Mechanics of Fluids

gives rise to waves on the surface. The formation of these waves requires
energy, and, since this energy must be derived from the motion, the ship
experiences an increased resistance to its passage through the liquid.
Waves on the surface of a liquid may be of two kinds. Those of the first
type are due to surface tension forces, and are known as capillary waves or
ripples; they are of little importance except for bodies that are small in size
compared with the waves. In the case of ship resistance it is waves of the
second type that are important. These result from the action of gravity on
the water that tends to accumulate around the sides of the hull. Usually two
main sets of waves are produced, one originating at the bow and the other
at the stern of the ship. These both diverge from each side of the hull, and
there are also smaller waves whose crests are perpendicular to the direction
of motion.
In the formation of waves, some water is raised above the mean level, while
some falls below it. When particles are raised work must be done against their
weight, so gravity, as well as viscosity, plays a part in the resistance to motion
of a ship. In a dimensional analysis of the situation the quantities considered
must therefore include the gravitational acceleration g. The complete list of
relevant quantities is therefore the total resistance force F, the velocity of the
ship u, the dynamic viscosity of the liquid µ, its density ϱ, some characteristic
length l to specify the size of the ship (the overall length, for example) and g.
As already indicated, surface tension forces are negligible (except for models
so small as to be most suitable as children’s bath-time toys). We assume that
the shape is specified, so that quoting a single length is sufficient to indicate
all other lengths for a particular design. The geometric similarity of a model
should extend to the relative roughness of the surfaces if complete similarity
is to be achieved. (Usually the depth of water is sufficiently large compared
with the size of the ship not to affect the resistance. Where this is not so,
however, the depth of the water must be included among the lengths that
have to satisfy geometric similarity. Likewise, the distance from a boundary,
such as a canal bank, might have to be considered.) The most suitable form
of the result of dimensional analysis is
F = ϱu 2 l 2 �
ϱul
φ
µ , u2
�
lg
that is
F = ϱu 2 l 2 φ{Re, (Fr) 2 } (5.19)
The total resistance therefore depends in some way both on the Reynolds
number and on the Froude number. We remember that dimensional analysis
tells us nothing about either the form of the function φ{ }, or how Re and
Fr are related within it. For complete similarity between a prototype and its
model the Reynolds number must be the same for each, that is
ϱpuplp
µp
= ϱmumlm
µm
(5.20)
Ship resistance 183