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.

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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