Practical Ship Hydrodynamics

esistance R is proportional to speed u:
R
D k
R0
D 2
u U0
Ship manoeuvring 165
k is the stopping constant. The index 0 denotes the values at the beginning
of the manoeuvre. If we assume a straight stopping track, the fundamental
equation of motion is:
m Ð u D ⊲u 2 C u 2 T ⊳
The mass m includes the hydrodynamic added mass m 00 for longitudinal motion
which may be estimated by the empirical formula:
m 00
m0
D
1
�
⊲L 3 /r⊳ 14
p
Here r is the volume displacement. uT D U0 T/R0 is the speed the ship
would have after a long time if the thrust T would be directed forward. The
above differential equation can be solved (by separation of variables) to yield:
1t D t2 t1 D m
�
arctan
kuT
u1
arctan
uT
u2
�
uT
The distance is given by multiplying the above differential equation by u D
ds/dt and solving again (by separation of variables) to yield:
1s D s2 s1 D m
2k ln
�
2
u1 C u 2 �
T
u 2 2 C u2 T
These two equations for 1t and 1s can be used to compute stepwise the
stopping process by splitting the process into time intervals where the thrust T
can be assumed to be constant. The reverse thrust is best determined by using
four-quadrant diagrams for the propellers, if these diagrams are available.
5.2.7 Jet thrusters
Transverse jet thrusters consist of a transverse pipe through the ship hull
located usually at the bow or at the stern. The pipe contains a screw propeller
which pumps water either to port or starboard thus creating a side thrust (and
moment). The purpose of a jet thruster is to increase manoeuvrability at low
speeds, allowing the ship to manoeuvre with no or less tug assistance. As
the rudder astern already supplies manoeuvring forces, jet thrusters are more
effective at the bow and usually placed there (‘bow thrusters’). Also stern jets
need to cope with potential collision problems in arranging both jet pipe and
propeller shaft. Jet thrusters may also serve as an emergency backup for the
main rudder. Backups for rudders are needed for ships with dangerous cargo.
Jet thrusters are less attractive for ships on long-distance routes calling at few
ports. The savings in tug fees may be less than the additional expense for fuel.
For ocean going ships, side thrusts of 0.08–0.12 kN per square metre underwater
lateral area are typical values. These values relate to zero forward speed