Test Case 4

--- Diffraction condition ---

DTMB5415 [Geometry]

Conditions

• Towing condition in head waves
• Without rudder
• Fixed at the dynamic trim
• sinkage: FP = -0.0031 L_PP , AP = -0.0007 L_PP
• Incident wave length λ = 1.5 L_PP and wave steepness: A_k = 0.025

Froude number ( Fn )	Reynolds number ( Rn )
0.28	4.861×106

Items and Remarks

Figure Number	Items	Remarks
	Integral variables: Coefficients of total resistance ( CT ) , heave force ( CH ) , pitching Moment ( CM ) 0th and 1st Amplitudes and 1st Phase for each hull force	experimental data is available. download(IIHR)
Fig. 4-1	Time histories of CT, CH, and CM	experimental data is available. CT (Ct_hist.dat) CH (Ch_hist.dat) CM (Cm_hist.dat)
Fig. 4-2	Wave-elevation ζT contours: t / Te = 0, 1/4, 1/2, 3/4
Fig. 4-3	Wave-elevation ζT contours: 0th amplitude
Fig. 4-4	Wave-elevation ζT contours: 1st amplitude
Fig. 4-5	Wave-elevation ζT contours: 1st phase
Fig. 4-6	u contours at x / LPP = 0.435 0.935, and t / Te = 0, 1/4, 1/2, 3/4
Fig. 4-7	v contours at x / LPP = 0.435 0.935, and t / Te = 0, 1/4, 1/2, 3/4
Fig. 4-8	w contours at x / LPP = 0.435 0.935, and t / Te = 0, 1/4, 1/2, 3/4
Fig. 4-9	Contours of 0th and 1st amplitudes and 1st phase of u at x / LPP = 0.435 0.935	ADOPTED(01/NOV/2004)
Fig. 4-10	Contours of 0th and 1st amplitudes and 1st phase of v at x / LPP = 0.435 0.935	ADOPTED(01/NOV/2004)
Fig. 4-11	Contours of 0th and 1st amplitudes and 1st phase of w at x / LPP = 0.435 0.935	ADOPTED(01/NOV/2004)

• Coordinate system for comparisons is fixed to midship FP on the undisturbed waterplane.
• All quantities are non-dimensionalized by density of water ( $\rho$ ), ship speed ( $U$ ), and length between perpendiculars ( $L_{PP}$ ):

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

  
  
  where $g$ is the gravitational acceleration and $\nu$ is the kinematic viscosity.
• All CFD predicted force coefficients should be reported using the provided ( Geometry page ) wetted surface area at rest ( $S_{0}$ ). Force coefficients are defined as follows:

  $$C_{T}(t) = \dfrac{F_{x}(t)}{\frac{1}{2} \rho U^2 S_{0}}, \quad C_{H}(t) =... ...2 S_{0}}, \quad C_{M}(t) = \dfrac{M_{y}(t)}{\frac{1}{2} \rho U^2 S_{0} L_{PP}}$$

  
  
  where $F_x, F_z, M_y$ are the total resistance, the heave force, and the pitching moment. The center of the pitching moment is $\left( x/L_{PP} =$ 0.5 $, z/L_{PP} = 0.031583 \right)$ .

• Symbols:
  

  $A$	Incident wave amplitude
  $A_k$	Wave steepness $A_k = \dfrac{2 \pi A}{\lambda}$
  $f_w$	Frequency of the incident wave, $\sqrt{\dfrac{g}{2 \pi \lambda}}$
  $f_e$	Encounter frequency, $f_w+\dfrac{U}{\lambda}$
  $T_e$	Encounter period $\dfrac{1}{f_e}$
  $t$	Time, at t=0 a crest of the incident wave is coincident with the FP of the ship
  $U$	Ship speed
  $u,v,w$	Velocity components in $x,y,z$ directions, respectively
  $\lambda$	Incident wave length
  $\zeta_T(x,y,t)$	Free surface elevation $\dfrac{z(x, y, t)}{L_{PP}}$

  
  

• Instructions

  $$x =$$

  $$\zeta_I (t) = \dfrac{A}{L_{PP}} \cos \left( 2 \pi f_e t + \gamma_I \right)$$

  $$\gamma_I t = 0 X ( X = C_T, C_H, C_M, \zeta_T )$$

  $$X_F (t) = \dfrac{X_0}{2} + \sum_{n=1}^N X_n \cos \left( 2 \pi n f_e t + \Delta \gamma_n \right)$$

  $$\Delta \gamma_n = \gamma_n - \gamma_I$$

  $$a_n = \dfrac{2}{T} \int_0^T X(t) \cos \left( 2 \pi n f_e t \right) dt \quad ( n = 0, 1, 2, \cdots )$$

  $$b_n = \dfrac{2}{T} \int_0^T X(t) \sin \left( 2 \pi n f_e t \right) dt \quad ( n = 1, 2, \cdots )$$

  $$X_n = \sqrt{a_n{}^2 + b_n{}^2}$$

  $$\gamma_n = \tan{}^{-1} \left( - \dfrac{b_n}{a_n} \right)$$

  $$X_n n \gamma_n$$