Case 2.11

Conditions:

• New data for T2015
• Towing condition in variable heading
• With rudder
• \( FR_{X Z \phi \theta} \)
• \( L_{PP} = 2.7 \) [m], g = 9.80665 [m/s²]
• \( \rho = 997.8858 \) [kg/m³], \( \nu = 9.6790 \times 10^{-7} \) [m²/s](\(21.5\) [\({}^\circ\)C])
• Six combined conditions:



No.	C0	C1	C2	C3	C4	C5
Speed [m/s]	\(1.34\)
Froude number (\(F_r\))	0.26
Reynolds number (\(R_e\))	\(3.738 \times 10^6\)
Wave encounter angle: \(\chi\) [deg]	Calm water	0(head sea)	45(bow sea)	90(beam sea)	135(quartering sea)	180(following sea)
Wave length: \(\lambda\) [m]		2.7
Wave height: \(H_s\) [m]		0.045

\(\zeta_s\) : wave amplitude, \(\zeta_s = \displaystyle{\frac{H_s}{2}}\)
\(k\) : wave number, \(\displaystyle{k=\frac{2 \pi}{\lambda}}\)


Note: The surge, heave, roll and pitch motions are given at center of gravity. The wave crest is at FP when \(t = 0\).
EFD data was taken by surge free mount system (mass-spring-damper system). Parameter setting of surge free mount is as follows:


Spring constant (K)	100 [N/m]
Damper coefficient (D)	10 [N/(m/s)]

Surge fixed computation is recommended for participants.

References:

Not yet available.

Requested computations:

Note:
a positive (+) surge(x) value is defined forward and a positive (+) heave (z) value is defined upwards.
a positive (+) roll(\(\phi\)) value is portside up and positive (+) pitch (\(\theta\)) value is defined bow up.

Submission Instructions:

• [Identifier] should be [Institute Name]-[Solver Name]. For example, if your institute is NMRI and solver is SURFv7, identifier should be NMRI-SURFv7.
• Identifier in the Figure should be [Institute Name]/[Solver Name]. For example, if your institute is NMRI and solver is SURFv7, identifier should be NMRI/SURFv7.
• All figures should be in black and white.
• Authors may change the contour levels for turbulence quantities for better qualitative comparison.

All quantities are non-dimensionalized by denstiy of water (\(\rho\)), ship speed (\(U\)), and length between parpendiculars (\(L_{PP}\)): \begin{align*} F_r = \frac{U}{\sqrt{g \cdot L_{PP}}}, \quad R_e = \frac{U \cdot L_{PP}}{\nu} \tag{1} \end{align*} where \(g\) is the gravitational acceleration and \(\nu\) is the kinematic viscosity.

Figure 1: Relationship between ship heading and wave encounter angle. (Arrow shows ship heading)

Remarks:

• Coordinate system for comparisons is fixed to \(x=0.0\) (FP) on the undisturbed water plane.
• The center of the pitching moment is same as the center of gravity.
• All CFD predicted force coefficients should be reported using the wetted surface area at rest ( \(S_0\,/\,{L_{PP}}^2\) ). Force coefficients are defined as follows: \begin{align*} C_T = \frac{R_T}{ \frac{1}{2} \rho U^2 S_0 } \tag{2} \end{align*} where \(R_T\) is the total resistance.

Fourier Series Instructions:

As a time reference, incident wave height at FP of the ship is defined as \begin{align*} \zeta_T (t) = \frac{\zeta_s}{L_{PP}} \cos ( 2 \pi f_e t + \gamma_I ) \tag{3} \end{align*} \(\gamma_I\) is the initial phase and is equal to be zero from the present definition of \(t=0\) below.
Fourier series for time history \(X\) (\(X=C_T\), \(x\), \(z\), \(\phi\), \(\theta\), and \(\zeta_T\)) are determined as follows: \begin{align*} X_F (t) &= \frac{X_0}{2} + \sum_{n=1}^N X_n \cos ( 2 n \pi f_e t + \Delta \gamma_n ) \tag{4} \\ \Delta \gamma_n &= \gamma_n - \gamma_I \tag{5}\\ a_n &= \frac{2}{T_e} \int_0^{T_e} X(t) \cos ( 2 n \pi f_e t ) dt \quad ( n = 0, 1, 2, \cdots ) \tag{6} \\ b_n &= \frac{2}{T_e} \int_0^{T_e} X(t) \sin ( 2 n \pi f_e t ) dt \quad ( n = 1, 2, \cdots ) \tag{7} \\ X_n &= \sqrt{ {a_n}^2 + {b_n}^2 } \tag{8} \\ \gamma_n &= tan^{-1} \left( - \frac{b_n}{a_n} \right) \tag{9} \end{align*} \(X_n\) is n-th harmonic amplitude and \(\gamma_I\) is the corresponding phase.

Symbols:

\(\zeta_s\) - Wave amplitude \(= \displaystyle{\frac{H_s}{2}} \)

\(f_e\) - Wave encounter frequency \(= f_w + \displaystyle{\frac{U}{\lambda} \cos \left(\frac{\pi}{180}\chi \right) }\)

\(f_w\) - Frequency of the incident wave \(=\sqrt{\displaystyle{\frac{g}{2 \pi \lambda}}} \)

\(t\) - Time, when\(t=0\), a crest of the incident wave is coincident at FP

\(T_e\) - Wave encounter period \(= \displaystyle{\frac{1}{f_e}} \)