# 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:

Table/Figure# Items EFD Data Submission Instruction
Data file Image Image files Sample + Tecplot layout file
2.11-1 Comparison of:
<in calm water>
• total resistance coefficient ($C_T \times 10^3$)
• sinkage ($z / L_{PP}$) and trim ($\theta$ [deg])

• total resistance coefficient ($C_T \times 10^3$), surge motion ($x / \zeta_s$), heave motion ($z / \zeta_s$), roll angle ($\phi / k \zeta_s$) and pitch angle ($\theta / k \zeta_s$)
• 0th, 1st, 2nd, 3rd and 4th harmonic amplitudes[-] and 1st, 2nd, 3rd and 4th harmonic phases [rad];
Refer to sample file for details Filename: [Identifier]_6conditions_2-11.xlsx (Excel file) [Identifier]_6conditions_2-11_20151112.xlsx
(updated on November, 13, 2015)
2.11-2-C1 Time histories of total resistance coefficient ($C_T$), surge motion ($x/\zeta_s$), heave motion ($z/\zeta_s$), roll motion($\phi/ k \zeta_s$) 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_2-11.png ( for $C_T$ )
[Identifier]_surge_T-his_C1_2-11.png ( for $x/\zeta_s$ )
[Identifier]_heave_T-his_C1_2-11.png ( for $z/\zeta_s$ )
[Identifier]_roll_T-his_C1_2-11.png ( for $\phi/ k \zeta_s$ )
[Identifier]_pitch_T-his_C1_2-11.png ( for $\theta/ k \zeta_s$ )
X-axis range:
$0.0 \le t / T_e \le 1.0$
Y-axis range:
$0.0066 \le C_T \le 0.0071$
$-0.03 \le \displaystyle{\frac{x }{\zeta}} \le 0.12$
$-1.0 \le \displaystyle{\frac{z }{\zeta}} \le 1.0$
$-0.4 \le \displaystyle{\frac{\phi }{k \zeta}} \le 0.4$
$-0.8 \le \displaystyle{\frac{\theta}{k \zeta}} \le 0.8$
Style:
CFD solid line
EFD open circles
Case2.11-2_20151112.zip
(updated on November, 13, 2015)
2.11-2-C2 Time histories of total resistance coefficient ($C_T$), surge motion ($x/\zeta_s$), heave motion ($z/\zeta_s$), roll motion($\phi/ k \zeta_s$) 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_2-11.png ( for $C_T$ )
[Identifier]_surge_T-his_C2_2-11.png ( for $x/\zeta_s$ )
[Identifier]_heave_T-his_C2_2-11.png ( for $z/\zeta_s$ )
[Identifier]_roll-his_C2_2-11.png ( for $\phi/ k \zeta_s$ )
[Identifier]_pitch_T-his_C2_2-11.png ( for $\theta/ k \zeta_s$ )
X-axis range:
$0.0 \le t / T_e \le 1.0$
Y-axis range:
$0.0058 \le C_T \le 0.0068$
$-0.15 \le \displaystyle{\frac{x }{\zeta}} \le 0.20$
$-1.5 \le \displaystyle{\frac{z }{\zeta}} \le 1.5$
$-1.6 \le \displaystyle{\frac{\phi }{k \zeta}} \le 0.8$
$-0.8 \le \displaystyle{\frac{\theta}{k \zeta}} \le 0.8$
Style:
CFD solid line
EFD open circles
2.11-2-C3 Time histories of total resistance coefficient ($C_T$), surge motion ($x/\zeta_s$), heave motion ($z/\zeta_s$), roll motion($\phi/ k \zeta_s$) 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_2-11.png ( for $C_T$ )
[Identifier]_surge_T-his_C3_2-11.png ( for $x/\zeta_s$ )
[Identifier]_heave_T-his_C3_2-11.png ( for $z/\zeta_s$ )
[Identifier]_roll_T-his_C3_2-11.png ( for $\phi/ k \zeta_s$ )
[Identifier]_pitch_T-his_C3_2-11.png ( for $\theta/ k \zeta_s$ )
X-axis range:
$0.0 \le t / T_e \le 1.0$
Y-axis range:
$0.0 \le C_T \le 0.008$
$-0.1 \le \displaystyle{\frac{x }{\zeta}} \le 0.1$
$-1.5 \le \displaystyle{\frac{z }{\zeta}} \le 1.5$
$-1.2 \le \displaystyle{\frac{\phi }{k \zeta}} \le 1.2$
$-0.15 \le \displaystyle{\frac{\theta}{k \zeta}} \le 0.15$
Style:
CFD solid line
EFD open circles
2.11-2-C4 Time histories of total resistance coefficient ($C_T$), surge motion ($x/\zeta_s$), heave motion ($z/\zeta_s$), roll motion($\phi/ k \zeta_s$) 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_2-11.png ( for $C_T$ )
[Identifier]_surge_T-his_C4_2-11.png ( for $x/\zeta_s$ )
[Identifier]_heave_T-his_C4_2-11.png ( for $z/\zeta_s$ )
[Identifier]_roll_T-his_C4_2-11.png ( for $\phi/ k \zeta_s$ )
[Identifier]_pitch_T-his_C4_2-11.png ( for $\theta/ k \zeta_s$ )
X-axis range:
$0.0 \le t / T_e \le 1.0$
Y-axis range:
$0.001 \le C_T \le 0.007$
$-0.8 \le \displaystyle{\frac{x }{\zeta}} \le 0.8$
$-0.8 \le \displaystyle{\frac{z }{\zeta}} \le 0.8$
$-4.5 \le \displaystyle{\frac{\phi }{k \zeta}} \le 4.5$
$-0.8 \le \displaystyle{\frac{\theta}{k \zeta}} \le 0.8$
Style:
CFD solid line
EFD open circles
2.11-2-C5 Time histories of total resistance coefficient ($C_T$), surge motion ($x/\zeta_s$), heave motion ($z/\zeta_s$), roll motion($\phi/ k \zeta_s$) 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_2-11.png ( for $C_T$ )
[Identifier]_surge_T-his_C5_2-11.png ( for $x/\zeta_s$ )
[Identifier]_heave_T-his_C5_2-11.png ( for $z/\zeta_s$ )
[Identifier]_roll_T-his_C5_2-11.png ( for $\phi/ k \zeta_s$ )
[Identifier]_pitch_T-his_C5_2-11.png ( for $\theta/ k \zeta_s$ )
X-axis range:
$0.0 \le t / T_e \le 1.0$
Y-axis range:
$0.0030 \le C_T \le 0.0055$
$-0.4 \le \displaystyle{\frac{x }{\zeta}} \le 0.4$
$-0.5 \le \displaystyle{\frac{z }{\zeta}} \le 0.3$
$-0.4 \le \displaystyle{\frac{\phi }{k \zeta}} \le 0.2$
$-0.5 \le \displaystyle{\frac{\theta}{k \zeta}} \le 0.5$
Style:
CFD solid line
EFD open circles

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 $= \displaystyle{\frac{1}{f_e}}$