Practical Ship Hydrodynamics

158 Practical Ship Hydrodynamics
5.2.3 Physical explanation and force estimation
In the following, forces due to non-zero rudder angles are not considered. If
the rudder at the midship position is treated as part of the ship’s body, only
the difference between rudder forces at the actual rudder angle υ and those
at υ D 0° have to be added to the body forces treated here. The gap between
ship stern and rudder may be disregarded in this case. Propeller forces and
hull resistance in straightforward motion are neglected here.
We use a coordinate system with origin fixed at the midship section on
the ship’s centre plane at the height of the centre of gravity (Fig. 5.1). The
x-axis points forward, y to starboard, z vertically downward. Thus the system
participates in the motions u, v,androf the ship, but does not follow the ship’s
heeling motion. This simplifies the integration in time (e.g. by a Runge–Kutta
scheme) of the ship’s position from the velocities u, v, r and eliminates several
terms in the force formulae.
r,
N
y, v, Y
x, u,
X
Figure 5.1 Coordinates x, y; direction of velocities u, v, r, forces X, Y, and moments K, N
Hydrodynamic body forces can be imagined to result from the change of
momentum (Dmass Ð velocity) of the water near to the ship. Most important in
manoeuvring is the transverse force acting upon the hull per unit length (e.g.
metre) in the x-direction. According to the slender-body theory, this force is
equal to the time rate of change of the transverse momentum of the water
in a ‘strip’ between two transverse planes spaced one unit length. In such a
‘strip’ the water near to the ship’s side mostly follows the transverse motion
of the respective ship section, whereas water farther from the hull is less
influenced by transverse ship motions. The total effect of this water motion
on the transverse force is the same as if a certain ‘added mass’ per length m 0
moved exactly like the ship section in transverse direction. (This approach is
thus similar to the strip method approach in ship seakeeping.)
The added mass m 0 maybedeterminedforanyshipsectionas:
m 0 D 1
2 Ð Ð T 2 x Ð cy
Tx is the section draft and cy acoefficient.cy may be calculated:
ž analytically if we approximate the actual ship section by a ‘Lewis
section’(conformal mapping of a semicircle); Fig. 5.2 shows such solutions
for parameters (Tx/B) andˇ D immersed section area/⊲B Ð Tx)
ž for arbitrary shape by a close-fit boundary element method as for ‘strips’ in
seakeeping strip methods, but for manoeuvring the free surface is generally
neglected
ž by field methods including viscosity effects
f, K