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ESTIMATIONOFROUGHNESSCOEFFICIENTS

INOPENCHANNELS

Nguyen, Thu Hien

Submitted in total fulfilment of the requirements

of the degree of Doctor of Philosophy

July 2006

Department of Civil and Environmental Engineering

The University of Melbourne


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Abstract

An accurate estimation of Mannings nroughness coefficient is of primary importance

in any hydraulic study involving open-channel flows. However, the estimation of this

coefficient for a natural channel is not a trivial task as it depends on many factors such

as surface roughness, vegetation condition, cross-sectional shape, channel irregularity

and flow conditions. The literature shows that there is no universal method of

selecting the nvalue. Further study in this area is still required to improve the quality

of the estimation of Mannings n.

This study focuses on aspects of the estimation of Manning's nin open channels using

flow measurements for both steady and unsteady flows. For steady flow, the method of

using two-point velocity data to estimate Mannings n is reinvestigated. Two new

formulae for estimating this coefficient are derived. The sensitivity analyses of the

relative error in n and the relative errors in measurements are theoretically and

experimentally investigated. The proposed formulae are applied and verified for a

number of rivers where the method is applicable. They are also compared with

different current empirical formulae. It is suggested that this method can be used as a

means to estimate the roughness coefficients for wide streams where two-point

velocity data are available.

For unsteady flow, the roughness coefficient(s) embedded in the momentum equation

needs to be estimated either by trial and error methods or by an automatic calibration

approach. The latter approach is also known as the inverse problemor the roughness

identification problem. This part of the study focuses on this approach where the main

study objectives are to investigate the modelling factors affecting the quality of the

identified roughness coefficient(s) and to extend the method to channels with

compound sections and varying roughness between the main channel and the

floodplain.

A roughness identification model for unsteady flow applicable to compound channels

is developed. The implicit finite difference Pressmann scheme is adopted to solve the

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Saint-Venant equations. The compound channel is treated as a divided channel section

in which for any depth the conveyance of the compound section is then the sum of the

main channel and floodplain conveyances. The algebraic equation system is linearised

and solved by using the double sweep algorithm. The objective function of least

square errors between observed and simulated data was chosen for this inverse

problem, which is solved using the Powell algorithm. Manning's n is considered for

two cases: constant and stage dependent.

Synthetic data are used to investigate the modelling factors affecting the quality of the

identified roughness coefficients and the performance of the model for compound

channels. This investigation shows that understanding these modelling factors is veryimportant in avoiding an unidentifiable inverse problem and to improve the quality of

the identified parameters. Several recommendations to improve the quality of the

identified roughness coefficient are made. For the compound channel case, the results

show that the model can identify both the roughness coefficients of the main channel

and the floodplain. The quality of the identified floodplain roughness coefficient is

usually poorer than the main channel. However, it can be improved considerably when

the depth on the floodplain becomes significant compared to the main channel depth.

Moreover, in cases when the cross-sections along the reach do not change much, an

alternative approach of identifying the conveyanceKis suggested.

The roughness identification model is applied to two natural rivers, the Goulburn River

in Victoria, Australia and the Duong River in the Red River delta, Vietnam. Depending

on the data availability, the characteristics of the cross-sections, the flow variation in

the main channel and the floodplain, the performance of the model is investigated and

several problems related to the roughness coefficients of the study reaches are

explored.

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Declaration

This is to certify that:

i. The thesis comprise only my original work towards the PhD except where

indicated in the Preface,

ii. Due acknowledgement has been made in the text to all other material used,

iii. The thesis is less than 100,000 words in length, exclusive of tables, maps,

bibliographies, appendices and footnotes.

Nguyen, Thu Hien

July 2006

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Preface

A substantial portion of the work described in this thesis has been published in the

papers listed below.

Nguyen, H. T., and Fenton, J. D. (2004). Identification of roughness in open

channels, Proceedings of the 6th International Conference on Hydro-Science and

Engineering, Brisbane, Australia, The University of Mississippi, May 31-June 3, 2004,

CD-ROM.

Nguyen, H. T., and Fenton J. D. (2004). Using two-point velocity measurements to

estimate roughness in streams, Proceedings of 4th Australian Stream Management

Conference, Launceston, Tasmania, 19-22 October 2004, 445-450.

Nguyen, H. T., and Fenton, J. D. (2005). Identification of roughness for flood routing

in compound channels,Proceedings of the 31th IAHR Congress on Water Engineering

for the future: Choices and Challenges, Seoul, Korea, September 11-16, 2005, CD-

ROM.

Nguyen, H. T., and Fenton, J. D. (2005). Identification of roughness in compound

channels, Proceedings of the International Congress on Modelling and Simulation -

Advances and Applications for Management and Decision Making MODSIM05,

Melbourne, Australia, December 12-15, 2005, CD-ROM.

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Acknowledgements

I gratefully acknowledge the excellent supervision and support provided by my

supervisors, Professor John D. Fenton and Associate Professor Roger Hughes. They

have saved no efforts in providing the support, supervision, understanding and advice

throughout the time of my study at the University of Melbourne. Special thanks and

appreciation also go to Professor Tom McMahon, Ass. Professor Nichola Haritos and

Dr. Andrew Western for their assistances and helpful discussions during my supervisor

was going for study leave.

I wish to extend my thanks to Mr. Joska Shepherd for his efforts in setting up the

experimental arrangements of the study. I also appreciate Dr. Yebegaeshet T. Zerihun

for his helpful advice regarding the use of the ADV instrument.

I also would like to thank Ms Barbara Dworakovski, Thiess Environmental Services,

Pty Ltd. for providing me the data for the rivers in Australia. Special thanks to Dr.

Tony Ladson for the roughness information of the rivers in Victoria, Australia. I owe

many thanks to Dr. Murray Hicks and staff in NIWA, New Zealand for the velocity

measurement data of the rivers in New Zealand. Special thanks to Dr. Mike

Stewardson for the topography data of the Goulburn River in Victoria, Australia. I also

would like to thank the staff of the Bureau of Meteorology and Hydrology and The

Institute of Water Resources Planning, Vietnam for their help in providing

hydrological and topographical data for the Duong River in Vietnam.

The financial support of Australian Development Scholarship (ADS) is gratefully

acknowledged.

I am indebted to my family: my parents and my brother. I thank them for their

continual support and encouragement throughout my years of education and I sincerely

dedicate this thesis to them. I extend deep gratitude to my husband and my two

daughters for their love, support, patience and understanding during the course of the

PhD study.

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Table of contents

Table of contents

Abstract. .................................................................................................iii

Declaration.................................................................................................. .v

Preface... ................................................................................................vi

Acknowledgements ...................................................................................vii

Table of contents.......................................................................................viii

List of Figures............................................................................................xii

List of tables ............................................................................................xvii

List of notations .........................................................................................xx

Chapter 1 Introduction ............................................................................1

1.1 Description of the study ...................................................................................... 1

1.1.1 General...................................................................................................... 1

1.1.2 Background of the research ...................................................................... 2

1.1.3 Scope and objectives................................................................................. 5

1.2 Research methodology ........................................................................................ 5

1.3 Outline of the thesis............................................................................................. 6

Chapter 2 Methods of estimating roughness coefficients in open

channels ...................................................................................9

2.1 Introduction ......................................................................................................... 9

2.2 Flow resistance in open channels ........................................................................ 9

2.3 Methods of estimating Manning's n for steady flow in open channels ............ 12

2.3.1 Direct measurement method ................................................................... 12

2.3.2 Using the tables or guidelines ................................................................. 14

2.3.3 Photographic comparisons ...................................................................... 15

2.3.4 Fitting calculated water surface profile to observed levels..................... 16

2.3.5 Field measurements of velocity distribution........................................... 16

2.3.6 Empirical formulae ................................................................................. 18

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Table of contents

2.3.7 Lump roughness coefficient model......................................................... 22

2.4 Estimating roughness coefficient for unsteady flow in open channels ............. 23

2.4.1 Trial and error method ............................................................................ 24

2.4.2 Automatic calibration methods ............................................................... 25

2.5 Equivalent roughness coefficient for compound channels................................ 28

2.6 Variation of Manning's nwith flow .................................................................. 30

2.7 Summary of key reviews................................................................................... 33

Chapter 3 Estimation of Roughness Coefficients Using Velocity Data

................................................................................................35

3.1 Introduction ....................................................................................................... 353.2 Background ....................................................................................................... 36

3.2.1 Velocity distribution in turbulent flow ................................................... 36

3.2.2 Current method using velocity data to estimate Mannings n ................ 39

3.3 Proposed method using velocity data to estimate Mannings n ........................ 41

3.3.1 Using average equivalent roughness....................................................... 41

3.3.2 Using average shear stress ...................................................................... 43

3.4 Sensitivity analysis ............................................................................................ 44

3.4.1 Sensitivity analysis.................................................................................. 44

3.4.2 Experimental work and analysis ............................................................. 46

3.5 Application to natural rivers.............................................................................. 57

3.5.1 Data......................................................................................................... 58

3.5.2 Results and discussion ............................................................................ 59

3.6 Comparing the performance of the proposed method with other empirical

equations............................................................................................................ 65

3.7 Summary and conclusion .................................................................................. 71

Chapter 4 Roughness Identification Model Development for

Unsteady Flow ......................................................................73

4.1 Introduction ....................................................................................................... 73

4.2 General discussion............................................................................................. 73

4.3 Hydraulic sub-model ......................................................................................... 76

4.3.1 Governing equations of 1-D unsteady flow in open channels ................ 76

4.3.2 Numerical methods for unsteady flow computation............................... 77

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Table of contents

4.3.3 Implicit Pressmanns scheme.................................................................. 81

4.3.4 Compound channel models..................................................................... 89

4.3.5 Identification of roughness functions...................................................... 94

4.3.6 Identification of conveyance function .................................................... 96

4.4 Optimisation sub-model .................................................................................... 97

4.4.1 Objective functions ................................................................................. 97

4.4.2 Optimisation methods ........................................................................... 101

4.5 Summary ......................................................................................................... 104

Chapter 5 Numerical experiments on the roughness identification

problem ...............................................................................1055.1 Introduction ..................................................................................................... 105

5.2 Generation of synthetic data............................................................................ 106

5.2.1 Cross-sectional data .............................................................................. 106

5.2.2 Flow data............................................................................................... 107

5.3 Model performance and preliminary results ................................................... 111

5.4 Modelling factors affecting the quality of the identified roughness

coefficient.. .................................................................................................. 115

5.4.1 Effect of weighting factor () ............................................................... 116

5.4.2 Effect of computational grid sizes ........................................................ 122

5.4.3 Effect of different combinations of boundary conditions ..................... 126

5.4.4 Effect of errors in initial conditions ...................................................... 139

5.5 Roughness identification in compound channels ............................................ 142

5.5.1 Model performance............................................................................... 142

5.5.2 Factors affecting the quality of the identified roughness coefficients in

compound channels............................................................................... 144

5.6 Identification of roughness functions.............................................................. 148

5.7 Identification of conveyance functions ........................................................... 154

5.8 Summary and conclusion ................................................................................ 159

Chapter 6 Application of the roughness identification problem to

natural rivers ......................................................................162

6.1 Introduction ..................................................................................................... 162

6.2 Goulburn River................................................................................................ 162

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Table of contents

6.2.1 General description of study area ......................................................... 162

6.2.2 Data....................................................................................................... 164

6.2.3 Application, results and discussions ..................................................... 166

6.3 Duong River .................................................................................................... 183

6.3.1 General description the study area ........................................................ 183

6.3.2 Data....................................................................................................... 185

6.3.3 Application, results and discussions ..................................................... 186

6.4 Summary and conclusion ................................................................................ 196

Chapter 7 Conclusions and Recommendations .................................199

7.1 Summary ......................................................................................................... 199

7.2 Conclusions ..................................................................................................... 200

7.3 Recommendations for further research ........................................................... 206

References ..........................................................................................209

Appendix A Measured and computed roughness coefficients using

velocity data of the 14 rivers................................................224

Appendix B Solution algorithm for a general river network ..................226

Appendix C Powells method..................................................................231

Appendix D Preliminary results of the identified roughness coefficients

using synthetic data ..............................................................235

Appendix E Data for the Goulburn River ...............................................240

Appendix F Data for the Duong River....................................................248

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List of Figures

List of Figures

Figure 3.1 Two-point velocity measurement at a cross-section ................................... 39

Figure 3.2 Theoretical relationship between relative errors in estimated n and relative

errors in x with different depths for a channel with 035.0=n ................... 45

Figure 3.3 Theoretical relationship between relative errors in estimated n and relative

errors inx for different roughness channels with a flow depth of 2 m.......... 46

Figure 3.4 The experimental set-up diagram (not to scale). ......................................... 47

Figure 3.5 The two roughness types were carried out in the experiment..................... 48

Figure 3.6 Velocity time series measurement at 7.1=z cm of the gravel bed flume of

8.5 cm water depth. ....................................................................................... 50

Figure 3.7 Average SNR and COR values and recorded flags for measurement at

cm of the gravel bed flume of 8.5 cm water depth. .......................... 507.1=z

Figure 3.8 Variation of velocity profiles at different locations along the flume .......... 51

Figure 3.9 Measured velocity profiles.......................................................................... 53

Figure 3.10 Experimental relationships between relative errors in xand relative errors

in estimated n and corresponding theoretical lines for wire mesh roughness

with different depths ..................................................................................... 56

Figure 3.11 Experimental relationships between relative errors in x and relative errors

in estimated n and corresponding theoretical lines for the gravel bed with

different depths ............................................................................................. 56

Figure 3.12 Experimental relationships between relative errors inx and relative errors

in estimated n and corresponding theoretical lines for the same depth with

different types of roughness.......................................................................... 57

Figure 3.13 Computed nobtain from velocity data using average equivalent roughness

versus measured n ......................................................................................... 62

Figure 3.14 Computed n obtain from velocity data using average shear stress versus

measured n .................................................................................................... 62

Figure 3.15 Computed nobtained from different equations versus measured n forthe 12 selected rivers..................................................................................... 68

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List of Figures

Figure 4.1 Roughness identification procedure ............................................................ 75

Figure 4.2 Numerical Pressmanns scheme.................................................................. 81

Figure 5.1 The cross-sections of the model channels................................................. 107

Figure 5.2 Effect of different parameter in Equation (5.1) on the size and shape of the

upstream hydrograph: (a) effect ofpt , (b) effect of pQ , (c) effect of ... 109

Figure 5.3 Noise contamination of the observed stages and discharges at the

observation gauge at the middle of the channel.......................................... 111

Figure 5.4 Variation of the mean of identified parameters with the number of samples

..................................................................................................................... 114

Figure 5.5 Identified roughness coefficient versus with different time steps for

synthetic data with 700=pQ m3/s, 8=pt hours, =n 0.035, =S 0.0004,

=x 1 km.................................................................................................... 118

Figure 5.6 Identified roughness coefficient versus with different space steps for

synthetic data with 700=pQ m3/s, 8=pt hours, =n 0.035, =0S 0.0004,

5.0=t hour ............................................................................................... 118

Figure 5.7 Identified roughness coefficient versus with different channel slopes for

synthetic data with 700=pQ m3/s, 8=pt hours, =n 0.035........................ 120

Figure 5.8 Identified roughness coefficient versus with different channel roughness

for synthetic data with 700=pQ m3/s, 8=pt hours, =0S 0.0004 ............. 120

Figure 5.9 Identified roughness coefficient versus with different peak discharges for

synthetic data with 8=pt hours, =n 0.035, =0S 0.0004............................ 121

Figure 5.10 Identified roughness coefficient versus with different time to peak

discharge for synthetic data with 700=pQ m3

/s, =n 0.035, =0S 0.0004 121Figure 5.11 Identified roughness coefficient versus grid sizes................................... 124

Figure 5.12 Sensitivity of computed discharge and stage hydrographs to the value of

roughness coefficient at the middle of the channel with different

combinations of boundary conditions ......................................................... 132

Figure 5.13 Variation of the means of the identified nand their confidence intervals of

95% with noise level 1.0= for the Q-Z combination.............................. 134

Figure 5.14 Variation of the means of the identified nand their confidence intervals of95% with noise level 1.0= for the Z-Z combination .............................. 135

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List of Figures


95% with noise level 1.0= for the Q-Qz combination ........................... 136


95% with noise level 1.0= for the Z-Qz combination............................ 137

Figure 5.17 Comparison of the computed stage hydrographs at the cross-section of

30 km with different combinations of boundary conditions................. 140=x

Figure 5.18 Identified nversus the starting time to calculate the objective function for

the case of the Q-Z combination................................................................. 141

Figure 5.19 Variation of identified n with the starting time to calculate objective

function and the slope of the channel (the errors in initial stage are +0.5 m)

..................................................................................................................... 141

Figure 5.20 The means of identified roughness coefficients and their range at a 95%

confidence level with different noise levels................................................ 144

Figure 5.21 The true roughness and identified roughness curves with noise-free data

..................................................................................................................... 150

Figure 5.22 The true roughness and identified roughness curves when data contained

noise (5 noisy data samples) ....................................................................... 151

Figure 5.23 The true roughness curves and identified roughness curves with noise-free

data .............................................................................................................. 152

Figure 5.24 The true roughness curves and identified roughness curves when data

contained noise (5 noisy data samples)....................................................... 153

Figure 5.25 The true conveyance and identified conveyance curves with noise-free

data .............................................................................................................. 155

Figure 5.26 The true conveyance and identified conveyance curves by using the

polynomial function when data contained noise (5 noisy data samples) .... 157

Figure 5.27 The true conveyance and identified conveyance curves by using the power

function when data contained noise (5 noisy data samples) ....................... 157

Figure 5.28 Observed and simulated hydrographs computed from the identified

roughness functions and the identified conveyance functions.................... 158

Figure 6.1 Study reach of the Goulburn River and the gauging stations along the reach

..................................................................................................................... 163

Figure 6.2 Observed and simulated hydrograph at different gauging stations for theflood event 21/07 - 06/08/1978 using the identified roughness coefficient 168

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List of Figures

Figure 6.3 Observed and simulated hydrograph at different gauging stations for the

flood event 04/09 - 25/09/1979 using the identified roughness coefficient 169

Figure 6.4 Verification of the identified roughness coefficient by the independent flood

event 27/09 - 09/10/1978 ............................................................................ 170

Figure 6.5 Verification of the identified roughness coefficient by the independent flood

event 29/07 - 05/08/1980 ............................................................................ 171

Figure 6.6 Identified roughness coefficient with different combinations of boundary

conditions and observed data types............................................................. 173

Figure 6.7 Observed and simulated hydrographs for the flood event 27/09

09/10/1978 using the identified roughness coefficient from Z-Z boundary

conditions and the observed stages at Shepparton Golf Club used in the

objective function........................................................................................ 174

Figure 6.8 Identified roughness curves with different flood events at the Shepparton

Golf Club gauge: (a) Linear function of stage, (b) Quadratic function of stage

..................................................................................................................... 176

Figure 6.9 Verification of the identified roughness coefficients by the independent

flood event 08/08 - 31/08/1978................................................................... 179

Figure 6.10 Verification of the identified roughness coefficients by the independent

flood event 25/06 05/07/1981 .................................................................. 180

Figure 6.11 Identified roughness coefficients obtained from different flood events . 181

Figure 6.12 Identified roughness function of the main channel with different flood

events at the Shepparton Golf Club gauge: (a) Linear function of stage, (b)

Quadratic function of stage......................................................................... 183

Figure 6.13 Duong River and the gauging stations along the river ............................ 185

Figure 6.14 Observed and simulated stage hydrograph at different gauging stations byusing the identified roughness coefficients obtained from the flood itself . 189

Figure 6.15 Observed and simulated stage hydrograph at different gauging stations by

using the average values of the identified roughness coefficients obtained

from the 1996 and 1997 floods ................................................................... 190

Figure 6.16 Identified roughness coefficients obtained from different flood events . 191

Figure 6.17 Identified floodplain roughness functions with different flood events at

Ben Ho gauge: (a) Linear function of stage, (b) Quadratic function of stage

..................................................................................................................... 192

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List of Figures

Figure 6.18 Variation of the roughness coefficient of the main channel with time

during the flood season ............................................................................... 195

Figure B.1 Model of a river network.......................................................................... 226

Figure C.1 Contours of the function values to show the progress of Powells method

..................................................................................................................... 233

Figure C.2 Flow chart for Powells method ............................................................... 234

Figure E.1 Ten representative cross-sections of the Goulburn River......................... 241

Figure E.2 The computational scheme of the Goulbourn River................................. 243

Figure E.3 Observed stages for flood event 21/07-06/08/1978.................................. 244







Figure E.10 Rating curves at Shepparton and Loch Garry gauges............................. 247

Figure F.1 Sixteen representation cross-sections of the Duong River........................ 249

Figure F.2 The computational scheme of the Duong River........................................ 251

Figure F.3 Observed stages and discharges for flood event 14/8-31/8/1995 ............. 252

Figure F.4 Observed stages and discharges for flood event 16/8-30/8/1996 ............. 252

Figure F.5 Observed stages and discharges for flood event 22/7-6/8/1997 ............... 253

Figure F.6 Observed stages and discharges for flood event 26/7-5/8/1998 ............... 253

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List of Tables

List of tables

Table 2.1 Comparison of calculated and estimated values of Mannings n (after

Sargent (1979))................................................................................................15

Table 2.2 Some empirical formulae in Stricklers form (all xd is computed in metres)

.........................................................................................................................19

Table 2.3 Some empirical formulae in Limerinos form (R and xd are in m) ..............20

Table 2.4 Some other empirical formulae .....................................................................21

Table 3.1 Characteristic data of experimental runs .......................................................53

Table 3.2 Computed roughness coefficients from the full velocity profiles .................54

Table 3.3 Brief description of the bed and banks for the selected rivers ......................60

4 Summary of the main hydraulic and roughness characteristics of the studiedTable 3.

Table 3.

Table 5.

Table 5.

Table 5.

Table 5.

Table 5.

Table 5.7 Different combinations of boundary conditions..........................................128

rivers ............................................................................................................61

Table 3.5 Mean relative error of the computed Manning's nfor the selected rivers.63

Table 3.6 Some applicable equations for the selected river ..........66

7 Range of the difference between and measured n and estimated n from

different equations and the corresponding values ofMAEandMRE.............70

Table 4.1 Classification of mathematical models (after Cunge (1975))........................74

Table 5.1 The parameters of the upstream hydrographs used in many test cases .......108

2 The range and average values of the identified nof 50 samples with different

noise levels (the true value is 0.035) .............................................................113

3 The means of the identified roughness coefficients of 10 samples and their

range with a 95% confidence level (the true value is 0.035).........................114

4 Effect of different grid sizes on the identified roughness coefficient (the true

value is 0.035) ...............................................................................................123

5 Relative errors in the identified roughness coefficient with different time

steps related to channel characteristics..........................................................125

6 Relative errors in the identified roughness coefficient with different time

steps related to the characteristics of flood events ........................................126

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List of Tables

Table 5.8 Identified roughness coefficient from difference boundary combinations (the

true value =n 0.035)......................................................................................129

Table 5.9 Identified roughness coefficient using a single rating curve as a downstream

boundary condition (the true value =n 0.035) ..............................................130

Table 5.10 Errors in the identified n with different combinations of boundary

conditions and different types of observed data at the middle cross-section of

the channel when data contained noise ( 1.0= ) .........................................133

Table 5.11 Means of identified roughness coefficients and their confidence intervals of

95% with different noise levels .....................................................................143

Table 5.12 Identified roughness coefficients for noise-free data with different peak

discharges (the true values of cn and fn are 0.028 and 0.042 respectively)145

Table 5.13 Errors in the identified roughness coefficients of the main channel and

floodplains when data contain noise ( 05.0= ) ...........................................146

Table 5.14 Identified roughness coefficients for noise-free data with different

observation intervals (the true values of cn and fn are 0.028 and 0.042

respectively) ..................................................................................................147

Table 5.15 Errors in the identified roughness coefficients of the main channel and

floodplain when data contain noise ( 05.0= ).............................................148

Table 5.16 Identified coefficients of roughness function when data contain noise ....150

Table 5.17 Computation times by using the roughness functions and the conveyance

functions ........................................................................................................156

Table 6.1 Observed peak dischargepQ and stage pZ

Z

of the selected flood events....165

Table 6.2 Different combinations of boundary conditions..........................................173

Table 6.3 Root mean square errors (RMSE) for different roughness models.............177

Table 6.4 Identified roughness coefficients and RMSEs with different flood events.182

Table 6.5 Observed peak dischargepQ and stage p of the selected flood events....186

Table 6.6 Identified roughnesses and RMSE with different flood events...................193

Table A.1 Measured and computed roughness coefficients (using velocity data) ......224

Table D.1 Identified roughness coefficients from different noisy data samples .........235

Table D.2 The calculated 2 for 50 samples of noise level = 0.05 .........................238


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List of Tables



.6 Identified roughness coefficients from different noisy data samples for theTable D

Table E.1 Thalweg elevation at each cros

Table F.1 Thalweg elevation at each cross-section in the Duong River .....................248

compound channel.........................................................................................239

s-section in the Goulburn River from

Shepparton to Loch Garry .............................................................................240

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List of Notations

List of notations

A= wetted area

s = water surface width of the inundated areaB

B = water surface of the flow

C = Chezys roughness coefficient

c = wave celerity

Cr= Courant number

x

s

D = flow depth

xE = relative error inx(xis ratio of the velocity at two-tenths the depth to that at eight-

tenths the depth)

n = relative error in estimated nusing velocity measurements from the experimentE

d= height of roughness

d = the bed material diameter such thatxpercent of material by weight is smaller

Fr = Froude number

f = Weisbachs roughness coefficient

g= acceleration due to gravitation

K= conveyance

k= equivalent sand roughness height

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List of Notations

L = reach length

l = characteristic length known as the mixing length

n= Manning's roughness coefficient

cn = roughness coefficient of the main channel

ncomp= computed roughness coefficient

e = equivalent roughness coefficientn

fn = roughness coefficient of the floodplain

nmeas= measured roughness coefficient that is determined using the direct method

P = wetted perimeter

Q = discharge

b = initial discharge of the inflow hydrograph function (for the synthetic data)Q

= friction slope

= bed slope

= water surface slope

pQ = peak discharge of the inflow hydrograph function (for the synthetic data)

q= lateral inflow per unit length of channel

R= hydraulic radius

Re = Reynolds number

kRe = roughness Reynolds number

fS

0S

wS

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List of Notations

u= velocity at a point

t = time variable in the Saint-Venant equation

p= time to peak discharge of the inflow hydrograph function (for the synthetic data)t

2.0u = velocity at two-tenths the depth

8.0u = velocity at eight-tenths the depth

= cross-sectional average shear velocity

= shear velocity

V= cross-sectional average velocity

= to that at eight-tenths the depth or

space variable in the Saint-Venant equation

= observed discharge or stage in the objective function

= simulated discharge or stage in the objective function

z= distance of a point from the solid surface

= constant of integration in the vertical velocity distribution equation

qu = the velocity component in x direction of the lateral inflow in the Saint-Venant

equations

*U

*u

x ratio of the velocity at two-tenths the depth

OY

SY

0z

= water stage/water surface elevation

= bed level at a cross-section the level that the discharge equal to zero (cease level)

= elevation of the floodplain

Z

bZ

f

Z

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List of Notations

Z0 = minimum water level starting to calculate the roughness function

= parameter of the inflow hydrograph function (for the synthetic data)

tor

ulated stage/discharge

onstant

of the synthetic data

= kinematic viscosity

= weighting coefficient in the objective function

MAE = Mean Absolute Error

MRE = Mean Relative Error

RMSE = Root Mean Square Error

= momentum correction fac

t = computational time step

x = computational space step

= weighting coefficient of Pressmanns scheme

= error between the observed and sim

= von Krmns turbulent c

= mass density of the fluid

= standard deviation to indicate the noise/error level

0 = shear stress on the bed of the flow in the channel.

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Chapter 1

Chapter 1

Introduction

1.1 Description of the study

1.1.1 General

The knowledge of the roughness/friction coefficient is very important throughout

water engineering. While flow resistance in full pipe flow is one of the most

extensively studied in hydraulic engineering, many difficulties still remain in the

estimation of roughness coefficients for this area in open channels.

In the case of full pipe flow, the friction coefficient depends on relative roughness and

Reynolds number (Nikuradses experiments). The well-known Moody diagram

(Moody 1944) to determine the friction coefficient was developed from the work by

Colebrook (1938-1939). It has been applied with considerable success for determining

pipe flow resistance. However, flow resistance in open channels has been more

difficult to quantify because it depends on many factors such as surface roughness,

cross-sectional shape, vegetation conditions, channel irregularity and flow conditions.

The common roughness coefficients used in practice are Mannings n, Darcy-

Weisbachs friction coefficient f and Chezys coefficient C. When using these

coefficients for open channel resistance, they are theoretically interchangeable and

equivalent. However, for a given type of rough boundary surface, thef or Cvaries with

flow depth whereas nremains nearly unchanged (Yen 2002). This is the reason for the

popularity of Mannings formula in the open channel hydraulic area. This has made

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Chapter 1 Introduction

Mannings roughness coefficient n a valuable tool for assessing the effects of open

channel flow resistance.

Over the years, many organisations working on water field have interested in

improving the estimation of roughness coefficients such as US Geological Survey

(USGS), US Army Corps of Engineering (USACE), Land and Water Australia (LWA)

and DSIR Marine and Freshwater in New Zealand. There have been numerous studies

that have made important contributions to open channel flow resistance. As mentioned

above, the value of Mannings ndepends on many factors. Although those factors are

identifiable, their individual contributions and their interactions to the value of

roughness coefficient are difficult, if not impossible, to quantify. To select a value ofManning's n,all of their effects are lumped together into a single coefficient. This is

really a matter of intangibles. At our present stage of knowledge, there is no exact

method of selecting the nvalue. There are many problems still remaining and yet the

amount of research is almost negligible. Further study in this area is still required.

1.1.2 Background of the research

At present, several methods for estimating Manning's n have been developeddepending on the data availability and application conditions.

For steady flow, when discharges, water stages, friction slopes and some cross-sections

are measured, by using the steady flow equation, the value of the roughness coefficient

can be directly determined. This method is known as the direct methodand considered

as a most accurate method to estimate n. It is described and used in Barnes (1967) and

Hicks and Mason (1991). However, this method is time consuming and costly.

Therefore, in current practice this coefficient is usually estimated from

qualitative/empirical methods and empirical formulae.

When flow and cross-section data or surface bed material data are not available, the

qualitative/empirical methods are usually used. For these methods, tables, guidelines

or photographs of channel reaches of known nvalues can be used to estimate n (e.g.

Chow 1959; French 1985; Urquhart 1975; Barnes 1967; Hicks and Mason 1991; Coon

1998). Although these methods are rather simple and can quickly obtain the

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preliminary value of n, they are highly uncertain and subjective. Using these methods,

different investigators can provide different values of this roughness coefficient.

Many empirical formulae obtained from experimental flume and natural channels to

estimate the roughness coefficient have been proposed. These are usually in Stricklers

form (see French 1985) or Limerinos form (e.g. Limerinos 1970; Bray 1979). For the

formulae in the Stricklers form, information on the particle size distribution curve of

surface bed material is required. These data are not usually available for many rivers.

For formulae in Limerinos form, besides information on the particle size distribution

curve of surface bed material, the cross-section data (in terms of hydraulic radius) is

also required.

Some others types of empirical formulae come indirectly from slope-area empirical

formulae and are combined with Manning's equation to obtain the value of Manning's

n (e.g. Lacey, 1946; Riggs, 1976; Jarrett, 1984; Dingman and Sharma, 1997). Usually,

these formulae require the information about cross-sections (in terms of hydraulic

radius) and water surface slope. However, their practical applicability and accuracy are

still questionable.

In many rivers, a simple method to measure stream flow is to measure velocity in

several verticals at two-tenths and eight-tenths the depth. From the logarithmic law of

velocity distribution, it can be seen that the velocity distribution depends on the

roughness height, which may be related to Mannings n. If the distribution is known

then the value of n can be determined without any other information such as surface

bed material or hydraulic characteristics and water surface/friction slope of a reach.

Chow (1959) and French (1985) described this method of estimating nfor wide roughchannels. However, this is not well established and yet to be applied in practice.

Hence, it is necessary to reinvestigate this method. If it can be shown satisfactory for

practical applications, it will provide an option of estimating Manning's n in wide

channels where two-point velocity observations have been made.

For unsteady flow, although the parameters estimated by the above methods can be a

satisfactory representation of steady behaviour, the values may not be satisfied in these

flow conditions because Mannings nmay vary with discharge or depth (Khatibi et al.

3


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1997). In this case, there seems to be no approach available to modellers other than to

estimate a roughness coefficient using numerical models. The value of the roughness

coefficient can be obtained either by using trial and error or automatic optimisation

methods.

The earlier method is a traditional one. It mainly involves visual comparisons between

observed data and simulated ones to estimate the values of parameters. It suffers from

subjectivity and inefficiency. To overcome subjectivity problems and to achieve

efficiency, the later methods may be applied. The problem of identifying the values of

roughness coefficient embedded in the unsteady flow equations is referred to as the

inverse problem or the roughness identification problem. These methods involve asystematic iterative procedure that seeks to uncover the optimum values of parameters

by minimising a chosen error criterion/objective function.

There is a wide range of objective functions and optimization methods used for the

inverse problem (e.g. Becker and Yeh 1972, 1973; Wiggert et al. 1976; Fread and

Smith 1978; Wormleaton and Karmegam 1984; Atanov et al. 1999; Ramesh et al.

2000; Ding et al.2004). However, studies on the roughness identification problem in

open channels are still sparse. They have not been sufficient to establish a clear

understanding of the factors affecting the identified parameters in open-channel

problems. At present, only some factors have been considered by Khatibi et al.(1997).

Some other factors related to the modelling problem, such as values of weighting

coefficient, computational grid sizes, boundary condition combinations and errors in

initial conditions have not been considered. Moreover, the applications of this

problem to natural rivers are limited and only the roughness coefficient of the main

channel has been considered Wormleaton and Karmegam (1984). In flood routing in

natural rivers, many channels have compound sections and the values of roughness

coefficients in main channel and floodplains are usually different. Therefore, it is

necessary to extend the method to overbank flow where floodplain roughness

coefficient will obviously have to be considered.

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1.1.3 Scope and objectives

This study will focus on problems related to the estimation of the roughness coefficient

(Manning's n) for one-dimensional flow in open channels. The study includes twomain parts. The first part is for steady flow and the second part is for unsteady flow.

For steady flow, the main objective is to reinvestigate and extend the method of

estimating Mannings nusing two-point velocity measurements. This work is based on

a theory of logarithmic velocity distribution in fully turbulent flow for rough channels.

Thus, it can be only applied to wide channels where velocity distribution following

logarithmic distribution is assumed throughout the depth.

For unsteady flow, this research concentrates on aspects related to the inverse problem

or the roughness identification problem. The governing equations of one-dimensional

unsteady flow can be derived from the principles of conservation of mass and

momentum. The resulting equations are hyperbolic, non-linear differential equations

known as the Saint-Venant equations. The roughness coefficient, as embedded in the

momentum equation, cannot be measured directly and therefore needs to be estimated

by optimisation methods. There are two main objectives here. The first one is toinvestigate modelling factors affecting the quality of the identified parameter. The

second one is to extend the roughness identification problem to compound channels

and to channels where the roughness varies with water stage.

1.2 Research methodology

In order to obtain the research objectives, the following research methodology has

been carried out:

For steady flow

Review the limitations of the current method;

Derive the equations to estimate Manning's n using two-point velocity data;

Set up the experiment to verify the sensitivity analysis;

Apply the proposed method to natural rivers; and

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Assess the performance of the proposed method by comparing the computed

Manning's nwith the measured nand with the values computed by the other

empirical formulae.

For unsteady flow

Briefly review the numerical methods for solving one-dimensional unsteady

flow in open channels and the optimisation methods applied to the inverse

problem in this area;

Develop an appropriate roughness identification model applicable to compound

channels;

Test the performance of the model using synthetic data for both a single

channel and a compound channel;

Investigate the effect of modelling factors affecting the quality of the identified

roughness coefficient by using synthetic data;

Apply the roughness identification model to some natural rivers, especially for

the rivers with inundated floodplains.

1.3 Outline of the thesis

This thesis consists of seven chapters. Following the introduction, Chapter 2 briefly

discusses flow resistance in open channels and factors affecting the roughness

coefficient in open channels. Then current methods of estimating the roughness

coefficient Mannings n in both steady and unsteady flows are reviewed. Their

advantages and limitations are also discussed. Several aspects of estimating the

roughness coefficients for compound channels including the main channel andfloodplain are raised.

In Chapter 3, the method of using two-point velocity data to estimate Manning's n in

open channels for steady flow is reinvestigated and extended. Firstly, the current

method is reviewed. Then proposed formulae for estimating the roughness coefficient

of the cross-section using velocity gauging data will be derived. The sensitivity

analysis in terms of the effect measurement errors in depths and velocities to the

relative errors in estimated nis investigated and verified using experimental tests. The

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proposed method of estimating Manning's n is then applied to 14 rivers in Australia

and New Zealand. The performance of the proposed method is assessed by comparing

the computed nwith the measured values of this roughness coefficient. The method is

also compared with some applicable empirical formulae.

Chapter 4 concentrates on developing an appropriate model for identifying roughness

coefficient Mannings n for unsteady flows in open channels. This model consists of

two sub-models, a hydraulic sub-model and an optimisation sub-model. Firstly, the

chapter provides the theory, background and numerical techniques for solution of the

Saint-Venant equations. Secondly, a hydraulic sub-model is developed for a non-

prismatic channel with a compound section, lateral inflow, and offstream storage.Thirdly, different optimisation techniques used to identify the roughness coefficient in

open channels are briefly reviewed. Then, a direct non-linear optimisation algorithm is

adopted for the optimisation sub-model. Finally, a roughness identification model is

developed by combining the hydraulic sub-model and the optimisation sub-model.

This model will be used to investigate factors affecting the quality of the identified

roughness coefficient and applied to some natural rivers in the following chapters.

Chapter 5 addresses the investigation of different modelling factors affecting the

quality of the identified roughness coefficient. Synthetic data is generated where the

observed data are considered for both cases with and without noise or errors. Firstly,

some numerical experiments are done to test the performance of the model (developed

in Chapter 4). Then by using the synthetic data, the roughness identification model is

applied to investigate the modelling factors that affect the quality of the identified

roughness coefficient. These factors include the weighting coefficient of the numerical

scheme, computational grid size, combinations of boundary conditions and errors in

the initial conditions. Moreover, the problem is extended to compound channels. Some

additional factors affecting the quality of the identified roughness coefficients in

compound channels are considered. Furthermore, the problem is also extended to the

case of roughness variation with the water stage where the roughness coefficient may

be formulated as a function of the water stage. Finally, an alternative conveyance

function is tested for prismatic channels or channels where cross-sections do not

change much along the channel.

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In Chapter 6, the application of the roughness identification model to natural rivers is

performed. The model is applied to two natural rivers. The first river is the Goulburn

River in Victoria, Australia and the second is the Duong River in the North of

Vietnam. Both of them have extensive floodplains. Depending on the data availability,

the characteristics of the cross-sections, and the flow variation in the main channel and

the floodplain, the performance of the model is investigated. Several aspects related to

the roughness variation with water stage and time are also explored.

Finally, Chapter 7 summarises the major conclusions from this thesis and outlines

recommendations for further research in this area.

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Chapter 2

Chapter 2

Methods of estimating roughness

coefficients in open channels

2.1 Introduction

This chapter focuses on the different methods of estimating the roughness coefficients

in open channels. Before reviewing these methods, firstly the flow resistance and

roughness coefficients used in open channels are briefly discussed. Then, current

methods of estimating the roughness coefficient (Mannings nfor this study) for both

steady and unsteady flows are reviewed. Their strengths and weaknesses are analysed.

For unsteady flow, the necessities of extending the roughness estimation problem to

compound channels and to the channels where the value of the roughness coefficient

varies with flows are then discussed. Finally, key points are summarised.

2.2 Flow resistance in open channels

Real fluid flows in general and water flow in particular are always subject to hydraulic

resistance and energy dissipation. For flow computation, the resistance effect is

involved in flow formulae in terms of resistance/roughness coefficients. The most

commonly used formulae are:

Chezys equation:

RSCV= (2.1)

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Chapter 2 Methods of Estimating Roughness Coefficients in Open Channels

Darcy-Weisbachs equation:

RS

f

gV

8= (2.2)

and Mannings equation:

2/13/21 SRn

V= (2.3)

y flows it is the friction/resistance slope ( );g e acceleration

From Equations (2.1), (2.2) and (2.3), the roughness coefficients can be related as:

where Vis the cross-sectional average velocity, n,fand C are the Manning, Weisbach

and Chezy roughness coefficients (or resistance coefficients), respectively; R is thehydraulic radius; Sis the slope: for uniform flow it is the bed slope ( 0S ) and for non-

uniform and unsteadfS is th

due to gravitation.

CRV 8ggnf

RgSf === 6/1 (2.4)

quation (2.3) can be rearrangedMannings e to give the friction slope:

23/42

2

3/4

2

K

QQ

R

QQn

R

VVnSf === (2.5)

where Q is the discharge; A is the wetted cross-section area; K is the conveyance

( nARK /3/2= ).

Chezys equation and Mannings equation are used widely for flow computation in

open channels. The Darcy-Weisbach equation is mainly used in the field of pipe flow

problems but has also been recommended to be used in open channel flows (Task

Force Committee of US Army Corps of Engineers 1963). From a theoretical point of

view, these three flow equations are equally appropriate for flow computation. The

primary difficulty in using any of these equations in practice is accurately estimating

an appropriate value of these roughness coefficients.

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Concerning the preference of using Mannings n, Darcy-Weisbachs f or Chezys C

coefficients for open channel resistance, in Equation (2.4) theoretically these

coefficients are interchangeable and equivalent. If the value of one resistance

coefficient is determined, the corresponding values of the other resistance coefficients

can be computed. However, the flow in natural rivers is usually fully rough and

turbulent. For rigid boundaries of open channels,fand Cvary with flow depth whereas

nremains nearly unchanged (French 1985 and Yen 2002). It can be argued that in view

of possible changing bed-forms and the composite-rough nature of a river, it is

unlikely ncan be a constant for different depth. For example, some field studies on the

roughness coefficients of some natural channels (Butler et al. 1978; Robbins 1976;

Sargent 1979) showed that n probably varies with flow depth. Nevertheless, the

popularity of the Manning formula in the open channel area may be an indication of

less variability of n as compared tof. This makes Mannings nbecome a valuable tool

to assess the effects of open channels flow resistance. It is also the reason why

Mannings equation is the most reliable and widely used in practice for flow

computation in open channels (Henderson 1966 and French 1985). Therefore, with the

consideration of estimating roughness coefficients in open channels, Mannings n is

chosen as the roughness parameter for the rest of this thesis. The symbol n and the

roughness coefficient are used interchangeable to indicate this roughness coefficient.

One of the difficulties involved in open channel flow computation is accurately

estimating an appropriate value of Manning's n. This is because its value depends on

many factors. The major factor is channel surface roughness, which is determined by

size, shape and distribution of the grains of material that line the bed and sides of the

channel (the wetted perimeter). Five other main factors are channel surface

irregularity, channel shape variation, obstructions, type and density of vegetation, and

degree of meandering (Cowan, 1956). Some additional factors that affect energy loss

in a channel are depth of flow, seasonal changes in vegetation, amount of suspended

material and bed load, and changes in channel configuration due to deposition and

scouring. All these factors were described in detail by Chow (1959). Rouse (1965)

critically reviewed hydraulic resistance in open channels on the basis of fluid

mechanics. He classified flow resistance into four main components: (i) surface

resistance, (ii) form resistance, (iii) wave resistance from free surface distortion, and

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(iv) resistance arising from flow unsteadiness. The effect of flow unsteadiness on

Manning's nwas also discussed by Sturm (2001) and Yen (2002).

Therefore, to select a value of Manning's n,we actually combine all of the effects of

many factors affecting the resistance to flow into a single coefficient. Furthermore, this

is really a matter of intangibles. As a result, there are a number of methods developed

to estimate Manning's n. In the next sections, the current methods of estimating the

value of this roughness coefficient are briefly described. Their strengths and

weaknesses are also discussed.

2.3 Methods of estimating Manning'sn

for steady flow inopen channels

The estimation of Manning's n for a given flow in a given channel reach involves a

substantial measure of engineering judgment. In practice, depending on the availability

of data, there are different methods of estimating this roughness coefficient in open

channels. These are discussed in the following subsections.

2.3.1 Direct measurement method

If the other parameters in Mannings formula can be evaluated by measurement or

observation, then the value of Manning's n can be calculated. This was the basis for

derivation of the roughness parameter (e.g. Barnes 1967; French 1985; Hick and

Mason 1991). According to Arnold et al.(1988) the ideal characteristics of a reach for

applying this method are:

it is straight; its length is at least five times its width;

it has uniform cross-sections or is converging;

its flow is contained without overflow; and

it has straight entrance and exit conditions, with no backwater effects.

The formula to calculate Manning's nby the direct method is:

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( ) ( ) ( )

( ) ( )

21

23/2

13/2

,1

2

,1),1(111

+

=

=

=

N

i ii

ii

N

i

iiiiNN

ARAR

L

hkhhZZ

Qn (2.6)

where Q is the water discharge, Nis the number of cross-sections (with theNth cross-

section being furthest upstream), Z is the water surface e vation,le is the

velocity head, V is average velocity,

gVh 2/2=

iih ,1 is the change in velocity head between

sections the 1i and i, k head loss coefficient of a contracting or expanding reach (kis

assumed to be equal to zero for uniform or contracting reaches and 0.5 for expanding

reaches (Hicks and Mason 1991)),Ais the wetted channel cross-sectional area,Ris the

hydraulic radius, and iiL ,1 is the distance between sections 1i and i.

The discharge can be derived from velocity measurements or rating curves. The cross-

sectional area and wetted perimeter (hence the hydraulic radius) can be derived from

stage observation and cross-sectional surveys (usually more than two cross-sections in

each case). In practice, a survey covering a length of 1 km or more along the channel

may be required to provide a reasonable estimate of the slope.

Manning's ndetermined from this method is usually considered as an accurate value. It

is considered as a measured Manning's n and used as a basis to assess the performance

of the other methods (e.g. Jarrett 1984; Sargent 1979; Coon 1998). However, this

method is time consuming and costly because it requires surveying not only the water

stages, discharges, and some cross-sections along the reach but also the slope.

stressed that care should be taken to

relate to a steady flow situation, that is,

Moreover, for applying this method, it should be

ensure as far as possible that the observations

to a situation in which conditions do not change significantly with time. Roughness

parameter values derived from data relating to unsteady flows are likely to be

erroneous.

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2.3.2 Using the tables or guidelines

When both flow and cross-section data and surface bed material data are not available,

usually the tables or guidelines to estimate Manning's nare used. Most text books onhydraulics and many other publications dealing with free-surface flow present tables

showing the values of roughness coefficients for channels of various descriptions.

Typical examples are the tables presented by Chow (1959), Henderson (1966) and

French (1985). For each kind of channel the minimum, normal and maximum values

of Manning's nare presented.

Urquahart (1975) provided a detailed guideline of how to estimate the value of

In general, this method is the most simple, rough and quick selection of the nvalue to

. However, such tables might be assumed only to offer the

asis for a preliminary estimation of the value of Manning's nin a given channel reach.

Moreover, in most cases, the extent to which the reliability of the tabulated values has

bee a

comparison between the computed Manning' with

the values estimate hows figures fo s of sim n. From

the table re signific ifferences values,

especiall

Manning's nfor a given channel. This method involves the selection of a basic nvalue

for uniform, straight and regular channel in a native material and then modifing this

value by adding correction coefficients. These coefficients are added by a critical

consideration of the effects of some other factors such as vegetation, channel

irregularity, obstructions and channel alignment. In this process, it is critical that each

factor be considered and evaluated independently. It is suggested that the turbulence of

the flow can be used as a measure or indicator of degree of retardation; i.e. factors

which induce a greater turbulence should also result in an increase in Manning's n.

be used in a given problem

b

n established is not explicitly stated. For example, Sargent (1979) made

s nof 5 rivers in United Kingdom

d from C r the channel ilar descriptio

it can be seen that there a ant d between these

y for the first two rivers.

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Table 2.1Comparison of calculated and estimated values of Mannings n(after

Sargent (1979))

Station Estimated n Calculated n

River Almond at Craigiehall 0.04 0.0183-0.0255

Water of Leith at Murrayfield 0.03-0.04 0.0193-0.0257

River Esk at Musselburgh 0.03 0.0265-0.0438

River Tyne at East Linton 0.035-0.050 0.0382-0.0490

River Tyne at Spilmersford 0.03-0.04 0.0254-0.0291

2.3.3 Photographic comparisons

In the absence of a satisfactory quantitative procedure to determine the Manning'sn for

a channel, this visual assistance method can be applied. The principle of this method is

that based on the photographs and geometric and hydraulic descriptions of a wide

range of channel reaches of known n values, the n value of a certain reach can be

adopted for a different reach having recognisably similar hydraulic characteristics. At

present, there are several documents available with photographs of a wide range of

withn values ranging from 0.024 to 0.075. More than 150 colour

photos from 78 streams in New Zealand with a wide range of mean flow (from 0.1 to

ith floodplains, it is necessary to refer to Arcement and

Schneider (1989).

typical channels, accompanied by brief descriptions of the channel conditions, thehydraulic parameters, and the corresponding n values. For example, Chow (1959)

presented 24 black and white photographs of channel reaches with n value ranging

from 0.012 to 0.150. Barnes (1967) presented approximately 100 colour photos from

50 natural channels

353 m3/s), slope (from 0.00001 to 0.042), and bed material (from silt to boulders and

bedrock) were also presented by Hicks and Mason (1991). A similar visual aid was

also employed by the U.S. Geological Survey (Arcement and Schneider 1989).

In these documents, the values of Manning's nof the channels were determined using

the direct measurement method, as discussed in Section 2.3.1. Measurements were

generally limited to flow below bankfull level. To estimate the roughness coefficients

for compound channels w

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Although this method is very simple and can quickly be used to obtain value of

In principle, estimation of the value of Manning's n by this method involves the

ethod was applied for a number of streams in

south-east Australia (e.g. Goyen 1982; Keller 1982; Mitchell 1982; Morris 1982;

measurements of velocity distribution

calculate the roughness coefficient from Mannings equation. However, with

the logarithmic law of velocit

Manning's nfor a channel, this method is subjective and in some cases, the accuracy is

limited. Coon (1998) showed that many channels appearing similar upon visual

inspection actually have unrecognisable differences that could substantially affect the

value of the roughness coefficient. For example, water surface slope is one of the

factors that can have an influence on the nvalues (e.g Bray 1979; Jarrett 1984;Riggs

1976; Coon 1998). However, it is impossible to accurately estimate the water surface

slope from a photograph.

2.3.4 Fitting calculated water surface profile to observed levels

calculation of water surface profiles using a range of assumed values of the roughness

coefficient. The water surface profiles are calculated at a given discharge for steady

gradually-varied flow. For this method, it is necessary that a reliable estimate of the

discharge should be available. The value of nyielding the water profile which best fits

the observed water levels then might be expected to represent a reasonable estimate of

the value of n for the channel. This m

Sheehan 1982).

For applying this method, the quantity of data required, and the effort and cost

involved in its acquisition may clearly be of significant magnitude. Moreover, because

the water profiles are calculated for steady cases, care should be taken when applying

to the cases of data which may relate to unsteady conditions (i.e. the use of the peak

discharge and the flood marks along a reach). In order to overcome this limitation,

unsteady flow models may be applied.

2.3.5 Field

The practical measurement of discharge is usually made by taking velocity

measurements in several verticals at two-tenths and eight-tenths the depths. Although

discharges are known, usually the slope is not known and hence one cannot directly

reference to y distribution (Keulegan 1938), it can be

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seen that the velocity distribution depends on the roughness height, which may be

related to Mannings n. If the distribution is known then the value of n can be

determined.

This concept was used by Boyer (1954) and Langbein (1940) for estimating the value

of n.The procedure for estimating nby using velocity measurements is presented by

Chow (1959) and French (1985). The formula to calculate Manning's nwas derived as:

)95.0(57.5

)1( 6/1

+

=x

Dxn (2.7)

where 8.02.0 / uux= , 2.0u and 8.0u the velocities at two-tenths and eight-tenths the

depth andDis the depth of flow calculated in metres.

This equation gives the value for n for a wide channel with logarithmic velocity

distribution. When this equati n is ap ied to actual streams, the value of Dmay be

taken as the mean depth. French and McCutcheon (1977) found that this method

provided reasonable estimates of n in a study of the Cumberland River, Tennessee.

However, French (1985) indicated that although the above equation provided a

reasonable estimate of n, additional verification of the method was still required. Chow

(1959) also remarked that if this method can be shown as sa

o pl

tisfactory for practical

determining the roughness coefficient in

ade.

more

general formulae to calculate Manning's n and verify the method by applying it to

anning's nis known.

applications, it will provide an easy way of

streams where the velocity observations have been m

In practice, the velocities are measured at many vertical lines at a certain cross-

sections, but which values of 2.0u and 8.0u should be used in Equation (2.7) to

calculate nis not clear. Also, the velocity distribution at a vertical line reflects only the

roughness characteristic at the vicinity of this vertical. It indicates that the formula for

calculating nhas not been well established. Therefore, it is necessary to derive

some natural rivers where the measured M

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2.3.6 Empirical formulae

There are disagreements between researchers regarding the conditions under which

S

value of coefficients a and b (French 1985). The most common material diameters

used in these empirical fo re d50, d75 and d90. However, there is no clear

explana Hey et al.1982). Simons and Senturk (1977)

found t alue of nare different from the ones computed by the Meyer-

Peter and Muller formula (1948) and they are not affected very much by variation of

d90. Also fr vers, Bray (1979) concluded that the

correla nis very low and that the selection of different

grain d not affect the result. In a plot of Mannings nversus d5 hese

rivers, Bray (1979) found that there was considerable scatter of data points and that the

Besides the methods described above, there are a number of empirical formulae

developed for estimating Manning's n. These formulae are usually based on the resultsobtained from experimental flumes or natural channels. They can be classified into

three main groups: the group with Stricklers form (Strickler 1923), the group with

Limerinos form (Limerinos 1970), and the group with other empirical equations

obtained from equating Mannings equation with other empirical flow formulae.

Empirical formulae with Stricklers form:

For this group, the value of n is associated with the surface roughness of a channel

boundary. These formulae are in the following form:

b

xadn= (2-8)

where xd is the bed material diameter such that x percent of material by weight is

smaller and aand bare constants. The earliest formula of this type was proposed by

Strickler in 1923 and is mentioned in many textbooks (e.g. Chow 1959; French 1985;

Henderson 1966). Most researchers have accepted the constant b to be equal to 1/6.

From the statistical analysis of 67 gravel bed rivers, Bray (1979) showed that the

variability of this coefficient is very low. Table 2.2 shows some formulae of different

authors in Stricklers form for estimating Manning's n.

tricklers experiment was conducted, the definition of dxand its unit, as well as the

rmulae we

tion or reasons for their selection (

hat the measured v

om a study of 67 gravel bed ri

tion coefficient between dxand

iameters does 0for t

18


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Stricklers formula gave a lower estimate of n. However, Hey et al.

that d90is a good indication of bed roughness.

Table pirical formulae in Stricklers form (all is com uted in metres)

Author(s) Equation

(1982) indicated

2.2Some em xd p

Strickler (1923) (cited in Yen 1991)61

500474.0 dn=

Keulegan (1938)61

500395.0 dn=

Henderson (1966)61

75038.0 dn=

Meyer-Peter and Muller (1948)61

900385.0 dn= 61

650416.0 dn= Irmay (1949)

61

900249.0 dn= 61

500594.0 dn=

Bray (1979)61

650569.0 dn= 61

900523.0 dn=

Raudkivi (1976)16.0

65041.0 dn= 61

50047.0 dn= Subramania (1982) and Garde and Raju (1978)

61Chen (1991a) 500397.0

dn=

Empirical formulae with Limerinos form:

This type of equations is derived from the existing semi-logarithmic formulations of

grave n ed in

the following form:

l bed rivers, where the Darcy-Weisbach friction coefficient fca be express

+=

xd

RCC

flog

121 (2.9)

and from that Manning's ncan be obtained by:

19


42/275


+

=

xd

RCC

Rn

log

113.0

21

6/1

(2.10)

whereRis the hydraulic radius, is the bed material diameter such thatxpercent of

material by weight is sm and are constants.

Table 2.3Some empirical formulae in Limerinos form (

xd

1C 2Caller and

R and are in m)xd

Author(s) Equation

Limerinos (1970)

+

=

84

61

log03.216.1

113.0

d

RRn

50d

+ log36.2248.0R

=61113.0 R

n

Bray (1979)

+

=

65

61

log28.2608.0

113.0

d

R

Rn

+

=61

log16.226.1

113.0

d

R

Rn

90

Griffiths (1981)

+

=

50

61

log98.176.0

113.0

d

R

Rn

Phillips and Ingersoll (1998)

+

=

50

log23.246.1

113.0

dR

Rn

61

Similar to Stricklers ifferent bed material

diameters in their formulae of Limerinos form equations. The most common

diameters employed in their formulae were d50, d84 and 2.3 shows some

empiri n the form of Limerinos equation. By comparing with formulae in

the Strickler form, Bray (1979) indicated that the equations in Limerinos formperform better because for their determination of Manning'sn is stage dependent.

form equations, different authors used d

d90. Table

cal formulae i

20


43/275


Using the power formulations, Manning's n is also estimated in a similar way (e.g.

Bray 1979; Charlton et al.1988; Chen 1991b; Griffiths 1981). Also, there are some

other g ulae proposed by other authors (e.g. Afzalimehr

and An .1988), where the friction coefficient is not only the

function of the relative roughness but also of the Froude number, a sediment mobility

parame l form factor that can improve the prediction of the

friction

Other empirical equations

In these equations,nwas not originally the subject of the empirical equations. These

le 2.4Some other empirical formulae

ravel bed semi-logarithmic form

ctil 1998; Colosimo et al

ter

thesisf_final1

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