This increases towards unity as v approaches V. In other words, the
less the velocity of the jets exceeds that of the ship, and therefore
the greater the area of the orifice of discharge, the greater is the
efficiency of the propeller.
In the "Waterwitch" v was about twice V. Hence in this case the
theoretical efficiency of the propeller, friction neglected, was about
2/3.
[Illustration: FIG. 165.]
§ 164. _Pressure of a Steady Stream in a Uniform Pipe on a Plane
normal to the Direction of Motion._--Let CD (fig. 165) be a plane
placed normally to the stream which, for simplicity, may be supposed
to flow horizontally. The fluid filaments are deviated in front of the
plane, form a contraction at A1A1, and converge again, leaving a mass
of eddying water behind the plane. Suppose the section A0A0 taken at a
point where the parallel motion has not begun to be disturbed, and
A2A2 where the parallel motion is re-established. Then since the same
quantity of water with the same velocity passes A0A0, A2A2 in any
given time, the external forces produce no change of momentum on the
mass A0A0A2A2, and must therefore be in equilibrium. If [Omega] is the
section of the stream at A0A0 or A2A2, and [omega] the area of the
plate CD, the area of the contracted section of the stream at A1A1
will be c_c([Omega] - [omega]), where c_c is the coefficient of
contraction. Hence, if v is the velocity at A0A0 or A2A2, and v1 the
velocity at A1A1,
v[Omega] = c_c v([Omega] - [omega]);
.:v1 = v[Omega]/c_c ([Omega] - [omega]). (1)
Let p0, p1, p2 be the pressures at the three sections. Applying
Bernoulli's theorem to the sections A0A0 and A1A1,
p0 v² p1 v1²
-- + -- = -- + ---.
G 2g G 2g
Also, for the sections A1A1 and A2A2, allowing that the head due to
the relative velocity v1 - v is lost in shock:--
p1 v1² p2 v² (v1 - v)²
-- + --- = -- + -- + ---------;
G 2g G 2g 2g
.: p0 - p2 = G(v1 - v)²/2g; (2)
or, introducing the value in (1),
G / [Omega] \²
p0 - p2 = -- ( ----------------------- - 1 ) v² (3)
2g \c_c ([Omega] - [omega]) /
Now the external forces in the direction of motion acting on the mass
A0A0A2A2 are the pressures p0[Omega]1 - p2[Omega] at the ends, and the
reaction -R of the plane on the water, which is equal and opposite to
the pressure of the water on the plane. As these are in equilibrium,
(p0 - p2)[Omega] - R = 0;
/ [Omega] \² v²
.: R = G[Omega] ( ----------------------- - 1 ) --; (4)
\c_c ([Omega] - [omega]) / 2g
an expression like that for the pressure of an isolated jet on an
indefinitely extended plane, with the addition of the term in
brackets, which depends only on the areas of the stream and the plane.
For a given plane the expression in brackets diminishes as [Omega]
increases. If [Omega]/[omega] = [rho], the equation (4) becomes
_ _
v² | / [rho] \² |
R = G[omega] -- |[rho] ( --------------- - 1 ) |, (4a)
2g |_ \c_c ([rho] - 1) / _|
which is of the form
R = G[omega](v²/2g)K,
where K depends only on the ratio of the sections of the stream and
plane.
For example, let c_c = 0.85, a value which is probable, if we allow
that the sides of the pipe act as internal borders to an orifice. Then
/ [rho] \²
K = [rho] ( 1.176 --------- - 1 ).
\ [rho] - 1 /
[rho] = K =
1 [infinity]
2 3.66
3 1.75
4 1.29
5 1.10
10 .94
50 2.00
100 3.50
The assumption that the coefficient of contraction c_c is constant for
different values of [rho] is probably only true when [rho] is not very
large. Further, the increase of K for large values of [rho] is
contrary to experience, and hence it may be inferred that the
assumption that all the filaments have a common velocity v1 at the
section A1A1 and a common velocity v at the section A2A2 is not true
when the stream is very much larger than the plane. Hence, in the
expression
R = KG[omega]v²/2g,
K must be determined by experiment in each special case. For a
cylindrical body putting [omega] for the section, c_c for the
coefficient of contraction, c_c([Omega] - [omega]) for the area of the
stream at A1A1,
v1 = v[Omega]/c_c([Omega] - [omega]); v2 = v[Omega]/([Omega] - [omega]);
or, putting [rho] = [Omega]/[omega],
v1 = v[rho]/c_c ([rho] - 1), v2 = v[rho]/([rho] - 1).