2.3.3. Hydrodynamic loads on bodies (Part I)

2.3.3.1. 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.

(2.447)\[\begin{aligned} & \boldsymbol{F}^1 = \int_{S_B^0} P_p^1 \boldsymbol{n}_p^0 ds = \boldsymbol{F}_H^0 + \boldsymbol{F}_H^1 + \boldsymbol{F}_D^1 \end{aligned}\]

(2.448)\[\begin{aligned} & \boldsymbol{M}^1 = \int_{S_B^0} P_p^1 \left( \boldsymbol{x}_G - \boldsymbol{x}_p \right) \times n_p^0 ds = \boldsymbol{M}_H^0 + \boldsymbol{M}_H^1 + \boldsymbol{M}_D^1 \end{aligned}\]

where sub-index \(H\) stands for hydrostatic loads and \(D\) stands for dynamic loads. The hydrostatic loads are split as follows:

(2.449)\[\begin{aligned} & \boldsymbol{F}_H^0 = -\int_{S_B^0} \rho gz\boldsymbol{n}_p^0 ds = \rho g\forall \end{aligned}\]

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

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

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

where \(\forall\) is the body displacement, and \(\overline{\overline{\boldsymbol{K}}}_H\) is the hydrostatic restoring matrix, which are obtained as follows:

(2.453)\[\begin{aligned} & ∀ = -\int_{S_B^0} z_p n_{pz} ds \end{aligned}\]

(2.454)\[\begin{aligned} & x_B = -\frac{1}{2∀} \int_{S_B^0} x_p^2 n_{px} ds \end{aligned}\]

(2.455)\[\begin{aligned} & y_B = -\frac{1}{2∀} \int_{S_B^0} y_p^2 n_{py} ds \end{aligned}\]

(2.456)\[\begin{aligned} & z_B = -\frac{1}{2∀} \int_{S_B^0} z_p^2 n_{pz} ds \end{aligned}\]

(2.457)\[\begin{aligned} & K_H (3,3) = \rho g \int_{S_B^0} n_{pz} ds \end{aligned}\]

(2.458)\[\begin{aligned} & K_H (3,4) = \rho g \int_{S_B^0} (y_p-y_G) n_{pz} ds \end{aligned}\]

(2.459)\[\begin{aligned} & K_H (3,5) = -\rho g \int_{S_B^0} (x_p - x_G) n_{pz} ds \end{aligned}\]

(2.460)\[\begin{aligned} & K_H (4,4) = \rho g \int_{S_B^0} (y_p - y_G)^2 n_{pz} ds + \rho g∀ (z_B - z_G) \end{aligned}\]

(2.461)\[\begin{aligned} & K_H (4,5) = -\rho g \int_{S_B^0} (x_p -x_G) (y_p - y_G) n_{pz} ds \end{aligned}\]

(2.462)\[\begin{aligned} & K_H (4,6) = -\rho g∀ (x_B - x_G) \end{aligned}\]

(2.463)\[\begin{aligned} & K_H (5,5) = \rho g \int_{S_B^0} (x_p - x_G)^2 n_{pz} ds + \rho g∀(z_B - z_G) \end{aligned}\]

(2.464)\[\begin{aligned} & K_H (5,6) = -\rho g∀ (y_B - y_G) \end{aligned}\]

where \(B\) stands for the body center of buoyancy, and \(G\) for the body center of gravity. The dynamic loads are computed as:

(2.465)\[\begin{aligned} & F_D^1 = -\int_{S_B^0} \rho \frac{\partial \varphi^1}{\partial t} \boldsymbol{n}_p^0 ds \end{aligned}\]

(2.466)\[\begin{aligned} \boldsymbol{M}_D^1 = -\int_{S_B^0} \rho \frac{\partial \varphi^1}{\partial t} \overrightarrow{\boldsymbol{G}^0 \boldsymbol{P}^0} \times \boldsymbol{n}_p^0 ds \end{aligned}\]

2.3.3.2. Second order loads

Up to second order loads can be split as:

(2.467)\[\begin{aligned} & \boldsymbol{F}^{1+2} = \boldsymbol{F}_H^0 + \boldsymbol{F}_H^1 + \boldsymbol{F}_H^2 + \boldsymbol{F}_D^1 + \boldsymbol{F}_D^2 \end{aligned}\]

(2.468)\[\begin{aligned} & \boldsymbol{M}^{1+2} = \boldsymbol{M}_H^0 + \boldsymbol{M}_H^1 + \boldsymbol{M}_H^2 + \boldsymbol{M}_D^1 + \boldsymbol{M}_D^2 \end{aligned}\]

Where the hydrostatic loads are:

(2.469)\[\begin{split}\begin{aligned} & \boldsymbol{F}_H^{1+2} = -\int_{S_B^0} \rho g (z_p + r_{pz}^{1+2} ) \boldsymbol{n}_p^1 ds = \boldsymbol{F}_H^0 + \boldsymbol{F}_H^1 + \boldsymbol{\theta}^1 \times \boldsymbol{F}_H^1 + \overline{\overline{\boldsymbol{K}}}_H \boldsymbol{\delta }^2 \\ & - \int_{S_B^0} \rho g \left( \overline{\overline{H}} \overrightarrow{\boldsymbol{R}^0 \boldsymbol{P}^0} \right)_z \overrightarrow{\boldsymbol{G}^0 \boldsymbol{P}^0} \times \boldsymbol{n}_p^0 ds \end{aligned}\end{split}\]

(2.470)\[\begin{split}\begin{aligned} & \boldsymbol{M}_H^{1+2} = -\int_{S_B^0} \rho g \left( z_p + r_{pz}^{1+2} \right) \overrightarrow{\boldsymbol{G}^0 \boldsymbol{P}^0} \times \boldsymbol{n}_p^1 ds = \\ & \boldsymbol{M}_H^0 + \boldsymbol{M}_H^1 + \boldsymbol{\theta}^1 \times \boldsymbol{M}_H^1 + \overline{\overline{\boldsymbol{K}}}_H \boldsymbol{\theta}^2 - \int_{S_B^0} \rho g \left( \overline{\overline{\boldsymbol{H}}} \overrightarrow{ \boldsymbol{R}^0 \boldsymbol{P}^0 } \right)_z \overrightarrow{\boldsymbol{G}^0 \boldsymbol{P}^0} \times \boldsymbol{n}_p^0 ds \end{aligned}\end{split}\]

Then:

(2.471)\[\begin{aligned} & \boldsymbol{F}_H^2 = \overline{\overline{\boldsymbol{K}}}_H \boldsymbol{\delta }^2 + \boldsymbol{\theta}^1 \times \boldsymbol{F}_H^1 - \int_{S_B^0} \rho g (\overline{\overline{\boldsymbol{H}}} \overrightarrow{\boldsymbol{R}^0 \boldsymbol{P}^0})_z \boldsymbol{n}_p^0 ds \end{aligned}\]

(2.472)\[\begin{aligned} & \boldsymbol{M}_H^2 = \overline{\overline{\boldsymbol{K}}}_H \boldsymbol{\theta}^2 + \boldsymbol{\theta}^1 \times \boldsymbol{M}_H^1 - \int_{S_B^0} \rho g (\overline{\overline{\boldsymbol{H}}} \overrightarrow{\boldsymbol{R}^0 \boldsymbol{P}^0})_z \overrightarrow{\boldsymbol{G}^0 \boldsymbol{P}^0} \times \boldsymbol{n}_p^0 ds \end{aligned}\]

On the other hand, dynamic loads up to second order are split in four components:

(2.473)\[\begin{aligned} & \boldsymbol{F}_D^{1+2} = \boldsymbol{F}_D^1 + \boldsymbol{F}_D^2 = \boldsymbol{F}_D^1 + \boldsymbol{F}_{D1}^2 + \boldsymbol{F}_{D2}^2 + \boldsymbol{F}_{D3}^2 + \boldsymbol{F}_{D4}^2 \end{aligned}\]

(2.474)\[\begin{aligned} & \boldsymbol{M}_D^{1+2} = \boldsymbol{M}_D^1 + \boldsymbol{M}_D^2 = \boldsymbol{M}_D^1 + \boldsymbol{M}_D1^2 + \boldsymbol{M}_D2^2 + \boldsymbol{M}_D3^2 + \boldsymbol{M}_D4^2 \end{aligned}\]

where

(2.475)\[\begin{aligned} & \boldsymbol{F}_{D1}^2 = -\rho \int_{S_B^0} \frac{\partial \varphi^2}{\partial t} \boldsymbol{n}_p^0 ds + \boldsymbol{\theta}^1 \times \boldsymbol{F}_D^1 \end{aligned}\]

(2.476)\[\begin{aligned} & \boldsymbol{F}_{D2}^2 = -\rho \int_{S_B^0} \left( \boldsymbol{r}_p^1 \cdot \nabla \left( \frac{\partial \varphi^1}{\partial t} \right) \right) \boldsymbol{n}_p^0 ds \end{aligned}\]

(2.477)\[\begin{aligned} & \boldsymbol{F}_{D3}^2 = -\frac{1}{2} \rho \int_{S_B^0} \left( \nabla \varphi^1 \cdot \nabla \varphi^1 \right) \boldsymbol{n}_p^0 ds \end{aligned}\]

(2.478)\[\begin{aligned} & \boldsymbol{F}_{D4}^2 = -\frac{1}{2} \rho g \int_{\Gamma_B^0} \left( \xi^1 - r_{pz}^1 \right)^2 \frac{\boldsymbol{n}_p^0}{\sqrt{1 - {n_{pz}^0}^2}} dl \end{aligned}\]

(2.479)\[\begin{aligned} & \boldsymbol{M}_{D1}^2 = -\rho \int_{S_B^0} \frac{\partial \varphi^2}{\partial t} \overrightarrow{\boldsymbol{G}^0 \boldsymbol{P}^0} \times \boldsymbol{n}_p^0 ds + \boldsymbol{\theta}^1 \times \boldsymbol{M}_D^1 \end{aligned}\]

(2.480)\[\begin{aligned} & \boldsymbol{M}_{D2}^2 = -\rho \int_{S_B^0} \left(r_p^1 \cdot \nabla \left( \frac{\partial \varphi^1}{\partial t} \right) \right) \overrightarrow{\boldsymbol{G}^0 \boldsymbol{P}^0} \times \boldsymbol{n}_p^0 ds \end{aligned}\]

(2.481)\[\begin{aligned} & \boldsymbol{M}_{D3}^2 = -\frac{1}{2} \rho \int_{S_B^0} \left( \nabla \varphi^1 \cdot \nabla \varphi^1 \right) \overrightarrow{\boldsymbol{G}^0 \boldsymbol{P}^0} \times \boldsymbol{n}_p^0 ds \end{aligned}\]

(2.482)\[\begin{aligned} & \boldsymbol{M}_{D4}^2 = -\frac{1}{2} \rho g \int_{\Gamma_{B}^0} \left( \xi^1 - r_{pz}^1 \right)^2 \overrightarrow{\boldsymbol{G}^0 \boldsymbol{P}^0} \times \frac{\boldsymbol{n}_p^0}{\sqrt{1 - {\boldsymbol{n}_{pz}^0}^2}} dl \end{aligned}\]

2.3.3.3. Mean drift loads

Being the first and second order responses harmonic and taking time average, the following relations holds when:

(2.483)\[\begin{aligned} \Big<F^1\Big>=0 \end{aligned}\]

(2.484)\[\begin{aligned} \Big<M^1\Big>=0 \end{aligned}\]

(2.485)\[\begin{aligned} & \Big<F_H^2\Big> = \Big<\boldsymbol{\theta}^1 \times \boldsymbol{F}_H^1\Big> - \Big<\int_{S_B^0} \rho g \left( \overline{\overline{\boldsymbol{H}}} \overrightarrow{\boldsymbol{R}^0 \boldsymbol{P}^0} \right)_z \boldsymbol{n}_p^0 ds \Big> \end{aligned}\]

(2.486)\[\begin{aligned} & \Big<M_H^2\Big> = \Big<\boldsymbol{\theta}^1 \times \boldsymbol{M}_H^1\Big> - \Big<\int_{S_B^0} \rho g \left( \overline{\overline{\boldsymbol{H}}} \overrightarrow{\boldsymbol{R}^0 \boldsymbol{P}^0} \right)_z \overrightarrow{\boldsymbol{G}^0 \boldsymbol{P}^0} \times \boldsymbol{n}_p^0 ds\Big> \end{aligned}\]

(2.487)\[\begin{aligned} & \Big<F_{D1}^2\Big> = \Big<\boldsymbol{\theta}^1 \times \boldsymbol{F}_D^1\Big> \end{aligned}\]

(2.488)\[\begin{aligned} & \Big<F_{D2}^2\Big> = - \rho \Big<\int_{S_B^0} \left( \boldsymbol{r}_p^1 \cdot \nabla \left( \frac{\partial \varphi^1}{\partial t} \right) \right) \boldsymbol{n}_p^0 ds\Big> \end{aligned}\]

(2.489)\[\begin{aligned} & \Big<F_{D3}^2\Big> = - \frac{1}{2} \rho \Big<\int_{S_B^0} \left( \nabla \varphi^1 \cdot \nabla \varphi^1 \right) \boldsymbol{n}_p^0 ds\Big> \end{aligned}\]

(2.490)\[\begin{aligned} & \Big<F_{D4}^2\Big> = - \frac{1}{2} \rho g \Big<\int_{\Gamma_B^0} \left( \xi^1 -r_{pz}^1 \right)^2 \frac{n_p^0}{\sqrt{1 - {\boldsymbol{n}_{pz}^0}^2}} dl\Big> \end{aligned}\]

(2.491)\[\begin{aligned} & \Big<M_{D1}^2\Big> = \Big<\boldsymbol{\theta}^1 \times \boldsymbol{M}_D^1\Big> \end{aligned}\]

(2.492)\[\begin{aligned} & \Big<M_{D2}^2\Big> = - \rho \Big<\int_{S_B^0} \left( r_p^1 \cdot \nabla \left(\frac{\partial \varphi^1}{\partial t} \right) \right) \overrightarrow{\boldsymbol{G}^0 \boldsymbol{P}^0} \times \boldsymbol{n}_p^0 ds\Big> \end{aligned}\]

(2.493)\[\begin{aligned} & \Big<M_{D3}^2\Big> = - \frac{1}{2} \rho \Big<\int_{S_B^0} \left( \nabla \varphi^1 \cdot \nabla \varphi^1 \right) \overrightarrow{\boldsymbol{}G^0 \boldsymbol{P}^0} \times \boldsymbol{n}_p^0 ds\Big> \end{aligned}\]

(2.494)\[\begin{aligned} & \Big<M_{D4}^2\Big> = - \frac{1}{2} \rho g \Big<\int_{\Gamma_B^0} \left( \xi^1 - r_{pz}^1 \right)^2 \overrightarrow{\boldsymbol{G}^0 \boldsymbol{P}^0} \times \frac{ \boldsymbol{n}_p^0 }{ \sqrt{ 1 - {\boldsymbol{n}_{pz}^0}^2}} dl\Big> \end{aligned}\]

Second order terms depending with non-zero time average depends on first order quantities. Hence, second order drifting loads only depends on the first order problem solution.