Case 2.10

Conditions:

• New data for T2015
• Towing condition in head waves
• With rudder
• \( FR_{Z \theta} \)
• \( L_{PP} = 6.0702 \) [m], g = 9.81 [m/s²]
• \( \rho = 998.63 \) [kg/m³], \( \nu = 1.14 \times 10^{-6} \) [m²/s]
• Six combined conditions:

No.	C0	C1	C2	C3	C4	C5
Speed [m/s]	\(2.017\)
Froude number (\(F_r\))	0.261
Reynolds number (\(R_e\))	\( 1.074 \times 10^7\)
Wave length: \(\lambda\) [m]	0.0	3.949	5.164	6.979	8.321	11.840
Wave height: \(H_s\) [m]	0.0	0.062	0.078	0.123	0.149	0.196

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


Note: The heave and pitch motions are given at the center of gravity and the wave crest is at FP when \(t=0\).

References:

Not yet available.

Requested computations:

Note: a positive (+) sinkage value is defined upwards and a positive (+) trim 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.

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 \(X\) (\(X=C_T\), \(z\), \(\theta\), 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 \(= H_s / 2 \)

\(f_e\) - Wave encounter frequency \(= f_w + U / \lambda \)

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

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

\(T_e\) - Wave encounter period \(= 1 / f_e \)