fluid_mechanics

540 Chapter 10 ■ Open-Channel Flow
to the observer. Viscous effects, which have been neglected in this discussion, will eventually damp
out such waves far upstream. Such flow conditions, V 6 c, or Fr 6 1, are termed subcritical.
On the other hand, if the stream is moving rapidly so that the flow velocity is greater than
the wave speed 1i.e., V 7 c2, no upstream communication with downstream locations is possible.
Any disturbance on the surface downstream from the observer will be washed farther downstream.
Such conditions, V 7 c or Fr 7 1, are termed supercritical. For the special case of V c or
Fr 1, the upstream propagating wave remains stationary and the flow is termed critical.
E XAMPLE 10.1
GIVEN At a certain location along the Rock River shown in
Fig. E10.1a, the velocity, V, of the flow is a function of the depth,
y, of the river as indicated in Fig. E10.1b. A reasonable approximation
to these experimental results is
V 5 y 2 3
(1)
where V is in ft/s and y is in ft.
FIND For what range of water depth will a surface wave on the
river be able to travel upstream?
14
SOLUTION
While the river travels to the left with speed V, the surface wave
travels upstream (to the right) with speed c 1g y2 1 2
relative to
the water (not relative to the ground). Hence relative to the stationary
ground, the wave travels to the right with speed
For the wave to travel upstream, c V 7 0 so that from Eq. 2,
or
F I G U R E E10.1a
c V 1g y2 12 5 y 2 3
132.2 fts 2 y2 12 5 y 2 3
132.2 y2 1 2 7 5 y 2 3
y 6 2.14 ft
(2)
(Ans)
COMMENT As shown above, if the river depth is less than
2.14 ft, its velocity is less than the wave speed and the wave can
travel upstream against the current. This is consistent with the fact
that if a wave is to travel upstream, the flow must be subcritical (i.e.,
Fr Vc 6 1). For this flow
Fr Vc 15 y 23 21g y2 1 2
5 y 1 6
132.2 fts 2 2 1 2
0.881 y 1 6
This result is plotted in Fig. E10.1c. Note that in agreement with
the above answer, for y 6 2.14 the flow is subcritical; the wave
can travel upstream.
V
Fr
V, ft/s
12
10
V = 5y 2/3
8
6
4
Measured values
2
0
0 1 2 3 4
y, ft
F I G U R E E10.1b
1.2
1
(2.14,1)
0.8
0.6
0.4
0.2
0
0 1 2 3 4
y, ft
F I G U R E E10.1c
The character of an open-channel flow may depend strongly on whether the flow is subcritical
or supercritical. The characteristics of the flow may be completely opposite for subcritical flow
than for supercritical flow. For example, as is discussed in Section 10.3, a “bump” on the bottom