River Engineering

John Fenton

Institute of Hydraulic and Water Resources EngineeringVienna University of Technology

June 20, 2011

Table of Contents

Table of Contents 2

1 Introduction 31.1 The nature of what we will and will not do – illuminated by some aphorisms and some people . 31.2 The problem of flow in a river . . . . . . . . . . . . . . . . . . . 4

2 River hydraulics 102.1 A note on terminology in the English language: . . . . . . . . . . . . . . 102.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . 102.3 The one-dimensional equations of hydraulics . . . . . . . . . . . . . . 122.4 Structures, controls, and boundary conditions . . . . . . . . . . . . . . 50

3 Measurement and analysis 623.1 Hydrometry and the hydraulics behind it . . . . . . . . . . . . . . . . 623.2 The analysis and use of stage and discharge measurements . . . . . . . . . . . 69

4 Computational hydraulics 80

5 Sediment transport 815.1 General . . . . . . . . . . . . . . . . . . . . . . . . . 815.2 Initiation of motion . . . . . . . . . . . . . . . . . . . . . . 815.3 Bedforms and alluvial roughness . . . . . . . . . . . . . . . . . . 815.4 Transport formulae . . . . . . . . . . . . . . . . . . . . . . 815.5 Unsteady aspects . . . . . . . . . . . . . . . . . . . . . . 81

6 River morphology 826.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 826.2 Planform . . . . . . . . . . . . . . . . . . . . . . . . 826.3 Longitudinal profile . . . . . . . . . . . . . . . . . . . . . 826.4 Bends . . . . . . . . . . . . . . . . . . . . . . . . . 826.5 Channel characteristics . . . . . . . . . . . . . . . . . . . . . 826.6 Bifurcations and confluences . . . . . . . . . . . . . . . . . . . 82

7 River engineering 837.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 837.2 Bed regulation . . . . . . . . . . . . . . . . . . . . . . . 837.3 Discharge control . . . . . . . . . . . . . . . . . . . . . . 837.4 Water level control . . . . . . . . . . . . . . . . . . . . . . 837.5 Water quality control . . . . . . . . . . . . . . . . . . . . . 837.6 River engineering for different purposes . . . . . . . . . . . . . . . . 83

References 84

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Chapter 1 Introduction

1.1 The nature of what we will and will not do – illuminated by some aphorismsand some people

"There is nothing so practical as a good theory" – stated in 1951 by Kurt Lewin (D-USA, 1890-1947): this isessentially the guiding principle behind these lectures. We want to solve practical problems, both in professionalpractice and research, and to do this it is a big help to have a theoretical understanding and a framework. We claimto aspire to deep connected simplicity rather than shallow disparate complexity.

"The purpose of computing is insight, not numbers" – the motto of a 1973 book on numerical methods forpractical use by the mathematician Richard Hamming (USA, 1915-1998?). That statement has excited the opinionsof many people (search any three of the words in the Internet!). However, in contradiction to Hamming’s assertion,numbers are often important in engineering, whether for design, control, or other aspects of the practical world.A characteristic of many engineers, however, is that they are often blinded by the numbers, and do not seek thephysical understanding that can be a valuable addition to the numbers. In this course we are not going to deal withmany numbers. Instead we will deal with the methods by which numbers could be obtained in practice, and willtry to obtain insight into those methods. Hence we might paraphrase simply: "The purpose of this course is insightinto the behaviour of rivers; with that insight, numbers can be often be obtained more simply, cheaply, and reliably.

"It is EXACT, Jane" – a story told to the lecturer by a botanist colleague. The most important river in Australia isthe Murray River, 2 375 km (Danube 2 850 km), maximum recorded flow 3 950m3s−1 (Danube at Iron Gate Dam:15 400m3s−1). It has many tributaries, flow measurement in the system is approximate and intermittent, there ishuge biological and fluvial diversity and irregularity. My colleague, non-numerical by training, had just seen thedemonstration by an hydraulic engineer of a computational model of the river. She asked: "Just how accurate isyour model?". The engineer replied intensely: "It is EXACT, Jane".

Nothing in these lectures will be exact. We are talking about the modelling of complex physical systems.

A further example of the sort of thinking that we would like to avoid: in the area of palaeo-hydraulics, someAustralian researchers made a survey to obtain the heights of floods at individual trees. This showed that the palaeo-flood reached a maximum height on the River Murray at a certain position of 18.01m (sic), Having measured thecross-section of the river, they applied the Gauckler-Manning-Strickler Equation to determine the discharge of theprehistoric flood, stated to be 7 686m3s−1 (sic) ...

William of Ockham (England, c1288-c1348): "Occam"’s razor is the principle that can be popularly stated as"when you have two competing theories that make similar predictions, the simpler one is the better." The term razorrefers to the act of shaving away unnecessary assumptions to get to the simplest explanation, attributed to 14th-century English logician and Franciscan friar, William of Ockham. The explanation of any phenomenon shouldmake as few assumptions as possible, eliminating those that make no difference in the observable predictionsof the explanatory hypothesis or theory. When competing hypotheses are equal in other respects, the principlerecommends selection of the hypothesis that introduces the fewest assumptions and postulates the fewest entitieswhile still sufficiently answering the question. That is, we should not over-simplify our approach.

In general, model complexity involves a trade-off between simplicity and accuracy of the model. Occam’s Razoris particularly relevant to modelling. While added complexity usually improves the fit of a model, it can make themodel difficult to understand and work with.

The principle has inspired numerous expressions including "parsimony of postulates", the "principle of simplicity",the "KISS principle" (Keep It Simple, Stupid). Other common restatements are:

Leonardo da Vinci (I, 1452–1519, world’s most famous hydraulician, also an artist) – his variant short-circuits theneed for sophistication by equating it to simplicity "Simplicity is the ultimate sophistication".

Wolfang A. Mozart (A, 1756–1791) "Gewaltig viel Noten, lieber Mozart", soll Kaiser Joseph II. über die erste dergroßen Wiener Opern, die "Entführung", gesagt haben, und Mozart antwortete: "Gerade so viel, Eure Majestät,als nötig ist." (Emperor Joseph II said about the first of the great Vienna operas, "Die Entführung aus dem Serail","Far too many notes, dear Mozart", to which Mozart replied "Your Majesty, there are just as many notes as are

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necessary"). The truthfulness of the story is questioned - Joseph was more sophisticated than that ...

Albert Einstein (D-USA,1879-1955): "Make everything as simple as possible, but not simpler."

Karl Popper (A-UK, 1902-1994) argued that a preference for simple theories need not appeal to practical oraesthetic considerations. Our preference for simplicity may be justified by his falsifiability criterion: We prefersimpler theories to more complex ones "because their empirical content is greater; and because they are bettertestable". In other words, a simple theory applies to more cases than a more complex one, and is thus more easilyfalsifiable. Popper coined the term critical rationalism to describe his philosophy. The term indicates his rejectionof classical empiricism, and of the classical observationalist-inductivist account of science that had grown out ofit. Logically, no number of positive outcomes at the level of experimental testing can confirm a scientific theory(Hume’s "Problem of Induction"), but a single counterexample is logically decisive: it shows the theory, fromwhich the implication is derived, to be false. For example, consider the inference that "all swans we have seen arewhite, and therefore all swans are white," before the discovery of black swans in Australia. Popper’s account ofthe logical asymmetry between verification and falsifiability lies at the heart of his philosophy of science. It alsoinspired him to take falsifiability as his criterion of demarcation between what is and is not genuinely scientific:a theory should be considered scientific if and only if it is falsifiable. This led him to attack the claims of bothpsychoanalysis and contemporary Marxism to scientific status, on the basis that the theories enshrined by them arenot falsifiable.

Falsifiability, however, seems not so important a principle when it comes to engineering. Another objection isthat it is not always possible to demonstrate falsehood definitively, especially if one is using statistical criteria toevaluate a null hypothesis. More generally it is not always clear, if evidence contradicts a hypothesis, that thisis a sign of flaws in the hypothesis rather than of flaws in the evidence. However, this is a misunderstanding ofwhat Popper’s philosophy of science sets out to do. Rather than offering a set of instructions that merely needto be followed diligently to achieve science, Popper makes it clear in The Logic of Scientific Discovery that hisbelief is that the resolution of conflicts between hypotheses and observations can only be a matter of the collectivejudgment of scientists, in each individual case.

Thomas Kuhn (USA, 1922-1996): In The Structure of Scientific Revolutions argued that scientists work in a seriesof paradigms, and found little evidence of scientists actually following a falsificationist methodology. Popper’sstudent Imre Lakatos (H-UK, 1922-1974) attempted to reconcile Kuhn’s work with falsificationism by arguingthat science progresses by the falsification of research programs rather than the more specific universal statementsof naive falsificationism.

Kuhn argued that as science progresses, explanations tend to become more complex before a paradigm shift offersradical simplification. For example Newton’s classical mechanics is an approximated model of the real world.Still, it is quite sufficient for most ordinary-life situations.

Popper seems to have anticipated Kuhn’s observations. In his collection Conjectures and Refutations: The Growthof Scientific Knowledge (Harper & Row, 1963), Popper writes, "Science must begin with myths, and with thecriticism of myths; neither with the collection of observations, nor with the invention of experiments, but with thecritical discussion of myths, and of magical techniques and practices. The scientific tradition is distinguished fromthe pre-scientific tradition in having two layers. Like the latter, it passes on its theories; but it also passes on acritical attitude towards them. The theories are passed on, not as dogmas, but rather with the challenge to discussthem and improve upon them."

Another of Popper’s students Paul Feyerabend (A-USA, 1924-1994) ultimately rejected any prescriptive method-ology, and argued that the only universal method characterizing scientific progress was "anything goes!"

The utility of deep simplicity compared with shallow complexity

In view of the above, the lecturer prefers to approach problems with the former, deep simplicity, rather than thelatter, shallow complexity. The following lecture notes will be a reflection of that. Unfortunately "deep" sometimeslooks complicated at first, which can be off-putting. However, once a principle or a practice is established orunderstood, it seems simple in retrospect.

1.2 The problem of flow in a river

The governing equations are the three-dimensional equations of mass and momentum conservation, the latter calledthe Navier-Stokes equations, which are partial differential equations that describe the distribution of velocity andpressure throughout a field of flow. They are valid at small (microscopic) scales and large (atmospheric andoceanic) scales. For most practical problems in water flow, the flow becomes unstable: paradoxically, the viscosity

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whose primary role is as a momentum diffusion or smoothing phenomenon, actually causes the flows to be unstableand to become turbulent. Most flows of interest to civil engineers are actually turbulent boundary layer flows.

1.2.1 Physical modelling

The problem of a turbulent flow with a free surface in the presence of irregular and moveable boundaries is socomplicated that one effective way of modelling it is to build a physical scale model, which reproduces most ofthe real phenomena. This has been the traditional way of much hydraulics. There are problems, however, in thatin a scale model, different physical properties of the fluid and the flow have different relative importances, and themodel is an approximate one. In the spirit of modelling, however, that is a legitimate approach and in hydraulics ithas often provided answers where otherwise none could be obtained.

1.2.2 Computational fluid mechanics

It is possible to apply computational fluid mechanics (CFD) to solve such problems, which usually means thenumerical solution in time and space of three-dimensional partial differential equations.

Direct numerical simulation (DNS) is the most basic method where the Navier-Stokes equations are solvednumerically in time, and the instability of the flow and the resulting turbulence is simulated. It captures all ofthe relevant scales of turbulent motion, so no model is needed for the smallest scales. This approach is extremelyexpensive, if not intractable, for complex problems on modern computing machines, hence the need for models torepresent the smallest scales of fluid motion.

Reynolds-averaged Navier–Stokes equations (RANS) is the oldest approach to turbulence modeling. The veloc-ities and pressure are written as a sum of slowly-varying plus rapidly-varying turbulent components. The equationsare averaged mathematically over a time scale large compared with the turbulent fluctuations, which gives equa-tions for the slowly-varying terms, but which introduces new apparent stresses known as Reynolds stresses. Itis necessary to introduce phenomenological models for such quantities. Such models include using an algebraicequation for the Reynolds stresses which include determining the turbulent viscosity, and depending on the levelof sophistication of the model, solving transport equations for determining the turbulent kinetic energy and dissi-pation. Models include k − ε (Spalding) and the Mixing Length Model (Prandtl).

Large eddy simulation (LES) is a technique in which the smaller eddies are filtered and are modelled using asub-grid scale model, while the larger energy carrying eddies are simulated. This method generally requires amore refined mesh than a RANS model, but a far coarser mesh than a DNS solution.

Vortex method is a grid-free technique for the simulation of turbulent flows. It uses vortices as the computationalelements, mimicking the physical structures in turbulence.

1.2.3 Mathematical/theoretical modelling

For most hydraulics problems the irregularity of geometry, the large size, and the lack of data available, mean thatthere is an imbalance between the little that is known about a problem and the sophisticated methods that can beapplied to simulate it. In this course we are going to take a different path, and use mathematical modelling. Abetter title might be "theoretical modelling". There is almost an inverse correlation between the sophistication ofa model and the understanding that emerges: simple models provide insight, complicated models provide manyresults, but insight is harder to obtain.

A mathematical model uses mathematical language to describe a system. Eykhoff in 1974 defined a mathematicalmodel as "a representation of the essential aspects of an existing system (or a system to be constructed) whichpresents knowledge of that system in usable form". For our purposes, that is a quite useful definition.

Often when engineers analyze a system to be controlled or optimized, they use a mathematical model. In analysis,engineers can build a descriptive model of the system as a hypothesis of how the system could work, or try toestimate how an unforeseeable event could affect the system. Similarly, in control of a system, engineers can tryout different control approaches in simulations.

Black-box and Clear-box modelling: Mathematical modelling problems are often classified into black box orwhite box (also called glass box or clear box) models, according to how much a priori information is availableof the system. A black-box model is a system of which there is no a priori information available. A clear-boxmodel is a system where all necessary information is available. Practically all systems are somewhere between theblack-box and clear-box models, for which the term grey box can be used.

Usually it is preferable to use as much a priori information as possible to make the model more accurate. Thereforethe clear-box models are usually considered easier, because if one has used the information correctly, then the

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model will behave correctly.

In black-box models one tries to estimate both the functional form of relations between variables and the numericalparameters in those functions. Using a priori information we could end up, for example, with a set of functions thatprobably could describe the system adequately. If there is no a priori information we would try to use functions asgeneral as possible to cover all different models. An often used approach for black-box models are neural networkswhich usually do not make assumptions about incoming data. The problem with using a large set of functions todescribe a system is that estimating the parameters becomes increasingly difficult when the amount of parameters(and different types of functions) increases. This is usually (but not always) true of models involving differentialequations.

Any model which is not pure clear-box contains some parameters that can be used to fit the model to the systemit describes. If the modelling is done by a neural network, the optimization of parameters is called training. Inmore conventional modelling through explicitly given mathematical functions, parameters are determined by curvefitting.

As the purpose of modelling is to increase our understanding of the world, the validity of a model rests not only onits fit to empirical observations, but also on its ability to extrapolate to situations or data beyond those originallydescribed in the model. One can argue that a model is worthless unless it provides some insight which goes beyondwhat is already known from direct investigation of the phenomenon being studied.

1.2.4 Nature of problems

Physical equations or

t

Input System Output

Physical parameters or

t

numerical coefficients

assumed equations

Figure 1.1: Nature of a physical system with input and output

Consider a physical system such as that shown in Figure 1.1, which is simplified here so as to show a single time-dependent input, and also just one output quantity. In Clear-box modelling, such as the hydraulic model of a river,the model equations are known to some extent, and the physical parameters which are used by those equations areknown, probably to a lesser extent. In a Black-box model, such as a neural network or a unit-hydrograph modelof a catchment, the equations of the system are assumed generic response equations, linear or nonlinear, which areexpressed in terms of numerical coefficients.

Simulation is when both the system and the inputs are given, and it is required to calculate the responses, whichmight be, for example include, a water level hydrograph at a station, such as at Hainburg an der Donau. A commonprocedure in hydraulics is to assume values for physical parameters such as resistance of the stream (which mightinclude, for example, telephoning your friend who worked on a similar river and asking her/him what they used forresistance ...). Rather more sophisticated is the numerical determination of physical (clear-box) or computational(black-box) parameters. This is the process of System Identification, where measured values of input and outputare used to determine what the system characteristics are. This is also known as the solution of an Inverse problem.After the execution of this phase, then simulation can be performed with rather more confidence.

Bibliography of useful referencesTables 1.1-1.4 show some of the many references available, some which the lecturer has referred to in these notesor in his work. The last column shows whether the source is the lecturer (JDF), the library in the Institute ofHydraulic Engineering (E222) or the University Library (UBTUW).

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Reference Comments

English

Chanson, H. (1999), The Hydraulics of Open Channel Flow,Arnold, London.

Good introduction, also sedi-ment aspects

UBTUWJDF

Chaudhry, M.H. (1993), Open-channel flow, Prentice-Hall. Very good readable but techni-cal book

E222JDF

Chow, V.T. (1959), Open-channel Hydraulics, McGraw-Hill,New York.

Classic, now dated, not so read-able

E222JDF

Francis, J.R.D. & Minton, P. (1984), Civil Engineering Hy-draulics, fifth edn, Arnold, London.

Good elementary introduction JDF

French, R.H. (1985), Open-Channel Hydraulics, McGraw-Hill,New York.

Wide general treatment E222JDF

Henderson, F.M. (1966), Open Channel Flow, Macmillan, NewYork.

Classic, high level, readable UBTUWJDF

Jain, S.C. (2001), Open-Channel Flow, Wiley. High level, but terse and read-able

JDF

Julien, P.Y. (2002), River Mechanics, Cambridge. Ditto — more applications tomorphology

E222UB-TUWJDF

Montes, S. (1998), Hydraulics of Open Channel Flow, ASCE,New York.

Encyclopaedic & Definitive JDF

Townson, J.M. (1991), Free-surface Hydraulics, Unwin Hyman,London.

Simple, readable, mathematical E222JDF

Vreugdenhil, C.B. (1989), Computational Hydraulics: An Intro-duction, Springer.

Simple introduction to compu-tational hydraulics

E222JDF

Deutsch

Forchheimer, Ph. (1930), Hydraulik, Teubner, Leipzig The Austrian and Internationalclassic of mathematical hy-draulics

UBTUW

Naudascher, E., (1992), Hydraulik der Gerinne und Gerin-nebauwerke, Springer Verlag, Wien, New York

UBTUW

Preißler, G., Bollrich, G., (1985) Technische Hydromechanik,VEB Verlag fur Bauwesen, Berlin

UBTUW

Table 1.1: Introductory and general references

Reference Comments

Boiten, W. (2000), Hydrometry, Balkema A modern treatment of rivermeasurement

JDF

Bos, M.G. (1978), Discharge Measurement Structures, secondedn, International Institute for Land Reclamation and Improve-ment, Wageningen.

Good encyclopaedic treatmentof structures

E128

Bos, M.G., Replogle, J.A. & Clemmens, A.J. (1984), Flow Mea-suring Flumes for Open Channel Systems, Wiley.

Good encyclopaedic treatmentof structures

Fenton, J.D. & Keller, R.J. (2001), The calculation of stream-flow from measurements of stage, Technical Report 01/6, Co-operative Research Centre for Catchment Hydrology, MonashUniversity.

Two level treatment - practicalaspects plus high level review oftheory

JDF

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

Standard readable presentationof structures

E222

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

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Reference Comments

Cunge, J.A., Holly, F.M. & Verwey, A. (1980), Practical Aspectsof Computational River Hydraulics, Pitman, London.

Thorough and reliable presenta-tion

JDF

Dooge, J.C.I. (1987), Historical development of concepts in openchannel flow, in G. Garbrecht, ed., Hydraulics and HydraulicResearch: A Historical Review, Balkema, Rotterdam, pp.205—230.

Interesting review JDF

Flood Studies Report (1975), Flood Routing Studies, Vol. 3,Natural Environment Research Council, London.

A readable overview E222JDF

Lai, C. (1986), Numerical modeling of unsteady open-channelflow, in B. Yen, ed., Advances in Hydroscience, Vol. 14, Aca-demic.

Good review UBTUWJDF

Liggett, J.A. (1975), Basic equations of unsteady flow, in K.Mahmood & V. Yevjevich, eds, Unsteady Flow in Open Chan-nels, Vol. 1, Water Resources Publications, Fort Collins, chapter2.

Readable overview JDF

Liggett, J.A. & Cunge, J.A. (1975), Numerical methods of so-lution of the unsteady flow equations, in K. Mahmood & V.Yevjevich, eds, Unsteady Flow in Open Channels, Vol. 1, Wa-ter Resources Publications, Fort Collins, chapter 4.

Readable overview JDF

Miller, W.A. & Cunge, J.A. (1975), Simplified equations of un-steady flow, in K. Mahmood & V. Yevjevich, eds, Unsteady Flowin Open Channels, Vol. 1, Water Resources Publications, FortCollins, chapter 5, pp. 183—257.

Readable JDF

Price, R.K. (1985), Flood Routing, in P. Novak, ed., Develop-ments in hydraulic engineering, Vol. 3, Elsevier Applied Science,chapter 4, pp. 129—173.

The best overview of theadvection-diffusion approxima-tion for flood routing

E222

Skeels, C.P. & Samuels, P.G. (1989), Stability and accuracyanalysis of numerical schemes modelling open channel flow, inC. Maksimovic & M. Radojkovic, eds, Computational Modellingand Experimental Methods in Hydraulics (HYDROCOMP ’89),Elsevier.

Review JDF

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

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Reference Notes

The full equations for wave propagation and flood routing

Cunge, J.A., Holly, F.M. & Verwey, A. ( 1980) Practical Aspects ofComputational River Hydraulics, Pitman, London.

The best explanation of this field

Liggett, J.A. (1975) Basic equations of unsteady flow, Unsteady Flowin Open Channels, K.Mahmood & V.Yevjevich (eds), Vol.1, Water Re-sources Publications, Fort Collins, chapter 2.

A little disappointing, but the nextbest explanation

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

The advection-diffusion approximation for flood routing

Price, R.K. (1985) Flood Routing, Developments in Hydraulic En-gineering, P.Novak (ed.), Vol.3, Elsevier Applied Science, chapter4,pp.129—173.

The best overview

Dooge, J.C.I. (1986) Theory of flood routing, River Flow Modellingand Forecasting, D.A. Kraijenhoff & J.R. Moll (eds), Reidel, chapter3,pp.39—65.

A good general study

Numerical methods — fundamentals

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

Noye, B.J. (1976) International Conference on the Numerical Simu-lation of Fluid Dynamic Systems, Monash University 1976, North-Holland, Amsterdam; Noye, B.J. (1981) Numerical solutions to partialdifferential equations, Proc. Conf. on Numerical Solutions of PartialDifferential Equations, Queen’s College, Melbourne University, 23-27August, 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) Computa-tional Techniques and Applications: CTAC 85, North-Holland, Ams-terdam.

All offer a simple introduction to fi-nite difference methods;

Smith, G.D. (1978) Numerical Solution of Partial Differential Equa-tions, Oxford Applied Mathematics and Computing Series, Second Edn,Clarendon, Oxford.

A more detailed introduction to fi-nite difference methods

Morton, K.W. & Baines, M. (1982) Numerical methods for fluid dy-namics, Academic; Morton, K. & Mayers, D. (1994) Numerical so-lution of partial differential equations : an introduction, Cambridge;Morton, K.W. (1996) Numerical solution of convection-diffusion prob-lems, Chapman and Hall, London; Richtmyer, R.P. & Morton, K.W.(1967) Difference Methods for Initial Value Problems, Second Edn, In-terscience, New York.

All are rather more comprehen-sive, describing some more generalmethods

Table 1.4: Useful references

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Chapter 2 River hydraulics

2.1 A note on terminology in the English language:

Throughout there will be use of apparently different terms for the single main quantity that we will be talking about– the stream. Let us be clear what is meant – see Table 2.1. The material in this course, which is called "riverengineering", is applicable to all of those, as is the field of "open channel hydraulics", or "Gerinnehydraulik" inGerman.

Open channel The generic term for a flow of water with a free surface

Stream Generic term, used in passing when a certain shortness & vagueness is necessary

Canal A constructed channel, generally prismatic and with a relatively firm boundary

Channel A generic term for an extended depression along which water passes

Waterway Stream on which boats navigate, may be natural or constructed

Natural streams (decreasing size)

River A natural channel that is large or medium-sized

Creek USA and AUS — a small river, but in England — a tidal inlet, rather different

Rivulet Small river

Brook Smaller stream

Beck Northern England — where the Angles and Saxons landed; from their word Bach

Ditch, Furrow Small channel, probably excavated

Gutter The channel at the side of a road pavement

Table 2.1: Terminology – various names for what is essentially the same thing

2.2 Summary

Initially we consider the two equations for mass conservation, where almost no essential approximations are nec-essary:

1. Conservation of mass in the flow in the channel: one of the few exact equations in hydraulics (the hydrostaticpressure equation is another one).

2. Conservation of mass of sediment – the Exner equation (Felix Maria von Exner-Ewarten , A, 1876-1930): alsoexact for uniform materials; the Wahl-Wiener who presents these lectures will defend this equation againstattacks by foreign barbarians during the last 10 years.

Then for momentum conservation, consider the hierarchy of river models, starting with the Navier-Stokes equa-tions, where the models become simpler as we pass down the list:

1. The Navier-Stokes equations: the three-dimensional partial differential equations of fluid mechanics; solvingthem is not feasible for most river problems.

2. Three-dimensional turbulent flow equations: where the turbulence is modelled. Solving them is only slightlymore feasible for most problems.

3. Conservation of moment of momentum, allowing for vertical variation of pressure and velocity: introduced bySteffler in Canada, but it has never been exploited. One could develop a 2-D river model in which secondarycurrents were described, and which could simulate differential erosion on both sides of the river.

4. Boussinesq approximation: essentially a one-dimensional model, but which allows for variation of pressureacross the flow due to curved streamlines.

5. Two-dimensional vertically-averaged river model: The real geometry of a meandering river can be included,

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but no secondary currents are possible and erosion predictions are unreliable.6. One-dimensional model using curvilinear co-ordinates: The long-wave equations for rivers that are curved in

plan – your lecturer ...7. One-dimensional model for straight streams: the long-wave equations, shallow-water equations, "Saint-Venant"

equations, the basis of most hydraulics. The solutions of these equations are much misunderstood, and someof the most elementary deductions and interpretations are wrong.

8. The low-inertia approximation: A rather simpler but still a good approximation9. Advection-diffusion flood routing10. Muskingum-Cunge routing11. Kinematic wave routing12. Black-box flood routing

Figure 2.1 shows how all the theories relate to each other. The arrows generally show the direction of increasingapproximation. The assumptions made in each case are shown as text without a box.

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Real stream

Boussinesq approximations,non-hydrostatic, can describetransition between sub- and

super-critical flow

One-dimensional long wave equationsfor curved streams, e.g. §2.3.1.3, eqn

(2.4)

One-dimensional long wave equations,eqns (2.22) and (2.23)

Characteristic formulation, §2.3.12,reveals speed of front of disturbances

Characteristic-basednumerical models

Finite-difference-based numericalmethods, Preissmann Box Scheme

Linearised Telegrapher’s equation, §2.3.13.1,eqn (2.84), reveals nature of propagation of

disturbances

Slow-change/slow-flow approximation, §2.3.8,eqn (2.67), simple equation & computational

method

Linearised, and slow-change/slow-flow,Advection-diffusion equation, §2.3.9, eqn(2.70), reveals nature of most disturbances

Two-dimensional equations, possiblyincluding moment of momentum, toinclude secondary flows, not yet

established

Assume pressure hydrostatic

Two-dimensional equations, nosecondary flows

Interchange of time & space differentiation,Muskingum-Cunge routing

Vertically averaged

Assume stream straight, no cross-stream variation

Assume small disturbances Assume flow and disturbances both slow

Both assumptions: small disturbances and slow flow

Assume diffusion small

Assume diffusion zero

Kinematic wave approximation

Assume stream curvature small

Figure 2.1: Hierarchy of one-dimensional open channel theories and approximations

2.3 The one-dimensional equations of hydraulics

Initially we will develop a model using the mass conservation equations for water and for soil, and for momen-tum conservation we will use approximation 7, which is relatively simple to obtain, and provides many useableresults from the one-dimensional long-wave equations. The approach is to consider the conservation of mass andstreamwise momentum along the stream. In developing the model for water and sediment flow in a river, we do notconsider details of motion in the plane across the stream. It will be found that this approach requires surprisinglyfew essential approximations – the model is actually a good one, rather better than is commonly believed.

There may be one surprise – there will be no mention of energy, other than to show how it contributes nothingextra, and that momentum is to be preferred.

Initially we make the traditional approximation that all rivers are straight. Later we will relax this and considermeandering streams.

12

In the past there has been little attention paid to defining co-ordinates or a control volume. Many derivations useas the fundamental space variable x some sort of distance along the channel bed, which is ambiguous for channelsof arbitrary shape, and what is worse, this curved co-ordinate ideally requires special treatment and considerationof the curvature of the axis. It is easier instead to use cartesian co-ordinates, for which we use x the horizontaldistance along the stream, y the horizontal transverse co-ordinate, and z the vertical, relative to some arbitraryorigin.

2.3.1 Mass conservation of water and soil

i∆x

∆xQ

A+∆A

Qb

Ab +∆Ab

x

y

z

AQ+∆Q

Ab

Qb +∆Qb

Figure 2.2: Elemental length of channel showing control volumes

Consider Figure 2.2, showing an elemental slice of channel of length ∆x with two stationary vertical faces acrossthe flow. It includes two different control volumes. The surface shown by solid lines includes the channel crosssection, but not the moveable bed, and is used for mass and momentum conservation of the channel flow. Thesurface shown by dotted lines contains the soil moving as bed load and extends down into the soil such that thereis no motion at its far boundaries. Each is modelled separately, subject to a mass conservation equation, and eachto a relationship that determines the flux.

The fluid in the channel might carry a suspended sediment load. Here, to good approximation, the concentrationof suspended sediment will be considered constant so that the density ρ of the fluid-solid suspension is constant,the same as for any fluid entering the control volume from rainfall, seepage, or tributaries, so that we can considerconservation of volume rather than of mass.

Physically-oriented derivation

This is a simpler and more intuitive derivation. Afterwards, a more mathematical derivation will be presented, asit provides something of a guide for when we consider momentum.

On the upstream vertical face at any instant, there is a volume flux (rate of volume flow) Q, and on the downstreamface Q+∆Q, so that

Net volume flow rate of fluid leaving across vertical faces = ∆Q =∂Q

∂x∆x+ terms in (∆x)2 .

If rainfall, seepage, or tributaries contribute an inflow volume rate i per unit length of stream, the volume flowrate of this other fluid entering the control volume is i∆x. If the sum of these two contributions is not zero, thenvolume of fluid is changing inside the elemental slice, so that the water level will change. The rate of changewith time t of fluid volume inside the control volume can be expressed most simply in terms of rate of change ofcross-sectional area as ∆x∂A/∂t. Equating this to the net rate of fluid entering the control volume, dividing by∆x and taking the limit as∆x→ 0 gives

∂A

∂t+

∂Q

∂x= i. (2.1)

Remarkably for hydraulics, this is almost exact. The only approximation has been that the channel is straight.

The second flow, the bed load, is composed of larger soil particles. The interstices between them are assumed to beoccupied by the same fluid as in the channel. In the absence of any detailed mechanism of transport or dilation, it

13

is assumed that the porosity of the bed load is constant. The composite bulk density of the bed load ρb is composedof the main solid constituent of the bed load, the water in the pores, and the suspended particles in the pore water,and it is assumed to be constant. The bed has a cross-sectional area Ab, the bulk volumetric flow rate is Qb, andthere is an inflow of mass rate mi per unit length, possibly due to deposition or erosion. Mass conservation iscalculated following the same reasoning for this control volume as for that of the channel, giving

∂Ab∂t

+∂Qb∂x

=mi

ρb. (2.2)

The volume transport rate used here is the bulk flow rate; it is related to the volume flow rate of solid matter Qs usedin transport formulae, by Qs = Qb (1− ϕ), where ϕ is the porosity of the bed load. This is a slight generalisationof Exner’s law.

Derivation using integral formulation with vectors

There is insight to be gained by repeating the derivation using a general integral formulation for the conservation ofany quantity. Here it will be considered initially for mass, later it will be used for momentum, when it is simpler touse this approach from the start. Consider the mass conservation equation for an arbitrarily moving and deformingcontrol surface CS and control volume CV (?, §§3.2 & 3.3):

ddt

ZCV

ρ dV

| z Total mass in CV

+

ZCS

ρur·n dS

| z Rate of flow of mass

across boundary

= 0, (2.3)

where t is time, dV is an element of volume, ur is the velocity vector of the fluid relative to the local element of thecontrol surface, which is possibly moving itself, n is a unit vector with direction normal to and directed outwards,and dS is an elemental area of the surface, as are shown on figure 2.3. The quantity urn = ur·n is the component

n

ururn = ur·n

dS

Figure 2.3: The normal component of fluid velocity relative to the control surface is urn = ur·n, hence the rate of volumetransport is ur·ndS

of relative velocity normal to the surface at any point. It is this velocity that is responsible for the transport of anyquantity across the surface, mass or momentum in our case.

Now consider the control surface to be the elemental slice in figure 2.2, bounded by two fixed vertical planestransverse to the channel, and bounded above by the free surface, which is free to move, and below by the soilsurface, which is similarly free to move. An element of volume in the first integral in equation (2.3) is dV =∆x dA, where dA is an element of cross-sectional area. The whole term becomes

ρ∆xd

dt

ZA

dA = ρ∆x∂A

∂t,

where it is necessary to use partial differentiation as A is a function of x as well. We have expressed the rate ofchange of mass in the control volume in terms of the rate of change of cross-sectional area.

Considering the second integral in equation (2.3), on the free surface we have chosen the control surface to coincidewith the free surface of the fluid, so that particles on the free surface remain on the free surface, and the relativenormal velocity urn = ur·n ≡ 0 everywhere on that surface. Similarly particles on the interface between waterand soil, or on any solid boundaries in the flow also define the control surface, and there is also no contribution tomass or momentum transport. We include the usually poorly-known inflow from rain, seepage or tributaries in anad hoc manner, by just writing the contribution as −ρ∆x i, negative as it is an inflow.

On the upstream face of the control surface, which is fixed in space and through which fluid moves, ur·n = −u ,

14

where u is the x component of fluid velocity, so that the contribution due to flow entering the control volume is

−ρZA

u dA = −ρQ,

where Q is the total volume flow rate or discharge in the channel. Similarly the downstream face contribution is+ρ (Q+∆Q), where we use a Taylor expansion to write it as

ρ (Q+∆Q) = ρ

µQ+

∂Q

∂x∆x+ . . .

¶.

Combining the contributions from the rate of change of mass in the control volume and the various contributionswhich cross the control surface, dividing by ρ∆x and taking the limit ∆x → 0 we have the unsteady massconservation equation

∂A

∂t+

∂Q

∂x= i,

the same as equation (2.1) obtained by simpler but less-general means.

Exactly the same procedure would be followed in the case of the bed-load, which would give equation (2.2).

Streams curved in plan

Centroid: n = n

Surface mid-point: n = nm

Centre of curvature: n = r = 1/κ

z

n

nL

nR

Figure 2.4: Section of curved river, showing important dimensions and axes

? have considered streams curved in plan (i.e. almost all rivers!) and obtained the result:³1− nm

r

´ ∂A

∂t+

∂Q

∂s= i, (2.4)

where nm is the transverse offset of the centre of the river surface from the curved streamwise reference axis s,which generally follows the path of the river and r is the radius of curvature of that axis. Usually nm is smallcompared with r, and the curvature term is a relatively small one, however for a highly meandering river there willbe an effect on flood propagation velocities.

Upstream Volume

The mass conservation equation (2.1) suggests the introduction of a function V (x, t)which is the volume upstreamof point x at time t, such that for the channel flow

∂V

∂x= A and

∂V

∂t= Q (x0, t) +

Z x

x0

i dx0 −Q. (2.5)

The derivative of volume with respect to distance x gives the area, as shown, while the time rate of change ofvolume upstream is given by the rate at which the volume is increasing due to inflow, minus the rate at whichvolume is passing the point and therefore no longer upstream. Substituting for A and Q into equation (2.1),shows that it is identically satisfied. The addition of Q (x0, t) has been suggested by Fatemeh Soroush (Personalcommunication, 2011), and gives a definition which is an improvement on an earlier one used by the lecturer. Inthe case of the bed load, similarly introducing the upstream volume of the bed load Vb(x, t) such that

∂Vb∂x

= Ab and∂Vb∂t

= Qb (x0, t) +

Z x mi

ρb(x0) dx0 −Qb, (2.6)

then the mass conservation equation (2.2) is identically satisfied. The use of these V and Vb will reduce the

15

total number of equations used from four to two, and enable the reduction of the combined problem to a singlethird-order differential equation.

Use of stage instead of cross-sectional area

We usually work in terms of water surface elevation stage η, which is easily measurable and which is practicallymore important. We make a significant assumption here, but one that is usually accurate: the water surface ishorizontal across the stream. Now, if the surface changes by an amount δη in an increment of time δt, then thearea changes by an amount δA = B δη, where B is the width of the stream surface. Taking the usual limit of smallvariations in calculus, we obtain ∂A/∂t = B ∂η/∂t, and the mass conservation equation can be written

B∂η

∂t+

∂Q

∂x= i. (2.7)

The discharge Q could be written as Q = UA, where U is the mean streamwise velocity over a section, andsubstituted into this. However, we believe the discharge to be more practical and fundamental than the velocity,and that will not be done here. When it comes to sediment transport, the velocity is important, but it can always beobtained from the discharge and the area.

2.3.2 Momentum conservation equation for channel flow

The equation

The conservation of momentum principle is now applied to the mixture of water and suspended solids in the mainchannel for a moving and deformable control volume (?, §§3.2 & 3.4):

ddt

ZCV

ρu dV+ZCS

ρu ur·n dS = P, (2.8)

where u is the fluid velocity, and P is the force exerted on the fluid in the control volume by both body forces,which act on all fluid particles, and surface forces which act only on the control surface. Note how similar the leftside is to the left side of the mass conservation equation (2.3). It was a conservation equation for mass, integratingthe mass per unit volume ρ throughout the control volume and its flux across the control surface. Equation (2.8)is a conservation equation for momentum, obtained by integrating the contributions of fluid momentum per unitvolume, the quantity ρu.

Contributions to force P

1. Body force: for the straight channel considered, the only body force acting is gravity; this force per unit mass isg, the gravitational acceleration vector. The total force on the fluid in the control volume is the integral of thiswith respect to mass, dm = ρdV , giving

RCV

ρg dV = ρgV . We will consider only the x-component of themomentum equation, which have chosen to be horizontal, as g only has a component in the−z direction, therewill be no contribution from gravity to our equation! The manner in which gravity acts is to cause pressuregradients in the fluid, giving rise to the following term, due to pressure variations around the control surface.

2. Surface forces: we separate these into pressure and shear forces, which act normally and tangentially to thesurface respectively.a. Pressure – the direction of the pressure force on the fluid at the control surface is −n, where n is the

outward-directed normal; its local magnitude is pdS, where p is the pressure and dS an elemental areaof the control surface, as in figure 2.3. Hence, the total pressure force is the integral − R

CSpndS. This

is difficult to evaluate for arbitrary control surfaces, as the pressure and the non-constant unit vector haveto be integrated over all the faces. A simpler derivation is obtained if the term is evaluated using Gauss’Divergence Theorem: Z

CS

pndS =

ZCV

∇pdV,

where ∇p = (∂p/∂x, ∂p/∂y, ∂p/∂z), the vector gradient of pressure. This has turned a complicatedsurface integral into a volume integral of a simpler quantity.

b. Shear forces: there is little that we can say that is exact about the shear forces. We will consider these whenwe make several practical assumptions.

16

Collecting terms

Substituting the force contributions into the momentum equation (2.8) and taking just the x component:

ddt

ZCV

ρudV

| z Unsteady term

+

ZCS

ρu ur·n dS

| z Fluid inertia term

= −Z

CV

∂p

∂xdV

| z Pressure gradient term

+ Horizontal component of shear forceon control surface| z

Horizontal shear force term

(2.9)

This expression, for an arbitrary control surface, has required no additional assumptions or approximations. Now,however, we consider the section of channel with the control surface as shown in figure 2.2. To obtain a useableexpression it is necessary to make hydraulic approximations for all terms after the first.

Hydraulic approximations

1. Unsteady termThe element of volume, as for the mass conservation equation, is dV = ∆x dA, and the term contribution canbe written

ρ∆xddt

ZA

u dA = ρ∆x∂Q

∂t, (2.10)

where the integralRAu dA has a simple and practical significance – it is just the discharge Q, so that the

contribution of the term can be written simply as shown, but where again it has been necessary to use thepartial differentiation symbol, as Q is a function of x as well. No additional approximation has been made inobtaining this term. It can be seen that the discharge Q plays a simple role in the momentum of the flow.

2. Fluid inertia termThe second term on the left of equation (2.8) is

RCS

ρu ur·n dS, has its most important contributions from

the stationary vertical faces perpendicular to the main flow. If there is also fluid entering or leaving fromrainfall, tributaries, or seepage, there are contributions over the other faces, which we shall include below in anapproximate manner. Considering these contributions separately:a. Stationary vertical faces: on the upstream face, ur·n = −u, giving the contribution to the term of−ρ R

Au2 dA. The downstream face at x + ∆x has a contribution of a similar nature, but positive, and

where all quantities have increased over the distance ∆x. The net contribution, the difference between thetwo, after neglecting terms like (∆x)2, can be written

ρ∆x∂

∂x

ZA

u2 dA.

In fact, the u2 term means that there are contributions from the turbulent fluctuations. If we write u = u+u0,where u is the mean value in time, and u0 is the turbulent fluctuation, then substituting into u2 and takingthe mean value in time, using the fact that u0 = 0 by definition, the term becomes

ρ∆x∂

∂x

ZA

³u2 + u02

´dA.

It can be shown that this is the only term in the momentum equation (2.9) where turbulence plays a role. Toevaluate this integral it is necessary to have a detailed knowledge of the flow and turbulence distribution overthe cross-section. As we have not solved the Navier-Stokes equations or the 3-D turbulent flow equations,that information is not available, and so here we make the first major hydraulic assumption. The crude butsimple approach is to assume that u2 and u02 are both proportional to the square of the mean velocity overthe section (Q/A)2, and to write the integral in terms of a Boussinesq coefficient β:Z

A

³u2 + u02

´dA ≈ β

Q2

A.

Traditionally the coefficient β was used to make some allowance for the non-uniformity of velocity distri-bution. ? suggested that it should be modified to include also the effects of turbulence as shown. A typicalvalue is β = 1.05. The contribution to the equation is then simply but approximately.

ρ∆x∂

∂x

µβQ2

A

¶. (2.11)

17

It will be shown below, that the whole term is of a relative magnitude that of the square of the Froude num-ber, and its effects are small in many situations. It is useful to retain the β, unlike many presentations thatimplicitly assume it to be 1.0, as it is a signal and reminder to us that we have introduced an approximation.

b. Lateral momentum contributions: the remaining contribution to momentum flux is from inflow such astributary streams, or, although less likely, from flow seeping in or out of the ground, or from rainfall. Inobtaining the mass conservation equation above, these were lumped together as an inflow i per unit length,such that the mass rate of inflow is ρ i∆x, (i.e. an outflow of −ρ i∆x) and if this inflow has a meanstreamwise velocity of ui before it mixes with the water in the channel, the contribution is

−ρ∆xβi i ui, (2.12)

where βi is the Boussinesq momentum coefficient of the inflow. The term is unlikely to be known accuratelyor to be important in most places.

3. Pressure gradient term – the hydrostatic approximationThe contribution is − R

CV∂p/∂x dV , the volume integral of the streamwise pressure gradient. The hydraulic

approximation has a problem, because we have not attempted to calculate the detailed pressure distributionthroughout the flow. However, in most places in most channel flows the length of disturbances is much greaterthan the depth, so that streamlines in the flow are only very gently curved, and the pressure in the fluid isaccurately given by the equivalent hydrostatic pressure, that due to a stationary column of water of the samedepth. Hence we write for a point of elevation z,

p = ρg ×Depth of water above point = ρg(η − z),

where η is the elevation of the free surface above that point. The quantity that we need is the horizontal pressuregradient ∂p/∂x = ρg ∂η/∂x, and so the streamwise pressure gradient is entirely due to the slope of the freesurface. The contribution is

−ZA

∂p

∂xdV ≈ −ρ∆x g

ZA

∂η

∂xdA ≈ −ρ∆x gA∂η

∂x, (2.13)

where any variation with y has been ignored, as the surface elevation usually varies little across the channel,and so ∂η/∂x is constant over the section and has been able to be taken outside the integral, which is thensimply evaluated.

4. Horizontal shear force termThis term is probably the least well-known of all, and yet it is important in the equation. Other presentationsassume that the slope of the channel is small, but that is unnecessary, and we shall not be forced to makeit. The results will be applicable to steep spillways, inter alia. However, we are about to adopt a significantapproximate model of the process of resistance to motion. ? recommended use of the Weisbach formulation forbed resistance, but that suggestion has been almost entirely ignored by the profession. The Gauckler-Manning-Strickler equation continues to be widely used, with the disadvantages noted by the Committee. Using theWeisbach formulation here provides additional insights into the nature of the equations and some convenientquantifications of the effects of resistance.Consider the equation for the shear force τ on a pipe wall e.g. ?:

τ =λ

8ρV 2, (2.14)

where λ is the dimensionless Weisbach resistance factor, and V is the spatial and temporal mean velocity alongthe pipe. If the pipe is uniform and long, then τ is constant everywhere, both around the perimeter of the flowas well as along the pipe. The channel is a more complicated situation.Consider a vertical rectangular elemental prism shown in figure 2.5. The bed has a slope angle of θ in the xdirection, which in many hydraulic problems is small. The transverse slope may be quite large however. Thetotal shear force is τ ∆s∆P , where s is locally downstream along the bed and∆P is an element of perimeter.This gives us

Shear force on prismatic element =λ

8ρV 2∆s∆P, (2.15)

in which the mean velocity tangential to the boundary V over the prism is assumed to be related to U , themean x-component of velocity in the elemental prism, by V = U/ cos θ, as suggested in the figure. Now, thehorizontal component of shear force, which is what we require, is obtained by multiplying the term (2.15) bycos θ = ∆x/∆s, giving

Horizontal shear force on prismatic element =λ

8ρV 2∆x∆P.

18

τ

U

V

θ

θ

∆x

∆s

(a) Side elevation (b) Looking down the channel

∆y

∆P

Slope ∂η/∂x

Figure 2.5: Element of channel showing dimensions and shear force from the bed

Summing around the perimeter, expressing it as an integral in perimeter distance, gives

Total horizontal shear force on control surface =ρ∆x

8

ZP

λV 2 dP.

Now we make some gross approximations. We replace V , the mean velocity parallel to the bed, by U/ cos θ,as suggested in the figure, and if we write 1/ cos2 θ = sec2 θ = 1 + tan2 θ the integral becomes

Total horizontal shear force on control surface =ρ∆x

8

ZP

λU2¡1 + tan2 θ

¢dP.

As we usually do not know the variation of λ or the distribution of U or even in many cases the variation of thebed topography to give θ, we write it as the total perimeter P multiplied by a product of the mean values of thethree terms in the integrand:Z

P

λU2¡1 + tan2 θ

¢dP ≈ λ

µQ

A

¶2 ³1 + S2

´P,

where the quantities shown with tildes are mean values for the section: λ is the mean value of the dimensionlessresistance factor around the perimeter, and the mean slope term S is the mean downstream slope evaluatedaround the wetted perimeter. In almost all river situations, where S is usually poorly known, it is so small thatit plays no role here. In situations such as on steep spillways, where it is important, it is actually well-defined.Collecting terms gives

Total horizontal shear force on control surface = −ρ∆x8

λQ |Q|A2

³1 + S2

´P,

where the Q2 term have been replaced by −Q |Q|, so that the direction of the stress is opposite to the flow,allowing also for tidal flow situations where the flow reverses:It is convenient for all subsequent work to introduce the symbol Λ:

Λ =λ

8

³1 + S2

´, (2.16)

so that the result can be written

Total horizontal shear force on control surface = −ρ∆xΛQ |Q|A2

P. (2.17)

Collection of terms and discussion

Now all contributions from the hydraulic approximations to terms in equation (2.9) are collected, using equations(2.10), (2.11), (2.12), (2.13), and (2.17), and bringing all derivatives to the left and others to the right, all dividedby ρ∆x, gives the momentum equation:

∂Q

∂t+

∂

∂x

µβQ2

A

¶+ gA

∂η

∂x= −ΛP Q |Q|

A2+ βi i ui (2.18)

19

At this stage the non-trivial assumptions in the derivation are stated, roughly in decreasing order of importance:

1. Resistance to flow is modelled by an empirical law, such that it is proportional to wetted perimeter and meanflow velocity squared. The law can incorporate finite effects of viscosity inasmuch as that affects the resistancecoefficient, but no attempt is made to incorporate viscous or turbulent shear terms in the momentum balance.The Navier-Stokes equations are not being used.

2. All surface variation is sufficiently long that the pressure throughout the flow is given by the hydrostatic pressurecorresponding to the depth of water above each point.

3. Effects of curvature of the stream course are ignored. Its effects are of the order of the ratio of the width of thestream to its radius of curvature.

4. In the momentum flux term the effects of both non-uniformity of velocity over a section and turbulent fluctu-ations are approximated by a momentum or Boussinesq coefficient. Nowhere else is any special treatment forturbulence necessary, so we have developed a model including turbulence.

5. Surface elevation η across the stream is constant.6. Suspended load concentration is a constant over the section at a particular time and place.

In the resistance term, the only place necessary, allowance has been made for the stream being on a finite slope –no limitation on slope has been required.

Expressing ∂A/∂x in terms of surface slope ∂η/∂x and mean bed slope S

A

B

∆η

y

z

∆Z

At xAt x+∆x

Figure 2.6: Two channel cross-sections separated by ∆x

In the momentum equation (2.18), if expanded, there would be a mixture of derivatives of area ∂A/∂x and surfaceelevation ∂η/∂x. We choose to work with the more practical quantity of surface elevation, and in so doing, areforced to consider the bottom geometry in greater detail.

The cross-section of a river in Figure 2.6 shows how ambiguous and possibly non-unique the concept of the“bottom” of the stream may be. In a distance ∆x the surface elevation may change by an amount ∆η as shown,so that the contribution to the increase in cross-section area ∆A is B × ∆η, where ∆η is usually negative asthe surface drops downstream. The change in the bed is ∆Z, which in general varies across the section, withcontribution to ∆A of − R

B∆Z dy, the area between the solid and dotted lines on the figure corresponding to

the bed at the two locations. The minus sign is because, if the bed drops away and ∆Z is negative, as usual, thecontribution to area increase is positive. Combining the two terms,

∆A = B∆η −ZB

∆Z dy (2.19)

In practice the precise details of the bed are rarely known. For the latter contribution, the integral of the change inbed elevation across the stream, we introduce the symbol S for the mean downstream bed slope across the sectionsuch that

S = − 1B

ZB

∂Z

∂xdy, (2.20)

where the negative sign has been introduced such that in the usual case when Z decreases with x, this meandownstream bed slope at a section is positive. In an important problem where bed details might be known, thiscan be evaluated. In the usual case where bed topography is poorly known, a reasonable local approximation orassumption is made. Using equations (2.19) and (2.20) we can write

∆A = B∆η +BS∆x,

20

where in a distance ∆x the mean bed level across the channel then changes by −S ×∆x under the water. In therare case where the sides of the stream are vertical diverging or converging walls, an extra term would have to beincluded. Taking the usual calculus limit, we obtain

∂A

∂x= B

µ∂η

∂x+S

¶, (2.21)

which might have been able to have been written down without the mathematical details.

2.3.3 Forms of the governing equations

In equation (2.18) we ignore the last term, the inflow momentum term βi i ui as it is small and poorly known.Expanding the term ∂

¡βQ2/A

¢/∂x, gives a term in ∂β/∂x which we also neglect, as we usually know little about

how β varies. There are now x-derivatives of both A and η in the equation, which are not independent. We useequation (2.21) to eliminate first one and then the other to give two alternative forms of the momentum equationgoverning flows and long waves in waterways. In both cases, we restate the corresponding mass conservationequation, using (2.1) and (2.7), to give the pairs of equations:

Formulation 1 – Long wave equations in terms of area A and discharge Q

Eliminating ∂η/∂x gives the equations in terms of A and Q:

∂A

∂t+

∂Q

∂x= i, (2.22a)

∂Q

∂t+ 2β

Q

A

∂Q

∂x+

µgA

B− β

Q2

A2

¶∂A

∂x= gAS − ΛP Q |Q|

A2. (2.22b)

Formulation 2 – Long wave equations in terms of stage η and discharge Q

Now eliminating ∂A/∂x, but retaining A in all coefficients, as it can be calculated in terms of η:

∂η

∂t+1

B

∂Q

∂x=

i

B, (2.23a)

∂Q

∂t+ 2β

Q

A

∂Q

∂x+

µgA− β

Q2B

A2

¶∂η

∂x= β

Q2B

A2S − ΛP Q |Q|

A2. (2.23b)

Nature of equations

Each of the two formulations is in the form of a pair of equations that can be written as a vector evolution equation

∂u

∂t+ C

∂u

∂x= r (u) ,

where u is the vector of unknowns, for example, [η,Q], C is a 2 × 2 matrix with algebraic coefficients, and r isthe vector of right side terms, due to inflow, slope, and resistance.

It can be shown that the system is hyperbolic, although this mathematical terminology seems not very useful for us.The implication of that is that solutions are of a wave-like nature. We will see that the behaviour of disturbances ismore complicated than we might expect or is often stated. This arises because the right sides are functions of thedependent variables, that we have written here as r (u). In particular we will see that a common interpretation ofthe system in terms of characteristics, with the solution that of travelling waves with simple properties, is incorrect.The solution is actually more complicated: disturbances travel at speeds which depend on their length, and showdiffusion as well.

Comparison with previous presentations

The questions arise, "how does this relate to other presentations of results?", and, "has the present approach actuallydone anything new?" It is strange, but there are almost no presentations in textbooks of the equations in thisstandard form. Most retain a term ∂

¡Q2/A

¢/∂x. As we have seen, the expansion of this is non-trivial. Textbooks

also present results in terms of a depth-like quantity h (called "the depth"), which is presumably only useful forcanal problems – where the bottom is flat, and the depth at the centre, can be defined. We will not deal withit, as boundary conditions are almost always expressed in terms of surface elevation η. Presentations able to berecommended include those of ? and ?.

The above presentation has given a couple of additions, which might be useful in practice:

1. The use of the Boussinesq momentum coefficient β, reminding us that the term is being modelled approxi-mately only. Some (?, ?) have included it, but also included equations without it. The lecturer’s claim that

21

it is better to include it, rather than implicitly assuming it to be 1.0000 might be viewed cynically, when laterin the lectures it will be suggested that terms in β are of magnitude of the Froude number, and can usually beneglected anyway – equivalent to making the β = 0 "approximation"!

2. The use of mean downstream bed slope S, as defined in equation (2.20), in the right sides of the momentumequations (2.22b) and (2.23b): this is the only way in which variable bed topography has entered explicitly.Textbook presentations that do not expand the term ∂

¡Q2/A

¢/∂x do not confront this problem. In prac-

tice, however, S is often be calculated very approximately as, for example, even the water surface or evensurrounding land elevation difference between two stations divided by the distance between them.

3. The resistance term: in equations (2.22b) and (2.23b) it appears as −ΛPQ |Q| /A2,which has a relativelysimple interpretation as a dimensionless coefficient multiplied by the perimeter (over which the resistanceacts), multiplied by the mean velocity in the stream, squared. This clearly follows from the approximatephysical law assumed. There is no limitation on slope: these are long wave approximations, not shallow slopeapproximations.

Alternative use of energy conservation

The question also arises – how would the use of energy conservation relate to this work? Consider the Energyconservation equation for the flow in the body of the channel (?, §3.6):

∂

∂t

ZCV

ρ edV +

ZCS

(p+ ρe) ur·ndS = Losses, (2.24)

where e is the internal energy per unit mass of fluid, which in hydraulics is the sum of potential and kinetic energiese = gz +

¡u2 + v2 + w2

¢/2. Proceeding in a manner similar to the momentum equation above, we obtain

gBη∂η

∂t+

α02

∂

∂t

µQ2

A

¶+

∂

∂x

µgηQ+ α1

Q3

2A2

¶= gHii− ΛeP Q2 |Q|

A3, (2.25)

where α0 and α1 are energy transmission coefficients defined by

αn =1

(Q/A)n+2

A

ZA

(u2 + v2 + w2) un dA,

Hi is the head of the incoming flow, and Λe is a dimensionless energy loss coefficient. Equation (2.25) has twotime derivative terms (which are clearly related to the rate of change of potential and kinetic energy at a section).It is not as simple as the momentum conservation equation, which is in terms of ∂Q/∂t. To eliminate ∂η/∂t and∂A/∂t we use two different forms of the mass conservation equation, then, neglecting inflow and ∂α1/∂x termsgives

α0∂Q

∂t+

α0 + 3α12

Q

A

∂Q

∂x+

µgA− α1

Q2B

A2

¶∂η

∂x= α1

Q2B

A2S − ΛeP Q |Q|

A2.

This can be compared with the momentum equation (2.23b) for no inflow:

∂Q

∂t+ 2β

Q

A

∂Q

∂x+

µgA− β

Q2B

A2

¶∂η

∂x= β

Q2B

A2S − ΛP Q |Q|

A2.

It can be seen that the structure of the two equations is the same, but they are slightly different in numericaldetail. The leading term of the energy formulation is α0 ∂Q/∂t; as α0 ≈ 1.05, this gives a slightly different timescale from the momentum equation. As the coefficients α0, α1 and β are all just greater than 1, the numericalcoefficients of the other terms are also just slightly different. In general, Λe is different from Λ. It is the assertionin these lectures, however, that the processes of energy dissipation, as given by Λe, are more complicated than theboundary stresses that give rise to the coefficient Λ. The momentum losses of the latter are a local phenomenon, todo with the local nature of the boundary and flow at the boundary only, whereas energy losses are more complicatedand more distributed, in boundary layers, separation zones, vortices, and turbulent and viscous decay. Accordingly,the coefficient Λ should be easier to quantify than Λe, and the momentum approach is to be preferred. It is alsopossible that the momentum response of the flow to boundary changes would be faster than energy processes,which consist of more complicated 3-D motions, and might take further downstream to adjust, as suggested infigure 2.7. The momentum model is simpler, and so, following Occam’s Razor, we are inclined to follow that path.

2.3.4 Hydraulic resistance –Weisbach-Chézy, Gauckler-Manning-Strickler

? started a well-known paper on resistance to flow with the words:

22

Small bed forms Large bed forms

Continuing adjustments

to energy cascade

Equilibrium velocity profileEquilibrium velocity profile

≈ 20 depths

Figure 2.7: Hypothesised faster response of momentum than energy, here at a change in roughness

For many centuries, if not millennia, the resistance to flow in open channels has claimed the attention ofhydraulic engineers.

And then:

In this period of wide-spread scientific development, a review of existing knowledge with regard to open-channel hydraulics might appear to be as lacking in timeliness as it is in glamour. Yet a glance at publica-tions on the subject during the past decade or so will reveal many a significant anomaly. Momentum andenergy analyses are at the same time over-simplified and confused with one another. Representation of pa-rameters by symbols is mistaken for empirical formulation of functions. Flow formulas are sometimes saidto involve the Froude number when they do not, and yet the Froude number is as frequently ignored when itis actually essential. Boundary texture and cross-sectional non-uniformity are often discussed without dis-tinction. Roughness measures are mistaken for resistance coefficients and are permitted to reflect not onlyviscous effects but even changes in depth. In a word, principles of mechanics that have proved useful inother phases of fluid motion are still not generally applied to problems of open-channel flow,which perforceremain subject to empirical solution – all too often without even the guidance of physical reasoning.

He then went on:

However, it need only be recalled that the Chézy equation was unheard of two centuries ago; the Manningtype of formula, one century; the Reynolds-number resistance diagram only a half century; and the Kármán-Prandtl relationships little more than a quarter century – yet all are in regular use today.

Were he alive today, he would despair, for the apparent convergence in development that he wrote about, has allseemed to fizzle out.

? wrote, warning about the use of the Gauckler-Manning-Strickler (G-M-S) formula:

Many engineers have become accustomed to using Manning’s n for evaluating frictional effects in openchannels. At the present stage of knowledge, if applied with judgement, both n and λ are probably equallyeffective in the solution of practical problems. The design engineer who prefers to use n in her/his com-putations should continue to do so, but he/she should recognize the limitations on her/his method ... It isbelieved that experimental measurements of friction in open channels over a wide range of conditions arebetter correlated and understood by the use of λ. Furthermore, λ is commonly used by engineers in manyother branches of engineering and probably provides the only basis for pooling all experience on frictionalresistance in both open and closed conduits. It is recommended, therefore, that engineering teachers andresearch workers emphasize the use of the friction factor λ ...

This has been almost completely ignored. What the Task Force was advocating was the use of the Weisbachapproach, which we shall now consider.

Dimensional analysis

Consider the dimensional quantities governing the problem of the shear stress on the boundary of a stream whichis straight: shear, where all quantities are taken to be section-averaged ones; the problem of variation aroundthe boundary is too difficult for us at this stage: mean shear stress τ , discharge Q, cross-sectional area A, wettedperimeterP , surface widthB, mean roughness size k, gravitational acceleration g, density ρ, and dynamic viscosity

23

μ. From the 9 variables and 3 dimensions, from the Buckingham Π Theorem there are 6 dimensionless quantities.If we use as repeating variables ρ, Q, and A, the remaining physical quantities whose importance we are toexamine, are as shown in the first column of table 2.2. The dimensionless results are shown in column 3. Howeverin most cases the physical significance of the terms is not clear. Multiplying by the factors in column 4 gives thedimensionless numbers in column 5, with more-easily-understood significance.

Physicalquantity

ΠDimensionlesswith ρ, Q,and A

Factor Result Significance

τ Π1τ

ρ(Q/A)21

τ

ρ(Q/A)2= Λ, as introduced in equations (2.14) and(2.16)

P Π2 P/√A Π3 BP/A Aspect ratio of section: roughly width/depth

B Π3 B/√A 1/Π2 B/P Top width / Bottom ”width”

k Π4 k/√A Π2 ε = k/(A/P ) Relative roughness: roughness size / mean depth

normal to bed

g Π5g√A

(Q/A)21/Π3

gA/B

(Q/A)2= 1/F 2 where F is the Froude number: effectsof gravity

μ Π6μ√A

ρQΠ2

νP

Q= 1/R where R is the Reynolds number: effectsof viscosity

Table 2.2: Dimensional analysis of resistance in a channel

Resistance coefficient Λ: The dimensional analysis has shown how important is the quantity Λ, that was intro-duced in equations (2.14) and (2.16). Traditionally, however, originating from pipe flows, the quantity consideredis λ = 8Λ, the Darcy/Weisbach coefficient. In pipe flows it is used as

∆H = λL

D

V 2

2g, (2.26)

where ∆H is the head loss in a pipe of length L and diameter D. (There is a point of view that g plays no role inconfined flow such as in a pipe and should not be used here – instead, g∆H should be considered as a unity, theenergy loss per unit mass of fluid, such that the total rate of energy loss in the pipe is ρQg∆H).

There is an interesting aside here, relating head loss and boundary shear stress. To do that, we actuallyuse the integral momentum theorem applied to the control volume in a length of uniform pipe of length Linclined at an angle θ between stations 1 and 2, where here it is easier to consider a control volume withfaces across the pipe rather than vertical as we did for the channel:

(p1 − p2)πD2

4| z Net force due to pressure

+ ρgπD2

4L sin θ| z

Component of weight forcealong pipe

= τπDL| z Force due to

boundary shear

.

As L sin θ = z1 − z2 we obtain

z1 +p1ρg−µz2 +

p2ρg

¶= ∆H =

4τL

ρgD, (2.27)

and so the quantity that we refer to as head loss has actually been obtained from momentum considerations.Comparing equations (2.26) and (2.27) we obtain equation (2.14)

τ =λ

8ρV 2.

It would probably have been more fundamental to have started by assuming τ = ΛρV 2, the use of λ/8came about because of using head and a circular pipe.

Relative roughness k/(A/P ): It can be seen that the relative roughness has appeared in the form k/(A/P ),

24

where we might think of the quantity A/P as the mean depth measured from the river bed, the important quantityin considering boundary resistance. Of course, many streams are wide, P ≈ B, such that A/P is equal to themean depth A/B.

Reynolds number R: Also in the last row of the table, the term which is the inverse of the Reynolds number, whatappears as UD in many definitions of Reynolds number, velocity multiplied by transverse dimension, in this caseis Q/P .

Remaining quantities: the remaining quantities Π2, Π3 andΠ5 are unimportant for pipe flows. For channel flowsit is possible to use as a basis the extensive experimental results obtained in the early 20th century for pipes.

Laminar flow | Transition Zone | Completely turbulent 0.1

0.05

Weisbachfrictionfactor

0.01

103 104 105 106 107 108

Reynolds Number R=UD/

0.07

0.02

0.01

Relativeroughness

0.001

0.0001

0.00001

Figure 2.8: The Moody diagram, showing graphically the solution of the Colebrook & White equation for λ as a function ofR with a parameter the relative roughness ε = d/D.

The most important experimental results for industrial pipes were obtained by Colebrook and White (?, ?). Theyobtained results for λ as a function of relative roughness ε = k/D, where D is pipe diameter (A/P = D/4 forcircular pipes) and Reynolds number R = UD/ν. They presented an equation connecting λ, ε, and R, which wasimplicit in λ, the Colebrook-White equation. In the pre-computer age, that implicit relation was difficult to solvenumerically. That has probably been the main reason that the approach has not been adopted. ? plotted the resultsfrom that equation, as shown in Figure 2.8.There are, however, explicit solutions that are not as well-known as theyshould be.

In channels, where the Reynolds number is usually so large that the flow is fully turbulent, ?obtained for on thebasis of a trapezoidal shape,

λ =1

2.032log−210

µ112.

k

A/P

¶≈ 1.3

ln2³112.

kA/P

´ , (2.28)

where the rounded factor of 1/12. comes from a recommendation from ?. This simple approximation could havebeen the basis for much hydraulics, as it gives an explicit formula for the resistance factor in terms of the relativeroughness k/(A/P ). Of course, determining the equivalent roughness k is a difficult problem, yet the knowledgeof the manner of variation with roughness gives us an important tool. This is plotted in figure 2.9. It is notablejust how little the resistance coefficient varies over a very wide range of roughnesses – over a 100-fold roughnessvariation the resistance varies by a factor of only 3.

A more general approximation has been obtained by ?, including the results from a number of open channelexperiments. Converting to natural logarithms here for simplicity, and rounding a coefficient ? gave the formulahere recommended for use in calculating the resistance coefficient:

λ =1.33

ln2µ

112.

kA/P +

³2.νPQ

´0.9¶ . (2.29)

25

0

0.02

0.04

0.06

1/1000 1/100 1/10

λ

Relative roughness k/(A/P )

Figure 2.9: Theoretical variation of λ with relative roughness

Yen stated that such a formula was applicable for ε = k/ (A/P ) < 0.05 and R = Q/Pν > 30 000. If one doesknow the effective roughness size k, this provides the best way of calculating the resistance coefficient.

2.3.5 Steady uniform flow

The simplest model of a river is where the channel is prismatic, with a constant bed and surface slope S0, and theflow is steady (∂/∂t = 0) and uniform (∂/∂x = 0). In equation (2.16) we introduced the symbol Λ, where fortypical streams, Λ = λ/8, such that in the long wave equations (2.22b) or (2.23b), the resistance term appearedas ΛPQ |Q| /A2. In this case of steady uniform flow, ∂η/∂x = −S = −S0, all terms except the gravity andresistance terms disappear from the momentum equation to give

ΛPQ2

A2= gAS0, (2.30)

giving the Weisbach-Chézy equation for the mean velocity U :

U =Q

A=

rg

Λ

A

PS0 = C

rA

PS0, (2.31)

where we have introduced the Chézy coefficient C =pg/Λ =

p8g/λ. The most notable features are that the

velocity varies likepA/P and like

√S0.

There is an interesting deduction from equation (2.31). We calculate the square of the Froude number (widely usedin open channel hydraulics) from it:

F 2 =U2

gA/B=

S0Λ

B

P,

and as for wide rivers B ≈ P we have the conclusion that F 2 ≈ S0/Λ, and as S0 is constant, and to firstapproximation Λ is constant, so that Froude number is roughly constant for any given stream, independent of theflow rate.

Gauckler-Manning-Strickler formula

The resistance formulation most widely used in practice is the Gauckler-Manning-Strickler approach. To relatethis to our formulation above, we consider its fundamental expression for a steady uniform flow:

U =Q

A=1

n

µA

P

¶2/3pS0 = kSt

µA

P

¶2/3pS0, (2.32)

in terms of the Manning coefficient n, which is dimensional, with units of L−1/3T, and the Strickler coefficientsimply kSt = 1/n. The use of the symbol kSt for the Strickler coefficient is an unfortunate typographical coinci-dence with the mean roughness size k. Comparing this equation (2.32) with the Weisbach-Chézy equation (2.31),the main difference is that here the velocity is modelled as varying like (A/P )2/3, rather than (A/P )1/2. For thetwo formulations (2.31) and (2.32) to agree, Λ can be expressed as

Λ =g

C2= gn2

µP

A

¶1/3=

g

k2St

µP

A

¶1/3. (2.33)

26

0.00

0.05

0.10

0.15

0.20

0.25

1/1000 1/100 1/10 1

n√g

k1/6

Relative roughness k/(A/P )

Yen, eqn (**)Bridging eqn (**)Strickler eqn with **=0.128

Figure 2.10: Theoretical variation of Manning’s n with relative roughness,showing its relative insensitivity to depth as measured by A/P

Now using this relation we obtain a theoretical expression for the variation of Manning’s n. Solving (2.33) andusing Λ = λ/8 gives

n =

sλ

8g

µA

P

¶1/6, (2.34)

into which we substitute Keulegan’s expression for λ, (2.28), or Yen’s equation (2.29) with infinite Reynoldsnumber to give the theoretical expression for the variation of the apparent Manning n with relative roughnessk/(A/P ):

n√g

k1/6≈ −0.40ln³112.

kA/P

´³k

A/P

´1/6 . (2.35)

This is plotted in figure 2.10, and it can be seen that over a wide range of relative roughnesses expected in nature,there is little variation of n√g/k1/6,leading to the widely-supposed justification for the G-M-S formula that thevalue of n is relatively invariant with depth. As an approximation, from the figure we might take the value ofn√g/k1/6 to be constant at 0.125, leading to the approximate formula for n:

n ≈ 0.125 k1/6

√g. (2.36)

Such an equation, with a numerical value of g implicitly substituted, is known as the Strickler equation, where kis taken to be the diameter of the bed material, such as d50, the diameter of the bed material such that 50% of thematerial by weight is smaller. If we substitute g = 9.8ms−2 we obtain n ≈ 0.040 d1/650 , where d50 is in metres,typical of the versions of the Strickler equation presented by ?, but where some confusion follows from using d50also in millimetres and feet.

This enables us to propose a formula by substituting the Strickler-type formula (2.36) into the G-M-S expression(2.32) to give the formula for discharge in terms of known channel quantities:

U =Q

A=

1

0.125 k1/6

µA

P

¶2/3pgS0. (2.37)

In real rivers, however, n varies with depth – it generally changes, as the nature of the roughness and the cross-section vary with depth. It is not a fundamental quantity – it was not obtained from the boundary friction, althoughequation (2.34) shows how it can be so related. It has no theoretical justification – it is an empirical approach whichwas convenient in a pre-computer age to to describe the problem approximately and conveniently, mimicking thebehaviour of the Weisbach form. If we accept that in more general problems the resistance coefficient varies, thenthere is no real reason to use Manning’s n rather than the more fundamental λ or Λ.

27

The G-M-S formula is widely used – and abused. The manner in which a value of n is adopted in practice is oftenmost unsatisfactory. Although equation (2.36) does provides a basis for calculating the resistance coefficient fromknowledge of the bed material, other more typical ad hoc approaches include:

• An Australian approach – using the telephone (“You did some work on River X years ago. What do you thinkn is on River Y (10km from X) for the reach between A and B?”).

• Tables of values in books on open channels, given channel conditions, for example pp116-123 of ?, ? forUS rivers, or ? for New Zealand Rivers. The photograph in figure 2.11 is of the Columbia River at Vernitain Washington, taken from Barnes. It has the lowest n = 0.024 of all the examples in the book. It is 500mwide and has boulders of diameter . . . well it doesn’t say. Pity, because it is the roughness size to river depthwhich determines the resistance characteristics. Presenting a picture is not much help – one has little idea ofthe underwater conditions and how variable they are.

Figure 2.11: Example of a figure from Barnes (1967)

Conveyance and Friction Slope

It is convenient to introduce the conveyance K (units of discharge L3T−1), so that the resistance term in themomentum equations appears as

−ΛP Q |Q|A2

= −gAQ |Q|K2

. (2.38)

From equation (2.33), the various definitions of K become

K =

rg

Λ

A3

P= C

rA3

P=1

n

A5/3

P 2/3= kSt

A5/3

P 2/3, (2.39)

such that in the form of equation (2.38) it is a convenient shorthand that includes the resistance coefficient ofwhatever law is being used, plus cross-sectional geometry terms.

A simple and important result for steady uniform flow is that

Q = K0

pS0, (2.40)

where the subscript 0 denotes the conveyance corresponding to the normal depth h0 of the uniform flow.

Many textbook presentations write the friction term in terms of a dimensionless quantity Sf = Q |Q| /K2, calledthe "friction slope", possibly better known as "resistance slope", so that the resistance term in the momentumequations appears as−gASf. It is also known as the "slope of the energy grade line", or the "head gradient", whichgives an uninformative and misleading picture, for in our momentum-based approach it is neither of those things.

Computations of steady uniform flow

As we know, steady flow does not change with time; uniform flow is where the depth does not change along thewaterway. For this to occur the channel properties also must not change along the stream, such that the channel isprismatic, and this occurs only in constructed canals. However in rivers if we need to calculate a flow or depth, itis common to use a cross-section which is representative of the reach being considered, and to assume it constantfor the approximate application of theory. This is the simplest problem we consider!

The Weisbach and Chézy equations and the Gauckler-Manning-Strickler forms (2.31) and (2.32) are formulae for

28

the discharge Q in terms of resistance coefficient, slope S0, area A, and wetted perimeter P :

Weisbach-Chézy : Q = A

r8g

λ

µA

P

¶1/2pS0 = A

r8g

λ

A

PS0 (2.41a)

G-M-S : Q =A

n

µA

P

¶2/3pS0 (2.41b)

in which both A and P are functions of the flow depth. Each equation show how flow increases with cross-sectionalarea and slope and decreases with wetted perimeter. If we know or can assume the resistance parameter, this isstraightforward. However if we have to calculate it from the Yen formula (2.29), which is in terms of Q, it has tobe done iteratively – see example 2.1 below.

Trapezoidal sections

γ

1P

h

B

W

Figure 2.12: Trapezoidal section showing important dimensions

Most canals are excavated to a trapezoidal section, and this is often used as a convenient approximation to rivercross-sections too. In many of the problems in this course we will consider the case of trapezoidal sections. Wewill introduce the terms defined in Figure 2.12: the bottom width is W , the depth is h, the top width is B, and theside slope, unusually defined to be the ratio of H:V dimensions is γ. From these the following important sectionproperties are easily obtained:

Top width : B =W + 2γh (2.42a)Area : A = h (W + γh) (2.42b)

Wetted perimeter : P =W + 2p1 + γ2h. (2.42c)

(Exercise: Obtain these relations).

The hydraulically-wide channel

Many rivers and canals are sufficiently wide that we can neglect the effects of the sides and for purposes of simplecalculations we say A ≈ Bh and P ≈ B, so that we model the section as a rectangle that is much wider than it isdeep, and it is called hydraulically-wide. We introduce the discharge per unit width q = Q/B, which is equal tothe integral of velocity over the depth. In this case equations (2.41a) and (2.41b) become

Weisbach-Chézy : q =

r8g

λh3/2

pS0 and (2.43a)

G-M-S : q =1

nh5/3

pS0, (2.43b)

and we see that q ∼ h1.5 or h1.67. An immediate deduction is, for example, that if a flow increases by a factor of10, the the depth increases by a factor of approximately 102/3 ≈ 4.6, while a 100-fold increase in flow means anincreased depth with a factor of 1002/3 ≈ 22.Computation of discharge

This is done by evaluating any of the equations. If the numerical value of the resistance coefficient is to becalculated from the Yen formula (2.29) it has to be done iteratively.

Example 2.1 Calculate the flow in a river of slope S0 = 10−3, approximated by a trapezoidal section with bottom width10m, side slopes 2 : 1, and depth 1m, if the mean roughness size is 1 cm, using

(a) the Yen formula (2.29) and the Weisbach-Chézy formula (2.41a) – we use the viscosity of water at 20C ν = 1 ×10−6m2s−1.

(b) the Strickler formula (2.36) and the G-M-S formula (2.41b) combined to give equation (2.37) above

With h = 1m, equation (2.42b) gives A = 1 (10 + 2× 1) = 12.0m2, equation (2.42c) gives P = 10 + 2√1 + 22 × 1 =

14. 47m.

29

(a) Weisbach

Firstly we calculate the λ, neglecting the viscosity term

λ =1.33

ln2 112.

kA/P + 2. νPQ

0.9=

1.33

ln2 112.

0.0112/14.47

= 0.0279

Calculate Q

Q = A8g

λ

A

PS0 = 12.0

8× 9.80.0279

12

14.470.001 = 18.3m3s−1

Repeat (i) and (ii), including that result in the viscosity term

λ =1.33

ln2 112.

0.0112/14.47

+ 2. 1×10−6×14.4718.3

0.9= 0.0280,

and we see that the viscosity correction is not worth incorporating.

(b) G-M-S/Strickler/Equation (2.37)

Q =A

0.125 k1/6A

P

2/3

gS0 =12

0.125× 0.011/612

14.47

2/3√9.8× 0.001 = 18.1m3s−1

Computation of normal depth

If the discharge, slope, and the appropriate roughness coefficient are known, equations (2.41a)-(2.41b) is a tran-scendental equation for the normal depth h, which can be solved by the methods for solving transcendental equa-tions. A simple method is to use direct iteration, whereby we arrange the equations in the form h = f(h), andsolve by repeated evaluation. For this to converge, f(h) must be a relatively slowly-varying function. Here wedevelop a procedure which seems to work well:

In the case of hydraulically-wide channels, we neglect the contributions of the side slopes so that neither the wettedperimeter P nor the breadth B vary much with depth h. Hence the quantity A(h)/h also does not vary stronglywith h. Hence we can rewrite the G-M-S expression:

Q =1

n

A5/3(h)

P 2/3(h)

pS0 =

√S0n

(A(h)/h)5/3

P 2/3(h)× h5/3,

where most of the variation with h is contained in the last term h5/3, and by solving for that term we can re-writethe equation in a form suitable for direct iteration

h =

µQn√S0

¶3/5× P 2/5(h)

A(h)/h,

where the first term on the right is a constant for any particular problem, and the second term is expected to bea relatively slowly-varying function of depth, so that the whole right side varies slowly with depth – a primaryrequirement that the direct iteration scheme be convergent and indeed be quickly convergent.

Here, in addition, we include the Strickler formula (2.36), which is equation (2.37) re-written

h =

µ0.125

Q√S0

k1/6√g

¶3/5× P 2/5(h)

A(h)/h. (2.44)

Experience with typical trapezoidal sections shows that this works well and is quickly convergent. However, it alsoworks well for flow in circular sections such as sewers, where over a wide range of depths the mean width does notvary much with depth either. For small flows and depths in sewers this is not so, and a more complicated methodsuch as the secant method might have to be used.

Example 2.2 Calculate the normal depth in the previous example 2.1, for the calculated discharge of 18.1m3s−1 using theG-M-S formulation, with equation (2.44). We pretend we do not know the answer of 1m and for an initial estimate we choose1.5m. We have A = h (10 + 2h), P = 10 + 4.472h, giving the scheme

h = 0.125Q√S0

k1/6√g

3/5

× (10 + 4.472h)2/5

10 + 2h

= 0.12518.1√0.001

0.011/6√9.8

3/5

× (10 + 4.472h)2/5

10 + 2h

= 4.125(10 + 4.472h)2/5

10 + 2h

30

and starting with h = 1.5we have the sequence of approximations: h = 0.979m, h = 1.002m, h = 1.001m, and the methodhas converged to this result, quite satisfactorily in its simplicity and speed (it did not converge to the exact result because thespecified discharge was rounded).

2.3.6 Steady gradually-varied flow

A common problem in river engineering is, for example, how far upstream water levels might be changed, andhence flooding possibly enhanced, due to downstream works such as the installation of a bridge or other obstacles.Far away from the obstacle or control, the flow may be uniform, but generally it is variable. The transition betweenconditions at the control or point of known level, and where there is uniform flow is described by the Gradually-Varied Flow Equation, which is an ordinary differential equation for the water surface height. The solution willapproach uniform flow if the channel is prismatic, but in general we can treat non-prismatic waterways also. Thesteady flow approximation is often used as a first approximation, even when the flow is unsteady, such as in floods.

The differential equation

Consider the mass conservation equation for steady flow, when ∂/∂t ≡ 0, and equation (2.1) becomes

dQ

dx= i,

with the solution obtained by integration

Q(x) = Q0 +

Z x

x0

idx,

where at an upstream station x0 the discharge is Q0, the extra discharge just being given by the integral of theinflow i.

The momentum equation (2.23b) for ∂Q/∂t = 0 and ∂Q/∂x = i, and assuming Q positive, becomesµgA− β

Q2B

A2

¶dη

dx= β

Q2B

A2S − ΛP Q2

A2− 2βQ

Ai, (2.45)

which is a first-order ordinary differential equation for η(x), provided we have evaluated Q(x), and that we knowhow the geometric quantities A and B depend on surface elevation at each point. This is the Gradually-VariedFlow Equation. The equation may be solved numerically using any of a number of methods available for solvingordinary differential equations which are described in books on numerical methods. These methods are usuallyaccurate and can be found in many standard software packages.

It is surprising that books on open channels do not recognise that the problem of numerical solution of thegradually-varied flow equation is actually a standard numerical problem, although practical details may makeit more complicated. Instead, such texts use methods such as the ”Direct step method” and the ”Standard stepmethod”, which can become complicated. There are several software packages such as HEC-RAS which use suchmethods, but solution of the gradually-varied flow equation is not a difficult problem to solve for specific problemsin practice if one recognises that it is merely the solution of a differential equation.

In sub-critical (relatively slow) flow, the effects of any control can propagate back up the channel, and so it is thatthe numerical solution of the gradually-varied flow equation also proceeds in that direction. On the other hand, insuper-critical flow, all disturbances are swept downstream, so that the effects of a control cannot be felt upstream,and numerical solution also proceeds downstream from the control.

No inflow:

If there is no inflow, i = 0 and Q = Q0, a constant, throughout. Dividing both sides of equation (2.45) by gAgives

dη

dx= F 2

βS − ΛP/B1− βF 2

=βS − ΛP/B1/F 2 − β

, (2.46)

where, unusually for lectures on flow with a free surface, it has taken us until now to define the Froude number

F 2 =Q2B

gA3=(Q/A)

2

g (A/B),

the ratio of the mean velocity Q/A squared to g times the mean depth A/B. In this course we call F 2 the Froudenumber, and not F , as the latter quantity occurs quite rarely, and F 2 expresses the real relative importance ofinertia terms.

31

The Gradually-Varied Flow Equation (GVFE) in the form of equation (2.46) seems simple – deceptively simple.For example, β can be taken as a constant; S might be a function of x, but we probably do not have enoughinformation to express it as a function of η; many open channels are much wider than their depth, and so P ≈B and P/B ≈ 1. This leaves most of the functional variation with η on the right side in the term 1/F 2 =gA3/Q2B in which, for practical river problems the dependence of A and B on the local elevation η is actuallyquite complicated.

Constant slope:

As a special case, consider a channel with a bed of constant slope S = S0. It is simpler to use as a variable thedepth of flow h, where h = η−Z, where Z is the elevation of the bed at a section, so that dZ/dx = −S0. Equation(2.46) becomes

dη

dx=

dh

dx+

dZ

dx=

dh

dx− S0 =

βS0 − ΛP/B1/F 2 − β

.

Solving for dh/dx and introducing the conveyance gives the GVFE for a prismatic canal of constant slope:

dh

dx=

S0 −Q2/K2

1− βF 2. (2.47)

Properties of gradually-varied flow and the governing equations

Normal depth h0

h(x)

Figure 2.13: Subcritical flow retarded by a gate, showing typical behaviour of the free surface and, if the channel is prismatic,decaying upstream to normal depth

• The equation and its solutions are important, in that they tell us how far the effects of a structure or works in oron a stream extend upstream or downstream.

• It is an ordinary differential equation of first order, hence one boundary condition must be supplied to obtainthe solution. In sub-critical flow, this is the depth at a downstream control; in super-critical flow it is the depthat an upstream control.

• If the channel is prismatic, far from the control, the flow is uniform, and the depth is said to be normal.

• In general the boundary depth is not equal to the normal depth, and the differential equation describes thetransition from the one to the other. The solutions look like exponential decay curves, and below we will showthat they are, to a first approximation. Figure 2.13 shows a typical sub-critical flow in a prismatic channel,where the depth at a control is greater than the normal depth.

• The differential equation is nonlinear, and the dependence on h is complicated, such that analytical solution isnot possible, and we will usually use numerical methods.

• However, a small-disturbance approximation can be made, the resulting analytical solution is useful in provid-ing us with some insight into the quantities which govern the extent of the upstream or downstream influence.

• If the flow approaches critical flow, when βF 2 → 1, then dh/dx→∞, and the surface becomes vertical. Thisviolates the assumption we made that the flow is gradually varied and the pressure distribution is hydrostatic.This is the one great failure of our open channel hydraulics at this level, that it cannot describe the transitionbetween sub- and super-critical flow.

Approximate analytical solution by linearising

Whereas the numerical solutions give us numbers to analyse, sometimes very few actual numbers are required,such as merely estimating how far upstream water levels are raised to a certain level, the effect of downstreamworks on flooding, for example. Here we introduce a different way of looking at a physical problem in hydraulics,

32

where we obtain an approximate mathematical solution so that we can provide equations which reveal to us moreof the nature of the problem than do numbers. This work was originally done by ?.

This is carried out by ”linearising” about the uniform flow in a prismatic channel, i.e. by considering smalldisturbances to that flow. Consider the water depth to be written

h(x) = h0 + h1(x), (2.48)

where we use the symbol h0 for the constant normal depth, and h1(x) is a relatively small departure of the surfacefrom that. We use the governing differential equation (2.47) (although our notation has obscured the fact that Fand K are functions of h). Substitute equation (2.48) into the equation and writing numerator and denominator asTaylor series in h1:

dh0dx

+dh1dx

=

¡S0 −Q2/K2

¢0+ h1

¡ddh

¡S0 −Q2/K2

¢¢0+ terms in h21

(1− βF 2)0 + h1¡ddh (1− βF 2)

¢0+ terms in h21

.

Now, as h0 is constant, dh0/dx = 0. Also, from equation (2.40),¡S0 −Q2/K2

¢0= 0 and the first term in the

numerator is zero. Now evaluating d¡S0 −Q2/K2

¢/dh = 2Q2/K3 dK/dh, and considering just the first term

top and bottom, neglecting all higher order powers of h1 as it is small, we find

dh1dx≈ h1

2Q2/K30 (dK/dh)0

1− βF 20= μ0 h1, (2.49)

where

μ0 =2S0

1− βF 20

1

K

dK

dh

¯0

, (2.50)

and where we have used Q2/K20 = S0.

Equation (2.49) is a linear differential equation which we can solve analytically by separation of variables, giving

h1 = Ceμ0x, and h = h0 + Ceμ0x, (2.51)

where C is a constant which would be evaluated by satisfying the boundary condition at the control, and where μ0is a constant decay rate given by equation (2.50).

This shows that the water surface is actually approximated by an exponential curve passing from the value of depthat the control to normal depth. As dK/dh is positive, and for subcritical flow 1 − βF 20 is also positive, equation(2.50) shows that μ0 is positive, and far upstream as x → −∞, the water surface decays to normal depth. Forsupercritical flow, 1− βF 20 < 0, μ0 is negative, and the water surface approaches normal depth downstream.

Now we obtain an approximate expression for the rate of decay μ0. From the Gauckler-Manning-Strickler formulafor a wide channel, a common approximation, we can show that K ∼ h5/3, dK/dh ∼ 5/3 × h2/3, and for slowflow βF 20 ¿ 1, we find

μ0 ≈10

3

S0h0

. (2.52)

The larger this number, the more rapid is the decay with x. The formula shows that more rapid decay occurs withsteeper slopes (large S0), and smaller depths (h0). Hence, generally the water surface approaches normal depthmore quickly for steeper and shallower streams, and the effects of a disturbance can extend a long way upstreamfor mild slopes and deeper water.

Let us use equation (2.52) to calculate the distance upstream that the disturbance decays by 1/2, that is, exp (μ0x) =0.5. We find

10

3

S0x

h0= ln 0.5 giving

x

h0=3 ln 0.5

10

1

S0.

For S0 = 10−4 and a stream 2m deep, the distance is 4 km. For the stream disturbance to decay to 1/16 = (1/2)4of the original, this distance is 4 × 4 km = 16 km. These are possibly surprising results, showing how far thebackwater effect extends.

Numerical solution of the gradually-varied flow equation

Consider the gradually-varied flow equation (2.47)

dh

dx=

S0 −Q2/K2

1− βF 2

33

where F 2(h) = Q2B(h)/gA3(h). The equation is a differential equation of first order, and to obtain solutionsit is necessary to have a boundary condition h = h0 at a certain x = x0, which will be provided by a control.The problem may be solved using any of a number of methods available for solving ordinary differential equationswhich are described in books on numerical methods.

The direction of solution is very important. For mild slope (sub-critical flow) cases the surface decays somewhatexponentially to normal depth upstream from a downstream control, whereas for steep slope (super-critical flow)cases the surface decays exponentially to normal depth downstream from an upstream control. This means that toobtain numerical solutions we will always solve (a) for sub-critical flow: from the control upstream, and (b) forsuper-critical flow: from the control downstream.

Euler’s method The simplest (Euler) scheme to advance the solution from (xi, hi) to (xi +∆xi, hi+1) is

xi+1 ≈ xi +∆xi, where∆xi is negative for subcritical flow, (2.53a)

hi+1 ≈ hi +∆xidh

dx

¯i

= hi +∆xiS0 −Q2/K2(hi)

1− βF 2(hi)+O

³(∆xi)

2´. (2.53b)

This is the simplest but least accurate of all methods – yet it might be appropriate for open channel problemswhere quantities may only be known approximately. One can use simple modifications such as Heun’s methodto gain better accuracy, or use Richardson extrapolation – or even more simply, just take smaller steps∆xi.

Richardson extrapolation There is an interesting method for obtaining more accurate solutions from compu-tational schemes for almost any physical problem. Applied to the Euler scheme in the present context, as thelocal truncation error is O

³(∆xi)

2´

, so that after a number of steps proportional to 1/∆xi, the actual solu-

tion has an error O³(∆xi)

1´

so that n = 1, a first-order scheme. In the present problem, if we use a constantspace step ∆ to obtain the first solution 1, then another constant space step half that ∆/2, requiring twice thenumber of steps, then r = 1/2, and it can be shown that a better estimate of the solution is

hi ≈ 2h2i(∆/2)− hi(∆), (2.54)

which is trivially applied to each or any of the steps. The notation h2i(∆/2) is intended to show that the samepoint in physical space is used; with half the step size it will now take twice the number of steps to reach thatpoint.

Heun’s method In this case the value of hi+1 calculated from Euler’s method, equation (2.53b), is used as a firstestimate of the depth at the next point, written h∗i+1, then the value of the derivative at that point

¡xi+1, h

∗i+1

¢is calculated. Heun’s method is then to use the mean slope over the step, the mean of the initial value andthat at the other end of the interval calculated by the Euler step. Then, the change over the step is calculated,multiplying that mean slope by the step length. That is,

hi+1 ≈ hi +∆xi2

Ãdh

dx

¯(xi,hi)

+dh

dx

¯(xi+1,h∗i+1)

!

= hi +∆xi2

µS0 −Q2/K2(hi)

1− βF 2(hi)+

S0 −Q2/K2(h∗i+1)1− βF 2(h∗i+1)

¶+O

³(∆xi)

3´. (2.55)

Now, the error of a single step is proportional to the third power of the step length and the error at any pointwill be proportional to the second power.

Neither of these two methods are presented in hydraulics textbooks as alternatives, yet they are simpleand flexible, and reveal the nature of what we are doing. The step ∆xi can be varied at will, to suit possibleirregularly spaced cross-sectional data. In many situations, where F 2 ¿ 1, we can ignore the βF 2 term in thedenominators, giving a notationally simpler scheme.

Predictor-corrector method – Trapezoidal method This is simply an iteration of the last method, wherebythe step in equation (2.55) is repeated several times, at each stage setting h∗i+1 equal to the updated value ofhi+1. This gives an accurate and convenient method, and it is surprising that it has not been used.

Higher-order methods One of the aims here has been to emphasise that all that is being done is to solvenumerically a differential equation, and any method can be used, for which reference can be made to any bookon numerical solution of ordinary differential equations. There are sophisticated methods such as high-orderRunge-Kutta methods and predictor-corrector methods. However, in the case of open channel hydraulics therewill usually be some variation of parameters along the channel that such sophistication is unnecessary.

34

1.6

1.8

2

2.2

2.4

2.6

-1000 -800 -600 -400 -200 0

Surfaceelevationη (m)

x (m)

Accurate: 4th order Runge-KuttaRK-4, n in error by 5%

Euler, eqn (2.53)Euler-Richardson, eqns (2.53) & (2.54)

Direct step, eqn (2.57)Modified direct step, eqn (2.58)

Heun, eqn (2.55)Trapezoidal, eqn (2.55)+

Figure 2.14: Backwater curve computed with various schemes; the dashed line is the surfacefor uniform flow.

-0.03

-0.02

-0.01

0

0.01

-800 -600 -400 -200 0

ElevationError(m)

x (m)

Figure 2.15: Errors for the different schemes; symbols as for Figure 2.14.

35

Comparison of schemes

To compare the performance of the various numerical schemes, Example 10-1 of ? was solved using each. Allquantities specified by Chow were converted to SI units and rounded to the numbers shown here: a flow of11.33m3 s−1 passes down a trapezoidal channel of gradient S0 = 0.0016, bed width 6.10m and channel sideslopes 0.5, g = 9.8m s−2, the quantity α or β = 1.10, and Manning’s n = 0.025. At x = 0 the flow is backedup to a depth of 1.524m, and the backwater curve was computed for 1000m. Results for the water surface profileare shown in Figure 2.14, while Figure 2.15 shows the errors. Some 10 computational steps were used for eachscheme.

The basis of accuracy is shown by the solid line, from a highly-accurate Runge-Kutta 4th order method (see, forexample, p372 of ?, ?, or almost any other book on numerical methods). It is not recommended here as a method,however, as it makes use of information from three intermediate points at each step, information which in non-prismatic channels is usually not available. The rest of the results are shown in reverse order of accuracy. Thedotted line is that with the same numerical method, but where the roughness n was changed by -5%, to give anidea of the effect of uncertainty in knowledge of that quantity. The maximum error, of about−3 cm, in the normaldepth, is greater than any of the other methods, so that a preliminary conclusion is that if the roughness is notknown to within 5%, almost certainly the case in practice, then any of the methods can be used.

It can be seen that Euler’s method, eqn (2.53), was the least accurate, as expected. As it is a first-order scheme,halving the step size would halve the error. Actually doing just that and then applying Richardson extrapolation,equation (2.54), gave the second most accurate of all the methods, with an error of about 1mm. The most accurateof all was the Trapezoidal method, namely using Heun’s method, equation (2.55) and iterating the final step. Allthe other methods, Heun’s, and the two inverse formulations, equations (2.57) and (2.58), gave errors intermediatebetween the two extremes. It is interesting that the two traditional methods were accurate, notably the traditionalinverse formulation over the modified version presented in this work; and the Trapezoidal method, the basis for theso-called "standard" method.

The results show the disadvantages of the inverse formulation (Direct Step), that the distance between compu-tational points becomes large as uniform flow is approached, and the points are at awkward distances. In thisexample relatively few steps were chosen (roughly 10) so that the numerical accuracies of the methods could bedistinguished visually. The computational effort was very small indeed.

In this problem the analytical solution (2.51) gave poor results. This was because the depth at the control was ratherlarger (50%) than the normal depth, and the linearisation adopted, for small departures from normal depth, was notaccurate. In general, however, it does give a simple approximate result for the rate of decay and how far upstreamthe effects of the control extend. For many practical problems, this accuracy and simplicity may be enough.

Traditional methods

Here we present methods for comparison as they are given in textbooks.

Total energy line

2 1

Sub-critical flow

αU22 /2g Sf∆x

αU21 /2g

h2

h1

∆x

S0∆x

Figure 2.16: Elemental section of waterway

Derivation of the gradually-varied flow equation using energy Consider the elemental section of waterwayof length ∆x shown in Figure 2.16. We have shown stations 1 and 2 in what might be considered the reverseorder, but for the more common sub-critical flow, numerical solution of the governing equation will proceedback up the stream. Considering stations 1 and 2:

Total head at 2 = H2

Total head at 1 = H1 = H2 −HL,

36

and we introduce the concept of the friction slope Sf which is the gradient of the total energy line such thatHL = Sf ×∆x. This gives

H1 = H2 − Sf∆x,

and if we introduce the Taylor series expansion for H1:

H1 = H2 +∆xdH

dx+ . . . ,

substituting and taking the limit∆x→ 0 gives

dH

dx= −Sf, (2.56)

an ordinary differential equation for the head as a function of x, and we use the approximation that the frictionslope is given by Q2/K2.

Direct step method Textbooks do present the Direct Step method, which is applied by taking steps in the heightand calculating the corresponding step in x. It has practical disadvantages, such that it is applicable only toprismatic sections, results are not obtained at specified points in x, and as uniform flow is approached the ∆xbecome infinitely large. However it is a surprisingly accurate method.

The reciprocal of equation (2.56) isdx

dH= − 1

Sf.

The numerical method as set out in textbooks is to approximate the differential equation (2.56) by the finitedifference expression

∆xi =∆Hbi

S0 −Q2/K2 (Hb)(2.57a)

=∆Hbi

S0 − 12Q

2¡K−2i +K−2i+1

¢ (2.57b)

where the overbar in equation (2.57a) indicates the mean of the friction slope at beginning and end of thecomputational interval, which finds its mathematical expression in equation (2.57b), where the shorthand Ki

has been used for K (Hbi).While this is a plausible approximation, it is not mathematically consistent. It is an apparent attempt to

develop a Trapezoidal method. Applying Heun’s method as formally presented in equation (2.55) automaticallyleads to the Trapezoidal scheme which in this case gives

xi+1 = xi +∆Hb,i

2

µ1

S0 −Q2/K2i

+1

S0 −Q2/K2i+1

¶+O

³(∆Hb,i)

3´, (2.58)

The term O (. . .) is a Landau order symbol, showing in this case that the local truncation error is proportionalto the third power of the step, which is a strong result and explains the accuracy of the method. Since the use ofa step size of∆Hb,i over the whole computational domain requires a number of steps proportional to 1/∆Hb,i,the global error in this case will be of order (∆Hb,i)

2, thus the global error, or accumulated error at the endof that integration interval will be of this order, so that halving the step should improve the global accuracy byabout a factor of 4.

In view of the method presented here, the method is no longer applicable only to prismatic sections, butthe practical disadvantages remain that results are not obtained at specified points in x, and as uniform flow isapproached the∆x become infinitely large.

Standard step method The nomenclature "standard" is not very descriptive. Presumably it refers to finding thesolution for η at specified values of x, rather than the other way round, for which the term "direct", as above,is even worse. This is an implicit method, requiring numerical solution of a transcendental equation at eachstep. It can be used for irregular channels, and is rather more general. In this case, the distance interval ∆x isspecified and the corresponding depth change calculated. In the Standard step method the procedure is to write

∆H = −Sf∆x,

and then write it asH2(h2)−H1(h1) = −∆x

2(Sf1 + Sf2) ,

37

for sections 1 and 2, where the mean value of the friction slope is used. This gives

αQ2

2gA22+ Z2 + h2 = α

Q2

2gA21+ Z1 + h1 − ∆x

2(Sf1 + Sf2) ,

where Z1 and Z2 are the elevations of the bed. This is a transcendental equation for h2, as this determines A2,P2, and Sf2. Solution could be by any of the methods we have had for solving transcendental equations, suchas direct iteration, bisection, or Newton’s method.

Although the Standard step method is an accurate and stable approximation, the lecturer considers it unnec-essarily complicated, as it requires solution of a transcendental equation at each step. It would be much simplerto use a simple explicit Euler or Heun’s method as described above.

2.3.7 Long wave equation in terms of volume upstream V

It is possible to simplify the usual approach of considering two long wave equations in two unknowns by intro-ducing a quantity that satisfies one of the equations exactly. We consider the (A,Q) formulation of the long waveequations, equations (2.22). In partial derivative terms we introduce the function V (x, t) which is the total volumeof fluid upstream of point x at time t, and use equations (2.5) in §2.3.1.4 so that we write:

∂V

∂x= A and (2.59a)

∂V

∂t= Q(x0, t) +

Z x

x0

i(x0) dx0 −Q, (2.59b)

where Q(x0, t) is the inflow at x0, the upstream end of the flow domain, which has been introduced by FatemehSoroush (2011, Personal Communication), improving on an earlier definition. With this addition V can now moreconsistently be understood as the total volume in the channel. The derivative of volume V with respect to x isclearly the local cross-sectional area, as shown, while the rate of change of V with time is equal to the rate atwhich flow is entering, Q(x0, t) +

R xx0i(x0) dx0 minus the discharge Q at point x which is the rate at which flow is

passing point x, and hence is no longer upstream.

Equations (2.59) can be solved to give

A =∂V

∂xand (2.60a)

Q = Q(x0, t) +

Z x

x0

i(x0) dx0 − ∂V

∂t, (2.60b)

so that both A and Q have been expressed as derivatives of V . Substituting into the mass conservation equation(2.22a):

∂

∂t

µ∂V

∂x

¶+

∂

∂x

µZ x

x0

i dx0 − ∂V

∂t

¶= i,

so that the left side becomes i, which equals the right side, and the mass conservation equation is identicallysatisfied! The momentum conservation equation (2.22b), with all signs reversed becomes:

∂2V

∂t2+ 2β

Q

A

∂2V

∂x∂t+

µβQ2

A2− gA

B

¶∂2V

∂x2= ΛP

¡R xi dx0 − ∂V/∂t

¢2(∂V/∂x)

2 − gAS +

Z x ∂i

∂tdx0 + 2β

Q

Ai. (2.61)

where for notational simplicity, single symbols Q and A have been retained in coefficients of derivatives. Themomentum equation has become a second-order partial differential equation in terms of the single variable V .This makes theoretical manipulations easier and provides insight into the nature of solutions of the equation. Thisis anticipated by the comment that the first three terms on the left actually look like a wave equation – the traditionalforms of equations (2.22) and (2.23) do not look like wave equations. However the single equation certainly doeslook complicated – it will be of most use when we consider approximations.

The case of no inflow i = 0 is rather simpler, and is adequate for our purposes. Equation (2.61) becomes

∂2V

∂t2+ 2β

Q

A

∂2V

∂x∂t+

µβQ2

A2− gA

B

¶∂2V

∂x2= ΛP

(−∂V/∂t)2(∂V/∂x)

2 − gAS. (2.62)

This equation, which is a single equation in a single unknown and could be used for flood routing and wavepropagation studies. Here we will use it first to obtain an approximate equation which is then solved to givesolutions that show the real behaviour of long waves.

38

2.3.8 The slow-change/slow-flow approximation

In many river and canal problems, such as flood propagation, change is slow and the flow is relatively slow suchthat the Froude number is small. In the momentum equations we now neglect the time derivatives and terms of theorder of the Froude number squared. This is usually a very good approximation.

Formulation 1 – area A

Consider Formulation 1 for the momentum equation (2.22b) in terms of area A,

gA

B

∂A

∂x= gAS − gA

Q |Q|K2

,

where we have written the resistance term using the conveyance K as introduced in equation (2.38). Restrictingour attention to uni-directional flows, such that Q |Q| = Q2 we find that the momentum equation becomes simply

Q = K

rS − 1

B

∂A

∂x, (2.63)

an explicit formula for discharge in terms of resistance and geometrical quantities. Substituting forQ from equation(2.63) into the corresponding mass conservation equation (2.22a) gives

∂A

∂t+

∂

∂x

ÃK

rS − 1

B

∂A

∂x

!= i, (2.64)

so that we have a single partial differential equation in the geometric quantity area A.

Formulation 2 – surface elevation η

If, instead, we use Formulation 2 for the momentum equation (2.23b) in terms of stage η the low-inertia approxi-mation is simply

Q = K

r−∂η∂x

, (2.65)

which seems surprisingly simple. We might have written this last equation down assuming that it was a plausibleengineering approximation – it is a surprise that it is actually an approximation to the momentum equation – andactually a very good one. Our only problem seems to be to get used to the minus sign under the square root; wejust have to remember that for uni-directional flow ∂η/∂x is always negative. Substituting into the correspondingmass conservation equation (2.23a) gives

∂η

∂t+1

B

∂

∂x

ÃK

r−∂η∂x

!=

i

B, (2.66)

another single partial differential equation in terms of elevation η. It is remarkable that these simple equations areaccurate enough for most routing purposes.

Formulation 3 – upstream volume V

If we were only given information in terms of water level variation at boundaries and only required water levelvariations as output, these equations (2.64) and (2.66) might be enough for our purposes. However, often infor-mation in the form of an inflow hydrograph in terms of Q(x0, t) is given, difficult to include in these. Now ourconcept of upstream volume V becomes useful. Here we can just substitute A = ∂V/∂x into equation (2.64) togive

∂2V

∂t∂x+

∂

∂x

ÃK

rS − 1

B

∂2V

∂x2

!= i,

and integrating with respect to x gives us the equation in terms of V :

∂V

∂t+K(Vx)

sS − 1

B(Vx)

∂2V

∂x2=

Z x

x0

i dx, (2.67)

where we have shown that both surface width B and conveyance K are functions of the cross-sectional area,which we write as A = ∂V/∂x = Vx. This Volume Routing equation might be useful in a range of hydrologicand hydraulic computations, replacing the solution of the long wave equations. It is a nonlinear partial differentialequation which is a single equation in a single variable. The only approximations that have been made is that F 2is small and variation is slow.

39

Boundary conditions:

At an upstream boundary x0 we might have a given inflow as a function of time Q(x0, t). From equation (2.60b),when x = x0 we have simply V (x0, t) = 0 for all t, which makes physical sense – there is never any volumeupstream of the upstream boundary.

At control points, such as a downstream structure we will usually have some relationship between Q and A suchas provided by weir formulae, or resistance formulae, which give ∂V/∂t as a function of ∂V/∂x there. As part ofthe solution we will have to differentiate numerically to give the latter and then integrate to give the updated valueof V . At open boundaries, where there is no control point, we may simply be able to apply the partial differentialequation as if it were an interior point.

Initial conditions:

To provide the initial conditions in the interior of the computational region, we consider the steady state formof equation (2.67), noting that ∂V/∂t is not zero for steady flow, as volume is continually passing, but instead∂V/∂t = −Q0 where Q0 is the steady flow (here for i = 0), giving:

d2V

dx2= B(Vx)

µS − Q20

K2(Vx)

¶. (2.68)

This is just a low-inertia version of the gradually-varied flow equation (2.46)

dη

dx= F 2

βS − ΛP/B1− βF 2

=βS − ΛP/B1/F 2 − β

.

Equation (2.68) is slightly more complicated than the conventional formulation, as we have to solve the second-order equation numerically, which is usually done by introducing a subsidiary variable dV/dx – which is A in ourformulation.

100

200

300

400

96 00 24 48 72

Q

(m3s−1)

Time (h)

(a) Steep slope 0.001InflowLong wave eqnsEqn (2.67) (obscured)

24 48 72 96Time (h)

(b) Gentle slope 0.0001InflowLong wave eqnsEqn (2.67) (obscured)

Figure 2.17: Inflow and computed outflow hydrographs for two model riverswith (a) a steep slope and (b) a gentle slope

Consider the two cases in figure 2.17 computed for a hypothetical trapezoidal channel with bottom width 40m,side slopes H:V 0.5, Manning’s n = 0.035, and with an inflow hydrograph with a base flow of 100m3s−1, time topeak 24 h, and peak flow of 400m3s−1. The figure shows the inflow and outflow hydrographs of a 100 km reachobtained from numerical solutions of the nonlinear long wave equations plus solutions of the slow change / slowflow equation, an approximation to the full equations. Part (a) of the figure is for a steep bed slope of 0.001, (witha Froude number of the initial flow F 2 ≈ 0.1 – and remember that F 2 is relatively independent of flow, so it would

40

hold for the flood peak too) (b) for a gentler slope of 0.0001, when F 2 ≈ 0.01. The accuracy of the approximationis remarkable, especially for the steeper slope and larger Froude number.

2.3.9 A further approximation – the advection diffusion equation

We have seen that in the case of a common situation in river engineering, where the change due to input is slow andwhere the Froude number of the flow is not large, that the equations simplify. Now we obtain a further approxima-tion by linearising, to show that the equations exhibit advection-diffusion behaviour, and simple deductions can bemade about the nature of wave propagation in waterways.

Advection-diffusion equation

This is a an approximation which was first obtained by Hayami in 1951 which gives considerable insight into thenature of flood and wave propagation in rivers. It can be obtained by linearising any of the low-inertia equations(2.64), (2.66) or (2.67) about a uniform flow of slope S0, to give an equation in terms of depth or discharge. Itis slightly more general to take the upstream volume formulation (2.67) and to linearise that. In fact, we alreadyhave the solution to hand, for if we ignore the inertial terms in the linear Telegrapher’s equation (2.84), simply byignoring second time derivative terms and those inertial terms with a coefficient β, giving

2σ

µ∂φ

∂t+ c

∂φ

∂x

¶− gA0

B0

∂2φ

∂x2= 0, (2.69)

where as before, φ is the perturbation upstream volume. This can be written

∂φ

∂t+ c|z

Kinematicwave speed

∂φ

∂x= ν|z

Diffusioncoefficient

∂2φ

∂x2(2.70)

such that the equation governing the propagation of floods and long waves for low Froude number is an advection-diffusion equation,where the coefficients c and ν are functions of the underlying uniform flow. The kinematic wavespeed is given by equation (2.86), but we simplify it here by neglecting variation of Λ with Q:

c =3

2U0

µ1− 1

3

A0Λ0P0

∂ (ΛP )

∂A

¯0

¶. (2.71)

It can be shown in terms of the conveyance that

c =dQ0dA0

=pS0

dK

dA

¯0

=

√S0B0

dK

dh

¯0

, (2.72)

where Q0 = K0

√S0,and dK/dh is the derivative of the conveyance with respect to the depth h.

The coefficient of the second derivative ν in equation (2.70) is a diffusion coefficient with units of L2T−1 and isgiven by

ν =gA0/B02σ

=gA0/B0 × U0

2gS0=

Q02B0S0

=K0

2B0√S0

, (2.73)

on using the uniform flow relationship Q0 = K0

√S0.

The behaviour of solutions

Solutions of equations with advection and diffusion terms are well known, and this makes possible a physicaldescription of wave motion in waterways. Solutions of the advection equation

∂φ

∂t+ c

∂φ

∂x= 0 (2.74)

are easily shown to be a waveform, any waveform, travelling in the positive x direction with velocity c. That is,φ = f(x−ct) is a solution of equation (2.74), where f() is any function. Also, solutions of the diffusion equation

∂φ

∂t= ν

∂2φ

∂x2(2.75)

have a simple mathematical structure. It can be shown that any disturbance of a certain wavelength is damped atan exponential rate where the exponent is inversely proportional to the square of the wavelength, such that shortdisturbances and irregularities are damped very quickly, and the process of diffusion is such as to smooth solutions.In the case of the advection-diffusion equation (2.70) the processes are combined, such that waves are advected ata speed c and the wave profile undergoes diffusion at the same time. This is illustrated in Figure 2.18, which is thesolution for an infinite channel with a single disturbance which is carried downstream and diffused.

41

Figure 2.18: Analytical solution of advection-diffusion equation

It is useful to get a feel for nature of wave propagation in waterways. Here we provide a simple tool for estimatingthe relative importance of diffusion. If we were to scale the advection-diffusion equation (2.70) such that it wasin terms of dimensionless variables X = x/L, where L is the length of reach, such that X goes from 0 to 1,and a dimensionless time scale T = tc/L,such that T = 1 is the time taken for a wave to transit the pool, thensubstituting into equation (2.70) gives

∂φ

∂T

dT

dt+ c

∂φ

∂X

dX

dx= ν

∂2φ

∂X2

µdX

dx

¶2, then

∂φ

∂T

c

L+ c

∂φ

∂X

1

L= ν

∂2φ

∂X2

1

L2, and multiplying through:

∂φ

∂T+

∂φ

∂X=

ν

cL

∂2φ

∂X2

and the importance of the diffusion term to the other terms is the dimensionless Péclet number ν/cL. This lookslike the inverse of a Reynolds number (which is correct – the Reynolds number is the inverse of a dimensionlessviscosity or diffusion number). Now we substitute in the approximations for a wide channel, where A0 = B0h0,where h0 is the depth, giving from equations (2.71) and (2.73):

A measure of the importance of diffusion =ν

cL=

U0A02B0S0

2

3U0 L≈ 13

h0S0L

.

This shows us the effects of diffusion very simply, as h0 is the depth of the stream, and S0L is the amount bywhich a stream drops over the reach of interest, we have

A measure of the importance of diffusion ≈ 13× Depth of stream

Drop of stream.

This shows a surprising result, that for streams which are steep and the water depth relatively shallow, such assteep mountain streams, the gravity and friction terms are in balance, and in this high friction limit, apparentlyparadoxically, the flood wave moves as a kinematic wave with little diminution. On the other hand, where slopesare gentle and friction less, such as in irrigation channels, the effects of diffusion are stronger. This is an unusualand unexpected result, and opposes the intuitive practical interpretation of the behaviour of waves in gently-slopingrivers and channels that they travel without a great deal of diminution due to friction. In fact, waves in a waterwaywith a mild slope may be markedly diminished in height and spread out much more in space and time.

We have shown that it is possible to formulate a low-inertia model such that it is a good approximation in mostcases of rivers and canals, and hence is worthy of further examination, as it is considerably simpler than the fullequations, both in presentation and numerical properties. It shows that waves in waterways with finite frictiontravel downstream at a speed roughly equal to 3/2 × Water speed, and not upstream and downstream at a speedgiven by c =

√g ×Mean depth, widely used for back of the envelope calculations. In reality, however, the

diffusion coefficient can be large, and the equation behaves more like the diffusion equation, where disturbancesare diminished and spread out in space as time passes. In many situations the effects of diffusion are dominant.

42

2.3.10 An even further approximation –Muskingum-Cunge routing

? proposed the scheme from Muskingum methods from hydrologic storage routing,

Q(x+ δ, t+∆) = C1Q(x, t+∆) + C2Q(x, t) + C3Q(x+ δ, t), (2.76)

where δ is a step size in x, ∆ is a step size in t, expressing the solution at one point of a computational rectanglein (x, t) space, at a later point and time, and the quantities are given by

C1 =∆− 2KX

2K (1−X) +∆, C2 =

∆+ 2KX

2K (1−X) +∆, and C3 =

−∆+ 2K (1−X)

2K (1−X) +∆, (2.77)

where K =δ

c, and X = 1

2

µ1− Q

BcS0δ

¶. (2.78)

If we take a bi-dimensional power series expansion of equation (2.76) with (2.77) and (2.78) about the point (x, t),we find that to lowest order it is actually satisfying the differential equation

∂Q

∂t+ c

∂Q

∂x+

ν

c

∂2Q

∂t ∂x= O (δ,∆) , (2.79)

then the scheme is a solution of a modified diffusion equation. It can be compared with the advection-diffusionequation (2.70), where we write it using not just the perturbed volume potential φ but the discharge Q:

∂Q

∂t+ c

∂Q

∂x− ν

∂2Q

∂x2= 0. (2.80)

Thus we see that the equation which Muskingum-Cunge routing is solving is (2.79) instead of (2.80). We can seehow the two are related. Consider the advection-diffusion equation in the low-diffusion approximation where νis small, where the equation becomes the kinematic wave equation, and so we can consider the operator ∂/∂t +c∂/∂x to be small, and we replace one derivative ∂/∂x in the second derivative by−1/c ∂/∂t, which immediatelygives us the left side of equation (2.79). Hence we see that the Muskingum-Cunge equation is actually a low-diffusion approximation to the advection-diffusion equation. The interchanging of time and space differentiationhas meant that one can solve the problem by simple wave-like interpolation methods, rather than by approximatingthe diffusion operator, however one has to solve a transcendental equation at each point and step. It is not surprisingthat in the lecturer’s experience it gives poor results for small slopes, where diffusion is large. The lecturer hasshown that the method contains significant diffusion, inherent in the approximation leading to equation (2.79),which is greatest for small slopes and faster variation.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 0.2 0.4 0.6 0.8 1

Ratio ofM-C valueto A-D value

Dimensionless diffusion-frequency coefficient N = a0σ/c20

Decay rateμM Ci

μADi

Wave speedcM C

cAD

Figure 2.19: Propagation speeds and downstream decay rates predicted by the Muskingum-Cungeapproximation compared with those from the Advection-Diffusion equation

Figure 2.19 shows propagation speed and downstream decay rate of this approximation. It is clear that its behaviouris less than wonderful.

43

2.3.11 The most approximate approximation of all – the Kinematic wave equation

Consider the advection-diffusion equation (2.70) in the limit where the diffusion can be ignored, for streams thatare shallow and steep, which gives equation (2.74)

∂φ

∂t+ c

∂φ

∂x= 0, (Kinematic wave equation)

which is the kinematic wave equation. Solutions of this are disturbances travelling without diffusion, with a localspeed given by the local kinematic wave speed. In fact, this equation was obtained by linearising about a uniformflow, so strictly speaking c should be considered a constant, which we will now denote by c0. In this case theequation has the exact solution φ(x, t) = f(x − c0t), where f is any function of the argument, so that anydisturbance travels without change. We can show this by considering the solution at a point x at a later time t+∆,where the solution is f (x− c0 (t+∆)). Writing the argument as f((x−c0∆)−c0t), we can see that the solutionat the later time φ (x, t+∆) = φ (x− c0∆, t), namely the solution that was originally upstream a distance c0∆,showing that the solution is a simple wave of translation. It would be possible to solve problems approximatelyby using a local value of c and using this "upwinding" form of solution, and writing the updated solution at x att + ∆ that which was upstream a disatnce c∆,where c now varies along the channel. This, however is a highlyapproximate method, and not generally to be recommended.

2.3.12 The characteristic formulation of the equations

xj

tn +∆

x+j

t

x

tn

(xj , tn +∆)

x−j

1

βU + C

1

βU − C

xj−1 xj+1

Figure 2.20: Part of (x, t) plane, showing information arriving from upstream and downstream at velocities βU ±C

The two long wave equations (2.23), which are partial differential equations, can be expressed as four ordinarydifferential equations. Two of the differential equations are for paths for x(t), a path known as a characteristic:

dx

dt= βU ± C, (2.81)

where U = Q/A is the mean fluid velocity in the waterway at that section and the velocity C is

C =

rgA

B+ U2

¡β2−β¢,

often described as the "long wave speed". It is, as equation (2.81) shows, actually the speed of the characteristicsrelative to the flowing water. There are two contributions of ±C, corresponding to both upstream and downstreampropagation of information. Two characteristics that meet at a point are shown on Figure 2.20. The "downstream"or "+" characteristic has a velocity at any point of βU + C. In the usual case where U is positive, both partsare positive and the term is large. As shown on the diagram, the "upstream" or "-" characteristic has a velocityβU−C, which is usually negative and smaller in magnitude than the other. Not surprisingly, upstream-propagatingdisturbances travel more slowly. The characteristics are curved, as all quantities determining them are not constant,but functions of the variable A, B, and Q.

The other two differential equations for η and Q can be established from the long wave equations:

B

µ−βQ

A± C

¶dη

dt+

dQ

dt= β

Q2B

A2S − ΛP Q |Q|

A2+ i

µ−βQ

A± C

¶, (2.82)

On each of the two characteristics given by the two alternatives of equation (2.81), each of these two equationsholds, taking the corresponding plus or minus signs in each case. To advance the solution numerically means that

44

the four differential equations (2.81) and (2.82) have to be solved over time, usually using a finite time step ∆.Figure 2.20 shows the nature of the process on a plot of x against t.

The usual computational problem is, for a time tn+1 = tn+∆, and for each of the discrete points xm, to determinethe values of x+m and x−m at which the characteristics cross the previous time level tn. From the information aboutη and Q at each of the computational points at that previous time level, the corresponding values of η+m, η−m, Q+m,and Q−m are calculated and then used as initial values in the two differential equations (2.82) which are then solvednumerically to give the updated values η (xm, tn+1) and Q (xm, tn+1), and so on for all the points at tn+1.

The advantage of characteristics has been believed to be that numerical schemes are relatively stable. The lectureris unconvinced that they are any more stable then simple finite difference approximations to the original partialdifferential equations, but this remains to be proved conclusively.

The use of characteristics has led to a widespread misconception in hydraulics where C is understood to be thespeed of propagation of waves. It is not – it is the speed of characteristics. If surface elevation were constanton a characteristic there would be some justification in using the term ”wave speed” for the quantity C, as distur-bances travelling at that speed could be observed. However as equation (2.82) holds, in general neither η (surfaceelevation – the quantity that we see), nor Q, is constant on the characteristics and one does not have observabledisturbances or discharge fluctuations travelling at C relative to the water. While C may be the speed of propaga-tion of information in the waterway relative to the water, it cannot properly be termed the wave speed as it wouldusually be understood.

2.3.13 The behaviour of floods and waves in rivers

Linearised equation – the Telegrapher’s equation

As in the analytical solution for steady flow of §2.3.6.3, here we linearise the equation by considering smallperturbations about a uniform flow of cross-sectional area A0 and discharge Q0 in a channel of constant slope,S = S0. We write V = A0x−Q0t+ εφ, where ε is a small parameter such that εφ is a small perturbation in thevolume potential about the uniform flow. In this case the two physical quantities A and Q are given by

A =∂V

∂x= A0 + ε

∂φ

∂x,

Q = −∂V∂t

= Q0 − ε∂φ

∂t

When we substitute into equation (2.62) and express all nonlinear terms as series we ignore terms in ε2 and higher.In fact, the linearised version of the left hand side can simply be written down, as the coefficients of the secondderivatives on the left become simply their values for the undisturbed uniform flow. The right side is more difficult.The quantity ΛP is written as the two-dimensional Taylor series

ΛP = Λ0P0 +∂ (ΛP )

∂A(A−A0) +

∂ (ΛP )

∂Q(Q−Q0) + . . .

= Λ0P0 +∂ (ΛP )

∂A

¯0

ε∂φ

∂x− P0

∂Λ

∂Q

¯0

ε∂φ

∂t+ . . . ,

where the resistance coefficientΛmay also be a function of the discharge in cases (a) where bed forms can develop,and (b) where it depends on the Reynolds number, as shown on the Stanton-Moody plot of the Colebrook-Whiteequation, figure 2.8. The velocity squared term becomes

ÃQ0 − ε∂φ∂tA0 + ε∂φ∂x

!2=

µQ0A0

¶2Ã1− εQ0

∂φ∂t

1 + εA0

∂φ∂x

!2= U20

µ1− 2 ε

Q0

∂φ

∂t− 2 ε

A0

∂φ

∂x

¶+O

¡ε2¢,

where we have introduced U0 = Q0/A0. After introducing the symbols

(ΛP )A =∂ (ΛP )

∂A

¯0

and ΛQ =∂Λ

∂Q

¯0

,

45

the whole term on the right of equation (2.62) becomesµΛ0P0 + ε (ΛP )A

∂φ

∂x− P0εΛQ

∂φ

∂t

¶U20

µ1− 2 ε

Q0

∂φ

∂t− 2 ε

A0

∂φ

∂x

¶− gS0

µA0 + ε

∂φ

∂x

¶= Λ0P0U

20 − gS0A0

+εΛ0P0U20

µ− 2

Q0

∂φ

∂t− 2

A0

∂φ

∂x

¶+εU20

µ(ΛP )A

∂φ

∂x− P0ΛQ

∂φ

∂t

¶−εgS0 ∂φ

∂x.

There are no terms at zeroeth order ε0 on the left side, so the term on the right at this order has to be satisfied itself,giving

U20 =g

Λ0

A0P0

S0, (2.83)

which is the uniform flow relationship. Now considering all the terms on the left and right of equation (2.62) at ε1and bringing them together, and occasionally using equation (2.83) to eliminate U20 , we obtain the equation

2σ

µ∂φ

∂t+ c

∂φ

∂x

¶+

∂2φ

∂t2+ 2βU0

∂2φ

∂x∂t− (C20 − β2U20 )

∂2φ

∂x2= 0, (2.84)

where it has been convenient to introduce the symbols whose significance is described immediately below. Thisis a Telegrapher’s equation, with a first-order "kinematic" part containing the effects of friction, and second ordercontributions corresponding to the wave dynamics. Such an equation was obtained by ? and ? for wide channels.

The symbols introduced are, although the names and significance are not yet obvious, and will be justified furtherbelow:

1. Decay rate σ:

σ =gS0U0

µ1 +

1

2

Q0Λ0

∂Λ

∂Q

¯0

¶=

sgS0Λ0A0/P0

µ1 +

1

2

Q0Λ0

∂Λ

∂Q

¯0

¶(2.85)

which has dimensions of T−1, and measures the rate at which disturbances decay; it increases with both slopeand roughness and decreases with depth. In river flow problems this can usually be simplified, as the Reynoldsnumber of the flow is so large that there is no longer any variation with it, and ∂Λ/∂Q = 0.

2. Kinematic wave speed c:

c =3

2U0

1− 13

A0Λ0P0

∂ (ΛP )

∂A

¯0

1 + 12

Q0Λ0

∂Λ

∂Q

¯0

, (2.86)

is the speed of waves when all dynamic terms, of the order of F 2 are unimportant, which will be proved below.This is a slight generalisation of the usual Kleitz-Seddon equation to allow for the resistance coefficient beinga function of discharge, which states that

c =dQ(A)

dA,

where the Q(A) is the discharge given by any of the usual resistance formulae.For convenience, the symbol θ is introduced such that

c = θU0, (2.87)

where θ is roughly 3/2 from equation (2.86), and its most important variation is caused by the change ofperimeter with area. The role of this term as a "kinematic wave speed" will first be established below.

3. Dynamic wave speed C0:

C0 =qgA0/B0 +

¡β2 − β

¢U20 , (2.88)

is the speed of waves when there is no friction or bed load, which will be shown below. This is the formula forthe speed of characteristics, but the term "wave speed" widely used for

pgA/B is not justified, as is just part

of the expression (2.88), and it will be seen that even C0 itself is the wave speed only in the limit of no friction.

46

Behaviour of solutions –modal analysis

We now obtain solutions of equation (2.84), which we believe to be a good model for small disturbances to theflow. Unfortunately simple deductions from the form of the partial differential equation seem not to be possible.However, we obtain useful insight by assuming that, as any disturbance can be written as a Fourier series, wecan consider just one term of that Fourier series and examine how it propagates, namely its speed of propagationand its decay rate as it travels. This is only possible because the equation is linear in φ. ? have done this for atwo-equation formulation.

It is simpler to write the variation in x in complex form, such that we represent all variation along the waterwayas the periodic wave exp (i kx), where i=

√−1 and k = 2π/L, where L is a wavelength we are considering. Theexponential can be expanded: exp (ikx) = cos kx+i sin kx, thereby revealing more its periodic form. We writefor the general solution:

φ = exp (ikx− μt) = exp (ikx) exp (−μt) , (2.89)

showing the periodic behaviour in x, and where the behaviour in time t is contained in the term μ. The real partof μ shows how the solution decays or grows in time, the imaginary part determines how the solution oscillates intime. We substitute (2.89) into (2.84) and find that it satisfies the equation exactly, giving a quadratic equation forμ:

μ2 − 2μ (σ + ikβU0) + 2ikcσ + k2¡C20 − β2U20

¢= 0

with solution:μ = σ + i kβU0 ± i

qk2C20 − σ2 + 2iσkv (2.90)

in which the substitution for c has been made:

c = βU0 + v, (2.91)

where v will be seen to be the speed of propagation of waves relative to the advection velocity βU0 in the limitof large resistance in the channel. These definitions of C0 in equation (2.88) and v here, are such that β has beensubsumed in them.

It is possible to express equation (2.90) in standard complex notation (i.e. Real+i× Imaginary). The symbol =±1 is introduced, +1 for downstream-propagating waves and −1 for upstream, and the symbol χ also introducedfor notational convenience

χ =

q(k2C20 − σ2)

2+ (2σkv)2.

It is insightful to consider real and imaginary parts. Substituting μ = μR+iμI into equation (2.89)

φ = exp (ikx− μt) = exp (ikx) exp (−μRt− iμIt)

= exp (−μRt) exp (ikx− iμIt) = exp (−μRt) exp³

ik³x− i

μIkt´´

,

and the role of the two parts becomes clear:

1. Real part: μR = < (μ) is decay rate of the disturbance. The real part of equation (2.90) gives

< (μ)σ

= 1− √2

qχ/σ2 + k2C20/σ

2 − 1, (2.92)

giving a formula for the rate at which disturbances decay. In most situations we expect this to be positive suchthat they are actually damped. The details of this are not as important as the physical result that the rate ofdecay depends on the wavelength in the form of k – different wavelength components decay at different rates.

2. Imaginary part: the propagation velocity of the wave is μI/k = = (μ) /k, and from equation (2.90), if we writethis as

= (μ)k

= βU0 + C,

showing that there is an advection velocity βU0 relative to which the waves propagate up- and down-stream atspeed C where

C

C0=

1√2

sχ

(kC0)2 + 1−

µσ

kC0

¶2, (2.93)

which is a function of the wavelength, expressed in terms of k = 2π/L. This means that for waves travellingon streams with resistance there is no such thing as a unique long wave speed, and waves of different lengthstravel at different speeds, contrary to many writings.

47

These propagation characteristics of the waves depend on just two dimensionless parameters σ/kC0 and kv/σ,each of which contains the wavenumber k. It is more revealing to eliminate that between them, giving the twoparameters σ/kC0 and v/C0:

1. The relative importance of resistance: σ∗ = σ/kC0 = σ/ω0, using the dynamic wave frequency ω0 = kC0,.For flows that are not fast, equation (2.85) gives

σ∗ =σ

kC0≈ L

2π

sS0Λ0A0

B0

A0

P0

≈ L√S0Λ0

2πD0,

where the symbol D0 has been introduced for the mean depth D0 = A0/B0 and where D0 ≈ A0/P0 for manychannels that are wide. We will refer to σ∗ as the resistance-slope-wavelength parameter, as it increases witheach of those quantities.

2. Vedernikov number (2.95), Ve = v/C0, which is a function of the Froude number from equations (2.87) and(2.91):

Ve =v

C0=

(θ − β)F0q1 +

¡β2 − β

¢F 20

, (2.94)

so that as θ ≈ 1.5 to 1.7, and β ≈ 1.05, for relatively slow flows, Ve is roughly 0.5F0 and we can think of it asa measure of the velocity of flow in the channel.

0

0.5

1

Dimensionlessdecayrate

< (μ0) /σ

Ve = 1/8, F0 ≈ 0.25Ve = 1/4, F0 ≈ 0.5

Ve = 1/2

Ve = 1

(a)Decay rate for downstream-travelling waves

0

0.5

1

0.01 0.1 1 10 100

Dimensionlesswave speed

C/C0

Resistance-slope-wavelength parameter σ∗ = σ/kC0

F0 ≈ 0.25, Ve = 1/8F0 ≈ 0.5, Ve = 1/4

Ve = 1/2

Ve = 1(b) Wave speed

Figure 2.21: Wave propagation parameters and their dependence on the resistance-slope-wavelength parameter and Ved-ernikov number: (a) Decay rate for downstream-travelling waves, (b) Wave speed

Now we can examine the nature of wave propagation by plotting the decay rate and propagation velocity as func-tions of these two parameters, given in figure 2.21. Part (a) shows the variation of the dimensionless decay rate< (μ) /σ for downstream-travelling waves from equation (2.92), while (b) shows the dimensionless wave speedC/C0 from equation (2.93).

Decay rate of waves:

Considering the decay rate, figure 2.21(a) shows that for small values of σ∗ the decay rate depends on the Ved-ernikov number Ve. This can be explored mathematically by writing equation (2.92) as a power series in σ∗, valid

48

when that is small, which gives< (μ0)σ

= 1− Ve +O¡σ2∗¢.

For downstream-propagating waves = +1, and the curves on the left side of figure (a) tend to this value, 1− Ve.For upstream propagating waves, = −1, the dimensionless decay rate is 1 + Ve, larger than for downstreamwaves.

Possibly the most important result seems to be that for Ve = 1, < (μ0) = 0, and there is no decay. This is theboundary of instability, and here can be established by performing simple algebraic operations for the stabilitycondition < (μ) 6 0 that leads to the criterion for stability

Ve =v

C0≤ 1, (2.95)

where v/C0 is the Vedernikov number Ve. (?) first obtained solutions like (2.90) for a wide rectangular channelwith β = 1, and obtained the equivalent criterion for stability that S0 ≤ 4Λ in present terms. Substituting into thisequation gives the condition for stability

The criterion v ≤ C0 for stability (equation 2.95)can be further examined. Using the definitions of v and C0 fromequations (2.88) and (2.91) and equation (2.94) gives for stability:

c2 − 2cβU0 + βU20 ≤ gD0.

The kinematic wave speed c is proportional to the mean fluid velocity U0, from equations (2.86) and (2.87),c0 = θU0, which gives the stability condition:

F 20 =U20gD0

≤ 1

θ2 − β(2θ − 1) . (2.96)

For a wide channel, the Weisbach and Chézy expressions have θ = 3/2, while if Gauckler-Manning-Strickler isused θ = 5/3, giving the stability criteria:

Weisbach− Chezy F 20 ≤1/2

9/8− β, (2.97)

Gauckler−Manning− Strickler F 20 ≤3/7

25/21− β. (2.98)

The denominator in the two cases goes to zero at β ≈ 1.13 and β ≈ 1.19 respectively. It has been suggested (?) thatβ incorporates turbulence effects as well, and a typical value might be greater than 1.05, such that the denominatorin both (2.97) and (2.98) is small and the limiting values of Froude number obtained might be large. These limitsare plotted in Figure 2.22. The stability depends strongly on the resistance law and the velocity distribution. If thechannel were of finite width, the value of θ would be smaller, and equation (2.96) predicts a larger limiting valueof F0 for instability. Above this value the wave system is unstable, and we have the development of Roll Waves.

Propagation speed of waves:

Now the possibly more important quantity of wave speed can be examined. Considering figure 2.21(b) it canbe seen that for small values of resistance-slope-wavelength, all waves travel at the same speed C0, however, ingeneral this is not the case. This contradicts one of the bases of much hydraulics, where it is widely stated andbelieved that all long waves travel at the same speed,

pgA0/B0. The figure shows that this is simply not the case,

and the propagation speed is a function of the resistance-slope-wavelength parameter and the Vedernikov number,and takes on very different values.

For small σ∗, equation (2.93) can be shown to have the solution

C

C0= 1− 1

2

¡1− V 2

e

¢σ2∗ +O

¡σ4∗¢,

and in the limit of very small σ∗ we get C = C0 =qgA0/B0 +

¡β2 − β

¢U20 from equation (2.88), so that the

speed is indeed independent of wavelength in this limit. In the case β = 1 we recover the widely-believed resultthat C =

pgA0/B0.

For large σ∗, however, equation (2.93) gives

C

C0= Ve

µ1 +

1− V 2e

2σ2∗+O

¡1/σ4∗

¢¶,

49

0

1

2

3

4

5

1 1.05 1.1 1.15 1.2

Froudenumber

F0

Velocity distribution parameter β

Stable

Unstable

θ = 3/2, Weisbach-Chezyθ = 5/3, Gauckler-Manning-Strickler

Figure 2.22: Stability limits for flows in wide channels and their dependence on velocity distribution parameter β and theexponent in the friction formula

as can be seen on the figure. For large values of resistance-slope-wavelength, we then have C ≈ C0Ve = v, suchthat the velocity of waves is βU0 ± v, which is the so-called kinematic wave speed c in the downstream case.

Both these results can be identified on the figure. The first is the traditional result, that waves propagate at the"long wave speed", C = C0 , where C0 is actually modified by velocity terms, equation (2.88), which we see hereis valid only for small σ∗. For larger values, the waves do not travel at C0, but at speeds intermediate between that,and in the limit, the kinematic wave speed C∗ = v∗, or, C0 = v0. This too, is widely known, but the general result,that wave speed depends on wave length seems also to be widely unknown.

Conclusions

The long wave equations have been widely interpreted as a simple hyperbolic system where disturbances travel ata constant wave speed. That is incorrect. In fact the system is an advective-diffusive-dispersive system, where thepropagation speeds of disturbances and their rates of decay all depend on the wave length of the disturbances. Thebehaviour is much more complicated than the simple hyperbolic inferences suggest.

Any numerical solution of the equations should give the correct behaviour;what is not correct are simple inferencesas to the behaviour based on the constant wave speed belief, and an accompanying acceptance that effects ofresistance are small. They are not. And, they are complicated.

2.4 Structures, controls, and boundary conditions

Generally the treatment of boundary conditions in the one-dimensional open channel equations is decidedly non-trivial.

2.4.1 Upstream boundary conditions

At the upstream boundary x = x0 we often have the condition that the inflow is known as a function of time,Q (x0, t) = Qin(t). Typical conditions at an upstream boundary are shown on the left side of figure 2.23. If wedo not use the method of characteristics, but instead some finite difference scheme, the solution at time t may beknown at a finite number of computational points x0, x1, . . . shown by the crosses, and it is required to calculatethe solution at time t+∆. If we use, for example, η and Q as the dependent variables, we still have to determinethe value of the other variable at the boundary, as only one condition can be specified in a sub-critical flow. We can

50

xi

t+∆

x+ x

t

(xi, t+∆)

x−

C0− C+ C−

dxdt = u− c0

dxdt = u+ c0

x0 x1

Figure 2.23: Computational points and typical characteristics on (x, t) axes

use the mass conservation equation (2.23a)∂η

∂t+1

B

∂Q

∂x=

i

B,

and from the point values of Q at t an estimate of ∂Q/∂x can be made and a value of ∂η/∂t calculated and usedto give a prediction of η (x0, t+∆). It is clear that only simple low-accuracy numerical approximations are beingmade. Interior values are updated using approximations to the two partial differential equations at the interiorpoints.

If the method of characteristics were to be used, then a characteristic such as C0− would be used, and for a finitedifference approximation of the characteristic equation such as (2.82), the value of η (x0, t+∆) calculated. Again,low-order numerical approximations would be made.

If the upstream volume were used as a dependent variable, then as generally ∂V/∂t =R x

i dx0−Q, at the upstreamboundary we obtain the ordinary differential equation dV (x0, t) /dt = −Qin(t), which can be solved by simpleor by higher order methods.

2.4.2 Downstream boundary conditions

If anything, the downstream condition is more difficult. If it were say, the sea with tides or a reservoir with aspecified surface level, η (L, t), that is relatively simple, and the other condition would be obtained from applicationof the momentum equation in terms of ∂Q/∂t there. Alternatively, the method of characteristics could be used,this time with a C+ characteristic. The third alternative of using V is rather less simple, as specification of η as afunction of time implies that A(L, t) can be calculated, giving the derivative ∂V/∂x as a function of time.

More complicated is the condition if there is a control at the downstream point, such as a weir or sluice, where Qis a known function of η. In this case it is necessary to use the momentum equation in terms of ∂Q/∂t to give anupdated approximate value of Q(L, t +∆) and the equation of the control solved to give the updated value of η.In the case of V this will be a value of ∂V/∂x.

At open boundaries, where an arbitrary end to the computational domain is proposed, the problems are greater,although we may simply be able to apply the partial differential equation as if it were an interior point, which is apoint that the lecturer has yet to resolve satisfactorily.

2.4.3 Hydraulic structures on channels

Hydraulic structures are anything that can be used to divert, restrict, stop, or otherwise manage the natural flow ofwater. They can be made from materials ranging from large rock and concrete to obscure items such as woodentimbers or tree trunks.

A dam, for instance, is a type of hydraulic structure used to hold water in a reservoir as potential energy, just asa weir is a type of hydraulic structure which can be used to pool water for irrigation, establish control of the bed(grade control) or, as a new innovative technique, to divert flow away from eroding banks or into diversion channelsfor flood control.

The main texts to which reference can be made are ?, French (1985?), Henderson (1966?), Novak (2001?).

Overshot gate - the sharp-crested weir

The conventional analysis of flow over a sharp-crested weir is one of the great misleading results in hydraulicengineering, which is widely known to be so, yet every textbook reproduces it. Instead of the actual pressure dis-tribution shown dashed in Figure 2.24, a hydrostatic distribution is assumed, shown dotted, which gives maximumpressure at the crest, whereas it is actually zero there, then Bernoulli’s law is applied, and then it is assumed thatthe fluid velocity is horizontal only (whereas it is actually vertical at the crest!). Results obtained for the horizon-

51

Assumed pressure distributionHypothetical actual pressure distribution

h

H

P

D

C

Figure 2.24: Side view of sharp-crested weir

0.0

0.2

0.4

0.6

0.8

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

z/h

u/√gh

Accurate computationsClassical theory

Figure 2.25: Horizontal velocity distribution over crest - simple theory and accurate computations

hence we obtain the well-known 3/2 law:q = C

√gH3/2.

If we now consider a finite width of channel B, finite width of weir b, and a finite apron height P , with now a totaldischarge Q we obtain

Q

bpgH3

= f(b/B,H/P, b/P ).

Following traditional forms we write the quantity on the right in terms of a discharge coefficientCD as

Q

bpgH3

=2

3CD

√2. (2.99)

It is a strong temptation to incorporate the 2/3 and the√2 in with the CD, as both are artefacts of the inappropriate

use of Bernoulli’s theorem. In 1924 the Schweizerischer Ingenieur- und Architekten-Verein (SIA), the Swiss Societyof Engineers and Architects, published a Standard containing the following expression for CD, which we have hereconverted from the dimensional expression presented in ? and a particular case presented by Rehbock:

CD =

Ã0.578 + 0.037

µb

B

¶2+ 0.615

6− 5 (b/B)20.047H (g/ν2)1/3 + 1.6

!Ã1 +

1

2

µb

B

¶4µH/P

1 +H/P

¶2!, (2.100)

in which the last term in the first bracket allows for real fluid effects. The dimensionless expression 0.047H¡g/ν2

¢1/3appeared as 1000H in the original, where 1000 is actually a dimensional one of units m−1. Here we have re-

52

expressed it in terms of the only possible combination of gravitational acceleration g and viscosity ν, assumingthat values of g = 9.8ms−2 and ν = 106 were used to arrive at the value of 1000m−1.

An approximate expression from equations (2.99) and (2.100) is Q = 0.6√gb (η1 − zc)

3/2 ,where η1 is the up-stream surface elevation and zc is the elevation of the crest.

Triangular weir

An analysis as misguided as the traditional one produces the result quoted by ? on p352, where some results of ?are quoted. The expression is

Q = CD

8

15

p2g tan

θ

2H2.5,

where CD is the coefficient of discharge, and where θ is the angle between the sides of the triangle. A typical resultfrom ? is that CD is roughly 0.58, which has been found to agree well with experiment (see ?). We prefer to writethe expression as

Q = 0.44 tanθ

2

√gH2.5,

suggesting that it is empirical.

Rapidly-varying flow problems

In the following sections we will be considering steady rapidly-varied flow. The horizontal distances are smallenough that we ignore effects of friction and slope, as they are roughly in balance anyway, and are not so importantfor these problems. The energy and momentum fluxes are:

Energy flux at a section : ρQ

µgη +

α

2

Q2

A2

¶= Constant,

Momentum flux at a section : ρA

µgh+ β

Q2

A2

¶= Constant,

where η is the elevation of the free surface and h is the depth of the centroid below the surface. As Q is usuallyconstant along the channel, we can usually consider the total head to be a constant:

η +α

2g

Q2

A2= Constant.

It is convenient to consider the Specific Head, the head relative to the bottom of a channel, defined by

H = h +αQ2

2g

1

A2(h), (2.101)

where h is the depth of flow and areaA is a function of h. As the total head is constant, and we can write η = Z+h,where Z is the elevation of the channel bottom, we have

H + Z = constant. (2.102)

Broad-crested weirs – critical flow as a control

A typical relationship for H(h) is plotted in figure 2.26(a), where both sub- and super-critical limbs of the curvecan be seen. Now consider a sub-critical flow at 1 such as that shown in part (b) of the figure, where the bottomof the stream rises by an amount ∆. As total head is constant, from equation (2.102) H1 + Z1 = H2 + Z2, andH2 = H1 −∆, and this is shown in part (a) (note that the change of ∆ is plotted horizontally!), and the depth halso decreases as shown, from h1 to h2.

Consider now the case where the step ∆ is high enough that the flow is brought to critical there, h2 = hc, wheresuch an hc is shown on figure 2.26(a). If the step were raised even more, then the depth of flow over the raised bedwould remain constant at hc and the upstream depth would have to increase so as to make the flow possible. Thestep is then acting as a weir, controlling the flow.

Consider the situation where the bed falls away after the horizontal section, such as on a spillway, as in figure 2.27.The flow upstream is subcritical, but the flow downstream is fast (supercritical). Somewhere between the two, theflow depth must become critical - the flow reaches its critical depth at some point on top of the weir, and the weirprovides a control for the flow. In this case, the total head upstream (the height of the upstream water surface abovethe sill) uniquely determines the discharge, and it is enough to measure the upstream surface elevation where theflow is slow and the kinetic part of the head negligible to provide a point on a unique relationship between that

53

Specific head H

Flowdepth

h

H = h+αQ2

2g

1

A2(h)hc

∆h2

h11

2

∆

1 2

h2h1

(a) (b)

Figure 2.26: (a) Head-surface relation for a particular discharge and cross-section, showing the change in depth if as in (b)the bottom rises by ∆

hc

hc

hc

Figure 2.27: Spillway as broad-crested weir

Small energy loss

hc

hc hc

h1

Figure 2.28: Broad-crested weir in a canal showing substantial recovery of energy

head over the weir and the discharge. No other surface elevation need be measured. Such a horizontal flow controlis called a broad-crested weir.

In recent years there has been a widespread development of broad-crested weirs placed in canals where the flow issubcritical both before and after the weir, but passes through critical on the weir, as shown in figure 2.28. There isa small energy loss after the flume. The advantage is that it is only necessary to measure the upstream head overthe weir.We can apply simply theory of critical flows to give the discharge per unit width q:

q =pgh3c =

qg¡23H1

¢3=q

827gH

3/21

where

H1 = h1 +q2

2gh21,

the head upstream. As H1 is a function of h1, the relationship for discharge q is not simply a 3/2 power law, butfor sufficiently low upstream Froude number the deviation will not be large. In practice the discharge for the weir

54

as a whole is written, introducing various discharge coefficients ?,

Q = 0.54CsCeCvb√gh

3/21 .

Free overfall

hbhc

Figure 2.29: Free overfall, showing location of critical depth hc and brink depth hb

Consider the free overfall shown in Figure 2.29. It can be shown that the discharge per unit width is q =

1.65√gh

3/2b .

Sluice gate or undershot gate

h1

h2D

Figure 2.30: Sluice gate for tutorial example

Now we consider the case where the gate is not drowned, but a stream of supercritical flow can exist for somedistance downstream. Applying the energy theorem between a point at section 1 and one at 2:

gh1 + α1

2

µq

h1

¶2= gh2 + α

1

2

µq

h2

¶2.

Solving for q we obtain:

q =

r2g

α

h1h2√h1 + h2

which it is easier to write in dimensionless form:qp2gh31

=

√αh2/h1p1 + h2/h1

. (2.103)

In practice we know the upstream depth h1 and the gate opening D, such that

h2 = CcD,

where Cc is the coefficient of contraction. A number of theoretical and experimental studies have been made ofthis, but the variation is not large. ? shows some of these but ends up recommending a constant value of 0.61.

Equation (2.103) is a convenient way of representation, as h1 is given, and so we find that the dimensionlessdischarge is simply a function of h2/h1, the depth ratio, or D/h1. In fact, the variation is rather linear – if weexpand the denominator of the right side of equation (2.103) using the binomial theorem, we obtain

qp2gh31

=√αh2h1

µ1− 1

2

h2h1+ . . .

¶,

and we find that to first order, for small gate openings,

q ≈p2gαh1 × h2,

and the result is almost obvious, that the discharge per unit width is given by a velocity corresponding to the fullupstream head of water multiplied by the downstream depth of flow.

55

Drowned sluice gate

hc

h1

h0D

h2

Figure 2.31: A drowned sluice gate

Consider Figure 2.31, a rather abstract representation of the highly turbulent flow just downstream of the gate,where the gate is drowned. In this case the analysis follows that suggested on p208 in ?. Applying the energytheorem between a point at section 1 and one at c:

gh1 +α

2

µq

h1

¶2= gh0 +

α

2

µq

hc

¶2, (2.104)

where we recognise that the pressure contribution is given by the hydrostatic force of depth h0 but that the velocitycontribution is due to flow in the jet only. Solving for q we obtain:

q =

r2g

αh1hc

sh1 − h0h21 − h2c

. (2.105)

And now we apply the momentum theorem between sections c and 2:

βq2

hc+ 1

2gh20 = β

q2

h2+ 1

2gh22. (2.106)

Again solving for q we obtain

q =

rg

2β

shch2 (h22 − h20)

h2 − hc. (2.107)

Now if we eliminate q between the two equations (2.105) and (2.107) we obtain the quadratic in h0:

h20 +4βh21hcαh2

(h2 − hc) (h1 − h0)

(h21 − h2c)− h22 = 0, (2.108)

and if we introduce the depth scale h4 defined by

h4 =β

αhc

1− hc/h2

1− (hc/h1)2, (2.109)

equation (2.108) becomesh20 + 4h4 (h1 − h0)− h22 = 0,

and the solution is :h0 = 2h4 +

q4h24 − 4h1h4 + h22. (2.110)

In practice we know the upstream and downstream depths h1 and h2 as part of the computations. If the gateopening is D, also presumed to be known, then

hc = CcD,

where Cc is the coefficient of contraction which we may assume to be 0.6. Hence we can calculate h4 from (2.109),h0 from (2.110), and the discharge per unit width from (2.105) or (2.107).

56

2.4.4 Interior conditions

Obstacles such as bridge piers

When water in a river or canal flows past an obstacle such as a bridge pier or woody debris, there are accompanyingmomentum and energy losses, and the water level is different on either side of the obstacle. In sub-critical flow thewater level is higher upstream, this backwater causing a greater risk of flooding. In super-critical flow, the level ishigher downstream.

The problem has not been researched as much as it might and there has been some ambiguity about the governingprinciples. As early as the 1850s energy conservation was used. In the early 20th century some large experimentalcampaigns were carried out, but results obtained were empirical. They were also widely variable, and it has beendifficult to use them for a particular application. In more recent times there has been a gradual shift towards usingmomentum conservation. The resistance force of an obstacle to a flow causes an immediate momentum loss to theflow. In comparison most energy losses are distributed, in boundary layers, shear layers, separation zones, vortices,and turbulent and viscous decay in the wake after the obstacle. Momentum loss is easier to measure, in the formof the force on an obstacle, and easier to model, in terms of drag coefficients.

Here we consider the backwater caused by finite obstacles, provided that they do not change the overall natureof the flow. This means, for example, that situations are excluded where an embankment partially blocks thewaterway and alters the flow from a generally one-dimensional one.

The main problem is in calculating the force on an obstacle. This work separates it into two parts, one due toviscous and form drag, the second, termed free surface resistance, is due to the variation of the free surface aroundthe obstacle, for which an approximate theory is developed. Applying momentum conservation gives an implicitequation for the backwater in terms of the force on the obstacle. Then approximate explicit formulae are obtainedwhich show considerable sensitivity to geometry and flow, such that it is not surprising that experimental resultshave shown much scatter.

Some implications for practice are discussed. A hierarchy of different levels of treatment is suggested, dependingon the importance of backwater and the importance of the project. It is suggested that in any investigation themagnitude of the backwater be calculated approximately using the approximate theory. If small it can be neglectedor added to the boundary resistance, for which formulae are given.

Figure 2.32(a) shows a typical problem of a sub-critical flow with a surface-piercing pier that has caused the meanwater level behind it to rise to some excess or backwater height∆η above the water level determined by conditionsfurther downstream. Far upstream the flow is unaffected. As it approaches the obstacle, the water depth graduallyincreases, until it reaches the value required to overcome the momentum loss at the obstacle. The flow dividesaround it so that there may be a stagnation point on the upstream face with a small mound just upstream. It is alocal effect only, so that a traditional slowly-varying one-dimensional approach to the hydraulics is still possibleuntil just before the mound, at a section denoted by 1 in figure 2.32. As the flow divides and accelerates aroundthe pier, the surface level drops, and waves may form. Generally the level becomes a minimum beside the pier,or a little way downstream. By the end of the pier, any momentum loss has already occurred, however mostenergy loss is still to occur in the wake region downstream. The flow has separated from the sides of the obstacle,leaving a relatively stagnant zone behind it. The mean water level just behind the pier may be lower than thatfurther downstream, however it recovers quickly. Section 2 on the figure is deemed to be at a point some littleway downstream where the flow can again be considered one-dimensional for hydraulics purposes. The solidline on the figure shows the actual water level on the sides of the obstacle, and on the axis of the obstacle, up-and down-stream. Its rapid variation shows the local dynamical effects near the obstacle. The dashed line showsthe streamwise variation of the mean surface level across the stream, which is not a visible quantity, but whichdetermines the overall momentum balance at a section.

Part (b) of figure 2.32 shows the physical idealisation adopted to calculate the backwater ∆η, assuming flow in ahorizontal prismatic frictionless channel with an arbitrary origin, common in other rapidly-varying flow problems.Defined as∆η = η1 − η2, it is positive for the more common case of sub-critical flow.

Drag force

The turbulent, three-dimensional, rapidly-varying free-surface flow near obstacles in practice is so complicated thatat the time of writing it is not yet feasible to calculate the forces on them from fluid-mechanical principles. The partof the force due to viscous and form drag TD can be estimated using the empirical expression from conventionalfluid mechanics for the drag on a bluff body (see for example, §7.6 of White 2003?): TD = 1

2ρCDu2a, where

ρ is fluid density, CD is drag coefficient, u is the local horizontal fluid velocity impinging on the object, and ais the projected area of the object normal to the flow direction. For values of CD reference can be made to mostintroductions to fluid mechanics, however they are almost all for the case of fully-immersed bodies with few shear

57

Surface if no obstacle: slowly-varying flow

Surface along axis and sides of obstacle

Mean of surface elevation across channel∆η

1 2

1 2

η1 η2

∆η

(a) The physical problem, longitudinal section showing backwater ∆ηat obstacle decaying upstream to zero

(b) The idealised problem, uniform channel with no friction or slope

Figure 2.32: A typical physical problem – flow past a bridge pier

or free-surface effects. The general backwater problem is more complicated as the incident flow velocity on abody depends on its position in the flow, whether near the bottom, surface, or extending across the whole flow. Theobstacle may be a compound body with different components, such as the piers and roadway of a bridge and eachelement may be subject to a different velocity. For example, the appropriate velocity for a vertical pier extendingover the whole depth would be the mean velocity in the flow; that for a bridge deck would be the surface velocity,while for a vane or block near the bed it would be rather smaller. All those could be plausibly combined linearly.Here for a single element a simple dimensionless factor γ is introduced here such that the drag force is written interms of the mean velocity in the channel Q/A:

TD =12ργCDa

Q2

A2. (2.111)

where the mean velocity incident on the body is given by u2 = γQ2/A2, where Q is the discharge in the channeland A its cross-sectional area. The coefficient γ depends on the body’s position in the flow; for a pier that extendsfrom top to bottom it could be calculated from knowledge of the turbulent logarithmic velocity distribution, orsimply lumped in with the drag coefficient.

Free surface resistance

When flow passes a body that penetrates or approaches the free surface, the pressure variations around it causesurface disturbances whose magnitudes depend on the Froude number. Near the front, the stagnation effect leadsto an increase in the water level locally so that there is a crest near the upstream face. Around the sides, wherethe water accelerates to pass around the body, a wave system originates, as seen in figure 2.32 for a fast flow.Part of the resistance of the obstacle, then, is determined by the shape of the free surface immediately aroundthe obstacle. This suggests that that component could be described as "free surface resistance" rather than theterm "wave resistance" used in naval architecture. Eventually the momentum possessed by the wave train passesinto the water by breaking and dissipation (except for any water that actually breaks onto the river banks) and isincorporated in the momentum of the downstream flow.

Here the effects of the free surface are assumed given by the net hydrostatic force on the obstacle as given by thevarying water level around it. In the momentum approach of HEC (2002?, equations 5-1 and 5-3) a similar modelis used. There the gradually-varied flow equations are solved numerically between the front and rear faces. Here it

58

is suggested that the variation is rapid enough that that is questionable and instead a lumped local approach can beused.

The horizontal hydrostatic force on an arbitrary curved surface is given by a surprisingly-simple expression (see,for example, White 2003?, §2.6, or almost any other book on elementary fluid mechanics). It is ρgah,where a is theprojected area surface perpendicular to the force direction (streamwise here) and h is the depth of the centroid ofthe projected area. The net force on the obstacle due to free surface effects is the difference between contributionson the upstream (US) and downstream (DS) projections:

TS = ρg¡ah¢

US − ρg¡ah¢

DS . (2.112)

For relatively small backwater the difference between the two contributions can be written as the first term of theTaylor series: ρg (ηUS − ηDS) ∂

¡ah¢/∂η, where η is the free surface elevation. The resulting expression can be

shown to beTS = ρga (ηUS − ηDS) , (2.113)

in terms of the water level difference between the upstream and downstream ends, which depends on the length ofthe disturbance around the body. For a slow flow when the wavelengths are short there may be several wave crestsand troughs around a pier, although they might be barely perceptible; for a faster flow the wavelength and heightare larger; in general the water level at the rear of the body is an oscillatory function of stream velocity, and so is thenet force on the body. This is well-known for the resistance of ships, which are designed to minimise such forces.For bluff objects such as bridge piers the effect should be rather more marked. With increasing fluid velocity, theoscillations should increase in height and length, until there is a single wave trough near the downstream face,such that the length of the pier is half the wavelength, when the force should show a local maximum. For highervelocities and wavelengths the trough would be further downstream from the pier and there should be no moreoscillations in the force until the flow approaches critical beside the pier.

A crude but simple approximation is now made. It is assumed that, in keeping with the wavelike nature of dis-turbances, the water surface around the pier is in the form of a cosine wave with an amplitude proportional to theoverall level difference ∆η between upstream and downstream such that η = η2+c∆η cos kx, where c is a di-mensionless factor, k is the wave number, and x has origin at the upstream face. It is expected that c will be 1 orslightly greater so as to agree with the actual free surface on the upstream face at x = 0. On the downstream facewhere x = L in reality there may be a finite level drop as water passes around the corner from the side of the pier,which is subsumed in the coefficient c here. According to this model the difference between upstream and down-stream levels ηUS− ηDS =c∆η (1− cos kL), where L is an effective obstacle streamwise length. Substituting intoequation (2.113), this means that

TS = ρgac (1− cos kL)∆η. (2.114)

For small waves on a slow flow one can imagine short periodic waves around the side of an obstacle, and a cosinewave may be quite a good approximation. For more dramatic flows, such as that shown in Figure 2.32, it will beless so, nevertheless we proceed with it as a model.

To determine k, if a dimensional analysis is performed for the idealised problem of small short two-dimensionalwaves of wavenumber k on a flow of speed U , the relationship is obtained that kU2/g = θ, where θ is a constant.Hence we write k = θg/U2 and use it in equation (2.114). Introducing the dimensionless symbol CS gives

TS = ρgaCS∆η, where

CS = c (1− cos kL) = c

µ1− cos θ

F 2L

h

¶, (2.115)

a function of the square of the Froude number F = U/√gh and L/h, the ratio of streamwise dimension to water

depth h at the obstacle. This shows the unexpected behaviour, that for small F , the argument of the cosine functionbecomes large and small variations in F will cause the cosine function and hence CS to oscillate between valuesof 0 when a crest is also near the downstream face, and 2c when a trough is there. It will be seen that this has thecapacity to explain fluctuations in experimental results.

Total resistance force

Combining equation (2.111) for the fluid dynamic drag and equation (2.115) for the free surface resistance givesthe total resistance force of an obstacle in a stream flow:

T = 12ργCDa1

Q2

A21+ ρga1CS∆η, (2.116)

in which a = a1 and A = A1, respectively the projected area of the obstacle perpendicular to the flow direction,and the flow cross-section, both given by the elevation of the free surface at the upstream section 1. Figure 2.32

59

shows how the depth and velocity at 1 are those most closely approximating the flow that actually impinges on thebody, rather than that at 2.

The coefficient CS as it appears in equation (2.116) is another form of resistance coefficient. What has customarilybeen measured or used in calculations, has been the apparent drag coefficient, which includes both fluid dynamicand free surface resistance, written here as C0D:

T = 12ργC

0Da

Q2

A2. (2.117)

In applying this, Gippel et al. (1996) used the values of a and A upstream at 1. On the other hand several peoplehave used the downstream conditions at 2. There is a practical reason for that, as in sub-critical flow problemsit is the depth there that is known, determined by conditions even further downstream. However, that can beincorporated in the analysis, and it seems that it is more rational to use the actual fluid velocity that impinges onthe body, given more accurately by the conditions at section 1, as in equation (2.116).

Application of momentum theorem

The momentum flux M at any section of gradually-varying open channel flow is, assuming a hydrostatic pressuredistribution,

M = ρ

µgAh+ β

Q2

A

¶. (2.118)

where g is gravitational acceleration, A is cross-sectional area of the channel, h is the depth of the centroid of thesection below the surface, such that Ah is the first moment of area of the section about a transverse axis at thewater level, and β is a Boussinesq coefficient allowing for non-uniformity of velocity over the section.

The conservation of momentum theorem states that M1 −M2 − T = 0. Substituting expression (2.118) for M at1 and 2 and the expression for total resistance force T from equation (2.116), and dividing by ρ gives

µgAh+ β

Q2

A

¶1

−µgAh+ β

Q2

A

¶2

−12γCDa1Q2

A21− ga1CS (η1 − η2) = 0. (2.119)

For sub-critical flow and known downstream conditions at point 2, and where both A1 and h1 are known functionsof η1, this is a transcendental equation for η1 that can be solved numerically for a given flow and channel cross-section. For super-critical flow, where the flow depth increases after the obstacle, one would instead calculate η2from a knowledge of upstream conditions at 1.

Approximate explicit solutions

Numerical solution of equation (2.119) is not difficult, and any root-finding method could be used, such as trial-and-error, bisection, the secant method or Newton’s method, although for the latter the derivative of the left sidewith respect to η1 is complicated. Instead, to obtain more physical insight it is useful to obtain approximate explicitsolutions, which will be done here.

Consider the physical quantities associated with the two water levels, at 1 and 2. The change of surface elevation∆η is usually small relative to the overall depth. The momentum equation (2.119) is now written for sub-criticalflow with all quantities at section 1 expressed as Taylor series in terms of the usually-known conditions at section2:

A1 = A2 +B2∆η +12B

02∆η

2 +O¡∆η3

¢, (2.120a)

Ah¯1 = Ah

¯2 +A2∆η +

12B2∆η

2 +O¡∆η3

¢(2.120b)

a1 = a2 +∆η b2 +O¡∆η2

¢, (2.120c)

where B is the surface width, B0 = dB/dη is its derivative with respect to surface elevation which is a function ofthe side slopes at the stream banks. The Landau order symbols O (. . .) denote the order of the neglected terms ineach case.

Substituting into the momentum equation (2.119) and using power series operations such as use of the binomial

60

theorem, gives the series in the dimensionless backwater∆η/ (A2/B2), where A2/B2 is the mean depth:

−ε2 + ∆η

A2/B2

µ1− βF 22 − σ2CS + ε2

µ2− b2

a2

A2B2

¶¶+µ

∆η

A2/B2

¶2µ12 + βF 22

µ1− B0

2A22B2

2

¶− σ2CS

A2b2B2a2

¶= O

¡∆η3,∆η2ε2

¢, (2.121)

where ε2 =12γCD σ2F

22 , a dimensionless expression of the force on the structure, σ2 = a2/A2 is the blockage

ratio of the obstacle were the surface level to be that at the downstream section 2; and F 22= Q2B2/gA32, the square

of the Froude number there.

First-order solution

It is insightful to consider the first-order solution, for which we recognise that the dimensionless backwater∆η/(A2/B2) and the dimensionless force ε2 are the same order of magnitude, so ignoring all second-order termsin equation (2.121) by taking just the terms linear in ε2 and∆η/(A2/B2), the lowest-order solution is obtained

∆η

A2/B2=

ε21− βF 22 − σ2CS

= 12γCD

a2A2

F 221− βF 22−σ2CS

. (2.122)

This is a generalisation of Fenton (2003?), an explicit approximation for the relative change of surface level in termsof the stream conditions and the geometry and resistance characteristics of the obstacle. For sub-critical Froudenumbers, such that βF 22+σ2CS < 1, the denominator is positive and so is the backwater∆η2. If βF 22+σ2CS → 1,the flow approaches choking, such that the backwater is large, and the theory breaks down.

Conclusions

A model has been proposed for the force on an obstacle due to surface variation around it and approximate formulaefor backwater have been obtained in terms of drag and free surface resistance coefficients. Experimental resultsagree with the predictions from this model, which shows how they may be more variable than would be the case forprototypes, which may explain why there are few results for practical applications. However the present work alsodoes not give results in the form of numerical values of resistance coefficients; it provides a basis for understanding.

A hierarchy of different levels of treatment is suggested, depending on the importance of backwater and the im-portance of the project. Initially an approximate solution to the problem can be used to estimate the backwaterrelative to the effects of boundary resistance. If the backwater is small, it can be neglected. If it is not so small,a correction can be made to the boundary resistance coefficient used, which is usually not known accurately any-way. If the backwater is relatively large, and the project important, there are two main implications. The first isthat in a numerical model the spatial discretisation must be changed and the obstacle has to be treated as a discon-tinuity of elevation in the stream. The second is that the problem of evaluating the magnitude of the backwater isdifficult, as there are few reliable results, and any that are available are for quite particular geometries. The bestsolution would be to perform load tests on a model obstacle which, it is suggested, can be done in a relatively shortlaboratory installation.

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Chapter 3 Measurement and analysis

3.1 Hydrometry and the hydraulics behind it

3.1.1 Definitions

In English, the traditional word used to describe the measurement of water levels and flow volumes is "Hydrogra-phy". That is a substantial ambiguity, for that word is also used for the measurement of water depths for navigationpurposes, has been so used since the great navigators of the eighteenth century, and organisations with names like"*** Hydrographic Service" are usually only concerned with the mapping of an area of sea and surrounding coastaldetail.

Here we are going to follow Boiten (2000?), who provides a refreshingly modern approach to the topic we areconcerned with, calling it ”Hydrometry” the ”measurement of water”, which in the past has received little research.Below, a practitioner will be called a hydrometrician, but the term hydrograph will be retained for a record, eitherdigital or graphical, of the variation of water level or flow rate with time.

There is a website with much information of a rather practical nature in the Water Measurement Manual onhttp://www.usbr.gov/pmts/hydraulics_lab/pubs/wmm/. It is remarkable, however, that a field so important hasreceived such little benefit from hydraulics research.

3.1.2 Water levels

Water levels are the basis for any river study. Most kinds of measurements, such as discharges, have to be relatedto river stages (the stage is simply the water surface height above some fixed datum). Both stage and dischargemeasurements are important. Often, however, the actual discharge of a river is measured rarely, and routine mea-surements are those of stage, which are related to discharge.

Water levels are obtained from gauges, either by direct observation or in recorded form. The data can serve severalpurposes:

• By plotting gauge readings against time, the hydrograph for a particular station is obtained. Hydrographs of aseries of years are used to determine duration curves, showing the probability of occurrence of water levels atthe station or from a rating curve, the probability of discharges.

• Combining gauge readings with discharge values, a relationship between stage and discharge can be deter-mined, resulting in a rating curve for the station.

• Apart from use in hydrological studies and for design purposes, the data can be of direct value for navigation,flood prediction, water management, and waste water disposal.

Methods

Most water level gauging stations are equipped with a sensor or gauge and a recorder. In many cases the waterlevel is measured in a stilling well, thus eliminating strong oscillations.

Staff gauge:

This is the simplest type, with a graduated gauge plate fixed to a stable structure such as a pile, bridge pier, or awall. Where the range of water levels exceeds the capacity of a single gauge, additional ones may be placed on theline of the cross section normal to the plane of flow.

Float gauge:

A float inside a stilling well, connected to the river by an inlet pipe, is moved up and down by the water level.Fluctuations caused by short waves are almost eliminated. The movement of the float is transmitted by a wirepassing over a float wheel, which records the motion, leading down to a counterweight.

Pressure transducers:

The water level is measured as an equivalent hydrostatic pressure and transformed into an electrical signal via a

62

semi-conductor sensor. These are best suited for measuring water levels in open water (the effect of short wavesdies out almost completely within half a wavelength down into the water), as well as for the continuous recordingof groundwater levels. They should compensate for changes in the atmospheric pressure, and if air-vented cablescannot be provided air pressure needs to be measured separately.

Bubble gauge:

This is a pressure actuated system, based on measurement of the pressure which is needed to produce bubblesthrough an underwater outlet. These are used at sites where it would be difficult to install a float-operated recorderor pressure transducer. From a pressurised gas cylinder or small compressor gas is led along a tube to some pointunder the water (which will remain so for all water levels) and bubbles constantly flow out through the orifice. Thepressure in the measuring tube corresponds to that in the water above the orifice. Wind waves should not affectthis.

Ultrasonic sensor:

These are used for continuous non-contact level measurements in open channels. The sensor points vertically downtowards the water and emits ultrasonic pulses at a certain frequency. The inaudible sound waves are reflected bythe water surface and received by the sensor. The round trip time is measured electronically and appears as anoutput signal proportional to the level. A temperature probe compensates for variations in the speed of sound inair. They are accurate but susceptible to wind waves.

Peak level indicators:

There are some indicators of the maximum level reached by a flood, such as arrays of bottles which tip and fillwhen the water reaches them, or a staff coated with soluble paint.

Presentation of results

Stage records taken along rivers used for hydrological studies, for design of irrigation works, or for flood protec-tion require an accuracy of 2 − 5 cm, while gauge readings upstream of flow measuring weirs used to calculatedischarges from the measured heads require an accuracy of 2− 5mm. These days almost all are telemetered to acentral site. There is a huge volume of electronic hydrometry data being sent around Victoria.

Hydrographs, rating tables, and stage relation curves are typical presentations of water level data:

• Hydrograph – when stage records or the discharges are plotted against time.

• Rating table – at many gauging stations water levels are measured daily or hourly, while discharges are mea-sured some times a year, using direct methods such as a propeller meter. From the corresponding water levelsfrom these, and possibly for others over years, a stage-discharge relationship can be built up, so that the routinemeasurement of stage can be converted to discharge. We will be considering these in detail.

• Stage relation curves – from the hydrographs of two or more gauging stations along the river, relationships canbe formulated between the steady flow stages. These can be used to calculate the surface slope between twogauges, and hence, to determine the roughness of the reach. Under unsteady conditions the relationship will bedisturbed. We will also be considering this later.

3.1.3 Discharge

Flow measurement may serve several purposes:

• information on river flow for the design and operation of diversion dams and reservoirs and for bilateral agree-ments between states and countries.

• distribution and charging of irrigation water

• information for charging industries and treatment plants discharging into public waters

• water management in urban and rural areas

• reliable statistics for long-term monitoring.

Continuous (daily or hourly) measurements are very useful.

There are many methods of measuring the rate of volume flow past a point, of which some are single measurementmethods which are not designed for routine operation; the rest are methods of continuous measurements.

Velocity area method (”current meter method”)

63

Figure 3.1: Traditional manner of taking current meter readings. In deeper water a boat is used.

The area of cross-section is determined from soundings, and flow velocities are measured using propeller currentmeters, electromagnetic sensors, or floats. The mean flow velocity is deduced from points distributed systemati-cally over the river cross-section. In fact, what this usually means is that two or more velocity measurements aremade on each of a number of vertical lines, and any one of several empirical expressions used to calculate the meanvelocity on each vertical, the lot then being integrated across the channel.

Calculating the discharge requires integrating the velocity data over the whole channel - what is required is thearea 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)

u dz dy, (3.1)

so that we integrate the velocity from the bed z = 0 to the surface z = h(y), where h is the local depth and whereour z is a local co-ordinate. Then we have to integrate these contributions right across the channel, for values ofthe 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 hydrometricians think of as calculatingthe mean velocity over the depth. Convention in hydrometry is that the mean velocity over a vertical can beapproximated by

u =1

2(u0.2h + u0.8h) , (3.2)

that is, the mean of the readings at 0.2 of the depth and 0.8 of the depth. ? has developed some families of methods.Consider the law for turbulent flow over a rough bed, which can be obtained from the expressions on p582 of ?:

u =u∗κln

z

z0, (3.3)

where u∗ is the shear velocity, κ = 0.4, ln() is the natural logarithm to the base e, z is the elevation above the bed,and z0 is the elevation at which the velocity is zero. (It is a mathematical artifact that below this point the velocityis actually negative and indeed infinite when z = 0 – this does not usually matter in practice). If we integrateequation (3.3) over the depth h we obtain the expression for the mean velocity:

u =1

h

hZ0

u dz =u∗κ

µln

h

z0− 1¶. (3.4)

Now it is assumed that two velocity readings are made, obtaining u1 at z1 and u2 at z2. This gives enoughinformation to obtain the two quantities u∗/κ and z0. Substituting the values for point 1 into equation (3.3) givesus one equation and the values for point 2 gives us another equation. Both can be solved to give the solution

u∗κ=

u2 − u1ln (z2/z1)

and z0 =

µzu21zu12

¶ 1u2−u1

. (3.5)

64

It is not necessary to evaluate these, for substituting into equation (3.4) gives a simple formula for the mean velocityin terms of the readings at the two points:

u =u1 (ln(z2/h)+1)− u2 (ln(z1/h)+1)

ln (z2/z1). (3.6)

As it is probably more convenient to measure and record depths rather than elevations above the bottom, leth1 = h− z1 and h2 = h− z2 be the depths of the two points, when equation (3.6) becomes

u =u1 (ln(1− h2/h)+1)− u2 (ln(1− h1/h)+1)

ln ((h− h2) / (h− h1)). (3.7)

This expression gives the freedom to take the velocity readings at any two points, and not necessarily at pointssuch as 0.2h and 0.8h. This might simplify streamgauging operations, for it means that the hydrometrician, aftermeasuring the depth h, does not have to calculate the values of 0.2h and 0.8h and then set the meter at those points.Instead, the meter can be set at any two points, within reason, the depth and the velocity simply recorded for each,and equation (3.7) applied. This could be done either in situ or later when the results are being processed. This hasthe potential to speed up hydrographic measurements.

If the hydrometrician were to use the traditional two points, then setting h1 = 0.2h and h2 = 0.8h in equation(3.7) gives the result

u = 0.4396u0.2h + 0.5604u0.8h ≈ 0.44u0.2h + 0.56u0.8h , (3.8)

whereas the conventional hydrographic expression is (see e.g. #7.1.5.3 of ?):

u = 0.5u0.2h + 0.5u0.8h . (3.9)

The nominally more accurate expression, equation (3.8), gives less weight to the upper measurement and more tothe lower. It might be useful, as it is just as simple as the traditional expression, yet is based on an exact analyticalintegration of the equation for a turbulent boundary layer.

This has been tested by taking a set of gauging results. A canal had a maximum depth of 2.6m and was 28m wide,and a number of verticals were used. The conventional formula (3.2), the mean of the two velocities, was accurateto within 2% of equation (3.8) over the whole range of the readings, with a mean difference of 1%. That errorwas always an overestimate. The more accurate formula (3.7) is hardly more complicated than the traditional one,and it should in general be preferred. Although the gain in accuracy was slight in this example, in principle it isdesirable to use an expression which makes no numerical approximations to that which it is purporting to evaluate.This does not necessarily mean that either (3.2) or (3.8) gives an accurate integration of the velocities which wereencountered in the field. In fact, one complication is where, as often happens in practice, the velocity distributionnear the surface actually bends back such that the maximum velocity is below the surface.

Integration of the mean velocities across the channel

The problem now is to integrate the readings for mean velocity at each station across the width of the channel.Here traditional practice seems to be in error – often the Mean-Section method is used. In this the mean velocitybetween two verticals is calculated and then multiply this by the area between them, so that, given two verticals iand i+ 1 separated by bi the expression for the contribution to discharge is assumed to be

δQi =1

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

This is not correct. From equation (3.1), the task is actually to integrate across the channel the quantity which isthe 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 thearea is a triangle. We let the water’s edge be i = 0 and the first internal point be i = 1, then the Mean-SectionMethod 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. Thesame happens at the other side. Contributions at these edges are not large, and in the middle of the channel

65

the formula is not so much in error, but in principle the Mean-Section Method is wrong and should not be used.Rather, the Trapezoidal rule should be used, which is just as easily implemented. In a gauging in which the lecturerparticipated, a flow of 19.60m3s−1 was calculated using the Mean-Section Method. Using the Trapezoidal rule,the flow calculated was 19.92m3s−1, a difference of 1.6%. Although the difference was not great, practitionersshould be discouraged from using a formula which is wrong.

An alternative approach

It is strange that only very local methods are used in determining the vertical velocity distribution. ? consideredvelocity distributions given by the more general law, assuming an additional linear and an additional quadraticterm in the velocity profile:

u =u∗κln

z

z0+ a1 z + a2z

2, (3.10)

and using a global approximation method, where a function was assumed which could describe all the velocityprofiles on all the verticals, and then this was fitted to the data. An example of the results is given in figure 3.2.

Figure 3.2: Cross-section of canal with velocity profiles and data points plotted transversely, showing fit by global function

Dilution methods

In channels where cross-sectional areas are difficult to determine (e.g. steep mountain streams) or where flowvelocities are too high to be measured by current meters dilution or tracer methods can be used, where continuityof the tracer material is used with steady flow. The rate of input of tracer is measured, and downstream, after totalmixing, the concentration is measured. The discharge in the stream immediately follows.

Integrating float methods

There is another rather charming and wonderful method which has been very little exploited. At the moment ithas the status of a single measurement method, however the lecturer can foresee it being developed as a contin-uing method, although some research has been attempted, and always difficulties (light, weather, etc.) provedinsurmountable.

Theory

Consider a single buoyant particle (a float, an orange, an air bubble), which is released from a point on the bed. Weassume that it has a constant rise velocity w. As it rises it passes through a variable horizontal velocity field u(z),where z is the vertical co-ordinate. The kinematic equations of the float are

dx

dt= u(z),

dz

dt= w.

Dividing the left and right sides, we obtain a differential equation for the particle trajectory

dz

dx=

w

u(z),

however this can be re-arranged as: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 on the bed,and where we have used a local vertical co-ordinate z with origin on the bed and where the fluid locally has a depth

66

h. The quantity on the left is important - it is the vertical integral of the horizontal velocity, or the discharge perunit width at that section. We can generalise the expression for variation with 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-ordinate direction 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 few approxi-mations:

Q = w

BZ0

L(y) dy.

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!

Ultrasonic flow measurement

Figure 3.3: Array of four ultrasonic beams in a channel

Mean velocity along beam path

Unfortunately, in all textbooks and International Standards a constant velocity is assumed - precisely what is beingsought to measure, and totally ignoring the fact that velocity varies along the path and indeed is zero at the ends!Here we include the variability of velocity in our analysis.

Consider a velocity vector inclined to the beam path at an angle α. If the velocity is u(s), showing that the velocitydoes, in general, depend on position along the beam, then the component along the path is u(s) cosα. Let c bethe speed of sound. The time dt taken for a sound wave to travel a distance ds along the path against the generaldirection of flow is dt = ds/ (c− u(s) cosα). If the path has total length L, then the total time of travel T1 isobtained by integrating to give

T1 =

T1Z0

dt =

LZ0

ds

c− u(s) cosα, (3.11)

and repeating for a traverse in the reverse direction:

T2 =

T2Z0

dt =

LZ0

ds

c+ u(s) cosα. (3.12)

67

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, (3.13)

where we have ignored terms which contain the square of the fluid velocity compared with the speed of sound.Evaluating gives

T1 =L

c+1

c2

LZ0

u(s) cosαds and T2 =L

c− 1

c2

LZ0

u(s) cosαds. (3.14)

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

LZ0

u(s) cosαds = 2L2T1 − T2

(T1 + T2)2 . (3.15)

It can be shown that the relative error of this expression is of order (u/c)2, where u is a measure of velocity. Asu ≈ 1ms−1 and c ≈ 1400ms−1 it can be seen that the error is exceedingly small. What we first need to computethe flow is the integral of the velocity component transverse to the beam path, for which we use the symbol Qz ,the symbol with subscript suggesting the derivative of discharge with respect to elevation:

Qz =

Z L

0

u(s) sinαds. (3.16)

Now we are forced to assume that the angle that the velocity vector makes with the beam is constant over the path(or at least in some rough averaged sense), and so for α constant, taking the trigonometric functions outside theintegral signs and combining equations (3.15) and (3.16) we obtain

Qz = 2 tanαL2 T1 − T2

(T1 + T2)2 . (3.17)

This shows how the result is obtained by assuming the angle of inclination of the fluid velocity to the beam isconstant, but importantly it shows that it is not necessary to assume that velocity u is constant over the beam path.Equation (3.17) is similar to that presented in Standards and trade brochures, and implemented in practice, butwhere it is obtained by assuming that the velocity is constant. It is fortunate that the end result is correct.

Vertical integration of beam data

The mean velocities on different levels obtained from the beam data are considered to be highly accurate, providedall the technical problems associated with beam focussing etc. are overcome, and the streamflow has a constantangle α to the beam. The problem remains to calculate the discharge in the channel by evaluating the verticalintegral of Qz, which, as shown by equation (3.16), is the integral along the beam of the velocity transverse to thebeam. The problem is then to evaluate the vertical integral of the derivative of discharge with elevation:

Q =

hZ0

Qz(z) dz, (3.18)

where in practice the information available is that Qz = 0 on the bottom of the channel z = 0 and the two tofour values of Qz which have been obtained from beam data, as well as the total depth h. It is in the evaluationof this integral that the performance of the trade and scientific literature has been poor. Several trade brochuresadvocate the routine use of a single beam, or maybe two, suggesting that that is adequate (see, for example, Boiten2000?, p141). In fact, with high-quality data for Qz at two or three levels, there is no reason not to use accurateintegration formulae. However, practice in this area has been quite poor, as trade brochures that the author hasseen use the inaccurate Mean-Section Method for integrating vertically over only three or four data points, whenits errors would be rather larger than when it is used for many verticals across a channel, as described previously.This seems to be a ripe area for research.

Acoustic-Doppler Current Profiling (ADCP) methods

In these, a beam of sound of a known frequency is transmitted into the fluid, often from a boat. When the soundstrikes moving particles or regions of density difference moving at a certain speed, the sound is reflected back andreceived by a sensor mounted beside the transmitter. According to the Doppler effect, the difference in frequencybetween the transmitted and received waves is a direct measurement of velocity. In practice there are many particles

68

in the fluid and the greater the area of flow moving at a particular velocity, the greater the number of reflectionswith that frequency shift. Potentially this method is very accurate, as it purports to be able to obtain the velocityover quite small regions and integrate them up. However, this method does not measure in the top 15% of the depthor near the boundaries, 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.

Electromagnetic methods

Signal probes

Coil for producing magnetic field

Figure 3.4: Electromagnetic installation, showing coil and signal probes

The motion of water flowing in an open channel cuts a vertical magnetic field which is generated using a large coilburied beneath the river bed, through which an electric current is driven. An electromotive force is induced in thewater and measured by signal probes at each side of the channel. This very small voltage is directly proportionalto the average velocity of flow in the cross-section. This is particularly suited to measurement of effluent, waterin treatment works, and in power stations, where the channel is rectangular and made of concrete; as well as insituations where there is much weed growth, or high sediment concentrations, unstable bed conditions, backwatereffects, or reverse flow. This has the advantage that it is an integrating method, however in the end recourse has tobe made to empirical relationships between the measured electrical quantities and the flow.

3.2 The analysis and use of stage and discharge measurements

Almost universally the routine measurement of the state of a river is that of the stage, the surface elevation ata gauging station, usually specified relative to an arbitrary local datum. While surface elevation is an importantquantity in determining the danger of flooding, another important quantity is the actual flow rate past the gaugingstation. Accurate knowledge of this instantaneous discharge - and its time integral, the total volume of flow - iscrucial to many hydrologic investigations and to practical operations of a river and its chief environmental andcommercial resource, its water. Examples include decisions on the allocation of water resources, the design ofreservoirs and their associated spillways, the calibration of models, and the interaction with other computationalcomponents of a network.

3.2.1 Stage discharge method

The traditional way in which volume flow is inferred is for a Rating Curve to be derived for a particular gaugingstation, which is a relationship between the stage measured and the actual flow passing that point. The measurementof flow is done at convenient times by traditional hydrologic means, with a current meter measuring the flowvelocity at enough points over the river cross section so that the volume of flow can be obtained for that particularstage, measured at the same time. By taking such measurements for a number of different stages and correspondingdischarges over a long period of time, a number of points can be plotted on a stage-discharge diagram, and a curvedrawn through those points, giving what is hoped to be a unique relationship between stage and flow, the RatingCurve, as shown in Figure 3.5. If the supposedly unique relationship between the flow rate and the stage is writtenQr(η), subsequent measurements of the surface elevation at some time t, such as an hourly or daily measurement,are then used to give the discharge:

Q(t) = Qr(η(t)).

It is assumed that for any stage reading, which is the routine periodic measurement, the corresponding dischargecan be calculated. This is what happens when the stage is read and telemetered to a central data managementauthority. From the rating curve for that stage, the corresponding discharge can be calculated. This is very widelyused and is the routine method of flow measurement. It begs a number of questions.

69

Discharge (m3s−1)

Stage

(m)

Figure 3.5: Rating curve (stage-discharge diagram) and data points to which it has been fitted

Qfalling Qrated Qrising Discharge

Stage

A measured stage value

Steady flow rating curve Actual flood event

Figure 3.6: Stage-discharge diagram showing the steady-flow rating curve and an exaggerated looped trajectory of a particu-lar flood event, which can be due to two effects: changing roughness and unsteady flow effects

There are several problems associated with the use of a Rating Curve:

• The assumption of a unique relationship between stage and discharge may not be justified.

• Discharge is rarely measured during a flood, and the quality of data at the high flow end of the curve might bequite poor.

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

• It has to describe a range of variation from no flow through small but typical flows to very large extreme floodevents.

• There are a number of factors which might cause the rating curve not to give the actual discharge, some ofwhich will vary with time. Factors affecting the rating curve include:– The channel changing as a result of modification due to dredging, bridge construction, or vegetation growth.– Sediment transport - where the bed is in motion, which can have an effect over a single flood event, becausethe effective bed roughness can change during the event. As a flood increases, any bed forms present will tendto become larger and increase the effective roughness, so that friction is greater after the flood peak than before,so that the corresponding discharge for a given stage height will be less after the peak. This will contribute to

70

a flood event showing a looped curve on a stage-discharge diagram as is shown on Figure 3.6.– Backwater effects - changes in the conditions downstream such as the construction of a dam or flooding inthe next waterway.– Unsteadiness - in general the discharge will change rapidly during a flood, and the slope of the water surfacewill be different from that for a constant stage, depending on whether the discharge is increasing or decreasing,also contributing to a flood event appearing as a loop on a stage-discharge diagram such as Figure 3.6.– Variable channel storage - where the stream overflows onto flood plains during high discharges, giving riseto different slopes and to unsteadiness effects.– Vegetation - changing the roughness and hence changing the stage-discharge relation.– Ice - which we will ignore.

Some of these can be allowed for by procedures which we will describe later.

The hydraulics of a gauging station

High flow

Low flow

Localcontrol

Distantcontrol

Channel control

Gaugingstation

Channel control

Flood

1

3

2

4

5

Larger bodyof water

Figure 3.7: Section of river showing different controls at different water levels with implications for the stage dischargerelationship at the gauging station shown

A typical set-up of a gauging station where the water level is regularly measured is given in Figure 3.7 whichshows a longitudinal section of a stream. Downstream of the gauging station is usually some sort of fixed controlwhich may be some local topography such as a rock ledge which means that for relatively small flows there is arelationship between the head over the control and the discharge which passes. This will control the flow for smallflows. For larger flows the effect of the fixed control is to ”drown out”, to become unimportant, and for some otherpart of the stream to control the flow, such as the larger river downstream shown as a distant control in the figure,or even, if the downstream channel length is long enough before encountering another local control, the sectionof channel downstream will itself become the control, where the control is due to friction in the channel, givinga relationship between 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. not changingwith time anywhere in the system) there would be a unique relationship between stage and discharge, howevercomplicated this might be due to various controls. In practice, the natures of the controls are usually unknown.

Something which the concept of a rating curve overlooks is the effect of unsteadiness, or variation with time. Ina flood event the discharge will change with time as the flood wave passes, and the slope of the water surface willbe different from that for a constant stage, depending on whether the discharge is increasing or decreasing. Figure3.7 shows the increased surface slope as a flood approaches the gauging station. The effects of this are shown onFigure 3.6, in somewhat exaggerated form, where an actual flood event may not follow the rating curve but will ingeneral follow the looped trajectory shown. As the flood increases, the surface slope in the river is greater than theslope for steady flow at the same stage, and hence, according to conventional simple hydraulic theory explainedbelow, more water is flowing down the river than the rating curve would suggest. This is shown by the dischargemarked Qrising 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 highlydynamic floods, and also since the maximum discharge and maximum stage do not coincide, the arrival time of thepeak discharge could be in error and may influence flood warning predictions. Similarly water-quality constituentloads could be underestimated if the dynamic characteristics of the flood are ignored, while the use of a dischargehydrograph derived inaccurately by using a single-valued rating relationship may distort estimates for resistancecoefficients during calibration of an unsteady flow model.

71

Rating curves – representation, approximation and calculation

7

8

9

10

11

12

13

14

0 500 1000 1500 2000 2500

Stage(m)

Discharge (Q m3s−1)

Figure 3.8: Rating curve using natural (Q, η) axes

Figure 3.8 shows the current rating curve for the Ovens River at Wangaratta in Victoria, Australia, where flowmeasurements have been made since 1891. There are a couple of difficulties with such a curve, including readingresults off for small flows, where the curve is locally vertical, and for high flows where it is almost horizontal.A traditional way of overcoming the difficulty of representing rating curves over a large range has been to uselog-log axes. However, this has no physical basis and has a number of practical difficulties, although it has beenrecommended by International Standards.Hydraulic theory can help here, for it can be used to show that the stage-discharge relationship will tend to show stage varying approximately like η ∼ Q1/2, for both cases:

1. Flow across a U-shaped (parabolic) weir, the approximate situation for low flow at a gauging station, when alocal control such as a rock ledge controls the flow, and

2. Uniform flow down a U-shaped (parabolic) waterway for large flows, when the local control is washed out andthe waterway acts more like a uniform flow governed by Manning’s law.

7

8

9

10

11

12

13

14

0 10 20 30 40 50

Stage(m)

√Q (m3s−1)1/2

Figure 3.9: The same rating curve as in figure 3.8 but using (√Q, η) axes

In these cases, both parts of the relationship would plot as (different) straight lines on¡√

Q,η¢

axes. In figure 3.9

72

we plot the results from the above figure on such a square root scale for the discharge, and we see that indeedat both small and large flows the rating curve is a straight line. This means that simpler procedures of numericalapproximation and interpolation could be used. Sometimes results have to be taken by extrapolating the curve. Ifthis has to be done, then linear extrapolation on the

¡√Q,η

¢axes might be reasonable, but it is still a procedure to

be followed with great caution, as the actual geometry for above-bank flows can vary a lot.

A number of such practical considerations are given in ?.

3.2.2 Stage-discharge-slope method

This is presented in some books and in International Standards, however, especially in the latter, the presentation isconfusing and at a low level, where no reference is made to the fact that underlying it the slope is being measured.Instead, the fall is described, which is the change in surface elevation between two surface elevation gauges andis simply the slope multiplied by the distance between them. No theoretical justification is provided and it ispresented in a phenomenological sense (see, for example, ?).

Although the picture in Figure 3.7 of the factors affecting the stage and discharge at a gauging station seemscomplicated, the underlying processes are capable of quite simple description. In a typical stream, where all wavemotion is of a relatively long time and space scale, the governing equations are the long wave equations, whichare a pair of partial differential equations for the stage and the discharge at all points of the channel in terms oftime and distance along the channel. One is a mass conservation equation, the other a momentum equation. Underthe conditions typical of most flows and floods in natural waterways, however, the flow is sufficiently slow thatthe equations can be simplified considerably. Most terms in the momentum equation are of a relative magnitudegiven by the square of the Froude number, which is U2/gD, where U is the fluid velocity, g is the gravitationalacceleration, and D is the mean depth of the waterway. In most rivers, even in flood, this is small, and theapproximation may be often used. For example, a flow of 1ms−1 with a depth of 2m has F 2 ≈ 0.05.In §2.3.8 we saw that under these circumstances, a surprisingly good approximation to the momentum equation ofmotion for flow in a waterway is the simple equation:

Q(t) = K(η(t))p−∂η/∂x(t), (3.19)

where we have shown that the slope of the free surface might be a function of time. This gives us a formula forcalculating the discharge Q which is as accurate as is reasonable to be expected in river hydraulics, provided weknow

1. the stage and the dependence of conveyance K on stage at a point from either measurement or the G-M-S orChézy’s formulae, and

2. the slope of the surface

Stage-conveyance curves

Equation (3.19) shows how the discharge actually depends on both the stage and the surface slope, whereas tra-ditional hydrography assumes that it depends on stage alone. If the slope does vary under different backwaterconditions or during a flood, then a better hydrographic procedure would be to gauge the flow when it is steady,and to measure the surface slope , thereby enabling a particular value of K to be calculated for that stage. If thiswere done over time for a number of different stages, then a stage-conveyance relationship could be developedwhich should then hold whether or not the stage is varying. Subsequently, in day-to-day operations, if the stageand the surface slope were measured, then the discharge calculated from equation (3.19) should be quite accurate,within the relatively mild assumptions made so far. All of this holds whether or not the gauging station is affectedby a local or channel control, and whether or not the flow is changing with time.

This suggests that a better way of determining streamflows in general, but primarily where backwater and unsteadyeffects are likely to be important, is for the following procedure to be followed:

1. At a gauging station, two measuring devices for stage be installed, so as to be able to measure the slope of thewater surface at the station. One of these could be at the section where detailed flow-gaugings are taken, andthe other could be some distance upstream or downstream such that the stage difference between the two pointsis enough that the slope can be computed accurately enough. As a rough guide, this might be, say 10 cm, sothat if the water slope were typically 0.001, they should be at least 100m apart.

2. Over time, for a number of different flow conditions the discharge Q would be measured using conventionalmethods such as by current meter. For each gauging, both surface elevations would be recorded, one becomingthe stage η to be used in the subsequent relationship, the other so that the surface slope Sη can be calculated.Using equation (3.19), Q = K(η)

pSη, this would give the appropriate value of conveyance K for that stage,

automatically corrected for effects of unsteadiness and downstream conditions.

73

3. From all such data pairs (ηi,Ki) for i = 1, 2, . . ., the conveyance curve (the functional dependence of K onη) would be found, possibly by piecewise-linear or by global approximation methods, in a similar way to thedescription of rating curves described below. Conveyance has units of discharge, and as the surface slope isunlikely to vary all that much, we note that there are certain advantages in representing rating curves on a plotusing the square root of the discharge, and it my well be that the stage-conveyance curve would be displayedand approximated best using (

√K, η) axes.

4. Subsequent routine measurements would obtain both stages, including the stage to be used in the stage-conveyance relationship, and hence the water surface slope, which would then be substituted into equation(3.19) to give the discharge, corrected for effects of downstream changes and unsteadiness.

If hydrography had followed the path described above, of routinely measuring surface slope and using a stage-conveyance relationship, the ”science” would have been more satisfactory. Effects due to the changing of down-stream controls with time, downstream tailwater conditions, and unsteadiness in floods would have been automat-ically incorporated, both at the time of determining the relationship and subsequently in daily operational practice.

However, for the most part slope has not been measured, and hydrographic practice has been to use rating curvesinstead. The assumption behind the concept of a discharge-stage relationship or rating curve is that the slope at astation is constant over all flows and events, so that the discharge is a unique function of stage Qr(η) where weuse the subscript r to indicate the rated discharge. Instead of the empirical/rational expression (3.19), traditionalpractice is to calculate discharge from the equation

Q(t) = Qr(η(t)), (3.20)

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

In comparison, equation (3.19), based on a convenient empirical approximation to the real hydraulics of the river,contains the essential nature of what is going on in the stream. It shows that, although the conveyance might bea unique function of stage which it is possible to determine by measurement, because the surface slope will ingeneral vary throughout different flood events and downstream conditions, discharge in general does not dependon stage alone.

3.2.3 Looped rating curves – correcting for unsteady effects in obtaining dischargefrom stage

In conventional hydrography the stage is measured repeatedly at a single gauging station so that the time derivativeof stage can easily be obtained from records but the surface slope along the channel is not measured at all. Themethods of this section are all aimed at obtaining the slope in terms of the stage and its time derivatives at a singlegauging station. The simplest and most traditional method of calculating the effects of unsteadiness has been theJones formula, derived by B. E. Jones in 1916 (see for example ?, ?). The principal assumption is that to obtain theslope, the x derivative 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 and Whitham,1955?) such that it obeys the partial differential equation:

∂h

∂t+ c

∂h

∂x= 0, (3.21)

where h is the depth and c is the kinematic wave speed. Solutions of this equation are simply waves travellingat a velocity c without change. The equation was obtained as the last of a series of approximations in Section2.3.11. The kinematic wave speed c is given by the derivative of flow with respect to cross-sectional area, theKleitz-Seddon law

c =1

B

dQr

dη=1

B

dK

dη

pS, (3.22)

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

pS, and the slope

pS is the mean slope

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

The Jones method assumes that the surface slope Sη in equation (3.19) can be simply related to the rate of changeof stage with time, assuming that the wave moves without change. Thus, equation (3.21) gives an approximationfor the surface slope: ∂h/∂x ≈ −1/c×∂h/∂t. We then have to use the simple geometric relation between surfacegradient and depth gradient, that ∂η/∂x = ∂h/∂x− S, such that we have the approximation

Sη = −∂η∂x

= S − ∂h

∂x≈ S +

1

c

∂h

∂t

74

and recognising that the time derivative of stage and depth are the same, ∂h/∂t = ∂η/∂t, equation (3.19) gives

Q = K

rS +

1

c

∂η

∂t(3.23)

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 theoryis accurate. In general, however, there will be a certain amount of diffusion observed, where the wave crest subsidesand the effects of the wave are smeared out in time.

To allow for those effects ? provided the theoretical derivation of two methods for calculating the discharge. Thederivation of both is rather lengthy. The first method used the full long wave equations and approximated thesurface slope using a method based on a linearisation of those equations. The result was a differential equation fordQ/dt in terms of Q and stage and the derivatives of stage dη/dt and d2η/dt2, which could be calculated fromthe record of stage with time and the equation solved numerically. The second method was rather simpler, and wasbased on the next best approximation to the full equations after equation (3.21). This gives the advection-diffusionequation

∂h

∂t+ c

∂h

∂x= ν

∂2h

∂x2, (3.24)

where the difference between this and equation (3.21) is the diffusion term on the right, where ν is a diffusioncoefficient (with units of L2T−1), given by

ν =K

2BpS.

Equation (3.24), is a consistent low-inertia approximation to the long wave equations, where inertial terms, whichare of the order of the square of the Froude number, which approximates motion in most waterways quite well.However, it is not yet suitable for the purposes of this section, for we want to express the x derivative at a point interms of time derivatives. To do this, we use a small-diffusion approximation, we assume that the two x derivativeson the right of equation (3.24) 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. Theresulting expression is:

∂h

∂t+ c

∂h

∂x=

ν

c2∂2h

∂t2,

and solving for the x derivative, we have the approximation

Sη = −∂η∂x

= S − ∂h

∂x≈ S +

1

c

∂h

∂t− ν

c3d2h

dt2,

and substituting into equation (3.19) gives

Q = Qr(η)

vuuuut 1|zRating curve

+1

cS

dη

dt| z Jones formula

− ν

c3S

d2η

dt2| z Diffusion term

(3.25)

where Q is the discharge at the gauging station, Qr(η) is the rated discharge for the station as a function of stage,S is the bed slope, c is the kinematic wave speed given by equation (3.22):

c =

pS

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 wherethe coefficient ν is the diffusion coefficient in advection-diffusion flood routing, given by:

ν =K

2BpS=

Qr

2BS. (3.26)

In equation (3.25) it is clear that the extra diffusion term is a simple correction to the Jones formula, allowing forthe subsidence of the wave crest as if the flood wave were following the advection-diffusion approximation, whichis a good approximation to much flood propagation. Equation (3.25) provides a means of analysing stage recordsand correcting for the effects of unsteadiness and variable slope. It can be used in either direction:

75

• If a gauging exercise has been carried out while the stage has been varying (and been recorded), the value ofQ obtained can be corrected for the effects of variable slope, giving the steady-state value of discharge for thestage-discharge relation,

• And, proceeding in the other direction, in operational practice, it can be used for the routine analysis of stagerecords to correct for any effects of unsteadiness.

The ideas set out here are described rather more fully in ?.

An example

A numerical solution was obtained for the particular case of a fast-rising and falling flood in a stream of 10 kmlength, of slope 0.001, which had a trapezoidal section 10m wide at the bottom with side slopes of 1:2, and aManning’s friction coefficient of 0.04. The downstream control was a weir. Initially the depth of flow was 2m,while carrying a flow of 10m3s−1. The incoming flow upstream was linearly increased ten-fold to 100m3s−1over 60 mins and then reduced to the original flow over the same interval. The initial backwater curve problemwas solved and then the long wave equations in the channel were solved over six hours to simulate the flood. Ata station halfway along the waterway the computed stages were recorded (the data one would normally have), aswell as the computed discharges so that some of the above-mentioned methods could be applied and the accuracyof this work tested.

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 (3.25)

Figure 3.10: Simulated flood with hydrographs computed from stage record using three levelsof approximation

Results are shown on Figure 3.10. It can be seen that the application of the diffusion level of approximation ofequation (3.25) has succeeded well in obtaining the actual peak discharge. The results are not exact however,as the derivation depends on the diffusion being sufficiently small that the interchange between space and timedifferentiation will be accurate. In the case of a stream such as the example here, diffusion is relatively large, andour results are not exact, but they are better than the Jones method at predicting the peak flow. Nevertheless, theresults from the Jones method are interesting. A widely-held opinion is that it is not accurate. Indeed, we see herethat in predicting the peak flow it was not accurate in this problem. However, over almost all of the flood it wasaccurate, and predicted the time of the flood peak well, which is also an important result. It showed that both beforeand after the peak the ”discharge wave” led the ”stage wave”, which is of course in phase with the curve showing

76

the flow computed from the stage graph and the rating curve. As there may be applications where it is enough toknow the arrival time of the flood peak, this is a useful property of the Jones formula. Near the crest, however, therate of rise became small and so did the Jones correction. Now, and only now, the inclusion of the extra diffusionterm gave a significant correction to the maximum flow computed, and was quite accurate in its prediction that thereal flow was some 10% greater than that which would have been calculated just from the rating curve. In thisfast-rising example the application of the unsteady corrections seems to have worked well and to be justified. It isno more difficult to apply the diffusion correction than the Jones correction, both being given by derivatives of thestage record.

3.2.4 The effects of bed roughness on rating curves

Introduction

5678910111213141516

0 5000 10000 15000 20000

Stage(m)

Discharge (Q m3s−1)

Figure 3.11: Flood trajectories for Station 41 on the Red River, showing raw data – correcteddata was everywhere obscured

We have considered the detailed records for 1995 and 1996 for the Red River, Viet Nam. The trajectories, plottedon Stage/Discharge axes on Figure 3.11, show considerable loopiness. In the expectation that the cause was theunsteady wave effects as described above, we applied the theory described above, assuming for simple purposes,a slope S0 = 0.0015, and using the recorded values of stage, depth, area, and breadth. Everywhere, no visiblecorrection was made, and the methods of the section above have failed to describe the loopiness. It seems that theloopiness in this case must be due to effects of the bed topography and friction changing with flow. This is themore common situation, and it is bed roughness changes which will more often be the cause of loopiness.

? have written on the nature of the relationship between stage and discharge when the river bed is mobile, whenbed forms may change, depending on the flow, such that in a flood there may be different bed roughness beforeand after, and the stage-discharge trajectory may show loopiness, as shown in Figure 3.11 above. This suggests adifferent approach to the relationship between stage and discharge observed in a river, when the river itself is thecontrol, rather than a hydraulic structure.

Let us consider the mechanism by which changes in roughness cause the flood trajectories to be looped, by consid-ering a hypothetical and idealised situation. In Figure 3.12 is shown a plot of Stage versus Discharge. The ratingcurves which would apply if the bedforms were held constant are shown – for a flat bed and for various increas-ing bedform roughness. In the left corner is a stage-time graph with three flood events, one small, one large, andone not yet completed. The points labelled O, A, ..., G are also shown on the flood trajectory, showing the actualrelationship between stage and discharge at each time.

O: flow is small, over a flat bed.

A: a small flood peak has arrived. The flow is not enough to change the nature of the bed, and the floodtrajectory follows the flat-bed rating curve back down to a smaller flow, and then back up to a larger flow.

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Stage

Stage

Time

Discharge

O

AE

B

C

GD

Large bedforms

Medium bedforms

Small bedforms

Flat bed

Flood trajectory

Rating curves — for constant roughness

F

O A B C D EF G

Figure 3.12: Rating curves for different bedforms and a looped flood trajectory

B: the bed is no longer stable and bed forms, with increasing roughness, start to form. Accordingly, the carryingcapacity of the channel is reduced and the stage increases.

C: the flood peak has arrived, and the bed forms continue to grow, so that a little time later the stage is amaximum.

D: the bedforms have continued to grow until here, although the flow is decreasing.

E: the flow has decreased much more quickly than the bed-forms can adjust, and the point is only a little belowthe rating curve corresponding to the largest bedforms.

F: over the intervening time, flow has been small and almost constant, however the time has been enough toreduce the bed-forms. Now another flood starts to arrive, and this time, instead of following the flat-bed curve,it already starts from a finite roughness, such as we seem to see in Figure 3.11.

G: after this, the history of the stage will still depend on the history of the flow and the characteristics of thebed-form rate of growth.

A theoretical approach

This process could be modelled mathematically. If we had an expression for the rate of growth of roughness as afunction of discharge, depth, and roughness itself, say,

dΩ

dt= f(Q,h,Ω), (3.27)

where Q is discharge, h is depth, and Ω is some measure of roughness, and if we had a channel friction law suchas Gauckler-Manning-Strickler, which we write in a general way:

Q = F (h,Ω, S), (3.28)

then if we assume that slope is constant, as it almost always is except when unsteady effects are important, thendifferentiating (3.28) with respect to time,

dQ

dt=

∂F

∂h

dh

dt+

∂F

∂Ω

dΩ

dt

=∂F

∂h

dh

dt+

∂F

∂Ωf(Q,h,Ω),

having substituted equation (3.27), giving us an ordinary differential equation for Q as a function of t provided weknow the stage hydrograph h(t), which is what is usually measured.

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In general the condition of the river bed, whether smooth, or with dunes, antidunes or standing waves, will dependon the time history of the flow. That is, the roughness now depends on the preceding conditions for a period oftime.

An empirical approach

This immediately suggests the concept of using some form of convolution, where the effects of preceding eventsare incorporated in an integral sense. In hydrology, this is most familiar as the unit hydrograph, see for example,Chapter 7 of ?. In the case of a river with data such as in Figure 3.11 we suggest the following nonlinear influencefunction, where the discharge Q at time t can be written as nonlinear functions of stage η at previous times t−∆,t− 2∆, etc.:

Q(t) = a00 +a01η(t) + a02η2(t) + . . .

+a11η(t−∆) + a12η2(t−∆) + . . .

+a21η(t− 2∆) + a22η2(t− 2∆) + . . .

+ . . . .

If only the first line of that equation had been taken, then that is a conventional rating curve, where the dischargeat t is assumed to be a function of the stage at t.

Now, the procedure would be to calculate the coefficients aij from a long data sequence, using least squaresmethods. This could then be then applied to future events so that a better prediction of the actual discharge couldbe obtained.

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Chapter 4 Computational hydraulics

4.0.5 Steady flow

4.0.6 Unsteady flow

Finite differences – Forwards-time-centred space

The slow-change/slow-flow approximation

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Chapter 5 Sediment transport

5.1 General

5.2 Initiation of motion

5.3 Bedforms and alluvial roughness

5.4 Transport formulae

5.5 Unsteady aspects

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Chapter 6 River morphology

6.1 Introduction

6.2 Planform

6.3 Longitudinal profile

6.4 Bends

6.5 Channel characteristics

6.6 Bifurcations and confluences

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Chapter 7 River engineering

7.1 Introduction

7.2 Bed regulation

7.3 Discharge control

7.4 Water level control

7.5 Water quality control

7.6 River engineering for different purposes

7.6.1 Flood control and drainage

7.6.2 Navigation

7.6.3 Hydropower

7.6.4 Water supply

7.6.5 Waste discharge

7.6.6 Crossings of other infrastructure

7.6.7 Soil conservation

7.6.8 Nature preservation and restoration

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ReferencesFrom Wikipedia:

UNESCO- und DIN-Definition

Die Hydrographie ist "die Wissenschaft und Praxis der Messung und Darstellung der Parameter, die notwendigsind, um die Beschaffenheit und Gestalt des Bodens der Gewässer, ihre Beziehung zum festen Land und denZustand und die Dynamik der Gewässer zu beschreiben." (Definition nach United Nations Economic and SocialCouncil, 1978)

Diese Definition ist auch in der DIN 18709-3 festgehalten. In der DIN werden die Bezeichnungen Gewässerver-messung, Seevermessung und Binnengewässervermessung einheitlich mit "hydrographic surveying" ins Englischeübertragen.

Englische IHO-Definition

Hydrography: That branch of applied sciences which deals with the measurement and description of the features ofthe seas and coastal areas for the primary purpose of navigation and all other marine purposes and activities –interalia- offshore activities, research, protection of the environment, and prediction services. (Definition: InternationalHydrographic Organisation, IHO) Eine neue Definition von „Hydrography“ wird momentan in der IHO diskutiert(April, 2009).

Österreicher ÖNORM-Definition

Die Hydrographie ist jener Teil der Hydrologie der sich mit der quantitativen Erfassung und Beschreibung desWasserkreislaufes auf, unter und über der Erdoberfläche und mit der Behandlung der damit zusammenhängendenFragen beschäftigt. (Ergänzende ÖNORM B 2400 zur ÖNORM EN ISO 772)

Diese wohl in Österreich und in der Schweiz bekannte Definition ist in Deutschland unüblich und trifft auch nichtdie Bedeutung des englischen Wortes "Hydrography".

Bifurcations and confluences – this should go after the structures part

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