Boats and Things That Float - Physics for Game Programmers - page 250

The ratio of the wake velocity to the boat velocity is called the wake fraction , w f .

vvv

v −

w b a

w ==

(9.8)

f

v

b b

The boat velocity and speed of advance of the propeller can be related using the wake
fraction.

v =− (1 w ) v

(9.9)

a f b

In looking at Equation (9.6), we see it doesn't really provide the velocity of the boat because
the wake velocity at a given point in time is also required. A variation of Equation (9.6) exists that
relates the boat velocity to the turnover rate of the propeller. The quantity, s a , in Equation (9.10)
is called the apparent slip ratio .

( )

v =− 1 s Pn (9.10)

b a

A final note about propeller geometry is that propellers generally have two, three, four, or
five blades. Having fewer blades increases the forces experienced by each blade, known as the
blade loading . If the blade loading is too high, the performance and even structural integrity of
the propeller can be compromised. Increasing the number of blades shifts the loading from the
blades to the engine, possibly preventing the engine from producing its theoretical maximum
power.

Thrust
The thrust of a boat due to the propellers can be analyzed in terms of power. The engine gener-
ates a certain power level, P e , which will typically not be constant, but will be a function of the
turnover rate of the engine. This power is transferred through shafts, gearboxes, and other
mechanical devices to the propeller. There are losses along the way, so the power that is avail-
able to the propeller, P p , is some fraction of the engine power.

PP = h

(9.11)

p s e

The shaft transmission efficiency , h s , in Equation (9.11) will generally have a value fairly
close to 1.
When the boat moves through the water a certain velocity, v b , a certain power level called
the effective power , P eff , is necessary to overcome the resistance, R , experienced by the boat at
that velocity. The effective power is equal to the resistance acting on the boat multiplied by the
boat velocity.

P = v (9.12)

eff b

The effective power can be related to the engine power using the coefficient of propulsive
efficiency , h p .

P = h P (9.13)

eff p e

Typical values for the coefficient of propulsive efficiency range from 0.5 to 0.7. 1
Equations (9.12) and (9.13) can be used to estimate the maximum velocity for a boat. If the overall