bae 6333 – fluvial hydraulics › media › lectures › 5 › 5... · bae 6333 – fluvial...
TRANSCRIPT
![Page 1: BAE 6333 – Fluvial Hydraulics › media › lectures › 5 › 5... · BAE 6333 – Fluvial Hydraulics Nonuniform Flow Chapter 4 (Graf, 1998) • Two Types of Nonuniform Flow: –Gradually](https://reader034.vdocuments.net/reader034/viewer/2022042402/5f128471bef5994e5157e74e/html5/thumbnails/1.jpg)
BAE 6333 – Fluvial Hydraulics
Nonuniform Flow
Chapter 4 (Graf, 1998)
![Page 2: BAE 6333 – Fluvial Hydraulics › media › lectures › 5 › 5... · BAE 6333 – Fluvial Hydraulics Nonuniform Flow Chapter 4 (Graf, 1998) • Two Types of Nonuniform Flow: –Gradually](https://reader034.vdocuments.net/reader034/viewer/2022042402/5f128471bef5994e5157e74e/html5/thumbnails/2.jpg)
• Two Types of Nonuniform Flow:
– Gradually Varied Flow – changing conditions extend
over a long distance
– Rapidly Varied Flow – changing flow conditions occur
abruptly
Nonuniform Flow
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Gradually Varied Flow
• Consider prismatic channel with steady,
non-uniform flow:
![Page 4: BAE 6333 – Fluvial Hydraulics › media › lectures › 5 › 5... · BAE 6333 – Fluvial Hydraulics Nonuniform Flow Chapter 4 (Graf, 1998) • Two Types of Nonuniform Flow: –Gradually](https://reader034.vdocuments.net/reader034/viewer/2022042402/5f128471bef5994e5157e74e/html5/thumbnails/4.jpg)
Saint-Venant Equations
• Combined equation of continuity and
equation of energy…
0
0
0
x
hU
x
Uh
x
AU
x
UA
x
Q
t
hB
x
Q
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Saint-Venant Equations
• Combined equation of continuity and
equation of energy…
h
ef
hh
ef
RC
AQSS
dx
dh
g
AQ
dx
d
g
U
RC
g
g
U
RfS
dx
dHS
dx
dz
dx
dH
dx
dz
dx
dh
g
U
dx
d
zhg
UH
2
22
2
2
2
2
2
)/(
2
)/(
24
18
24
1
2
2
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Saint-Venant Equations
• Combined equation of continuity and
equation of energy…
dx
dh
dh
dA
gA
Q
dx
dA
Ag
Q
hAg
Q
dx
d
g
AQ
dx
d
hA
Prismatic:
RC
AQSS
dx
dh
g
AQ
dx
d
h
ef
3
2
3
2
2
22
2
22
2
2)(22
)/(
)(f
)/(
2
)/(
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Saint-Venant Equations
• Combined equation of continuity and
equation of energy…
3
2
2
2
3
2
2
2
2
2
3
2
1
/1
1
/
)/(
gA
BQ
SRC
AQ
S
gA
BQ
RC
AQS
dx
dh
Prismatic:
RC
AQS
dx
dh
dx
dhB
gA
Q
fh
f
h
f
h
f
![Page 8: BAE 6333 – Fluvial Hydraulics › media › lectures › 5 › 5... · BAE 6333 – Fluvial Hydraulics Nonuniform Flow Chapter 4 (Graf, 1998) • Two Types of Nonuniform Flow: –Gradually](https://reader034.vdocuments.net/reader034/viewer/2022042402/5f128471bef5994e5157e74e/html5/thumbnails/8.jpg)
Wide and Rectangular Channel
• Further simplify for wide and rectangular
channel…
– Normal Depth:
f
nfn
fnn
fnn
SC
qhShCq
ShChq
B
ShBhCUAQ
2
23322
)(
![Page 9: BAE 6333 – Fluvial Hydraulics › media › lectures › 5 › 5... · BAE 6333 – Fluvial Hydraulics Nonuniform Flow Chapter 4 (Graf, 1998) • Two Types of Nonuniform Flow: –Gradually](https://reader034.vdocuments.net/reader034/viewer/2022042402/5f128471bef5994e5157e74e/html5/thumbnails/9.jpg)
Wide and Rectangular Channel
• Further simplify for wide and rectangular
channel…
– Critical Depth:
g
qhghhq
ghh
q
ghBh
Q
ghA
Q
gD
UFr
ccc
ccccch
2322
1
![Page 10: BAE 6333 – Fluvial Hydraulics › media › lectures › 5 › 5... · BAE 6333 – Fluvial Hydraulics Nonuniform Flow Chapter 4 (Graf, 1998) • Two Types of Nonuniform Flow: –Gradually](https://reader034.vdocuments.net/reader034/viewer/2022042402/5f128471bef5994e5157e74e/html5/thumbnails/10.jpg)
Wide and Rectangular Channel
• Using Chezy Equation…
3
3
3
2
23
2
3
2
232
2
3
2
22
2
1
1
1
1
)(1
1
1
1
h
h
h
h
S
gh
q
SCh
q
Sdx
dh
Bhg
BQ
SChB
Q
S
gA
BQ
hSCA
Q
Sdx
dh
c
n
f
f
f
f
f
f
f
![Page 11: BAE 6333 – Fluvial Hydraulics › media › lectures › 5 › 5... · BAE 6333 – Fluvial Hydraulics Nonuniform Flow Chapter 4 (Graf, 1998) • Two Types of Nonuniform Flow: –Gradually](https://reader034.vdocuments.net/reader034/viewer/2022042402/5f128471bef5994e5157e74e/html5/thumbnails/11.jpg)
Wide and Rectangular Channel
• Using Manning’s Equation…
3
3/10
3
2
23
2
1
1
1
1
h
h
h
h
S
gh
q
SCh
q
Sdx
dh
c
n
f
f
f
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Critical Slope
• Bed slope that results in uniform flow at
critical depth (yn = yc) for a given discharge
– Combine critical flow equation with uniform
flow equation:
h
c
ch
fh
BRC
gAS
B
AgSRC
B
gA
A
QSRCU
2
2
2
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Critical Slope
• If Sf < Sc for a given Q and C, then hn > hc:
– Mild Slope
– Uniform flow corresponding to this normal
depth will be subcritical (fluvial)
• If Sf > Sc for a given Q and C, then hn < hc:
– Steep Slope
– Uniform flow corresponding to this normal
depth will be supercritical (torrential)
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Forms of Water Surface
• Water surface profiles for the possible
cases encountered in open-channel flow…
– First classification based on bed slope, Sf:
• Sf = 0 (Horizontal Slope): H
• Sf < 0 (Adverse Slope): A
• Sf > 0 (Mild Slope, Steep Slope, or Critical Slope):
M, S, or C
– Figure 4.2 in Graf
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Water-Surface Profiles • M1 –
– Curves goes downstream towards a horizontal
tangent
– Upstream of a dam or weir, pier, at junctures of
certain bed slopes
Finnemore and Franzini, Fluid Mechanics
Note: y = h, yo = hn
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Water-Surface Profiles • M2 –
– Curves goes downstream towards the critical depth
– Upstream of an increase in bed slope and upstream
or a hydraulic drop
Finnemore and Franzini, Fluid Mechanics
Note: y = h, yo = hn
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Water-Surface Profiles • M3 –
– Curves goes downstream towards the critical depth
where it terminates at a hydraulic jump
– Occurs when supercritical flow enters a mild channel
and after a change in bed slope from steep to mild
Finnemore and Franzini, Fluid Mechanics
Note: y = h, yo = hn
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Water-Surface Profiles • S1 –
– Curves begins at critical depth of a hydraulic jump
and terminates as a tangent to a horizontal line
– Upstream of a dam or weir and at a juncture of certain
bed slopes
Finnemore and Franzini, Fluid Mechanics
Note: y = h, yo = hn
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Water-Surface Profiles • S2 –
– Takes place in transition between critical depth and
uniform flow (usually very short)
– Occurs downstream of a sudden increase in bed
slope and downstream of an enlargement
Finnemore and Franzini, Fluid Mechanics
Note: y = h, yo = hn
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Water-Surface Profiles • S3 –
– Takes place in transition between supercritical flow
and uniform flow and approaches a tangent
– Occurs downstream of a gate, when the flow is below
the normal depth and when the bed slope is reduced
Finnemore and Franzini, Fluid Mechanics
Note: y = h, yo = hn
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Water-Surface Profiles • Critical Slope – C
– C1 – Curve is horizontal and occurs at a juncture of certain bed
slopes and upstream of a dam (weir)
– C2 - ?????
– C3 – Curve is horizontal and occurs when the bed slope is
reduced to critical slope and downstream of a sluice gate when
the flow is below normal depth
3
3/10
3
2
23
2
1
1
1
1
h
h
h
h
S
gh
q
SCh
q
Sdx
dh
c
n
f
f
f
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Water-Surface Profiles • Horizontal Slope – H (Sf=0, hn is infinite)
– H1 – not established because hn is infinite
– H2 and H3 correspond to M2 and M3 when the channel bed
becomes horizontal
– H2 is encountered at a hydraulic drop
– H3 is encountered when supercritical flow enters into a horizontal
channel
3
3/10
3
2
23
2
1
1
1
1
h
h
h
h
S
gh
q
SCh
q
Sdx
dh
c
n
f
f
f
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Water-Surface Profiles • Adverse Slope – A (Sf<0, hn does not exist)
– A1 – not established because hn does not exist
– A2 and A3 correspond to H2 and H3
– A2 is encountered at a juncture of certain bed slopes
– H3 is encountered when supercritical flow enters into an adverse
channel
3
3/10
3
2
23
2
1
1
1
1
h
h
h
h
S
gh
q
SCh
q
Sdx
dh
c
n
f
f
f
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The flow in a 15-ft wide rectangular channel that has
a constant bottom slope is 1400 cfs. A computation
using Manning’s equation indicates that the normal
depth is 6.0 ft. At a certain section the depth of flow
in the channel is 2.8 ft. Does the depth increase,
decrease, or remain the same as one proceeds
downstream from this section?
Example
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Example 4.A – Graf
A trapezoidal channel with bottom width of
7.0 m and side slopes of m = 1.5 conveys
Q = 28 m3/s with a bed slope of 0.0010 (n
= 0.025). The channel is terminated by a
sudden drop of the channel bed.
Determine what type of water-surface
profile is to be expected.
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Computation of Water Surface
Profiles
• Integration of dh/dx equation from
earlier…
– Method of successive approximations
– Method of direct integration
– Method of graphical integration
3
2
2
2
3
2
2
2
1
/1
1
/
gA
BQ
SRC
AQ
S
gA
BQ
RC
AQS
dx
dh fh
f
h
f
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Computation of Water Surface
Profiles
• Method of successive approximation:
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Computation of Water Surface
Profiles
• Method of successive approximation:
22
1
2
122
2
1
2
2
2
2
2
22
11
2
1
2
)/(
)/(
2
)/(
ii
ii
h
fii
h
f
h
ef
AAg
Qxx
RAC
QShh
Ad
g
Qdx
RC
AQSdh
RC
AQSS
dx
dh
g
AQ
dx
d
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Computation of Water Surface
Profiles
• Three forms of this equation:
sManningxxR
UnSHHb
ChezyxxRC
USHHa
Channels:Prismatic
AAg
Qxx
RAC
Qzhzh
AAg
Qxx
RAC
QShh
ChannelsAll
ii
h
fisis
ii
h
fisis
ii
ii
h
ii
ii
ii
h
fii
')3(
)3(
11
2)2(
11
2)1(
:
13/4
22
1
12
2
1
22
1
2
122
2
1
22
1
2
122
2
1
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Computation of Water Surface
Profiles • If you arbitrarily select Dx, solve for the variations in the
flow depth, Dh – – Standard Step Method (Method of Reaches)
• If you arbitrarily select Dh, solve for the variations in the distance, Dx –
– Direct Step Method (Method of Depth Variation)
• Before you use either method: 1. Establish control points (known relationship between flow depth
and discharge)
2. Computations proceed upstream for subcritical flow, Fr<1 and downstream for supercritical flow, Fr>1
3. When you are closer to the critical depth (curvature of water surface more pronounced), you must use smaller steps
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Direct Step Method
(Explicit Method)
• Use known flow depth, hi, at xi
• Select hi+1 (should be very close to hi)
• Calculate xi+1 using finite-difference equations
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Standard Step Method
(Implicit Method)
• Use known flow depth, hi, at xi
• Select xi+1
• Guess value of hi+1
– Calculate average C, A, Rh, and U with corresponds
to the average flow depth
– Use difference equations given above to calculate hi+1
– Use calculated hi+1 as new guess
– Continue with successive approximations until the
calculated hi+1 matches previously calculated value
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Method of Direct Integration
• Note that integration is a direct solution
method:
– You can proceed from one section to another
whatever the distance between the sections
– In Methods of Successive Approximations,
you must use small distances to avoid
computational inaccuracy
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Method of Direct Integration
• Chow (1959):
– N = hydraulic exponent for the conveyance
(function of cross-section and type of friction
coefficient)
– M = second hydraulic exponent
M
c
N
n
f
h
h
h
h
Sdx
dh
1
1
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Method of Direct Integration
• Chow (1959):
8.433)(
3.50.2253
2)(
Mdh
dB
B
AB
A
hhM
Ndh
dPRB
A
hhN h
dhh
h
Sdx
dhdhh
h
nN
MNM
c
N
f
n
n
11
11
1
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Method of Direct Integration
• Back to N and M:
– Assume trapezoidal channel…
– See equations 4.26a on page 196 of Graf
mdh
dBmhbB
mdh
dPmhbP
22
1212 22
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Method of Direct Integration
• Chow (1959):
1
1.4,11
1.4,11
11
1
11
11
1
/
00
0 0
11
MN
NJ
TableJN
Jd
N
JdIntegralSecond
TableNdd
IntegralFirst
dh
hd
S
hxx
dhh
h
Sdx
JN
JN
MN
NN
N
MNM
c
Nii
f
nii
nN
MNM
c
N
f
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Method of Direct Integration
• Chow (1959):
1
11
1
11111 ,,
,,
1
D
ii
ii
i
i
M
n
ciii
f
ni
ii
i
i
M
n
ciii
f
ni
xxx
JN
J
h
hN
S
hx
JN
J
h
hN
S
hx
i
i
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Example 4.A – Graf
A trapezoidal channel with bottom width of 7.0 m
and side slopes of m = 1.5 conveys Q = 28 m3/s
with a bed slope of 0.0010 (n = 0.025). The
channel is terminated by a sudden drop of the
channel bed. Calculate and plot the profile
upstream from the drop using:
(i) method of direct integration (Chow)
(ii) direct step method
(iii) standard step method
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