Page 190 - Introduction to Naval Architecture

176 RESISTANCE

Putting this into words, the wave-making resistances of geometri-
cally similar forms will be in the ratio of their displacements when
their speeds are in the ratio of the square roots of their lengths. This
has become known as Fronde's law of comparison and the quantity
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V/(gL)°' is called the Froude number. In this form it is non-
dimensional. If g is omitted from the Froude number, as it is in the
presentation of some data, then it is dimensional and care must be
taken with the units in which it is expressed. When two geometrically
similar forms are run at the same Froude number they are said to be
run at corresponding speeds.
The other function in the total resistance equation, /}, determines
the frictional resistance. Following an analysis similar to that for the
wave-making resistance, it can be shown that the frictional resistance of
geometrically similar forms will be the same if:

This is commonly known as Rayleigh's law and the quantity VL/v is
called the Reynolds' number. As the frictional resistance is proportional
to the square of the length, it suggests that it will be proportional to
the wetted surface of the hull. For two geometrically similar forms,
complete dynamic similarity can only be achieved if the Froude
number and Reynolds' number are equal for the two bodies. This
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would require V/(gL)°' and VL/v to be the same for both bodies.
This cannot be achieved for two bodies of different size running in
the same fluid.

THE FROUDE NOTATION

In dealing with resistance and propulsion Froude introduced his own
notation. This is commonly called the constant notation or the circular
notation. The first description is because, although it appears very odd
to modern students, it is in fact a non-dimensional system of
representation. The second name derives from the fact that in the
notation the key characters are surrounded by circles.
Froude took as a characteristic length the cube root of the volume of
displacement, and denoted this by U. He then defined the ship's
geometry with the following: