Open Channel Flow

CHAPTER I
Introduction

Open channel flow occurs where ever the flow proceeds with the
liquid surface exposed to constant pressure.

 In practice this pressure is the atmospheric pressure, and the

flow proceeds with free surface (exposed to the atmosphere).

Thus open channel flow may occur regardless of the type of

conduit in which it is occurring i.e. an open channel flow may
exist in a pipe, if it is flowing partially full.

 In practice flow in sewers, canals, streams and gutters is

exposed to atmospheric pressure and hence is an example of
open channel flow.
Intro
The longitudinal profile of the free surface in an open
channel flow defines the hydraulic gradient and
determines the cross-sectional area of flow, as is shown in
Figure 1.1.

It also necessitates the introduction of an extra variable,

the stage, to define the position of the free surface at any
point in the channel.

In consequence, problems in open channel flow are more

complex, and the solutions are more varied, making the
study of such problems both interesting and challenging.
Intro
 Flow classification

Recalling that flow may be steady or unsteady

and uniform or non-uniform, the major
classifications applied to open channels are as
follows:

Steady uniform flow, in which the depth is

constant, both with time and distance.

 the gravity forces are in equilibrium with the

resistance forces.
Flow classification

Steady non-uniform flow, in which the depth varies

with distance, but not with time.

The flow may be either

a) gradually varied or
b) rapidly varied.

Type (a) requires the joint application of energy

and frictional resistance equations.
Type (b) requires the application of energy and
momentum principles.
Flow classification
Unsteady flow, in which the depth varies with both time and distance (unsteady
uniform flow is very rare).

This is the most complex flow type, requiring the solution of energy, momentum
and friction equations through time.
Types of Open Channel

 Channels where flow occurs under free surface can

either be natural, such as rivers and streams, or artificial.

 Artificial channels comprise all man-made channels,

including irrigation and navigation canals, spillway
channels, sewers, culverts and drainage ditches.

 They are normally of regular cross-sectional shape and

bed slope, and as such are termed prismatic channels.
Their construction materials are varied, but commonly
used materials include concrete, steel and earth.
Types…

 The surface roughness characteristics of these

materials are normally well defined within
engineering tolerances.
 In consequence, the application of hydraulic theories
to flow in artificial channels will normally yield
reasonably accurate results. Various terms are used
to refer to channels built under different conditions.
  A prismatic channel is characterized by unvarying
cross section, constant bottom slope, and relatively
straight alignment.
Types
 Canal: a channel built on ground, i.e excavated to the desired shape and
slope with or without lining, usually having a mild slope. The lining could
be made of concrete, stone masonry, cement, wood or bituminous
material.

 Flume: a channel built (or supported) above the ground to convey fluid
from one point to another. In the field flumes are made of concrete,
wood, sheet metal or masonry. Laboratory flumes are usually made of
wood, metal, glass or a composite of these materials.

 Chute: is a channel of steep slopes. If the change in elevation in the

direction of flow occurs in a relatively short distance the channel is called
a drop.

 Culvert: is a relatively short and usually buried conduit that is commonly

used for drainage purposes, as in highways and embankments. Open
channel prevails whenever the culvert is flowing partially full.
Types
 In contrast, natural channels are normally very irregular in shape,
and their materials are diverse.

 The surface roughness of natural channels changes with time,

distance and water surface elevation.

 Therefore, it is more difficult to apply hydraulic theory to natural channels

and obtain satisfactory results.

 Many applications involve man-made alterations to natural channels

(e.g. river control structures and flood alleviation measures).

 Such applications require an understanding not only of hydraulic

theory, but also of the associated disciplines of sediment transport,
hydrology and river morphology.
 Definition sketch of geometric channel
properties

 Depth (y) - the vertical distance

of the lowest point of a channel
section from the free surface;
 Stage (h) - the vertical distance
of the free surface from an
arbitrary datum;
 Area (A) - the cross-sectional
•Surface width (B) - the width of the
area of flow normal to the
channel section at the free surface;
direction of flow;
•Hydraulic radius (R) - the ratio of area
to wetted perimeter (A / P);  Wetted perimeter (P) - the
•Hydraulic mean depth (Dm) - the ratio length of the wetted surface
of area to surface width (A / B). measured normal to the
direction of flow;
Table 1.1 Geometric properties of some
common prismatic channels
Velocity distribution in open
channels
 The measured velocity in an open channel will always vary
across the channel section because of friction along the
boundary.

 Neither is this velocity distribution usually axisymmetric (as it

is in pipe flow) due to the existence of the free surface. It
might be expected to find the maximum velocity at the free
surface where the shear force is zero but this is not the case.

 The maximum velocity is usually found just below the surface.

 WHY?????
velocity distribution in open channels
Determination of energy and
momentum coefficients
 To determine the values of α and β the
velocity distribution must have been
measured (or be known in some way).

 In irregular channels where the flow may be

divided into distinct regions α may exceed 2
and should be included in the Bernoulli
equation.
compound channel with three regions of flow

 dA
V13 A1  V23 A2  V33 A3  Vi Ai
3
u 3

 3   3
V A 3
V ( A1  A2  A3 ) V  Ai

Q V1 A1  V2 A2  V3 A3
V   
V Ai i

A A1  A2  A3  A i

 dA V12 A1  V22 A2  V32 A3  Vi Ai

2
u 2

   2
2
V A 2
V ( A1  A2  A3 ) V  Ai
Uniform flow and the Development of
Friction formulae
 When uniform flow occurs gravitational
forces exactly balance the frictional
resistance forces which apply as a shear force
along the boundary (channel bed and walls).

forces on a channel length in uniform flow

…Friction formulae

 gravity force = ρgAL sin 

 and the boundary shear force resolved in the
direction of flow is
 shear force = τoPL , where P is the wetted
perimeter
 In uniform flow these balance, i.e.
 τoPL = ρgAL sin 
…Friction formulae

 Considering a channel of small slope, (as

channel slopes for uniform and gradually
varied flow seldom exceed about 1 in 50) then
 Sin  ≈ tan  = So
 So
gAS o
o   gRS o
P
The Chezy equation

 For a state of rough turbulent flow, which is the

predominant type of flow in open channel, it was
experimentally verified that the shear force is
proportional to the flow velocity squared i.e.
τo α V2
τo = KV2
Substituting into the above equation
g
V  RS o
K
Or grouping the constants together as
one equal to C
V  C RS 0
This is the Chezy equation and the C the
"Chezy C"
 Table 1.2: Selected values of C.

Type of channel bed Mean value of C

Smooth cement 90
Well-laid brickwork 70
Cement concrete 70
Natural channel ( in good condition) 35
Natural channel ( in bad condition 25
Chezy

 The value of C can also be estimated using

the Ganguillet and Kutter formula, which has
been developed based on measurements in
open channels of various types.

0.00281 1.811
41.65  
C S n
 0.00281 n
1   41.65  
 S  R
where n is known as Kutter’s n. The above formula gives C in British units,
which could be converted into metric units (exercise)
The Manning equation

 Many studies have been made on the

evaluation of C for different natural and
manmade channels. Today most practicing
engineers use some form of these
relationships to give C: R1/ 6
C
n
Substituting Chezy’s equation in to the above
formula gives velocity of uniform flow:

R 2/3
So 1 A5 / 3
V Q
n P 2 / 3 S o1 / 2
n
a few typical values of Manning's n

ERA’S DDM 2000

Conveyance

 Channel conveyance, K, is a measure of the carrying

capacity of a channel. The K is really an agglomeration
of several terms in the Chezy or Manning's equation:

Q  AC RS o A5 / 3
So, K  ACR1/ 2 
Q  KS o1 / 2 nP 2 / 3
Use of conveyance may be made when
calculating discharge and stage in compound
channels and also calculating the energy and
momentum coefficients in this situation.
Computations in uniform flow
 We can use Manning's formula for discharge to calculate steady
uniform flow.
 Two calculations are usually performed to solve uniform flow
problems.
1. Discharge from a given depth
2. Depth for a given discharge
 In steady uniform flow the flow depth is know as normal depth.
 As we have already mentioned, and by definition, uniform flow
can only occur in channels of constant cross-section (prismatic
channels) so natural channel can be excluded.
 However we will need to use Manning's equation for gradually
varied flow in natural channels - so application to natural/irregular
channels will often be required.
Optimal Shape of Cross-Section
 The most hydraulically-efficient shape of channel is the one which
can pass the greatest quantity of flow for any given area or,
equivalently, the smallest area for a given quantity of flow.

 From Manning’s formula and the corresponding expression for

quantity of flow we see that this occurs for the minimum hydraulic
radius or, equivalently, for the minimum wetted perimeter.

 A semi-circle is the most hydraulically-efficient of all channel cross-

sections.

 However, hydraulic efficiency is not the only consideration and one

must also consider, for example, fabrication costs, excavation and,
for loose granular linings, the maximum slope of the sides.
The application of Energy Equation
for rapidly varied flow
 Rapid changes in stage and velocity occur whenever there is a
sudden change in cross-section, a very steep bed-slope or some
obstruction in the channel.

 This type of flow is termed rapidly varied flow.

 Typical examples are flow over sharp-crested weirs and flow

through regions of greatly changing cross-section (Venturi flumes
and broad-crested weirs).

 Rapid change can also occur when there is a change from super-
critical to sub-critical flow (see later) in a channel reach at a
hydraulic jump.
The application of Energy Equation

 In these regions the surface is highly curved and

the assumptions of hydro static pressure
distribution and parallel streamlines do not
apply.

 However it is possibly to get good approximate

solutions to these situations yet still use the
energy and momentum concepts outlined
earlier. The solutions will usually be sufficiently
accurate for engineering purposes.
The energy (Bernoulli) equation
 The figure below shows a length of channel inclined
at a slope of  and flowing with uniform flow.

p V 2
Recalling the Bernoulli equation, 
g 2 g
 z  cons tan t
Bernoulli
 And assuming a hydrostatic pressure distribution we can write
the pressure at a point on a streamline, A say, in terms of the
depth d (the depth measured from the water surface in a
direction normal to the bed) and the channel slope.
pA =ρgy2
y2
In terms of the vertical distance d   y1 cos 
cos  So, p A  gy1 cos 2 
y 2  y1 cos 2 

So the pressure term in the above Bernoulli pA

 y1 cos 2 
equation becomes g
As channel slope in open channel are very small (1:100 pA
 y1
= θ = 0.57o and cos2θ = 0.9999) so unless the channel is g
unusually steep

And the Bernoulli equation becomes V 2

y zH
2g
Flow over a raised hump - Application of
the Bernoulli equation

 Steady uniform flow is interrupted by a raised bed level as

shown. If the upstream depth and discharge are known we
can use Bernoulli’s equation and the continuity equation to
give the velocity and depth of flow over the raised hump.
Flow over a raised hump
 Applying Bernoulli’s equation between sections 1 and 2 (assuming
a horizontal rectangular channel z1 = z2 and taking α =1.0)
V12 V 22
y1   y2   z
2g 2g
Using continuity equation V1A1 = V2A2 = Q
Q
V1 y1  V 2 y 2  q ,Where q is the flow per unit width
B
Substituting this into Bernoulli’s equation we have: y1  q2 q2
 y2   z
2 gy12 2 gy 22
 q2 
Rearranging: 2 gy 23  y 22  2 gz  2 gy1  2

  q2  0

 y1 

Thus we have a cubic function with the only unknown being the
downstream depth, y2. There are three solutions to this but only one is
correct for this situation. We must find out more about the flow before we
can decide which it is.
Specific Energy
 Specific energy, Es, is
defined as the energy of the
flow with reference to the  Fig. 1.10 Specific energy
channel bed as the datum. (definition sketch)
 The concept of specific
energy was first introduced
by Bakmenteff (1918).
 With reference to the figure
the total energy of flow
with respect to the channel
bottom is given by
p1 V12
E s1  ( y  y1 )   V12
g 2 g  y
2g
Specific Energy…
 Thus the specific energy at an open channel section is equal to the sum of
the flow depth and the velocity head.
 In the above equation V1 denotes the velocity of flow at the point of
interest, in the figure above at point 1. In practice it is easier to use the
average velocity of flow at the section and speak about the specific energy
of the flow at a section.
 How ever the velocity of flow changes from point to point with in the flow
and as a result the specific energy changes from stream line to stream line.
It is common to use the average velocity of flow with a correction factor.
 The specific energy computed using the average velocity is taken to apply
for all points in the section, i.e. is taken as the specific energy of the
section. For steady flow this can be written in terms of discharge Q
 (Q / A) 2 q 2
Es  y  Es  y 
2g 2 gy 2

For a rectangular channel of width b, Q/A = q/y 2 q 2

( E s  y) y   cons tan t
2g
cons tan t
( E s  y) 
y2
Specific Energy…

 It can be observed that the specific energy is a

function of depth of flow, y, only. If one plots
the depth of flow as ordinate against the
specific energy for a constant Q, the energy
diagram is obtained, which is a very useful
curve in open channel hydraulics.
 Flow over a raised hump – revisited:
Application of the Specific energy equation.

 The specific energy equation

may be used to solve the raised
hump problem. The figure shows
the hump and stage drawn
alongside a graph of Specific
energy Es against y.
  
 The Bernoulli equation was
applied earlier to this problem
and the equation from that
example may be written in terms
of specify energy:
 Es1 = Es2 +∆z
Flow over a raised hump – revisited

 These points are marked on the figure. Point A on the curve

corresponds to the specific energy at section 1 in the channel, but
Point B or Point B' on the graph may correspond to the specific
energy at point 2 in the channel.

 All point in the channel between point 1 and 2 must lie on the
specific energy curve between point A and B or B'.

 To reach point B' then this implies that Es1 - Es2 > ∆z which is not
physically possible. So point B on the curve corresponds to the
specific energy and the flow depth at section 2.
Critical, Sub-critical and super critical flow

 The specific energy change with depth was plotted above for
a constant discharge Q, it is also possible to plot a graph with
the specific energy fixed and see how Q changes with depth.
For a fixed discharge:

1. The specific energy is a minimum, Esc, at

depth Yc, This depth is known as critical
depth.
2. For all other values of Es there are two
possible depths. These are called alternate
depths. For
 subcritical flow y > yc
 supercritical flow y < yc
For a fixed Specific energy :
 The discharge is a maximum at critical depth, Yc.
 For all other discharges there are two possible depths of flow for a
particular Es i.e. there is a sub-critical depth and a super-critical
depth with the same Es.
 An equation for critical depth can be obtained by setting the
differential of Es to zero: 2
 (Q / A)
Es  y 
2g
dE s Q 2 d  1  dA
 0  1  
dy 2 g dA  A 2  dy
Since δA=Bδy, in the limit
Q 2
dA/dy = B and 0  1 B c 2 Ac3
2g
Q 2 Bc
1
gAc3
For a fixed Specific…

 For a rectangular channel Q = qb, B = b and A

= by, and taking α = 1 this equation becomes
1/ 3
 q2 
yc    as Vc yc = q
 g 
 
Vc  gy c

Substituting this in to the specific energy equation

Vc2 y
E sc  y c   yc  c
2g 2
2
y c  E sc
3
Variation of the Discharge with depth for a given specific
energy value
The Froude number
V
 The Froude number is defined FN 
gDm
for channels as:
 Its physical significance is the Inertial Force
FN2 
ratio of inertial forces to Gravitational Force

gravitational forces squared

 It can also be interpreted as Water Velocity
FN 
Wave Velocity
the ratio of water velocity to
wave velocity
Froude…
 This is an extremely useful non-dimensional number in open-channel hydraulics.
 Its value determines the regime of flow - sub, super or critical, and the direction in which
disturbances travel
 Fr < 1 sub-critical
 water velocity > wave velocity
 upstream levels affected by downstream controls
 Fr = 1 critical
 Fr > 1 super-critical
 water velocity < wave velocity
 upstream levels not affected by downstream controls
The Hydraulic jump

 The hydraulic jump is an important feature in open

channel flow and is an example of rapidly varied
flow.

 A hydraulic jump occurs when a super-critical flow

and a sub-critical flow meet.

 The jump is the mechanism for the two surfaces to

join. They join in an extremely turbulent manner
which causes large energy losses.
Hydraulic jump
 Because of the large energy losses the energy or specific energy equation
cannot be use in analysis, the momentum equation is used instead.
 Resultant force in x- direction = F1 - F2
 Momentum change = M2 – M1
 F1- F2 =M2 –M1
 Or for a constant discharge
 F1 +M1 =F2 +M2 = constant
Application of the momentum equation to a hydraulic jump.
Hydraulic jump

 For a rectangular channel this may be

evaluated using y2
y F2  g y2b
F1  g 1 y1b 2
2
M 1  QV1 M 2  QV 2

Q Q
 Q  Q
y1b y2b

Substituting for these and rearranging gives

y1  2 
y2   1  8FN 1  1
2  
or
y2  2 
y1   1  8FN 2  1
2  
 Hydraulic jump
 So knowing the discharge and either one of the depths
on the upstream or downstream side of the jump the
other - or conjugate depth - may be easily computed.
 More manipulation with the above equation and the
specific energy give the energy loss in the jump as

 y 2  y1 3
E 
4 y1 y 2

These are useful results and which can be used in gradually varied flow
calculations to determine water surface profiles.

In summary, a hydraulic jump will only occur if the upstream flow is

super-critical.

The higher the upstream Froude number the higher the jump and the
greater the loss of energy in the jump.
 CHAPTER FIVE
5.0 RAPIDLY VARIED FLOW
 Characteristics of RVF
 STEADY NON-UNIFORM FLOW
 Pronounced curvature of streamlines.
 Abrupt change of flow profile (virtually broken)
 Resulting the state of high turbulance
 Example: Hydraulic Jump

 RVF vs GVF
 In view of contrast with UF & GVF the following characteristics should be noted.
 Pronounced curvature  hydrostatic pressure distribution can not be assumed
 Rapid variation in flow regime takes place in a very short distance
 Effect of boundary friction is comparatively small, which would play a primary role in a
GVF
 In RVF the velocity-distribution coefficients  and  are much greater than unity and can
not be accurately determined.
 Flow is actually confined by separation zones as well as solid boundaries. (Because profiles
could be broken).
 Approach to the problem
 The theory that assumes:
 Parallel flow,
 Hydrostatic distribution of pressure- Does not apply in RVF computation.
 For RVF of continuous flow profile a mathematical equation can be established,
 Approach to the solution of such equation
 Graphical method (e.g. flow-net analysis)
 Numerical method (e.g. method of relaxation)
 No satisfactory general solution has yet been obtained

 Practical approach
 No general solution yet been found
 Various RVF phenomena are treated as isolate cases
 Each with own semi-empirical/empirical treatment
 Experimental results are used empirically
 Flow interpreted qualitatively using energy principle, momentum principle, geometry
plus sometimes dimensional analysis
 Three isolated cases
 Flow over spillway
 Hydraulic jump
 Flow under gate
 Flow over spillways
 Definition:
 Spillway: is a structure over or through a dam for discharging flood flows; overflow
channel; opening built into a dam or the side of a reservoir to release (to spill) excess
floodwater.
 SHARP-CRESTED WEIR (SCW) VS BROAD CRESTED WEIR (BCW)
 BCW
 Overflow structure with horizontal crest
 above which the deviation from a hydrostatic pressure distribution
 because of centripetal acceleration may be neglected.
  stream-lines are parallel and straight
 Criteria 0.5  H1/L0.07
 If 0.07 H1/L the energy loss above the crest can not be neglected
 0.5  H1/L so that the hydrostatic pressure distribution can be assumed
 L = length of the weir crest in the direction of flow,
 H1 total energy head over the weir crest
 SCW
 Overflow structure (H1/L > 15)
 The crest length in the direction of the flow is short enough not to influence the H-Q
relationship of a weir
 In practice, 0.002m L so that even at a minimum head of 0.03m the nappe is
completely free from the weir body after passing the weir  no adhered nappe can
occur
 An air pocket beneath the nappe form from which a quantity of air is removed
continuously by the over falling jet.
 Therefore, Precaution is required not to ensure that the pressure in the air pocket is not
reduced. Otherwise resulting undesirable effects:
 Owing to the increase of the under pressure the curvature of the over falling jet will
increase, causing increase of the discharge coefficient
 Irregular supply of air to the pocket will cause vibration of the jet resulting an
unsteady flow
 SCW is the simplest form of overflow spillway
 Motto:
 Spillways must discharge the peak flow under smallest possible head.
 Negative pressure on the crest must be limited to avoid danger of cavitation on the
crest or vibration of the structure.
 Theoretically, there should be atmospheric pressure on the crest

 Round-Crested overflow spillway

 Designed in conformity with the shape of the low surface of the flow nappe
over a sharp-crested weir
 shape of the flow-nappe is interpreted by the principle of the projectile
Derivation of Nappe Profile over Sharp Crested Weir by the Principle of
Projectile.
Derivation of Nappe Profile over Sharp Crested Weir by the
Principle of Projectile.
 Flow Over Spillways
 Sharp-Crested weir vs Broad-crested weir
 Sharp crested weir - Simplest form of over flow spillway
 Round-Crested overflow Spillway
  designed in conformity with the shape of the low surface of the flow nappe
over a sharp crested weir.
 Shape of flow nappe is interpreted by the principle of the projectile.
 Let Vo = the velocity at pt-x = 0,  is angle of inclination of the velocity Vo with the
horizontal
 Horizontal velocity = Vo Cos  - constant and the only force acting on the nappe is
gravity.
 Horizontal distance traveled is time
s t
 S  vt
 X = Vo t Cos V= t (1)

 In same time t, the particle will travel1 a vertical distance y (taking y is positive downward)
y   Vo t Sin   gt 2
 (2)2
 Move the origin up so that it coincides with the peak.
1
y   Vo ts in   gt 2  C '
2
 x 
t   
 V Cos 
 Eliminating t from 1 & 2, from 12  o 
 x  1  x 
y   Vo   Sin   g    C '
 V0 Cos  2  Vo Cos  
hv Dividing each term by the total head H above the crest
H
Y X gH C'
  tan    HX 2 
H H 2Vo2 Cos 2 H

 gH C'
 Let
A , 2 CosB=
2 - tan, & C 
2V 0  H
2
Y X X
 A   B C
 Then, H H
  H

 = General equation for the lower surface of the nappe in dimension less term
 Since the horizontal velocity component is constant, the vertical thickness of the nappe T may be
assumed constant and T
D 
H
 Adding a term to the above equation the
2 general equation for the upper surface of
the nappe is Y X X
 A   B CD
H H
  H

 These equations are quadratic hence, the nappe Surfaces are theoretically parabolic.
 Several experimental studies on the nappe over a sharp-crested weir have been made.
 Reputed works has been done by US Bureau of Reclamation, they developed the following
equations for the constants in
hv the general nappe equations.hv  hv 
2
hv
H H 1.568   0.892  0.127
 A = - 0.425 + 0.25 B = 0.411 - 1.603 - H H
hv
 H
 C = 0.150 – 0.45 D = 0.57 – 0.02 (10m) 2 exp (10m)
 Where:
 hv = the velocity
hv head of the approach flow
 m= - 0.208 H
 For high weirs, the velocity of approach is relatively small and can be ignored (hv  0)
 A = 0.425
 B = 0.055
 C = 0.150
 D = 0.559
 Experimental data have indicated that these equations are not valid When,
X hv
 H < 0.5 and that > 0.2
H

 i.e., Additional data for verification are required

 For < 0.5, The pressure with in the nappe in the Vicinity of the weir crest is > Patm
 because of the convergence of the streamlines.
 Consequently, forces other than gravity are acting on the nappe,
 which makes the principle of the projectile invalid.
 N.B: The above theory and equations apply only if the approach flow is sub critical.
 For Supercritical flow, or Fr >1, the nappe profile becomes essentially a function of the
Froude number rather than a function of the boundary geometry as described above.
 Aeration of the Nappe
 In the preceding discussion the over falling nappe is considered a crated; i.e., The upper
and lower nappe surfaces are subject to full atmospheric pressure.
 In practice,
 - Usually insufficient aeration below the nappe occurs due to removal of air by over
falling set.
 Effects of reduction of pressure
 Increase in pressure difference on the spillway itself
 Change in the shape of the nappe for which the spillway crest is designed
 Increase in discharge, sometimes accompanied by fluctuation or pulsation of the
nappe, which may be very objectionable if the weir or spillway is used for measuring
purposes.
 Unstable performance of the hydraulic model
 Crest Shape of Overflow Spillways
 Earliest shapes were based on a simple parabola designed to fit the trajectory of the falling nappe (the
equation for the lower surface of the nappe).
 Bazin’s made comprehensive laboratory investigation for nappe shapes.
 the used of Bazin’s data in design will produce a crest shape that conincides with lower surface of as
aerated nappe over a sharp-crested weir.
 Such a profile is known as Bazin profile Advantage
 Should cause no negative pressure on the crest (the presence of negative pressure will lead to
danger of cavitation damage).
 In selecting a suitable profile avoidance of negative pressure should be considered an objective,
 along with such other factors as maximum hydraulic efficiency, practicability, stability &
economy.

 Extensive experiments on the shape of the nappe over-sharp crested weir were conducted
by U. S Bureau of Reclamation; including Bazin’s,
 The Bureau has developed coordinates of the nappe surface for various slope faced weirs
 On the basis of the Bureau data, The U.S Army. Corps of Engineers has developed several
standard shapes at its Waterways Experimental Station.
 Such shapes designed as the WES standard spillway shapes, can be expressed by the
following equation:-
n 1

X n  K Hd Y
  X and Y are Coordinates of the crest profile with the origin at the highest point of the
crest.
 Hd is the design head excluding the velocity head of the approach flow
 K & n are parameters depending on the slope of the upstream face. values of k & N are
given as flows:
 Slope of upstream face k n
 Vertical 2.000 1.850
 3 :1 (V = H) 1.936 1.836
 3:2 1.939 1.810
 3:3 1.873 1.776

 For intermediate slopes: approximate value of k and n may be obtained by plotting the
above values against the corresponding slopes and interpolating
 from the plot the required values for any given slope within the plotted range.
 The upstream face of the spillway crest may some times be designed to set back, as shown
by the dashed lines
 Discharge of WES Spillway
 The discharge over a spillway can be computed by an equation in the form of SCW/BCW
 Q = CLHe1.5
 He- the total energy head on the crest, including the velocity head in the approach canal.
 The effect of the approach velocity is negligible when height h of the spillway is greater
than 1.33Hd (h > 1.33 Hd), where the design head exclude the approach velocity head.
 Under this condition, i.e. h/Hd > 1.33, He = Hd can be taken (the approach velocity head is
negligible) and the coefficient of discharge C has been found to be C = 2.21 (if is in ft C He =
4.03)

 Discharge of Sharp Crested weir Q  C L H 1.5

 Discharge formula over sharp-crested weir can be expressed in the general form as:
 Where C = discharge Coefficient.
 L = effective length of the weir crest H = is the
measured head above the crest; excluding velocity head
 Effective length may be computed as
 L = L' - 0.1NH
 Where L1 = is the measured length of the crest
 N = number of contractions (Eg. piers for gates)
 N = 2 for two end contractions
 N = 1 for one end contraction
 N = 0 for no contraction
 According to a well-known Rehbock formula
 C = 3.27 + 0.40 Where h – is the height of weir.
 N.B - This equation holds up to = 5 but can be extended to = 10 with fair approximation.
- For > 15 the weir becomes a sill, and a critical section immediately upstream from the sill controls the
discharge.
 The critical depth of the section is approximately equal to H+h.
 By the critical depth – discharge relationship, it can be shown that the coefficient C is: -
C = 5.68 H  H 
1.5 H
1   X
h  h  h
h
 The transition between weir and sill (between = 10 & 15), however, has not yet been clearly defined.
 Experiments have shown that the coefficient C remains approximately constant for sharp-crested
weir under varying heads if the nappe is aerated.
 Weir: Low river dam used to raise the upstream water level,
 built across a stream to control raise or diver) the flow of water.
 Measuring weirs are across a stream for the purpose of measuring the flow.
 Sill: horizontal overflow section of an irrigation check or measuring structure
 used for under water of structure across a river or canal.
 Hydraulic Jump
 The theory of jump developed is for horizontal or slightly inclined channels
 in which the weight of water in the jump has little effect upon the jump behavior
 and hence is ignored in the analyses.
 The results thus obtained however can be applied to most channels encountered in engineering
problems.
 For channels of large slope, the weight effect of water in the jump may become so pronounced that it
must be included in the analysis.
 Practical Applications
 To dissipate energy in water flowing over a dam, weir and other hydraulic structure
 and thus prevent scouring d/s from the structure.
 To recover head or raise the water level on the d/s side of a measuring flume and thus maintains high
water level in the channel for water distribution purposes.
 To increase weight on the apron and reduce uplift pressure by raising the water depth on the apron.
 To increase the discharge of a sluice gate by holding sack tail water, thus preventing drawn jump.
 To mix chemical used for water purification.
 To aerate water for city water supplies
 Jump in Horizontal Rectangular channel
 For supercritical flow in a horizontal rectangular channel, the energy of flow is dissipated through
frictional resistance along the channel,
 resulting in a decrease in velocity and an increase in depth in the direction of flow.
 A hydraulic jump will form in the channel if the Froud Number Fr1 of the flow, the flow depth y1, and a
drown stream depth y2 satisfy the following equation:

 y2 1
  1  8F1  1
2

y1 2 
 This has been verified with experiments
 Types of Jump
 Hydraulic Jumps on horizontal floor are of several distinct types. They can be conveniently classified
according to Froud Number Fr1 of the incoming flow as follows.
 Fr1 =1 critical flow no jump can form
 1< Fr1 < 1.7 the water surface shows undulation (undular jump)
 1.7 < Fr1 < 2.5 a series of small rollers develop on the surface of the jump,
but the d/s water surface remains smooth.
The velocity throughout is fairly uniform, and the energy loss low.
The jump is called weak jump.
 2.5 < Fr1 < 4.5 there is an oscillating jet entering the jump bottom to surface and back again with no
periodicity.
Each oscillation produces a large wave of irregular period which, very common in canals,
can travel for miles doing unlimited damage to earth banks and ripraps.
This jump is called Oscillating Jump.
 4.5 < Fr1 < 9.0 steady Jump:-
 The down stream extremity of the surface roller and the point at which the high-velocity jet tends to
leave the flow occur at practically the same vertical section. The action and position of this jump
are least sensitive to variation in tail-water depth.
The jump is well balanced and the performance is at its best.
The energy dissipation ranges from 45 to 70%.
 Fr > 9.0 Strong jump:-
The high-velocity jet grabs intermittent slugs of water rolling down the front face of the
jump,
generating waves down-stream and a rough surface can prevail.
The jump action is rough but effective since the energy dissipation may reach 85%.
 N.B. It should be noted that the ranges of the Froude Number given above for the various
types of jump are not clear-cut but overlap to a certain extent depending on local
conditions.

 Basic characteristics of the Jump

 Energy Loss: the loss of energy in the jump is equal to the difference in specific energy
before and after the jump.
E  E1  E2 
 y2  y1 
3

4 y1 y 2
E
Re lative loss : the ratio
E1

 Efficiency: the ratio of the specific energy after the jump to that before the jump is defined
as the efficiency of the jump.

E2

 2
3
8F1  1 2  4 F1  1
2

E1 2
8F1 2  F1
2
 
 The relative loss is equal to , E2
1
E1
 this also is a dimensionless function. of Fr1.
 Height of Jump:- the difference between the depths after and before the jump.
 Hj = y2 – y1
 Expressing each term as a ratio with respect to initial specific energy. y2
hj
 y1
 h1 y2 y
 1 E1
E1
 E1
E1 E1 E1
 Where is the relative height,
 is the relative initial depth, and
2
hj 1  8 F1 3
 2
 is the relative sequence depth. E1 F1  2

 All these ratios can be shown to be dimensionless function of F1.

 For Example,

 Length of Jump:
 The length of a jump (also length of stilling basin) is empirically given as,
L= K(y2-y1)
 Where, k – is a coefficient derived from laboratory and filed experiment. 4.5 < k < 5.5 where the lower k = 4.5 applies
of Fr2 > 10 and the higher k=5.5 for Fr2 < 3.
 Gates
Flow inunder
canals are mainly used as water level regulators.
Gates
 Sometimes, gates are used as discharge regulator (measuring device).
 They are under-shot or underflow structures. Example slice gate, radial gate roller gate
 The design of underflow gate focuses on head-discharge relationship (Q-H).
 The objective is to minimize head loss; this means that the gate has to be lifted out off the
water for design discharge.
 The other concern of the design is the pressure distribution over the gate as a function of
opening and gate form.
 The H-Q relationship for gate depends on the shape and dimension of the control section
and the resulting curvature of the streamlines.
 For gated structures the control section is defined by the vena contract, being the smallest
cross section just down steam of the gate.
 In the vena contract, streamlines are straight and parallel.
 In gate flow 3 flow types can be distinguished.

 1) Free flow: the opening is relatively small (h1/a >2), and the contraction of the steam-
lines in vertical direction is strong.
 The down stream water level (h2) won’t affect the flow underneath the gate and a
hydraulics jump will occur down stream of the vena contra.
 The discharge depends up on the gate opening the upstream water level and the
contraction coefficient.
 Submerged flow: the d/s water level influences the flow underneath the gate. The
hydraulic jump is drowned and the jet underneath the gate is submerged. The discharge
depends upon the upstream and downstream water level and the gate opening.
 The boundary between free and submerged flow is a sharp one, which can be cleanly found
from the gate opening and the two water levels.
 Weir flow: on off gate

 The equation for a free flow underneath a sharp edged gate is:

Q  Cd Ba 2gh1
 Cd = discharge coefficient
 B = Width of gate opening
 a = height of gate opening
 h1 = upstream water depth
Cc
Cd 
 The discharge coefficient Cd a
1  Cc
h1

 Where,
 CC = Contraction coefficient of the jet depending on the shape of the gate and
 d = diameter of the rounded bottom edge
 For d
 4.7  Cc  0.99 Rounded edged gates 
a
 For submerged flow, some equations include the difference between the upstream and
downstream depths and others use the upstream water level only. The general equation is
offer given as. Q  C Ba 2gh 2 1

 Where,
 a = vertical opening of the gate( a< 0.67h1)
 h1 = Upstream water depth
 B = Effective width of the opening
 C2 = discharge coefficient. h h
a  0.67 * h1 or 1  0.67 1  1  1.5
 For values of a athe discharge follows from the equation for a broad-

crested weir is: 3

Q  1.7 * B * H 2

.
SHARP CRESTED WEIR
BROAD CRESTED WEIR
Sketch of
a Venturi
flume
Irrigation Canal
Grand Canal of China
Beijing and Hangzhou
Hydroelectric Power Plant
SPILLWAY