Hydraulic Structures: Fourth Edition

(Amelia) #1
whereis the increased value of the Coriolis coefficient reflecting the
high degree of turbulence and uneven velocity distribution; 25 for
3 Fr 1 10, while 1.
From equations (5.11) and (5.12) we obtain

 1  1 4( 1). (5.13)


Equation (5.13) shows that the efficiency of energy dissipation in the jump
itself within the stilling basin decreases with the Froude number, leaving
up to 50% of the energy to be dissipated downstream of the basin at low
Froude numbers (Section 5.3.3).
The hydraulic jump entrains a substantial amount of air additional
to any incoming aerated flow. A constant air concentration throughout
the jump (C ̄ 1 C ̄ 2 ) results in a lower height of the jump than for the
case without air, while for C ̄ 1 0 and C ̄ 2  0 (which is a more realistic
assumption) a slightly higher y 2 is needed than for no air (Naudascher,
1987). Thus the main significance of the presence of air in the jump region
is the requirement of higher stilling basin side walls due to the higher depth
of flow (equation (4.33)). The effect of air entrainment by hydraulic jumps
on oxygen concentration in the flow is briefly discussed in Section 9.1.7.
The highly turbulent nature of the flow in the hydraulic jump induces
large pressure fluctuations on the side walls and particularly on the floor of
the basin which, in turn, could lead to cavitation. Using a cavitation
number,, in the form (p 
2
)1/2/(1/2 V^21 ) (equation (4.14)), where pis the
deviation of the instantaneous pressure pfrom the time-averaged pressure
p ̄(pf(t) can be obtained from pressure transducer records), the relation-
ship between andx/y 1 (wherexis the distance from the toe of the jump)
for a free and submerged jump at Fr 1 5 is shown in Fig. 5.5 (Narayanan,
1980; Locher and Hsu, 1984). Assuming the length of the jump to be
approximately 6(y 2 y 1 ), the hydrostatic pressure at the point of maximum
pressure fluctuations, i.e. in a free jump at x/y 1 12, will be gy ̄, with

y ̄y 1  12 y 1  3 y 1.

For0.05, cavitation will occur if

p 0  gy ̄ 0.05k pv 0 (5.14)

wherekp/(p  
2
)1/21(1k5).
The value of kcan be computed from equation (5.14) and assuming,
for example, a normal distribution of pressure fluctuations, the intermit-
tency factor, i.e. the proportion of time for which kis exceeded (the
probability of occurrence of cavitation) can be computed (worked example

V^21



2

y 2 y 1

6(y 2 y 1 )

1 (1 8 Fr^21 )1/2

[ 3 (1 8 Fr^21 )1/2]^3

e 5

e4,5

e 4

e4,5

252 ENERGY DISSIPATION

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