Page 227 - Introduction to Naval Architecture

212 PROPULSION

It is assumed that the water is initially, and finally, at pressure p 0. At
the actuator disc it receives an incremental pressure increase dp. The
water is initially at rest, achieves a velocity aV a at the disc, goes on
accelerating and finally has a velocity bV & at infinity behind the disc.
The disc is moving at a velocity V^ relative to the still water. Assuming
the velocity increment is uniform across the disc and only the column
of water passing through the disc is affected:

Since this mass finally achieves a velocity bV a, the change of momentum
in unit time is:

Equating this to the thrust generated by the disc:

The work done by the thrust on the water is:

This is equal to the kinetic energy in the water column,

Equating this to the work done by the thrust:

That is half the velocity ultimately reached is acquired by the time the
water reaches the disc. Thus the effect of a propulsor on the flow
around the hull, and therefore the hull's resistance, extends both
ahead and astern of the propulsor.
The useful work done by the propeller is equal to the thrust
multiplied by its forward velocity. The total work done is this plus the
work done in accelerating the water so: