Open channel hydraulics

Open Channel Hydraulics

To My Family

Open Channel

Hydraulics

A. Osman Akan

AMSTERDAM
BOSTON
HEIDELBERG
LONDON
NEW YORK
OXFORD

PARIS
SAN DIEGO
SAN FRANCISCO
SINGAPORE
SYDNEY
TOKYO

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First edition 2006

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06 07 08 09 10 10 9 8 7 6 5 4 3 2 1

Contents

Preface ix

Acknowledgments xi

CHAPTER

1 Fundamentals of open-channel flow 1

1.1 Geometric elements of open channels 1

1.2 Velocity and discharge 2

1.3 Hydrostatic pressure 2

1.4 Mass, momentum and energy transfer in open-channel flow 7

1.4.1 Mass transfer 7

1.4.2 Momentum transfer 7

1.4.3 Energy transfer 8

1.5 Open-channel flow classification 10

1.6 Conservation laws 11

1.6.1 Conservation of mass 11

1.6.2 Conservation of momentum 12

1.6.3 Conservation of energy 14

1.6.4 Steady flow equations 17

1.6.5 Steady spatially-varied flow equations 18

1.6.6 Comparison and use of momentum and energy equations 20

Problems 20

References 23

CHAPTER

2 Energy and momentum principles 24

2.1 Critical flow 24

2.1.1 Froude number 24

2.1.2 Calculation of critical depth 25

2.2 Applications of energy principle for steady flow 28

2.2.1 Energy equation 28

2.2.2 Specific energy diagram for constant discharge 31

2.2.3 Discharge diagram for constant specific energy 40

2.2.4 Specific energy in rectangular channels 41

2.2.5 Choking of flow 45

2.3 Applications of momentum principle for steady flow 47

2.3.1 Momentum equation 47

2.3.2 Specific momentum diagram for constant discharge 49

2.3.3 Discharge diagram for constant specific momentum 53

2.3.4 Hydraulic jump 54

2.3.5 Specific momentum in rectangular channels 58

2.3.6 Hydraulic jump in rectangular channels 61

2.3.7 Choking and momentum principle 63

Problems 64

References 66

CHAPTER

3 Normal flow 67

3.1 Flow resistance 67

3.1.1 Boundary layer and flow resistance 68

3.1.2 The Darcy–Weisbach equation 70

3.1.3 The Chezy equation 71

3.1.4 The Manning formula 72

3.2 Normal flow equation 74

3.3 Normal depth calculations in uniform channels 76

3.4 Normal depth calculations in grass-lined channels 80

3.5 Normal depth calculations in riprap channels 83

3.6 Normal flow in composite channels 86

3.7 Normal flow in compound channels 88

Problems 92

References 96

CHAPTER

4 Gradually-varied flow 97

4.1 Classification of channels for gradually-varied flow 98

4.2 Classification of gradually-varied flow profiles 99

4.3 Significance of Froude number in gradually-varied flow calculations 101

4.4 Qualitative determination of expected gradually-varied flow profiles 104

4.5 Gradually-varied flow computations 110

4.5.1 Direct step method 111

4.5.2 Standard step method 118

4.6 Applications of gradually-varied flow 121

4.6.1 Locating hydraulic jumps 121

4.6.2 Lake and channel problems 124

4.6.3 Two-lake problems 130

4.6.4 Effect of choking on water surface profile 134

4.7 Gradually-varied flow in channel systems 141

vi
Contents

4.8 Gradually-varied flow in natural channels 144

Problems 151

References 156

CHAPTER

5 Design of open channels 157

5.1 General design considerations 157

5.2 Design of unlined channels 159

5.2.1 Maximum permissible velocity method 159

5.2.2 Tractive force method 163

5.2.3 Channel bends 172

5.3 Design of channels with flexible linings 174

5.3.1 Design of channels lined with vegetal cover 175

5.3.2 Design of riprap channels 179

5.3.3 Temporary flexible linings 186

5.4 Design of rigid boundary channels 188

5.4.1 Experience curve approach 189

5.4.2 Best hydraulic section approach 191

5.4.3 Minimum lining cost approach 192

5.5 Channel design for non-uniform flow 194

Problems 197

References 198

CHAPTER

6 Hydraulic structures 200

6.1 Flow measurement structures 200

6.1.1 Sharp-crested weirs 200

6.1.2 Broad-crested weirs 207

6.1.3 Flumes 209

6.2 Culverts 212

6.2.1 Inlet control flow 214

6.2.2 Outlet control flow 220

6.2.3 Sizing of culverts 225

6.3 Overflow spillways 225

6.3.1 Shape for uncontrolled ogee crest 226

6.3.2 Discharge over an uncontrolled ogee crest 227

6.3.3 Discharge over gate-controlled ogee crests 230

6.4 Stilling basins 232

6.4.1 Position of hydraulic jump 232

6.4.2 Hydraulic jump characteristics 238

6.4.3 Standard stilling basin designs 239

6.5 Channel transitions 244

Contents
vii

6.5.1 Channel transitions for subcritical flow 244

6.5.2 Channel transitions for supercritical flow 252

Problems 261

References 264

CHAPTER

7 Bridge hydraulics 266

7.1 Modeling bridge sections 266

7.1.1 Cross-section locations 266

7.1.2 Low-flow types at bridge sites 269

7.1.3 Low-flow calculations at bridge sites 269

7.1.4 High-flow calculations at bridge sites 284

7.2 Evaluating scour at bridges 294

7.2.1 Contraction scour 296

7.2.2 Local scour at piers 303

7.2.3 Local scour at abutments 308

Problems 311

References 314

CHAPTER

8 Introduction to unsteady open-channel flow 315

8.1 Governing equations 315

8.2 Numerical solution methods 318

8.2.1 Explicit finite difference schemes 319

8.2.2 Implicit finite difference schemes 321

8.2.3 Special considerations 338

8.2.4 Channel systems 341

8.3 Approximate unsteady-flow models 342

8.3.1 Diffusion-wave model for unsteady flow 342

8.3.2 Finite difference equations 343

8.3.3 Solution of finite difference equations 344

8.4 Simple channel-routing methods 347

8.4.1 The Muskingum method 347

8.4.2 The Muskingum–Cunge method 351

Problems 357

References 358

Index 361

viii
Contents

Preface

This book was conceived as a textbook for undergraduate seniors and first-year

graduate students in civil and environmental engineering. However, I am

confident the book will also appeal to practising engineers. As a registered

professional engineer, and having taught a number of graduate courses over the

years attended by full-time engineers, I am familiar with what is needed in the

engineering practice.

The students are expected to have had a fluid mechanics course before

studying this book. Chapter 1 presents a review of fluid mechanics as applied to

open-channel flow. The conservation laws are revisited, and the equations of

continuity, momentum, and energy are derived. In Chapter 2, the applications

of the energy and momentum principles are discussed along with the problem of

choking in steady flow. It is also demonstrated that the hydraulic behavior of

open-channel flow can be very different under the subcritical and supercritical

conditions. Also, the phenomenon of hydraulic jump is introduced. Chapter 3 is

devoted to normal flow. A brief description of flow resistance formulas is first

provided in relation to the boundary layer theory, and then the normal flow

calculations for uniform, grass-lined, riprap, composite, and compound channels

are presented. Chapter 4 deals with water surface profile calculations for

gradually-varied flow. I realize that this can be a difficult subject at first, since the

boundary conditions needed to calculate a water surface profile depend on the

type of the profile itself. Therefore, in Chapter 4, I have attempted to emphasize

how to identify the flow controls, predict the profile, and formulate a solution

accordingly. Once the solution is correctly formulated, the numerical calculations

are easily performed. Chapter 5 is devoted to the hydraulic design of different

types of open channels. Several charts are provided to facilitate the lengthy

trial-and-error procedures we often need. Chapter 6 discusses various flow￾measurement structures, culverts, spillways, stilling basins, and channel transi￾tions. Chapter 7 is devoted to bridge hydraulics. First the flow calculations are

discussed in the vicinity of bridge sections, then the contraction and local scour

phenomena are described, and finally empirical equations are given to estimate

the total bridge scour. The subject of unsteady open-channel flow, by itself, could

be an advanced-level graduate course. Therefore, no attempt is made in this

book to cover this subject thoroughly. However, while Chapter 8 is only an

introduction to unsteady flow, it includes enough information to help a student to

develop an implicit finite difference model. Simpler channel routing schemes are

also discussed.

I mean to give the students a solid background on the fundamental principles

and laws of open-channel flow in this book. However, the book also includes

numerous detailed, worked-out examples. Where applicable, these examples are

enriched with underlying arguments derived from the basic laws and principles

discussed in earlier sections.

I believe that the first five chapters provide adequate material for an

undergraduate open-channel hydraulics course for civil and environmental

engineering students. Selected sections from Chapter 6 can also be included

instead of Chapter 5. It is suggested that all eight chapters be covered if the book

is used for a graduate course. However, in that event, less time should be spent

on the first three chapters.

Most of the equations adopted in the book are dimensionally homogeneous,

and can be used in conjunction with any consistent unit system. The unit-specific

equations are clearly identified.

Various design procedures are included in the book. These procedures

heavily rely upon the available experimental and field data, such as the allowable

shear stress for earthen channels or various coefficients for bridge scour

equations. The reader should understand that all this empirical information is

subject to change as more effort is devoted to open-channel studies. Also, for

real-life design problems, the reader is urged to review the references cited since

it is impossible to include all the details, assumptions, and limitations of the

procedures that can be found only in the design manuals. Moreover, obviously,

local manuals and ordinances should be followed for designing hydraulic

structures where available.

x
Preface

Acknowledgments

I am thankful to Professor Cahit C¸|ray, who introduced me to the fascinating

subject of open-channel hydraulics when I was an undergraduate student at

Middle East Technical University. I attended the University of Illinois for my

graduate studies, and received my MS and PhD degrees under the supervision of

Ben C. Yen, from whom I learned so much. Dr Yen, a gentleman and scholar,

remained my friend, teacher, and mentor until he passed away in 2001. He always

has a warm place in my heart. I only hope that he would be proud if he saw this

book published. I am indebted to Ven Te Chow and F. M. Henderson for their

earlier books on open-channel hydraulics, which I studied as a student. I still use

these books frequently for reference. I have learned from the work of many other

authors and colleagues that I cannot enumerate here, and I am grateful to all.

I would like to thank John Paine for reviewing parts of chapter 5 and for his

suggestions. I would like to thank my students for pointing out some errors when

the draft manuscript was used as a course-pack. I also would like to thank Old

Dominion University for the institutional support I received during the

preparation of this book. Old Dominion University is a wonderful institution

for students to learn and for faculty to teach and conduct research.

I am most indebted to my wife, Gu¨zin, and my son, Doruk, for all the

happiness, love, inspiration, and support they have given me throughout this

project and always.

1 Fundamentals of

open-channel flow

Open channels are natural or manmade conveyance structures that normally

have an open top, and they include rivers, streams and estuaries. An important

characteristic of open-channel flow is that it has a free surface at atmospheric

pressure. Open-channel flow can occur also in conduits with a closed top, such as

pipes and culverts, provided that the conduit is flowing partially full. For

example, the flow in most sanitary and storm sewers has a free surface, and is

therefore classified as open-channel flow.

1.1 GEOMETRIC ELEMENTS OF OPEN CHANNELS

A channel section is defined as the cross-section taken perpendicular to the main

flow direction. Referring to Figure 1.1, the geometric elements of an open

channel are defined as follows:

Flow depth, y Vertical distance from the channel bottom to the

free surface.

Depth of flow section, d Flow depth measured perpendicular to the

channel bottom. The relationship between

d and y is d ¼ y cos
. For most manmade

and natural channels cos

1.0, and

therefore y
d. The two terms are used

interchangeably.

Top width, T Width of the channel section at free surface.

Wetted perimeter, P Length of the interface between the water

and the channel boundary.

Flow area, A Cross-sectional area of the flow.

Hydraulic depth, D Flow area divided by top width, D ¼ A/T.

Hydraulic radius, R Flow area divided by wetted perimeter, R ¼ A/P.

Bottom slope, S0 Longitudinal slope of the channel bottom,

S0 ¼ tan

sin
.

Table 1.1 presents the relationship between various section elements. A similar,

more detailed table was previously presented by Chow (1959).

1.2 VELOCITY AND DISCHARGE

At any point in an open channel, the flow may have velocity components in

all three directions. For the most part, however, open-channel flow is assumed

to be one-dimensional, and the flow equations are written in the main flow

direction. Therefore, by velocity we usually refer to the velocity component in

the main flow direction. The velocity varies in a channel section due to the

friction forces on the boundaries and the presence of the free-surface. We use

the term point velocity to refer to the velocity at different points in a channel

section. Figure 1.2 shows a typical distribution of point velocity, v, in a

trapezoidal channel.

The volume of water passing through a channel section per unit time is called the

flow rate or discharge. Referring to Figure 1.3, the incremental discharge, dQ,

through an incremental area, dA, is

dQ ¼ vdA ð1:1Þ

where v ¼ point velocity.

Then by definition,

Q ¼

Z

A

dQ ¼

Z

A

vdA ð1:2Þ

where Q ¼ discharge.

In most open-channel flow applications we use the cross-sectional average velocity,

V, defined as

V ¼ Q

A ¼ 1

A

Z

A

vdA ð1:3Þ

1.3 HYDROSTATIC PRESSURE

Pressure represents the force the water molecules push against other molecules

or any surface submerged in water. The molecules making up the water are in

T

P

A y d

Water surface

Channel bottom

FIGURE 1.1 q

Definition sketch for

section elements

2
1 Fundamentals of open-channel flow