Water Flow in Open Channels -...
74
1 1 Water Flow in Open Channels The Islamic University of Gaza Faculty of Engineering Civil Engineering Department Hydraulics - ECIV 3322 Chapter 6
Transcript of Water Flow in Open Channels -...
1 1
Water Flow in Open Channels
The Islamic University of Gaza
Faculty of Engineering
Civil Engineering Department
Hydraulics - ECIV 3322
Chapter 6
2 2
Introduction
An open channel is a duct in which the liquid flows
with a free surface.
Open channel hydraulics is of great importance in civil
engineers, it deals with flows having a free surface,
for example:
• Channels constructed for water supply, irrigation,
drainage, and
• Sewers, culverts, and
• Tunnels flowing partially full; and
• Natural streams and rivers.
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Pipe Flow and Open Channel Flow
Pipe Flow
• The liquid completely fills the pipe and flow under pressure.
• The flow in a pipe takes place due to difference of pressure (pressure gradient),
• The flow in a closed conduit is not necessarily a pipe flow.
Open Channel Flow
• Flow takes place due to the slope of the channel bed (due to gravity).
• The flow must be classified as open channel flow if the liquid has a free surface.
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Pipe Flow Open Channel Flow
5
For Pipe flow (Fig. a):
• The hydraulic gradient line (HGL) is the sum of the elevation and the
pressure head (connecting the water surfaces in piezometers).
• The energy gradient line (EGL) is the sum of the HGL and velocity
head.
• The amount of energy loss when the liquid flows from section 1 to
section 2 is indicated by hL.
For open channel flow (Fig. b):
• The hydraulic gradient line (HGL) corresponds to the water surface
line (WSL); where it subjected to only atmospheric pressure which is
commonly referred to as the zero pressure reference.
• The energy gradient line (EGL) is the sum of the HGL and velocity
head.
• The amount of energy loss when the liquid flows from section 1 to
section 2 is indicated by hL. For uniform flow in an open channel, this
drop in the EGL is equal to the drop in the channel bed.
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6.1 Classifications of Open Channel Flow
Classification based on the time criterion:
1. Steady Flow (time independent)
(discharge and water depth do not change with time)
2. Unsteady Flow (time dependent)
(discharge and water depth at any section change with time)
Classification based on the space criterion:
1. Uniform flow (are mostly steady)
(discharge and water depth remains the same at every section in the channel)
2. Non-uniform Flow
(discharge and water depth change at any section in the channel)
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Non-uniform flow is also called varied flow ( the flow
in which the water depth and or discharge change
along the length of the channel), it can be further
classified as:
• Gradually varied flow (GVF) where the depth of the
flow changes gradually along the length of the channel.
• Rapidly varied flow (RVF) where the depth of flow
changes suddenly over a small length of the channel.
8
a) Uniform flow are mostly steady
b) Unsteady uniform flows
are very rare in nature
c) Steady varied flow
(over a spillway crest)
d) Unsteady varied flow (flood wave)
e) Unsteady varied flow (tidal surge)
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10
6.2 Uniform Flow in Open Channel
Uniform flow in an open channel must satisfy the following main features:
1. The water depth y, flow area A, discharge Q, and the velocity distribution V at all sections throughout the entire channel length must remain constant.
2. The slope of the energy gradient line (Se), the water surface slope (Sws), and the channel bed slope (S0) are equal.
Se = Sws = S0
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This is possible when the gravity force (W sin θ)θ)θ)θ) component equal the resistance to the flow (Ff)
⇒
=⇒=
0
2
0..)( S
P
A
KVPLKVALS
γγ
alityproportionoftcnsK
channelofareaunitperforceresisting
PLKVPLF
ALSALW
S
endsatforcescHydrostatiFF
FFFW
f
f
tan
,
)(
sin)(sin
tansin
221
0sin
0
2
0
0
0
21
=
=
==
==
==
==
=−−+
τ
τ
γθγθ
θθ
θ
ehSRCV =
tconsChezyK
C tan==γ
RadiusHydraulicP
ARh
==
The Chezy Formula
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• Rh = hydraulic radius or hydraulic mean depth
P
A
perimeterwetted
areawettedflowofareaRh
==)(
• C = Chezy coefficient (Chezy’s resistance factor), m1/2/s, varies in
relation of both the conditions of channel and flow.
• Manning derived the following empirical relation:
where n = Manning’s coefficient for the channel roughness
See the next table for typical values of n.
CnRh=
1 1 6/
13
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Manning’s formula
• Substituting into Chezy equation, we obtain the Manning’s formula for
uniform flow:
ehSR
nV
3/21=
ehSRA
nVAQ
3/21==OR
• Q in m3/sec,
• V in m/sec,
• Rh in m,
• Se in (m/m),
• n is dimensionless
Where:
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Example 1
1.5m
3.0m
2
1
Open channel of width = 3m as shown, bed slope = 1:5000,
d=1.5m find the flow rate using Manning equation, n=0.025.
( )( )
sVAQ
V
P
AR
P
A
SRn
V
h
eh
/m 84.49538.0
m/s 538.050001927.0
025.0
1
927.0708.9
9
9.708 35.132
m 95.1935.0
1
3
3
2
22
2
3
2
=×==
=×=
===
=++=
=×+×=
=
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Example 2
The cross section of an open channel is a trapezoid with a bottom width of 4 m and side slopes 1:2, calculate the
discharge if the depth of water is 1.5 m and bed slope =
1/1600. Take Chezy constant C = 50.
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Example 2
Open channel as shown, bed slope = 69:1584, find the flow
rate using Chezy equation, take C=35.
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( ) ( )
sVAQ
V
P
AR
P
A
SRCV
h
eh
/m 84.11352.1627.0
m/s 7.01584
69.0917.035
917.018.177
52.162
m 177.18 04.552.28.166.38.115072.0
m 52.16215072.06.32
52.272.08.1652.2
2
04.552.2
3
2222
2
=×==
=×=
===
=++++++=
=×+××
+×+×
=
=
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6.3 Hydraulic Efficiency of open channel sections
Based on their existence, an open channel can be natural or artificial:
• Natural channels: such as streams, rivers, valleys, etc.
These are generally irregular in shape, alignment and
roughness of the surface.
• Artificial channels: built for some specific purpose, such
as irrigation, water supply, wastewater, water power
development, and rain collection channels. These are
regular in shape and alignment with uniform roughness of
the boundary surface.
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Based on their shape, an open channel can be prismatic or non-prismatic:
• Prismatic channels: the cross section is uniform and the bed slop is constant.
• Non-prismatic channels: when either the cross section or the slope (or both) change, the channel is referred to as non-prismatic. It is obvious that only artificial channel can be prismatic.
• The most common shapes of prismatic channels are rectangular, parabolic, triangular, trapezoidal and circular; see the next figure.
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Most common shapes of prismatic channels
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• Most economical section is called the best hydraulic
section or most efficient section as the discharge,
passing through a given cross-sectional area A, slope of
the bed S0 and a resistance coefficient, is maximum.
• Hence the discharge Q will be maximum when the
wetted perimeter P is minimum.
PconstS
P
ACASRCAVAQ
eeh
1*.====
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Economical Rectangular Channel
D,BA ×= BD2P +=
D
A 2DP +=
0 dD
dP=
222 2 0 2 D
DB
D
A
D
A
dD
dP==⇒=
−=D
B 2 =⇒
2
B D =
P should be minimum for a given area;
D
D
DD
DD
DB
DB
P
ARh
4
2
22
2
2
2
=+×
=+×
==
So, the rectangular channel will be most economical when either:
the depth of the flow is half the width, or
the hydraulic radius is half the depth of flow.
2
DRh
=
24
Economical Trapezoidal Channel
D)Dn(BA +=
212 nDBP ++=
Dn D
A B −=
212 nD ) Dn D
A(P ++−=
0 dD
dP= ⇒=++−−= 0 12 2
2nn
D
A
dD
dPn
D
An12
2
2 +=+
D
DnBn
D
DnD)(Bn
2 12
2
2 +=+
+=+
2
Dn2Bn1D2 +
=+
or
P B B nD B nD= + + = +2 2 ( )
RA
P
B nD D
B nDh = =
++
( )
( )2R
Dh =2
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Other criteria for economic Trapezoidal section
• When a semi-circle is drawn with the trapezoidal center,
O, on the water surface and radius equal to the depth of
flow, D, the three sides of the channel are tangential to
the semi-circle”.
• To prove this condition, using the figure shown, we have:
OF OM B n DB
n D= = + = +sin ( ) sin ( ) sinα α α1
22
2
sinα =+
OF
Bn D
2using triangle KMN, we have:
sinα = =+
MK
MN
D
D n12
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OF
B nD
n=
+
+
( )2
12
D nB nD
12
2
2+ =+
using equation to replace the numerator , we obtain:
OFD n
n=
+
+
1
1
2
2OF D=
Thus, if a semi-circle is drawn with O as center and radius
equal to the depth of flow D, the three sides of a most
economical trapezoidal section will be tangential to the semi-
circle.
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The best side slope for Trapezoidal section
when n =1
3
ο=θ⇒ 60
B D n n= + −2 1 2( )⇒+
=+2
21 2 DnBnD
⇒+=++= )(22 DnBDnBBP BA
DnD= −
A
Dn D D n n− = + −2 1 2( )
DA
n n
2
22 1
=+ −
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Now, from equations: P B nD= +2( )
BA
DnD= −
PA
D= 2
PA
DA n n
2 2 24 4 2 1= = + −( ) ( )squaring both sides
⇒= 0dn
dP2 4 1 2 12
1
2PdP
dnA n n= + −
−[( ) * ( ) ]
2
11
2
n
n+= ⇒+=⇒ 22
14 nn n =1
3
1 3
1n= = tanθ ⇒ =θ ο
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The best side slope is at 60o to the horizontal, i.e.; of all trapezoidal
sections a half hexagon is most economical. However, because of
constructional difficulties, it may not be practical to adopt the most
economical side slopes
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Circular section
αα
2sin8
4
22 ddA −=
drP αα 2 ==
In the case of circular channels, the area of the flow cannot be
maintained constant. Indeed, the cross-sectional area A and the wetted
perimeter P both do not depend on D but they depend on the angle αααα.
Referring to the figure shown, we
can determine the wetted
perimeter P and the area of flow
A as follows:
Thus in case of circular channels, for most economical section, two
separate conditions are obtained:
1. Condition for maximum discharge, and
2. Condition for maximum velocity.
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1. Condition for Maximum Discharge for Circular Section:
α ο= 154 D d= 0 95.
(Using Manning’s formula) α ο= 151 D d= 094.
2. Condition for Maximum Velocity for Circular Section:
(Using the Chezy formula)
Q AV AC R S CA
PSh= = =
3
Q C SA
P2 2
3
=
dQ
dα= 0
V C R S CA
PSh= = V C S
A
P
2 2=
dV
dα= 0
α ο= 128 75. D d= 081.
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Variation of flow and velocity with depth in circular pipes
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6.4 Energy Principle in Open Channel Flow
g
VyZEnergyTotal2
2
++=
The total energy of a flowing liquid per unit weight is given by:
If the channel bed is taken as the datum, then the total energy per unit
weight will be:
g
VyE
specific2
2
+=
Specific energy (Es) of a flowing liquid in a channel is defined as
energy per unit weight of the liquid measured from the channel bed as
datum. It is a very useful concept in the study of open channel flow.
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E yV
gE Es p k= + = +
2
2Ep = potential energy of flow = y
Ek = kinetic energy of flow =
g
V
2
2
E yQ
g As = +
2
22Valid for any cross section
Specific Energy Curve:
It is defined as the
curve which shows the
variation of specific
energy (Es ) with depth
of flow y.
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Specific Energy Curve (Rectangular channel)
Consider a rectangular channel in which a constant discharge
q = discharge per unit width = = constant ( since Q and B are constants)
Q
B
⇒=×
==y
q
yB
Q
A
QV
kpsEE
yg
qyE +=+=
2
2
2pE
EK
EP Es
yc
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Sub-critical, critical, and supercritical flow
The criterion used in this classification is what is known by Froude number, Fr, which is the measure of the relative effects of inertia forces to gravity force:
h
rDg
VF =
T
A
Width SurfaceWater
Area) (Wetted Flow of Area==hD
T
T
Flow Fr
Sub-critical Fr < 1
Critical 1 = Fr
Supercritical Fr >1
gA
TQFr 3
22 =
V = mean velocity of flow of water,
Dh = hydraulic depth of the channel
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cyE ⇒min
E y and ys1 1 1⇒ '
Referring to the energy curve, the following features can be observed:
1. The depth of flow at point C is referred to as critical depth, yc.
(It is defined as that depth of flow of liquid at which the specific energy is
minimum, The flow that corresponds to this point is called
critical flow (Fr = 1.0).
2. For values of Es greater than Emin , there are two corresponding depths.
One depth is greater than the critical depth and the other is smaller then
the critical depth, for example;
These two depths for a given specific energy are called the alternate depths.
y yc>
y yc<
3. If the flow depth
In this case Es increases as y increases.
In this case Es decreases as y increases.
the flow is said to be sub-critical (Fr < 1.0).
4. If the flow depth the flow is said to be super-critical (Fr > 1.0).
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38
Critical depth, yc for rectangular channel
⇒=0dy
dE
2
2
2 yg
qyE
s+=
Critical depth, yc , is defined as that depth of flow of liquid at which the
specific energy is minimum, Emin,
The mathematical expression for critical depth is obtained by differentiating
energy equation with respect to y and equating the result to zero;
d
dyy
q
g y
q
g y( ) ( )+ = +
−=
2
2
2
3212
20
1 02
3
32
− = ⇒ =q
g yy
q
g
31
2
=g
qyc
39
Critical velocity, Vc for rectangular channel
,2
3
g
qyc
=
y
q
yB
Q
A
QV =
×==
Vq
yc
c
=
OR ⇒=g
yVy cc
c
22
3 V g yc c= ×
r
c
c Fyg
V==
×1
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Minimum Specific Energy in terms of critical depth
E yq
g yc
c
min = +2
22
g
qyc
2
3 =E y
yc
cmin = +
2
Eyc
min =3
2y
Ec =2
3
minOR
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Critical depth, yc , for Non- Rectangular Channels
⇒=0dy
dEs 0)(
2
21)
2(
3
2
2
2
=−=+dy
dA
Ag
Q
Ag
Qy
dy
d
(constant discharge is assumed) 1 02
3− =Q
g A
dA
dy( )
dA/dy = the rate of increase of area with respect to y = T (top width).
OR
⇒=− 013
2
Ag
TQ Q
g
A
T
2 3
= condition must be satisfied for the flow at the critical depth.
Recalling that ⇒=T
AD
h
Q
gA Dh
22=
The equation may also be written in terms of velocity � V
g
Dh2
2 2=
The velocity head is equal to one-half the hydraulic depth for critical flow.
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⇒+=2
2
2 Ag
QyE
sE y
A
Ts = +
2This equation represents
the critical state
E yA
Tc c= +
1
2( )OR
The general equation for the specific energy in critical state applicable to channels of all shapes.
Trapezoidal section
Circular section Triangle section
Rectangular section
EB ny y
B nyc
c c
c
=+
+
( )
( )
3 5
2 2
Ed d
c = − +−
21
16
2 2( cos )
( sin )
sinα
α αα
E yc c=5
4
2
3c
c
yE =
43
Constant Specific Energy
The specific energy was varied and the discharge was assumed to be
constant. Let us now consider the case in which the specific energy is
kept constant and the discharge Q is varied.
⇒=−2
2
2 Ag
QyE
sQ A g E ys= −2 ( )
Q A g E y gA E gA ys s2 2 2 22 2 2= − = −( ) ( )
The discharge will maximum if dQ
dy= 0
QdQ
dyg E A
dA
dyg y A
dA
dyAs= − +
2 2 2 2
2( ) ( )
dA/dy = T 02)2(2)2(22=−−⇒ gAyATgATEg
s
44
4 4 2 0E T yT As − − =
⇒=− AyETs)(2 E y
A
Ts = +
2
but E yQ
g As = +
2
22
⇒+=+T
Ay
Ag
Qy
222
2 Q
g
A
T
2 3
=
Thus for a given specific energy, the discharge in a given channel is a
maximum when the flow is in the critical state. The depth corresponding
to the maximum discharge is the critical depth.
45
6.5 Hydraulic Jump
• A hydraulic jump occurs when flow changes from a supercritical flow
(unstable) to a sub-critical flow (stable).
• There is a sudden rise in water level at the point where the hydraulic
jump occurs.
• Rollers (eddies) of turbulent water form at this point. These rollers cause
dissipation of energy.
•A hydraulic jump occurs in practice at the toe of a dam or below a sluice gate
where the velocity is very high.
46
General Expression for Hydraulic Jump:
In the analysis of hydraulic jumps, the following assumptions are made:
(1) The length of hydraulic jump is small. Consequently, the loss of head
due to friction is negligible.
(2) The flow is uniform and pressure distribution is due to hydrostatic
before and after the jump.
(3) The slope of the bed of the channel is very small, so that the
component of the weight of the fluid in the direction of the flow is
neglected.
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Location of hydraulic jump
Generally, a hydraulic jump occurs when the flow changes from
supercritical to subcritical flow.
The most typical cases for the location of hydraulic jump are:
1. Jump below a sluice gate.
2. Jump at the toe of a spillway.
3. Jump at a glacis.
(glacis is the name given to sloping floors provided in hydraulic structures.)
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•The net force in the direction of flow = the rate of change of moment in that direction
)( 12 VVg
Q−=
γ
The net force in the direction of the flow, neglecting frictional resistance and the
component of weight of water in the direction of flow,
R = F1 - F2 .
Therefore, the impulse-moment yields
F FQ
gV V1 2 2 1− = −
γ( )
Where F1 and F2 are the pressure forces at section 1 and 2, respectively.
γ γγ
A y A yQ
gV V1 1 2 2 2 1− = −( )
γ γγ
A y A yQ
g A A1 1 2 2
2
2 1
1 1− = −( )
Q
gAA y
Q
gAA y
2
11 1
2
22 2+ = +
y = the distance from the water surface to the centroid of the flow area
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Comments:
• This is the general equation governing the hydraulic jump for any
shape of channel.
• The sum of two terms is called specific force (M). So, the equation can
be written as:
M1 = M2
• This equation shows that the specific force before the hydraulic jump
is equal to that after the jump.
Q
gAA y
Q
gAA y
2
11 1
2
22 2+ = +
50
Hydraulic Jump in Rectangular Channels
A B y1 1= yy
11
2= A B y2 2= y
y2
2
2=
Q
gByBy
y Q
gB yBy
y2
1
11
2
2
22
2 2+ = +( )( ) ( )( )
Q
gAA y
Q
gAA y
2
11 1
2
22 2+ = +
B=
q
g
y y
y y
y y22 1
1 2
22
12
2
−
=
−
2 2
1 2 2 1
q
gy y y y= +( )
y y y yq
g2 12
221
220+ − =
using
, we get
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yy y q
g y2
1 1
2 2
12 2
2= − +
+
yy y q
g y1
2 2
2 2
22 2
2= − +
+
This is a quadratic equation, the solution of which may be written as:
where y1 is the initial depth and y2 is called the conjugate depth. Both are called
conjugate depths.
These equations can be used to get the various characteristics of hydraulic jump.
y
y
q
g y
2
1
2
13
1
21 1
8= − + +
y
y
q
g y
1
2
2
23
1
21 1
8= − + +
52
yq
gc3
2
=
++−=
3
11
2 8112
1
y
y
y
yc
y
y
y
y
c1
2 2
31
21 1 8= − + +
( )21
1
2 8112
1F
y
y++−=
y
yF1
2221
21 1 8= − + +
1
1
1yg
VF=
FV
g y2
2
2
=
But for rectangular channels, we have
Therefore,
These equations can also be written in terms of Froude’s number as:
53
H E E EL = = −∆ 1 2
E yq
g ys = +
2
22
H yq
g yy
q
g yL = + − +
1
2
12 2
2
23
2 2
Due to the turbulent flow in hydraulic jump, a dissipation (loss) of energy
occurs:
Where, E = specific energy
For rectangular channels:
hence,
Head Loss in a hydraulic jump (HL):
After simplifying, we obtain
21
3
12
4
)(
yy
yyHE
L
−==∆
54
h y yj = −2 1
L hj j≅ 6
Height of hydraulic jump (hj):
The difference of depths before and after the jump is known as the
height of the jump,
Length of hydraulic jump (Lj):
The distance between the front face of the jump to a point on the
downstream where the rollers (eddies) terminate and the flow becomes
uniform is known as the length of the hydraulic jump. The length of the
jump varies from 5 to 7 times its height. An average value is usually
taken:
55
6.6 Gradually Varied Flow
• Non-uniform flow is a flow for which the depth of flow is varied.
• This varied flow can be either Gradually varied flow (GVF) or
Rapidly varied flow (RVF).
• Such situations occur when:
- control structures are used in the channel or,
- when any obstruction is found in the channel,
- when a sharp change in the channel slope takes place.
56
Classification of Channel-Bed Slopes
The slope of the channel bed is very important in determining the
characteristics of the flow.
Let
• S0 : the slope of the channel bed ,
• Sc : the critical slope or the slope of the channel that
sustains a given discharge (Q) as uniform flow at the critical
depth (yc).
• yn is is the normal depth when the discharge Q flows as
uniform flow on slope S0.
57
S S or y yc n c0 = =
S S or y yc n c0 < >
S S or y yc n c0 > <
S0 0 0= .
S negative0 =
The slope of the channel bed can be classified as:
1) Critical Slope C : the bottom slope of the channel is equal to the critical slope.
2) Mild Slope M : the bottom slope of the channel is less than the critical slope.
3) Steep Slope S : the bottom slope of the channel is greater than the critical slope.
4) Horizontal Slope H : the bottom slope of the channel is equal to zero.
5) Adverse Slope A : the bottom slope of the channel rises in the direction of the
flow (slope is opposite to direction of flow).
58
59
Classification of Flow Profiles (water surface profiles)
• The surface curves of water are called flow profiles (or water surface
profiles).
• The shape of water surface profiles is mainly determined by the slope of
the channel bed So.
• For a given discharge, the normal depth yn and the critical depth yc
may be calculated. Then the following steps are followed to classify the
flow profiles:
1- A line parallel to the channel bottom with a height of yn is drawn and is
designated as the normal depth line (N.D.L.)
2- A line parallel to the channel bottom with a height of yc is drawn and is
designated as the critical depth line (C.D.L.)
3- The vertical space in a longitudinal section is divided into 3 zones
using the two lines drawn in steps 1 & 2 (see the next figure)
60
4- Depending upon the zone and the slope of the bed, the water profiles
are classified into 13 types as follows:
(a) Mild slope curves M1 , M2 , M3 .
(b) Steep slope curves S1 , S2 , S3 .
(c) Critical slope curves C1 , C2 , C3 .
(d) Horizontal slope curves H2 , H3 .
(e) Averse slope curves A2 , A3 .
In all these curves, the letter indicates the slope type and the subscript
indicates the zone. For example S2 curve occurs in the zone 2 of the
steep slope.
61
Flow Profiles in Mild slope
Flow Profiles in Steep slope
62
Flow Profiles in Critical slope
Flow Profiles in Horizontal slope
Flow Profiles in Adverse slope
63
Dynamic Equation of Gradually Varied Flow
Objective: get the relationship between the water surface slope and other
characteristics of flow.
The following assumptions are made in the derivation of the equation
1. The flow is steady.
2. The streamlines are practically parallel (true when the variation in
depth along the direction of flow is very gradual). Thus the hydrostatic
distribution of pressure is assumed over the section.
3. The loss of head at any section, due to friction, is equal to that in the
corresponding uniform flow with the same depth and flow
characteristics. (Manning’s formula may be used to calculate the slope
of the energy line)
4. The slope of the channel is small.
5. The channel is prismatic.
6. The velocity distribution across the section is fixed.
7. The roughness coefficient is constant in the reach.
.
64
H Z yV
g= + +
2
2
dH
dx
dZ
dx
dy
dx
d
dx
V
g= + +
2
2
Consider the profile of a gradually varied flow in a small length dx of an open channel the channel as shown in the figure below.
The total head (H) at any section is given by:
Taking x-axis along the bed of the channel and differentiating the equation with
respect to x:
65
− = − + +
S S
dy
dx
d
dx
V
gf 0
2
2
dy
dx
dy
dy
d
dx
V
gS S f+
= −2
02
dy
dx
d
dy
V
gS S f1
2
2
0+
= −
dy
dx
S S
d
dy
V
g
f=
−
+
0
2
12
• dH/dx = the slope of the energy line (Sf).
• dZ/dx = the bed slope (S0) .
Therefore,
Multiplying the velocity term by dy/dy and transposing, we get
or
This Equation is known as the dynamic equation of gradually varied flow. It gives the variation of depth (y) with respect to the distance along the bottom of
the channel (x).
66
dy
dx
S S
Q T
g A
f=−
−
0
2
31
dy
dx
dE dx
Q T
g A
=
−
/
1
2
3
The dynamic equation can be expressed in terms of the discharge Q:
The dynamic equation also can be expressed in terms of the specific energy E :
67
dy
dx= 0
dy
dxpositive=
dy
dxnegative=
Depending upon the type of flow, dy/dx may take the values:
The slope of the water surface is equal to the bottom slope. (the water surface is parallel to the channel bed)
or the flow is uniform.
The slope of the water surface is less than the bottom slope (S0) . (The water surface rises in the direction of flow) or the
profile obtained is called the backwater curve.
The slope of the water surface is greater than the bottom slope. (The water surface falls in direction of flow) or the
profile obtained is called the draw-down curve.
(a)
(c)
(b)
68
Notice that the slope of water surface with respect to horizontal (Sw) is different
from the slope of water surface with respect to the bottom of the channel (dy/dx).
A relationship between the two slopes can be obtained:
Sbc
ab
cd bd
abw = = =−
sinφ
Scd
ad
cd
ab0 = = ≈sinθ
• Consider a small length dx of the open channel.
• The line ab shows the free surface,
• The line ad is drawn parallel
to the bottom at a slope of S0 with the horizontal.
• The line ac is horizontal.
Let θ be the angle which the bottom makes with the horizontal. Thus
The water surface slope (Sw) is given by
69
dy
dx
bd
ad
bd
ab= ≈
dx
dySS
w−=0
dy
dxS Sw= −0
The slope of the water surface with respect to the channel bottom is given by
This equation can be used to calculate the water surface slope with respect to horizontal.
70
Water Profile Computations (Gradually Varied Flow)
• Engineers often require to know the distance up to which a surface
profile of a gradually varied flow will extend.
• To accomplish this we have to integrate the dynamic equation of
gradually varied flow, so to obtain the values of y at different locations
of x along the channel bed.
•The figure below gives a sketch of calculating the M1 curve over a
given weir.
71
Direct Step Method
• One of the most important method used to compute the water profiles is
the direct step method.
• In this method, the channel is divided into short intervals and the
computation of surface profiles is carried out step by step from one section
to another.
For prismatic channels:
Consider a short length of channel, dx , as shown in the figure.
dx
72
S dx yV
gy
V
gS dxf0 1
12
222
2 2+ + = + +
S dx E E S dxf0 1 2+ = +
dxE E
S S f=
−
−2 1
0
Applying Bernoulli’s equation between section 1 and 2 , we write:
or
or
where E1 and E2 are the specific energies at section 1 and,
respectively.
This equation will be used to compute the water profile curves.
73
12 yy >
12yy <
VnR S f1 12 3
1
1= /
VnR S f2 22 3
2
1= /
SS S
fm
f f=
+1 2
2
The following steps summarize the direct step method: 1. Calculate the specific energy at section where depth is known. For example at section 1-1, find E1, where the depth is known (y1). This
section is usually a control section.
2. Assume an appropriate value of the depth y2 at the other end of the small
reach. Note that:
if the profile is a rising curve and,
3. Calculate the specific energy (E2) at section 2-2 for the assumed depth (y2).
4. Calculate the slope of the energy line (Sf) at sections 1-1 and 2-2 using Manning’s formula
and
And the average slope in reach is calculated
if the profile is a falling curve.
74
L dxE E
S S fm1
2 1
0,2 = =
−
−L
E E
SS Sf f
1 22 1
01 2
2
, =−
−+
nnLLLL
,13,22,1....... −+++=
5. Compute the length of the curve between section 1-1 and 2-2
or
6. Now, we know the depth at section 2-2, assume the depth at the next
section, say 3-3. Then repeat the procedure to find the length L2,3.
7. Repeating the procedure, the total length of the curve may be obtained.
Thus
where (n-1) is the number of intervals into which the channel is divided.
