Mechanics of Fluids

The condition for dynamic similarity of flows of this type is that the
magnitude ratio of inertia to gravity forces should be the same at corresponding
points in the systems being compared. The pressure forces, as in
the previous case where viscous forces were involved, are taken care of by
the requirement that the force polygon must be closed. The magnitude of the
inertia force on a fluid particle is, as shown in Section 5.3.1, proportional
to ϱu 2 l 2 where ϱ represents the density of the fluid, l a characteristic length
and u a characteristic velocity. The gravity force on the particle is its weight,
that is, ϱ (volume) g which is proportional to ϱl 3 g where g represents the
acceleration due to gravity. Consequently the ratio
|Inertia force|
|Gravity force| is proportional to ϱl2u2 ϱl3 u2
=
g lg
In practice it is often more convenient to use the square root of this ratio so
as to have the first power of the velocity. This is quite permissible: equality
of u/(lg) 1/2 implies equality of u 2 /lg.
Ratios of forces arising in dynamic similarity 165
The ratio u/(lg) 1/2 is known as the Froude number after William Froude Froude number
(1810–79), a pioneer in the study of naval architecture, who first introduced
it. Some writers have termed the square of this the Froude number, but the
definition Froude number = u/(lg) 1/2 is now more usual.
Dynamic similarity between flows of this type is therefore obtained by
having values of Froude number (based on corresponding velocities and corresponding
lengths) the same in each case. The boundaries for the flows must,
of course, be geometrically similar, and the geometric scale factor should be
applied also to depths of corresponding points below the free surface.
Gravity forces are important in any flow with a free surface. Since the
pressure at the surface is constant (usually atmospheric) only gravity forces
can under steady conditions cause flow. Moreover, any disturbance of the
free surface, such as wave motion, involves gravity forces because work must
be done in raising the liquid against its weight. The Froude number is thus a
significant parameter in determining that part of a ship’s resistance which is
due to the formation of surface waves.
5.3.3 Dynamic similarity of flow with surface tension forces acting
In most examples of flow occurring in engineering work, surface tension
forces are negligible compared with other forces present, and the engineer
is not often concerned with dynamic similarity in respect to surface tension.
However, surface tension forces are important in certain problems such as
those in which capillary waves appear, in the behaviour of small jets formed
under low heads, and in flow of a thin sheet of liquid over a solid surface.
Here the significant force ratio is that of |Inertia force| to |Surface tension
force|. Again, pressure forces, although present, need not be separately considered.
The force due to surface tension is tangential to the surface and
has the same magnitude perpendicular to any line element along the surface.
If the line element is of length �l then the surface tension force is γ(�l)