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Case 2.10

Conditions:

References:

Not yet available.

Requested computations:

Table/Figure# Items EFD Data Submission Instruction
Data file Image Image files Sample + Tecplot layout file
2.10-1 Comparison of:
<in calm sea>
total resistance coefficient (\(C_T \times 10^3\))
sinkage (\(z / L_{PP}\)) and trim (\(\theta\) [deg])

<in head waves>
total resistance coefficient (\(C_T \times 10^3\))
0th, 1st, 2nd, 3rd and 4th harmonic amplitudes[-] and 1st, 2nd, 3rd and 4th harmonic phases [rad];
heave motion (\(z / \zeta_s\)) and pitch angle (\(\theta / k \zeta_s\))
0th, 1st, 2nd, 3rd and 4th harmonic amplitudes[-] and 1st, 2nd, 3rd and 4th harmonic phases [rad];
wave amplitude (\(\zeta_T / L_{PP}\))
1st harmonic amplitude
Refer to sample file for details Filename: [Identifier]_6conditions_2-10.xls (Excel file) [Identifier]_6conditions_2-10_20150914.xlsx
(updated on September, 14, 2015)
2.10-2-C1 Time histories of total resistance coefficient (\(C_T\)), heave motion (\(z / \zeta_s\)) and pitch angle (\(\theta / k \zeta_s \)), which are reconstructed from the Fourier Series (not the raw signals)
(test condition 1)
Refer to sample file for details Filename:
[Identifier]_CT_T-his_C1_2-10.png ( for \(C_T\) )
[Identifier]_heave_T-his_C1_2-10.png ( for \(z / \zeta_s \) )
[Identifier]_pitch_T-his_C1_2-10.png ( for \(\theta / k \zeta_s \) )
X-axis range:
\( 0.0 \le \displaystyle{\frac{t}{T_e}} \le 1.0 \)
Y-axis range:
\( -0.003 \le C_T \le \color{blue}0.012 \)
\( \color{blue}-1.0 \color{red} \le \displaystyle{\frac{z}{\zeta_s}} \le 0.5 \)
\( \color{blue}-0.15 \color{red} \le \displaystyle{\frac{\theta}{k \zeta_s}} \le 0.05 \)
Style:
CFD solid line
EFD open circles
Case2.10-2.zip
(updated on September, 14, 2015)
2.10-2-C2 Time histories of total resistance coefficient (\(C_T\)), heave motion (\(z / \zeta_s\)) and pitch angle (\(\theta / k \zeta_s \)), which are reconstructed from the Fourier Series (not the raw signals)
(test condition 2)
Refer to sample file for details Filename:
[Identifier]_CT_T-his_C2_2-10.png ( for \(C_T\) )
[Identifier]_heave_T-his_C2_2-10.png ( for \(z / \zeta_s \) )
[Identifier]_pitch_T-his_C2_2-10.png ( for \(\theta / k \zeta_s \) )
X-axis range:
\( 0.0 \le \displaystyle{\frac{t}{T_e}} \le 1.0 \)
Y-axis range:
\( -0.008 \le C_T \le \color{blue}0.016 \)
\( \color{blue} -1.0 \color{red} \le \displaystyle{\frac{z}{\zeta_s}} \le 0.5 \)
\( -0.5 \le \displaystyle{\frac{\theta}{k \zeta_s}} \le 0.5 \)
Style:
CFD solid line
EFD open circles
2.10-2-C3 Time histories of total resistance coefficient (\(C_T\)), heave motion (\(z / \zeta_s\)) and pitch angle (\(\theta / k \zeta_s \)), which are reconstructed from the Fourier Series (not the raw signals)
(test condition 3)
Refer to sample file for details Filename:
[Identifier]_CT_T-his_C3_2-10.png ( for \(C_T\) )
[Identifier]_heave_T-his_C3_2-10.png ( for \(z / \zeta_s \) )
[Identifier]_pitch_T-his_C3_2-10.png ( for \(\theta / k \zeta_s \) )
X-axis range:
\( 0.0 \le \displaystyle{\frac{t}{T_e}} \le 1.0 \)
Y-axis range:
\( -0.010 \le C_T \le 0.020 \)
\( -2.0 \le \displaystyle{\frac{z}{\zeta_s}} \le 2.0 \)
\( -2.0 \le \displaystyle{\frac{\theta}{k \zeta_s}} \le 2.0 \)
Style:
CFD solid line
EFD open circles
2.10-2-C4 Time histories of total resistance coefficient (\(C_T\)), heave motion (\(z / \zeta_s\)) and pitch angle (\(\theta / k \zeta_s \)), which are reconstructed from the Fourier Series (not the raw signals)
(test condition 4)
Refer to sample file for details Filename:
[Identifier]_CT_T-his_C4_2-10.png ( for \(C_T\) )
[Identifier]_heave_T-his_C4_2-10.png ( for \(z / \zeta_s \) )
[Identifier]_pitch_T-his_C4_2-10.png ( for \(\theta / k \zeta_s \) )
X-axis range:
\( 0.0 \le \displaystyle{\frac{t}{T_e}} \le 1.0 \)
Y-axis range:
\( -0.020 \le C_T \le 0.040 \)
\( -2.0 \le \displaystyle{\frac{z}{\zeta_s}} \le 2.0 \)
\( -2.0 \le \displaystyle{\frac{\theta}{k \zeta_s}} \le 2.0 \)
Style:
CFD solid line
EFD open circles
2.10-2-C5 Time histories of total resistance coefficient (\(C_T\)), heave motion (\(z / \zeta_s\)) and pitch angle (\(\theta / k \zeta_s \)), which are reconstructed from the Fourier Series (not the raw signals)
(test condition 5)
Refer to sample file for details Filename:
[Identifier]_CT_T-his_C5_2-10.png ( for \(C_T\) )
[Identifier]_heave_T-his_C5_2-10.png ( for \(z / \zeta_s \) )
[Identifier]_pitch_T-his_C5_2-10.png ( for \(\theta / k \zeta_s \) )
X-axis range:
\( 0.0 \le \displaystyle{\frac{t}{T_e}} \le 1.0 \)
Y-axis range:
\( -0.040 \le C_T \le 0.050 \)
\( -2.0 \le \displaystyle{\frac{z}{\zeta_s}} \le 2.0 \)
\( -2.0 \le \displaystyle{\frac{\theta}{k \zeta_s}} \le 2.0 \)
Style:
CFD solid line
EFD open circles

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

Submission Instructions:

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:

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 \(= 1 / f_e \)