Mechanics of Fluids

442 Flow with a free surface
Fig. 10.24 Various types of
hydraulic jump in
rectangular channels.
Fig. 10.25
gradually diminishing in size. Such a jump is known as an undular jump (see
Fig. 10.24). For larger values of h2/h1 and Fr1, however, the jump is direct,
that is, the surface does rise fairly abruptly.
At the wave front there is a roller (rather like an ocean wave about to
break on the shore). This results from the upper layers of the wave tending
to spread over the oncoming rapid stream. The frictional drag of the rapid
stream penetrating underneath, and the transfer of momentum from the
lower layers by eddies, however, prevent the upper layers moving upstream.
When h2/h1 is between about 3.0 and 5.5, oscillations may be caused
which result in irregular waves being transmitted downstream. For values of
h2/h1 between 5.5 and 12 the jump is stable and a good dissipator of energy.
The length of the jump (i.e. the horizontal distance between the front of the
jump and a point just downstream of the roller) is usually of the order of
five times its height.
These figures refer only to channels of rectangular section. In other
channels the shape of the jump is often complicated additionally by
cross-currents.
10.10.1 The force applied to obstacles in a stream
The techniques used to study a hydraulic jump may also be used to calculate
the force experienced by an obstacle placed in a moving stream.
Such an obstacle may be a pier of a bridge, a large block placed on the
bed of the stream, a buoy or an anchored boat. If the obstacle illustrated
in Fig. 10.25 exerts of force F on the liquid (in the upstream direction)
then, with the assumptions of a horizontal bed, uniform velocity over
the cross-section, uniform depth across the width and negligible boundary