Final Project Sub

7/30/2019 Final Project Sub

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BACKWATER CALCULATIONSCOMPARISON OF DIFERENT

NUMERICAL METHODS (DATA ANALYSIS AND NUMERICAL

MODELLING)

Submitted by

AJALA OLUKUNLE MOSES

For the

MSc in Civil Engineering

LONDON SOUTH BANK UNIVERSITY

Faculty of Engineering, Science and the

Built Environment (FESBE)

November 2010


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Backwater Calculations

Abstract Page ii

Abstract

The most common occurrence of gradually varied flow is the backwater created by channels,

storm sewer inlets, or channel constrictions. For these conditions, the flow depth needs to be

greater than the normal depth in the channel and the water surface profile should be computed

using backwater techniques.

There are many reasons why it is necessary to compute water-surface profiles. For instance

the stage of a river may need to be determined for a given discharge; or it may be necessary to

calculate Manning's "n" for a cross section. The main objective of using backwater techniques

for computation is to determine the shape of the flow profile. There are many methods for

analysing water surface profile, two of which would be compared in this thesis namely:

Direct step method : first suggested by Polish engineer Charnomskii in 1914 and then

by Husted in 1926 (Chow 1959)

ISIS software packaging for river modelling

In this thesis, three different channels (rectangular, trapezoidal and circular) would be

analysed. All channels have different geometrical cross-section area to enable a fair

comparison of the methods, based on the assumption that the flow is gradually varied flow

and the channel is prismatic.

To begin with, calculating the coordinate of the water surface profile using direct step method

is an iterative process achieved by choosing a range of flow depths, beginning at the

downstream end, and proceeding incrementally up to the point of normal flow depth (Chow


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Abstract Page iii

1959). This method is best accomplished by the use of spread sheet (see Table 1, 4 and 5).

This table is then used to compute the flow profile required (see Figs. 4-1, 4-3 and 4-5)

Subsequently ISIS analyses were performed for all the channels to confirm the findings of the

direct step method and the water surface profile computed. Both methods showed similarities

in the water surface profiles for all the three channels as expected.


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Acknowledgements Page iv

Acknowledgements

There are lots of people that I have to thank after one year of Master Study, hard work and

unforgettable experiences.

First I would like to thank my project supervisor, Dr. Stephen Mitchell, for his guidance,

support, word of encouragement, invaluable suggestions and contributions throughout this

project.

A special regard again goes to Dr. Stephen Mitchell for his enormous time, effort and

assistance with regards to advice on presentation of the work. And I wish you good luck with

your new job.

Also, my deep respect goes to the entire staff of Engineering Department. My greatest regards

goes to you Mr Abdul Rahim for your massive support throughout my studies in London

South Bank University without you I don't think my master program would have been a

reality. Claudia I have nothing for you but love and forever you guys will remain in my heart.

Appreciation is also extended to my family and friends: Mr and Mrs Ajala, Mr and Mrs Toye,

Gavin Okoror, Jummy Ibidapo-Obe, Mr and Mrs Akanbi, Niyi Akanbi, Funke Alimi. I

sincerely appreciate your support and words of encouragement.

My deepest gratitude goes to my best friend Gmoney, thank you for rocking my world over

the last few yearsyou are the best.

A special thanks to Sylvia Okuku, remember the time when I needed someone to just push me

to reach for this goal? You are heaven sent!!

My greatest pleasure to Mrs Madu for your word of support (have you handed in your

project?). Also, my deepest feeling goes to Yvonne Madu for your word of encouragement.

Without you, I would not have been capable to bring this to an end.

Additionally, I wish to thank my line manager Mark Vara, Joe Price and Craig for not giving

up on me on shift changing. It was a pleasure to work under your supervision.

Lastly but most importantly, I thank God for making the completion of my Master degree a

reality, and my parents Mr and Mrs Ajala, thank you for bringing me up in the way of the

Lord, without which this would not be possible. Love you loads!!!

People like this are essential to make a project possible.

Ajala Olukunle Moses

26

th

October, 2010


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List of Figures Page v

Table of Contents

List of Tables ............................................................................................................................ iix

Common Notations .................................................................................................................... x

CHAPTER ONE ........................................................................................................................ 1

Introduction ................................................................................................................................ 1

1.1 Background .................................................................................................................. 1

1.2 Aim and Objectives ........................................................................................................ 6

1.3 Scope of the Thesis ........................................................................................................... 6

1.4 Thesis Layout ................................................................................................................... 6

CHAPTER TWO........................................................................................................................ 8

Literature Review ....................................................................................................................... 8

2.1 Development of Backwater Equation ............................................................................... 8

2.2 Backwater Calculations with Ferut ................................................................................. 14

2.3 Backwater Computation for Transcritical River flows ................................................... 16

2.3.1 Energy and Momentum Equation ............................................................................ 18

2.3.2 Discrete Solution ...................................................................................................... 21

2.4 Conclusion ...................................................................................................................... 25

CHAPTER THREE .................................................................................................................. 27

3.1 Methods of Computation ................................................................................................ 27

3.2 Choosing a Method ......................................................................................................... 27

3.3 Direct Step Method ......................................................................................................... 28

3.4 Computation of Backwater Profile by the Direct Step Method ...................................... 32

3.4.1 Rectangular Channel ................................................................................................ 32


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List of Figures Page vi

MODEL 1 ......................................................................................................................... 32

3.4.2 Trapezoidal Channel ................................................................................................ 33

MODEL 2 ......................................................................................................................... 33

3.4.3 Circular Channel ...................................................................................................... 34

MODEL 3 ......................................................................................................................... 34

3.5 Data's Analysis Computation Procedures ..................................................................... 35

3.6 Building the Model in ISIS ............................................................................................. 37

3.7 Translate the descriptive cross section geometry to ISIS format ................................... 44

3.8 Backwater Profile Outputs by the ISIS Method ............................................................. 45

3.8.1 Rectangular Channel ................................................................................................ 45

3.8.2 Trapezoidal Channel ................................................................................................ 45

3.8.3 Circular Channel ...................................................................................................... 46

CHAPTER FOUR .................................................................................................................... 47

Results and Discussion ............................................................................................................. 47

4.1 Analysis of Results for Direct Step Method and ISIS Method ....................................... 47

CHAPTER FIVE ...................................................................................................................... 61

Conclusions and Recommendations ......................................................................................... 61

5.1 Conclusions .................................................................................................................... 61

5.2 Different Between the Direct Step Method and ISIS Method ........................................ 62

5.3 Similarities ...................................................................................................................... 63

5.4 Recommendations ........................................................................................................ 64

References and Bibliography ................................................................................................... 65

APPENDIX .............................................................................................................................. 67


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List of Figures Page vii

List of Figures

Figure 1-1: Outlet Control Headwater for Channel with Free Surface (after Hydraulics

Manual (2009)) 1

Figure 2-1: The smooth drop inlet experiment photographFlow from the bottom right to top

left (after Chanson (2009)). ...................................................................................................... 12

Figure 2-2: Comparison between experimental data and backwater calculations (after

Chanson (2009)). ...................................................................................................................... 13

Figure 2-3: Free-surface measurement (DARCY and BAZIN 1865) (after Chanson (2009)). 13

Figure 2.4: Definition sketch of non-uniform flow (after Chow (1959))................................. 17

Figure 2-5: Calculation intervals above which the scheme is numerically stable as a function

of flow depth and Froude number for a hydraulically wide section (after Beffa (1996)) ........ 23

Figure 2-6: Calculation flow depths in a Transcritical channel using different flow equations

(after Beffa (1996)) .................................................................................................................. 23

Figure 3-1: Channel reach for the derivation of step methods ................................................. 29

Figure 3-2: Upstream cross section for the rectangular channel .............................................. 32

Figure 3-4: A schematic representation of a single channel with eight sections and two

boundary conditions (after Batica (2009), Fu (2009)) ............................................................. 38

Figure 3-5: Simple schematic diagram of a Flow-Time data entry form (after Batica (2009),

Fu (2009)) ................................................................................................................................. 39

Figure 3-6: Connecting a river unit to a QTBDY unit entry form (after Batica (2009), Fu

(2009)) ...................................................................................................................................... 39

Figure 3-7: Schematic specification of cross section data form (after Batica (2009), Fu (2009))

.................................................................................................................................................. 40


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List of Figures Page viii

Figure 3-8: The Stage-Time data entry form for the downstream boundary condition (after

Batica (2009), Fu (2009)). ........................................................................................................ 41

Figure 3-9: Run form interface (after Batica (2009), Fu (2009)). ............................................ 42

Figure 3-10: Notification of the successful completion of a simulation (after Batica (2009), Fu

(2009)). ..................................................................................................................................... 42

Figure 3-11: The progress of the unsteady simulation in ISIS (after Batica (2009), Fu (2009)).

.................................................................................................................................................. 43

Figure 3-12: Simple schematic for conversion of cross-section geometry to ISIS format (after

Batica (2009), Fu (2009)). ........................................................................................................ 44

Figure 3-13: The Cross-Section Data for Section 30 ............................................................... 45



Figure 4-1: An S2 Flow Profile Computed by the Direct Step Method for Rectangular

Channel ..................................................................................................................................... 50

Figure 4-2: Elevation vs. Nodal Label Output by the ISIS Method for Rectangular Channel A

.................................................................................................................................................. 51

Figure 4-3: An M1 Flow Profile Computed by the Direct Step Method for Trapezoidal

Channel ..................................................................................................................................... 55

Figure 4-4: Elevation vs. Nodal Label Output by the ISIS Method for Trapezoidal Channel . 56

Figure 4-5: An S2 Flow Profile Computed by the Direct Step Method for Circular Channel . 59

Figure 4-6: Elevation vs. Nodal Label Computed by the ISIS Method for Circular Channel . 60


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List of Tables Page ix

List of Tables

Table 2-1: Computed Flow Depth in a Rectangular Channel Using Momentum Equation

(2.10), Energy Equation (2.11), and Reduced Momentum Equation (2.12) (after Beffa (1996))

.................................................................................................................................................. 24

Table 1: Computation of Flow Profile by the Direct Step Method for a Rectangular Channel49

Table 2: Critical Depth and Normal Depth Computation by Direct Step Method ................... 52

Table 3: Result Values for Froude Number and Velocity for Both Direct Step and ISIS

Method ..................................................................................................................................... 53

Table 4: Computation of Flow Profile by the Direct Step Method for a Trapezoidal Channel 54

Table 5: Computation of Flow Profile by the Direct Step Method for a Circular Channel ..... 58


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Common Notations Page x

Common Notations

A AREA OF FLOW: (m2)

b WIDTH' OF RECTANGULAR CHANNEL: (m)

Da DIAMETER OF CONDUIT: (m)

D DEPTH OF FLOW: (m)

Fr FROUDE NUMBER

H TOTAL HEAD

y CHANGE IN WATER SURFACE ELEVATION :( m)

g ACCELERATION OF GRAVITY ( 9.81m/s2)

K CONVEYANCE: (m3/s)

L BACKWATER LENGHT: (m)

n MANNING'S ROUGHNESS COEFFICIENT

P WETTED PERIMETER :( m)

Q DISCHARGE: (m3/s)

R HYDRAULIC RADIUS = (A/P): (m)

SF FRICTION SLOPE

S0 BED SLOPE

T TOP WITH OF CHANNEL: (m)

V VELOCITY: (m/s)

Y FLOW DEPTH: (m)

yc CRITICAL DEPTH: (m)

yn NORMAL DEPTH :( m)

Z SIDE SLOPE OF A CHANNEL


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AJALA OLUKUNLE MOSES Page 1

CHAPTER ONE

Introduction

1.1BackgroundA backwater condition (or subcritical flow) exists if there are flow restrictions that raise the

water level above the normal depth within a given channel reach. As such, the backwater

profile must be computed to verify that the channel's capacity is adequately designed, if

backwater conditions are found to exist for the design flow. Analysis and computation of

backwater profile in open channels are important from the point of view of safe and optimal

design and operation of any hydraulic structure.

A typical configuration considered as a prototype case study for this thesis is shown in Fig. 1-

1. The hydraulics profiles change with flow depth along the length of the channel if free

surface flow occurred in a channel. It is compulsory to calculate the backwater profile based

on the outlet depth H0.

Figure 1-1: Outlet Control Headwater for Channel with Free Surface (after Hydraulics

Manual (2009))


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Where:

HWoc = headwater depth due to outlet control (m)

hva = velocity head of flow approaching the channel entrance (m)

hvi = velocity head in the entrance (m)

he = entrance head loss (m)

hf = friction head losses ( m)

So = culvert slope ( m/m)

L = culvert length ( m)

Ho = depth of hydraulic grade line just inside the Channel at outlet (m)

hvo = velocity head inside culvert at outlet (m)

hTW = velocity head in tailwater (m)

ho = exit head loss (m)

dc = critical depth

du = normal depth

A backwater calculation is used to analyse the capacity of a channel systems to convey the

required design flow. In this case, channel system structures must established to contain the

hydraulic grade line as shown in Fig. 1-1 above for the specific flow rate. Direct step method

is used to compute a simple backwater profile (hydraulic grade line) in a channel for the

purpose of verifying adequate capacity of the flow. This method incorporates a re-

arrangement of the Manning's formula (Eq. 1.6) which expressed in terms of friction slope

(i.e. the gradient of the head line in m). The friction slope is used to establish the head loss in

each channel section due to barrel friction, which will be combined with other head losses to


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obtain water surface elevation at all points along the channel system (as discussed in chapter

3). The general equation that this thesis is based upon is shown below:

Gradually Varied Flow Equation:

21 FrSS

dx

dy fo

(1.1)

Where:

So = Bottom slope, positive in the downward direction

Sf= Friction slope, positive in the downward direction

y = Water depth, measured from culvert bottom to water surface

x = Longitudinal distance, measured along the culvert bottom

Fr = Froude number

In English Units the Manning's Equation Form is:

2/13/249.1 SRn

v (1.2)

Where:

v = Average cross section velocity in (m/s)

R= Hydraulic radius, (Wetted Area / Wetted Perimeter in m)

n = Manning's coefficient (dimensionless)values developed through experimentation.

S = Slope of the channel in (m)

If velocity is known, the discharge (Q) can then be computed as
http://www.fsl.orst.edu/geowater/FX3/help/8_Hydraulic_Reference/Froude_Number_and_Flow_States.htmhttp://www.fsl.orst.edu/geowater/FX3/help/8_Hydraulic_Reference/Froude_Number_and_Flow_States.htmhttp://www.fsl.orst.edu/geowater/FX3/help/8_Hydraulic_Reference/Froude_Number_and_Flow_States.htm


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Q = AV (1.3)

2/13/249.1 SARnQ (1.4)

Where Q is the discharge in m3/s

For uniform flow, Q is referred to as Normal discharge

The above equation can also be re-arranged such that:

2/13/2

SnQAR (1.5)

The Friction Slope is approximated From Manning's Equation Above:

3

4

22

R

VnSf

(1.6)

Where:

Sf= Friction slope, positive in the downward direction

n = Manning's roughness coefficient (various values of n are included in the appendix)

V = Average cross section velocity

= Constant equal to 1.49 for English units and 1.00 for SI units

R = Hydraulic radius (Area / Wetted Perimeter)

InGeneral, the gradually varied flow will be the only flow to be discussed in this thesis. For

the gradually varied flow condition, the depth of flow must be established through a water

surface profile analysis. The basic principles in water surface profile analysis are where:

Water surface approaches the uniform depth line asymptotically,


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Water surface approaches the critical depth line at a finite angle,

Subcritical flow is controlled from a downstream location, and

Supercritical flow is controlled from an upstream location.

There are three general methods for determining flow profiles in prismatic channels (channel with

unvarying cross section, constant bottom slope, and relatively straight alignment) for backwater

computation:

1. The Direct Step method

2. The Standard Step method

3. Direct integration

These three methods make use of the energy equation to compute the water surface profile.

The direct integration and direct step method analyse straight prismatic channel sections while

the standard step method analyses nonprismatic channels (when the cross section, alignment,

and/ or bottom slope change along the channel). The direct integration method solves the

varied flow equation to determine the length of reach between successive depths. This method

is not commonly used unless sufficient profiles and length of channel are involved to

guarantee the amount of pre computational preparation.

In this thesis, the direct step method and ISIS program for river modeling will be used to

analyze the water surface profile in three different channels namely: rectangle, trapezoidal and

circular channels. These methods would be used to determine the length of reach between

successive depths by solution of the energy and friction equation that will be written for end

section of the reach.


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1.2 Aim and Objectives

The purpose of this thesis is to investigate the changes in water surface profile with flow

depth in three different channels (rectangle, trapezoidal and circular). This will be done by

adopting Manning's equation to develop backwater calculations by using the direct step

method. For comparison, ISIS software packages for river modelling would be developed to

model a similar water surface profile. The results from both investigations (data analysis and

numerical modelling) will be compared to check that the methods conform together.

1.3 Scope of the Thesis

This thesis covers backwater calculations in an open channel flow. It gives necessary

background theories on gradually varied flow and water surface profile in open channels. It

does not attempt to give elaborate information on backwater calculation in any other media

other than open channels; neither does it examine deeply on various forms of water profiles.

It was considered reasonably appropriate to carry out calculation analysis studies for the

backwater calculations on rectangular, trapezoidal and circular channels.

1.4 Thesis Layout

The present thesis is organised into six chapters and a few appendices. Some of these chapters

are based on further development from previous chapters, but in general, all of them can be

read independently as self-contained entities.

Chapter 2 presents in-depth information about the research carried out in the last decade on

the backwater calculation equations.


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Chapter 3 shows different methods of computation of flow profiles in prismatic channels,

analyses of basic equations for gradually varied flow and procedures of using direct step

method and ISIS program for analysis.

Chapter 4 gives the computation results for both the direct step method and ISIS analysis.

Chapter 5 presents the discussions based on the calculated results in chapter 4, with respect to

the conclusions of the thesis, they are presented in detail at the end of each chapter and the

most important concepts are summarised in chapter 6.


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CHAPTER TWO

Literature Review

The backwater calculation is not an easy task to carry out without solid background

knowledge on the theme. However in the past decades, this subject has attracted the attention

of many researchers around the world. This chapter summarizes the most important

developments and results found in the specialized research literature on development of

backwater equation.

2.1 Development of Backwater Equation

Belanger (1828) developed the backwater equation with following assumptions in other to

calculate the free-surface profile of gradually varied flow in an open channel.

A steady flow

A one-dimensional flow motion

A gradual variation of the wetted surface with distancex along the channel

Frictional losses that are discharge

A hydrostatic pressure distribution

Backwater equation (Eq. 2.1) was derived by Belanger from momentum considerations in a

method somehow comparable to the modern development of normal flow conditions

(Henderson 1966; Chanson 1999, 2004) obtained

0cossin 32

2 AbVaVA

Pd

gA

Qw (2.1)


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Where:

between the bed and the horizontalx = longitudinal distance positive downstream

d = flow depth measured normal to the invert

A = cross-sectional area

Pw = wetted perimeter

Q = discharge

In Belanger (1828) Eq. (16) corresponds to Eq. (2.1) above, the equation is now

rewritten in a more conventional form as a differential equation as:

0cossin 322

AbVaVAPd

gA

Qw

(2.2)

Belanger used equations (2.1) and (2.2) to estimate the frictional losses using the prony

formula to produce:

gV

D

fbVaVDx

H

HH 2

4 22

(2.3)

w

HP

AD

4

Where:

H = total head


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DH = hydraulic diameter

a and b are constant

Numerous values were proposed for the coefficient of a and b. BELANGER (1828) used a =

4.44499 10-5

and b =3.093140 10-4

(in SI units) that were estimated by Johann EYELWEIN

(Chanson 1999, 2004). The values of a and b in Prony formula and Belanger (1828) are with

the same accuracy as reported. The right term in Eq. (2.3) above corresponded to the

traditional expression of the head losses in terms of the Darcy-Weisbach frictional factor f.

Where Sf is denoted as the friction slope (Sf = -H/x), and S0 as the bed slope (S0 = sin),

continuity equation may be combined with the Belanger's backwater Eq. (2.1) to yield

fSSg

Vd

x

0

2

2cos

(2.4)

Eq. (2.4) is similar to the modern expressions of backwater equation (Henderson 1966;

Montes 1998; Chanson 2004). Backwater equation is expressed in general form by

Chanson as

fSSx

A

Ag

Q

xd

x

d

03

2

sincos

(2.5)

(Chanson 1999)

The main differences between Belanger's Eq. (2.4) and Eq. (2.5) above are the Coriolis

coefficient () or kinetic energy correction coefficient and the non constant bed slope

term. Based on the Belanger's developed equation (Chanson 2009) no further assumption

was made and is basically the same to the modern forms of the backwater equation


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adopted by today's hydraulic engineers. BELANGER introduced the kinetic energy

correction coefficient t in a later development of the backwater equation. Eq. (2.1) was

tested for a non-prismatic smooth drop inlet. The experimental facility and the comparison

between the experimental observation with Eq. (2.1) are shown in Fig 2-1 and 2.2 in

which the flow resistant was calculated using Prony formula (left hand side of Eq. (2.3),

with equation (2.4) in which the friction slope was calculated in term of the Darcy friction

factor, and with Eq. (2.5). All of this calculation was performed using the step method,

distance calculated from depth (Chanson 2009).

The experimental data are plotted together with the bed elevation Zo and sidewall profile,

and the results show very little differences between data and calculations. But overall they

all correlate well with the computations (Fig 2-2) despite the difficult geometry and the

crude nature of the prony formula. BELANGER (1828) computations give the same

results to modern estimates. But JeanBaptiste BELANGER integrate backwater

calculations manually without the uses of neither computer nor calculator, nor even slide

rule and this explains the common usage of PRONY's simplified formula at the time

(BROWN 2002).

Jean-Baptiste BELANGER selected know water depth to integrate backwater equation

and calculating manually the distance in between. This was done by integrating two water

depth limits BELANGER (1828). The method is known as step method distance

calculated from depth (HENDERSON 1966; CHANSON 2004) today or the direct step

method.

The originality of BELANGER's 1828 work proved the successful development of the

backwater equation for steady one-dimensional gradually-varied flows in an open channel


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(Chanson 2009). BELANGER's work was drawn out the fundamental assumption and

worked out an equation of gradually varied in open channel flows, which was derived

from momentum equation that is still in use today, but for the flow resistant model.

Jean-Baptiste BELANGER also introduced further modern concepts:

The step method

Distance calculated from depth and the critical flow conditions

Jean-Baptiste BELANGER's technique of numerical integration was ahead of his time, when

there was neither computer nor electronic calculator which makes him investigated further.

(Chanson 2009)

Figure 2-1: The smooth drop inlet experiment photograph Flow from the bottom right

to top left (after Chanson (2009)).


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Figure 2-2: Comparison between experimental data and backwater calculations (after

Chanson (2009)).

Figure 2-3: Free-surface measurement (DARCY and BAZIN 1865) (after Chanson

(2009)).


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BELANGER investigated further on the two singularities of the water equation. One

matched up with uniform equilibrium flow condition S0 = Sffor which the normal depth is

equal to the flow depth achieved the normal depth expression of PRONY (1804):

BELANGER (1828)

sin

4

)( 2 HD

bVaV

(2.6)

The second singularity of the backwater equation corresponded to x/d = 0 which gives:

1cos 3

2

d

A

Ag

Q

(2.7)

Eq. (2.7) corresponded to the critical flow condition in a channel of non-prismatic channel

with hydrostatic pressure distribution. In a case of a channel of regular cross-section, i.e. a

prismatic rectangular open channel, Eq. (2.7) gives a typical result: V2=g d cos

(LIGGET 1993; CHANSON 2006).

2.2 Backwater Calculations with Ferut

Ferut (1952) was a first-generation British electronic computer that was installed at the

University of Toronto in 1952 (Campbell 2009). The role of Manchester pioneer, which

carried the computer from Manchester to Canada on its Voyage across the Atlantic in April

1952, cannot be left without a proper prologue, long before the construction of the St

Lawrence Seaway begin in 1954. In the early 1950s, severe planning was underway

concerning St. Lawrence Seaway and Power Project, but before construction can commence,

a lengthy series of backwater calculations was required to predict upriver changes to the water


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profile. The calculations were made much complicated by the numerous islands along the

route and 99 specific backwater cases were identified (Gotlieb 1960).

When Ferut was discovered the first real problem was to learn to write programs. Despite four

years spent developing UTEC, almost no one in the Computation Centre (or for that matter,

anywhere in the world) had much experience programming electronic computers. The

computer Centre turned to Christopher Strachey for immediate help. There were several

problems for Strachey in preparing the backwater calculations on Ferut. The problems were:

1.

Numerical scale: the Ferut was a fixed-point machine with no hardware facilities for

floating-point arithmetic.

2. The data from the 267 stations along the St. Lawrence River far exceeded the primary

and secondary storage capability of Ferut.

Although it was Strachey's mission to teach the Computation Centre staff how to program, he

received most of the credit for the backwater program and most of Strachey hand written

notes, routines, and plans survived (Strachey 1953). McFarlane verified that the computer

and computer program were producing proper results, most of the character backwater cases

were carried out by hand on a desk-top calculator. The hand and computer results were

compared with favourable results: the water level profile was nearly identical, mostly through

the relevant 29 km section (McFarlane 1960).

The comparison of Strachey and McFarlane's work show that the entire set of backwater

computations would have taken 20 person-years of time, if done manually. However, it took

Ferut just 500 h of machine time, with all the calculations finished in May 1953 (Gotlieb

1960). Construction of the St. Lawrence Seaway and Power Project did not begin until 1954


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and by then it was no longer an all-Canadian enterprise. Just as Ferut had been printing the

final backwater results in 1953, the white house publicly announced that it would seek to

convince Congress and the Americans that the Canadians were capable or ready to begin

construction, but all signs north of the border were of a country anxious to begin (Mabee

1961).

President Eisenhower signed the Wiley-Dondero Seaway Act in May 1954, authorizing US

participation in the St. Lawrence Seaway and Power Project, and another four month before

ground was broken in August. When completed in 1959 it was officially opened by Her

Majesty Queen Elizabeth II and President Dwight D Eisenhower (Campbell 2009).

2.3 Backwater Computation for Transcritical River flows

The construction of the St. Lawrence Seaway and Power Project gives an engineer the

solution limit that the upstream backwater computation can be extended to Froude number

exceeding one (Beffa 1996). It was discovered that backwater computation can be applied to

model steady flows in non-uniform channels (Fig. 2.4).The method can also be used to

analyse slightly unsteady flows (e.g. water surface profiles for flood waves or even dam break

waves). For subcritical flows the computation is more simple, whereas it becomes more

complicated for flows which change from one state to another (i.e. transcritical flows). In this

type of flow, the direction of the computation varied from upstream to downstream or vice

versa depending on the Froude number (Molinas and Yang 1985).


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Figure 2.4: Definition sketch of non-uniform flow (after Chow (1959))

This practice is well suitable for well-defined hydraulic jumps (e.g. in abrupt changes of bed

slope) but is more complex to deal with if the flow conditions oscillate between subcritical

and supercritical from one cross section to the other. This always happens in steep natural

channels where supercritical flows may occur for short distance and varies between

subcritical and supercritical (Trieste 1992; Wahl 1993). For near critical flow conditions, no

hydraulics jump is formed but standing waves occur with less energy dissipation than in the

formal practice which is well suitable for well defined hydraulics jumps. Analysis of the

discrete equation for steady flow open-channel flows shows that the solution of the upstream

backwater computation can be extended to Froude numbers exceeding one. An iterative

method is proposed that allows for solution of either the momentum equation or the energy up

to a limiting Froude number. The method is especially useful for transcritical flows in natural

rivers where the flow oscillates between subcritical and supercritical, and changing the

direction of calculation would be impracticable.


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Adopting a different iteration process, Chawdhary and Samuels (1990) found that the

backwater computation can be applied for transcritical flows without altering the direction of

the calculation (Beffa 1996). The energy and momentum equation shows that the limitation

on Froude number depends on the energy looses and the size of the calculation interval. These

limits are analyzed and a procedure is proposed, for higher Froude numbers, which minimizes

divergence from a solution based on the complete flow equation shows below.

2.3.1 Energy and Momentum Equation

Let's assume hydrostatic pressure distribution, the flow through a cross section may be

characterized either by the total energy head

2

2

2gA

QH

(2.7)

Where:

Q = discharge;

A = wetted area;

= velocity distribution coefficient;

g = gravitational;

= water surface level above data or by the specific force

yAgA

QM

2(2.8)


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Where

= momentum coefficient

y = distance from the water surface to the centroid of the cross section.

The energy equation for steady non-uniform channel flow basically defined equilibrium

between the gradient of the energy head and energy slope SH, i.e.

02 2

2

HH SgAQ

dx

d

Sdx

dH

(2.9)

Where x = longitudinal dimension of the channel. The energy slope accounts for bed friction

and local energy losses, i.e., SH = SF + Se. For bed frictional in turbulent flows the quadratic

friction law can be applied

2

K

QSf (2.10)

With the conveyance factors K as a function of bed roughness and channel geometry. On

using Manning's formula the conveyance factor reads3/21ARnK , with Manning's n and the

hydraulic radius R. Eddy losses due to channel expansion and contraction can be related to the

gradient of the velocity head

2

2

2gA

Q

dx

dcS ee

(2.11)

Where ce is the empirical energy loss coefficient. For contraction ce is set to 0.1 and 0.3 for

expansion (CHOW 1959).


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In the case of shocks and for lateral inflows and outflows the momentum equation should be

applied (Henderson 1966). The momentum equation expresses a balance between forces

acting on the flow and, also, account for external force due to channel expansion and

contraction (Cunge et al. (1980)). It can be expressed as:

01

2

fS

dx

d

A

Q

dx

d

gA

(2.12)

Combined energy momentum equations can be used for backwater calculations. Using a box

scheme where the variables are located at cross sections on either side of the computation

interval, the discrete form of Eq. (2.9) may be written as

22

02

2

2

2

22

1

K

Q

K

Qx

A

Q

A

Q

g

cr

o

o

o

oo

eE (2.13)

Where the values with subscript o = downstream cross section; the vales without o =

downstream cross section; x = distance between the cross section. The arithmetic mean for

frictional slope is used and re is the residual that is equal to zero for a converged solution. A

discrete form of the momentum equation (2.12) may be written as

22

0

22

2

1

K

Q

K

Qx

A

Q

A

Q

gAr o

o

o

o

om

E (2.14)

With the mean wetted area AAA om 2/1 ; and residual rm = 0.


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2.3.2 Discrete Solution

The standard backwater analysis for subcritical flow carry on to an upstream direction from a

given water level at a downstream section, equations (2.13) or (2.14) may possibly be applied

to estimate the solution at the next section. Making the water level as the independent

variable, an iteration Newton scheme can be obtained from the local Taylor expansion of the

residual:

02

O

rrr (2.15)

yielding:

2

O

r

r(2.16)

The initial guess for the water level should be above or equal to the critical water level for the

given discharge for the change in the water level to obtain a new estimate. Assuming that the

variation of the coefficient and are small, the derivative of the energy residual from (2.13)

will be:

xKK

Qw

gA

Qc

re

E

3

2

3

2

11 (2.17)

Where Aw = surface width. The first right hand side is equal to the square of the

Froude number 322 gAwQF ; thus, for critical flow (i.e. F = 1), equation (2.17) reduce

to

x

K

K

Q

c

re

E

3

2

(2.18)


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Given that the conveyance of a wide section increases with increased water level, the

derivatives of equation (2.18) might remain negative for subcritical flow. This allows the

continuation of the computation until 0 Er , where the maximum value of no solution is

available for the upstream procedure. Using equation (2.17) we obtain the requirement for the

allowable Froude number:

2/1

3

3

1

x

K

wK

gAcF e

(2.19)

For a hydraulically wide section (with R h, where h is the mean depth h = A / w) and using

manning's equation (2.19) can be rewritten as:

2

3/4

2 5

311

gn

h

Fcx e

(2.20)

Equation (2.20) is illustrated in Fig 2-5 for a channel contraction (i.e. )1.0ec . Therefore,

for a given depth, there is a particular computation interval, x, above which the iterative

scheme is unconditionally stable due to friction term. After this interval, the upstream method

is applicable only for flows up to a certain Froude number.

Fig 2-6 shows the results for the reduced momentum equation. Compared to the complete

momentum equation, the variations in the depths are diminished for the whole channel.

Clearly, the reduced equation should be used only if the full equation cannot be solved. The

results of the computation are also shown in Table 2-1 for the momentum equation and the

energy equation, together with the bed levels to allow a comparison with other computation

scheme.


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Figure 2-5: Calculation intervals above which the scheme is numerically stable as a

function of flow depth and Froude number for a hydraulically wide section (after Beffa

(1996))

Figure 2-6: Calculation flow depths in a Transcritical channel using different flow

equations (after Beffa (1996))


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Table 2-1: Computed Flow Depth in a Rectangular Channel Using Momentum Equation

(2.10), Energy Equation (2.11), and Reduced Momentum Equation (2.12) (after Beffa

(1996))


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2.4 Conclusion

Jean-Baptiste BELANGER's contributions were remarkable and have influenced other

leading hydraulic engineers including BRESSE (1860), DARCY and BAZIN (1865), BARRE

de SAINT VENANT (1871), and BOUSSINESQ (1877). In the 1820s, Jean-Baptiste

BELANGER (1790-1874) succeeded on the method to calculate gradually-varied open

channel flow properties for steady flow condition. BELANGER success article are in use for

the treatment of the stationary hydraulic jump, at the moment called the Belanger equation.

But at the time of computation, he applied the wrong basic principle at the time. The error

was discovered and corrected ten years later and the correct solution was first published by

BELANGER in 1841 Chanson (2009). The uniqueness of BELANGER's (1828) work paved

way for development of the backwater calculation for steady one-dimensional gradually-

varied flow in an open channel. His technique of numerical integration was ahead of time

schedule, when there was neither computer nor electronic calculator.

In the early 1950s, a lengthy series of backwater calculation was required to predict upriver

changes to the water profile. It was estimated that these calculations would have taken 20

person-years to complete by using BELANGER (1828) work, but in 1952 and 1953 Ontario

Hydro was able to make use of the first modern electronic computer in Canada- the Ferut at

the University of Torontoto complete the work in about eight months. These were the first

major calculations carried out on any electronic computer in Canada, and helped prove that an

all Canadian navigation route was possible. Despite the significance of the computer's

assistant, Ferut's role was not widely revealed until 1960, when a short series of semi-

celebratory articles appeared in the Engineering Journal explained how the backwater results

had been derived (Gotlieb 1960; McFarlane 1960).


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Overall, this chapter reviews the conceptual/theoretical dimension and the methodological

concept of the literature review on backwater calculations and discovers research questions or

hypotheses that are worth researching in later chapters.


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CHAPTER THREE

3.1 Methods of Computation

The computation of gradually-varied flow profiles involves the solution of the dynamic

equation of gradually varied flow. The main objective of the computation is to determine the

shape of the flow profile.

This chapter would describe two well established computations (direct step method and ISIS

modelling) adopted for this thesis.

3.2 Choosing a Method

The choice of an appropriate method for computing water profiles for data and numerical

modelling was based on these characteristics:

The channel reach

The type of flow hydrograph and the thesis objective

However, quantitative information on the variation of the flow depth and flow velocity along

a channel is required in many engineering applications (Less Hamill 1995). Estimation of the

extent of inundation in such a case is possible only by performing computations to determine

the flow depths. Thus quantitative knowledge of flow depths and velocities is essential when

conducting backwater calculations. These computations, generally known as gradually varied

flow computations determine the water-surface elevations along the channel length for

specified:

Discharge

Flow depth at any one location


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The manning roughness coefficient

Longitudinal profile of the channel and Channel cross-sectional parameters

Generally, all the numerical procedures mentioned earlier are based on the numerical solution

of the non-linear first-order ordinary differential equation for GVF and the direct application

of the algebraic energy equation, using certain approximation. These methods for gradually

varied flow computation are presented in the following section.

3.3 Direct Step Method

Since prismatic channel is the only channel that will be discussed for this thesis, the direct

step method is a simple step applicable to it. In general, direct step method is characterized by

dividing the channel into short reaches and carrying the computation step by step from one

end of the reach to the other and computing the profile upstream (Chow 1959).

Calculating the coordinates of the water surface profile using this method is an iterative

process achieved by choosing a range of flow depths, established at the downstream end of

the channel, and proceeding incrementally up to the point of interest or to the point of normal

flow depth. This is best accomplished by the use of spread sheet table (see chapter 4).

To illustrate analysis of a single reach, consider the following diagram below


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Figure 3-1: Channel reach for the derivation of step methods

Equating the total head at the two end sections 1 and 2, the following may be written:

xSg

Vy

g

VyxS f

22

2

2

22

2

1

110 (3.1)

Where,

x = distance between cross sections (m)

y1,y

2= depth of flow (m) at cross sections 1 and 2

V1, V

2= velocity (m/s) at cross sections 1 and 2

1,

2= energy coefficient at cross sections 1 and 2

So = bottom slope

1 2


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Sf

= friction slope = (n2

V2

)/ (2.22R1.33

)

g = acceleration due to gravity, (9.81 m/sec2

)

If the specific energy E at any cross section is defined as:

g

VyE

2

2

(3.2)

Assuming = 1= 2where is the energy coefficient that correlates with the non-uniform

distribution of velocity over the channel cross section. Combining and rearranging Equations

3.1 and 3.2 to solve for x:

ff SS

E

SS

EEx

00

12

(3.3)

The average value of Sf is denoted by

fS . When the manning formula is used, the friction

slope is expressed by:

3/4

22

22.2 R

VnSf (3.4)

From the above equations, the Manning roughness coefficient "n" can be related to the Darcy

and Chezy coefficient by:

6

1

2

1

093.0 Rfn (3.5)

The roughness coefficient used for computation in this thesis can be found in Appendix A1.


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From Fig. 3-1 above, the channel slope, Manning's "n" and energy coefficient , together with

water surface elevation y2, the value of x were calculated for arbitrarily chosen values of y1.

The coordinate defining the water surface profile was obtained from the cumulative sum of

x and corresponding values of y. The normal flow depth, y n was initially calculated from

Manning's Equation to establish the upper limit of the backwater effect and then the critical

flow depth calculated to determine the flow profile type (Chow 1959).

The direct step method adopted for this thesis is based on Eq. (3.3), which was used to

analyse the water profile in three different channels as shown in the examples below.

In order to prove the visibilities of the backwater calculations in open channels by the

theoretical formula above, three different channel models will be analysed based on the direct

step method.


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3.4 Computation of Backwater Profile by the Direct Step Method

3.4.1 Rectangular Channel

MODEL 1

Figure 3-2: Upstream cross section for the rectangular channel

A proposed rectangular cross section channel above (Fig 4.1) where b = 6 m, So = 0.100, and

n = 0.0117 carrying a discharge of 4.48 m3/s to compute the backwater profile created by a

dam which backs up the water to a depth of 7 m immediately behind the dam.

As mentioned earlier the normal flow depth, yn will first be calculated from manning's Eq

(1.4) above to establish the upper limit of the backwater effect.

Following the solution in Appendix B1, the critical depth and normal depths were found to be

yc = 0.38 m and yn = 0.12 m, respectively. The computation of water surface profile by means

of Eq. (3.3) is given in Table 1 for various values of y varying from 7 m.

The computed profile is plotted as shown in Fig. 4-1 (see chapter four)

7m

2m

2m

Datum 2m

Water Surface

6m


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3.4.2 Trapezoidal Channel

MODEL 2

22

6m

Datum 2.5m3m

Water Surface

Y

Figure 3-1: Upstream cross section for the trapezoidal channel

A trapezoidal channel in Fig 3-3 where b = 6 m, z = 2 m, So = 0.0016, and n = 0.025 carrying

a discharge of 11.6 m3/s. Table 2 shows the computation of backwater profile created by a

dam which backs up the water depth of 4 m immediately behind the dam. The upstream end

of the profile assumed at a depth equal to 1% greater than the normal depth.

Following the same procedure adopted for the rectangular channel, the critical depth and

normal depths were found to be yc = 0.317 m and yn = 0.90 m respectively in Appendix B2.

Since yn is greater than yc, let the flow start form a depth greater than yn (Table 4) and the

computed profile is plotted as shown in Fig. 4-3.


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3.4.3 Circular Channel

MODEL 3

For the circular channel the following data were assumed for water surface computation:

Discharge Q = 1 m/s3

Width B = 6 m

Manning's n = 0.015

Bottom Slope S0 = 0.0016

Diameter = 5.5 m

The table in Appendix B3 for geometric elements for a circular section was used to determine

the critical and normal depths which was found to be yc = 0.33m and yn = 0.16m respectively.

With the data given above, the direct step computations are carried out as shown in Table 5

and computed profile is plotted as shown in Fig 4-5.


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3.5 Data's Analysis Computation Procedures

The values in each column of Tables 1 and 4 are explained as follows:

Column 1 (channel bottom):- a trial value is first entered in this column; this will be verified

or rejected on the basic of the computation made in the remaining columns of the table. And

the next steps are imitative by adding the previous assumed value to product of bottom slope

and x in col. 16.

Column 2 (water - surface elevation):- For the first step the elevation must be given or

assumed. Since the first channel bottom elevation assumed is 293 m and the height of the

water 7 m, the first entry is 300 m (Table 1). When the trial value in the second step has been

verified, it becomes the basic for the verification of the trial value in the next step and so on.

Column 3 (depth of flow in m):- arbitrarily assigned values ranging from 7 to 0.116 m (Table

1).

Column 4 (water area in m2

):- corresponding to the depth, y, in column 3).

Column 5 (wetted perimeter in m):- calculated based on the formula given for various

geometric elements of channel sections in the appendix.

Column 6 (mean velocity (m/s)):- obtained by dividing 4.48 m/s3

by the water area in column

2 of Table 1.

Column 7(Froude number):-"F" = 2/1)/(gyV.

Column 10:- (hydraulic radius in m) corresponds to y in column 3 (Area/ Wetted perimeter).

Column 11:- (velocity head in m)

Column 12 (frictional slope):- computed by Eq. (3.4) with n = 0.017 and V as given in

column 6 and R3/4

in column 10.

Column13:- difference between the bottom slope 0.017 and frictional slope.


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Column 14:- average of the difference between the bottom slope and frictional slope just

computed in column 11 and that of the previous step.

Column 15:- length of the reach in m between the consecutive steps, computed by Eq. (3.3).

Column 16:- this is equal to the cumulative sum of the value in col. 15 computed for previous

steps

Column 17:- elevation of the total head in m, this is computed by adding the calculated

bottom elevation in col. 1 to normal flow depth yn calculated and that of previous steps, which

is found in column 17.

The above procedures were applied to both rectangular and trapezoidal channel. For circular

channel, different procedure is introduced because of the shape of the channel. The

computation is arranged in a form similar to that used for the rectangular channel, but the Eq.

(3.6) is used to determine angles for different reach in circular channel.

aD

y21cos2 1

(3-6)

Where:

= Angle in radians

Da = Diameter in m

y = Water depth in m

For the entire table prepared, the assumed depth is considered correct when the resulting

value of x entered in column 17 agrees with the length of the reach in column 2 (as shown in

Tables 1 and 4). It should be noted that the depth of flow computed in this thesis has been

carried to more decimal places than would be necessary for practical purposes.


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3.6 Building the Model in ISIS

An ISIS model constructed for this thesis used a number of different hydraulic units, which

can be assumed as a building block that is connected together in Fig. 3-4. The ISIS has the

following types of unit as a minimum:

Upstream boundary: to represent the flow entering the model

Downstream boundary: to represent the downstream water level

River section: to represent the river channel

Each unit mentioned above contains model data appropriate to the unit type, e.g. a River

Section unit contains cross-section geometry and roughness data for the river, these also

contain one or more node labels in other to be able indentify the unit and to be able to define

the connectivity with other units in the model.

The following description explains brief ways of building a model representing a single

rectangular channel in Fig. 3-2.

The single channels modelled for the rectangular channel contained: upstream boundary

condition, downstream boundary condition and 30 sections to represent a channel as shown in

Appendix C1. Fig. (3.4) shows a very simple channel with eight cross sections by which the

rectangular model was developed and other channels modelling were based upon.


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Figure 3-4: A schematic representation of a single channel with eight sections and two

boundary conditions (after Batica (2009), Fu (2009))

QTBDY is shorthand for the discharge (Q) Time series boundary condition (for the upstream

inflow to the channel). HTBDY represents the water level time series boundary condition for

the downstream end of the channel.

In order to build a rectangular channel in Fig. 3-2, these steps were followed:

1. Start a new (blank) model in ISIS (> File > New)

2. Insert a Flow-Time Boundary (QTBDY) unit to represent an upstream boundary by

either click on the Flow-Time Boundary picture button or select > Edit > Insert >

Boundaries > Hydrographs > Flow/Time from the main menu. And label it in upper

case as SECT 1, click 'OK' and save the file with a proper name.

3. Specify the data for the boundary condition by double clicking the newly inserted unit

to display the data entry screen for the boundary. Insert 4.48 m3/s for both the peak

flow and base flow and time peak is 12 hours into the Flow-Time table. Remember to

make sure that the unit of the time is hours (by defaults it is "seconds").

QTBDY

HTBDY


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Figure 3-5: Simple schematic diagram of a Flow-Time data entry form (after Batica

(2009), Fu (2009))

4. Insert a river section unit by clicking on the river section picture button or select >

Edit > Insert > Channels > River > River Section from the main menu. When the

Node label editor dialog appears, enter the label with the same label as SECT 1 and

click OK.

Figure 3-6: Connecting a river unit to a QTBDY unit entry form (after Batica (2009), Fu

(2009))

5. To connect the SECT 1 labels together click 'Yes'. The river section unit is located

immediately downstream of the Flow-Time Boundary (hence the node labels of these

units are the same.).


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6. To enter the cross section data. Double click on the newly inserted units to display the

data entry screen for the river section. Enter the cross section data into the table,

leaving the rest of the parameters in the table at their default values. The cross section

data's entered for the rectangular channel are giving in Appendix C1

Note: The cross-section geometry has to be translated to the format recognisable by

ISIS. As shown in Fig. 3-12

Figure 3-7: Schematic specification of cross section data form (after Batica (2009), Fu

(2009))

7. To specify the distance between the cross-sections. Enter 6 meters for the 'distance to

next section' field.

8. Specify the cross-section data for all the cross-sections. The same procedures were

repeated 30 times to insert 30 river sections, (SECT 1 to SECT 30). The last section

have a 'distance to next section = 0', this is used by ISIS to recognise it as the last

section of a channel reach.

9. Specify the information for the downstream boundary condition. For the last channel

section a HTBDY boundary condition unit is inserted in a similar manner as the first


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QTBDY boundary condition unit ( by clicking the picture icon , and prescribing the

time series data in the table, for example a constant water level of 2 meters was

inserted).

10. To visualise the channel that we have just build. This is done by clicking on the

visualiser picture icon or select > Tool > Visualiser from the main menu.

Figure 3-8: The Stage-Time data entry form for the downstream boundary condition

(after Batica (2009), Fu (2009)).

11.Click the 'Flow simulation' icon or select > Run > flow simulation. A flow

simulation run form interface will appear.


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Figure 3-9: Run form interface (after Batica (2009), Fu (2009)).

In this report we will only deal with the 'Time' tab which has a button names 'run' on the

bottom left side. There are five simulation types displaying on the 'Time' tab. The 'Steady

(Direct)' and the 'Unsteady (Fixed Timestep)' options are used for this report.

12.Running a steady simulation. Select 'Steady (Direct)' simulation type and click on

'Run' button.

Figure 3-10: Notification of the successful completion of a simulation (after Batica

(2009), Fu (2009)).


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13.Running and unsteady simulation. Select 'Unsteady (Fixed Timestep)' simulation type.

Enter the 'start time' and 'finish time'. The time spans for simulation are within the

range of the shortest time span of the boundary conditions which are defined in the

model set-up step. Also, the time step and save interval were defined.

14. Click on the 'Run' button after prescribing the parameters for the unsteady simulation.

Figure 3-11: The progress of the unsteady simulation in ISIS (after Batica (2009), Fu

(2009)).

15. After finishing the unsteady simulation, the summary outputs can be viewed in the

form of plain text as shown in Appendix C4, C5 and C6.


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3.7 Translate the descriptive cross section geometry to ISIS format

Figure 3-12: Simple schematic for conversion of cross-section geometry to ISIS format

(after Batica (2009), Fu (2009)).

River section geometry is specified by pairs of X and a Y value, where X is the distance

across the cross-section and Y is the bed elevation.


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3.8 Backwater Profile Outputs by the ISIS Method

For the results comparison, 'three channels' (rectangular, trapezoidal and circular) that were

analysed by direct step method earlier were also modelled with ISIS software packages for

river modelling. Some of the outputs for computed cross-sections are given below:

3.8.1 Rectangular Channel

Figure 3-13: The Cross-Section Data for Section 30

3.8.2 Trapezoidal Channel



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3.8.3 Circular Channel



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CHAPTER FOUR

Results and Discussion

4.1 Analysis of Results for Direct Step Method and ISIS Method

Tables 1, 4 and 5 below represent the computation of the flow profile by the direct step

method for three different model channels. In order to establish the upper limit of the

backwater effect, the normal flow depth, yn, was first calculated from manning's equation as

mentioned previously (Appendix B1 B3). In Table 1 below, the depth of flow (D) is

assumed and entered in col. 3 at each step. The assumed depth is considered correct when the

resulting values of normal depth line elevation (NDL) entered in column 17 agrees with the

water surface depth in column 2. These values are correct water-surface elevation because of

their agreement.

To check the comparison of two different numerical methods adopted for our calculations.

The computed water surface profile is plotted for the three channels as shown in Fig (4.1),

(4.3) and (4.5) by using Table 1, 4 and 5. Also, ISIS model was also used to analyse the same

geometrical channel based on the same data's for all the channels. The direct step method

computed flow profile Fig (4.1) and (4.3) is practically identical with that obtained by ISIS

method in Fig (4.2) and (4.4) respectively. This comparison highlighted the similarity in

performance of the two methods for the analysis.

For simplicity, the prismatic channel is considered for this thesis, and Eq. (4.1) below with

reference to the derivation of the gradually-varied flow equation in Chow textbook is used for

the discussion.


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2

2

0

/1

/1

ZZ

KKS

x

y

c

n

(4.1)

The values of K and Z in this equation are assumed to be increase or decrease with the depth

y. This is true for all open- channel sections (Chow 1959). For a backwater curve xy / is

positive; thus, Eq. (4-1) gives two possible cases:

1. 0/1 2 KKn and 0/12 ZZc

2. 0/1 2 KKn and 0/12 ZZc

Since the values of K and Z increase or decrease continuously with the depth y, the first case

indicates y > yn and y > yc.

From Table 2 below, y = 7 m, yc = 0.38 m and yn = 0.12 m (Rectangular channel). Case 1

proves the flow profile of the three channels. The first case indicates y > yn and y > yc. As y

> yn, the flow type for both rectangular and trapezoidal channel are subcritical.

Fig. (4-1), Fig. (4-3) and (4-5) correlated well with ISIS profile computed in Fig (4-2), (4-4)

and (4-6) respectively.


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Table 1: Computation of Flow Profile by the Direct Step Method for a Rectangular Channel

Col. 1 Col. 2 Col. 3 Col.4 Col.5 Col.6 Col.7 Col.8 Col.9 Col.10 Col.11 Col.12 Col.13 Col.14 Col.15 Col.16 Col.17

CHANNEL BOTTOM WATER SURFACE D (m) A(m2) W(m) V (m/s) F (1 -F

2) (1 -F

2)AVG R (m) V

2/2g SF (SO - SF ) (SO - SF)(AVG) X (m) SX) (m) NDL ELEVATION

293 300.000 7.000 42.000 20.00 0.107 0.013 1.000 0 2.100 0.001 0.000001 0.1000 0.0000 0 0 293.116

293.00 299.500 6 .500 39.000 19.00 0.115 0.014 1.000 1.000 2.053 0.001 0.000001 0.1000 0.1000 2 .000 2.000 293.116

293.20 299.200 6 .000 36.000 18.00 0.124 0.016 1.000 1.000 2.000 0.001 0.000002 0.1000 0.1000 2 .000 3.999 293.316

293.40 298.900 5 .500 33.000 17.00 0.136 0.018 1.000 1.000 1.941 0.001 0.000002 0.1000 0.1000 1 .999 5.999 293.516

293.60 298.600 5 .000 30.000 16.00 0.149 0.021 1.000 1.000 1.875 0.001 0.000003 0.1000 0.1000 1 .999 7.998 293.716

293.80 298.300 4 .500 27.000 15.00 0.166 0.025 0.999 0.999 1.800 0.001 0.000004 0.1000 0.1000 1 .999 9.997 293.916

294.00 298.000 4.000 24.000 14.00 0.187 0.030 0.999 0.999 1.714 0.002 0.000005 0.1000 0.1000 1.999 11.995 294.116

294.20 297.700 3.500 21.000 13.00 0.213 0.036 0.999 0.999 1.615 0.002 0.000007 0.1000 0.1000 1.998 13.993 294.316

294.40 297.399 3.000 18.000 12.00 0.249 0.046 0.998 0.998 1.500 0.003 0.000010 0.1000 0.1000 1.997 15.990 294.515

294.60 297.399 2.800 16.800 11.60 0.267 0.051 0.997 0.998 1.448 0.004 0.000013 0.1000 0.1000 1.996 17.986 294.715

294.80 297.399 2.600 15.600 11.20 0.287 0.057 0.997 0.997 1.393 0.004 0.000015 0.1000 0.1000 1.994 19.980 294.915

295.00 297.398 2.400 14.400 10.80 0.311 0.064 0.996 0.996 1.333 0.005 0.000019 0.1000 0.1000 1.993 21.973 295.114

295.20 297.397 2.200 13.200 10.40 0.339 0.073 0.995 0.995 1.269 0.006 0.000024 0.1000 0.1000 1.991 23.964 295.313

295.40 297.396 2.000 12.000 10.00 0.373 0.084 0.993 0.994 1.200 0.007 0.000032 0.1000 0.1000 1.988 25.952 295.512

295.60 297.395 1 .800 10.800 9.60 0.415 0.099 0.990 0.992 1.125 0.009 0.000043 0.1000 0.1000 1 .984 27.936 295.711

295.79 297.394 1.600 9.600 9.20 0.467 0.118 0.986 0.988 1.043 0.011 0.000059 0.0999 0.0999 1.977 29.913 295.910

295.99 297.391 1.400 8.400 8.80 0.533 0.144 0.979 0.983 0.955 0.014 0.000087 0.0999 0.0999 1.967 31.880 296.107

296.19 297.388 1.200 7.200 8.40 0.622 0.181 0.967 0.973 0.857 0.020 0.000137 0.0999 0.0999 1.949 33.829 296.304

296.38 297.383 1.000 6.000 8.00 0.747 0.238 0.943 0.955 0.750 0.028 0.000236 0.0998 0.0998 1.914 35.743 296.499

296.57 297.374 0.800 4.800 7.60 0.933 0.333 0.889 0.916 0.632 0.044 0.000465 0.0995 0.0996 1.839 37.581 296.690

296.76 297.358 0.600 3.600 7.20 1.244 0.513 0.737 0.813 0.500 0.079 0.001128 0.0989 0.0992 1.639 39.220 296.874

296.92 297.322 0.400 2.400 6.80 1.867 0.942 0.112 0.424 0.353 0.178 0.004037 0.0960 0.0974 0.871 40.092 297.038

297.01 297.125 0.116 0.696 6.23 6.437 6.034 - 35.409 -17.649 0.112 2.112 0.222635 -0.1226 -0.0133 264.674 304.766 297.125


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288.000

290.000

292.000

294.000

296.000

298.000

300.000

302.000

0 2.000 3.999 5.999 7.998 9.997 11.995 13.993 15.990 17.986 19.980 21.973 23.964 25.952 27.936 29.913

ELEVATIO

N(m)

DISTANCE UPSTREAM (m)

WATER SURFACE PROFILE

NDL WATER SURFACE CHANNEL BOTTOM

Figure 4-1: An S2 Flow Profile Computed by the Direct Step Method for Rectangular Channel


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Figure 4-2: Elevation vs. Nodal Label Output by the ISIS Method for Rectangular Channel A


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Table 2: Critical Depth and Normal Depth Computation by Direct Step Method

For rectangular channel, as y > yc > yn, the subcritical flow must occur in a steep channel.

While, trapezoidal channel with y > yn > yc, the subcritical flow must occur in a mild channel.

The corresponding flow for circular channel is supercritical and it occur in a steep channel as

yc > yn > y. Table 2 results proves the flow profile computed by ISIS program for all the three

channels (rectangular, trapezoidal and circular) by correlating the profiles type computed by

direct step method.

As shows in Fig (4.1) and (4.2) for rectangular channel, the control section is at the upstream;

water will enter the channel at the critical depth and thereafter flow at a depth less than y c but

greater than yn, since yc > yn. The flow profile is of the S2 (Appendix A2)

The ISIS flow profile Fig. (4.4) shows that flow starts with a depth greater than yn, the flow

profile is of the M1 which is similar to the direct step method computation for trapezoidal

channel. It should be noted that, when the depth approaches the normal depth, the increment

Channel Type Starting

depth (y)

(m)

Normal

depth

(yn)

(m)

Critical

depth (yc)

(m)

Flow Type Profiles

Type

Rectangular 7 0.12 0.38 y > yn

Subcritical

flow

Steep

Trapezoidal 4 0.82 0.32 y > yn

Subcriticalflow

Mild

Circular 0.10 0.16 0.33 y < yn

Supercritical

flow

Steep


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area varies so greatly with the changes in y value that it becomes difficult for the fact that yn is

greater than yc .

Figs (4.1)(4.6) show the maximum water levels in each of the channels for both the direct

step and ISIS method. The three channels clearly show the extent of the accuracy of both

methods adopted. It is only comparison between Fig (4.3) and (4.4) that any different between

the two methods can possibly be seen, but this different is only slightly, which might due to

some little errors in channel modelling or method used.

Table 3: Result Values for Froude Number and Velocity for Both Direct Step and ISISMethod

RECTANGULAR

CHANNEL

FROUDE

NUMBERVELOCITY (m/s)

Direct Step Method 0.94 1.87

ISIS Method 0.91 1.83

TRAPEZOIDAL

CHANNEL

FROUDE




CIRCULAR

CHANNEL

FROUDE




Table 3 shows the similarities between the Froude number and velocity computed for both the

direct step method and ISIS method. These values also prove the validity of both methods as

the differences is just 2%, which is insignificant. Also, the rate at which the Froude number

varies with velocity for the direct step method (Table 1, 3 and 5) is equivalent to ISIS method

(Appendix C1, C5 and C8).


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Table 4: Computation of Flow Profile by the Direct Step Method for a Trapezoidal Channel

Col. 1 Col. 2 Col. 3 Col.4 Col.5 Col.6 Col.7 Col.8 Col.9 Col.10 Col.11 Col.12 Col.13 Col.14 Col.15 Col.16 Col.17

CHANNEL BOTTOM WATER SURFACE y(m) A(m2) W(m) V (m/s) F (1 -F

2) (1 -F

2)AVG R (m) V

2/2g SF (SO - SF ) (SO - SF)(AVG) x (m) Sx (m) NDL

266.00 270.00 4.00 56.00 23.89 0.207 0.033 0.999 0 2.34 0.0002 0.000004 0.00160 0 0.0 0 266.8

266.00 269.75 3.75 50.63 22.77 0.229 0.038 0.999 0.999 2.22 0.0003 0.000005 0.00159 0.001596 156.5 156.5 266.8

266.25 269.75 3.50 45.50 21.65 0.255 0.044 0.998 0.998 2.10 0.0003 0.000007 0.00159 0.001594 156.6 313.1 267.0

266.50 269.75 3.25 40.63 20.53 0.286 0.051 0.997 0.998 1.98 0.0004 0.000009 0.00159 0.001592 156.7 469.8 267.3

266.75 269.75 3.00 36.00 19.42 0.322 0.059 0.996 0.997 1.85 0.0005 0.000013 0.00159 0.001589 156.9 626.6 267.5

267.00 269.75 2.75 31.63 18.30 0.367 0.071 0.995 0.996 1.73 0.0007 0.000018 0.00158 0.001584 157.1 783.7 267.8

267.25 269.75 2.50 27.50 17.18 0.422 0.085 0.993 0.994 1.60 0.0009 0.000027 0.00157 0.001577 157.5 941.2 268.0

267.51 269.76 2.25 23.63 16.06 0.491 0.105 0.989 0.991 1.47 0.0012 0.000041 0.00156 0.001566 158.2 1099.4 268.3

267.76 269.76 2.00 20.00 14.94 0.580 0.131 0.983 0.986 1.34 0.0017 0.000064 0.00154 0.001548 159.3 1258.7 268.5

268.01 269.76 1.75 16.63 13.83 0.698 0.168 0.972 0.977 1.20 0.0025 0.000107 0.00149 0.001514 161.3 1420.0 268.8

268.27 269.77 1.50 13.50 12.71 0.859 0.224 0.950 0.961 1.06 0.0038 0.000192 0.00141 0.001451 165.6 1585.6 269.0

268.54 269.79 1.25 10.63 11.59 1.092 0.312 0.903 0.926 0.92 0.0061 0.000377 0.00122 0.001316 176.0 1761.6 269.3

268.82 269.82 1.00 8.00 10.47 1.450 0.463 0.786 0.844 0.76 0.0107 0.000848 0.00075 0.000988 213.7 1975.3 269.6

269.16 269.98 0.82 6.26 9.67 1.852 0.653 0.574 0.680 0.65 0.0175 0.001721 -0.00012 0.000316 538.4 2513.7 269.9


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264.00

265.00

266.00

267.00

268.00

269.00

270.00

271.00

156.5 313.1 469.8 626.6 783.7 941.2 1099.4 1258.7 1420.0 1585.6 1761.6 1975.3 2513.7

ELEVATION(m)

DISTANCE UPSTREAM (m)

WATER SURFACE PROFILE

NDL WATER SURFACE CHANNEL BOTTOM

Figure 4-3: An M1 Flow Profile Computed by the Direct Step Method for Trapezoidal Channel


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Figure 4-4: Elevation vs. Nodal Label Output by the ISIS Method for Trapezoidal Channel


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Note: The vales taken for the results analysis in Table 3 above were taken from computed of

flow table for direct step methods for the three channels and output results from ISIS method,

highlighted in red (Appendix C).

The similarities of water surface profiles produced by both methods are explained better by

ISIS results output. The unsteady graphical interface (Appendix C3, C6 and C9) proves the

accuracy of all the assumed vales used for our geometrical modelling for both methods.

Moreover, the unsteady interface graphs show the stability of our models. Because, the values

of model conveyance and iterations are between tolerant boundaries and also both discharge

hydrographs have the same shape. Also the maximum inflow is equal to the maximum

outflow except for the trapezoidal channel which is beyond the scope of this thesis.


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Table 5: Computation of Flow Profile by the Direct Step Method for a Circular Channel

(Radian)

y ( m) ( 1- 2y /Da) cos-1

(1-2y/Da) 2cos-1

(1-2y/Da) A (m2) W (m) V (m/s) F (1 -F

2) (1 -F

2)AVG R (m) V

2/2g SF SO - SF (SO - SF)(AVG) x (m) Sx (m) z ( m) y + z ( m) yn + z (m) yc + z (m)

0.20 0.93 0.38 0.767 0.277 2.111 3.615 2.581 -5.661 0 0.131 0.6661 0.0000882 0.00151 0.00000 0.0 0.00 0.0000 0.20 0.17 0.33

0.45 0.84 0.58 1.160 0.920 3.191 1.086 0.517 0.733 -2.464 0.288 0.0602 0.0000228 0.00158 0.00154 -398.9 -398.89 0.6382 1.09 0.80 0.97

0.70 0.75 0.73 1.459 1.760 4.013 0.568 0.217 0.953 0.843 0.439 0.0165 0.0000109 0.00159 0.00158 133.1 -265.80 0.4253 1.13 0.59 0.760.95 0.65 0.86 1.714 2.740 4.715 0. 365 0. 120 0.986 0. 969 0.581 0. 0068 0.0000065 0.00159 0.00159 152.3 - 113.51 0.1816 1. 13 0. 35 0.51

1.20 0.56 0.97 1.944 3.830 5.346 0.261 0.076 0.994 0.990 0.716 0.0035 0.0000044 0.00160 0.00159 155.2 41.70 -0.0667 1.13 0.10 0.26

1.45 0.47 1.08 2.157 5.005 5.931 0.200 0.053 0.997 0.996 0.844 0.0020 0.0000032 0.00160 0.00160 156.0 197.65 - 0.3162 1.13 -0.15 0.01

1.70 0.38 1.18 2.358 6.248 6.485 0.160 0.039 0.998 0.998 0.963 0.0013 0.0000025 0.00160 0.00160 156.2 353.84 - 0.5661 1.13 -0.40 -0.24

1.95 0.29 1.28 2.551 7.542 7.016 0.133 0.030 0.999 0.999 1.075 0.0009 0.0000020 0.00160 0.00160 156.3 510.11 - 0.8162 1.13 -0.65 -0.49

2.20 0.20 1.37 2.739 8.874 7.532 0.113 0.024 0.999 0.999 1.178

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