Practical Ship Hydrodynamics

246 Practical Ship Hydrodynamics
Once a potential has been determined, the forces can be determined by
direct pressure integration on the wetted hull. The forces are corrected by the
hydrostatic forces at rest. (The hydrostatic x force and y moment should be
zero, but are non-zero due to discretization errors. The discretization error is
hoped to be reduced by subtracting the value for the hydrostatic force):
� �
f1 D pn1 dS psn1 dS
S
�
S0
�
f3 D pn3 dS psn3 dS
S
�
S0
�
f5 D p⊲zn1 xn3⊳ dS ps⊲zn1 xn3⊳ dS
S
S0
S is the actually wetted surface. S0 is the wetted surface of the ship at rest.
ps D gz is the hydrostatic pressure, where is the density of water. p is
the pressure determined from Bernoulli’s equation:
p D 2 ⊲V 2
⊲r ⊳ 2 ⊳ gz
The force in the x direction, f1, is the (negative) wave resistance. The nondimensional
wave resistance coefficient is:
��
CW D f1
2 V2 �
S
The z force and y moments are used to adjust the position of the ship. We
assume small changes of the position of the ship. 1z is the deflection of the
ship (positive, if the ship surfaces) and 1 is the trim angle (positive if bow
immerses) (Fig. 7.2).
q
z
x
f 5
f 3
f 1
Figure 7.2 Coordinate system; x points towards bow, origin is
usually amidships in still waterline; relevant forces and moment
For given 1z and 1 , the corresponding z force and y moment (necessary
to enforce this change of position) are:
� �
�� �
f3
1z
D
1
f5
� AWL Ð Ð g AWL Ð Ð g Ð xWL
AWL Ð Ð g Ð xWL IWL Ð Ð g
AWL is the area, IWL the moment of inertia, and xWL the centre of the still waterplane.
IWL and xWL are taken relative to the origin which we put amidships.
Inversion of this matrix gives an equation of the form:
� � � �� �
1z a11 a12 f3
D
1
a21 a22
f5
The coefficients aij are determined once in the beginning by inverting the
matrix for the still waterline. Then during each iteration the position of the ship