RiverHydraulics

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December 10, 2005
421-423 – River hydraulics
John Fenton
Department of Civil and Environmental Engineering
University of Melbourne, Victoria 3010, Australia
Abstract
This elective is a course on a topic which is of increasing importance in Australia. The nature of
flows in rivers, their measurement, the calculation of flows and flood propagation, the knowledge of
the factors affecting water quality, and the stability of rivers and how to ensure this are described
in this course. We will see how some of the fundamentals have been glossed over, and some very
simple improvements can be made to traditional practice. It is hoped that students taking this course
will develop a deep understanding of the processes at work in rivers. Some additional notes will be
distributed on the geomorphology of rivers.
Throughout these lectures, both in approximations to wave motion in waterways, and in the transport
of pollutants, we will encounter the physical process of diffusion. An introduction to diffusion is
given in Appendix A.1, but is not for examination.
Table of Contents
References . . . . . . . . . . . . . . . . . . . . . . . 6
1. Introduction . . . . . . . . . . . . . . . . . . . . . 8
2. Hydrography/Hydrometry . . . . . . . . . . . . . . . . 8
2.1 Water levels . . . . . . . . . . . . . . . . . . . 8
2.2 Discharge . . . . . . . . . . . . . . . . . . . 9
2.3 The analysis and use of stage and discharge measurements . . . . 16
3. The propagation of waves in waterways . . . . . . . . . . . . 24
3.1 Mass conservation equation . . . . . . . . . . . . . . 25
3.2 Momentum conservation equation . . . . . . . . . . . . 26
3.3 The nature of the propagation of long waves and floods in rivers . . . 29
3.4 A new low-inertia approach – Volume routing . . . . . . . . 33
4. Computational hydraulics . . . . . . . . . . . . . . . . 37
4.1 The advection equation . . . . . . . . . . . . . . . 37
4.2 The diffusion equation . . . . . . . . . . . . . . . 41
4.3 Advection-diffusion combined . . . . . . . . . . . . . 42
5. Water quality . . . . . . . . . . . . . . . . . . . . 43
5.1 Useful sources for further reading . . . . . . . . . . . . 43
5.2 Water quality characteristics . . . . . . . . . . . . . . 44
5.3 Types of pollutant . . . . . . . . . . . . . . . . . 44
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421-423 – River hydraulics John Fenton
5.4 Mass balance concepts . . . . . . . . . . . . . . . 45
5.5 Impacts of human works . . . . . . . . . . . . . . . 45
5.6 Transport processes . . . . . . . . . . . . . . . . 45
5.7 Tools for problem solving . . . . . . . . . . . . . . 46
5.8 A simple river model – organic wastes and self purification . . . . 47
5.9 Salinity in rivers . . . . . . . . . . . . . . . . . 53
6. Turbulent diffusion and dispersion . . . . . . . . . . . . . . 56
6.1 Diffusion and dispersion in waterways . . . . . . . . . . . 57
6.2 Dispersion . . . . . . . . . . . . . . . . . . . 58
6.3 Non-dimensionalisation – Péclet number and Reynolds number –
viscosity as diffusion . . . . . . . . . . . . . . . . 59
7. Sediment motion . . . . . . . . . . . . . . . . . . . 60
7.1 Incipient motion . . . . . . . . . . . . . . . . . 61
7.2 Relationships for fluvial quantities . . . . . . . . . . . . 62
7.3 Dimensional similitude . . . . . . . . . . . . . . . 62
7.4 Bed forms . . . . . . . . . . . . . . . . . . . 63
7.5 Mechanisms of sediment motion . . . . . . . . . . . . 63
Appendix A On diffusion and von Neumann stability analyses . . . . . . . 65
A.1 The nature of diffusion . . . . . . . . . . . . . . . 65
A.2 Examining stability by the Fourier series (von Neumann’s) method . . 69
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421-423 – River hydraulics John Fenton
Useful references
Tables 1.1-1.4 show some of the many references available, some which the lecturer has referred to in
these notes or in his work. For most book references, The University of Melbourne Engineering Library
Reference Numbers are given.
Reference Comments Engng Lib. no.
Chanson, H. (1999), The Hydraulics of Open Channel Flow, Arnold,
London.
Good technical book,
moderate level, also
sediment aspects
627.042 CHAN
Chaudhry, M. H. (1993), Open-channel flow, Prentice-Hall. Good technical book 627.042 CHAU
CF/DIBM
Chow, V. T. (1959), Open-channel Hydraulics, McGraw-Hill, New
York.
Classic, now dated, not
so readable
532.54 CH
Francis, J. & Minton, P. (1984), Civil Engineering Hydraulics, fifth edn,
Arnold, London.
Good elementary intro-
duction
532.002462 FRAN
French, R. H. (1985), Open-Channel Hydraulics, McGraw-Hill, New
York.
Wide general treatment 627.1 FREN
Henderson, F. M. (1966), Open Channel Flow, Macmillan, New York. Classic, high level,
readable
532.54 HEND
Jain, S. C. (2001), Open-Channel Flow, Wiley. High level, but terse
and readable
Julien, P. Y. (2002), River Mechanics, Cambridge. A readable but high-
level work
Montes, S. (1998), Hydraulics of Open Channel Flow, ASCE, New
York.
Encyclopaedic 627.042 MONT
Townson, J. M. (1991), Free-surface Hydraulics, Unwin Hyman, Lon-
don.
Simple, readable, math-
ematical
627.042 TOWN
Vreugdenhil, C. B. (1989), Computational Hydraulics: An Introduc-
tion, Springer.
Simple introduction
to computational
hydraulics
627.015118 VREU
Table 1.1 : Introductory and general references
Reference Comments Engng Lib. no.
Boiten, W. (2000), Hydrometry, Balkema A modern treatment of
river measurement
627.0287 BOIT
Bos, M. G. (1978), Discharge Measurement Structures, second edn, In-
ternational Institute for Land Reclamation and Improvement, Wagenin-
gen.
Good encyclopaedic
treatment of structures
627.042 DISC
Bos, M. G., Replogle, J. A. & Clemmens, A. J. (1984), Flow Measuring
Flumes for Open Channel Systems, Wiley.
Good encyclopaedic
treatment of structures
627.042 BOS
Fenton, J. D. & Keller, R. J. (2001), The calculation of streamflow from
measurements of stage, Technical Report 01/6, Co-operative Research
Centre for Catchment Hydrology, Monash University.
Two level treatment -
practical aspects plus
high level review of
theory
Novak, P., Moffat, A. I. B., Nalluri, C. & Narayanan, R. (2001), Hy-
draulic Structures, third edn, Spon, London.
Standard readable pre-
sentation of structures
627 HYDR
Table 1.2 : Books on practical aspects, flow measurement, and structures
3
421-423 – River hydraulics John Fenton
Reference Comments Engng Lib. no.
Cunge, J. A., Holly, F. M. & Verwey, A. (1980), Practical Aspects of
Computational River Hydraulics, Pitman, London.
Thorough and reliable
presentation
627.125 CUNG
Dooge, J. C. I. (1987), Historical development of concepts in open chan-
nel flow, in G. Garbrecht, ed., Hydraulics and Hydraulic Research: A
Historical Review, Balkema, Rotterdam, pp. 205–230.
Interesting review f 627 HYDR
Fenton, J. D. (1996), An examination of the approximations in river and
channel hydraulics, in Proc. 10th Congress, Asia-Pacific Division, Int.
Assoc. Hydraulic Res., Langkawi, Malaysia, pp. 204–211.
A modern mathemati-
cal view
Flood Studies Report (1975), Flood Routing Studies, Vol. 3, Natural
Environment Research Council, London.
A readable overview f 551.4890941 FLOO
Lai, C. (1986), Numerical modeling of unsteady open-channel flow, in
B. Yen, ed., Advances in Hydroscience, Vol. 14, Academic.
Good review, a bit
dated
CARM Centre En 532
ADVA
Liggett, J. A. (1975), Basic equations of unsteady flow, in K. Mahmood
& V. Yevjevich, eds, Unsteady Flow in Open Channels, Vol. 1, Water
Resources Publications, Fort Collins, chapter 2.
Readable overview 532.54 MAHM : v.1
Liggett, J. A. & Cunge, J. A. (1975), Numerical methods of solution of
the unsteady flow equations, in K. Mahmood & V. Yevjevich, eds, Un-
steady Flow in Open Channels, Vol. 1, Water Resources Publications,
Fort Collins, chapter 4.
Readable overview 532.54 MAHM : v.1
Miller, W. A. & Cunge, J. A. (1975), Simplified equations of unsteady
flow, in K. Mahmood & V. Yevjevich, eds, Unsteady Flow in Open
Channels, Vol. 1, Water Resources Publications, Fort Collins, chapter 5,
pp. 183–257.
Readable 532.54 MAHM : v.1
Price, R. K. (1985), Flood Routing,in P. Novak, ed., Developments
in hydraulic engineering, Vol. 3, Elsevier Applied Science, chapter 4,
pp. 129–173.
The best overview of
the advection-diffusion
approximation for flood
routing
627 DEVE : V.3
Skeels, C. P. & Samuels, P. G. (1989), Stability and accuracy analysis of
numerical schemes modelling open channel flow, in Cˇ. Maksimovic´ &
M. Radojkovic´, eds, Computational Modelling and Experimental Meth-
ods in Hydraulics (HYDROCOMP ’89), Elsevier.
Review
Zoppou, C. & O’Neill, I. C. (1982), Criteria for the choice of flood
routing methods in natural channels, in Proc. Hydrology and Water
Resources Symposium, Melbourne, pp. 75–81.
Readable overview
Table 1.3 : References on flood & wave propagation – theoretical and computational
4
421-423 – River hydraulics John Fenton
Reference Notes Library and Number
General
Chin (2000) A good introduction Engin 627 CHIN
Martin & McCutcheon (1999) A good book, being both introductory and
encyclopaedic, concentrating on the hy-
draulic engineering aspects
Engin f 627.042 MART
McGauhey (1968) A complete descriptive (non-mathematical)
presentation, which is interesting.
Eng 628.1 McGau
Fundamental processes of mixing and dispersion
Fischer, List, Koh, Imberger & Brooks
(1979)
A comprehensive and standard reference BioMed 628.39 MIXI, EarthSci 628.39
MIXI
Holly (1985) Also fundamental, but shorter Engin 627 DEVE : V.3
Streeter, Wylie & Bedford (1998) A good simpler introduction (Chapter 9) Engin 620.106 STRE
Rutherford (1994) Engin 551.483 RUTH
Csanady (1973) Engin 532.7 CSAN, Maths 531.163 C89
Numerical methods – fundamentals
Noye (1976), Noye (1981), Noye (1984),
Noye & May (1986)
All offer a simple introduction to finite dif-
ference methods
Engin 620.1064072 INTE, Maths 515.353
C76, Baill 515.35 COMP, Biomed
551.4600724 NUME
Smith (1978) A more detailed introduction to finite differ-
ence methods
Maths 515.353 Sm57
Richtmyer & Morton (1967), Morton &
Baines (1982), Morton & Mayers (1994),
Morton (1996)
All are rather more comprehensive, describ-
ing some more general methods
EarthSci 530.155625 RICH, En-
gin 620.106015194 NUME, Maths
532.050151535 MORT, Maths 515.353
MORT
Zoppou & Knight (1997) Analytical solutions to the advection-
diffusion equation where the coefficients are
not constant
Numerical methods – application to environmental modelling
Sauvaget (1985) A simple review Engin 627 DEVE : V.3
The nature of diffusion
Fischer et al. (1979) Very clear - already recommended above BioMed 628.39 MIXI, EarthSci 628.39
MIXI
Jost (1960, page 25; 1964) A leisurely and clear introduction Chem 541.341 JOS
Borg & Dienes (1988) A simple and clear introduction Physics: 530.41 BORG
Widder (1975) A more mathematical approach Maths 515.353 W633
The full equations for wave propagation and flood routing
Cunge, Holly & Verwey (1980) The best explanation of this field Engin 627.125 CUNG
Liggett (1975), Liggett & Cunge (1975) A little disappointing, but the next best ex-
planation
Engin, 532.54 MAHM : v.1
The advection-diffusion approximation for flood routing
Price (1985) The best overview Engin 627 DEVE : V.3
Dooge (1986) A good general study
Sivapalan, Bates & Larsen (1997)
Others
Pasmanter (1988) Estuaries and tidal flows
Kobus & Winzelbach (1989) Groundwater Engin 628.114 INTE
Table 1.4 : Useful references
5
421-423 – River hydraulics John Fenton
References
Australian Standard (1990) Measurement of water flow in open channels, number AS 3778, Standards
Australia.
Australian Standard 3778.3.1 (2001) Measurement of water flow in open channels - Velocity-area meth-
ods - Measurement by current meters and floats, Standards Australia, Sydney.
Boiten, W. (2000) Hydrometry, Balkema.
Borg, R. J. & Dienes, G. J. (1988) An Introduction to Solid State Diffusion, Academic.
Chin, D. A. (2000) Water-Resources Engineering, Prentice Hall.
Chow, V. T. (1959) Open-channel Hydraulics, McGraw-Hill, New York.
Collett, K. O. (1978) The present salinity position in the River Murray basin, Proc. Royal Society of
Victoria 90(1), 111–123.
Csanady, G. T. (1973) Turbulent Diffusion in the Environment, Reidel, Dordrecht.
Cunge, J. A., Holly, F. M. & Verwey, A. (1980) Practical Aspects of Computational River Hydraulics,
Pitman, London.
Dooge, J. C. I. (1986) Theory of flood routing, River Flow Modelling and Forecasting, D. A. Kraijenhoff
& J. R. Moll (eds), Reidel, chapter 3, pp. 39–65.
Elmore, H. L. & Hayes, T. W. (1960) Solubility of atmospheric oxygen in water, J.Sanitary Div. ASCE
86(SA4), 41–53.
Fenton, J. D. (1999) Calculating hydrographs from stage records, in Proc. 28th IAHR Congress, 22-27
August 1999, Graz, Austria, published as compact disk.
Fenton, J. D. (2002) The application of numerical methods and mathematics to hydrography, in Proc.
11th Australasian Hydrographic Conference, Sydney, 3 July - 6 July 2002.
Fenton, J. D. & Abbott, J. E. (1977) Initial movement of grains on a stream bed: the effect of relative
protrusion, Proc. Roy. Soc. Lond. A 352, 523–537.
Fenton, J. D. & Keller, R. J. (2001) The calculation of streamflow from measurements of stage, Techni-
cal Report 01/6, Cooperative Research Centre for Catchment Hydrology, Melbourne.
Feynman, R. P. (1985) Surely you’re joking, Mr. Feynman! : adventures of a curious character, Norton,
New York.
Fischer, H. B., List, E. J., Koh, R. C. Y., Imberger, J. & Brooks, N. H. (1979) Mixing in Inland and
Coastal Waters, Academic.
French, R. H. (1985) Open-Channel Hydraulics, McGraw-Hill, New York.
Goldsmith, E. & Hildyard, N. (1992) The Social and Environmental Effects of Large Dams, Wadebridge
Ecological Centre, Camelford, Cornwall, UK.
Henderson, F. M. (1966) Open Channel Flow, Macmillan, New York.
Herschy, R. W. (1995) Streamflow Measurement, Second Edn, Spon, London.
Holly, F. M. (1985) Dispersion in rivers and coastal waters – 1. Physical principles and dispersion
equations, Developments in Hydraulic Engineering, P. Novak (ed.), Vol. 3, Elsevier, London,
chapter 1.
Jost, W. (1960) Diffusion in Solids, Liquids, Gases, Academic, New York.
Jost, W. (1964) Fundamental Aspects of Diffusion Processes, Angewandte Chemie Int. Edn 3, 713–722.
Keiller, D. & Close, A. (1985) Modelling salt transport in a long river system, in Proc. 21st Congress
IAHR, Melbourne, Vol. 2, pp. 324–328.
6
421-423 – River hydraulics John Fenton
Kobus, H. E. & Winzelbach, W. (1989) Contaminant Transport in Groundwater, Balkema, Rotterdam.
Liggett, J. A. (1975) Basic equations of unsteady flow, Unsteady Flow in Open Channels, K. Mahmood
& V. Yevjevich (eds), Vol. 1, Water Resources Publications, Fort Collins, chapter 2.
Liggett, J. A. & Cunge, J. A. (1975) Numerical methods of solution of the unsteady flow equations,
Unsteady Flow in Open Channels, K. Mahmood & V. Yevjevich (eds), Vol. 1, Water Resources
Publications, Fort Collins, chapter 4.
Lighthill, M. J. & Whitham, G. B. (1955) On kinematic waves. I: Flood movement in long rivers, Proc.
R. Soc. Lond. A 229, 281–316.
Martin, J. L. & McCutcheon, S. C. (1999) Hydrodynamics and Transport for Water Quality Modeling,
Lewis, Boca Raton.
McGauhey, P. H. (1968) Engineering Management of Water Quality, McGraw-Hill, New York.
Morgan, A. E. (1971) Dams and other disasters: a History of the Army Corps of Engineers, Porter
Sargent, Boston.
Morton, K. (1996) Numerical solution of convection-diffusion problems, Chapman and Hall, London.
Morton, K. & Baines, M. (1982) Numerical methods for fluid dynamics, Academic.
Morton, K. & Mayers, D. (1994) Numerical solution of partial differential equations : an introduction,
Cambridge.
Noye, B. J. (1976) International Conference on the Numerical Simulation of Fluid Dynamic Systems,
Monash University 1976, North-Holland, Amsterdam.
Noye, B. J. (1981) Numerical solutions to partial differential equations, Proc. Conf. on Numerical
Solutions of Partial Differential Equations, Queen’s College, Melbourne University, 23-27 August,
1981, B. J. Noye (ed.), North-Holland, Amsterdam, pp. 3–137.
Noye, B. J. (1984) Computationaltechniques for differential equations, North-Holland, Amsterdam.
Noye, J. & May, R. L. (1986) Computational Techniques and Applications: CTAC 85, North-Holland,
Amsterdam.
Pasmanter, R. A. (1988) Deterministic diffusion, effective shear and patchiness in shallow tidal flows,
Physical Processes in Estuaries, J. Dronkers & W. van Leussen (eds), Springer, Berlin.
Price, R. K. (1985) Flood Routing, Developments in hydraulic engineering, P. Novak (ed.), Vol. 3,
Elsevier Applied Science, chapter 4, pp. 129–173.
Richtmyer, R. P. & Morton, K. W. (1967) Difference Methods for Initial Value Problems, Second Edn,
Interscience, New York.
Rutherford, J. C. (1994) River Mixing, Wiley, Chichester.
Sauvaget, P. (1985) Dispersion in rivers and coastal waters – 2. Numerical computation of dispersion,
Developments in Hydraulic Engineering, P. Novak (ed.), Vol. 3, Elsevier, London, chapter 2.
Schlichting, H. (1968) Boundary-Layer Theory, Sixth Edn, McGraw-Hill, New York.
Sivapalan, M., Bates, B. C. & Larsen, J. E. (1997) A generalized, non-linear, diffusion wave equation:
theoretical development and application, J. Hydrology 192, 1–16.
Smith, G. D. (1978) Numerical Solution of Partial Differential Equations, Oxford Applied Mathematics
and Computing Series, Second Edn, Clarendon, Oxford.
Streeter, V. L., Wylie, E. B. & Bedford, K. W. (1998) Fluid Mechanics, Ninth Edn, WCB/McGraw-Hill.
Widder, D. V. (1975) The Heat Equation, Academic, New York.
Yalin, M. S. & Ferreira da Silva, A. M. (2001) Fluvial Processes, IAHR, Delft.
Zoppou, C. & Knight, J. H. (1997) Analytical solutions for advection and advection-diffusion equations
with spatially variable coefficients, J. Hydraulic Engng 123(2), 144–148.
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421-423 – River hydraulics John Fenton
1. Introduction
At the conclusion of this unit, students should be able to describe the nature of flow and floods in
streams, understand the basis of computational methods for rivers, the common means of measurement
of streamflow, the fundamentals of water quality in rivers, and fluvial processes and fluvial morphology.
2. Hydrography/Hydrometry
Boiten (2000) provides a refreshingly modern approach to this topic, calling it ”Hydrometry” the ”mea-
surement of water”, which in the past has received little research. In particular, the Australian Standard
(1990) is a very poor document, providing little practical assistance.
2.1 Water levels
Water levels are the basis for any river study. Most kinds of measurements, such as discharges, have to be
related to river stages (the stage is simply the water surface height above some fixed datum). Both stage
and discharge measurements are important. Often, however, the actual discharge of a river is measured
rarely, and routine measurements are those of stage, which are related to discharge.
Water levels are obtained from gauges, either by direct observation or in recorded form. The latter is
now much more likely in the Australian water industry. The data can serve several purposes:
• By plotting gauge readings against time, the hydrograph for a particular station is obtained. Hy-
drographs of a series of years are used to determine duration curves, showing the probability of
occurrence of water levels at the station or from a rating curve, the probability of discharges.
• Combining gauge readings with discharge values, a relationship between stage and discharge can
be determined, resulting in a rating curve for the station.
• Apart from use in hydrological studies and for design purposes, the data can be of direct value for
navigation, flood prediction, water management, and waste water disposal.
2.1.1 Methods
Most water level gauging stations are equipped with a sensor or gauge and a recorder. In many cases the
water level is measured in a stilling well, thus eliminating strong oscillations.
Staff gauge: This is the simplest type, with a graduated gauge plate fixed to a stable structure such
as a pile, bridge pier, or a wall. Where the range of water levels exceeds the capacity of a single gauge,
additional ones may be placed on the line of the cross section normal to the plane of flow.
Float gauge: A float inside a stilling well, connected to the river by an inlet pipe, is moved up and
down by the water level. Fluctuations caused by short waves are almost eliminated. The movement of
the float is transmitted by a wire passing over a float wheel, which records the motion, leading down to
a counterweight.
Pressure transducers: The water level is measured as an equivalent hydrostatic pressure and trans-
formed into an electrical signal via a semi-conductor sensor. These are best suited for measuring water
levels in open water (the effect of short waves dies out almost completely within half a wavelength down
into the water), as well as for the continuous recording of groundwater levels. They should compensate
for changes in the atmospheric pressure, and if air-vented cables cannot be provided air pressure needs
to be measured separately.
Bubble gauge: This is a pressure actuated system, based on measurement of the pressure which
is needed to produce bubbles through an underwater outlet. These are used at sites where it would be
8
421-423 – River hydraulics John Fenton
difficult to install a float-operated recorder or pressure transducer. From a pressurised gas cylinder or
small compressor gas is led along a tube to some point under the water (which will remain so for all
water levels) and bubbles constantly flow out through the orifice. The pressure in the measuring tube
corresponds to that in the water above the orifice. Wind waves should not affect this.
Ultrasonic sensor: These are used for continuous non-contact level measurements in open channels,
and are widely used in the Australian irrigation industry. The sensor points vertically down towards the
water and emits ultrasonic pulses at a certain frequency. The inaudible sound waves are reflected by the
water surface and received by the sensor. The round trip time is measured electronically and appears as
an output signal proportional to the level. A temperature probe compensates for variations in the speed
of sound in air. They are accurate but susceptible to wind waves.
Peak level indicators: There are some indicators of the maximum level reached by a flood, such as
arrays of bottles which tip and fill when the water reaches them, or a staff coated with soluble paint.
2.1.2 Presentation of results
Stage records taken along rivers used for hydrological studies, for design of irrigation works, or for flood
protection require an accuracy of 2−5 cm, while gauge readings upstream of flow measuring weirs used
to calculate discharges from the measured heads require an accuracy of 2 − 5mm. These days almost
all are telemetered to a central site. There is a huge volume of electronic hydrometry data being sent
around Victoria.
Hydrographs, rating tables, and stage relation curves are typical presentations of water level data:
• Hydrograph – when stage records or the discharges are plotted against time.
• Rating table – at many gauging stations water levels are measured daily or hourly, while discharges
are measured some times a year, using direct methods such as a propeller meter. From the corre-
sponding water levels from these, and possibly for others over years, a stage-discharge relationship
can be built up, so that the routine measurement of stage can be converted to discharge. We will be
considering these in detail.
• Stage relation curves – from the hydrographs of two or more gauging stations along the river,
relationships can be formulated between the steady flow stages. These can be used to calculate
the surface slope between two gauges, and hence, to determine the roughness of the reach. Under
unsteady conditions the relationship will be disturbed. We will also be considering this later.
2.2 Discharge
Flow measurement may serve several purposes:
• information on river flow for the design and operation of diversion dams and reservoirs and for
bilateralagreements between states and countries.
• distribution and charging of irrigation water
• information for charging industries and treatment plants discharging into public waters
• water management in urban and rural areas
• reliable statistics for long-term monitoring.
Continuous (daily or hourly) measurements are very useful.
There are many methods of measuring the rate of volume flow past a point, of which some are single
measurement methods which are not designed for routine operation; the rest are methods of continuous
measurements.
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421-423 – River hydraulics John Fenton
2.2.1 Velocity area method (”current meter method”)
The area of cross-section is determined from soundings, and flow velocities are measured using pro-
peller current meters, electromagnetic sensors, or floats. The mean flow velocity is deduced from points
distributed systematically over the river cross-section. In fact, what this usually means is that two or
more velocity measurements are made on each of a number of vertical lines, and any one of several
empirical expressions used to calculate the mean velocity on each vertical, the lot then being integrated
across the channel.
Calculating the discharge requires integrating the velocity data over the whole channel - what is required
is the area integral of the velocity, that is Q =
R
u dA. If we express this as a double integral we can
write
Q =
Z
B
Z(y)+h(y)Z
Z(y)
udz dy, (2.1)
so that we integrate the velocity from the bed z = 0 to the surface z = h(y), where h is the local depth
and where our z is a local co-ordinate. Then we have to integrate these contributions right across the
channel, for values of the transverse co-ordinate z over the breadth B.
Calculation of mean velocity in the vertical
The first step is to compute the integral of velocity with depth, which hydrographers think of as calcu-
lating the mean velocity over the depth. Convention in hydrography is that the mean velocity over a
vertical can be approximated by
u =
1
2
(u0.2h + u0.8h) , (2.2)
that is, the mean of the readings at 0.2 of the depth and 0.8 of the depth. Fenton (2002) has developed
some families of methods which are based more on rational methods. Consider the law for turbulent
flow over a rough bed, which can be obtained from the expressions on p582 of Schlichting (1968):
u =
u∗
κ
ln
z
z0
, (2.3)
where u∗ is the shear velocity, κ = 0.4, ln() is the natural logarithm to the base e, z is the elevation
above the bed, and z0 is the elevation at which the velocity is zero. (It is a mathematical artifact that
below this point the velocity is actually negative and indeed infinite when z = 0 – this does not usually
matter in practice). If we integrate equation (2.3) over the depth h we obtain the expression for the mean
velocity:
u¯ =
1
h
hZ
0
u dz =
u∗
κ
µ
ln
h
z0
− 1
¶
. (2.4)
Now it is assumed that two velocity readings are made, obtaining u1 at z1 and u2 at z2. This gives
enough information to obtain the two quantities u∗/κ and z0. Substituting the values for point 1 into
equation (2.3) gives us one equation and the values for point 2 gives us another equation. Both can be
solved to give the solution
u∗
κ
=
u2 − u1
ln (z2/z1)
and z0 =
µ
zu21
zu12
¶ 1
u2−u1
. (2.5)
It is not necessary to evaluate these, for substituting into equation (2.4) gives a simple formula for the
mean velocity in terms of the readings at the two points:
u¯ =
u1 (ln(z2/h)+1)− u2 (ln(z1/h)+1)
ln (z2/z1)
. (2.6)
As it is probably more convenient to measure and record depths rather than elevations above the bottom,
10
421-423 – River hydraulics John Fenton
let h1 = h− z1 and h2 = h− z2 be the depths of the two points, when equation (2.6) becomes
u¯ =
u1 (ln(1− h2/h)+1)− u2 (ln(1− h1/h)+1)
ln ((h− h2) / (h− h1))
. (2.7)
This expression gives the freedom to take the velocity readings at any two points, and not necessarily
at points such as 0.2h and 0.8h. This might simplify streamgauging operations, for it means that the
hydrographer, after measuring the depth h, does not have to calculate the values of 0.2h and 0.8h and
then set the meter at those points. Instead, the meter can be set at any two points, within reason, the
depth and the velocity simply recorded for each, and equation (2.7) applied. This could be done either
in situ or later when the results are being processed. This has the potential to speed up hydrographic
measurements.
If the hydrographer were to use the traditional two points, then setting h1 = 0.2h and h2 = 0.8h in
equation (2.7) gives the result
u¯ = 0.4396u0.2h + 0.5604u0.8h ≈ 0.44u0.2h + 0.56u0.8h , (2.8)
whereas the conventional hydrographic expression is (see e.g. #7.1.5.3 of Australian Standard 3778.3.1
2001):
u¯ = 0.5u0.2h + 0.5u0.8h . (2.9)
The nominally more accurate expression, equation (2.8), gives less weight to the upper measurement
and more to the lower. It might be useful, as it is just as simple as the traditional expression, yet is based
on an exact analytical integration of the equation for a turbulent boundary layer.
This has been tested by taking a set of gauging results. A canal had a maximum depth of 2.6m and
was 28m wide, and a number of verticals were used. The conventional formula (2.2), the mean of the
two velocities, was accurate to within 2% of equation (2.8) over the whole range of the readings, with
a mean difference of 1%. That error was always an overestimate. The more accurate formula (2.7) is
hardly more complicated than the traditional one, and it should in general be preferred. Although the
gain in accuracy was slight in this example, in principle it is desirable to use an expression which makes
no numerical approximations to that which it is purporting to evaluate. This does not necessarily mean
that either (2.2) or (2.8) gives an accurate integration of the velocities which were encountered in the
field. In fact, one complication is where, as often happens in practice, the velocity distribution near the
surface actually bends back such that the maximum velocity is below the surface.
Fenton (2002) then considered velocity distributions given by the more general law, assuming an addi-
tional linear and an additional quadratic term in the velocity profile:
u =
u∗
κ
ln
z
z0
+ a1 z + a2z
2, (2.10)
and by taking readings at four depths, enough information is obtained to obtain the solution for u¯.
Methods and computer code for this were presented. Also, in something of a departure, a global approx-
imation method was used, where a function was assumed which could describe all the velocity profiles
on all the verticals, and then this was fitted to the data. An example of the results is given
Figure 2-1. Cross-section of canal with velocity profiles and data points plotted transversely, showing fit by global
function
11
421-423 – River hydraulics John Fenton
Integration of the mean velocities across the channel:
The problem now is to integrate the readings for mean velocity at each station across the width of the
channel. Here traditional practice seems to be in error – often the Mean-Section method is used. In this
the mean velocity between two verticals is calculated and then multiply this by the area between them,
so that, given two verticals i and i+ 1 separated by bi the expression for the contribution to discharge is
assumed to be
δQi =
1
4
bi (hi + hi+1) (ui + ui+1) .
This is not correct. From equation (2.1), the task is actually to integrate across the channel the quantity
which is the mean velocity times the depth. For that the simplest expression is the Trapezoidal rule:
δQi =
1
2
bi (ui+1 hi+1+ui hi)
To examine where the Mean-Section Method is worst, we consider the case at one side of the channel,
where the area is a triangle. We let the water’s edge be i = 0 and the first internal point be i = 1, then
the Mean-Section Method gives
δQ0 =
1
4
b0u1h1,
while the Trapezoidal rule gives
δQ0 =
1
2
b0u1h1,
which is correct, and we see that the Mean-Section Method computes only half of theactual contribution.
The same happens at the other side. Contributions at these edges are not large, and in the middle of the
channel the formula is not so much in error, but in principle the Mean-Section Method is wrong and
should not be used. Rather, the Trapezoidal rule should be used, which is just as easily implemented. In
a gauging in which the lecturer participated, a flow of 1693 Ml/d was calculated using the Mean-Section
Method. Using the Trapezoidal rule, the flow calculated was 1721 Ml/d, a difference of 1.6%. Although
the difference was not great, practitioners should be discouraged from using a formula which is wrong.
In fact the story is rather more scandalous, because at least one ultrasonic method uses the Mean-Section
Method for integrating vertically over only three or four data points, when its errors would be rather
larger.
In textbooks one does find an approximate method known as the Mid-Section Method, which takes as
the elemental contribution
δQi = ui hi × 1
2
(bi + bi+1) .
When the individual contributions are summed this becomes the Trapezoidal Rule.
2.2.2 Slope area method
This is widely used to calculate peak discharges after the passage of a flood. An ideal site is a reach of
uniform channel in which the flood peak profile is defined on both banks by high water marks. From this
information the slope, the cross-sectional area and wetted perimeter can be obtained, and the discharge
computed with the Gauckler-Manning-Strickler (G-M-S) formula or the Chézy formula. To do this
however, roughness coefficients must be known, such as Manning’s n in the G-M-S formula
Q =
1
n
A5/3
P 2/3
p
S¯,
where A is the area, P the wetted perimeter, and S¯ the slope.
2.2.3 Dilution methods
In channels where cross-sectional areas are difficult to determine (e.g. steep mountain streams) or where
12
421-423 – River hydraulics John Fenton
flow velocities are too high to be measured by current meters dilution or tracer methods can be used,
where continuity of the tracer material is used with steady flow. The rate of input of tracer is mea-
sured, and downstream, after total mixing, the concentration is measured. The discharge in the stream
immediately follows.
2.2.4 Integrating float methods
There is another rather charming and wonderful method which has been very little exploited. At the
moment it has the status of a single measurement method, however the lecturer can foresee it being
developed as a continuing method.
Theory
Consider a single buoyant particle (a float, an orange, an air bubble), which is released from a point
on the bed. We assume that it has a constant rise velocity w. As it rises it passes through a variable
horizontal velocity field u(z), where z is the vertical co-ordinate. The kinematic equations of the float
are
dx
dt
= u(z),
dz
dt
= w.
Dividing the left and right sides, we obtain a differential equation for the particle trajectory
dz
dx
=
w
u(z)
,
however this can be re-arranged as:
hZ
0
u(z) dz =
LZ
0
w dx = wL,
where the particle reaches the surface a distance L downstream of the point at which it was released on
the bed, and where we have used a local vertical co-ordinate z with origin on the bed and where the fluid
locally has a depth h. The quantity on the left is important - it is the vertical integral of the horizontal
velocity, or the discharge per unit width at that section. We can generalise the expression for variation
with y, across the channel, to write
hZ
Z(y)
u(y, z) dz = wL(y),
where Z(y) is the z co-ordinate of the bed. Now, if we integrate across the channel, in the co-ordinate
direction y, the integral of the left side is the discharge Q:
Q =
BZ
0
hZ
Z(y)
u(y, z) dz dy = w
BZ
0
L(y) dy,
where B is the total width of the channel. Hence we have an expression for the discharge with very few
approximations:
Q = w
BZ
0
L(y) dy.
13
421-423 – River hydraulics John Fenton
If we were to release bubbles from a pipe across the bed of the stream, on the bed, then this is
Q = Bubble rise velocity× area on surface between bubble path pattern and line of release.
This is possibly the most direct and potentially the most accurate of all flow measurement methods!
2.2.5 Ultrasonic flow measurement
Figure 2-2. Array of four ultrasonic beams in a channel
This is a method used in the irrigation industry in Australia, but is also being used in rivers in the USA.
Consider the situation shown in the figure, where some three or four beams of ultrasonic sound are
propagated diagonally across a stream at different levels. The time of travel of sound in one direction
is measured, as is the time in the other. The difference can be used to compute the mean velocity along
that path, i.e. at that level. These values then have to be integrated in the vertical.
Mean velocity along beam path
Unfortunately, in all textbooks and International and Australian Standards a constant velocity is assumed
- precisely what is being sought to measure, and totally ignoring the fact that velocity varies along the
path and indeed is zero at the ends! Here we include the variability of velocity in our analysis.
Consider a velocity vector inclined to the beam path at an angle α. If the velocity is u(s), showing that
the velocity does, in general, depend on position along the beam, then the component along the path is
u(s) cosα. Let c be the speed of sound. The time dt taken for a sound wave to travel a distance ds along
the path against the general direction of flow is dt = ds/ (c− u(s) cosα). If the path has total length L,
then the total time of travel T1 is obtained by integrating to give
T1 =
T1Z
0
dt =
LZ
0
ds
c− u(s) cosα , (2.11)
and repeating for a traverse in the reverse direction:
T2 =
T2Z
0
dt =
LZ
0
ds
c+ u(s) cosα
. (2.12)
Now we expand the denominators of both integrals by the binomial theorem:
T1 =
1
c
LZ
0
µ
1+
u(s)
c
cosα
¶
ds and T2 =
1
c
LZ
0
µ
1− u(s)
c
cosα
¶
ds, (2.13)
where we have ignored terms which contain the square of the fluid velocity compared with the speed of
14
421-423 – River hydraulics John Fenton
sound. Evaluating gives
T1 =
L
c
+
1
c2
LZ
0
u(s) cosα ds and T2 =
L
c
− 1
c2
LZ
0
u(s) cosα ds. (2.14)
Adding the two equations and solving for c and re-substituting we obtain
LZ
0
u(s) cosα ds = 2L2
T1 − T2
(T1 + T2)
2 . (2.15)
It can be shown that the relative error of this expression is of order (u¯/c)2, where u¯ is a measure of
velocity. As u¯ ≈ 1m s−1 and c ≈ 1400m s−1 it can be seen that the error is exceedingly small. What
we first need to compute the flow is the integral of the velocity component transverse to the beam path,
for which we use the symbol Qz , the symbol with subscript suggesting the derivative of discharge with
respect to elevation:
Qz =
Z L
0
u(s) sinα ds. (2.16)
Now we are forced to assume that the angle that the velocity vector makes with the beam is constant
over the path (or at least in some rough averaged sense), and so for α constant, taking the trigonometric
functions outside the integral signs and combining equations (2.15) and (2.16) we obtain
Qz = 2 tanαL
2 T1 − T2
(T1 + T2)
2 . (2.17)
This shows how the result is obtained by assuming the angle of inclination of the fluid velocity to the
beam is constant, but importantly it shows that it is not necessary to assume that velocity u is constant
over the beam path. Equation (2.17) is similar to that presented in Standards and trade brochures, and
implemented in practice, but where it is obtained by assuming that the velocity is constant. It is fortunate
that the end result is correct.
Vertical integration of beam data
The mean velocities on different levels obtained from the beam data are considered to be highly accurate,
provided all the technical problems associated with beam focussing etc. are overcome, and the stream-
flow has a constant angle α to the beam. The problem remains to calculate the discharge in the channel
by evaluating the vertical integral of Qz, which, as shownby equation (2.16), is the integral along the
beam of the velocity transverse to the beam. The problem is then to evaluate the vertical integral of the
derivative of discharge with elevation:
Q =
hZ
0
Qz(z) dz, (2.18)
where in practice the information available is that Qz = 0 on the bottom of the channel z = 0 and the
two to four values of Qz which have been obtained from beam data, as well as the total depth h. It is
in the evaluation of this integral that the performance of the trade and scientific literature has been poor.
Several trade brochures advocate the routine use of a single beam, or maybe two, suggesting that that is
adequate (see, for example, Boiten 2000, p141). In fact, with high-quality data for Qz at two or three
levels, there is no reason not to use accurate integration formulae. However, practice in this area has
been quite poor, as trade brochures that the author has seen use the inaccurate Mean-Section Method
for integrating vertically over only three or four data points, when its errors would be rather larger than
when it is used for many verticals across a channel, as described previously. This seems to be a ripe area
for research.
15
421-423 – River hydraulics John Fenton
2.2.6 Acoustic-Doppler Current Profiling methods:
In these, a beam of sound of a known frequency is transmitted into the fluid, often from a boat. When
the sound strikes moving particles or regions of density difference moving at a certain speed, the sound
is reflected back and received by a sensor mounted beside the transmitter. According to the Doppler
effect, the difference in frequency between the transmitted and received waves is a direct measurement
of velocity. In practice there are many particles in the fluid and the greater the area of flow moving
at a particular velocity, the greater the number of reflections with that frequency shift. Potentially this
method is very accurate, as it purports to be able to obtain the velocity over quite small regions and
integrate them up. However, this method does not measure in the top 15% of the depth or near the
boundaries, and the assumption that it is possible to extract detailed velocity profile data from a signal
seems to be optimistic. The lecturer remains unconvinced that this method is as accurate as is claimed.
2.2.7 Electromagnetic methods
The motion of water flowing in an open channel cuts a vertical magnetic field which is generated using
a large coil buried beneath the river bed, through which an electric current is driven. An electromotive
force is induced in the water and measured by signal probes at each side of the channel. This very small
voltage is directly proportional to the average velocity of flow in the cross-section. This is particularly
suited to measurement of effluent, water in treatment works, and in power stations, where the channel
is rectangular and made of concrete; as well as in situations where there is much weed growth, or
high sediment concentrations, unstable bed conditions, backwater effects, or reverse flow. This has the
advantage that it is an integrating method, however in the end recourse has to be made to empirical
relationships between the measured electrical quantities and the flow.
Coil for producing magnetic fieldSignal probes Coil for producing magnetic fieldSignal probes
Figure 2-3. Electromagnetic installation, showing coil and signal probes
2.2.8 Flow measuring structures
These are often bound up with control and regulatory functions, as well as measurement. We will not
treat them in this course.
2.3 The analysis and use of stage and discharge measurements
2.3.1 Stage discharge method
Almost universally the routine measurement of the state of a river is that of the stage, the surface eleva-
tion at a gauging station, usually specified relative to an arbitrary local datum. While surface elevation is
an important quantity in determining the danger of flooding, another important quantity is the actual flow
rate past the gauging station. Accurate knowledge of this instantaneous discharge - and its time integral,
the total volume of flow - is crucial to many hydrologic investigations and to practical operations of a
river and its chief environmental and commercial resource, its water. Examples include decisions on the
allocation of water resources, the design of reservoirs and their associated spillways, the calibration of
models, and the interaction with other computational components of a network.
16
421-423 – River hydraulics John Fenton
Qfalling Qcalculated Qrising Discharge
Stage
A measured stage value
Steady flow rating curve Actual flood event
Figure 2-4. Stage-discharge diagram showing the steady-flow rating curve and an exaggerated looped trajectory of
a particular flood event.
The traditional way in which volume flow is inferred is for a rating curve to be derived for a particular
gauging station, which is a relationship between the stage measured and the actual flow passing that
point. The measurement of flow is done at convenient times by traditional hydrologic means, with
a current meter measuring the flow velocity at enough points over the river cross section so that the
volume of flow can be obtained for that particular stage, measured at the same time. By taking such
measurements for a number of different stages and corresponding discharges over a period of time, a
number of points can be plotted on a stage-discharge diagram, and a curve drawn through those points,
giving what is hoped to be a unique relationship between stage and flow, the rating curve, as shown in
Figure 2-4. This is then used in the future so that when stage is routinely measured, it is assumed that
the corresponding discharge can be obtained from that curve, such as the discharge Qcalculated shown in
the figure for a particular value of stage.
There are several problems associated with the use of a Rating Curve:
• The assumption of a unique relationship between stage and discharge is, in general, not justified.
• Discharge is rarely measured during a flood, and the quality of data at the high flow end of the
curve might be quite poor.
• It is usually some sort of line of best fit through a sample made up of a number of points - sometimes
extrapolated for higher stages.
• It has to describe a range of variation from no flow through small but typical flows to very large
extreme flood events.
• There are a number of factors which might cause the rating curve not to give the actual discharge,
some of which will vary with time. Factors affecting the rating curve include:
– The channel changing as a result of modification due to dredging, bridge construction, or vege-
tation growth.
– Sediment transport - where the bed is in motion, which can have an effect over a single flood
event, because the effective bed roughness can change during the event. As a flood increases,
any bed forms present will tend to become larger and increase the effective roughness, so that
friction is greater after the flood peak than before, so that the corresponding discharge for a
given stage height will be less after the peak. This will contribute to a flood event showing a
17
421-423 – River hydraulics John Fenton
looped curve on a stage-discharge diagram as is shown on Figure 2-4.
– Backwater effects - changes in the conditions downstream such as the construction of a dam or
flooding in the next waterway.
– Unsteadiness - in general the discharge will change rapidly during a flood, and the slope of
the water surface will be different from that for a constant stage, depending on whether the
discharge is increasing or decreasing, also contributing to a flood event appearing as a loop on
a stage-discharge diagram such as Figure 2-4.
– Variable channel storage - where the stream overflows onto flood plains during high discharges,
giving rise to different slopes and to unsteadiness effects.
– Vegetation - changing the roughness and hence changing the stage-discharge relation.
– Ice - which we can ignore –this is Australia, after all.
Some of these can be allowed for by procedures which we will describe later.
High water
Low water
Local controlGauging station
Distant control
Flood
Channel control
⊗
Figure 2-5. Section of river showing different controls at different water levels and a flood moving downstream
A typical set-up of a gauging station where the water level is regularly measured is given in Figure 2-5
which shows a longitudinal section of a stream. Downstream of the gauging station is usually some
sort of fixed control which may be some local topography such as a rock ledge which means that for
relatively small flows there is a relationship between the head over the control and the discharge which
passes. This will control the flow for small flows. For larger flows the effect of the fixed control is to
”drown out”, to become unimportant, and for some other part of the stream to control the flow, such as
the larger river downstream shown as a distant control in the figure, or even, if the downstream channel
length is long enough before encountering another local control, the section of channel downstream
will itself become the control, where the control is due to friction in the channel, giving a relationship
between the slope in the channel, the channel geometry and roughness and the flow. There may be
more controls too, but however many there are, if the channel were stable, and the flow steady (i.e. not
changing with time anywhere in the system) there would be a unique relationship between stage and
discharge, however complicated this might be due to various controls. In practice, the natures of the
controls are usually unknown.
Something which the concept of a rating curve overlooks is the effect of unsteadiness, or variation with
time. In a flood event the discharge will change with time as the flood wave passes, and the slope of
the water surface will be different from that for a constant stage, depending on whether the discharge is
increasing or decreasing. Figure 2-5 shows the increased surface slope as a flood approaches the gauging
station. The effects of this are shown on Figure 2-4, in somewhat exaggerated form, where an actual
flood event may not follow the rating curve but will in general follow the looped trajectory shown. As
the flood increases, the surface slope in the river is greater than the slope for steady flow at the same
stage, and hence, according to conventional simple hydraulic theory explained below, more water is
flowing down the river than the rating curve would suggest. This is shown by the discharge marked
18
421-423 – River hydraulics John Fenton
Qrising obtained from the horizontal line drawn for a particular value of stage. When the water level is
falling the slope and hence the discharge inferred is less.
The effects of this might be important - the peak discharge could be significantly underestimated during
highly dynamic floods, and also since the maximum discharge and maximum stage do not coincide,
the arrival time of the peak discharge could be in error and may influence flood warning predictions.
Similarly water-quality constituent loads could be underestimated if the dynamic characteristics of the
flood are ignored, while the use of a discharge hydrograph derived inaccurately by using a single-valued
rating relationship may distort estimates for resistance coefficients during calibration of an unsteady flow
model.
The use of slope as well as stage
Although the picture in Figure 2-5 of the factors affecting the stage and discharge at a gauging station
seems complicated, the underlying processes are capable of quite simple description. In a typical stream,
where all wave motion is of a relatively long time and space scale, the governing equations are the long
wave equations, which are a pair of partial differential equations for the stage and the discharge at all
points of the channel in terms of time and distance along the channel. One is a mass conservation equa-
tion, the other a momentum equation. Under the conditions typical of most flows and floods in natural
waterways, however, the flow is sufficiently slow that the equations can be simplified considerably. Most
terms in the momentum equation are of a relative magnitude given by the square of the Froude number,
which is U2/gD, where U is the fluid velocity, g is the gravitational acceleration, and D is the mean
depth of the waterway. In most rivers, even in flood, this is small, and the approximation may be often
used. For example, a flow of 1 ms−1 with a depth of 2 m has F 2 ≈ 0.05. Under these circumstances,
a surprisingly good approximation to the momentum equation of motion for flow in a waterway is the
simple equation:
∂η
∂x
+ Sf = 0, (2.19)
where η is the surface elevation, x is distance along the waterway and Sf is the friction slope. The usual
practice is to use an empirical friction law for the friction slope in terms of a conveyance function K, so
that we write
Sf =
Q2
K2
, (2.20)
in which Q is the instantaneous discharge, and where the dependence of K on stage at a section may be
determined empirically, or by a standard friction law, such as the Gauckler-Manning-Strickler formula
or Chézy’s formula:
G-M-S: K =
1
n
A5/3
P 2/3
or Chézy: K = C
A3/2
P 1/2
, (2.21)
where n and C are Manning’s and Chézy’s coefficients respectively, while A is cross-sectional area and
P is wetted perimeter, which are both functions of depth and x, as the cross-section usually changes
along the stream. In most hydrographic situations K would be better determined by measurements of
flow and slope rather than by these formulae as they are approximate only and the roughness coefficients
are usually poorly known.
Even though equation (2.20) was originally intended for flow which is both steady (unchanging in time)
and uniform (unchanging in space), it has been widely accepted as the governing friction equation in
more generally unsteady and non-uniform flows. Hence, substituting (2.20) into (2.19) gives us an
expression for the discharge, where we now show the functional dependence of each variable:
Q(t) = K(η(t))
q
Sη(t), (2.22)
where we have introduced the symbol Sη = −∂η/∂x for the slope of the free surface, positive in the
downstream direction, in the same way that we use the symbol Sf for the friction slope. This gives us
an expression for the discharge at a point and how it might vary with time. Provided we know
19
421-423 – River hydraulics John Fenton
1. the stage and the dependence of conveyance K on stage at a point from either measurement or the
G-M-S or Chézy’s formulae, and
2. the slope of the surface,
we have a formula for calculating the discharge Q which is as accurate as is reasonable to be expected
in river hydraulics.
Equation (2.22) shows how the discharge actually depends on both the stage and the surface slope,
whereas traditional hydrography assumes that it depends on stage alone. If the slope does vary under
different backwater conditions or during a flood, then a better hydrographic procedure would be to
gauge the flow when it is steady, and to measure the surface slope , thereby enabling a particular value
of K to be calculated for that stage. If this were done over time for a number of different stages, then
a stage-conveyance relationship could be developed which should then hold whether or not the stage
is varying. Subsequently, in day-to-day operations, if the stage and the surface slope were measured,
then the discharge calculated from equation (2.22) should be quite accurate, within the relatively mild
assumptions made so far. All of this holds whether or not the gauging station is affected by a local or
channel control, and whether or not the flow is changing with time.
If hydrography had followed the path described above, of routinely measuring surface slope and us-
ing a stage-conveyance relationship, the ”science” would have been more satisfactory. Effects due to
the changing of downstream controls with time, downstream tailwaterconditions, and unsteadiness in
floods would have been automatically incorporated, both at the time of determining the relationship and
subsequently in daily operational practice. However, for the most part slope has not been measured, and
hydrographic practice has been to use rating curves instead. The assumption behind the concept of a
discharge-stage relationship or rating curve is that the slope at a station is constant over all flows and
events, so that the discharge is a unique function of stage Qr(η) where we use the subscript r to indicate
the rated discharge. Instead of the empirical/rational expression (2.22), traditional practice is to calculate
discharge from the equation
Q(t) = Qr(η(t)), (2.23)
thereby ignoring any effects that downstream backwater and unsteadiness might have, as well as the
possible changing of a downstream control with time.
In comparison, equation (2.22), based on a convenient empirical approximation to the real hydraulics
of the river, contains the essential nature of what is going on in the stream. It shows that, although the
conveyance might be a unique function of stage which it is possible to determine by measurement, be-
cause the surface slope will in general vary throughout different flood events and downstream conditions,
discharge in general does not depend on stage alone.
2.3.2 Stage-conveyance curves
The above argument suggests that ideally the concept of a stage-discharge relationship be done away
with, and replaced by a stage-conveyance relationship. Of course in many, even most, situations it might
well be that the surface slope at a gauging station does vary but little throughout all conditions, in which
case the concept of a stage-discharge relationship would be accurate. In most situations it is indeed the
case that there is little deviation of results from a unique stage-discharge relationship.
The use of slope in determining flow
There is a considerable amount of hydraulic justification for using equation (2.22).
Q(t) = K(η(t))
q
Sη(t), (2.24)
It could not be claimed that this is a theoretical justification, as they are based on empirical friction laws
but, based on the cases studied above, the incorporation of slope appears to give a superior and more
fundamental description of the processes at work, and handles both long-term effects due to downstream
20
421-423 – River hydraulics John Fenton
conditions changing and short-term effects due to the flow changing.
This suggests that a better way of determining streamflows in general, but primarily where backwater
and unsteady effects are likely to be important, is for the following procedure to be followed:
1. At a gauging station, two measuring devices for stage be installed, so as to be able to measure
the slope of the water surface at the station. One of these could be at the section where detailed
flow-gaugings are taken, and the other could be some distance upstream or downstream such that
the stage difference between the two points is enough that the slope can be computed accurately
enough. As a rough guide, this might be, say 10 cm, so that if the water slope were typically 0.001,
they should be at least 100m apart.
2. Over time, for a number of different flow conditions the discharge Q would be measured using
conventional methods such as by current meter. For each gauging, both surface elevations would
be recorded, one becoming the stage η to be used in the subsequent relationship, the other so that
the surface slope Sη can be calculated. Using equation (2.22), Q = K(η)
p
Sη, this would give the
appropriate value of conveyanceK for that stage, automatically corrected for effects of unsteadiness
and downstream conditions.
3. From all such data pairs (ηi,Ki) for i = 1, 2, . . ., the conveyance curve (the functional dependence
of K on η) would be found, possibly by piecewise-linear or by global approximation methods, in a
similar way to the description of rating curves described below. Conveyance has units of discharge,
and as the surface slope is unlikely to vary all that much, we note that there are certain advantages
in representing rating curves on a plot using the square root of the discharge, and it my well be that
the stage-conveyance curve would be displayed and approximated best using (
√
K, η) axes.
4. Subsequent routine measurements would obtain both stages, including the stage to be used in the
stage-conveyance relationship, and hence the water surface slope, which would then be substituted
into equation (2.22) to give the discharge, corrected for effects of downstream changes and unsteadi-
ness.
The effects of varying roughness
Notes to be added
Attempting to include unsteady effects
In conventional hydrography the stage is measured repeatedly at a single gauging station so that the
time derivative of stage can easily be obtained from records but the surface slope along the channel is
not measured at all. The methods of this section are all aimed at obtaining the slope in terms of the
stage and its time derivatives at a single gauging station. The simplest and most traditional method of
calculating the effects of unsteadiness has been the Jones formula, derived by B. E. Jones in 1916 (see
for example Chow 1959, Henderson 1966). The principal assumption is that to obtain the slope, the x
derivative of the free surface, we can use the time derivative of stage which we can get from a stage
record, by assuming that the flood wave is moving without change as a kinematic wave (Lighthill and
Whitham, 1955) such that it obeys the partial differential equation:
∂h
∂t
+ c
∂h
∂x
= 0, (2.25)
where h is the depth and c is the kinematic wave speed. Solutions of this equation are simply waves
travelling at a velocity c without change. The equation will be obtained as one of a consistent series of
approximations in Section 3. The kinematic wave speed c is given by the derivative of flow with respect
to cross-sectional area, the Kleitz-Seddon law
c =
1
B
dQr
dη
=
1
B
dK
dη
p
S¯, (2.26)
where B is the width of the surface and Qr is the steady rated discharge corresponding to stage η, and
where we have expressed this also in terms of the conveyance K, where Qr = K(η)
√
S¯, and the slope
21
421-423 – River hydraulics John Fenton
√
S¯ is the mean slope of the stream. A good approximation is c ≈ 5/3×U , where U is the mean stream
velocity.
The Jones method assumes that the surface slope Sη in equation can be simply related to the rate of
change of stage with time, assuming that the wave moves without change. Thus, equation (2.25) gives
an approximation for the surface slope: ∂h/∂x ≈ −1/c × ∂h/∂t. We then have to use the simple
geometric relation between surface gradient and depth gradient, that ∂η/∂x = ∂h/∂x− S¯, such that we
have the approximation
Sη = −
∂η
∂x
= S¯ − ∂h
∂x
≈ S¯ + 1
c
∂h
∂t
and recognising that the time derivative of stage and depth are the same, ∂h/∂t = ∂η/∂t, equation
(2.22) gives
Q = K
r
S¯ +
1
c
∂η
∂t
(2.27)
If we divide by the steady discharge corresponding to the rating curve we obtain
Q
Qr
=
r
1 +
1
cS¯
∂η
∂t
(Jones)
In situations where the flood wave does move as a kinematic wave, with friction and gravity in balance,
this theory is accurate. In general, however, there will be a certain amount of diffusion observed, where
the wave crest subsides and the effects of the wave are smeared out in time.
To allow for those effects Fenton (1999) provided the theoretical derivation of two methods for cal-
culating the discharge. The derivation of both is rather lengthy. The first method used the full long
wave equations and approximated the surface slope using a method based on a linearisation of those
equations. The result was a differential equation for dQ/dt in terms of Q and stage and the deriva-
tives of stage dη/dt and d2η/dt2, which could be calculated from the record of stage with time and the
equation solved numerically. The second method was rather simpler, and was based on the next best
approximation to thefull equations after equation (2.25). This gives the advection-diffusion equation
∂h
∂t
+ c
∂h
∂x
= ν
∂2h
∂x2
, (2.28)
where the difference between this and equation (2.25) is the diffusion term on the right, where ν is a
diffusion coefficient (with units of L2T−1), given by
ν =
K
2B
√
S¯
.
Equation (2.28), to be studied in Section 3, is a consistent low-inertia approximation to the long wave
equations, where inertial terms, which are of the order of the square of the Froude number, which
approximates motion in most waterways quite well. However, it is not yet suitable for the purposes of
this section, for we want to express the x derivative at a point in terms of time derivatives. To do this, we
use a small-diffusion approximation, we assume that the two x derivatives on the right of equation (2.28)
can be replaced by the zero-diffusion or kinematic wave approximation as above, ∂/∂x ≈ −1/c×∂/∂t,
so that the surface slope is expressed in terms of the first two time derivatives of stage. The resulting
expression is:
∂h
∂t
+ c
∂h
∂x
=
ν
c2
∂2h
∂t2
,
and solving for the x derivative, we have the approximation
Sη = −
∂η
∂x
= S¯ − ∂h
∂x
≈ S¯ + 1
c
∂h
∂t
− ν
c3
d2h
dt2
,
22
421-423 – River hydraulics John Fenton
and substituting into equation (2.22) gives
Q = Qr(η)
vuuuut 1|{z}Rating curve +
1
cS¯
dη
dt| {z }
Jones formula
− ν
c3S¯
d2η
dt2| {z }
Diffusion term
(2.29)
where Q is the discharge at the gauging station, Qr(η) is the rated discharge for the station as a function
of stage, S¯ is the bed slope, c is the kinematic wave speed given by equation (2.26):
c =
√
S¯
B
dK
dη
=
1
B
dQr
dη
,
in terms of the gradient of the conveyance curve or the rating curve, B is the width of the water surface,
and where the coefficient ν is the diffusion coefficient in advection-diffusion flood routing, given by:
ν =
K
2B
√
S¯
=
Qr
2BS¯
. (2.30)
In equation (2.29) it is clear that the extra diffusion term is a simple correction to the Jones formula,
allowing for the subsidence of the wave crest as if the flood wave were following the advection-diffusion
approximation, which is a good approximation to much flood propagation. Equation (2.29) provides a
means of analysing stage records and correcting for the effects of unsteadiness and variable slope. It can
be used in either direction:
• If a gauging exercise has been carried out while the stage has been varying (and been recorded),
the value of Q obtained can be corrected for the effects of variable slope, giving the steady-state
value of discharge for the stage-discharge relation,
• And, proceeding in the other direction, in operational practice, it can be used for the routine analysis
of stage records to correct for any effects of unsteadiness.
The ideas set out here are described rather more fully in Fenton & Keller (2001).
An example
A numerical solution was obtained for the particular case of a fast-rising and falling flood in a stream of
10 km length, of slope 0.001, which had a trapezoidal section 10m wide at the bottom with side slopes of
1:2, and a Manning’s friction coefficient of 0.04. The downstream control was a weir. Initially the depth
of flow was 2m, while carrying a flow of 10m3 s−1. The incoming flow upstream was linearly increased
ten-fold to 100m3 s−1 over 60 mins and then reduced to the original flow over the same interval. The
initial backwater curve problem was solved and then the long wave equations in the channel were solved
over six hours to simulate the flood. At a station halfway along the waterway the computed stages were
recorded (the data one would normally have), as well as the computed discharges so that some of the
above-mentioned methods could be applied and the accuracy of this work tested.
Results are shown on Figure 2-6. It can be seen that the application of the diffusion level of approxi-
mation f equation (2.29) has succeeded well in obtaining the actual peak discharge. The results are not
exact however, as the derivation depends on the diffusion being sufficiently small that the interchange
between space and time differentiation will be accurate. In the case of a stream such as the example here,
diffusion is relatively large, and our results are not exact, but they are better than the Jones method at
predicting the peak flow. Nevertheless, the results from the Jones method are interesting. A widely-held
opinion is that it is not accurate. Indeed, we see here that in predicting the peak flow it was not accurate
in this problem. However, over almost all of the flood it was accurate, and predicted the time of the flood
peak well, which is also an important result. It showed that both before and after the peak the ”discharge
wave” led the ”stage wave”, which is of course in phase with the curve showing the flow computed from
the stage graph and the rating curve. As there may be applications where it is enough to know the arrival
time of the flood peak, this is a useful property of the Jones formula. Near the crest, however, the rate
of rise became small and so did the Jones correction. Now, and only now, the inclusion of the extra
23
421-423 – River hydraulics John Fenton
0
10
20
30
40
50
60
70
80
90
100
0 1 2 3 4 5 6
Flow
(m3 s−1)
Time (hours)
Actual flow
From rating curve
Jones formula
Eqn (2.29)
Figure 2-6. Simulated flood with hydrographs computed from stage record using three
levels of approximation
diffusion term gave a significant correction to the maximum flow computed, and was quite accurate in
its prediction that the real flow was some 10% greater than that which would have been calculated just
from the rating curve. In this fast-rising example the application of the unsteady corrections seems to
have worked well and to be justified. It is no more difficult to apply the diffusion correction than the
Jones correction, both being given by derivatives of the stage record.
2.3.3 Slope-Stage-Discharge Method
This is essentially the method which has been proposed earlier, incorporating the effects of slope. It is
presented in some books and in International and Australian Standards, however, especially in the latter,
the presentation is confusing and at a low level, where no reference is made to the fact that underlying
it the slope is being measured. Instead, the fall is described, which is the change in surface elevation
between two surface elevation gauges and is simply the slope multiplied by the distance between them.
No theoretical justification is provided and it is presented in a phenomenological sense (see, for example,
Herschy 1995).
An exception is Boiten (2000), however even that presentation loses sight of the pragmatic nature of de-
termining a stage-conveyance relationship with equation (2.22), and instead uses the Gauckler-Manning-
Strickler formula in its classical form
Q =
1
n
A5/3
P 2/3
p
Sη,
where it is assumed that the discharge must be given using these precise geometrical quantities of A and
P . It is rather more pragmatic to determine K(η) by measurements.
3. The propagation of waves in waterways
We now spend some time deriving the full equations for unsteady non-uniform flow. The fundamental
assumption we make is that the flow is slowly varying along the channel. The mathematics uses a
number of concepts from vector calculus, however we find that we can obtain general equations very
powerfully, and the assumptions and approximations (actually very few!) are clear.
24
421-423 – River hydraulics John Fenton
3.1 Mass conservation equation
Top of Control Volume
Water surface
y
z
x
q
Q+∆Q
Q
∆x
Figure 3-1. Element of non-prismatic waterway showing control volume extended into the air
Consider the elemental section of thickness∆x of non-uniform waterway shown in Figure 3-1, bounded
by two vertical planes parallel to the y − z plane. Consider also the control volume made up of this
elemental section, but continued into the air such that the bottom and lateral boundaries are the river
banks, and the upper boundaryis arbitrary but never intersected by the water.
The Mass Conservation equation in integral form is, written for a control volume CV bounded by a
control surface CS,
∂
∂t
Z
CV
ρ dV
| {z }
Total mass in CV
+
Z
CS
ρu.nˆ dS
| {z }
Rate of flow of mass across boundary
= 0,
where t is time, dV is an element of volume, u is the velocity vector, nˆ is a unit vector with direction
normal to and directed outwards from the control surface such that u.nˆ is the component of velocity
normal to the surface at any point, and dS is the elemental area of the control surface.
As the density of the air is negligible compared with the water, the domain of integration in the first
integral reduces to the volume of water in the control volume, and considering the elemental slice,
dV = ∆xdA , where dA is an element of cross-sectional area, the term becomes
ρ∆x
∂
∂t
Z
A
dA = ρ∆x
∂A
∂t
Now considering the second integral, on the upstream face of the control surface, u.nˆ = −u , where u
is the x component of velocity, so that the contribution due to flow entering the control volume is
−ρ
Z
A
udA = −ρQ.
Similarly the downstream face contribution is
+ρ (Q+∆Q) = +ρ
µ
Q+
∂Q
∂x
∆x
¶
.
On the boundaries which are the banks of the stream, the velocity component normal to the boundary
is very small and poorly-known. We will include it in a suitably approximate manner. We lump this
contribution from groundwater, inflow from rainfall, and tributaries entering the waterway, as a volume
25
421-423 – River hydraulics John Fenton
rate of q per unit length entering the stream. The rate at which mass enters the control volume is ρq∆x
(i.e. an outflow of−ρq∆x). Combining the contributions from the rate of change of mass in the CV and
the net contribution across the two faces, and dividing by ρ∆x we have the unsteady mass conservation
equation
∂A
∂t
+
∂Q
∂x
= q. (3.1)
Remarkably for hydraulics, this is an almost-exact equation - the only significant approximation we have
made is that the waterway is straight! If we want to use surface elevation as a variable in terms of surface
area, it is easily shown that in an increment of time δt if the surface changes by an amount δη, then the
area changes by an amount δA = B × δη, from which we obtain ∂A/∂t = B × ∂η/∂t, and the mass
conservation equation can be written
B
∂η
∂t
+
∂Q
∂x
= q. (3.2)
The assumption that the waterway is straight has almost universally been made. Fenton & Nalder (1995)1
have considered waterways curved in plan (i.e. most rivers!) and obtained the result (cf. equation 3.1):³
1− nm
r
´ ∂A
∂t
+
∂Q
∂s
= q,
where nm is the transverse offset of the centre of the river surface from the curved streamwise reference
axis s, and r is the radius of curvature of that axis. Usually nm is small compared with r, and the
curvature term is a relatively small one. It can be seen that if it is possible to choose the reference axis
to coincide with the centre of the river viewed in plan, then nm = 0 and curvature has no effect on this
equation. This choice of axis is not always possible, however, as the geometry of the river changes with
surface height.
3.2 Momentum conservation equation
We can write the Momentum Conservation Equation in integral form, similarly to the Mass Conservation
Equation:
∂
∂t
Z
CV
ρu dV
| {z }
Total momentum in CV
+
Z
CS
ρ uu.nˆ dS
| {z }
Rate of flow of momentum across boundary
= P
whereP is the force exerted on the fluid in the control volume by both body and surface forces, the latter
including shear forces and pressure forces. This is a vector equation.
There are two main contributions to P, forces due to pressure and shear stresses. The pressure term can
be written−
R
CS
pnˆ dS, the negative sign showing that the local force acts in the direction opposite to the
outward normal. Substituting these contributions into the momentum equation:
∂
∂t
Z
CV
ρu dV +
Z
CS
ρuu.nˆ dS = Shear force−
Z
CS
pnˆ dS
The last term is very difficult to evaluate for non-prismatic waterways, as the pressure and the non-
constant unit vector have to be integrated over all the submerged faces of the control surface. A much
simpler derivation is obtained if the term is evaluated using Gauss’ Divergence Theorem:Z
CS
pnˆ dS =
Z
CV
∇p dV
1 Fenton, J. D. & Nalder, G. V. (1995), Long wave equations for waterways curved in plan, in Proc. 26th
Congress IAHR, London, Vol. 1, pp. 573–578.
26
421-423 – River hydraulics John Fenton
where ∇p = (∂p/∂x, ∂p/∂y, ∂p/∂z), the vector gradient of pressure. This has turned a complicated
surface integral into a simple volume integral. It can be understood by considering the fluid to be divided
up into a number of elemental parallelepipeds, the pressure force on one elemental face being cancelled
by its force on the other.
Now taking the x component of the vector momentum equation we obtain:
∂
∂t
Z
CV
ρudV
| {z }
(a)
+
Z
CS
ρuu.nˆ dS
| {z }
(b)
= Horizontal shear force| {z }
(c)
−
Z
CV
∂p
∂x
dV
| {z }
(d)
,
and we now make hydraulic approximations for these terms. The first two are obtained in the same
manner as for the mass conservation equation.
(a) Unsteady term: The first term is
∂
∂t
Z
CV
ρudV = ρ
∂
∂t
Z
A
udA×∆x = ρ∂Q
∂t
∆x.
(b) Momentum flux term: There are two parts to this. The first is from contributions on solid
boundaries and the air boundary due to flow seeping in or out of the ground or from rainfall or tributaries.
They are lumped together as an inflow q per unit length, such that the mass rate of inflow is ρq∆x, (i.e.
an outflow of −ρq∆x) and if this inflow has a streamwise velocity of uq before it mixes with the water,
the contribution is
−ρq∆xuq. (3.3)
The main one is the contribution of momentum due to fluid crossing the control surface:Z
U/S & D/S Face
ρuu.nˆ dS = −ρ
Z
U/S Face
u2 dA+ ρ
Z
D/S Face
u2 dA. (3.4)
Almost never do we know the precise velocity distribution over the face. We introduce an empirical
quantity β such that the effects of both non-uniformity of velocity over a section and turbulent fluctua-
tions are approximated by Z
A
u2dA = β
Q2
A
, (3.5)
where β is the momentum coefficient or Boussinesq coefficient. If the velocity were constant and steady
over the section β would have a value of 1, which is the usual approximation to a term which otherwise
would be very difficult to evaluate. We will retain it, however, its effects are small in many situations as
shown below, as it is always associated with terms which are of relative magnitude that of the square of
the Froude number. Hence we write the contribution (3.4) asZ
CS
ρuu.nˆ dS = −ρβU2A
¯¯
U/S Face + ρβU
2A
¯¯
D/S Face
=
∂
∂x
¡
ρβU2A
¢×∆x
= ρβ
∂
∂x
µ
Q2
A
¶
×∆x.
(c) Shear force term: The shear forces are tangential to the boundary of the stream at each point,
and in the general case of a non-prismatic waterway, the geometry over which they act is complicated,
27
421-423 – River hydraulics John Fenton
not particularly well-known, and the flow structure is less well-known. Following the usual convention
in river hydraulics, a convenient empirical expression is adopted instead. The approximation is made
here that:
Horizontal component of shear force = Weight of fluid× (−Sf )
= −ρg ×A×∆x× Sf
where Sf is a small dimensionless quantity, which in derivations based on energy is the energy gradient.
Here we think of it as an empirical coefficient relating the horizontal component of the friction force to
the total gravitational force of the fluid in the control volume, and we will call it the friction slope. The
negative sign is introduced because in the usual case where flow is in the +x direction, the shear force is
in the other direction. Later we will assume that it can be given by the G-M-S formula, where the local
and instantaneous depth and discharge are used.
(d) Pressure gradient term This is
−
Z
CV
∂p
∂x
dV.
The approximation we now make, common throughout almost all open-channel hydraulics, is the ”hy-
drostatic approximation”, that pressure