Transcript Slide 1

ENGR 691 – 73: Introduction to Free-Surface Hydraulics in Open Channels
Lecture 04:
Nonuniform Flow
Yan Ding, Ph.D.
Research Assistant Professor, National Center for Computational
Hydroscience and Engineering (NCCHE), The University of Mississippi,
Old Chemistry 335, University, MS 38677
Phone: 915-8969; Email: [email protected]
Course Notes by: Mustafa S. Altinakar and Yan Ding
Lecture 4.
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
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Outline
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Transition Between Subcritical and Critical Flow
Introduction to Hydraulic Jump
Gradually Varied Flow (Governing Equations)
Forms of water surface (Channels on Mild Slope, Critical Slope,
Steep Slope, Adverse Slope, Horizontal Slope)
Control Points
Computation of Water Surface (Method of successive
Approximations; Method of Direct Integration; Method of
Graphical Integration)
Rapidly Varied Flow (Weirs; Spillways; Hydraulic Drop;
Underflow Gates; Hydraulic Jump)
Transitions (Channel with variable Bed Floor; Channel of variable
Width; Oblique Jump)
Lateral Inflow
Lecture 4.
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
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Transitions between subcritical and critical flow
Transition
from subcritical to supercritical flow
So  Sc
So  Sc
So  Sc
When the flow changes from subcritical
to supercritical the water surface lowers
gradually from a higher depth to a lower
depth by passing through critical depth.
In the region where the flow changes from
subcritical to critical flow, a gradually
varied flow takes place.
Lecture 4.
Transition
from supercritical to subcritical flow
So  Sc
When the flow changes from supercritical
to subcritical the water surface rapidly
increases from a supercritical depth to
subcritical depth. This sudden increase is
called a rapidly varied flow.
The rapidly varied flow may be preceded
by a gradually varied flow region where
the flow depth rises but stays below
critical depth.
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
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Introduction to Hydraulic Jump
h
h
hhj
h2
h2
conjugate depths
or
sequent depths
hc
h1
V1
M1  M 2
M
q
h1
V2
h1
h2
M
2
q
h

gh 2
q  V1h1  V2 h2
F  F
Momentum equation
p1
2
Lecture 4.
hhj
Specific Energy
Equation of continuity
with
H s2
H s1
Specific Momentum
2
alternate depths
hc
h
Fp1   1 B
2
 Fp2  Q V2 V1 
2
h
Fp 2   2 B
2
Hs
V2
Hs  h 
2g
Things to remember:
• Conjugate depths or
sequent depths (on Specific
Momentum Curve)
• Alternate depths (on
Specific Energy Curve)
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
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Introduction to Hydraulic Jump
2
2
q q
h1
h2

B 
B  Q   
2
2
 h2 h1 
2
2
h1
h2
 q q

 q   
2
2   h2 h1 
2
2
 q2
h1
h2
q2 


 

2
2  gh2 gh1 
2
2
h1
q 2 h2
q2



2 gh1
2
gh2
M1  M 2
By combining momentum equation and continuity equation, on gets:
h2 1 
2
  1  8Fr1  1

h1 2 
where
Lecture 4.
Fr1 
V1
gh1
h1 1 
2
  1  8Fr2  1

h2 2 
or
and
Fr2 
V2
gh2
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
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Gradually Varied Flow Equation
hf
2
U1 / 2g
U 2 / 2g
2
U 2 / 2g
Q
h1
Consider the steady non uniform flow in a channel.
We wish to develop an equation for the variation of
the water surface h(x), i.e. longitudinal water surface
profile.
For this, we will consider the equation of energy:
H
h
z
z1
h2
Q / A  H
U2
zh
zh
2g
2g
z2
and the equation of continuity:
2
Q  UA
ref. line
L
Differentiate the energy equation with respect to x to get:
2
dz dh d  Q / A  dH




dx dx dx  2 g  dx
  So
Assuming that the head loss can be expressed using Chezy equation, we have:
  Se
Se
2

Q / A

C 2 Rh
2
2

d  Q / A  dh
Q / A
 So  


dx  2 g  dx
C 2 Rh
Lecture 4.
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
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Gradually Varied Flow Equation
hf
2
U1 / 2g
U 2 / 2g
2
U 2 / 2g
Q
h1
Consider the steady non uniform flow in a channel.
We wish to develop an equation for the variation of
the water surface h(x), i.e. longitudinal water surface
profile.
For this, we will consider the equation of energy:
H
h
z
z1
h2
Q / A  H
U2
zh
zh
2g
2g
z2
and the equation of continuity:
2
Q  UA
ref. line
L
Differentiate the energy equation with respect to x to get:
2
dz dh d  Q / A  dH




dx dx dx  2 g  dx
  So
Assuming that the head loss can be expressed using Chezy equation, we have:
  Se
Se
2

Q / A

C 2 Rh
2
2

d  Q / A  dh
Q / A
 So  


dx  2 g  dx
C 2 Rh
Lecture 4.
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
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Gradually Varied Flow Equation
Note that for a prismatic channel the flow area is only a function of the flow depth, A = f(h) :
We can, therefore, write:
2
d  Q / A  Q 2  2 dA 
Q 2  dA dh 
Q 2  dh 
Q 2 B dh

 3 
   3 B



dx  2 g  2 g  A3 dx 
gA  dh dx 
gA  dx 
gA3 dx
B
2

Q / A

Substitute this expression back into the previous equation to get:
gA
Q / A
dh dh
B

 So   2
dx dx
C Rh
2
dh
 So
dx
By rearranging the terms, we obtain a differential equation describing the variation
of flow depth with distance, i.e. the equation for longitudinal water surface profile:
It is important to note that when:
dh
0
dx
the water surface profile equation reduces to
Lecture 4.
Q / A
2
 U  C Rh So
2
2
U  C Rh So
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
1
Q / A2
C 2 Rh S o
2

Q / A
1
gA / B
i.e. Chezy
equation for
uniform flow
8
Gradually Varied Flow Equation
dh
0
dx
The flow depth remains constant and is equal to normal depth (uniform flow)
dh
0
dx
The flow depth increases in the direction of flow
dh
0
dx
The flow depth decreases in the direction of flow
Consider again the equation for longitudinal water surface profile:
For
2

Q / A
1
gA / B
1
U2
gDh
1  Fr 2
dh
 So
dx
1
Q / A2
C 2 Rh S o
2

Q / A
1
gA / B
the denominator becomes zero and we have:
dh

dx
We can, therefore conclude that, at critical flow (Fr = 1 and h = hc), the water surface profile is perpendicular to bed.
Fr  1
The normal is equal to critical depth, hn = hc , when:
Uniform flow
Lecture 4.
1
Q / A2
2
C Rh So
2

Q / A
 0 1
gA / B
C 2 Rh So  gA / B
dh
0
dx
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
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Review of the Notion of Critical Flow
Consider the flow cases below (for all cases channel cross
section characteristics are the same):
hn
Critical slope is the bed slope
when normal depth, hn, is equal to
critical depth, hc.
hn
hc
hc  hn
hc
So  Sc
So  Sc
So  Sc
hn  hc
hn  hc
hn  hc
Fr  1
Fr  1
Fr  1
When flow is critical, we have:
Fr 
U
1
gDh
Fr 
Since the flow is also uniform, Chezy equation holds:
Q
If Manning-Strickler is used:
Lecture 4.
2
2
A2 4 / 3
gA3
Q  2 Rh Sc 
n
B
2
gA3
Q 
B
2
Q  CA Rh So
gA3
Equating two expressions, we have:
Q  C A Rh S c 
B
gA
Sc  2
The expression for critical discharge is obtained as:
C BRh
2
1
A
A g
B
Q2 B
1
gA3
1/ 2
1/ 2
note that we have changed So to Sc.
gAn2
Sc 
4/3
BRh
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
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Gradually Varied Flow Equation in Terms of Conveyance
The equation for gradually varied flow can also be written using the notion of conveyance:
Remember the
definition of
conveyance:
when using Manning Strickler
K (h)  C A Rh
when using Chezy
1/ 2
Q / A2
Kn (h)2 
Q2
So
2
1
 CAR 
1/ 2
h
2
C 2 B Rh
So
gA
K (h)
Now consider the term in the nominator of
gradually varied flow equation:

when the flow
is uniform, in
either case we
can write:
or
K n ( h) 
Q
1/ 2
So
2
C 2 A2 Rh So
Q2 B
Q2 B

gA3 C 2 A2 Rh 2 So
gA3
Q2 B

gA / B
gA3
Consider the term in the denominator of
gradually varied flow equation:
Q2B

gA3
Q  Kn (h) So
1/ 2
A 2/3
K (h)  Rh
n
Q2 B

gA3
2
K n So
K 2 Sc
1
Sc
Q / A2
Q2
Q2


C 2 Rh So C 2 A2 Rh S o CAR 1/ 2
h

2

2
2
K S
K
 n2 o  n2
K
So K So
2
The gradually varied flow equation can therefore be written as:
Lecture 4.
K 
1  n 
dh
 K 
 So
2
dx
 K n  So
1 

 K  Sc
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
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Special forms of Gradually Varied Flow Equation: Wide Channel
Let us now consider a wide rectangular channel. The Chezy equation can be written as:
QC AR
1/ 2
h
1/ 2
o
S
C Ah
1/ 2
n
1/ 2
o
S
hn 
Q  C B hn So
2
The critical depth in a rectangular channel is given by:
2
2
3
3
hc 
3
q2
C 2 So
q2
g
Using these expressions and assuming that the Chezy coefficient C does not depend on depth h, the gradually varied
flow equation can be written as:
3
h 
1  n 
dh
h
 So
3
dx
 hc 
1  
h
This equation is known as equation of Bresse
named after the French scientist J.A.C. BRESSE (1822-1883),
who developed it first.
1/ 2
A 2 / 3 1/ 2 Bhn 2 / 3 1/ 2 B h5 / 3 So
hn So 
If we use Manning-Strickler, we have: Q  hn So 
n
n
n
h5 / 3 
Qn
qn
 1/ 2
1/ 2
BSo
So
10 / 3
In this case equation of Bresse becomes:
Lecture 4.
h 
1  n 
dh
h
 So
3
dx
 hc 
1  
h
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
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Gradually Varied Flow: Forms of Water Surface
Before we present all possible gradually varied flow profiles, let us take a look at the general properties of such curves:
• The water surface profile approaches asymptotically to uniform depth hn.
• The water surface profile is orthogonal to the critical depth line, when h = hc.
Water surface profiles are classified according to the bed slope.
So  Sc
Channel on Mild slope
So  Sc
Channel on Steep slope
S
type profile
So  Sc
Channel on Critical slope
C
type profile
So  0
Channel on Horizontal slope
H
type profile
So  0
Channel on Adverse slope
A
type profile
So  0
M
type profile
For each
profile type
several
possibilities
are
distinguished.
These are
called
branches.
In studying gradually varied water surface profiles we should also keep in mind that:
• In subcritical flow (Fr < 1), the perturbations travel both upstream and downstream. The water surface profiles for
subcritical flow are controlled by a downstream control section.
• In supercritical flow (Fr > 1), the perturbations travel only downstream. The water surface profiles for
supercritical flow are controlled by an upstream control section.
Lecture 4.
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
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Gradually Varied Flow: Forms of Water Surface
Convention for numbering branches:
• When the water surface profile is
higher than both the normal depth
and the critical depth, the branch is
numbered as type 1,
• the water surface profile is between
the normal and critical depths, the
branch is numbered as type 2,
• the water surface profile is lower
than both the normal depth and the
critical depth, the branch is
numbered as type 3,
Lecture 4.
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
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Gradually Varied Flow: Forms of Water Surface
Channel on Mild slope
So  0
and
So  Sc
hn  hc
M-type profiles
h  hn  hc
hn  h  hc
hn  hc  h
Fr  1
Fr  1
Fr  1
dh
0
dx
dh
0
dx
dh
0
dx
Branch M1
Towards upstream the profile
approaches asymptotically
normal depth, towards
downstream the curve tends to
become horizontal.
Encountered:
• Upstream of a weir or a dam
• Upstream of a pier
• Upstream of certain bed slope
changes points
Lecture 4.
Branch M2
Towards upstream the profile
approaches asymptotically
normal depth, towards
downstream the curve
decreasingly tends to critical
depth.
Encountered:
• Upstream of an increase in
bed slope
• Upstream of a free drop
structure
Branch M3
Towards downstream the
profile approaches increasingly
to critical depth.
Encountered:
• When a supercritical flow
enters a mild channel
• After a change in slope from
steep to mild
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
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Gradually Varied Flow: Forms of Water Surface
Channel on Steep slope
So  0
and
So  Sc
hn  hc
S-type profiles
h  hc  hn
hc  h  hn
hc  hn  h
Fr  1
Fr  1
Fr  1
dh
0
dx
dh
0
dx
dh
0
dx
Branch S1
Towards upstream the profile
approaches asymptotically
normal depth, towards
downstream the curve tends to
become horizontal.
Encountered:
• Upstream of a weir or a dam
• Upstream of a pier
• Upstream of certain bed slope
changes points
Lecture 4.
Branch S2
Towards upstream the profile
approaches asymptotically
normal depth, towards
downstream the curve
decreasingly tends to critical
depth.
Encountered:
• Upstream of an increase in
bed slope
• Upstream of a free drop
structure
Branch S3
Towards downstream the
profile approaches increasingly
to critical depth.
Encountered:
• When a supercritical flow
enters a mild channel
• After a change in slope from
steep to mild
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
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Gradually Varied Flow: Forms of Water Surface
Channel on Critical slope
So  0
and
So  Sc
hn  hc
C-type profiles
h  hc  hn
h  hc  hn
Fr  1
Fr  1
dh
0
dx
dh
0
dx
Branch C1
The water surface profile is
horizontal, when Chezy
equation is used.
Encountered:
• Upstream of a dam/weir
• At certain bed slope change
locations
Lecture 4.
Branch C2
There is no physically possible
C2 profile.
Branch C3
The water surface profile is
horizontal, when Chezy
equation is used.
Encountered:
• When a supercritical flow
enters a mild channel
• After a change in slope from
steep to mild
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
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Gradually Varied Flow: Forms of Water Surface
Channel on Horizontal slope
So  0
hn  
H-type profiles
Branch H1
Normal depth becomes infinite
and is meaningless.
Consequently, H1 profile is not
possible.
  h  hc
  hc  h
Fr  1
Fr  1
dh
0
dx
dh
0
dx
Branch H2
Similar to M2 profile
Encountered:
• Upstream of a free drop
structure
Lecture 4.
Branch H3
Similar to M3 profile
Encountered:
• When a supercritical flow
enters a horizontal channel
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
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Gradually Varied Flow: Forms of Water Surface
Channel on Adverse slope
So  0
hn  
H-type profiles
Branch A1
Normal depth becomes infinite
and is meaningless.
Consequently, A1 profile is not
possible.
  h  hc
  hc  h
Fr  1
Fr  1
dh
0
dx
dh
0
dx
Branch A2
Similar to H2 profile
(parabolic)
Encountered:
• Upstream of a certain bed
slope change location
Lecture 4.
Branch A3
Similar to H3 profile
(parabolic)
Encountered:
• When a supercritical flow
enters a channel with adverse
slope
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
19
Gradually Varied Flow: Notion of Control Section
Note that the passage from subcritical flow to
supercritical flow occurs with a smooth surface.
On the other hand, when the flow passes from
supercritical flow to subcritical flow, a sudden
increase in the water depth is observed. On the
figure this is indicated by HJ, which means
hydraulic jump. We will study hydraulic jump
in more detail later.
Photograph from Ohio
University's Fluid
Mechanics Laboratory.
Athens, Ohio USA
http://www.lmnoeng.com/Channels/HydraulicJump.htm
Control point, as the name implies, is the point
that controls the water surface profile. At a
control point we can generally write an
expression between discharge and depth. Thus,
it can be used as boundary condition for
calculating the water surface profile.
Lecture 4.
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
20
Critical Depth as Control Section and Other Uses of Critical Depth
In open channel flow locally a critical flow situation may exist for certain situations, such as slope
change from mild to steep, free fall (drop structure), and excessive contraction, etc.
Changing from a mild slope to a steep slope
(passage from subcritical flow to supercritical
flow).
Subcritical flow at a free overfall.
In fact, the critical depth takes place about
3 to 4 times hc upstream of the brink (due
to curvature of streamlines). The depth at
the brink is approximately equal to:
hb  0.71 hc
The cases of critical flow due to excessive contraction and a high positive step will be studied later.
In open channel flow critical section is a valuable tool because, knowing the geometry of the section,
one can write the relationship between flow depth and discharge.
Due to this property, critical condition is sometimes forced at a point in the channel. Then the discharge
can be obtained by measuring the flow depth.
Lecture 4.
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
21
Computation of Gradually Varied Flow
Several methods are available for computing gradually varied water surface profiles:
1.
The most obvious is to solve the differential equation of gradually varied flow, equation of Bresse, using a
numerical method, such as 4th order Runge-Kutta method. This method is called method of direct integration.
3
Equation of Bresse
using Chezy equation:
h 
1  n 
dh
 h   f  x, h 
 So
x
3
dx
 hc 
1  
h
4th order Runge-Kutta method formula can be written as:
where:
10 / 3
Equation of Bresse
using ManningStrickler equation:
hx  x  hx 
h 
1  n 
dh
h
 So
 f x, hx 
3
dx
 hc 
1  
h
x
k1  2k2  2k3  k4 
6
x
Coordinate along the channel length. The origin can be arbitrarily placed at any location.
hx
Flow depth at location x. All flow parameters at this location are known.
hx  x Flow depth at location x+x. This is the unknown flow depth we are calculating.
x
x 

k3  f  x 
, hx 
k2 
k1  f x, hx 
2
2


x
x 

k2  f  x 
, hx 
k1 
2
2


k4  f x  x, hx  x k3 
Computations should start from a point where all flow parameters are known (such as a control point) and
proceed upstream if the flow is subcritical and downstream if the flow is supercritical.
Lecture 4.
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
22
Computation of Gradually Varied Flow
Several methods are available for computing gradually varied water surface profiles:
2.
The second possibility is to use directly the energy equation to compute the water surface profile by employing an
iterative procedure. This approach is called method of successive approximations.
This method can be applied in two ways:
2.1 The open channel reach under study is divided into sub-reaches at known intervals starting from a control
point where all the hydraulic parameters are known. Based on the depth at the known point the depth at the
next station is computed. This method is called method of reaches (Stand Step Method in Open-Channel
Flow, MH Chaudhry).
2.2 A control point where all the hydraulic parameters are known is identified. The depth at that station, h, is
known. We choose another depth h+h, and compute where this depth will be along the channel. This
method is called method of depth variation (Direct-Step Method, MH Chaudhry).
In this course, we will study only the method of reaches.
Please refer to the textbook and other references for more information on other methods that can be used for
computation of water surfaces.
Lecture 4.
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
23
Computation of Gradually Varied Flow: Method of Reaches
Consider the gradually varied flow shown in the figure.
 x Se
We have divided the reach under study into smaller
sub reaches of length x. We also define the cross
sections i, i+1, i+2, ….. etc.
We will assume that the geometric properties of the
channel (A, P, B, Rh, Dh) at each cross section can be
calculated by knowing the depth.
We will also assume that the depth at cross section i
is known. We would like to calculate the depth at
cross section i+1.
Let us write the equation of energy Bernoulli
equation) between two cross sections i and i+1 :
2
2
U
U
zi  hi  i  zi 1  hi 1  i 1  x Se
2g
2g
zi  zi1   Hs
i

dz dH s

 Se
dx
dx
Lecture 4.

 H si1  x Se
So 
2
2
U
U
zi  hi  i  zi 1  hi 1  i 1  x Se
2g
2g
zi  zi 1   H s
x
dH s
 Se
dx
i
 H si1
x
S
e

zi 1  zi   H s
x
i 1
 H si
x
S
e
dH s
 So  Se
dx
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
24
Computation of Gradually Varied Flow
Therefore, when using the method of reaches, we will be solving this ordinary differential equation:
The basic equation we are using is:
2
 x Se
2
U
U
zi  hi  i  zi 1  hi 1  i 1  x Se
2g
2g
Hi  H i 1  x Se
Since depth hi , invert elevation zi and the discharge Q
are known, we can calculate the left side of the
equation, i.e. the total energy head, Hi directly.
Let us now assume a depth hi+1 . Since the invert
elevation zi and the discharge Q are known, we can
also calculate the total energy head, Hi+1 directly.
Now the question is weather the assumed that is the correct depth. This can be easily done. If the assumed depth
hi+1 is correct, then, the difference between the total heads Hi and Hi+1 should be equal to x Se.
The energy gradient can be calculated using either the equation of Chezy or Manning Strickler:
Q / A
2
Chezy equation:
Lecture 4.
Se 
2
Manning Strickler equation:
C Rh
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
Se
2

Q / A n 2

Rh
4/3
25
Computation of Gradually Varied Flow
Since the hydraulic parameters are varying from cross section i to i+1,
we may want to use the average value of the energy gradient:
Note also that :
Se 
S ei  S ei1
2
x  xi 1  xi
We should therefore check that:
 Se  Sei1
H i  H i 1  xi 1  xi   i
2

Hi  H i 1  x Se or



is satisfied.
If the above equation is not satisfied, a new value should be assumed for hi+1 and the computations must be carried out
again.
All these calculation can easily be carried out on a spread sheet.
If there are singular losses between the two cross sections i and i +1, this should also be taken into account. Then the
equation becomes:
Q / A
U2
H i  H i 1  x Se  K
 x Se  K
2g
2g
2
Again considering average values we can write:
Consequently:
Lecture 4.
 Se  Sei1
H i  H i 1  xi 1  xi   i
2

Q / A2
2g
1 Q / Ai   Q / Ai 1 

2g
2
2
2
2
2

1 Q / Ai   Q / Ai 1 
  K
2g
2

Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
26
Computation of Gradually Varied Flow
A trapezoidal channel having a bottom width of b = 7.0m and side slopes of m = 1.5, conveys a discharge of Q = 28m3/s.
The channel has a constant bed slope of So = 0.001. The Manning friction coefficient for the channel is n = 0.025m-1/3s.
The channel terminates by a sudden drop of the bed.
1. Determine the type of water surface profile to be expected.
2. Calculate the water surface profile for a reach length of 3200m.
The computation of gradually varied flow equations can be easily carried out on a spreadsheet using Goal Seek function
Lecture 4.
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
27
Rapidly Varied Flow at Channel Transitions
A transition is a change in the channel geometry over a relatively short distance. The change can be contraction or
expansion of the section, or a change in the section cross section geometry (say from rectangular to trapezoidal), or an
abrupt rise or drop of the channel bed. In designing transition, the attention must be paid to create minimum amount of
disturbance to the flow.
https://www.fhwa.dot.gov/engineering/hydraulics/pubs/06086/hec14ch06.cfm#fig096
The figure shows, typical designs for channel transition from a rectangular cross section to a trapezoidal cross section.
Lecture 4.
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
28
Use of Specific Energy to study Rapidly Varied Flow at Channel Transitions
h
Curve plotted for a constant Q
Specific Energy
2
1
U / 2g
U2
Q2
Hs  h 
h
2g
2 gA2
h1
Alternate depths
h1  hc
hc
hc
h2
2
Uc / 2g
2
U 2 / 2g
h2  hc
Es  H s
Supercritical flow
Es or H s
Specific energy curve is an extremely useful tool for analyzing various flow situations. In the following
slides we will learn how the specific energy curve can be used to analyze various flow situations in
channel transitions (flow over a positive or negative step, flow through a contraction or expansion).
Lecture 4.
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
29
Rapidly Varied Flow at Channel Transitions
In the following pages we will study in detail the rapid change of water surface at four types of channel transitions
under both subcritical and supercritical conditions:
1.
Q
Subcritical flow over a positive step
z
2.
Side view
Supercritical flow over a positive step
Q
3.
Subcritical flow over a negative step
z
4.
Supercritical flow over a negative step
5.
Subcritical flow through a contraction
6.
Supercritical flow through a contraction
7.
Subcritical flow through an expansion
8.
Side view
Q
Top view
Q
Top view
Supercritical flow through an expansion
Lecture 4.
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
30
Rapidly Varied Flow: Subcritical Flow over a Positive Step
q  Q / B  const.
h
h
q  Q / B  const.
2
2
U1 / 2g
U 2 / 2g
hc
H s1 h1
h2
z
H s1  z
 z
Hs
H s2
h2
H s1  z
Hs
H s2
H s1
Assume that the head loss due to the step is negligible (the energy grade line remains parallel to the bed).
2
2
U
U
H s1  z  h1  1  z  h2  2  H s2
2g
2g
Lecture 4.
h1 U1 B  h2 U 2 B  Q
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
31
Rapidly Varied Flow: Supercritical Flow over a Positive Step
q  Q / B  const.
h
h
q  Q / B  const.
2
U 2 / 2g
2
U1 / 2g
H s2
hc
H s1
z
h2 h
1
H s1  z
 z
Hs
h2
H s1  z
Hs
H s2
H s1
Assume that the head loss due to the step is negligible (the energy grade line remains parallel to the bed).
2
2
U
U
H s1  z  h1  1  z  h2  2  H s2
2g
2g
Lecture 4.
h1 U1 B  h2 U 2 B  Q
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
32
Rapidly Varied Flow: Subcritical Flow over a Negative Step
q  Q / B  const.
h
h
q  Q / B  const.
2
U 2 / 2g
2
U1 / 2g
H s1
h2
h1
h2
H s1
 z
Hs
H s2
z
H s2  H s1  z
Hs
H s2
Assume that the head loss due to the step is negligible (the energy grade line remains parallel to the bed).
2
2
U
U
H s1  z  h1  1  z  h2  2  H s2
2g
2g
Lecture 4.
h1 U1 B  h2 U 2 B  Q
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
33
Rapidly Varied Flow: Supercritical Flow over a Negative Step
q  Q / B  const.
h
h
q  Q / B  const.
2
U 2 / 2g
2
U1 / 2g
H s1
h1 h
2
H s2
H s1
 z
Hs
z
H s2  H s1  z
h2
Hs
H s2
Assume that the head loss due to the step is negligible (the energy grade line remains parallel to the bed).
2
2
U
U
H s1  z  h1  1  z  h2  2  H s2
2g
2g
Lecture 4.
h1 U1 B  h2 U 2 B  Q
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
34
Top view
Rapidly Varied Flow: Subcritical Flow through a Contraction
h
Side view
B2
B1
q2  Q / B2
2
U1 / 2g
H s1 h1
h2
hc1
q1  Q / B1
Hs
H s1
h
2
U 2 / 2g
H s2
hc 2
H s2  H s1
Hs
Assume that the head loss due to contraction is negligible (the energy grade line remains parallel to the bed).
2
2
U
U
H s1  h1  1  h2  2  H s2
2g
2g
Lecture 4.
h1 U1 B1  h2 U 2 B2  Q
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
35
Top view
Rapidly Varied Flow: Supercritical Flow through a Contraction
h
Side view
B2
B1
q2  Q / B2
2
U1 / 2g
h
2
U 2 / 2g
H s1
hc1
H s2
q1  Q / B1
h1
Hs
H s1
hc 2
h2
H s2  H s1
Hs
Assume that the head loss due to contraction is negligible (the energy grade line remains parallel to the bed).
2
2
U
U
H s1  h1  1  h2  2  H s2
2g
2g
Lecture 4.
h1 U1 B1  h2 U 2 B2  Q
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
36
Top view
Rapidly Varied Flow: Subcritical Flow through an expansion
B2
B1
q1  Q / B1
2
U1 / 2g
h
2
U 2 / 2g
Side view
h
h2
H s1 h1
hc1
H s2
hc 2
q2  Q / B2
Hs
H s1
H s2  H s1
Hs
Assume that the head loss due to contraction is negligible (the energy grade line remains parallel to the bed).
2
2
U
U
H s1  h1  1  h2  2  H s2
2g
2g
Lecture 4.
h1 U1 B1  h2 U 2 B2  Q
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
37
Top view
Rapidly Varied Flow: Supercritical Flow through an Expansion
B2
B1
q1  Q / B1
2
U1 / 2g
h
2
U 2 / 2g
Side view
h
H s1
H s2
h1 hc1
q2  Q / B2
Hs
H s1
hc 2
h2
H s2  H s1
Hs
Assume that the head loss due to contraction is negligible (the energy grade line remains parallel to the bed).
2
2
U
U
H s1  h1  1  h2  2  H s2
2g
2g
Lecture 4.
h1 U1 B1  h2 U 2 B2  Q
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
38
Rapidly Varied Flow: Special Case of Choked Flow due to a High Positive Step
By subtracting z from Hs1, we
cannot fall back onto the specific
energy curve
h
q  Q / B  const.
h
q  Q / B  const.
At one point on the step the flow
goes through the critical depth of
the cross section on the step.
final energy line
initial energy line
final water surface
initial water surface
H s1
h1 f
h2  hc2
h1i
hc1
z
z
H s1  z
Hs
H s2  H sc2
Hs
H s1
If the step is too high, subtracting z from Hs1, we cannot fall back onto the specific energy curve. The flow is said to be choked.
The step is too high. The water accumulates upstream of the step until it can pass over it by going through critical flow over the
step. Same equations hold. However, now h1 is also an unknown. Condition of critical flow over the step provides the third
equation needed for the analysis.
H s1 f
U1 f
2
2
U
 z  h1 f 
 z  h2  2  H s2
2g
2g
Lecture 4.
h1 f U1 f B  h2 U2 B  Q
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
h2  hc2
39
Top view
Rapidly Varied Flow: Special Case of Choked Flow due to too much Contraction
B2
B1
h
final energy line
initial energy line
With the available energy Hs1, we
cannot cut the specific energy
curve of the contracted section
q2  Q / B2
h
At one point on the contracted
section the flow goes through the
critical depth of that cross section.
Side view
final water surface
initial water surface
H s1 h1 f
h1i
hc1
q1  Q / B1
Hs
H s1
H s2
h2  hc2
H s2  H sc2
Hs
If the step is contracted too much, with the specific energy Hs1 we cannot cut the specific energy curve of the contracted section.
The flow is said to be choked. The section is contracted too much. The water accumulates upstream of the contraction until it
can pass a discharge of Q to the downstream by going through critical flow of the contracted section. Same equations hold.
However, now h1 is also an unknown. Condition of critical flow at the contracted section provides the third equation needed for
the analysis.
H s1 f
U1 f
2
2
U
 z  h1 f 
 z  h2  2  H s2
2g
2g
Lecture 4.
h1 f U1 f B  h2 U2 B  Q
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
h2  hc2
40
Energy Losses For Subritical Flow in Open Channel Transitions
Head losses at contractions and expansions can be calculated using the following expressions:
H Lc
 U 2 2 U 12 

 Cc 

2 g 
 2g
U1
H Le
U2
Contraction
 U 2 2 U 12 

 Ce 

2 g 
 2g
U1
U2
Expansion
Taken from USACE (1994)
U.S. Army Corps of Engineers, 1994. “Hydraulic Design of Flood Control Channels,” Engineering and Design Manual, EM 1110-2-1601,
July 1991, Change 1 (June 1994).
Lecture 4.
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
41
Rapidly Varied Flow: Hydraulic Jump
Hydraulic jump is a natural
phenomenon that occurs when
supercritical flow is forced to
become subcritical.
The passage from supercritical
flow to subcritical takes place
with a sudden rise of the flow
depth accompanied by a very
turbulent motion that may
entrain air into the flow.
To derive the equation governing hydraulic jump in a channel (see figure above), we will make use of momentum and
continuity equations simultaneously.
Consider a control volume, which comprises the hydraulic jump. The upstream cross section of the control volume is in
supercritical flow and the downstream section is in subcritical flow. Forces acting on this control volume are the weight
of the fluid, W, the upstream and downstream pressure forces, FP1 and FP2 respectively, and the friction force, Ff. The
momentum equation can be written as:
F
x
 FP1  FP2  W sin   Ff  Q U2  U1 
Assuming a rectangular channel, we have:
Lecture 4.
A1  h1B
A2  h2 B
q Q/B
FP1  
h1
A1
2
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
FP2  
h1
A2
2
42
Rapidly Varied Flow: Hydraulic Jump
Using these expressions and neglecting the component of weight and friction forces, the momentum equation becomes:
2
2
q q
h1
h2

B 
B  Q   
2
2
 h2 h1 
2
2
2
 q2
h1
h2
q2 




2
2  gh2 gh1 
2
2
h1
h2
 q q

 q   
2
2
  h2 h1 
2
h1
q 2 h2
q2



2 gh1
2
gh2
Note that the left and right hand side of the equation
represent the specific momentum, which is defined as:
q 2 h2
M

gh 2
Let us now make use of equation of continuity to
write the momentum equation as:
2

1 2
U1
h 
2
h2  h1 
h1 1  1 
2
2
 h2 


2
Divide both sides by (h2 – h1) to get:
U
h2  h2 h1  2h1 1  0
g
2
Only the positive root of the above quadratic
equation is physically meaningful:
2
U
Fr1  1
gh1
2
Lecture 4.
2
and
2
h1
U
 2h1 1
4
g
h2 1 
2
  1  8Fr1  1

h1 2 
Written in dimensionless form, the above
equation becomes:
where
1
h2   h1 
2
U
Fr2  2
gh2
2
2
or
h1 1 
2
  1  8Fr2  1

h2 2 
This equation is called the equation of Bélanger in honor
of the French scientist who developed it for the first time.
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
43
Rapidly Varied Flow: Hydraulic Jump
Note that for a hydraulic jump on larger slopes, the weight of the fluid cannot be neglected. In this case, the equation of
Bélanger for hydraulic jump becomes:
h2 1 
2
  1  8  HJ Fr1  1

h1 2 
where
 HJ  100.027 
as given by Rajaratnam.
 is in degrees
Hydraulic jumps are classified according to the approach flow Froude
number.
https://www.fhwa.dot.gov/engineering/hydraulics/pubs/06086/hec14ch06.cfm#fig096
Lecture 4.
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
44
Rapidly Varied Flow: Hydraulic Jump
Photos from (Dr. H. Chanson): http://www.uq.edu.au/~e2hchans/undular.html
Fr1  1.6
Energy loss across the hydraulic jump:
2
2
3

U1  
U 2  h2  h1 
   h2 

hhj  H s1  H s2   h1 
 

2
g
2
g
4h1h2

 

Length of the hydraulic jump:
5
Lhj
h2  h1 
7
Classification of hydraulic jumps:
Fr1  1.7
Undular jump
1.7  Fr1  2.5
Weak jump
2.5  Fr1  4.5
Oscillating jump
4.5  Fr1  9.0
Steady jump
Fr1  9.0
Strong jump
Lecture 4.
jump type generally
preferred in engineering
applications
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
45
Use of Hydraulic Jump in Hydraulic Engineering
Hydraulic jump is used for dissipating the energy of
high speed flow which may harm the environment if
released in an uncontrolled way.
The hydraulic jump, should take place in a area where the
bottom is protected (for example by a concrete slab or large
size rocks). If the jump takes place on erodible material the
formation of the erosion hole may endanger even the
foundations of the structure.
In real engineering projects measures are taken to
ensure that the hydraulic jump takes place in the
area with a protected bottom. This is achieved by
creating a stilling basin with the use of chute
blocks, baffle piers, and end sill, etc.
Lecture 4.
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
46
Examples of the Use of Hydraulic Jump in Hydraulic Engineering
http://www.engineering.uiowa.edu/~cfd/gallery/images/hyd8.jpg
Lecture 4.
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
47
Design of Stilling Basins
USBR Type I Stilling Basin
USBR Type IV Stilling Basin
Lecture 4.
USBR Type II Stilling Basin
SAF Stilling Basin
USBR Type III Stilling Basin
Pillari’s Stilling Basin
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
48
Books on Design of Stilling Basins
Hydraulic Design of Stilling
Basins and Energy Dissipators
by A. J. Peterka, U.S.
Department of the Interior,
Bureau of Reclamation
Lecture 4.
Energy Dissipators and
Hydraulic Jump
by Willi H. Hager
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
49
Positioning of a Hydraulic Jump
Draw the downstream subcritical flow profile starting from a control section at the downstream.
Draw the upstream supercritical flow profile starting from a control section at the upstream.
Draw the conjugate depth curve for the upstream supercritical flow profile.
For a hydraulic jump with zero length the jump is a vertical water surface between A’ and Z’.
If we wish to take into account the length of the jump for each point on the conjugate depth curve, draw a line parallel
to the bed. The length of the line should be equal to the length of the jump, i.e. 3 to 5 times the height difference
between the conjugate depth and the water depth. The tips of these lines are joined to obtain a translated conjugate
depth curve which takes into account the length of the jump. The intersection of the downstream profile with the
translated conjugate depth gives the downstream end of the jump.
Thus, the jump takes place between A and Z.
Lecture 4.
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
50
Oblique Hydraulic Jump
Consider again the specific energy curve.
h
Curve plotted for a constant Q
2
Specific Energy
U1 / 2g
U2
Hs  h 
2g
h1
Alternate depths
h1  hc
hc
hc
h2  hc
h
h2
2
Uc / 2g
2
U 2 / 2g
Es  H s
Supercritical flow
Es or H s
Es or H s
It can be seen that when the flow is supercritical, a small variation in depth (say h) causes a large variation in kinetic
energy and, thus the specific energy (Es or Hs).
Therefore, in supercritical flow, a transition, such as a change in width or a change in direction, will provoke an abrupt
variation of flow depth and stationary, stable gravity waves will appear on the free surface.
Lecture 4.
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
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Oblique Hydraulic Jump
Referring to the figure on the left,
consider the case of a supercritical
deflected by a side wall making an angle
q with the approach channel.
A standing wave front forms making an
angle b with the approach channel
direction. This is called an oblique
(hydraulic) jump. Note that this is
somewhat different than a classical
hydraulic jump due to the fact that the
flow is still subcritical downstream of
the wave front:
U
U
Fr1  1  1  1
gh1 c1
Fr2 
and
The continuity equation
in the direction normal
to the wave front gives:
2
Neglecting the bottom friction, the momentum equation
in the direction normal to the wave front gives:
Froude numbers using the velocity components normal
to the oblique wave front are defined as:
Lecture 4.

2
h
h
 Fn   21   22  q U 2n  U1n
U 1n
Fr 
gh1
n
1
and
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
U2
U
 2 1
gh2 c2
h1U1  h2U 2

U 2n
Fr 
gh2
n
2
52
Oblique Hydraulic Jump
Note that in the direction tangent to the
wave front no momentum change takes
place. The equation of momentum in
tangential direction becomes:
 F  0  q U
t
t
2
 U1t
This clearly shows that:

U1t  U 2t
Geometric considerations allow us to
write:
U1n  U1 sin b
U1n
U 
tan b
U 2n
U 
tanb  q 
t
1
Combining continuity and momentum equations, one obtains
the equation for change of depth across an oblique jump:
In which the Froude number normal to the wave front is defined as:
Lecture 4.
t
2
 
h2 1 
  1  8 Fr1n
h1 2 
Fr1n 
U 2n  U 2 sinb  q 
2
 1

U1n
U sin b
 1
 Fr1 sin b
gh1
gh1
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
53
Oblique Hydraulic Jump
The angle b of the wave front can be
expressed as:

1 1 h2  h2
  1
sin b 
Fr1 2 h1  h1

Note that, for small variations of depth,
thus for gradual transitions, one gets:
sin b 
1
c
 1
Fr1 U1
Using equation of continuity and
geometric relationships, the equation for
the oblique jump can also be written as:
h2 U1n
tan b
 n 
h1 U 2 tanb  q 
Combining two equations for change of
depth across an oblique jump, we can write:
 
2
tan b  1  8 Fr1n  3 


tanq 
2
2 tan2 b  1  8 Fr1n  1
 
These derivations were originally carried out by Ippen (1949). He also experimentally verified the relationship above
which gives the relationship between q and b for contracting channels (only).
Lecture 4.
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
54
Oblique Hydraulic Jump
The relation ship between q and b is plotted on the left.
Following observations can be made:
• For all Froude numbers there exists a maximum value
for the angle of deflection, qmax.
• For all values q smaller than qmax, two values of b are
possible. However, since the analysis is made for the
case the flow remains supercritical after the jump, i.e.
Fr2 > 1, we should consider only the values on the left
side (solid lines).
It is important to note that, any perturbation created by one wall will be reflected by the other wall and so on. To
study this behavior, we will consider two cases:
• Asymmetrically converging channel, and
• Symmetrically converging channel.
Lecture 4.
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
55
Reflection of Oblique Jumps in an Asymmetrical Channel Contraction
Consider the channel on the
left. The left wall is deflected
into the flow by an angle q,
while the right wall remains
straight.
 
Fr1
2
tan b1  1  8 Fr1n  3 


tanq 
2
2 tan2 b1  1  8 Fr1n  1
b1
h2
tan b1

h1 tanb1  q 
h2  h1
Fr2
2
tan b 2  1  8 Fr2n  3 


tanq 
2
2 tan2 b 2  1  8 Fr2n  1
b2  b1
h3
tan b 2

h2 tanb 2  q 
h3  h2
Fr3
n 2
tan b 3 
3

tanq 
2 tan2 b 3  1  8 Fr3n
b3  b 2
h3
tan b 3

h2 tanb 3  q 
h3  h2
 
 
 
1  8Fr   3 
 
2

1
Fr1  Fr2  1
Fr2  Fr3  1
Fr3  Fr4  1
… and so on. If the contracting channel is sufficiently long, the wave reflection continues until the flow finally becomes
subcritical.
Lecture 4.
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
56
Reflection of Oblique Jumps in an Asymmetrical Channel Contraction
Consider the channel on the
left. The left wall is deflected
into the flow by an angle q,
while the right wall remains
straight.
 
Fr1
2
tan b1  1  8 Fr1n  3 


tanq 
2
2 tan2 b1  1  8 Fr1n  1
b1
h2
tan b1

h1 tanb1  q 
h2  h1
Fr2
2
tan b 2  1  8 Fr2n  3 


tanq 
2
2 tan2 b 2  1  8 Fr2n  1
b2  b1
h3
tan b 2

h2 tanb 2  q 
h3  h2
Fr3
n 2
tan b 3 
3

tanq 
2 tan2 b 3  1  8 Fr3n
b3  b 2
h3
tan b 3

h2 tanb 3  q 
h3  h2
 
 
 
1  8Fr   3 
 
2

1
Fr1  Fr2  1
Fr2  Fr3  1
Fr3  Fr4  1
… and so on. If the contracting channel is sufficiently long, the wave reflection continues until the flow finally becomes
subcritical.
Lecture 4.
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
57
Reflection of Oblique Jumps in an Symmetrical Channel Contraction
View of Supercritical Flow in Curved Transition:
(a) Experimental Visualization of Standing Wave Patterns, after Ippen and Dawson (1951);
(b) Computer Visualization Based on 2D, Shock-Capturing, Numerical-Model Predictions
Causon D. M., C. G. Mingham and D. M. Ingram (1999), Advances in
Calculation Methods for Supercritical Flow in Spillway Channels, ASCE,
Journal of Hydraulic Engineering, Vol. 125, No. 10, pp. 1039-1050.
Lecture 4.
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
58
Designing a Symmetrical Channel Contraction for Supercritical Flow
A good channel contraction design
for supercritical flow should reduce
or eliminate the undesirable cross
wave pattern. This can be achieved
by choosing a linear contraction
length LT, thus by choosing a
contraction angle q’, such that the
positive waves emanating from
points A and A’, due to converging
walls, arrive directly at points D
and D’, where negative waves are
generated due to diverging walls.
Such a design is shown in the
figure on the left.
The choice of the angle q’ depends
on the approach Froude number,
Fr1, and the contraction ratio B3/B1.
Based on continuity equation, and assuming that the flow remains
supercritical in the contracted section, i.e. Fr3 > 1, we can write:
From geometric considerations , we also have the relationship:
B3 h1 U1  h1 

 
B1 h3 U 3  h3 
LT 
3/ 2
Fr1
Fr3
B1  B3
2 tan q
The angle q’ to satisfy these two equations is calculated using an iterative procedure.
Lecture 4.
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
59
Gradually Varied Flow with Lateral Inflow
We will now consider the case of gradually varied flow with
lateral inflow. In the most general case, the discharge added
subtracted laterally affects both to the mass and the momentum
of the flow.
Consider the gradually varied flow with lateral inflow as shown
on the left.
dQ
  q  x 
dx
The continuity equation
with lateral flow becomes:
 q represents lateral discharge which can be positive if a
discharge is added, or negative if a discharge subtracted.
We can also write:
q 
dQ
dA
dU
U
A
dx
dx
dx
Equation of the momentum states that the sum of all forces is equal to the change in momentum:
F
x
 FP  W sin   Ff   QU 
Let us now analyze all the terms in this equation one by one.
Lecture 4.
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
60
Gradually Varied Flow with Lateral Inflow
The net hydrostatic pressure force will be:
F
P
  zP A   zP  dh A    dh A
zP is the distance from the free surface to the
centroid of the flow area A:
The weight of the water prism between
two sections that are dx apart is:
W sin    sin  P A dx   So A dx
The rightmost side assumes that  is small.
The friction force can be written as:
The change in the momentum can be
written as:
Note that
Ff   o P dx   Se A dx
since
 o   Rh Se
 QU    Q  dQU  dU    QU    q dxU cos 
q dx  dQ
Note also that the lateral flow is entering or leaving the channel at an angle  and with velocity Uℓ.
Let us insert above expression the equation of momentum, and simplify:
  dh A   So A dx   Se A dx   Q  dQU  dU    QU    q dxU cos 
 dh  So  Se  dx 
Lecture 4.
 Q  dQU  dU   QU   q dxU  cos 


A
A
A
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
61
Gradually Varied Flow with Lateral Inflow
Q  dQU  dU   QU   q dxU  cos 
dh
 So  Se  
dx
gAdx
gAdx
gAdx
dh
1
QU  QdU  UdQ  dQdU  QU   q U  cos 
 So  Se  
dx
gAdx
gAdx
gA
dh
QU QdU UdQ QU  q U  cos 
 So  Se  




dx
gAdx gAdx gAdx gAdx
gA
dh
QdU UdQ q U  cos 
 So  Se  


dx
gAdx gAdx
gA
dh
U  dU 1 dQ  q U  cos 
 So  Se   


dx
g  dx A dx 
gA
Recalling that:
q dx  dQ
and
U  dU 
This is the equation of free surface for a steady
gradually varied flow with lateral inflow, which is also
called a steady spatially varied flow.
Q  dQ
A  dA
Simplifying also second order terms AdA and dQdA, the spatially varied flow equation can be written as:
dh

dx
So  Se   q 2

Q
1



U
cos

l

2
gA
 gA

2
Q / A
1
gA / B
Lecture 4.
It can be verified that, this equation reduces to gradually varied
flow formula when lateral flow is zero, qℓ = 0.
Spatially varied flow equation can also be solved using the same
methods for solving gradually varied flow equation.
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
62
Example of Structures for Spatially Varied Flow: Side Channel Spillway
http://www.firelily.com/stuff/hoover/flood.control.html
Side channel spillway of Hoover Dam in Nevada as seen from the reservoir side.
http://www.tornatore.com/joel/pics/index.php?op=dir&directory=20040227
Side channel spillway of Hoover Dam in Nevada.
Lecture 4.
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
63
Example of Structures for Spatially Varied Flow: Side Channel Spillway
http://www.hprcc.unl.edu/nebraska/sw-drought-2003-photos1.html
Side channel spillway of Hoover Dam in Nevada, looking downstream.
Lecture 4.
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
64
Quiz No 1(5 minutes)
A steep channel is connected to a mild channel as
shown in figure. Both channels have a rectangular
cross section. The following data is given:
hn1
hn2
So1
hc
S o2
Q  6m3 / s
hc  0.612m
B1  B2  4.0m
So1  0.01
So1  0.001
n1  n2  0.012m1/ 3s
hn1  0.383m
hn2  0.818m
The flow in steep channel is steady and uniform. Determine in which channel, steep or mild, the hydraulic jump will
take place.
Lecture 4.
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
65
Quiz No 1(5 minutes): Solution
A steep channel is connected to a mild channel as
shown in figure. Both channels have a rectangular
cross section. The following data is given:
hn1
hn2
So1
B1  B2  4.0m
hc
So1  0.01
n1  n2  0.012m1/ 3s
S o2
hn1  0.383m
hc  0.612m
So1  0.001
Q  6m3 / s
hn2  0.818m
The flow in steep channel is steady and uniform. Determine in which channel, steep or mild, the hydraulic jump will
take place.
Solution: Assume that the steady uniform flow continues all the way down to the point where the slope becomes mild.
Let us see if there is a jump at that point what would be the conjugate depth.
h1cj 1 
2
  1  8Fr1  1

h1 2 
h1cj 
U1 
1 
2
h1  1  8 Fr1  1

2 
h1cj  0.92m  hn2  0.818m
Lecture 4.
Q
6

 3.91m / s
B hn1 4.0  0.383
h1cj 

Fr1 
U
3.91

 2.02
ghn1
9.81 0.383

1
0.383  1  8  2.02 2  1  0.92 m
2
A jump taking place at the point of slope change will be too strong. It can jump higher
than the normal depth in the channel. Therefore, the flow continues into the mild
channel without a jump and creates an M3 type profile up to a depth whose conjugate
depth is equal to the uniform flow depth of the mild channel.
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
66
Quiz No 2 (5 minutes)
Consider the channel on the left with the following
data:
h1
hc
Q
z
hc


h1  0.288m
H s1  0.301
hc  0.129m
z  0.12m
H sc  0.193
Determine if the flow is choked due to the positive step. What will be the flow depth over the step?
Lecture 4.
Engr 691-73 Introduction to Free-Surface Hydraulics in Open Channels
67
Quiz No 2 (5 minutes): Solution
Consider the channel on the left with the following
data:
h1
hc
Q
z
hc


h1  0.288m
H s1  0.301
hc  0.129m
z  0.12m
H sc  0.193
Determine if the flow is choked due to the positive step. What will be the flow depth over the step?
Solution: Assuming no singular energy losses due to the step, the energy grade line remains at the same level. Over
the step, the energy is reduced by an amount z.
H s1  z  0.301 0.12  0.181m  H sc  0.193m
Thus the flow is choked. The flow will go through the critical depth over the step, i.e.
Lecture 4.
h2  hc  0.129m
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