fluid_mechanics

10.6 Rapidly Varied Flow 559
as 1see Eq. 5.852
where
is obtained from Eqs. 10.23 or 10.25 as
(12
or
Thus, from Eq. 1,
or
h L
p d gQh L gby 1 V 1 h L
jump, the upstream and downstream values of E are different.
b = width = 100 ft
Spillway apron
V 1 = 18 ft/s
y 2
V 2
h L ay 1 V 2 1
2g b ay 2 V 2 2
2g b c 0.60 ft 118.0 ft s2 2
2132.2 fts 2 2 d
c 3.19 ft 13.39 ft s2 2
2132.2 fts 2 2 d
h L 2.26 ft
p d 162.4 lbft 3 21100 ft210.60 ft2118.0 fts212.26 ft2
1.52 10 5 ft # lbs
p d 1.52 105 ft # lbs
55031ft #
277 hp
lbs2hp4
(Ans)
COMMENTS This power, which is dissipated within the
highly turbulent motion of the jump, is converted into an increase
in water temperature, T. That is, T 2 7 T 1 . Although the power
dissipated is considerable, the difference in temperature is not
great because the flowrate is quite large.
By repeating the calculations for the given flowrate Q 1
A 1 V 1 b 1 y 1 V 1 100 ft 10.6 ft2118 fts2 1080 ft 3 s but with
various upstream depths, y 1 , the results shown in Fig. E10.7b are
obtained. Note that a slight change in water depth can produce a
considerable change in energy dissipated. Also, if y 1 7 1.54 ft
the flow is subcritical ( Fr 1 6 1) and no hydraulic jump can occur.
The hydraulic jump flow process can be illustrated by use of the
specific energy concept introduced in Section 10.3 as follows. Equation
10.23 can be written in terms of the specific energy,
E y V 2 2g, as E 1 E 2 h L , where E 1 y 1 V 12 2g
5.63 ft and E 2 y 2 V 2 22g 3.37 ft. As is discussed in
Section 10.3, the specific energy diagram for this flow can be obtained
by using V qy, where
q q 1 q 2 Q b y 1V 1 0.60 ft 118.0 fts2
d , hp
y, ft
1000
800
600
400
200
4
3
2
1
y 1 = 0.60 ft
(0.60 ft, 277 hp)
0
0 0.2 0.4 0.6 0.8
y 1 , ft
(2')
(a)
F I G U R E E10.7
(b)
(2)
(1.54 ft, 0 hp)
1 1.2 1.4 1.6
q = 10.8 ft 2 /s
h L = 2.26 ft
0
0 1 2 3 4 5 6
E 2 = 3.37 E 1 = 5.63
E, ft
(c)
Downstream
obstacles
(1)
Thus,
10.8 ft 2 s
E y
q2
2gy y 110.8 ft2 s2 2
2 2132.2 fts 2 2y y 1.81
2 y 2
where y and E are in feet. The resulting specific energy diagram
is shown in Fig. E10.7c. Because of the head loss across the
In going from state 112 to state 122 the fluid does not proceed
along the specific energy curve and pass through the critical
condition at state 2¿. Rather, it jumps from 112 to 122 as is represented
by the dashed line in the figure. From a one-dimensional
consideration, the jump is a discontinuity. In actuality, the jump
is a complex three-dimensional flow incapable of being represented
on the one-dimensional specific energy diagram.
The actual structure of a hydraulic jump is a complex function of Fr 1 , even though the depth
ratio and head loss are given quite accurately by a simple one-dimensional flow analysis 1Eqs. 10.24
and 10.252. A detailed investigation of the flow indicates that there are essentially five types of
surface and jump conditions. The classification of these jumps is indicated in Table 10.2, along with
sketches of the structure of the jump. For flows that are barely supercritical, the jump is more like
a standing wave, without a nearly step change in depth. In some Froude number ranges the jump is