Open channel hydraulíc for engineers
The subject of Open Channel Hydraulics for Engineers, also called Applied Hydraulics, is a subject required not only for Hydraulic Engineering students but also for other engineering fields involved, such as Construction Engineering, Transportation Engineering and Environmental Engineering. It follows the previous subject named Fluid Mechanics. The knowledge of open-channel hydraulics, which is
essential to the design of many hydraulic structures, has made advances by leaps and bounds.
OPEN CHANNEL HYDRAULICS FOR ENGINEERS
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CANTHO UNIVERSITY
MHO 5/6 project
Civil and Mechanical Engineering
OPEN CHANNEL HYDRAULICS
FOR ENGINEERS
LECTURE NOTE
PREPARED BY
LE ANH TUAN
DELFT, 2003
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OPEN CHANNEL HYDRAULICS FOR ENGINEERS
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To my wife Hoang Nga, my son Anh Tu and my daughter Hoang Ngan,
and all my closed friends ...
You are lovely rivers flowing in my dreams ...
LA. Tuan
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OPEN CHANNEL HYDRAULICS FOR ENGINEERS
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PREFACE
The subject of Open Channel Hydraulics for Engineers, also called Applied
Hydraulics, is a subject required not only for Hydraulic Engineering students but also
for other engineering fields involved, such as Construction Engineering,
Transportation Engineering and Environmental Engineering. It follows the previous
subject named Fluid Mechanics. The knowledge of open-channel hydraulics, which is
essential to the design of many hydraulic structures, has made advances by leaps and
bounds. The practical importance of this topic in the water resources development
together with the challenge posed by the variety and complexity of its problems, has
created for a long time the need for more comprehensive and detailed treatment of this
subject. As a result, many excellent texts have been written, not only in English, but
also in other languages. The best of these texts have imposed a logical and coherent
structure in the study of the subject, and created a tradition that present texts
consciously follow.
This lecture note is prepared as a reference document for the subject. It emphasizes
the dynamics of the open-channel flow, by attempting to provide a complete
framework of the basic equations of motion of the fluid, which are used as building
blocks for the treatment of many practical problems. The structure of the document,
with seven chapters totally, follows a logical sequence from a description and
classification of Fluid Mechanics and Open Channel Flows, as reviewed in Chapter 1.
A development of the basic equation of motion for uniform flow is encountered in
Chapter 2. Coming to Chapter 3, the fruitful concepts of specific energy and hydraulic
jumps are introduced and developed. Chapter 4 presents a variety of non-uniform
flows and applications of drawing water-surface profiles. Spatially-varied flow, found
at spillways and weirs is considered in Chapter 5. Transitions and energy dissipators
are discussed in Chapter 6. Finally, in Chapter 7, unsteady flow in open channels is
introduced generally and an introduction to the method of characteristics is presented.
Writing on the subject matter of this lecture note, grateful use has been made of the
authoritative texts and treatises on the subject by Ven Te Chow (1973), Henderson
(1966), Sergio Montes (1998) and Hubert Chanson (1999), to which frequent
references were made.
Developing this lecture note is part of the activities within the MHO 5/6 project. This
document could not have been completed without the enthusiastic support and advices
of Professor Dr. Henri L. Fontijn, Head of the Fluid Mechanics Laboratory of the
Faculty of Civil Engineering & Geosciences (CiTG), Delft University of Technology.
He spent numerous hours of his time reading and advising on the typed texts.
Special acknowledgments are due to the faculty and staff members of CiTG, CICAT,
Delft University of Technology and my colleagues at CanTho University, who
encouraged me in many ways.
Finally, I would like to thank all my family members and friends, whose help in
loving support is most gratefully acknowledged.
LE ANH TUAN
Delft University of Technology, the Netherlands, September 2003
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CONTENTS
Preface ............................................................................................................... iii
Contents ............................................................................................................. iv
List of symbols ................................................................................................... vii
Chapter 1: INTRODUCTION ........................................................................ 1
1.1. Review of fluid mechanics .................................................................... 1
1.1.1. Fluid mechanics .......................................................................... 1
1.1.2. Hydrostatics ................................................................................ 3
1.1.3. Continuity equation ..................................................................... 4
1.1.4. Types of flow .............................................................................. 4
1.1.5. Bernoulli’s equation .................................................................... 5
1.1.6. Euler’s equation .......................................................................... 5
1.1.7. Flow through orifices, mouthpieces and pipes .............................. 5
1.1.8. Flow through open channel .......................................................... 6
1.2. Structure of the course .......................................................................... 7
1.2.1. Objectives of the course ............................................................... 7
1.2.2. Historical note for the course........................................................ 7
1.2.3. Structure of the course.................................................................. 8
1.3. Dimensional analysis ............................................................................. 10
1.3.1. Fundamental dimensions .............................................................. 10
1.3.2. Dimensional homogeneity............................................................ 11
1.3.3. Principles of Dimensional Homogeneity ...................................... 12
1.3.4. Buckingham’s - theorem ............................................................ 14
1.3.5. Limitations of dimensional analysis ............................................. 16
1.4. Similarity and models ........................................................................... 16
1.4.1. Advantages of model analysis ...................................................... 16
1.4.2. Hydraulic similarity ..................................................................... 17
1.4.3. Geometric similarity..................................................................... 17
1.4.4. Kinematic similarity..................................................................... 18
1.4.5. Dynamic similarity....................................................................... 18
1.4.6. Technique of hydraulic modelling ................................................ 19
1.4.7. Developments in hydraulic model testing ..................................... 20
1.4.8. Undistorted models ...................................................................... 20
1.4.9. Comparison of an undistorted model and the prototype ................ 21
1.4.10. Distorted models ........................................................................ 22
1.4.11. Advantages and disadvantages of distorted models..................... 23
1.4.12. Comparison of a distorted model and its prototype ..................... 23
Chapter 2: UNIFORM FLOW ....................................................................... 25
2.1. Introduction .......................................................................................... 25
2.1.1. Definition .................................................................................... 25
2.1.2. Momentum analysis .................................................................... 27
2.2. Basic equations in uniform open channel flow ...................................... 28
2.2.1. Chezy’s formula........................................................................... 28
2.2.2. Manning’s formula....................................................................... 30
2.2.3. Discussion of factors affecting f and n.......................................... 32
2.3. Most economical cross-section ............................................................. 32
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2.3.1. Concept ....................................................................................... 32
2.3.2. Conditions for maximum discharge ............................................. 32
2.3.3. Problems of uniform-flow computation........................................ 36
2.4. Channel with compound cross-section .................................................. 38
2.5. Permissible velocity against erosion and sedimentation ......................... 40
Chapter 3: HYDRAULIC JUMP .................................................................... 46
3.1. Introduction .......................................................................................... 46
3.2. Specific energy ..................................................................................... 47
3.2.1. Specific energy ........................................................................... 47
3.2.2. Critical depth and critical velocity ............................................... 48
3.2.3. Types of flows ............................................................................ 49
3.3. Depth of hydraulic jump ....................................................................... 51
3.3.1. Concept ....................................................................................... 51
3.3.2. Water rise in hydraulic jump ....................................................... 51
3.3.3. Energy loss due to hydraulic jump ............................................... 53
3.3.4. Hydraulic jump features .............................................................. 54
3.4. Types of hydraulic jump ....................................................................... 55
3.4.1. Criterion for a critical state-of-flow ............................................. 55
3.4.2. Types of hydraulic jump .............................................................. 58
3.5. Hydraulic jump formulas in terms of Froude-number ............................ 61
3.5.1. Momentum-transfer curve ............................................................ 61
3.5.2. Direct hydraulic jump................................................................... 62
3.5.3. The initial depth and the sequent depth ........................................ 62
3.5.4. Energy loss ................................................................................. 64
3.5.5. Efficiency ................................................................................... 66
3.5.6. Height of jump ............................................................................ 66
3.5.7. Length of jump ............................................................................ 66
3.6. Submerged hydraulic jump ................................................................... 67
3.6.1. Definition .................................................................................... 67
3.6.2. Flow in submerged jump ............................................................. 68
Chapter 4: NON-UNIFORM FLOW .............................................................. 70
4.1. Introduction .......................................................................................... 70
4.1.1. General ....................................................................................... 70
4.1.2. Accelerated and Retarded flow .................................................... 71
4.1.3. Equation of non-uniform flow ..................................................... 73
4.2. Gradually-varied steady flow ................................................................ 75
4.2.1. Backwater calculation concept .................................................... 75
4.2.2. Equation of gradually-varied flow ............................................... 75
4.3. Types of water surface profiles ............................................................. 77
4.3.1. Classification of flow profiles ..................................................... 77
4.3.2. Sketching flow profiles ............................................................... 79
4.3.3. Prismatic channel with a change in slope...................................... 82
4.3.4. Composite flow profiles with various controls ............................. 83
4.4. Drawing water surface profiles ............................................................. 84
4.4.1. Direct-step method ...................................................................... 84
4.4.2. Direct numerical integration method ............................................ 88
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Chapter 5: SPILLWAYS ................................................................................ 90
5.1. Introduction .......................................................................................... 90
5.2. General formula .................................................................................... 91
5.3. Sharp-crested weir ................................................................................ 93
5.3.1. Experiments on sharp-crested rectangular weirs .......................... 93
5.3.2. Other types of sharp-crested weirs used for flow measurement..... 96
5.4. The overflow spillway .......................................................................... 98
5.4.1. The spillway crest ....................................................................... 98
5.4.2. The spillway face ........................................................................ 100
5.4.3. The spillway toe ........................................................................... 101
5.5. Broad-crested weir ................................................................................ 104
5.5.1. Introduction ................................................................................. 104
5.5.2. Broad-crested weir discharge formula ......................................... 105
5.5.3. Undular weir flow and discharge coefficients .............................. 105
Chapter 6: TRANSITIONS AND ENERGY DISSIPATORS ....................... 107
6.1. Introduction ........................................................................................... 107
6.2. Expansions and Contractions ................................................................ 108
6.2.1. The transition problem ................................................................ 108
6.2.2. Expansions and Contractions ....................................................... 108
6.3. Drop structures ..................................................................................... 114
6.3.1. Introduction ................................................................................. 114
6.3.2. Free overfall ................................................................................ 114
6.3.3. The head of the overfall ............................................................... 116
6.3.4. The base of the overfall ............................................................... 118
6.3.5. The drop structure ....................................................................... 119
6.4. Stilling basins ....................................................................................... 121
6.4.1. Concept of stilling basin .............................................................. 121
6.4.2. Simple stilling basin design for canals.......................................... 121
6.4.3. Specially designed stilling basins ................................................ 124
6.5. Other types of energy dissipators .......................................................... 128
6.5.1. Stepped spillways......................................................................... 128
6.5.2. Bucket-type and Ski-Jump ........................................................... 129
Chapter 7: UNSTEADY FLOW ..................................................................... 130
7.1. Introduction ........................................................................................... 130
7.2. The equations of motion ....................................................................... 131
7.2.1. Derivation of Saint-Venant equations .......................................... 131
7.2.2. The equations of motion .............................................................. 131
7.3. Solutions to the unsteady-flow equations .............................................. 135
7.3.1. Characteristic differential equations ............................................ 135
7.3.2. Initial condition............................................................................ 138
7.3.3. The simple-wave problem............................................................. 140
7.3.4. Numerical solution of the characteristic differential equations...... 143
7.4. Positive and negative waves; Surge formation ....................................... 145
REFERENCES .................................................................................................. 147
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LIST OF SYMBOLS
A cross-sectional area [m2]
b (average) width of channel [m]
B open-channel free surface width [m]
C Chezy coefficient [m½s-1]
Cc contraction coefficient
Cd discharge coefficient
c natural wave speed [m/s]
cf friction coefficient
D (circular) pipe diameter [m]; hydraulic depth [m]
E mean specific-energy head [m]
F force [N]; momentum transfer per unit width [Nm-1]
Fr Froude-number
f friction coefficient according to Darcy-Weisbach
g gravity constant [m/s2]
H local total energy-head [m]
height of water above crest of weir
HD design head, (i.e. head over spillway crest) [m]
h flow depth, measured perpendicular to channel bed [m]
hc critical water depth [m]
he equilibrium flow depth [m]
hn normal depth at which flow is uniform [m]; hn = he
ho observed water depth [m]
ib slope of channel bed
ic slope of critical-depth line
ie slope of energy grade line
if friction slope
io observed slope; bed slope
ie arithmetic mean slope of the energy grade line
L length (of channel, or pipe or weir) [m]
Lb length of stilling basin [m]
Lcrest crest length in flow direction [m]
n resistance coefficient in flow, called Manning's constant [m-1/3s]
P wetted perimeter [m]; weir height [m]
Pe equilibrium wetted perimeter [m]
p pressure [Pa]
q discharge per meter width [m2/s]
Q total volume discharge [m3/s]
R hydraulic radius [m]
Re Reynolds-number
s flow direction; coordinate along stream line
S (bed) slope; slope of energy gradient line
t time [s]
V depth-averaged or mean flow velocity [m/s]
Vo approach velocity to weir [m/s]
W channel bottom width [m]; top width of flow area [m]; water weight in
channel over a length L [N]
x Cartesian coordinate [m]
y Cartesian coordinate [m]
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z Cartesian coordinate [m]; altitude or elevation,
measured positive upwards [m]
zb change of bottom elevation between two cross-sections [m]
zcrest spillway crest elevation [m]
zo reference elevation [m]; bed elevation [m]
Greek symbols
velocity distribution coefficient; angle (of slope)
deflection angle
h change in flow depth [m]
E change in specific-energy head [m]
H energy-head loss, i.e. change in total energy-head [m]
L length [m]
p pressure difference [Pa]
s small distance along the flow direction [m]
V change in flow velocity [m/s]
zo change in bed elevation [m]
specific weight [N/m3]
dynamic viscosity [Pa.s]
kinematic viscosity [m2/s]
channel slope; angle of channel bed relative to horizontal
density [kg/m3]
surface tension [N/m]
shear stress [Pa]
o mean longitudinal shear stress acting over perimeter [Pa]
Subcripts
c critical flow conditions
conj conjugate flow property
des design flow conditions
e equilibrium flow
f friction
i characteristics of section {i} (in numerical integration process);
running index
j jump
l lateral
m model
max maximum
min minimum
p prototype
r ratio of prototype to model characteristics; roller
x x-component
y y-component
z z-component
1 upstream flow conditions
2 downstream flow conditions
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Abbreviations
LHS left-hand side of equation
sp.gr. specific gravity
sp.wt. specific weight (N/m3)
SI International System of Units (Système International d'unités)
SAF Saint Anthony Falls Hydraulics Laboratory
SIA Swiss Society of Engineers
RHS right-hand side of equation
USBR United States Bureau of Reclamation
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