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Linear seakeeping and added resistance analysis by means of body-fixed coordinate system

Abstract

This paper presents an alternative formulation of the boundary value problem for linear seakeeping and added resistance analysis based on a body-fixed coordinate system. The formulation does not involve higher-order derivatives of the steady velocity potential on the right-hand side of the body-boundary condition, i.e., the so-called m j -terms in the traditional formulation when an inertial coordinate system is applied. Numerical studies are made for a modified Wigley I hull, a Series 60 ship with block coefficient 0.7, and the S175 container ship for moderate forward speeds where it is thought appropriate to use the double-body flow as basis flow. The presented results for the forced heave and pitch oscillations, motion responses, and added resistance in head-sea waves show good agreement with experiments and some other numerical studies. A Neumann–Kelvin formulation is shown to give less satisfactory results, in particular for coupled heave and pitch added mass and damping coefficients.

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Correspondence to Yan-Lin Shao.

Appendices

Appendix 1

The transformation matrix \( {\mathbf{R}}_{b \to i}^{{}} \) that transforms the representation of a vector in the body-fixed oxyz system to its representation in the inertial OXYZ system is defined as

$$ {\mathbf{R}}_{b \to i} = \left[ {\begin{array}{*{20}c} {c_{6} c_{5} } & {c_{6} s_{5} s_{4} - s_{6} c_{4} } & {c_{6} s_{5} c_{4} + s_{6} s_{4} } \\ {s_{6} c_{5} } & {s_{6} s_{5} s_{4} + c_{6} c_{4} } & {s_{6} s_{5} c_{4} - c_{6} s_{4} } \\ { - s_{5} } & {c_{5} s_{4} } & {c_{5} c_{4} } \\ \end{array} } \right]. $$
(26)

Here \( s_{i} = \sin (\alpha_{i} ) \) and \( c_{i} = \cos (\alpha_{i} ) \) with i = 4, 5, 6, \( {\mathbf{R}}_{b \to i} \) can be approximated as

$$ {\mathbf{R}}_{b \to i} = {\mathbf{I}}_{33} + {\mathbf{R}}_{b \to i}^{(1)} + {\mathbf{R}}_{b \to i}^{(2)} + O(\varepsilon^{3} ) $$
(27)

with

$$ {\mathbf{I}}_{33} = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} } \right], $$
(28)
$$ {\mathbf{R}}_{b \to i}^{(1)} = \left[ {\begin{array}{*{20}c} 0 & { - \alpha_{6}^{(1)} } & {\alpha_{5}^{(1)} } \\ {\alpha_{6}^{(1)} } & 0 & { - \alpha_{4}^{(1)} } \\ { - \alpha_{5}^{(1)} } & {\alpha_{4}^{(1)} } & 0 \\ \end{array} } \right], $$
(29)
$$ {\mathbf{R}}_{b \to i}^{(2)} = \left[ {\begin{array}{*{20}c} 0 & { - \alpha_{6}^{(2)} } & {\alpha_{5}^{(2)} } \\ {\alpha_{6}^{(2)} } & 0 & { - \alpha_{4}^{(2)} } \\ { - \alpha_{5}^{(2)} } & {\alpha_{4}^{(2)} } & 0 \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} { - \frac{1}{2}[(\alpha_{5}^{(1)} )^{2} + (\alpha_{6}^{(1)} )^{2} ]} & {\alpha_{4}^{(1)} \alpha_{5}^{(1)} } & {\alpha_{4}^{(1)} \alpha_{6}^{(1)} } \\ 0 & { - \frac{1}{2}[(\alpha_{4}^{(1)} )^{2} + (\alpha_{6}^{(1)} )^{2} ]} & {\alpha_{5}^{(1)} \alpha_{6}^{(1)} } \\ 0 & 0 & { - \frac{1}{2}[(\alpha_{4}^{(1)} )^{2} + (\alpha_{5}^{(1)} )^{2} ]} \\ \end{array} } \right]. $$
(30)

Appendix 2

The linear force \( \vec{F}^{(1)} \), linear moment \( \vec{M}^{(1)}, \) and the added resistance \( \vec{F}^{(0)}, \) defined with respect to the inertial coordinate system, are written, respectively, as

$$ \vec{F}^{(1)} = \iint\limits_{\text{SB}} {\left[ {p^{(1)} \vec{n}^{(0)} + \,p^{(0)} \vec{n}^{(1)} } \right]{\text{d}}s} + \int\limits_{{{\text{CW}}_{0} }} {p^{(0)} \eta^{(1)} \vec{n}^{(0)} /\sqrt {1 - n_{3}^{2} } {\text{d}}l} , $$
(31)
$$ \vec{M}^{(1)} = \iint\limits_{\text{SB}} {\left\{ {p^{(1)} \vec{m}^{(0)} + p^{(0)} \vec{m}^{(1)} } \right\}\,{\text{d}}s}\; + \int\limits_{{{\text{CW}}_{0} }} {p^{(0)} \eta^{(1)} \vec{m}^{(0)} /\sqrt {1 - n_{3}^{2} } {\text{d}}l} . $$
(32)
$$ \begin{aligned} \vec{F}^{(0)} & = \overline{{\iint\limits_{\text{SB}} {\left[ {p^{(1)} \vec{n}_{{}}^{\left( 1 \right)} + p^{(0)} \vec{n}^{(2)} } \right]{\text{d}}s}}} + \iint\limits_{\text{SB}} {\left[ { - \rho \left( {\frac{\partial }{\partial t} - U\frac{\partial }{\partial x}} \right)\left( {\phi_{I}^{(2)} + \vec{x}^{(1)} \cdot \nabla \phi_{I}^{(1)} } \right)\vec{n}^{(0)} } \right]{\text{d}}s} \\ & \qquad + \overline{{\int\limits_{{{\text{CW}}_{0} }} {{{\left[ {\frac{1}{2}\rho g(\eta^{(1)} )^{2} n_{1}^{(0)} + p^{(0)} \eta^{(1)} n_{1}^{(1)} } \right]} \mathord{\left/ {\vphantom {{\left[ {\frac{1}{2}\rho g(\eta^{(1)} )^{2} n_{1}^{(0)} + p^{(0)} \eta^{(1)} n_{1}^{(1)} } \right]} {\sqrt {1 - n_{3}^{2} } {\text{d}}l}}} \right. \kern-\nulldelimiterspace} {\sqrt {1 - n_{3}^{2} } {\text{d}}l}}} }}. \\ \end{aligned} $$
(33)

Here CW0 is the mean waterline. \( \vec{r} \) is the position vector of a point on the ship surface relative to the center with respect to which the moments are defined. \( \vec{n}^{(0)} = (n_{1} ,n_{2} ,n_{3} ) \) and \( \vec{n}^{(k)} = {\mathbf{R}}_{b \to i}^{(k)} \vec{n}^{(0)} \) (k = 1, 2) are the components of the normal vector on the body. \( n_{1}^{(1)} \) is the first component of \( \vec{n}^{(1)}. \) \( \vec{m}^{(0)} = \vec{r} \times \vec{n}^{(0)} \) and \( \vec{m}^{(1)} = {\mathbf{R}}_{b \to i}^{(1)} (\vec{r} \times \vec{n}^{(0)} ) \) are the generalized normal vectors. \( \phi_{I}^{(k)} \) (k = 1, 2) is the kth order velocity potential of the incident wave described in the inertial coordinate system. \( p^{(0)} \) and \( p^{(1)} \) are pressure components giving zeroth-order and first-order contributions to the forces and moments. They are defined, respectively, as

$$ p^{(0)} = - \rho \left\{ {gz - U\phi_{x}^{(0)} + \frac{1}{2}\nabla \phi^{(0)} \cdot \nabla \phi^{(0)} } \right\}. $$
(34)
$$ p^{(1)} = - \rho \left\{ {\phi_{t}^{(1)} - U\phi_{x}^{(1)} - \left( {\vec{\dot{\xi }}^{(1)} + \vec{\omega }^{(1)} \times \vec{r}' + \vec{U}'^{(1)} - \nabla \phi^{(1)} } \right) \cdot \nabla \phi^{(0)} + g\left( {\xi_{3}^{(1)} + y\alpha_{4}^{(1)} - x\alpha_{5}^{(1)} } \right)} \right\} $$
(35)

\( \vec{U}'^{(1)} \) has been defined in Eq. 10.

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Shao, YL., Faltinsen, O.M. Linear seakeeping and added resistance analysis by means of body-fixed coordinate system. J Mar Sci Technol 17, 493–510 (2012). https://doi.org/10.1007/s00773-012-0185-y

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Keywords

  • Seakeeping
  • Added resistance
  • Body-fixed coordinate system
  • m j -Terms
  • Boundary element method