Test Case 1b.4:Pure Yaw

Conditions

• Captive pure yaw motion in still water
  
• Pitch and heave free; roll fixed
  
• With rudder
  
• With Propeller

Prescribed PMM motions:

• Sway motion : $\eta_{PMM}=-2S_{mm}\sin\left(2\pi \omega t\right)$
• Sway velocity : $v_{PMM}=-2\left(2\pi \omega \right)S_{mm}\cos\left(2\pi \omega t\right)$
• Sway acceleration : $\dot{v}_{PMM}=2\left(2\pi \omega \right)^2S_{mm}\sin\left(2\pi \omega t\right)$
• Heading angle : $\Psi=-\Psi_0\cos\left(2\pi \omega t\right)$
• Yaw rate : $r_{PMM}=\Psi_0\left(2\pi \omega \right)\sin\left(2\pi \omega t\right)$
• Yaw acceleration : $\dot{r}_{PMM}=\Psi_0\left(2\pi \omega \right)^2\cos\left(2\pi \omega t\right)$
• Non-dimensionalized yaw velocity : $r'=\max(r_{PMM})*L_{pp}/U$

where $2S_{mm}$ is the sway amplitude [m] and $\Psi_0$ is the yaw motion amplitude [rad] $\omega$ is the frequency of PMM rotations [Hz].

$F_n$ [-]	$R_n$ [-]	$L_{PP}$ [m]	$U_c$ [m/s]	$n$ [rps]	$\delta$ [deg]	$2S_{mm}$[m]	$\Psi_0$ [deg]	$\omega$ [Hz]	$r'$ [-]
$0.142$	$4.6 \times 10^6$	$5.52$	$1.047$	$8.59$	$0.0$	$1.0$	$13.36$	$0.0389$	$0.30$

Items and Remarks

Figure Number	Items	Remarks
Fig 1b.4-1	Time series of non-dimensionalized X-force ($X'$)	To be compared with experimental results.
Fig 1b.4-2	Time series of non-dimensionalized Y-force ($Y'$)	To be compared with experimental results.
Fig 1b.4-3	Time series of non-dimensionalized Y-aw-moment ($N'$)	To be compared with experimental results.
Fig 1b.4-4	Time series of non-dimensionalized Thrust force ($T'_x$)	To be compared with experimental results.
Fig 1b.4-5	Time series of non-dimensionalized Rudder X-force ($R'_x$)	To be compared with experimental results.
Fig 1b.4-6	Time series of non-dimensionalized Rudder Y-force ($R'_y$)	To be compared with experimental results.

• Coordinate system for comparisons is fixed to midship on the undisturbed water plane. ( see Figure 1)
  
• Froude number $F_n$ and Reynolds number $R_n$ are defined using towing carriage speed ( $U_c$ ) and length between perpendiculars ( $L_{PP}$ ):

  $$F_n = \dfrac{U_c}{\sqrt{g \cdot L_{PP}}}, \quad R_n = \dfrac{U_c \cdot L_{PP}}{\nu}$$

  
  
  where $g$ is the gravitational acceleration and $\nu$ is the kinematic viscosity.
  
• $n$ is the propeller revolution rate [rps] and $\delta$ is rudder angle [deg].
  

• All CFD predicted force coefficients should be reported using the provided ship length $L_{PP}$, mean draft $T_m$ and ship speed $U$. Force coefficients are defined as follows:

  $$X' = \dfrac{X+M(\dot{u}-rv-X_Gr^2-Y_G\dot{r})}{\frac{1}{2} \rho U^2 ... ...Y' = \dfrac{Y+M(\dot{v}+ru-Y_Gr^2+X_G\dot{r})}{\frac{1}{2} \rho U^2 L_{PP} T_m}$$

  $$N' = \dfrac{N+I_z\dot{r}+M(X_G(\dot{v}+ru)-Y_G(\dot{u}-rv))}{\frac{1}{2} \rho U^2 L_{PP}{}^2 T_m}$$

  $$T'_x = \dfrac{T}{\frac{1}{2} \rho U^2 L_{PP} T_m}, \quad R'_x = \dfrac... ...\rho U^2 L_{PP} T_m}, \quad R'_y = \dfrac{Y_R}{\frac{1}{2} \rho U^2 L_{PP} T_m}$$

  
  
  where $M$ is mass of a ship and $r$ is the yaw rate. $N'$ is around the origin of the coordinates.
  

Figure 1: Coordinate system for hydrodynamic forces and moment for dynamic PMM