# 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:

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:

• [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 $= 1 / f_e$