fenton (2008 course) river hydraulics

November 14, 2008

River hydraulics

John Fenton

Abstract

This elective describes the nature of flows in rivers, their measurement, the calculation of flows andflood propagation, the knowledge of the factors affecting water quality, and the stability of rivers. Wewill see how some of the fundamentals have been glossed over, and some very simple improvementscan be made to traditional practice. It is hoped that students taking this course will develop a deepunderstanding of the processes at work in rivers.

Throughout these lectures, both in approximations to wave motion in waterways, and in the transportof pollutants, we will encounter the physical process of diffusion. An introduction to diffusion isgiven in Appendix A.1, but is not for examination.

Table of Contents

References . . . . . . . . . . . . . . . . . . . . . . . 6

1. Introduction . . . . . . . . . . . . . . . . . . . . . 8

2. Hydrography/Hydrometry . . . . . . . . . . . . . . . . 82.1 Water levels . . . . . . . . . . . . . . . . . . . 82.2 Discharge . . . . . . . . . . . . . . . . . . . 92.3 The analysis and use of stage and discharge measurements . . . . 16

3. The propagation of waves in waterways . . . . . . . . . . . . 243.1 Mass conservation equation . . . . . . . . . . . . . . 253.2 Momentum conservation equation . . . . . . . . . . . . 263.3 The nature of the propagation of long waves and floods in rivers . . . 293.4 A new low-inertia approach – Volume routing . . . . . . . . 33

4. Computational hydraulics . . . . . . . . . . . . . . . . 374.1 The advection equation . . . . . . . . . . . . . . . 374.2 The diffusion equation . . . . . . . . . . . . . . . 414.3 Advection-diffusion combined . . . . . . . . . . . . . 42

5. Water quality . . . . . . . . . . . . . . . . . . . . 435.1 Useful sources for further reading . . . . . . . . . . . . 435.2 Water quality characteristics . . . . . . . . . . . . . . 435.3 Types of pollutant . . . . . . . . . . . . . . . . . 435.4 Mass balance concepts . . . . . . . . . . . . . . . 445.5 Impacts of human works . . . . . . . . . . . . . . . 44

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River hydraulics John Fenton

5.6 Transport processes . . . . . . . . . . . . . . . . 455.7 Tools for problem solving . . . . . . . . . . . . . . 465.8 A simple river model – organic wastes and self purification . . . . 465.9 Salinity in rivers . . . . . . . . . . . . . . . . . 53

6. Turbulent diffusion and dispersion . . . . . . . . . . . . . . 566.1 Diffusion and dispersion in waterways . . . . . . . . . . . 576.2 Dispersion . . . . . . . . . . . . . . . . . . . 576.3 Non-dimensionalisation – Péclet number and Reynolds number –

viscosity as diffusion . . . . . . . . . . . . . . . . 59

7. Sediment motion . . . . . . . . . . . . . . . . . . . 597.1 Incipient motion . . . . . . . . . . . . . . . . . 607.2 Relationships for fluvial quantities . . . . . . . . . . . . 617.3 Dimensional similitude . . . . . . . . . . . . . . . 627.4 Bed forms . . . . . . . . . . . . . . . . . . . 627.5 Mechanisms of sediment motion . . . . . . . . . . . . 63

Appendix A On diffusion and von Neumann stability analyses . . . . . . . 65A.1 The nature of diffusion . . . . . . . . . . . . . . . 65A.2 Examining stability by the Fourier series (von Neumann’s) method . . 69

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Useful referencesTables 1.1-1.4 show some of the many references available, some which the lecturer has referred to inthese notes or in his work.

Reference CommentsChanson, H. (1999), The Hydraulics of Open Channel Flow, Arnold, London. Good technical book, moderate level,

also sediment aspectsChaudhry, M. H. (1993), Open-channel flow, Prentice-Hall. Good technical bookChow, V. T. (1959), Open-channel Hydraulics, McGraw-Hill, New York. Classic, now dated, not so readableFrancis, J. & Minton, P. (1984), Civil Engineering Hydraulics, fifth edn, Arnold,London.

Good elementary introduction

French, R. H. (1985), Open-Channel Hydraulics, McGraw-Hill, New York. Wide general treatmentHenderson, F. M. (1966), Open Channel Flow, Macmillan, New York. Classic, high level, readableJain, S. C. (2001), Open-Channel Flow, Wiley. High level, but terse and readableJulien, P. Y. (2002), River Mechanics, Cambridge. A readable but high-level workMontes, S. (1998), Hydraulics of Open Channel Flow, ASCE, New York. EncyclopaedicTownson, J. M. (1991), Free-surface Hydraulics, Unwin Hyman, London. Simple, readable, mathematicalVreugdenhil, C. B. (1989), Computational Hydraulics: An Introduction,Springer.

Simple introduction to computationalhydraulics

Table 1.1 : Introductory and general references

Reference CommentsAustralian Standard 3778 (1990), Australian Standard - Measurement of waterflow in open channels, Standards Association of Australia, Homebush.

Could be much better than it is

Boiten, W. (2000), Hydrometry, Balkema A modern treatment of river measure-ment

Bos, M. G. (1978), Discharge Measurement Structures, second edn, Interna-tional Institute for Land Reclamation and Improvement, Wageningen.

Good encyclopaedic treatment of struc-tures

Bos, M. G., Replogle, J. A. & Clemmens, A. J. (1984), Flow Measuring Flumesfor Open Channel Systems, Wiley.

Good encyclopaedic treatment of struc-tures

Fenton, J. D. & Keller, R. J. (2001), The calculation of streamflow from mea-surements of stage, Technical Report 01/6, Co-operative Research Centre forCatchment Hydrology, Monash University.

Two level treatment - practical aspectsplus high level review of theory

Novak, P., Moffat, A. I. B., Nalluri, C. & Narayanan, R. (2001), HydraulicStructures, third edn, Spon, London.

Standard readable presentation of struc-tures

Table 1.2 : Books on practical aspects, flow measurement, and structures

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Reference CommentsCunge, J. A., Holly, F. M. & Verwey, A. (1980), Practical Aspects of Computa-tional River Hydraulics, Pitman, London.

Thorough and reliable presentation

Dooge, J. C. I. (1987), Historical development of concepts in open channel flow,in G. Garbrecht, ed., Hydraulics and Hydraulic Research: A Historical Review,Balkema, Rotterdam, pp. 205–230.

Interesting review

Fenton, J. D. (1996), An examination of the approximations in river and channelhydraulics, in Proc. 10th Congress, Asia-Pacific Division, Int. Assoc. HydraulicRes., Langkawi, Malaysia, pp. 204–211.

A modern mathematical view

Flood Studies Report (1975), Flood Routing Studies, Vol. 3, Natural Environ-ment Research Council, London.

A readable overview

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

Liggett, J. A. (1975), Basic equations of unsteady flow, in K. Mahmood &V. Yevjevich, eds, Unsteady Flow in Open Channels, Vol. 1, Water ResourcesPublications, Fort Collins, chapter 2.

Readable overview

Liggett, J. A. & Cunge, J. A. (1975), Numerical methods of solution of theunsteady flow equations, in K. Mahmood & V. Yevjevich, eds, Unsteady Flow inOpen Channels, Vol. 1, Water Resources Publications, Fort Collins, chapter 4.

Readable overview

Miller, W. A. & Cunge, J. A. (1975), Simplified equations of unsteady flow, inK. Mahmood & V. Yevjevich, eds, Unsteady Flow in Open Channels, Vol. 1,Water Resources Publications, Fort Collins, chapter 5, pp. 183–257.

Readable

Price, R. K. (1985), Flood Routing, in P. Novak, ed., Developments in hydraulicengineering, Vol. 3, Elsevier Applied Science, chapter 4, pp. 129–173.

The best overview of the advection-diffusion approximation for flood rout-ing

Skeels, C. P. & Samuels, P. G. (1989), Stability and accuracy analysis of numer-ical schemes modelling open channel flow, in C. Maksimovic & M. Radojkovic,eds, Computational Modelling and Experimental Methods in Hydraulics (HY-DROCOMP ’89), Elsevier.

Review

Zoppou, C. & O’Neill, I. C. (1982), Criteria for the choice of flood routingmethods in natural channels, in Proc. Hydrology and Water Resources Sympo-sium, Melbourne, pp. 75–81.

Readable overview

Table 1.3 : References on flood wave propagation – theoretical and computational

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Reference NotesGeneralChin (2000) A good introductionMartin & McCutcheon (1999) A good book, being both introductory and

encyclopaedic, concentrating on the hy-draulic engineering aspects

McGauhey (1968) A complete descriptive (non-mathematical)presentation, which is interesting.

Fundamental processes of mixing and dispersionFischer, List, Koh, Imberger & Brooks (1979) A comprehensive and standard referenceHolly (1985) Also fundamental, but shorterStreeter, Wylie & Bedford (1998) A good simpler introduction (Chapter 9)Rutherford (1994)Csanady (1973)Numerical methods – fundamentalsNoye (1976), Noye (1981), Noye (1984), Noye & May (1986) All offer a simple introduction to finite dif-

ference methodsSmith (1978) A more detailed introduction to finite differ-

ence methodsRichtmyer & Morton (1967), Morton & Baines (1982), Morton & Mayers (1994), Morton(1996)

All are rather more comprehensive, describ-ing some more general methods

Zoppou & Knight (1997) Analytical solutions to the advection-diffusion equation where the coefficients arenot constant

Numerical methods – application to environmental modellingSauvaget (1985) A simple reviewThe nature of diffusionFischer et al. (1979) Very clear - already recommended aboveJost (1960, page 25; 1964) A leisurely and clear introductionBorg & Dienes (1988) A simple and clear introductionWidder (1975) A more mathematical approachThe full equations for wave propagation and flood routingCunge, Holly & Verwey (1980) The best explanation of this fieldLiggett (1975), Liggett & Cunge (1975) A little disappointing, but the next best ex-

planationThe advection-diffusion approximation for flood routingPrice (1985) The best overviewDooge (1986) A good general studySivapalan, Bates & Larsen (1997)OthersPasmanter (1988) Estuaries and tidal flowsKobus & Winzelbach (1989) Groundwater

Table 1.4 : Useful references

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ReferencesAustralian 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 ofVictoria 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. ASCE86(SA4), 41–53.

Fenton, J. D. (1999) Calculating hydrographs from stage records, in Proc. 28th IAHR Congress, 22-27August 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 relativeprotrusion, Proc. Roy. Soc. Lond. A 352, 523–537.

Fenton, J. D. & Keller, R. J. (2001) The calculation of streamflow from measurements of stage,Technical Report 01/6, Cooperative Research Centre for Catchment Hydrology, Melbourne. http://www.catchment.crc.org.au/pdfs/technical200106.pdf

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 andCoastal 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, WadebridgeEcological 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 dispersionequations, 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 CongressIAHR, Melbourne, Vol. 2, pp. 324–328.

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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 ResourcesPublications, 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, PorterSargent, 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 NumericalSolutions 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) Computational techniques 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, Journal of Hydrology 192, 1–16.

Smith, G. D. (1978) Numerical Solution of Partial Differential Equations, Oxford Applied Mathematicsand 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 equationswith spatially variable coefficients, Journal of Hydraulic Engineering 123(2), 144–148.

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1. IntroductionAt the conclusion of this unit, students should be able to describe the nature of flow and floods instreams, understand the basis of computational methods for rivers, the common means of measurementof streamflow, the fundamentals of water quality in rivers, and fluvial processes and fluvial morphology.

2. Hydrography/HydrometryBoiten (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 levelsWater levels are the basis for any river study. Most kinds of measurements, such as discharges, have to berelated to river stages (the stage is simply the water surface height above some fixed datum). Both stageand discharge measurements are important. Often, however, the actual discharge of a river is measuredrarely, 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 isnow 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 ofoccurrence 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 canbe 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 fornavigation, 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 thewater 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 suchas 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 anddown by the water level. Fluctuations caused by short waves are almost eliminated. The movement ofthe float is transmitted by a wire passing over a float wheel, which records the motion, leading down toa 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 waterlevels in open water (the effect of short waves dies out almost completely within half a wavelength downinto the water), as well as for the continuous recording of groundwater levels. They should compensatefor changes in the atmospheric pressure, and if air-vented cables cannot be provided air pressure needsto be measured separately.

Bubble gauge: This is a pressure actuated system, based on measurement of the pressure whichis needed to produce bubbles through an underwater outlet. These are used at sites where it would be

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difficult to install a float-operated recorder or pressure transducer. From a pressurised gas cylinder orsmall compressor gas is led along a tube to some point under the water (which will remain so for allwater levels) and bubbles constantly flow out through the orifice. The pressure in the measuring tubecorresponds 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 thewater and emits ultrasonic pulses at a certain frequency. The inaudible sound waves are reflected by thewater surface and received by the sensor. The round trip time is measured electronically and appears asan output signal proportional to the level. A temperature probe compensates for variations in the speedof 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 asarrays 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 floodprotection require an accuracy of 2−5 cm, while gauge readings upstream of flow measuring weirs usedto calculate discharges from the measured heads require an accuracy of 2 − 5mm. These days almostall are telemetered to a central site. There is a huge volume of electronic hydrometry data being sentaround 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 dischargesare 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 relationshipcan be built up, so that the routine measurement of stage can be converted to discharge. We will beconsidering 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 calculatethe surface slope between two gauges, and hence, to determine the roughness of the reach. Underunsteady conditions the relationship will be disturbed. We will also be considering this later.

2.2 DischargeFlow measurement may serve several purposes:

• information on river flow for the design and operation of diversion dams and reservoirs and forbilateral agreements 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 singlemeasurement methods which are not designed for routine operation; the rest are methods of continuousmeasurements.

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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 pointsdistributed systematically over the river cross-section. In fact, what this usually means is that two ormore velocity measurements are made on each of a number of vertical lines, and any one of severalempirical expressions used to calculate the mean velocity on each vertical, the lot then being integratedacross the channel.

Calculating the discharge requires integrating the velocity data over the whole channel - what is requiredis the area integral of the velocity, that is Q =

Ru dA. If we express this as a double integral we can

write

Q =

ZB

Z(y)+h(y)ZZ(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 depthand where our z is a local co-ordinate. Then we have to integrate these contributions right across thechannel, 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 avertical 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 developedsome families of methods which are based more on rational methods. Consider the law for turbulentflow over a rough bed, which can be obtained from the expressions on p582 of Schlichting (1968):

u =u∗κln

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 elevationabove the bed, and z0 is the elevation at which the velocity is zero. (It is a mathematical artifact thatbelow this point the velocity is actually negative and indeed infinite when z = 0 – this does not usuallymatter in practice). If we integrate equation (2.3) over the depth h we obtain the expression for the meanvelocity:

u =1

h

hZ0

u dz =u∗κ

µln

h

z0− 1¶. (2.4)

Now it is assumed that two velocity readings are made, obtaining u1 at z1 and u2 at z2. This givesenough information to obtain the two quantities u∗/κ and z0. Substituting the values for point 1 intoequation (2.3) gives us one equation and the values for point 2 gives us another equation. Both can besolved to give the solution

u∗κ=

u2 − u1ln (z2/z1)

and z0 =

µzu21zu12

¶ 1

u2−u1. (2.5)

It is not necessary to evaluate these, for substituting into equation (2.4) gives a simple formula for themean 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,

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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 necessarilyat points such as 0.2h and 0.8h. This might simplify streamgauging operations, for it means that thehydrographer, after measuring the depth h, does not have to calculate the values of 0.2h and 0.8h andthen set the meter at those points. Instead, the meter can be set at any two points, within reason, thedepth and the velocity simply recorded for each, and equation (2.7) applied. This could be done eitherin situ or later when the results are being processed. This has the potential to speed up hydrographicmeasurements.

If the hydrographer were to use the traditional two points, then setting h1 = 0.2h and h2 = 0.8h inequation (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 3778.3.12001):

u = 0.5u0.2h + 0.5u0.8h . (2.9)

The nominally more accurate expression, equation (2.8), gives less weight to the upper measurementand more to the lower. It might be useful, as it is just as simple as the traditional expression, yet is basedon 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 andwas 28m wide, and a number of verticals were used. The conventional formula (2.2), the mean of thetwo velocities, was accurate to within 2% of equation (2.8) over the whole range of the readings, witha mean difference of 1%. That error was always an overestimate. The more accurate formula (2.7) ishardly more complicated than the traditional one, and it should in general be preferred. Although thegain in accuracy was slight in this example, in principle it is desirable to use an expression which makesno numerical approximations to that which it is purporting to evaluate. This does not necessarily meanthat either (2.2) or (2.8) gives an accurate integration of the velocities which were encountered in thefield. In fact, one complication is where, as often happens in practice, the velocity distribution near thesurface 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∗κln

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 profileson 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 globalfunction

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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 thechannel. Here traditional practice seems to be in error – often the Mean-Section method is used. In thisthe 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 isassumed to be

δQi =1

4bi (hi + hi+1) (ui + ui+1) .

This is not correct. From equation (2.1), the task is actually to integrate across the channel the quantitywhich is the mean velocity times the depth. For that the simplest expression is the Trapezoidal rule:

δQi =1

2bi (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, thenthe Mean-Section Method gives

δQ0 =1

4b0u1h1,

while the Trapezoidal rule gives

δQ0 =1

2b0u1h1,

which is correct, and we see that the Mean-Section Method computes only half of the actual contribution.The same happens at the other side. Contributions at these edges are not large, and in the middle of thechannel the formula is not so much in error, but in principle the Mean-Section Method is wrong andshould not be used. Rather, the Trapezoidal rule should be used, which is just as easily implemented. Ina gauging in which the lecturer participated, a flow of 1693 Ml/d was calculated using the Mean-SectionMethod. Using the Trapezoidal rule, the flow calculated was 1721 Ml/d, a difference of 1.6%. Althoughthe 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-SectionMethod for integrating vertically over only three or four data points, when its errors would be ratherlarger.

In textbooks one does find an approximate method known as the Mid-Section Method, which takes asthe 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 ofuniform channel in which the flood peak profile is defined on both banks by high water marks. From thisinformation the slope, the cross-sectional area and wetted perimeter can be obtained, and the dischargecomputed with the Gauckler-Manning-Strickler (G-M-S) formula or the Chézy formula. To do thishowever, roughness coefficients must be known, such as Manning’s n in the G-M-S formula

Q =1

n

A5/3

P 2/3

pS,

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


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 streamimmediately follows.

2.2.4 Integrating float methods

There is another rather charming and wonderful method which has been very little exploited. At themoment it has the status of a single measurement method, however the lecturer can foresee it beingdeveloped as a continuing method.

Theory

Consider a single buoyant particle (a float, an orange, an air bubble), which is released from a pointon the bed. We assume that it has a constant rise velocity w. As it rises it passes through a variablehorizontal velocity field u(z), where z is the vertical co-ordinate. The kinematic equations of the floatare

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:

hZ0

u(z) dz =

LZ0

w dx = wL,

where the particle reaches the surface a distance L downstream of the point at which it was released onthe bed, and where we have used a local vertical co-ordinate z with origin on the bed and where the fluidlocally has a depth h. The quantity on the left is important - it is the vertical integral of the horizontalvelocity, or the discharge per unit width at that section. We can generalise the expression for variationwith y, across the channel, to write

hZZ(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-ordinatedirection y, the integral of the left side is the discharge Q:

Q =

BZ0

hZZ(y)

u(y, z) dz dy = w

BZ0

L(y) dy,

where B is the total width of the channel. Hence we have an expression for the discharge with very fewapproximations:

Q = w

BZ0

L(y) dy.

13


Figure 2-2. Array of four ultrasonic beams in a channel

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

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 arepropagated diagonally across a stream at different levels. The time of travel of sound in one directionis measured, as is the time in the other. The difference can be used to compute the mean velocity alongthat 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 thepath 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 thatthe velocity does, in general, depend on position along the beam, then the component along the path isu(s) cosα. Let c be the speed of sound. The time dt taken for a sound wave to travel a distance ds alongthe 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 =

T1Z0

dt =

LZ0

ds

c− u(s) cosα, (2.11)

and repeating for a traverse in the reverse direction:

T2 =

T2Z0

dt =

LZ0

ds

c+ u(s) cosα. (2.12)

Now we expand the denominators of both integrals by the binomial theorem:

T1 =1

c

LZ0

µ1+

u(s)

ccosα

¶ds and T2 =

1

c

LZ0

µ1− u(s)

ccosα

¶ds, (2.13)

where we have ignored terms which contain the square of the fluid velocity compared with the speed of

14


sound. Evaluating gives

T1 =L

c+1

c2

LZ0

u(s) cosαds and T2 =L

c− 1

c2

LZ0

u(s) cosαds. (2.14)

Adding the two equations and solving for c and re-substituting we obtain

LZ0

u(s) cosαds = 2L2T1 − 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 ofvelocity. As u ≈ 1m s−1 and c ≈ 1400m s−1 it can be seen that the error is exceedingly small. Whatwe 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 withrespect to elevation:

Qz =

Z L

0u(s) sinαds. (2.16)

Now we are forced to assume that the angle that the velocity vector makes with the beam is constantover the path (or at least in some rough averaged sense), and so for α constant, taking the trigonometricfunctions outside the integral signs and combining equations (2.15) and (2.16) we obtain

Qz = 2 tanαL2 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 thebeam is constant, but importantly it shows that it is not necessary to assume that velocity u is constantover the beam path. Equation (2.17) is similar to that presented in Standards and trade brochures, andimplemented in practice, but where it is obtained by assuming that the velocity is constant. It is fortunatethat 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 channelby evaluating the vertical integral of Qz, which, as shown by equation (2.16), is the integral along thebeam of the velocity transverse to the beam. The problem is then to evaluate the vertical integral of thederivative of discharge with elevation:

Q =

hZ0

Qz(z) dz, (2.18)

where in practice the information available is that Qz = 0 on the bottom of the channel z = 0 and thetwo to four values of Qz which have been obtained from beam data, as well as the total depth h. It isin 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 isadequate (see, for example, Boiten 2000, p141). In fact, with high-quality data for Qz at two or threelevels, there is no reason not to use accurate integration formulae. However, practice in this area hasbeen quite poor, as trade brochures that the author has seen use the inaccurate Mean-Section Methodfor integrating vertically over only three or four data points, when its errors would be rather larger thanwhen it is used for many verticals across a channel, as described previously. This seems to be a ripe areafor research.

15


Signal probes

Coil for producing magnetic field

Figure 2-3. Electromagnetic installation, showing coil and signal probes

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. Whenthe sound strikes moving particles or regions of density difference moving at a certain speed, the soundis reflected back and received by a sensor mounted beside the transmitter. According to the Dopplereffect, the difference in frequency between the transmitted and received waves is a direct measurementof velocity. In practice there are many particles in the fluid and the greater the area of flow movingat a particular velocity, the greater the number of reflections with that frequency shift. Potentially thismethod is very accurate, as it purports to be able to obtain the velocity over quite small regions andintegrate them up. However, this method does not measure in the top 15% of the depth or near theboundaries, and the assumption that it is possible to extract detailed velocity profile data from a signalseems 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 usinga large coil buried beneath the river bed, through which an electric current is driven. An electromotiveforce is induced in the water and measured by signal probes at each side of the channel. This very smallvoltage is directly proportional to the average velocity of flow in the cross-section. This is particularlysuited to measurement of effluent, water in treatment works, and in power stations, where the channelis rectangular and made of concrete; as well as in situations where there is much weed growth, orhigh sediment concentrations, unstable bed conditions, backwater effects, or reverse flow. This has theadvantage that it is an integrating method, however in the end recourse has to be made to empiricalrelationships between the measured electrical quantities and the flow.

2.2.8 Flow measuring structures

These are often bound up with control and regulatory functions, as well as measurement. We will nottreat 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 isan important quantity in determining the danger of flooding, another important quantity is the actual flowrate 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 ariver and its chief environmental and commercial resource, its water. Examples include decisions on theallocation of water resources, the design of reservoirs and their associated spillways, the calibration ofmodels, and the interaction with other computational components of a network.

The traditional way in which volume flow is inferred is for a rating curve to be derived for a particular

16


Qfalling Qrated 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 ofa particular flood event

gauging station, which is a relationship between the stage measured and the actual flow passing thatpoint. The measurement of flow is done at convenient times by traditional hydrologic means, witha current meter measuring the flow velocity at enough points over the river cross section so that thevolume of flow can be obtained for that particular stage, measured at the same time. By taking suchmeasurements for a number of different stages and corresponding discharges over a period of time, anumber 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 inFigure 2-4. This is then used in the future so that when stage is routinely measured, it is assumed thatthe corresponding discharge can be obtained from that curve, such as the discharge Qcalculated shown inthe 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 thecurve might be quite poor.

• It is usually some sort of line of best fit through a sample made up of a number of points - sometimesextrapolated for higher stages.

• It has to describe a range of variation from no flow through small but typical flows to very largeextreme 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 floodevent, 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 thatfriction is greater after the flood peak than before, so that the corresponding discharge for agiven stage height will be less after the peak. This will contribute to a flood event showing alooped 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 orflooding in the next waterway.

17


– Unsteadiness - in general the discharge will change rapidly during a flood, and the slope ofthe water surface will be different from that for a constant stage, depending on whether thedischarge is increasing or decreasing, also contributing to a flood event appearing as a loop ona 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 flow

Low flow

Localcontrol

Distantcontrol

Channel control

Gaugingstation

Channel control

Flood

1

32

4

5

Larger bodyof water

Figure 2-5. Section of river showing different controls at different water levels with implications for the stagedischarge relationship at the gauging station shown

A typical set-up of a gauging station where the water level is regularly measured is given in Figure 2-5which shows a longitudinal section of a stream. Downstream of the gauging station is usually somesort of fixed control which may be some local topography such as a rock ledge which means that forrelatively small flows there is a relationship between the head over the control and the discharge whichpasses. 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 asthe larger river downstream shown as a distant control in the figure, or even, if the downstream channellength is long enough before encountering another local control, the section of channel downstreamwill itself become the control, where the control is due to friction in the channel, giving a relationshipbetween the slope in the channel, the channel geometry and roughness and the flow. There may bemore controls too, but however many there are, if the channel were stable, and the flow steady (i.e. notchanging with time anywhere in the system) there would be a unique relationship between stage anddischarge, however complicated this might be due to various controls. In practice, the natures of thecontrols are usually unknown.

Something which the concept of a rating curve overlooks is the effect of unsteadiness, or variation withtime. In a flood event the discharge will change with time as the flood wave passes, and the slope ofthe water surface will be different from that for a constant stage, depending on whether the discharge isincreasing or decreasing. Figure 2-5 shows the increased surface slope as a flood approaches the gaugingstation. The effects of this are shown on Figure 2-4, in somewhat exaggerated form, where an actualflood event may not follow the rating curve but will in general follow the looped trajectory shown. Asthe flood increases, the surface slope in the river is greater than the slope for steady flow at the samestage, and hence, according to conventional simple hydraulic theory explained below, more water isflowing down the river than the rating curve would suggest. This is shown by the discharge markedQrising obtained from the horizontal line drawn for a particular value of stage. When the water level isfalling the slope and hence the discharge inferred is less.

The effects of this might be important - the peak discharge could be significantly underestimated during

18


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 theflood are ignored, while the use of a discharge hydrograph derived inaccurately by using a single-valuedrating relationship may distort estimates for resistance coefficients during calibration of an unsteady flowmodel.

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 stationseems 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 longwave equations, which are a pair of partial differential equations for the stage and the discharge at allpoints 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 naturalwaterways, however, the flow is sufficiently slow that the equations can be simplified considerably. Mostterms 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 meandepth of the waterway. In most rivers, even in flood, this is small, and the approximation may be oftenused. 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 thesimple 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 usualpractice is to use an empirical friction law for the friction slope in terms of a conveyance function K, sothat 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 bedetermined empirically, or by a standard friction law, such as the Gauckler-Manning-Strickler formulaor Chézy’s formula:

G-M-S: K =1

n

A5/3

P 2/3or 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 andP is wetted perimeter, which are both functions of depth and x, as the cross-section usually changesalong the stream. In most hydrographic situations K would be better determined by measurements offlow and slope rather than by these formulae as they are approximate only and the roughness coefficientsare 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 inmore generally unsteady and non-uniform flows. Hence, substituting (2.20) into (2.19) gives us anexpression for the discharge, where we now show the functional dependence of each variable:

Q(t) = K(η(t))qSη(t), (2.22)

where we have introduced the symbol Sη = −∂η/∂x for the slope of the free surface, positive in thedownstream direction, in the same way that we use the symbol Sf for the friction slope. This gives usan expression for the discharge at a point and how it might vary with time. Provided we know1. 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,

19


we have a formula for calculating the discharge Q which is as accurate as is reasonable to be expectedin 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 underdifferent backwater conditions or during a flood, then a better hydrographic procedure would be togauge the flow when it is steady, and to measure the surface slope , thereby enabling a particular valueof K to be calculated for that stage. If this were done over time for a number of different stages, thena stage-conveyance relationship could be developed which should then hold whether or not the stageis 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 mildassumptions made so far. All of this holds whether or not the gauging station is affected by a local orchannel 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 tothe changing of downstream controls with time, downstream tailwater conditions, and unsteadiness infloods would have been automatically incorporated, both at the time of determining the relationship andsubsequently in daily operational practice. However, for the most part slope has not been measured, andhydrographic practice has been to use rating curves instead. The assumption behind the concept of adischarge-stage relationship or rating curve is that the slope at a station is constant over all flows andevents, so that the discharge is a unique function of stage Qr(η) where we use the subscript r to indicatethe rated discharge. Instead of the empirical/rational expression (2.22), traditional practice is to calculatedischarge from the equation

Q(t) = Qr(η(t)), (2.23)

thereby ignoring any effects that downstream backwater and unsteadiness might have, as well as thepossible changing of a downstream control with time.

In comparison, equation (2.22), based on a convenient empirical approximation to the real hydraulicsof the river, contains the essential nature of what is going on in the stream. It shows that, although theconveyance 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 awaywith, and replaced by a stage-conveyance relationship. Of course in many, even most, situations it mightwell be that the surface slope at a gauging station does vary but little throughout all conditions, in whichcase the concept of a stage-discharge relationship would be accurate. In most situations it is indeed thecase 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))qSη(t), (2.24)

It could not be claimed that this is a theoretical justification, as they are based on empirical friction lawsbut, based on the cases studied above, the incorporation of slope appears to give a superior and morefundamental description of the processes at work, and handles both long-term effects due to downstreamconditions changing and short-term effects due to the flow changing.

This suggests that a better way of determining streamflows in general, but primarily where backwaterand 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

20


the slope of the water surface at the station. One of these could be at the section where detailedflow-gaugings are taken, and the other could be some distance upstream or downstream such thatthe stage difference between the two points is enough that the slope can be computed accuratelyenough. 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 usingconventional methods such as by current meter. For each gauging, both surface elevations wouldbe recorded, one becoming the stage η to be used in the subsequent relationship, the other so thatthe surface slope Sη can be calculated. Using equation (2.22), Q = K(η)

pSη, this would give the

appropriate value of conveyance K for that stage, automatically corrected for effects of unsteadinessand downstream conditions.

3. From all such data pairs (ηi,Ki) for i = 1, 2, . . ., the conveyance curve (the functional dependenceof K on η) would be found, possibly by piecewise-linear or by global approximation methods, in asimilar 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 advantagesin representing rating curves on a plot using the square root of the discharge, and it my well be thatthe 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 thestage-conveyance relationship, and hence the water surface slope, which would then be substitutedinto 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 thetime derivative of stage can easily be obtained from records but the surface slope along the channel isnot measured at all. The methods of this section are all aimed at obtaining the slope in terms of thestage and its time derivatives at a single gauging station. The simplest and most traditional method ofcalculating the effects of unsteadiness has been the Jones formula, derived by B. E. Jones in 1916 (seefor example Chow 1959, Henderson 1966). The principal assumption is that to obtain the slope, the xderivative of the free surface, we can use the time derivative of stage which we can get from a stagerecord, by assuming that the flood wave is moving without change as a kinematic wave (Lighthill andWhitham, 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 wavestravelling at a velocity c without change. The equation will be obtained as one of a consistent series ofapproximations in Section 3. The kinematic wave speed c is given by the derivative of flow with respectto cross-sectional area, the Kleitz-Seddon law

c =1

B

dQr

dη=1

B

dK

dη

pS, (2.26)

where B is the width of the surface and Qr is the steady rated discharge corresponding to stage η, andwhere we have expressed this also in terms of the conveyance K, where Qr = K(η)

√S, and the slope√

S is the mean slope of the stream. A good approximation is c ≈ 5/3×U , where U is the mean streamvelocity.

The Jones method assumes that the surface slope Sη in equation can be simply related to the rate ofchange of stage with time, assuming that the wave moves without change. Thus, equation (2.25) gives

21


an approximation for the surface slope: ∂h/∂x ≈ −1/c × ∂h/∂t. We then have to use the simplegeometric relation between surface gradient and depth gradient, that ∂η/∂x = ∂h/∂x− S, such that wehave 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

rS +

1

c

∂η

∂t(2.27)

If we divide by the steady discharge corresponding to the rating curve we obtain

Q

Qr=

r1 +

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, wherethe 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 longwave equations and approximated the surface slope using a method based on a linearisation of thoseequations. 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 theequation solved numerically. The second method was rather simpler, and was based on the next bestapproximation to the full 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 adiffusion 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 waveequations, where inertial terms, which are of the order of the square of the Froude number, whichapproximates motion in most waterways quite well. However, it is not yet suitable for the purposes ofthis section, for we want to express the x derivative at a point in terms of time derivatives. To do this, weuse 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 resultingexpression 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

c3d2h

dt2,

22


and substituting into equation (2.22) gives

Q = Qr(η)

vuuuut 1|zRating 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 functionof 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-diffusionapproximation, which is a good approximation to much flood propagation. Equation (2.29) provides ameans of analysing stage records and correcting for the effects of unsteadiness and variable slope. It canbe 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-statevalue of discharge for the stage-discharge relation,

• And, proceeding in the other direction, in operational practice, it can be used for the routine analysisof 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 of10 km length, of slope 0.001, which had a trapezoidal section 10mwide at the bottom with side slopes of1:2, and a Manning’s friction coefficient of 0.04. The downstream control was a weir. Initially the depthof flow was 2m, while carrying a flow of 10m3 s−1. The incoming flow upstream was linearly increasedten-fold to 100m3 s−1 over 60 mins and then reduced to the original flow over the same interval. Theinitial backwater curve problem was solved and then the long wave equations in the channel were solvedover six hours to simulate the flood. At a station halfway along the waterway the computed stages wererecorded (the data one would normally have), as well as the computed discharges so that some of theabove-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 notexact however, as the derivation depends on the diffusion being sufficiently small that the interchangebetween 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 atpredicting the peak flow. Nevertheless, the results from the Jones method are interesting. A widely-heldopinion is that it is not accurate. Indeed, we see here that in predicting the peak flow it was not accuratein this problem. However, over almost all of the flood it was accurate, and predicted the time of the floodpeak well, which is also an important result. It showed that both before and after the peak the ”dischargewave” led the ”stage wave”, which is of course in phase with the curve showing the flow computed fromthe stage graph and the rating curve. As there may be applications where it is enough to know the arrivaltime of the flood peak, this is a useful property of the Jones formula. Near the crest, however, the rateof rise became small and so did the Jones correction. Now, and only now, the inclusion of the extra

23


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 flowFrom rating curve

Jones formulaEqn (2.29)

Figure 2-6. Simulated flood with hydrographs computed from stage record using threelevels of approximation

diffusion term gave a significant correction to the maximum flow computed, and was quite accurate inits prediction that the real flow was some 10% greater than that which would have been calculated justfrom the rating curve. In this fast-rising example the application of the unsteady corrections seems tohave worked well and to be justified. It is no more difficult to apply the diffusion correction than theJones 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 ispresented 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 underlyingit the slope is being measured. Instead, the fall is described, which is the change in surface elevationbetween 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

pSη,

where it is assumed that the discharge must be given using these precise geometrical quantities of A andP . It is rather more pragmatic to determine K(η) by measurements.

3. The propagation of waves in waterwaysWe now spend some time deriving the full equations for unsteady non-uniform flow. The fundamentalassumption we make is that the flow is slowly varying along the channel. The mathematics uses anumber of concepts from vector calculus, however we find that we can obtain general equations verypowerfully, and the assumptions and approximations (actually very few!) are clear.

24


3.1 Mass conservation equation

x

z

z

∆x

Q+∆Q

Q

i∆x

Top of control volume

Water surface

Figure 3-1. Elemental length of channel showing control volume extended into the air

Consider the elemental section of thickness∆x of non-uniform waterway shown in Figure 3-1, boundedby two vertical planes parallel to the y − z plane. Consider also the control volume made up of thiselemental section, but continued into the air such that the bottom and lateral boundaries are the riverbanks, and the upper boundary is arbitrary but never intersected by the water.

The Mass Conservation equation in integral form is, written for a control volume CV bounded by acontrol surface CS,

∂

∂t

ZCV

ρ dV

| z Total mass in CV

+

ZCS

ρu.n 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, n is a unit vector with directionnormal to and directed outwards from the control surface such that u.n is the component of velocitynormal 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 firstintegral 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

ZA

dA = ρ∆x∂A

∂t

Now considering the second integral, on the upstream face of the control surface, u.n = −u , where uis the x component of velocity, so that the contribution due to flow entering the control volume is

−ρZA

udA = −ρQ.

Similarly the downstream face contribution is

+ρ (Q+∆Q) = +ρ

µQ+

∂Q

∂x∆x

¶.

On the boundaries which are the banks of the stream, the velocity component normal to the boundaryis very small and poorly-known. We will include it in a suitably approximate manner. We lump thiscontribution from groundwater, inflow from rainfall, and tributaries entering the waterway, as a volumerate of q per unit length entering the stream. The rate at which mass enters the control volume is ρq∆x

25


(i.e. an outflow of−ρq∆x). Combining the contributions from the rate of change of mass in the CV andthe net contribution across the two faces, and dividing by ρ∆x we have the unsteady mass conservationequation

∂A

∂t+

∂Q

∂x= q. (3.1)

Remarkably for hydraulics, this is an almost-exact equation - the only significant approximation we havemade is that the waterway is straight! If we want to use surface elevation as a variable in terms of surfacearea, it is easily shown that in an increment of time δt if the surface changes by an amount δη, then thearea changes by an amount δA = B × δη, from which we obtain ∂A/∂t = B × ∂η/∂t, and the massconservation 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):³1− nm

r

´ ∂A

∂t+

∂Q

∂s= q,

where nm is the transverse offset of the centre of the river surface from the curved streamwise referenceaxis s, and r is the radius of curvature of that axis. Usually nm is small compared with r, and thecurvature term is a relatively small one. It can be seen that if it is possible to choose the reference axisto coincide with the centre of the river viewed in plan, then nm = 0 and curvature has no effect on thisequation. This choice of axis is not always possible, however, as the geometry of the river changes withsurface height.

3.2 Momentum conservation equationWe can write the Momentum Conservation Equation in integral form, similarly to the Mass ConservationEquation:

∂

∂t

ZCV

ρu dV

| z Total momentum in CV

+

ZCS

ρ uu.n 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 latterincluding 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 canbe written−

RCS

pn 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

ZCV

ρu dV +

ZCS

ρuu.n dS = Shear force−ZCS

pn 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 muchsimpler derivation is obtained if the term is evaluated using Gauss’ Divergence Theorem:Z

CS

pn dS =

ZCV

∇p dV

where ∇p = (∂p/∂x, ∂p/∂y, ∂p/∂z), the vector gradient of pressure. This has turned a complicated

1 Fenton, J. D. & Nalder, G. V. (1995), Long wave equations for waterways curved in plan, in Proc. 26thCongress IAHR, London, Vol. 1, pp. 573–578.

26


surface integral into a simple volume integral. It can be understood by considering the fluid to be dividedup into a number of elemental parallelepipeds, the pressure force on one elemental face being cancelledby its force on the other.

Now taking the x component of the vector momentum equation we obtain:

∂

∂t

ZCV

ρu dV

| z (a)

+

ZCS

ρuu.n dS

| z (b)

= Horizontal shear force| z (c)

−Z

CV

∂p

∂xdV

| z (d)

,

and we now make hydraulic approximations for these terms. The first two are obtained in the samemanner as for the mass conservation equation.

(a) Unsteady term: The first term is

∂

∂t

ZCV

ρudV = ρ∂

∂t

ZA

udA×∆x = ρ∂Q

∂t∆x.

(b) Momentum flux term: There are two parts to this. The first is from contributions on solidboundaries 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:ZU/S & D/S Face

ρuu.n dS = −ρZ

U/S Face

u2 dA+ ρ

ZD/S Face

u2 dA. (3.4)

Almost never do we know the precise velocity distribution over the face. We introduce an empiricalquantity β such that the effects of both non-uniformity of velocity over a section and turbulent fluctua-tions are approximated by Z

A

u2dA = βQ2

A, (3.5)

where β is the momentum coefficient or Boussinesq coefficient. If the velocity were constant and steadyover the section β would have a value of 1, which is the usual approximation to a term which otherwisewould be very difficult to evaluate. We will retain it, however, its effects are small in many situations asshown below, as it is always associated with terms which are of relative magnitude that of the square ofthe Froude number. Hence we write the contribution (3.4) asZ

CS

ρuu.n dS = −ρβU2A¯U/S Face + ρβU2A

¯D/S Face

=∂

∂x

¡ρβU2A

¢×∆x

= ρβ∂

∂x

µQ2

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,not particularly well-known, and the flow structure is less well-known. Following the usual convention

27


in river hydraulics, a convenient empirical expression is adopted instead. The approximation is madehere 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 tothe total gravitational force of the fluid in the control volume, and we will call it the friction slope. Thenegative sign is introduced because in the usual case where flow is in the+x direction, the shear force isin the other direction. Later we will assume that it can be given by the G-M-S formula, where the localand instantaneous depth and discharge are used.

(d) Pressure gradient term This is

−Z

CV

∂p

∂xdV.

The approximation we now make, common throughout almost all open-channel hydraulics, is the ”hy-drostatic approximation”, that pressure is given by

p ≈ ρg × height of water above = ρg (η − z) , (3.6)

which is the pressure at a point of elevation z where the free surface directly above has elevation η. Thisis the expression obtained in hydrostatics, where the fluid is not moving. It is an excellent approximationexcept where the flow is strongly curved, such as where there are short waves on the flow, or near astructure which disturbs the flow locally.

Differentiating with respect to x, equation (3.6) gives

∂p

∂x= ρg

∂η

∂x,

such that the pressure gradient is determined by the downstream slope of the free surface, which weassume is constant across the stream. Substituting into the pressure term we obtain

−Z

CV

∂p

∂xdV = −ρg ∂η

∂x×A×∆x

Collecting terms: Combining our four contributions to the momentum equation, including the inflowterm from equation (3.3) and dividing by ρ∆x we obtain

∂Q

∂t+ β

∂

∂x

µQ2

A

¶= −gASf − gA

∂η

∂x+ quq.

Expanding:

∂Q

∂t+ 2β

Q

A

∂Q

∂x− β

Q2

A2∂A

∂x= −gASf − gA

∂η

∂x+ quq.

In both mass and momentum equations we have derivatives of area. It is more convenient to use thesurface elevation η. We can show that for a small change in surface elevation δη, δA = B δη, so that, asthe bed does not move,

∂A

∂t= B

∂η

∂t

For the x derivative we must also allow for the fact that the bed elevation also depends on x.

The cross-section of a river in Figure ?? shows how ambiguous and possibly non-unique the concept ofthe “bottom” of the stream may be. In a distance ∆x the surface elevation may change by an amount

28


A

B

∆η

y

z

∆Z

At xAt x+∆x

Figure 3-2. Two channel cross-sections separated by∆x

∆η as shown, so that the contribution to the increase in cross-section area∆A is B ×∆η, where∆η isusually negative as the surface drops downstream. If the change in the bed is∆Z as shown in the figure,in general this varies across the section, and so the contribution to∆A is −

RB∆Z dy, the area between

the solid and dotted lines on the figure corresponding to the bed at the two locations. The minus sign isbecause if the bed drops away and∆Z is negative, as usual, the contribution to area increase is positive.Combining the two terms,

∆A = B ×∆η −ZB∆Z dy

In practice the precise details of the bed are rarely known, and it is convenient to introduce ∆Z, themean change in bed level across the section. Hence we have

∆A = B ×∆η −B ×∆Z.

Now we express this in terms of the mean bed slope across the stream S. In a distance∆x the mean bedlevel across the channel then changes by ∆Z = −S ×∆x under the water. We use the convention forbed slope that a downwards-sloping bed, the usual situation, has a positive value of bed slope. The totalincrease of area is∆A = B × S ×∆x+B ×∆η, giving

∂A

∂x= B

µ∂η

∂x+S

¶.

Substituting these relations gives the momentum equation governing flows and long waves in waterwayswith Q and η as dependent variables:

∂Q

∂t+ 2β

Q

A

∂Q

∂x+

µgA− β

Q2B

A2

¶∂η

∂x= β

Q2B

A2S − gASf + quq. (3.7)

3.3 The nature of the propagation of long waves and floods in riversEquations (3.2) and (3.7) are often called the “Saint-Venant” equations. They form a pair of partialdifferential equations for the surface elevation η and discharge Q at any point in a stream x at any time t.These equations are used to simulate wave motions in rivers and canals, notably the propagation of floodwaves and the routine simulation of irrigation channel operations. There is a software industry whichspecialises in numerical solutions.

There have been a number of different interpretations of the nature of the propagation of long wavesand floods in rivers, as described by these equations. Some quite misleading results have been obtained.We will now consider these, as they are often considered in research papers as well as more practicalapplications like the choice of software or, more importantly, and understanding of the real nature oflong wave propagation.

3.3.1 The use of characteristics – a view as waves travelling up and downstream with little

29


diminution

It can be shown mathematically that solutions of these two partial differential equations can be expressedas four ordinary differential equations. Two of the differential equations are for x as a function of t wherein this case x(t) is the position of a mathematically convenient path known as a characteristic. The firsttwo equations are

dx

dt= β

Q

A±C,

where the first part βQ/A, is simply β times the mean fluid velocity in the waterway at that section. Thesecond part C is the quantity

C =

rgA

B+

Q2

A2¡β2−β

¢, (3.8)

which is more important. The two velocities ±C, are relative to the water, corresponding to both up-stream and downstream propagation of information in sub-critical flow. The result in equation (3.8) hasincluded the momentum coefficient β. If we assume that velocity is constant over the section such thatβ = 1 we obtain the traditional result that C =

pgA/B, where A/B is the mean depth.

It can be shown that on characteristics in given by equation (3.8), the differential equations to be satisfiedare:

B

µ−βQ

A± C

¶dη

dt+

dQ

dt= β

Q2B

A2S − gASf + q

µuq − β

Q

A± C

¶− Q2

A

dβ

dx. (3.9)

taking the corresponding plus or minus signs in each case.

The use of characteristics has led to a widespread misconception in hydraulics and C is usually referredto as the speed of propagation of waves. It is not – it is the speed of propagation of information alongcharacteristics. Some of that information will appear as wave motion at that speed, but not all. Thefamiliar and widely-quoted result C =

pgA/B (which from equation (3.8) implicitly contains the

β = 1 or β = 0 approximations) has led to a common misconception in hydraulics such that C has oftenbeen referred to as the speed of propagation of waves. It is not – it is the speed of characteristics. Ifsurface elevation were constant on a characteristic there would be some justification in using the term”wave speed” for the quantity C, as disturbances travelling at that speed could be observed. Howeveras equation (3.9) holds in general, neither η, surface elevation, nor Q, is constant on the characteristicsand one would not have observable disturbances or discharge fluctuations travelling at C relative to thewater. While C may be the speed of propagation of information in the waterway relative to the water, itcannot properly be termed the wave speed as it would usually be understood.

3.3.2 The low inertia approximation – diffusion routing and nature of wave propagation inwaterways

Equations (3.2) and (3.7) (the long wave or Saint-Venant equations) are used to simulate wave motionsin rivers and canals, notably the propagation of flood waves and the routine simulation of irrigationchannel operations. There is a software industry which specialises in numerical solutions. We haveshown that mathematically the solutions of these equations look like two families of waves, propagatingat two different velocities, one upstream, the other downstream, with differing amounts of diffusivedissipation. The upstream propagating waves show rather more dissipation. The propagation in bothdirections is important in situations where transients are rapid, such as in hydro-electric supply canals.In most situations, however, the flow velocity is relatively small, such that a large simplification ispossible. We now show that the first two terms in equation (3.7), relative to the others, are of the orderof magnitude of F 2, and can be ignored.

Firstly consider ∂Q/∂t. The discharge Q is of an order of magnitude V ×W ×D, where V is a typicalvelocity, W is a typical width, and D is a typical mean depth. The time scale of motion is given by L/V ,where L is a typical length scale of the wave motion down the waterway, and our velocity scale gives a

30


measure of how quickly it is swept past. Hence,

∂Q

∂tis of a scale

V ×W ×D

L/V=

V 2WD

L.

Now we examine the scale of the term gA∂η/∂x, and here we assume that the vertical scale of ourdisturbances is the vertical scale of the channel D, giving

gA∂η

∂xis of a scale g ×W ×D × D

L=

gWD2

L.

Now we compare the magnitudes of the two terms ∂Q/∂t : gA∂η/∂x and we find that the ratio of thetwo terms is of a scale

V 2WD

L× L

gWD2=

V 2

gD, the scale of the Froude number squared.

Hence, possibly to our surprise, we find that the relative magnitude of the term ∂Q/∂t is roughly F 2,and in many flows in rivers and canals this is a small quantity and terms of this size can be ignored.Examining the second term in equation (3.7) it might be more obvious that it too is also of order F 2.If we neglect both such terms of order F 2, making the ”low-inertia” approximation (and neglect thepoorly-known inflow term), we find that the momentum equation (3.7) can simply be approximated by

∂η

∂x+ Sf = 0, (3.10)

which expresses the fact that, even in a generally unsteady situation, the surface slope and the frictionslope are the same magnitude. Now we use an empirical friction law for the friction slope Sf in termsof conveyance K, so that we write

Sf =Q2

K2,

where the dependence of K on depth at a section would be given by the G-M-S formula. Substitutingthis into (3.10) gives us an accurate approximation for the discharge in terms of the slope:

Q = K

r−∂η∂x

, (3.11)

even in a generally unsteady flow situation, provided the Froude number is sufficiently small. (Note that∂η/∂x is always negative in situations where this theory applies!). This provides us with a good methodof measuring the discharge - if we can calibrate a gauging station to give the conveyance as a functionof surface height, then by measuring the surface slope we can get the discharge.

At this point it is easier to introduce the local depth h such that if Z is the local elevation of the bottom,

η = Z + h and∂η

∂x=

∂Z

∂x+

∂h

∂x=

∂h

∂x− S0, where S0 = −

∂Z

∂x,

in which case equation (3.11) can be written

Q = K

rS0 −

∂h

∂x, (3.12)

Now we eliminate the discharge Q from the equations by simply substituting equation (3.12) into themass conservation equation (3.2), noting that as the bed does not move, ∂η/∂t = ∂h/∂t, to give thesingle partial differential equation in the single variable h: ,

∂h

∂t+1

B

∂

∂x

ÃK

rS0 −

∂h

∂x

!= 0 (3.13)

The conveyance is usually expressed as a function of roughness and of the local depth h, so that we can

31


perform the differentiation in equation (3.13) and we assume that if the variation of the local depth issmall compared with the overall slope so that |∂h/∂x| ¿ S0 we can write

∂h

∂t+

√S0B

dK

dh| z Propagation velocity

∂h

∂x=

K

2B√S0| z

Diffusion coefficient

∂2h

∂x2(3.14)

We now have a rather simpler single equation in a single unknown. This is an advection-diffusionequation, and the nature of it is rather clearer than our original pair of equations. It has solutions whichpropagate at the propagation velocity shown. We write the equation as

∂h

∂t+ c

∂h

∂x= ν

∂2h

∂x2, (3.15)

where c is a propagation speed, the kinematic wave speed, and ν is a diffusion coefficient (with units ofL2T−1), which are given by

c =

√S0B

dK

dhand ν =

K

2B√S0

. (3.16)

As K = 1/n × A5/3(h)/P 2/3(h), we can differentiate to give the kinematic wave speed c for anarbitrary section. However for the purposes of this course we can consider a wide rectangular channel(h ¿ B) such that A ≈ Bh and P ≈ B such that K = 1/n × B h5/3. Differentiating, dK/dh =1/n× 5/3×B h2/3, and substituting into (3.16) we obtain

c =5

3×√S0n

h2/3.

Now, the velocity of flow in this waterway is U = 1/n × (A/P )2/3√S0, which for our wide-channel

approximation is U ≈√S0/n × h2/3, and so we obtain the approximate relationship for the speed of

propagation of disturbances in a wide channel:

c ≈ 53U, (3.17)

which is a very simple expression: the speed of propagation of disturbances is approximately 123 timesthe mean speed of the water.

In equation (3.8) above it was stated that there is a velocity at which information travels in a waterway,

C =

rgA

B+

Q2

A2¡β2−β

¢.

In all text books this is presented for β = 1 such that C =pgA/B, where A/B is the mean depth of

the water. It is indeed the order of magnitude of the speed at which waves do move over essentially stillwater, but is widely used, incorrectly, to estimate the speed of disturbances in rivers and canals. It iscalled the dynamic wave speed, and part of waves do travel at this speed. However, provided the Froudenumber is small, such that F 2 ¿ 1, equation (3.17) is a good approximation to the speed at which thebulk of disturbances propagate.

Now to consider the effects of diffusion, if we examine the real or simulated propagation of waves instreams, the apparent motion is of waves propagating at the kinematic wave speed, but showing markeddiminution in size as they propagate. We consider as a test case, a pool 7km long, bottom width of7m, batter slopes of 1.5:1, a longitudinal slope of 0.0001, a depth of 2.1m at the downstream gate, andManning’s n = 0.02. We consider a base flow of 10 m3s−1 increased smoothly (a Gaussian function oftime) by 25% up to a maximum of 12.5 m3s−1 and back down to the base flow over a period of aboutthree hours. A computer program simulated conditions in the canal. The results presented in Figure 3-3show that over the length of only 7km the peak discharge has decreased by about 50% and has spreadout considerably in time - the wave propagation is not just a simple translation. In fact, the peak takesabout 55 minutes to traverse the pool, whereas using the dynamic wave speed that time would be 30

32


10

11

12

0 5 10 15 20

Dis

char

ge (c

u.m

/s)

Time (hours)

InflowOutflow

Figure 3-3. Inflow and outflow hydrographs for a section of a waterway, showing effects of diffusion

minutes, while simply using the kinematic wave speed it would be 140 minutes. The wave has diffusedconsiderably, showing that simple deductions based on a wave speed are only part of the picture. Thisdifference might be important for flood warning operations.

It is important to find out more about the real nature of wave propagation in waterways. Here we providea simple tool for estimating the relative importance of diffusion. If we were to scale the advection-diffusion equation (3.15) such that it was in terms of a dimensionless variable x/L which would beof order of magnitude 1, then the ratio of the importance of the diffusion term to the advection termcan be shown to be ν/cL. This looks like the inverse of a Reynolds number (which is correct – theReynolds number is the inverse of a dimensionless viscosity or diffusion number). Now we substitute inthe approximations for a wide channel, giving:

A measure of the importance of diffusion =ν

cL=

K

2B√S0× B√

S0K 0(h)L

=K/K 0(h)

2S0L, and as K ∝ h5/3 this gives

≈ 3

10× h

S0L.

This is a useful result, for it shows us the effects of diffusion very simply, as h is the depth of the stream,and S0L is the amount by which a stream drops over the reach of interest. Dropping the factor of 3/10,as our arguments are order-of-magnitude at best:

A measure of the importance of diffusion ≈ Depth of streamDrop of stream

.

For the example above, this is about 2.4, showing that diffusion is as important as advection, and re-minding us that the problem is not the simple translation of a wave.

This result is also interesting in considering different types of streams – a steep shallow mountain streamwill show little diffusion, whereas a deep gently sloping stream will have marked diffusion!

3.4 A new low-inertia approach – Volume routingWe have introduced several approximations in deriving the advection-diffusion equation. Here we use a

33


transformation of variables which enables us to use a single dependent variable in low-inertia routing.

Derivation of equation: Consider the volume of fluid upstream of a point x at a time t, denotedby V (x, t). From simple calculus, the derivative of volume with respect to distance x gives the cross-sectional area: ∂V/∂x = A, and as the time rate of change of V at a point is equal to the total rateupstream at which the volume is increasing, which is

Rx q dx less Q, the volume rate which is passing

the point, we have ∂V/∂t =Rx q dx−Q. Substituting the relations

A = ∂V/∂x and Q =

Zxq dx− ∂V/∂t (3.18)

into the mass conservation equation (3.1):

∂A

∂t+

∂Q

∂x= q

shows that it is identically satisfied! This might have been expected, as the equation is a mass conserva-tion equation, and hence for an incompressible fluid it is a volume conservation equation. We have beenable to express both A and Q in terms of a single variable V . Now we go on to use this in the simplifiedmomentum equation (3.10):

∂η

∂x+ Sf = 0.

Firstly, the derivative of the cross-sectional area can be related to the derivative of the stage by

∂A

∂x= B

µ∂η

∂x+ S

¶,

so that the simplified momentum equation becomes

1

B

∂A

∂x= S − Sf , (3.19)

such that if we use the frictional law in general form Sf = Q2/K2:

1

B

∂A

∂x+

Q2

K2− S = 0, (3.20)

and substituting for Q and A in terms of V , from equation (3.18) and as both breadth B and conveyanceK can be written as functions of area we obtain the single equation in the single variable

∂V

∂t+K(Vx)

sS − 1

B(Vx)

∂2V

∂x2=

Zxq dx, (3.21)

in which the only approximation relative to the long wave equations has been that we ignore terms ofO(F 2), such that it will be accurate for F 2 ¿ 1. This equation, which we term the Volume RoutingEquation, might be useful in a range of hydrologic and hydraulic computations, replacing the solutionof the long wave equations. It is a nonlinear partial differential equation which is a single equation ina single variable. From it, deductions can be made about the nature of wave propagation in waterways,which are not as misleading as those from the characteristic formulation of the long wave equations.

Relation to conventional advection-diffusion equations: In the formulation of (3.21) it is not ob-vious that the volume routing equation is of an advection-diffusion nature. We show that here. Considersmall perturbations about a uniform flow of area A0 and discharge Q0:

V = A0x−Q0t+ εv(x, t),

34


where ε is a small quantity. Immediately we have

∂V

∂t= −Q0 + ε

∂v

∂t,

∂V

∂x= A0 + ε

∂v

∂x, and

∂2V

∂x2= ε

∂2v

∂x2,

giving

K(Vx) = K

µA0 + ε

∂v

∂x

¶= K (A0) +

dK

dA

¯0

× ε∂v

∂x+ . . . ,

from a Taylor series expansion about the uniform flow, which has introduced this advective term. We dothe same for B(Vx) but recognise that we only need it to lowest order (it multiplies a Vxx term):

B(Vx) = B (A0) + . . . ,

and now we consider the termsS − 1

B(Vx)

∂2V

∂x2=

sS

µ1− 1

SB(Vx)

∂2V

∂x2

¶

=

sS

µ1− 1

SB(A0)× ε

∂2v

∂x2

¶=pS

µ1− 1

2

1

SB(A0)× ε

∂2v

∂x2+ . . .

¶from the binomial theorem. Substituting all these linearising approximations into (3.21) with q = 0,

−Q0 + ε∂v

∂t+

µK (A0) +

dK

dA

¯0

× ε∂v

∂x+ . . .

¶×pS

µ1− 1

2

1

SB(A0)× ε

∂2v

∂x2+ . . .

¶= 0,

and multiplying out and dropping all terms in ε2 and higher, we get

−Q0 +K (A0)pS + ε

∂v

∂t+pS

µdK

dA

¯0

× ε∂v

∂x

¶+pSK (A0)

µ−12

1

SB(A0)× ε

∂2v

∂x2

¶= 0.

Now, however, as Q0 = K (A0)√S, the first terms cancel, and we are left with all terms of the order of

magnitude ε, which we divide through by to give

∂v

∂t+

µpSdK

dA

¯0

¶| z

c0

∂v

∂x=

K (A0)

2√SB(A0)| z ν0

∂2v

∂x2= 0,

which is precisely the advection-diffusion equation that we obtained previously when we assumed a baseuniform flow from the outset.

Initial conditions: In general for such problems as we might want to use this for, such as real rivers,the initial conditions are important. If we consider the steady state form of equation (3.21) we obtain thecondition (for no inflow) (note that ∂V/∂t is not zero for steady flow, as volume is continually passing,but ∂V/∂t = −Q0 where Q0 is the initial steady flow. Then we have (more easily from equation 3.20):

d2V

dx2= B(Vx)

µS − Q20

K2(Vx)

¶. (3.22)

which is a second-order ordinary differential equation which we have to solve numerically. Note thatthis is just a low-inertia version of the gradually-varied flow equation

dh

dx=

S − Sf1− βF 2

.

This is a bit messier than the conventional formulation, as we have to solve the second-order equationnumerically, which is usually done by introducing a subsidiary variable dV/dx – which is A in ourformulation!

35


Boundary conditions: At an upstream boundary x0 we might have a given inflow as a function oftime Q(x0, t), which, from equation (3.18) gives us what dV/dt is there, such that we have also to solvethe ordinary differential equation dV (x0, t)/dt = −Q(x0, t) there as part of the solution, which is rela-tively simple. At control points, such as a downstream structure we will usually have some relationshipbetween Q and A such as provided by weir formulae, which gives ∂V/∂t as a function of ∂V/∂x there.As part of the solution we will have to differentiate numerically to give the latter and then integrate togive the updated value of V . At open boundaries, where there is no control point, we may simply be ableto apply the partial differential equation as if it were an interior point, although if finite differences werebeing used to evaluate the spatial derivatives a different formula in terms of points to one side wouldhave to be used.

Some results and insights into the real and simulated nature of wave motion: Consider as atest case a pool 7km long, bottom width of 7m, batter slopes of 1.5:1, a longitudinal slope of 0.0001, atarget depth of 2.1m at the check gate, and Manning’s n = 0.02. To perform the simulation we adoptthe general conditions of their Test 2-1, with an initial flow of 10m3 s−1. The inflow was increasedby 25% in 15 minutes. We developed a program to solve the volume routing equation using similarapproximation methods to the full model, with cubic spline approximation along the canal and simpleEuler forward time stepping but with Richardson extrapolation. We embedded it in a full model of thelong wave equations. The two programs simulated conditions in the canal for several hours, with anovershot weir at the check.

10

11

12

0 0.5 1 1.5 2 2.5 3

Dis

char

ge (c

u.m

/s)

Time (hours)

InflowDynamic modelVolume routing

Figure 3-4. Discharge hydrographs showing inflow and the outflow calculated from the full equations (3.2) and(3.7) and from the volume flow routing equation (3.21).

0

1

2

3

4

5

6

7

8

0 0.5 1 1.5 2 2.5 3

Rel

ativ

e w

ater

leve

l (cm

)

Time (hours)

Dynamic modelVolume routing

Figure 3-5. Variation of water level at the gate calculated from the full equations (3.2) and (3.7) and from thevolume flow routing equation (3.21).

To demonstrate the behaviour of the canal initially the flow from the headworks was increased uniformly

36


over 15 minutes and a constant downstream gate opening was maintained. After the flow and surfacelevel started to increase downstream, the check gate was brought up to the required full flow in 15 minsunder idealised control, and thereafter required delivering precisely the required increased flow. Figure3-4 shows the resulting flow hydrographs. It can be seen that in this canal with a relatively mild slope andmoderate friction that the outflow hydrograph is very different from the inflow hydrograph. The effectsof the diffusion-like term with the second derivative in the volume routing equation (3.21) are strong.The time when the dynamic model first showed some effect downstream (about 1/2 hour) correspondedclosely to the calculated travel time of a dynamic wave, showing that the forerunner of the motion was adynamic wave. However, the bulk of the motion is a relatively slow-moving kinematic-diffusion wave,which the approximate model closely predicts. Until the downstream gate was opened at 1.5 hours, onlyabout half of the increased flow had arrived. Calculations based on the kinematic wave speed showed anexpected travel time of about 2.25 hours, and from the figure it is clear that this is a more representativetravel time for the whole increase of flow. What is also obvious, of course, is how efficacious the openingof the downstream gate was in bringing the flow up to required levels. Relying on the slow movementof the flow transients is not enough.

In our simulation, as described, we suddenly opened the gate and thereafter maintained the desired flow.Under such conditions, what might now be a concern is the behaviour of the water level at the gate. Thisis shown in Figure 3-5. It can be seen that there is an initial period, corresponding to a time of rapidchanges in the flow, when there was a noticeable disagreement between the two models. However, theapproximate model did describe well the main feature of the flow, the increase of flow and surface heightas the slow-moving kinematic wave approached. After the gate opened suddenly, as one would expect,the level was quickly drawn down, and again this is described by the approximate model. The volumerouting method seems to be capable of simulating the behaviour of the pool, with some errors whererapid flow changes occur, but the overall behaviour of the movement of water masses and surface levelbehaviour are described satisfactorily. Numerical simulation using the volume routing equation (3.21)is very much simpler than that using the full equations.

4. Computational hydraulics

4.1 The advection equationWe will consider some special cases of the advection-diffusion equation, as they provide us with anumber of insights. Consider the equation with no diffusion, known as the advection equation.:

∂φ

∂t+ u (x, t)

∂φ

∂x= 0, (4.1)

where φ (x, t) is some passive scalar, and u (x, t) is a velocity, possibly a wave speed.

A typical problem is to solve the advection equation when we know φ (x, 0), that is, the distribution ofφ at some initial time, and we also know what φ (0, t) is, namely how it is varying at a boundary. Wewant to obtain the solution for all x and t.

4.1.1 Exact solution for constant velocity

In the case of a constant velocity u(x, t) = U , the equation has a simple analytical solution φ(x, t) =f(x−Ut), where the function f(t) is given by the history of φ at the upstream boundary, f(t) = φ(0, t),and to obtain the value at any general place and time (x, t) we just substitute f(x − Ut). The solutioncorresponds to a simple ”wave” travelling at a speed of U . We can easily verify that this is the solution,for

∂φ

∂t=

df

d(x− Ut)× ∂(x− Ut)

∂t= −U × f 0(x− Ut)

∂φ

∂x=

df

d(x− Ut)× ∂(x− Ut)

∂x= f 0(x− Ut),

37


where f 0(x − Ut) = df(x − Ut)/d(x − Ut). Substituting these values into equation (4.1) shows thatit is satisfied exactly. Figure 4-1 shows the exact solution of a triangular wave being advected with nodiffusion.

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2

x

Figure 4-1. Exact solution of advection equation for triangular wave

4.1.2 An advective numerical scheme

In situations where the velocity is not constant, then numerical solutions have to be made. It is rare thatsuch a simple equation has to be solved numerically, but here we include numerical schemes as mod-els for rather more complicated problems. The previous exact solution scheme suggests the followingscheme:

φ(x, t+∆) = φ(x− u(x, t)∆, t) +O(∆2),

where the O(∆2)means that neglected terms are of the order of∆2. This is an advective scheme, whichattempts to build in the nature of the solution. It can be interpreted as

To obtain the solution at some point x at a later time t +∆, take the known value of the velocityat (x, t), namely u(x, t), and at a distance upstream given by this velocity times the time step,interpolate the value.

In the case of a constant velocity u(x, t) = U this would be exact, for the value at (x, t+∆) is preciselythat which was upstream at (x− U∆, t). However, if the velocity is variable, it is not exact, and errorsare proportional to the square of the time step.

Such advective schemes are to much to be preferred in fluid mechanics, hydraulics, and hydrology.Schemes which do not incorporate the advective nature of the solution can have some very unpleasantcharacteristics, as we now demonstrate.

4.1.3 The simplest and most obvious finite difference scheme: Forward Time, Centre Space(FTCS)

Finite difference approximations to derivatives are used throughout engineering to provide numericalsolutions of partial differential equations. Here, instead of using the advective scheme above we adoptthe typical types of approximations to the derivatives used in finite difference approximation. Using theforward time and centre space approximations, we substitute into the advection equation (4.1):

φ (x, t+∆)− φ (x, t)

∆| z Forward approximation to ∂φ/∂t

+ u (x, t)φ (x+ δ, t)− φ (x− δ, t)

2δ| z Centre approximation to ∂φ/∂x

= 0,

38


and rearranging gives the FTCS scheme for computing the updated value at (x, t+∆):

φ (x, t+∆) = φ (x, t)− u (x, t)∆

2δ[φ (x+ δ, t)− φ (x− δ, t)] , (4.2)

so that the scheme can be represented as ”calculate the centre difference approximation ∂φ/∂x ≈(φ (x+ δ, t) − φ (x− δ, t))/2δ, and then calculate the change in value at x by calculating the distanceu∆ and the change u∆× ∂φ/∂x”. This can be interpreted as in Figure 4-2.

δ+xxδ−x

Approximation to derivative at x

Δu

Updated value

Figure 4-2. Interpretation of FTCS scheme showing how φ(x, t+∆) will be greater than φ(x, t)

We have deliberately drawn this such that the quantity has a maximum at x. This more clearly showsthat when the solution is updated, the value at t +∆ will be greater than the previous maximum. Thissuggests that the scheme will be unstable, as maxima will grow. This phenomenon is well-known innumerical methods for solving partial differential equations.

Figure 4-3 shows such a numerical solution for an initially triangular distribution for C = u∆/δ = 0.75,the same problem as in Figure 4-1, but here solved numerically. The parameter C is an important one incomputational hydraulics, the Courant Number, which expresses how far the solution should be advectedin a single time step relative to the space step. In this case, the solution should be carried 3/4 of a spacestep in a time step. We have found that this simple and obvious scheme is unstable, and is unable to beused at all, as was suggested by Figure 4-2.

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2

x

Figure 4-3. Unstable numerical solution with FTCS scheme and C = 0.75

4.1.4 Summary of schemes for the advection equation

39


Figure 4-4. Physical representation of three computational schemes for solving the advection equation.

The advection equation as presented in equation (4.1) is

∂φ

∂t+ u (x, t)

∂φ

∂x= 0, (A1)

and we have shown that solutions to this show the behaviour of a travelling wave. Scheme 1 belowapproximates this behaviour. We will see, however, that most schemes (such as 3 and 4 below) do notbuild this behaviour in, but rather just approximate the time and space derivatives.

In Figure 4-4 are shown the physical natures of common approximation schemes for the advectionequation. At a certain time step t three computational values at x− δ, x, and x+ δ are shown by solidcircles. At the next computational point time t+∆ three updated values of the solution at x are shown,according to three schemes described below, each taking information from x− u∆.

1. Advection scheme: The advection scheme

φ(x, t+∆) = φ(x− u(x, t)∆, t), (A2)

incorporates the travelling wave behaviour, saying ”interpolate the solution at a point x−u∆, and thatwill be the updated value of φ at x at time t+∆”. An interpolation of the points is shown by the solidcurve. The value of φ at (x − u∆, t) is obtained from the interpolation, which, from equation (A2),gives the value at (x, t + ∆).This updated solution is shown in Figure 4-4 by an open circle with asolid line. It can be shown that the scheme is always stable as the scheme mimics the real physicalbehaviour expected, of a translating wave. The interpolation can be done by any scheme – a simpleone here would be to fit a quadratic to the three solid points shown. The lecturer prefers using cubicsplines, which are a very powerful way of using a series of cubics to do the approximating.

2. Forward time-stepping (FT) schemes: Most other schemes do not exploit the travelling-wave na-ture of the solutions, but rather just approximate all the derivatives of the partial differential equation.Forward time stepping schemes all approximate the time derivative as shown in equation (4.2):

φ (x, t+∆)− φ (x, t)

∆+ u∆

∂φ

∂x(x, t) ≈ 0,

but they vary as to how the space derivative is evaluated. This can be re-written as the scheme

φ(x, t+∆) ≈ φ(x, t)− u∆∂φ

∂x(x, t). (A3)

This scheme can be interpreted as ”the change in φ is equal to −u∆ times the approximation to thederivative”, or, ”travel along the line with gradient that of the approximation back a distance u∆, andthat is the updated value”. It is interesting that this is the first-order Taylor expansion in x of thepotentially-exact scheme, equation (A2). Now we consider two such schemes. These are traditionallymuch more common throughout computational hydraulics than advection schemes. One wonders why.

3. FTCS and other schemes which approximate the derivative accurately: Here we consider a fam-ily of schemes which approximate the derivative in x accurately, including the ”Forwards-Time-Centred-Space” scheme considered in §4.1.3. While that spatial approximation is accurate, the time-

40


stepping scheme is only first-order, and not particularly accurate. To evaluate the derivative accu-rately a high-order scheme using splines or Fourier series or a centred space scheme could be usedas in §4.1.3. For our purposes here it doesn’t matter which scheme is used. In Figure 4-4 at x thelong-dashed line shows such an accurate approximation to the local gradient of the curve, and thelong-dashed circle shows the solution taken from such a scheme. It can be seen that the updated solu-tion, near the function maximum that we are examining, is actually higher, and the wave has grown inmagnitude. This suggests that such a scheme is unstable – which we can show mathematically is thecase. In fact, all such schemes are unuseable for any value of u∆ because in Figure 4-4 it can be seenthat the tangent is always above the interpolating function!

4. FTBS scheme: Finally we consider a simple Forwards Time Backwards Space scheme, where thederivative is approximated by a backwards difference approximation, as shown in Figure 4-4. Theshort-dashed line shows the backwards difference approximation to the gradient, and the correspond-ing updated point is the short-dashed circle. It can be seen that the solution is now lower than theaccurate advection solution. This shows the phenomenon of numerical diffusion, due to such a poorlevel of approximation. To a lesser degree it occurs in many computational schemes.If we were free to choose u∆ = δ the solution would be exact, as the point we would update fromis the exact solution at x − δ. However, u is usually a function of time and space and this cannot besatisfied at all points. If we were to take u∆ > δ, then as can be seen on Figure 4-4 the gradient lineis now above the exact solution, and the scheme would be unstable.It is common for this limitation to occur in computational schemes. It is called the Courant-Friedrichs-Lewy criterion, and introduces the Courant number C:

C =u∆

δ6 1 for stability,

whose essential meaning is ”for stability, the computational wave in a single time step should not travelmore than a single space step”.

4.2 The diffusion equationThe diffusion equation, obtained when the advective velocity of the medium is zero, is

∂φ

∂t= ν

∂2φ

∂x2, (Diffusion Equation)

and is well-known to describe many physical quantities in nature, including the flow of heat.

The commonly-used Forward Time Centre Space scheme: The best-known numerical schemeis where the time derivative in the diffusion equation is approximated by a forward difference, and thediffusive term by a centre-difference expression. We obtain

φ (x, t+∆)− φ (x, t)

∆= ν

φ (x+ δ, t)− 2φ (x, t) + φ (x− δ, t)

δ2,

which gives the scheme

φ (x, t+∆) = Dφ (x− δ, t) + (1− 2D) φ (x, t) +Dφ (x+ δ, t) , (4.3)

in which D is the computational diffusion number D = ν∆/δ2. This is widely used, notably in civil en-gineering, to solve the consolidation equation in Geomechanics, which is simply the diffusion equation.

It is shown in the Appendix that this has a conditional stability, such that

D =ν∆

δ26 1

2

for stability. In fact, in most cases this criterion is not that which governs the time step, rather, it isaccuracy considerations. Insight into this is gained by considering Figure 4-5 which shows the results

41


for an initial concentration of 1 at the centre point and 0 for all others (a finite equivalent to Figure A-4).Taking a value of D = 1/8 and performing four steps (to νt/δ2 = 1/2) of the scheme (4.3) gives thediffused but peaked shape shown, which we have seen above is in accordance with what we expect. Inone step of the limiting case D = 1/2 to the same point in time the solution has ”snapped through”far too much and gives a physically nonsensical result shown. Clearly, accuracy rather than stabilitydetermines the desirable step size. Of course, there are many other schemes which could be tried.

Figure 4-5. Results for solving the diffusion equation numerically

4.3 Advection-diffusion combinedWe consider some of the fundamentals and some possible methods.

4.3.1 Forward Time Centred Space scheme

For the full advection-diffusion equation we obtain

φ (x, t+∆)− φ (x, t)

∆+ u(x, t)

φ (x+ δ, t)− φ (x− δ, t)

2δ= ν

φ (x+ δ, t)− 2φ (x, t) + φ (x− δ, t)

δ2,

which gives the scheme

φ (x, t+∆) =

µD +

1

2C

¶φ (x− δ, t) + (1− 2D) φ (x, t) +

µD − 1

2C

¶φ (x+ δ, t) ,

in which C is the Courant number C = u∆/δ, expressing the relative amount by which the solution isadvected in a time step, and D is the computational diffusion number D = ν∆/δ2, expressing the effectof diffusion.

Performing a Von Neumann stability analysis (such as outlined in the Appendix), after considerabledifficulty, it can be shown that for stability, two criteria are obtained. The first is a limitation on thecomputational number D:

ν∆

δ26 1

2,

which is independent of the flow velocity, and is the same as we obtained above for pure diffusion aswell. The second becomes

u2∆

ν6 2,

and it can be seen how difficult and strange the behaviour of advection and diffusion can make numericalschemes. To satisfy the first criterion, the time step allowed is inversely proportional to diffusion, themore diffusion, the smaller the time step, which feels reasonable. However, to satisfy the second crite-rion, the allowable time step is proportional to the amount of diffusion, thus, strangely, the less diffusion

42


there is, the smaller is the time step allowed for stability, and in the limit of vanishing diffusion, thescheme is unconditionally unstable, as we have already discovered!

4.3.2 A simple advection-oriented scheme

The previous results suggested that the combination of advection and diffusion can be difficult to com-pute. Many of these difficulties are overcome if the advective nature of solutions are incorporated. Asimple scheme which the lecturer advocates is simply using the advective nature, writing the scheme

φ (x, t+∆) =

µ1 + ν∆

∂2

∂x2

¶φ (x− u∆, t)

= φ (x− u∆, t) + ν∆∂2φ

∂x2(x− u∆, t) .

This can be interpreted as ”interpolate to find the value of φ upstream a distance u∆ as well as its secondderivative there, and combine them as shown to give the updated value”.

Stability analysis: Using the von Neumann stability analysis (see the Appendix), we obtain astability criterion which is similar to that for the pure diffusion equation. By incorporating advection”exactly” we have overcome any difficulties with the combination of advection and diffusion as wefound above. If we used the FTCS scheme for the diffusion part we would obtain the same criterion asfor the pure diffusion case.

4.3.3 An advection-oriented scheme – the Holly-Preissmann method

This is described in some of the references, such as Sauvaget (1985), but is rather too complicated topresent here.

5. Water quality

5.1 Useful sources for further readingTable 1.4 shows some of the many references available, some which the lecturer has referred to in thesenotes or in his work. Those at the head of each section tend to be more important, accessible etc., andfor books the University of Melbourne Library and Reference Numbers are given. The details are givenin the References section at the end of these notes.

5.2 Water quality characteristicsThere are many characteristics which can be used to describe the quality of water in a river:

Physical: temperature, total dissolved solids (TDS), suspended solids, turbidity, colour and odour.Inorganic: pH, hardness, conductivity, nutrients, heavy metals and trace elements.Organic: biochemical oxygen demand (5 day and ultimate BODu), chemical oxygen demand.Gases: oxygen O2, carbon dioxide CO2, hydrogen sulphide H2S.Biological: micro-organisms including Escherichia coli (E-coli), pathogenic organisms and algae.

This list is not exhaustive. Acceptable standards for water quality vary depending on whether water is tobe used for domestic, agricultural or industrial purposes; or whether the aim is to maintain the biologyof a river.

5.3 Types of pollutantThe following is substantially taken from Fischer et al. (1979):

43


The following list is arranged in order of hazard, starting with the least dangerous, however for Aus-tralian conditions, what is least hazardous might be among the most environmentally damaging, namelycommon salt, and we have modified some comments.

Natural inorganic salts and sediments: These materials are not toxic in small concentrations andonly become possible pollutants in excessive doses. In the case of common salt in Australia, this is avery important case.

Waste heat: Once-through cooling systems for electricity generating plants use water for carryingaway large quantities of low-grade waste heat. Aquatic life can be severely affected – for example theextensive investigations for minimising effects of waste heat from Huntly Power Station on the WaikatoRiver in New Zealand.

Organic wastes: Domestic sewage containing ecosystem materials such as carbon, nitrogen, andphosphorous, can cause bad smells and nuisances. After treatment and dispersion (of varying degrees) itis considered acceptable to assimilate these materials into large water bodies. The biochemical oxygendemand (BOD) may be sufficiently reduced so that it can be satisfied by the natural dissolved oxygen inthe water body.

Trace metals: These, often heavy metals such as lead, mercury, and cadmium, are naturally presentin small amounts, but wastewater can have high concentrations.

Synthetic organic chemicals: These are slow to degrade in the environment and are often bio-accumulated in the food chain. Wastewaters may have high initial dilution, but the food chain canmultiply the concentration by several orders of magnitude. Biological processes can do the opposite ofthe physical process of turbulent mixing which reduces the concentration.

Radioactive materials: Long term storage of these is necessary, and the possibility of leaking intowater must be guarded against severely. In the northern hemisphere, salt mines are often chosen. Thenorthern hemisphere also likes the look of dry parts of the southern hemisphere.

Chemical and biological warfare agents These are designed to be toxic at very small doses. Theyhave a habit of returning to damage the agents who used them (Agent Orange).

5.4 Mass balance conceptsRegulatory systems usually distinguish between point and non-point sources of pollutants. A pointsource is usually the discharge from a structure which is designed for the outflow of wastewater. Excep-tions include the accidental spill of oil from a ship and the release of radioactive wastes from a powerplant. Most laws and regulations for water pollution control concern point sources. It is not usuallypossible to treat pollution from non-point sources. Examples include the runoff of salts and nutrientsfrom agriculture, soil erosion, acid rain, and street drainage.

5.5 Impacts of human worksA partial list of damage caused by traditional approaches is:

• Human-made reservoirs may cause deterioration in water quality because of summertime thermalstratification associated with oxygen depletion in the lower layers.

• The reservoirs raise the head on groundwater which may cause it to seep to the surface and bringpollutants.

• Diversion of water for various uses or to other watersheds reduces river flow and its ability toprovide flushing and to provide a satisfactory environment for organisms.

44


• Conveyances such as canals can transport large quantities of dissolved salts, sediment, nutrientsand parasites to places that would not otherwise receive such doses.

• Agricultural drainage systems may greatly accelerate the leaching of nutrients and salts from theland into natural hydrologic systems.

• Breakwaters for harbours interfere with natural circulation which could otherwise carry away pol-lutants.

• Estuarine modifications can radically change circulation patterns with dire consequences for flush-ing of pollutants

• Most coastal protection works of a ”hard ” nature cause environmental damage somewhere, evenif they perform the local task they were required. Sea-walls and groynes often lead to the loss ofsand from beaches, an environmental problem of another kind.

Sometimes it can be very difficult to turn around traditional culture, especially where that other ubiq-uitous substance money is concerned. However tradition and personal interest are powerful factors aswell. Morgan (1971, Monash University: Hargrave-Andrew Library 624 M847D) describes the en-trenched culture of the US Army Corps of Engineers, responsible for many works in the US. Goldsmithand Hildyard (1992, UniMelb Engin f 333.731 GOLD) describe at length the effects of dams around theworld.

5.6 Transport processes

Advection: Transport by an imposed current system, as in a river or coastal waters.

Convection: Vertical transport induced by hydrostatic instability, such as the vertical flow above aplain on a hot day.

Diffusion (molecular): The scattering of particles by random molecular motions, which may bedescribed by Fick’s law and the classical diffusion equation. Viscosity is an effect of molecular diffusion.

Diffusion (turbulent): The random scattering of particles by turbulent motion, considered roughlyanalogous to molecular diffusion, but with ”eddy” diffusion coefficients, much larger than moleculardiffusion coefficients.

Shear: The movement of fluid at different velocities at different positions.

Dispersion: The scattering of particles or a cloud of contaminant by the combined effects of shearand transverse diffusion (to be explained later).

Mixing: Diffusion or dispersion as described above; turbulent diffusion in buoyant jets and plumes;any process which causes one parcel of water to mingle with or be diluted by another.

Evaporation: The transport of water vapour from a water or soil surface to the atmosphere.

Radiation: The flux of radiant energy, such as at a water surface.

Particle settling: The sinking (or rising) of particles having densities different from the ambientfluid, such as sand grains or dead plankton.

Particle entrainment: The picking up of particles such as sand or organic detritus from the bed of awater body by turbulent flow past the bed.

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For pollutant analysis, fluctuations and irregularities are just as important as the mean flows.

5.7 Tools for problem solving

5.7.1 Order of magnitude analysis

Fischer et al. (1979, p12) write: ”For any mixing problem a skilful analyst should be able to work out arough approximation for the solution within a fraction of an hour!”. Such a process is known as ”order ofmagnitude” analysis. It is reminiscent of the wager that the physicist Richard Feynman (Feynman 1985)used to make, that he could solve any physical problem to within 10% in one minute! Such problem-solving skills are important for environmental engineers. Problems can be broken up into sub-models.Much time and effort which might otherwise be wasted can be saved by this process.

5.7.2 Numerical techniques

A numerical calculation can be no better than the validity of the underlying approximations made whenrepresenting a complex process by mathematical equations. It is still necessary to make assumptionsbased on judgement and insight, and to leave out those processes which have little effect on the results.

A wide variety of numerical methods can be applied to mixing problems, which usually means thenumerical solution of differential equations, whether ordinary or partial. Typically, finite differenceor finite element methods are used, although the fundamental equation for transport by a flow wherediffusion occurs, the advection-diffusion equation, shows a number of surprises and pathologies whennumerical solution is attempted. Stochastic (”Monte Carlo” methods) may be used to simulate diffusion,which is the like simulating the like. In general, the methods which are most robust and successful arethose where the solution method incorporates or mimics the nature of solutions.

5.7.3 Hydraulic models

These are necessary in some situations where the physical problem is sufficiently complicated. Also,they reveal physical phenomena which the relatively simple equations used cannot reveal. Variousphenomena revealed by experiments include: large scale vortices, internal waves and hydraulic jumps(caused by fluid stratification), multilayer shear flows, blocking, doubly-diffusive convection, etc..

5.7.4 Field studies

In cases of dispersion of pollutants it is often difficult to model all the physical scales adequately. Fieldexperiments are very important. Sometimes Lagrangian-type experiments, such as following drogues ordrifters (or in the case of a famous experiment, two floating pieces of parsnip) to track flow trajectoriesand dispersion. However, tracking is difficult, and fixed-location apparatus gives data at a reasonablecost.

5.7.5 Mixed approaches

For large complicated situations a careful interweaving of all the above is probably the best way toproceed. Each piece of a problem should be done in the most practical way – bearing in mind theaphorism that ”there is nothing quite so practical as a good theory”. The final synthesis of a variedapproach will undoubtedly be better than by depending on only one approach. Each different approachilluminates the others.

5.8 A simple river model – organic wastes and self purification

5.8.1 Dissolved oxygen and Biochemical Oxygen Demand (BOD)

The presence of dissolved oxygen in river water is essential for maintaining plant and animal life ina river. When wastes (such as domestic sewage) containing organic material enter a river, they are

46


decomposed (oxidised) by millions of bacteria. These depend for their survival on a good supply ofoxygen in the water. This natural purification process reduces the dissolved oxygen level in the riveruntil the rate of oxygen absorption from the air (and from oxygen-producing aquatic plants) exceeds therate of oxygen demand from the waste.

Dissolved Oxygen (O2): The concentration of dissolved oxygen is a measure of the health of a riverand low oxygen concentration is the most important indicator of the presence of easily decomposed or-ganic waste material. Continuous recording of the O2 content can yield as much significant informationas a stage recording gauge.

Saturation concentration, Cs, of dissolved oxygen depends on temperature, and for fresh water, Elmore& Hayes (1960) developed the formula

Cs = 14.652− 0.41022T + 0.0079910T 2 − 0.000077774T 3 (5.1)

where Cs is given in mg l−1 or g m−3 and T is the stream temperature in oC. Some typical values givenby this formula are shown in Table 5-1. Values of Cs decrease with increasing salinity (total dissolvedsolids) and for seawater, values are approximately 20% less than the values given here.

Table 5-1. Dependence of saturation concentration on temperature

T (oC) 5 10 15 20 25Cs 12.79 11.27 10.03 9.02 8.18

Since oxygen is essential to aquatic life it is important to set standards for the minimum level of dissolvedoxygen necessary to maintain the biology a particular river reach. A value of 5 g m−3 might be typical,although for colder rivers, such as trout streams, the standard level could be increased to 6 g m−3.

Biochemical Oxygen Demand (BOD): The impact of a pollutant on a river is normally measured asan oxygen demand (usually BOD) which is a gross measure of the concentration of oxidisable organicmaterial. Most organic materials are biodegradable and the amount of oxygen used in the metabolism ofbiodegradable organics is known as the “biochemical oxygen demand or BOD”. The units of BOD aremg l−1 or g m−3.

There are two phases of the BOD reaction. The first involves the oxidation of carbonaceous organicmaterial, with the end products being carbon dioxide (CO2), ammonia (NH3) and water (H2O). Thesecond phase includes the biological oxidation of ammonia to nitrate. This second (or nitrification)phase begins much later than the first phase and the oxygen demand is lower than the normal process ofre-aeration in nature. These are shown in Figure 5-1. Unless specified otherwise, BOD values denote ameasure of oxygen required for oxidation of carbonaceous organic material only.

Traditionally the standard BOD test has been the 5 day BOD, (BOD5). This test gives a measure ofthe amount of oxygen required over five days by bacteria involved in oxidising a sample of a particularwaste under laboratory conditions at 20oC.

Although the 5 day, 20oC BOD (BOD5) is widely used, it has a number of serious deficiencies. The 5day period does not usually correspond to the point where all of the waste has been consumed. It wasadopted at the beginning of the century in the United Kingdom where rivers do not have a flow time tothe sea greater than five days, and summer temperatures are less than 20oC.

In practice the ultimate BOD, BODu, is of most interest. Sometimes BOD20 is used to estimate BODu.

5.8.2 Biological self purification – the oxygen sag model of Streeter & Phelps

Biological self-purification implies that a body of water such as a river is automatically able to absorband remove the organic compounds of effluents or wastes. Here the properties of the water revert to theconditions upstream of the waste input point after a certain length of flow or travel time. The methodsof analysis used to determine the capacity of a river to assimilate organic pollution essentially began

47


Figure 5-1. Typical oxygen demand curve of aerobic decomposition of organic matter (after McGauhey, 1968)

with the simple classical model of Streeter and Phelps in 1925. In the Streeter-Phelps model, the BODand dissolved oxygen profiles along a river are based on the assumption that there are only two majorprocesses taking place:

• bacterial oxidation of the organic matter, which is responsible for the removal of BOD and oxygen;and

• reaeration at the water surface.

Firstly we consider the processes at work.

Deoxygenation in Rivers: The oxygen in rivers is depleted by the bacterial oxidation of suspendedand dissolved organic matter discharged into them. (There can also be oxygen depletion by the organ-isms found in the benthic - bottom - deposits on a stream bed, but this aspect will be neglected here).The amount of oxygen required to stabilise a waste is normally measured by a BOD test, and BOD istherefore the primary source of oxygen depletion in a river.

Here we assume a simple first order model, that the rate of uptake of oxygen (rate of deoxygenation) isproportional to the amount of organic matter available:

d

dt(BODr) = −k1 (BODr) (5.2)

where BODr¡g m−3

¢is the BOD remaining at time t, a measure of the content of easily degradable

organic substances, and k1(d−1) is the coefficient giving the rate of BOD deoxygenation.

Typical values of k1 are 0.2 to 0.35 d−1. It depends on the state of the waste material and the degreeof treatment. For example, for raw municipal waste water k1 might typically be 0.35, while for treatedwaste water k1 might equal 0.25.

The values of k1 are also temperature dependent. The temperature effect on k is usually expressed as

k1 (T ) = k1(20) θT−20

where T is the temperature and k1(20) is the value at 20oC. The values of θ range from 1.03 to 1.05 inthe range of 15o to 35oC.

Solving the differential equation (5.2) by separation of variables, we obtain

BODr = Ae−k1t,

48


where A is a constant. We obtain this from the initial condition such that when t = 0, BODr = BODu,the initial value, giving

BODr = BODu e−k1t. (5.3)

In the standard laboratory test, values of the oxygen used at the end of specified intervals of time aredetermined. Using equation (5.3), and noting that oxygen used is given by BODu − BODr, the valuesof k1 and BODu can be determined from laboratory data.

We introduce the symbol rD for the rate of deoxygenation, which will be a negative quantity. Fromequations (5.2) and (5.3):

rD =d

dt(BODr) = −k1 (BODr) = −k1BODu e−k1t (5.4)

Reoxygenation in Rivers: The main sources of oxygen replenishment in river water are absorptionfrom the atmosphere and photosynthesis of aquatic plants and algae. The term reaeration is used todescribe the atmospheric absorption of oxygen. The rate of reaeration is proportional to the dissolved-oxygen deficiency. The rate of reaeration, rR, can be expressed as

rR = k2 (Cs − CO2) (5.5)

where rR = rate of reaeration, g m−3d−1k2 = reaeration constant, d−1Cs = dissolved-oxygen saturation concentration, g m−3CO2

= dissolved-oxygen concentration, g m−3.

The re-aeration constant can be estimated from the characteristics of the stream and an appropriateformula such as the one suggested by O’Connor and Dobbins:

k2 =294pνO2

U

h3/2, (5.6)

where νO2= kinematic viscosity of oxygen (temperature dependent), m2s−1,

U = stream flow velocity, m s−1, andh = average depth of flow, m.

If, for example, we have a stream with U = 1m s−1 and h = 2 m, and noting that νO2= 1.5 ×

10−5m2s−1 at 20oC, k2 = 0.40 d−1.

The values of νO2at temperatures other than 20oC are given by the relationship

νO2(T ) = νO2

(20)× 1.037T−20. (5.7)

The Oxygen-Sag Model: Here we ignore the effects of dispersion until later. In this way we caninterchange between time and space and assume that space x and time t are simply related by the meanvelocity of flow in the river. The deoxygenation and reoxygenation processes will now be considered inan oxygen mass balance equation, which we now develop, assuming rather unrealistically, that the riverand waste are completely mixed where the waste enters the river and that wastes discharged into theriver are distributed uniformly over the cross-section. In general this assumption is only true at distanceswell downstream of the waste discharge input point.

Initial Mixing: the initial concentration, C0, of a constituent in the river-waste mixture at x = 0 is givenby

C0 =QrCr + qwCw

Qr + qw(5.8)

49


Figure 5-2. Longitudinal section of river of cross-sectional area A with oxygen fluxes crossing boundary of controlsurface. Deoxygenation is internal.

x

t

1

U

),0( 0t

x

Ux /

),( tx

Figure 5-3. Plot on (x, t) axes showing the characteristic (path) a parcel of fluid follows

where C0 = initial concentration of constituent at point of discharge, g m−3Cr = concentration of constituent in river before mixing, g m−3Cw = concentration of constituent in wastewater, g m−3.Qr = river discharge, m3s−1qw = wastewater discharge, m3s−1

Oxygen Balance: Consider, steady uniform one-dimensional flow in a prismatic river channel, a waste-water input discharge with complete mixing within the channel cross-section, and no longitudinal dis-persion, as shown in Figure 5-2. Consider now the mass balance (mass conservation) of the oxygen inthe river for an incremental volume given by A∆x as shown in the figure:

∂CO2

∂t×A∆x| z

Accumulation

= QCO2| z Inflow

− (QCO2+Q

∂CO2

∂x∆x)| z

Outflow

+ rD ×A∆x| z Deoxygenation

+ rR ×A∆x| z Reoxygenation

. (5.9)

Before substituting rD and rR and attempting to solve, it is very helpful to consider the trajectory orpath along which a parcel of mixture takes and then to solve the partial differential equation along thattrajectory. Figure 5-3 shows such a path on (x, t) axes, where the parcel sets out at a time t0. If themean velocity U = Q/A in the stream is constant, the trajectory is a straight line, starting at t = t0, withgradient 1/U , and the equation of the trajectory is

t = t0 +x

U. (5.10)

At time t, the time over which deoxygenation has taken place is t− t0,and so we modify equation (5.4),

50


replacing t by t− t0:

rD = −k1BODu e−k1(t−t0). (5.11)

Substituting equations (5.11) and (5.5), dividing through by A∆x and re-arranging, we have the partialdifferential equation for the dissolved oxygen concentration of a parcel of mixture which started at x = 0at t = t0:

∂CO2

∂t+ U

∂CO2

∂x+ k2CO2

= −k1BODu e−k1(t−t0) + k2Cs. (5.12)

Now, as we want to consider x and t related by the equation of the trajectory, the two are no longerindependent. We enter a general discussion for combinations of derivatives as in equation (5.12), whichoccur throughout fluid mechanics.

The material or advective derivative: At a fixed point in space the rate of change of a quantityφ such as temperature or even the velocity vector, is ∂φ/∂t. However, even in a steady flow field fluidparticles experience a different apparent rate of change by moving to a position where φ has a differentvalue. If a fluid element at x at time t is at x+ δx at time t+ δt, we can write the Taylor expansion forthe value of φ at the new point:

φ (x+ δx, t+ δt) = φ (x, t) + δt∂φ

∂t+ δx

∂φ

∂x+ higher order terms,

hence the rate of change of φ at the fluid particle, denoted by Dφ/Dt is

Dφ

Dt=

lim

δt→ 0

φ (x+ δx, t+ δt)− φ (x, t)

δt=

∂φ

∂t+

lim

δt→ 0

δx

δt

∂φ

∂x

=∂φ

∂t+ U

∂φ

∂x.

The quantity Dφ/dt is the Advective Material Derivative, the time rate of change experienced by a fluidparticle. The first term is the temporal rate of change, which is zero in steady flow, while the secondis the advective contribution, which exists because the particle moves to a point where the velocity isdifferent – it is being transported through a field where there is a gradient in φ at a velocity U . In a fluidflow the rate of change at a particle is different from that at a point.

Solution: Immediately we recognise that we can write equation (5.12) as

DCO2

Dt+ k2CO2

= −k1BODu e−k1(t−t0) + k2Cs, (5.13)

an ordinary differential equation forCO2. For solution we will impose the initial condition thatCO2

(0, t) =C0(t0).

Equation (5.13) is in a form suitable for use of an integrating factor, such that if we multiply both leftand right sides by the quantity I = exp k2 (t− t0), then it can be written:

D

Dt

³CO2

ek2(t−t0)´= −k1BODu e(k2−k1)(t−t0) + k2Cse

k2(t−t0).

Integrating both sides with respect to t (for mathematical solution the advective derivative has the sameproperties as an ordinary derivative):

CO2ek2(t−t0) = − k1

k2 − k1BODu e

(k2−k1)(t−t0) + Csek2(t−t0) +B,

where B is a constant of integration. Satisfying the initial condition gives

C0(t0) = −k1

k2 − k1BODu + Cs +B,

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which can be solved for B and substituted into the solution to give

CO2= C0(t0) e

−k2(t−t0)+Cs

³1− e−k2(t−t0)

´+

k1k2 − k1

BODu

³e−k2(t−t0) − e−k1(t−t0)

´. (5.14)

In this form it is still applicable to the trajectory described above. We can write it such that it is applicableto any (meaningful) value of x and t by substituting t− t0 = x/U and C0(t0) = C0(t− x/U), giving

CO2(x, t) = C0(t− x/U) e−k2x/U + Cs

³1− e−k2x/U

´+

k1k2 − k1

BODu

³e−k2x/U − e−k1x/U

´.

(5.15)This is a generalised version of the Streeter-Phelps oxygen sag equation. In this form the exponentialfunctions on the right are functions of the independent variable x, and time t appears explicitly in onlyone place. To obtain the value of the concentration at a particular place x and time t one evaluates thefunctions of x and computes the corresponding initial value of concentration which was introduced ata time t − x/U , i.e. prior to time t by an amount x/U , the travel time taken to get to the point (x, t)required.

Equation (5.15) seems to have advantages over conventional presentations of the Streeter-Phelps oxygensag equation, which is presented in textbooks only for the case of constant C0 and with time t as anindependent variable – even in the case where the whole problem is steady. The oxygen-sag equation,with its two exponential decay rates, is as shown in Figure 5-4. Biological decomposition begins imme-diately following waste input and uses oxygen from the river with a time scale of k1and a space scaleof k1/U . Atmospheric reaeration is proportional to the dissolved-oxygen deficit and therefore increaseswith increasing deficit giving a gradual mobilisation of the reaeration process with time and space scalesof k2 and k2/U . Eventually a critical point xc is reached when waste decomposition rate equals therate of atmospheric re-aeration. Downstream of this point the rate of re-aeration exceeds the rate ofdecomposition and the dissolved oxygen level begins to increase until finally saturation conditions arereached.

This model is a simple but commonly used one for modelling oxygen resources in a river. It neglects theeffects of algae and sludge deposits on the oxygen balance, as well as, more importantly, all effects ofdispersion due to natural processes including turbulence in the stream. Later we will spend some timeexamining these.

Figure 5-4. Characteristic oxygen-sag curve obtained using the Streeter-Phelps equation

52


Critical dissolved oxygen deficit: The critical or maximum dissolved-oxygen deficit Dc will occurwhen dCO2

/dt = 0. This can be found from equation (5.13) with this condition, leading to

CO2= Cs −

k1k2BODu e

−k1(t−t0) = Cs −k1k2BODu e

−k1x/U .

To find when this occurs we would have to differentiate the solution, equation (5.15) with respect to x.For a constant C0 we obtain

x =U

k2 − k1log

µk2k1

µk2k1− 1¶(Cs − C0)

BODu

¶.

Exercise: Verify that equation (5.15) satisfies(a) the required boundary condition CO2

(0, t) = C0(t), and(b) the original differential equation (5.12) (after substituting t − t0 = x/U on the right side).Hint: you will need the result from the theory of partial differentiation that

∂

∂tC0(t− x/U) = C 00(t− x/U) and

∂

∂xC0(t− x/U) = − 1

UC 00(t− x/U),

where C 00(t− x/U) indicates the ordinary derivative of the function with respect to the argumentshown.

5.9 Salinity in riversSalinity refers to the concentration of total dissolved solids (TDS) in water and is perhaps historicallythe most important water quality characteristic because of its relationship to agriculture. It is a grosschemical characteristic and can be expressed either in terms of:

• Concentration in g m−3 ≡mg l−1 ≡ ppm (parts per million) or

• Electrical Conductivity of the water in MicroSiemens cm−1 at 25oC, designated by μS cm−1 andknown as “EC” units.

The relationship between EC and TDS is given approximately by

1 μS cm−1 or EC unit ≡ 0.64 mg l−1 of total dissolved solids.

5.9.1 Tolerance Levels

Salinity limits have been established for various uses of river water. Although there is no sharp divisionbetween acceptable and unacceptable salinities, it is widely recognised that economic losses increasewith increasing salinity. According to the former Rural Water Commission, Victorian waters can beclassified as follows:

Class 1: TDS 0-175 ppm – low salinity water which can be used with most crops on most soils, withlittle likelihood that a salinity problem will develop.

Class 2: TDS 175-500 ppm – medium salinity water which can be used with plants of medium salttolerance without adopting special practices for salinity control.

Class 3: TDS 500-1500 ppm – high salinity water which cannot be used on soils with restricteddrainage, and even with adequate drainage, the salt tolerance of any plants to be irrigated must betaken into account.

Class 4: TDS 1500-3,500 ppm – very high salinity water not suitable for irrigation under ordinaryconditions.

Class 5: TDS above 3,500 ppm – extremely high salinity water to be used in emergencies.

These TDS levels are only approximate and concentration of particular salts within the total dissolved

53


solids can be important.

5.9.2 Human Consumption

The suitability of water for human consumption depends not only on the salinity level but on the presenceof organisms indicative of harmful pollution. The World Health Organisation has published internationalstandards for drinking water which recommend an upper limit of salinity of 1500 ppm. The limit ingeneral use in Victoria is 2000 ppm if no other water is available. However the desirable maximumsalinity is 835 EC units (530 g l−1 or ppm) according to the Engineering and Water Supply Departmentof South Australia.

5.9.3 Septic Tanks

Bacterial action in septic tanks can be maintained at very high levels of salinity.

5.9.4 Sources of Salinity in Rivers

The salinity of river water can derive from a number of sources.

• Mineral salts are released by rock weathering, collect in the soil, and are washed into streams bynatural runoff.

• Rainfall can carry small concentrations of oceanic salts.

• Saline groundwater can be seeping into a river.

• Where a saline water table rises to within capillary range of the surface accumulations of salt willoccur in a river catchment and will be eventually washed into the river system. The rise in the watertable could be the result of excessive land clearance or unsatisfactory irrigation practices.The salinity level in a river increases in the downstream direction as tributaries add their contri-butions of salt. although high salinity levels occur naturally in some rivers, particularly duringlow flow periods, these levels are increased wherever human activities in the river catchments haveresulted in rising water table levels.

5.9.5 The Murray River

The Murray River has been chosen to illustrate the variation of salinity along a river since it is our majorriver. It supplies water for extensive irrigation areas and water to augment the water supplies of mostSouth Australian towns and cities.

Figure 5-5 shows two longitudinal profiles of salinity along the Murray River from Hume Dam to theriver mouth in South Australia. Both profiles recorded in the early 1970s, indicate that salinity is lowuntil the Loddon River confluence is reached, when highly saline water from Barr Creek enters theMurray. Barr Creek is the main drain of a surface drainage network serving about 125,000 ha of salt-affected farmland in the Kerang region. It was the biggest point source of salt along the river and causeda marked salinity jump. (In recent years measures have been taken to restrict the highly saline flows ofBarr Creek entering the Murray).

The next major input is from the Wakool River (whose main source of salt is from groundwater seepage).Its effect can be masked by significant flows from the Murray which divert down the Edward River anddilute Wakool flows before they enter the Murray. Further downstream, the inflow of the MurrumbidgeeRiver provides a significant diluting effect.

Near Merbein, and in the 20km between Mildura and Merbein, major salt inputs occur as a result ofseepage of highly saline groundwater. Inputs of highly saline groundwater are the dominant source ofsalt input in the lower reaches of the Murray River. The salinity of the groundwater derives from the lateTertiary Era when a gulf of sea extended eastwards into the Murray Basin as far as Swan Hill, coveringmuch of today’s Mallee Region. When the sea began to retreat about one million years ago it left soiland groundwater salty.

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Figure 5-5. Longitudinal salinity profiles in the Murray River, after Collett (1978)

The longitudinal profiles shown in the figure illustrate salinity levels which are of great concern to SouthAustralia.

5.9.6 Measures for Salinity Mitigation in Rivers

Three important approaches can be identified:

• Restricting the entry of saline water to the river, particularly during critical periods. This couldinvolve for example restricting the entry of highly saline water from tributaries (as in the case ofBarr Creek in the Murray River example) or lowering the water table levels near points where salinegroundwater enters a river.

• Providing flushing flows of good quality water to dilute the concentration of dissolved solids ina river during critical periods. One important effect of river regulation is the improved salinitysituation in low flow periods, particularly in drought years.

• Improving water management in irrigation districts, particularly with respect to drainage and con-trol of water table levels. This aspect is outside the scope of these notes.

5.9.7 Modelling of Salt Transport in Rivers

The impact of salinity mitigation schemes and river operating policies for reducing salinity can be ex-amined using an appropriate numerical model describing salt transport down the river system. Sincesalt transport is totally dependent on the movement of water through the system, it is first necessaryto predict the velocity field before solving for the distribution of salt concentration. a one-dimensionalapproach is usually adopted for the model.

One approach is to start from the partial differential equations governing the flow and the distribution ofsalt concentration, and then to solve these numerically by converting them into algebraic equations fordiscrete points or elements. A finite-difference approach is usually adopted. Care must be exercised in

55


the choice of an appropriate discretisation scheme, since numerical methods can introduce “numericaldiffusion” which may distort the solution.

The basic one-dimensional flow equations are the Saint-Venant or Long Wave equations, which we willnow develop. These are widely used for studies of the movement of water in waterways, including flowsand floods in rivers, and the delivery of water in irrigation channel systems.

5.9.8 Further Reading

The articles by Collett (1978) and Keiller & Close (1985) are interesting.

6. Turbulent diffusion and dispersionWe consider flows which are reasonably steady (albeit turbulent) and have uniform geometry. For thesecases, we can use a Fickian-type of law for diffusion due to turbulence. Initially as an introduction, letus consider a local velocity field which is turbulent. We will only consider the x component, so that wewrite

u = u+ u0,

where u is the time-mean velocity at a point, and u0 is the fluctuating velocity, such that the time meanof u0 is zero, u0 = 0. Similarly, we write an expression for concentration of a pollutant

c = c+ c0,

and we want to calculate the flux of the pollutant, uc

uc =¡u+ u0

¢ ¡c+ c0

¢= u c+ u c0 + c u0 + u0c0.

However, we do not want to study the fluctuating values of this, and so we take the time mean:

uc = u c+ uc0 + cu0 + u0c0

= u c+ u c0 + c u0 + u0c0

= u c+ u0c0,

and so we have the possibly-surprising result that the mean of uc is not the mean of u times the meanof c, but contains the extra turbulent transport flux u0c0. Of course, there are other such quantitiesin the equations of hydraulics – notably quantities such as u02, u0v0, u0w0, etc. in the equations ofmotion. Quantities such as the latter three are known as the Reynolds stresses. One of the most importantproblems in fluid mechanics is to model these turbulent quantities in terms of the mean quantities of theflow.

The turbulent transport fluxes such as u0c0 have several attributes:

• They range in size from fluctuations the size of the flow geometry down to the smallest turbulentscale.

• They are random – individual fluctuations are not repeatable, but the statistical characteristics ofthose fluctuations are.

• The random nature of the flow results in intermittent but large spatial gradients in variables aspackets of fluid with quite different characteristics are brought into contact by the turbulence. Thisenhances molecular diffusion.

• The continuous introduction of packets of high and low concentration by the turbulence will re-sult in an overall dilution of zones of high concentration via turbulent mixing. This is turbulentdiffusion, which is rather more important than molecular diffusion in river flows.

The energy to create turbulence is typically put into the system at scales the size of the overall geometry.

56


Viscosity is the agent ultimately acting so as to dissipate the turbulent energy in the form of heat. Itoperates on the smallest possible scales of the flow field – at a molecular level. In between the cre-ation and dissipation turbulent scales there is an orderly cascade of turbulent energy by which energyis nonlinearly transmitted from the large to small turbulent fluctuations or scales by a continuously de-creasing sequence of turbulent eddies. This has been tersely described by L. F. Richardson, the father ofcomputational fluid mechanics, paraphrasing Jonathan Swift:

Big whirls have little whirls that feed on their velocity,And little whirls have lesser whirls and so on to viscosity - in the molecular sense.

6.1 Diffusion and dispersion in waterwaysWith the exception of free jets such as smoke or exhaust stacks in the atmosphere, most flows encounterand are affected by boundaries. These cause the flow field and its turbulence characteristics to be differ-ent in all three co-ordinate directions. It is anticipated that the eddy diffusivities will be different in eachdirection. The main objective here is to be able to predict the eddy diffusivities in turbulent channel flowand the importance of their differences in magnitude on turbulent mixing in the channel.

6.1.1 Eddy diffusivities

Gradients of momentum and transport are sharper in the vertical and the transverse direction. An ap-proximation obtained by Elder for the mean vertical eddy diffusivity εz is

εz = 0.067u∗D,

where D is the depth, and u∗ is the shear velocity,

u∗ =

rτ0ρ≈pgDS0,

where τ0 is the mean boundary shear stress.

For the transverse eddy diffusivity εy Fischer et al. (1979) suggest

εz ≈ 0.15u∗D,

with a similar result for the longitudinal diffusivity εx:

εx > 0.15u∗D.

The coefficients vary, depending on the channel or estuary type.

6.2 DispersionAn important feature of turbulent diffusion in waterways is that the velocity distribution is not uniformacross the stream. This has the ability to separate individual particles much more quickly than if just tur-bulent processes were at work, and for the effective diffusivity to be considerably greater. This is shownin Figure 6-1. Part (a) shows a hypothetical stream where the velocity is constant across the section.A puff of pollution introduced uniformly across the section is then carried downstream, and turbulentdiffusion plays the role that we see, whereby the concentration distribution is gradually diminished andwidened, according to the principles we have already studied. The top figure shows an instantaneoussnapshot of the pollution cloud, and in the next part of the figure the mean concentration distributionacross the stream is plotted, which will closely approximate a Gaussian. In Part (b) of the figure, how-ever, we see what happens when there is a realistic velocity profile. Some particles near the side do nottravel as far as those in the middle, and the cloud of pollution becomes increasingly distorted, and theextent of the cloud can quickly become large. Now, taking the mean across the stream we see the resultsin the bottom part of the figure, where the mean concentration is now much lower, as is the lateral extent

57


(a) Hypothetical uniform velocity profile

(b) Realistic velocity profile

Plan

Plan

Figure 6-1. The role of the velocity distribution in enhancing dispersion

of the cloud. The mean diffusivity has been considerably increased by the velocity profile.

This phenomenon was first described by G. I. Taylor in 1953. This artifact of spatial averaging is knownas dispersion. The basic operation involved is to obtain a partial differential equation which is derived interms of area-averaged variables. By considering the sort of control volume we used for the derivationof the oxygen-sag equation, Figure 5-2. We can easily show, generalising our original analysis to non-uniform distributions over the section (but with no de-oxygenation or reaeration), that

∂

∂t

ZA

c dA+∂

∂x

ZA

cu dA = 0.

Now, in a similar sense to what we did for time-fluctuating quantities we let

c = ec+ c00,

where ec is the spatial average over a section, and c00 is the amount by which the concentration varies

from that average. Writing a similar expression for u, and substituting, we obtainZA

c dA = ecA, andZA

cu dA = eceuA+ gc00u00A,

giving∂

∂t(ecA) + ∂

∂x(eceuA) + ∂

∂x

³gc00u00A´= 0.

However, eu is simply the mean velocity as we have already used it eu = U = Q/A,and if we expand theterms in this equation and subtract the mass conservation equation

∂A

∂t+

∂

∂x(euA) = 0,

then we obtain∂ec∂t+ eu ∂ec

∂x+

∂

∂x

³gc00u00´= 0,

58


where we have neglected an area derivative in the fluctuating component term (area changes relativelyslowly).

Taylor postulated (see e.g. Fischer et al. 1979) that gc00u00 can be modelled as

gc00u00 = −εT∂ec∂x

,

where εT is the dispersion coefficient. Substituting, and neglecting derivatives of this quantity, we obtain

∂ec∂t+ U

∂ec∂x

= εT∂2ec∂x2

,

the advection-diffusion equation, but where the coefficient of diffusion is the dispersion coefficient εT .

Fischer et al. (1979) and French (1985) give a detailed method for estimating εT . There is a simpleempirical formula:

εT = 0.011U2T 2

u∗D,

where T is the top width, as used previously. This expression is ”correct to within a factor of four”,which is considered acceptable ... For a narrow channel, a typical value (p423 of Streeter et al. 1998) fora mean flow velocity of 0.1 m s−1 is 2.75 m2s−1, and for a wide channel a typical value is 275 m2s−1.

6.3 Non-dimensionalisation – Péclet number and Reynolds number –viscosity as diffusion

Consider a solution of the advection-diffusion equation over a physical lengthL. If we non-dimensionalisesuch that x∗ = x/L, then the computational domain becomes (0, 1), which is convenient. If we alsonon-dimensionalise time with respect to t∗ = t× L/U , then we obtain

U

L

∂φ

∂t∗+

U

L

∂φ

∂x∗=

κ

L2∂2φ

∂x2∗,

and multiplying through we obtain

∂φ

∂t∗+

∂φ

∂x∗= Pe

∂2φ

∂x2∗, where Pe =

κ

UL.

The Péclet number Pe is a dimensionless number which expresses the relative importance of diffusionto advection. It has appeared here as a result of a non-dimensionalisation. Clearly we could use this toexpress rather more concisely solutions to the advection-diffusion equation, as it is the only parametergoverning the solution.

7. Sediment motionThe grains forming the boundary of an alluvial stream have a finite weight and finite resisting ability,including cohesion and coefficient of friction. They can be brought into motion if the forces due to fluidmotion acting on a sediment particle are greater than the resisting forces. Often this is expressed in termsof disturbing and resisting stresses on the bed of the stream. If the shear stress τ acting at a point on theflow boundary is greater than a certain critical value τ cr then grains will be removed from that region,and the bed is said to scour there. We introduce the concept of a relative tractive force at a point, τ/τ cr.If this is slightly greater than 1, only the grains forming the uppermost layer of the flow boundary canbe detached and transported. If τ/τ cr is greater than 1, but less than a certain amount, then grains aretransported by deterministic jumps in the neighbourhood of the bed. This mode of grain transport isreferred to as bed-load. If the ratio is large, then grains will be entrained into the flow and will be carrieddownstream by turbulence. This transport mechanism is known as suspended-load. The total transport

59


rate is the sum of the two.

The simultaneous motion of the transporting fluid and the transported sediment is a form of two-phaseflow. We can write all the variables which should dominate the problem of the removal and transport ofparticles:

ρ Density of water ML−3ρs Density of solid particles ML−3ν Kinematic viscosity of water L2T−1φ Diameter of grain Lg Gravitational acceleration LT−2h Depth of flow Lτ Shear stress of water on bed ML−1T−2

As we have 7 such quantities and 3 fundamental dimensions involved, there are 4 dimensionless numberswhich can characterise the problem. In fact, it is convenient to replace g by g0 = g (ρs/ρ− 1), theapparent submerged gravitational acceleration of the particles, and to replace τ by the shear velocityu∗ =

pτ/ρ. Convenient dimensionless variables, partly found from physical considerations, which

occur are

Θ =u2∗g0φ

=τ

(ρs − ρ) gφ, roughly the ratio of the shear force on a particle to its submerged weight

R∗ =u∗φ

ν, roughly the ratio of fluid inertia forces to viscous forces on the grain

G =ρsρ, the specific gravity of the bed material, and

φ

h, the ratio of grain size to water depth

Two important quantities here are R∗ which is the grain Reynolds number, andΘ the Shields parameter,which can be thought of as a dimensionless stress.

7.1 Incipient motionIn the 1930s Shields conducted a number of experiments in Berlin and found that there was a narrowband of demarcation between motion and no motion of bed particles, corresponding to incipient motion.He represented these on a figure of Θ versus R∗. A slight problem with this is that the fluid velocity (inthe form of shear velocity) occurs in both quantities. It is more reasonable to introduce the dimensionlessgrain size (see p7 of Yalin & Ferreira da Silva 2001):

δ =

µR2∗Θ

¶1/3= φ

µg0

ν2

¶1/3.

Here we consider what δ means. If we take a common value of G = 2.65, plus g = 9.8m s−2,ν = 10−6m2 s−1(for 20C), then we obtain δ ≈ φ × 25000 in units of metres. If φ is specified interms of millimetres then we have δ ≈ 25φ, and so for a range of particle sizes we have

δ 0.1 1 10 100 1000φ (mm) 0.004 0.04 0.4 4 40

Figure 7-1 shows a representation of Shields’ results, using δ for the abcissa. Instead of the experimentalresults we use a formula by Yalin which is an approximation to the results for incipient motion, givingthe critical value Θcr:

Θcr = 0.13 δ−0.392 e−0.015 δ

2

+ 0.045³1− e−0.068 δ

´. (7.1)

Above the line, for larger values ofΘ (and hence larger velocities or smaller and lighter grains), particles

60


0.01

0.1

1

0.1 1 10 100 1000

Dimensionlessshear stress

Θ

Dimensionless grain size δ

Motion

No motion

All beds random on thescale of the particles

Beds flat in laboratories

Beds random in nature

Yalin’s approximation to Shields’ data, eqn (7.1)Bagnold’s conjecture for random beds

Figure 7-1. Incipient motion diagram

will be entrained into the flow. Below the line, particles should be stable. For small particles thereappears to be a linear relationship (on these log-log axes), while for large particles the critical shearstress, based on Shields’ laboratory experiments, approaches a constant value of about 0.045. In betweenthere is a dip in the curve, with a minimum at about φ = 14, corresponding to a grain size of 0.35mm,about a fine sand.

Bagnold (personal communication) suggested that there was probably no fluid mechanical reason forthat, but that there is an implicit scale effect in the diagram, an artificial geometric effect, and suggestedthat the Shields diagram has been widely misinterpreted. He suggested that for experiments with smallparticles, while the overall bed may have been flattened, individual small grains may sit on top of othersand may project into the flow, so that the assemblage is random on a small scale. For large particles(gravel, boulders, etc.) in nature, they too are free to project into the flow, however in the experimentswhich determined the Shields diagram, the bed was made flat by levelling the tops of the large particles.Hence, there is an artificial scale effect, and if one were only to consider random beds of particles whichare free to project into the flow above their immediate neighbours, while the bed might level on a scalemuch larger than the particles. Fenton & Abbott (1977) followed Bagnold’s suggestion and examinedthe effect of protrusion of particles into the stream. Although they did not obtain definitive results, theywere able to recommend that for large particles the value of Θcr was more like 0.01 than 0.045, whichseems to be an important difference, the factor of 1/4 requiring a fluid velocity for entrainment intothe flow of randomly-placed particles to be about half that of the sheltered case. A curve representingBagnold’s hypothesis, as partly borne out by the experiments, is shown on Figure 7-1.

7.2 Relationships for fluvial quantitiesNow we consider some simple relations to relate the above results to real physical quantities. The shearvelocity u∗ is a very convenient quantity indeed. If we consider the steady uniform flow in a channel,then the component of gravity force down the channel on a slice of length ∆x is ρgA∆xS0. Howeverthe shear force resisting the gravity force is τ × P ×∆x. Equating the two we obtain

τ = ρgA

PS0.

61


In this work it is sensible only to consider the wide channel case, such that A/P = h, the depth, giving

τ = ρghS0,

or in terms of the shear velocity:

u∗ =

rτ

ρ=pghS0,

and so in terms of the dimensionless stress:

Θ =u2∗g0φ

=ghS0g0φ

=S0 h

(G− 1)φ,

giving a result which can be used in association with the Shields diagram or equation (3.15) to give theslope, depth, or particle size at which entrainment will occur.

7.3 Dimensional similitudeIn experiments with sediment transport, as in other areas of fluid mechanics, it is desirable to have thesame dimensionless numbers governing both experimental and full-scale situations. In this case wewould like the dimensionless particle size AND the dimensionless shear stress each to have the samevalues in both model and full scale. Using the subscript m for model and no subscript for the full scalesituation, we then should have

δm = δ such that φm

µg0mν2

¶1/3= φ

µg0

ν2

¶1/3,

but as gravitational acceleration g and viscosity ν can be assumed to be the same in each, we can write

φm (Gm − 1)1/3 = φ (G− 1)1/3 .

This means thatφmφ=

µG− 1Gm − 1

¶1/3. (7.2)

Obviously if we use the same material for the model as for the full scale, Gm = G, and we have φm = φ.But if we use a lighter material in the model we will have to have φm > φ.

Also we require the same dimensionless shear stress for the model and the full scale:

S0m hm(Gm − 1)φm

=S0 h

(G− 1)φ.

However, if we eliminate particle size using the previous relation (7.2) then we obtain

S0m hmS0 h

=(Gm − 1)2/3(G− 1)2/3

.

We can see that if we use the same material, the right side is unity and we have S0m hm = S0 h, so that ifwe use a model scale of, say, hm/h = 1/10, then we have S0m = 10S0, which is fortunate, as shallowslopes are difficult to arrange in a laboratory. On the other hand, if we use a rather lighter material inthe laboratory, as is often the case, then we will require S0m hm < S0 h, and possibly end up requiringa laboratory slope similar to that in the field, which might be very difficult to arrange.

7.4 Bed formsIn most practical situations, sediments behave as non-cohesive materials, and the fluid flow can distortthe bed into various shapes. The interaction process is complex. At low velocities the bed does not move.With increasing flow velocity the inception of movement occurs. The basic bed forms encountered are

62


ripples (usually of heights less than 0.1m), dunes, flat bed, standing waves, and antidunes. At highvelocities chutes and step-pools may form. Typical bed forms are summarised in Figure 7-2 below.

Figure 7-2. Bedforms and the Froude numbers at which they occur (after Richardson and Simons)

7.5 Mechanisms of sediment motionThe amount of sediment passing through a given stream cross-section is a critical factor in

• determining the amount of sediment deposited in a downstream reservoir, and thus the useful lifeof a dam or project

• investigating the stability of the streambed and stream banks

• influencing the water quality to be used for irrigation, water supply, and recreational considerations

• affecting the navigable depth of a stream

• determining the water level during floods

• studying the potential effects on stream ecology such as fisheries as well as other biological species

• investigating the movement of pollutants attached to sediment particles

7.5.1 Bed load

the movement of sediment particles along the streambed in the process of rolling, sliding, and/or salta-tion, when the flow turbulence picks up a particle from the bed and they then fall back to the bed,somewhere further downstream.

There are no readily-accepted formulae for the movement of gravel, as the process of armouring mayor may not take place, which gives widely variable results. For sand, however, there are several popularequations.

Einstein’s bed load equation:

63


7.5.2 Suspended load

the movement of sediment particles that are supported by the turbulent motion in the streamflow.

7.5.3 Bed-load rate of transport – Bagnold’s formula

The volumetric rate of transport qsb per unit width is given by

qsb =βub(τ − τ cr)

(ρs − ρ) g,

where β is a function of δ, and ub is the flow velocity in the vicinity of the bed. In the case of a roughturbulent flow, β ≈ 0.5. This formula is preferred, as it is simple, as accurate as any, and reflects themeaning of the bed-load rate. However the calculation of ub seems a little arbitrary.

64


Appendix A. On diffusion and von Neumann stability analyses

A.1 The nature of diffusion

A.1.1 A discrete physical analogy

n m = -4 -3 -2 -1 0 1 2 3 40 • 1 •1 • 1/2 • 1/2 •2 • 1/4 • 1/2 • 1/4 •3 • 1/8 • 3/8 • 3/8 • 1/8 •4 • 1/16 • 1/4 • 3/8 • 1/4 • 1/16 •

Figure A-1. Array of pins showing probabilities with which a ball will pass through a particular gap

The process of diffusion occurs because of a continuous process of random particle movements. We willnow model that, but instead of a multi-dimensional medium where random movements of any magnitudecan occur, we will consider a one-dimensional medium where a single particle is free to make a seriesof discrete movements of the same magnitude, either to left or right. Consider a ball dropped betweenthe two pins at the apex of the pyramid of pins shown in Figure A-1. The probability that it will passbetween the first two pins is 1. It will hit the pin beneath it and will roll to left or right, with an equalprobability. The next lower set of pins in the discrete physical analogy is equivalent to the next timestage in the real physical situation. The relative probabilities of dropping down each gap is (1, 1), withthe actual probabilities obtained by dividing by the sum of those relative probabilities, giving (1/2, 1/2).Similarly at the next level of pins, the probabilities are (1, 2, 1), giving (1/4, 1/2, 1/4), and at the nextlevel, (1, 3, 3, 1), or relatively, (1/8, 3/8, 3/8, 1/8) and so on, familiar to anybody who has expanded,say, (a+ b)3. The pattern is obvious – in fact those probabilities are given by the binomial coefficients,which we can express as

Ω(m,n) =n!¡

n+m2

¢!¡n−m2

¢!

1

2n, (A-1)

for the nth row of pins, starting from n = 0 at the first row where all balls pass through the same gap,and the mth gap where m goes from −n to n in steps of 2, avoiding the places occupied by the pins.

A remarkable result is now used, Stirling’s approximation for the factorial function:

k! ≈ kke−k√2πk, for large k.

Using this it can be shown Borg & Dienes (1988) that equation (A-1) can be approximated by

Ω(m,n) ≈r2

πne−m

2/2n, (A-2)

and we see that at the nth row of pins, the shape of the probability distribution is approximated bythe Gaussian function e−m

2/2n which occurs throughout statistics. This has a characteristic bell-shape.Note that variation with m (the horizontal passage position or co-ordinate) varies like e−γm

2 , and thecoefficient γ = 1/2n, such that it decreases with n, showing how the lateral extent increases with n(”time”). Figure A-2 shows a comparison between the binomial distribution, equation (A-1) and theGaussian distribution (A-2). We now wish to relate this to real diffusion processes and quantify thephysical situation rather more. We suppose that a particle of a diffusing substance can either move toleft or right a distance l in a time τ . Table A-1 shows the calculation of the mean square displacementobtained from the discrete model we considered above. The last column demonstrates the empiricalrelation for the mean square displacement:

∆2 =1

N

X∆2 = t× l2

τ,

65


0

0.1

0.2

0.3

0.4

0.5

-10 -8 -6 -4 -2 0 2 4 6 8 10

Prob

abili

ty

m

BinomialApproximation

Figure A-2. Probability distributions of particle positions at successive levels

Time Possibilities No. of Σ∆2 ∆2 = 1NΣ∆

2

t for∆ possibilities N0 0 1 0 0τ ±l × 1 2× 1 2l2 l2

2τ 0× 2,±2l × 1 2× 2 8l2 2l2

3τ ±l × 3,±3l × 1 2× 4 24l2 3l2

Table A-1. Mean square displacement∆2 for random walk as a function of the diffusion time

showing that it is proportional to time (”n” in the above analogy). If we set

l2

2τ= κ,

and call this quantity (of dimension L2T−1) the coefficient of diffusion, then we obtain

∆2 = 2κt,

similar to an expression obtained by Einstein in his paper on Brownian motion in 1905.

A.1.2 Fick’s law

x0 l

Figure A-3. Diffusion model with more particles crossing the plane at x = l/2 to the right than to the left

66


Now we obtain the basic law governing diffusion. Instead of one particle at each of the different physicallocations we postulated above, we consider a different number of particles at each. If a particle moves adisplacement l in a characteristic time τ , the mean velocity of motion is l/τ . There are probabilities of±1/2 that it will go either way. If the number of particles is n0 at x = 0, then the total flux of particles(number of particles times velocity) which move from x = 0 to x = l is

J+ =1

2n0

l

τ.

Similarly, if there are n1 particles at x = l, then the flux of particles in the other direction is

J− = −1

2n1

l

τ.

The total flux in the +x direction at x = l/2 is

J = J+ + J− = −(n1 − n0)

l

l2

2τ.

We have suggested that the quantity l2/2τ is the diffusion coefficient, a constant for a particular mediumand diffusing molecules. If we denote the difference in the concentration of particles by ∆c = n1 − n0over the distance∆x = l, we obtain

J = −∆c∆x

κ.

In the limit∆x→ 0 we obtain Fick’s law for the diffusion flux:

J = −κ ∂c∂x

.

Hence we see that the flux of molecules is proportional to the gradient of the concentration – but thatthis does not come from any special physical property of the concentration, but merely because wherethere are many particles there will be more travelling away from that region than are travelling to thatregion. This is the essence of the diffusion process. In Brownian motion and diffusion in solids therandom motion comes from the molecular natures of the constituents. In waterways the random motionimparted to individual elements is due to the turbulence in the water.

A.1.3 Exact solutions of the diffusion equation

The diffusion equation has a general solution which is exact for an infinite domain (−∞,+∞):

φ =1√4πνt

exp

µ− x2

4νt

¶. (A-3)

The factor 4π is there such that the total integral of this solution between −∞ and +∞ is 1. Thesolution at time t → 0 is an infinitely large and infinitesimally narrow concentration distribution. Thusthe solution corresponds to the concentration distribution due to a point source of pollution. Solutionsfor successive times are shown in Figure A-4, which shows how the solution extends out to±∞ as timeprogresses.

Another well-known analytical method for solution of the diffusion equation is the use of Fourier series,where the medium is of finite length. It can be shown by the method of separation of variables that asingle term in the series which satisfies the diffusion equation is

φ = exp(−νj2k2t) sin jkx,

where k = 2π/L, where L is the length of the medium, and j is an integer which goes from 1 to∞. It is clear that the shorter components (larger j) decay very much more quickly than the longerones. With this, we have the familiar behaviour that discontinuities are smoothed out very quickly.Consider the situation shown in Figure A-5, which shows a discontinuity between two regions wherethe concentration is initially different. A solution was obtained in the form of a Fourier series with

67


x

Figure A-4. General solution for κt = 1/4, 1/2, 1, 2, 4, 8

coefficients exponentially decaying with time. The total length of the domain is L. Results are shownare for the initial condition νt/L2 = 0, then for νt/L2 = 1/100, for 9 subsequent steps with a timeinterval of ν∆t/L2 = 1/20, and finally νt/L2 → ∞, the straight line. It is clear how the diffusionworks very quickly initially, but then proceeds more slowly. Notice how we can express our parameternon-dimensionally in this form.

x

Figure A-5. Diffusion of a discontinuity

68


A.2 Examining stability by the Fourier series (von Neumann’s) method

A.2.1 Pure advection

We introduce a general method for examining the stability of approximations to differential equations.Suppose that the solution to the difference equation (4.2) can be written

φ (x, t) = A (t) eikx,

where i =√−1, and as eikx = cos kx + i sin kx, the variation in x is a single wave with wavelength

L = 2π/k. This is not as arbitrary as it appears at first, as we can in theory represent any (periodic)variation in x as a Fourier series, and as we consider linear equations with constant coefficients only, wecan just restrict ourselves to a single term in the Fourier series such as this one. We now want to workout what the solution for A(t) is, as would be obtained by the numerical method. Substituting into ourFTCS computational scheme (4.2):

φ (x, t+∆) = φ (x, t)− u∆

2δ(φ (x+ δ, t)− φ (x− δ, t)) ,

which gives

A (t+∆) eikx = A (t) eikx − u∆

2δ

³A (t) eik(x+δ) −A (t) eik(x−δ)

´,

and dividing through by A (t) eikx gives

A (t+∆)

A (t)= 1− C

2

³e+ikδ − e−ikδ

´= 1− iC sin kδ

We consider the magnitude of the amplification factor |A (t+∆) /A (t)| (the factor by which the am-plitude of the solution changes in a single time step) by multiplying by the complex conjugate to give:¯

A (t+∆)

A (t)

¯2= (1− iC sin kδ) (1 + iC sin kδ)

= 1 + C2 sin2 kδ.

Now the criterion for stability is that the amplitude ratio should be less than or equal to one, that is,¯A (t+∆)

A (t)

¯≤ 1,

however this can never be, as C2 sin2 kδ > 0, and we conclude that the scheme is unconditionallyunstable – for any values of space or time steps, as we experienced above.

Methods such as this can be used to analyse rather more complicated schemes which do not have thesame simple physical interpretation we saw in Figure 4-2.

A.2.2 Von Neumann stability analysis of FTCS scheme with diffusion

Let φ = A(t) eikx. Substituting into equation (4.3) and dividing through as before, we obtain

A (t+∆)

A (t)= 1 +D

³e−ikδ − 2 + e+ikδ

´= 1− 2D (1− cos kδ)

This is a purely real quantity. Stability will be assured if (A(t+∆)/A(t))2 6 1. This gives thecondition, after factorising:

D (1− cos kδ) (D (1− cos kδ)− 1) 6 0.

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The first factor is positive, the second factor is positive or zero for all kδ, (i.e. for all possible wavelengthsgiven by k) so that for stability the last factor must be negative. That is,

D (1− cos kδ)− 1 6 0,

giving

D 6 1

1− cos kδ .

Over all possible values of kδ, the minimum value of the right side is 1/2, giving the criterion

D =ν∆

δ26 1

2

for stability. This is the value usually quoted in Geomechanics applications.

A.2.3 Exact solutions for combined advection-diffusion

In fact, because of the simplicity of the effects of advection, this can often be included almost trivially.Consider the advection diffusion equation (3.15). If we introduce another advective co-ordinate X =x− ut, such that it travels with the flow, we try the solution

φ(x, t) = Φ(X, t).

We have, from the chain rule for partial differentiation,

∂φ

∂t=

∂Φ

∂t+

∂Φ

∂X

∂X

∂t=

∂Φ

∂t− u

∂Φ

∂X,

∂φ

∂x=

∂Φ

∂X

∂X

∂x=

∂Φ

∂x, and

∂2φ

∂x2=

∂2Φ

∂x2,

so that the advection equation becomes

∂Φ

∂t= ν

∂2Φ

∂X2,

and we can see that a solution of the diffusion equation in X will be a solution of the advection-diffusionequation in x. This is all right if we do not have to impose any boundary conditions, so it can provideinsight but may not be practically very useful. We can use the solution from equation (A-3) to providethe solution on (−∞,+∞) of the advection-diffusion equation:

φ =1√4πνt

exp

Ã−(x− ut)2

4νt

!.

This solution is plotted on Figure A-6, and shows the behaviour of solutions of the advection-diffusionequation. Note that the time levels for the curves are not equally spaced, but are as given in the captionto Figure A-4.

x

Figure A-6. Exact analytical solution of advection-diffusion equation

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fenton (2008 course) river hydraulics

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