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CEE 4476b Environmental Hydraulics Design
Instructor: Dr. Andrew Binns Email: abinns2@uwo.ca Office: CMLP 1302 Phone: (519) 661-2111 ext. 88633
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Week 2, Lecture 4 Monday, January 14, 2013
CEE 4476b: Environmental hydraulics design
Today
s lecture
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Discharge diagram ã
Contractions and expansions with head loss ã
Critical depth in non-rectangular channels ã
Examples
Discharge diagram
ã
In addition to transitions in channel bottom elevation, transitions in channel width
b
can also be analyzed by the specific energy (E) concept
Transition from channel width
b
1
to channel width
b
2
Note:
For rectangular channels, it is no longer true that the flow rate per unit width (
q
=
Q
/
b
) remains constant as we saw in the last lecture
Discharge diagram
ã
With negligible energy loss, the energy equation simply states:
But this requires that the flow regime move from one specific energy curve to another that is appropriate for the new value of
q
Discharge diagram
ã
Shown graphically:
Discharge diagram
ã
Shown graphically:
Discharge diagram
ã
Alternative approach:
–
Quantity that remains constant is not
q
, but
E
in a width contraction
ã
Neglecting energy losses ã
Assuming a horizontal channel bottom
Discharge diagram
ã
Discharge function for a given specific energy
E
1
:
q
2
=
2
gy
2
(
E
1
y
)
q
=
y
2
g
(
E
1
y
)
()
1/2
Discharge diagram
ã
There is clearly a unique functional relationship between discharge per unit width
q
and depth
y
for a constant value of specific energy
q
=
y
2
g
(
E
1
y
)
()
1/2
Discharge diagram
ã
Unique relationship between
q
and
y
for constant specific energy
E
Discharge diagram
ã
Decrease in
y
from 1 to 2 corresponds to an increase in
q
in the discharge diagram
Discharge diagram
ã
From the above equation, we can determine the maximum value of the discharge function by setting
dq
/
dy
= 0 ã
Which is the same relationship we determined for critical depth in a rectangular channel
q
=
y
2
g
(
E
1
y
)
()
1/2
y
=
(2/3)
E
1
Discharge diagram
ã
Critical depth
y
c
is not only the depth of minimum specific energy for constant
q
–
But also the depth of maximum discharge for a given specific energy
ã
Note:
The position of point 1 determines the available specific energy and establishes a single discharge diagram for that value of
E
because
y
=
E
when
q
= 0 in the discharge diagram
y
=
(2/3)
E
1
Discharge diagram
ã
Note:
The position of point 1 determines the available specific energy and establishes a single discharge diagram for that value of
E
because
y
=
E
when
q
= 0 in the discharge diagram
Discharge diagram
ã
Note:
The position of point 1 determines the available specific energy and establishes a single discharge diagram for that value of
E
because
y
=
E
when
q
= 0 in the discharge diagram
Choking in contractions
ã
Choking can be caused in a contraction by decreasing the width to a value such that the available
E
no longer is sufficient to pass the flow through the contraction without an increase in upstream depth

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