The flow in Figure 2.23 can go from point A to C and then either back to D or down to E
depending on the downstream boundary conditions. An increase in slope of the bed downstream
from C and no separation would allow the flow to follow the line A to C to E. Similarly the flow
can go from B to C and back to E or up to D depending on boundary conditions. Figure 2.23 is
drawn with the side boundary forming a smooth streamline. If the contraction were due to bridge
abutments, the upstream flow would follow a natural streamline to a vena contracta, but then
downstream, the flow would probably separate. Tranquil approach flow could follow line A-C but
the downstream flow probably would not follow either line C-D or C-E but would have an
undulating hydraulic jump. There would be interaction of the flow in the separation zone and
considerable energy would be lost. If the slope downstream of the abutments was the same as
upstream, then the flow could not be sustained with this amount of energy loss. Backwater would
occur, increasing the depth in the constriction and upstream, until the flow could go through the
constriction and establish uniform flow downstream.
2.6.4 Transitions With Super Critical Flows
Contractions and expansions in rapid flows produce cross wave patterns similar to those
observed in curved channels (Ippen 1950 and Chow 1959). The cross waves are symmetrical
with respect to the centerline of the channel. Ippen and Dawson (1951) have shown that in order
to minimize the disturbance downstream of a contraction, the length of the contraction should be:
W1 - W2
L=
(2.150)
2 tan θ
where W is the channel width and the subscripts 1 and 2 refer to sections upstream and
downstream from the contraction. The contraction angle is θ and should not exceed 12. This
requires a long transition and should not be attempted unless the structure is of primary
importance. A model study should be used to determine transition geometry where a hydraulic
jump is not desired. If a hydraulic jump is acceptable, the inlet structure can be designed using
the procedure in HEC-14, Chapter 4B.
For an expansion, Rouse et al. (1951) found experimentally that the most satisfactory boundary
form is given by:
3/2
1 x
w
1
W Fr
=
+
(2.151)
W1 2 1 1
2
where x is the longitudinal distance measured from the start of the expansion or outlet section
and w is the lateral coordinate measured from the channel centerline. A boundary developed
from this equation diverges indefinitely. Therefore, for practical purposes, the divergent walls are
followed by a transition to parallel lines. A satisfactory straight transition can be created by flaring
the walls so that tan θ = 1/3 Fr. This criteria recommended by Blaisdel (1949) avoids creating an
abrupt expansion.
2.43
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