsymbol{n}_p^0 ds \end{aligned}\]