fluid_mechanics

10.6 Rapidly Varied Flow 557
and the energy equation 1Eq. 5.842
V10.11 Hydraulic
jump in a river
y 1 V 2 1
2g y 2 V 2 2
2g h L
(10.23)
The head loss, h L , in Eq. 10.23 is due to the violent turbulent mixing and dissipation that occur
within the jump itself. We have neglected any head loss due to wall shear stresses.
Clearly Eqs. 10.21, 10.22, and 10.23 have a solution y 1 y 2 , V 1 V 2 , and h L 0. This
represents the trivial case of no jump. Since these are nonlinear equations, it may be possible that
more than one solution exists. The other solutions can be obtained as follows. By combining Eqs.
10.21 and 10.22 to eliminate V 2 we obtain
which can be simplified by factoring out a common nonzero factor y 1 y 2 from each side to give
where
2
y 1
2 y2 2
2 V 1y 1
g aV 1y 1
V
y 1 b V 2 1y 1
1y
2 gy 1 y 2 2
2
a y 2
2
b a y 2
b 2 Fr 2
y 1 y 1 0
1
Fr 1 V 1 1gy 1
is the upstream Froude number. By using the quadratic formula we obtain
The depth ratio
across a hydraulic
jump depends on
the Froude number
only.
y 2
y 1
1 2 11 21 8Fr 1 2 2
Clearly the solution with the minus sign is not possible 1it would give a negative y 2y 1 2. Thus,
y 2
1 y 1 2 11 21 8Fr 1
2 2
(10.24)
This depth ratio, y 2y 1 , across the hydraulic jump is shown as a function of the upstream Froude number
in Fig. 10.16. The portion of the curve for Fr 1 6 1 is dashed in recognition of the fact that to have a
hydraulic jump the flow must be supercritical. That is, the solution as given by Eq. 10.24 must be
restricted to Fr 1 1, for which y 2y 1 1. This can be shown by consideration of the energy equation,
Eq. 10.23, as follows. The dimensionless head loss, h L y 1 , can be obtained from Eq. 10.23 as
h L
1 y 2
Fr 1 2
2
y 1 y 1 2 c 1 ay 1
b d
y 2
(10.25)
where, for given values of Fr 1 , the values of y 2y 1 are obtained from Eq. 10.24. As is indicated in
Fig. 10.16, the head loss is negative if Fr 1 6 1. Since negative head losses violate the second law
4
3
__
y 2
y1
__
y 2
y1
2
or
h L __ y1
1
No jump
possible
h L __ y1
0
–1
0 1 2 3 4
Fr 1 =
V 1 ______
√gy 1
F I G U R E 10.16 Depth ratio and dimensionless
head loss across a hydraulic jump as a function of
upstream Froude number.