Practical Aspects Computational River Hydraulics

r

Editorial Board Main Editon

L J. Mostertman (Co-ordinating), International Institute foc Hydraulic and Environmental Engineering, Delft

W. L Moore, University of Texas at Austin E.Mosonyi, Universitit Karlsruhe

Advisory Editors

E. P. Evans, Sir William Halcrow and Partners, Swindon E. M. l.aurenson, Monash University

Y. Maystre, Ecole Poly technique Pederale de Lausanne O. Rescher, Technische Universitat Wien

Advisor for Computational Hydraulics Series M. B. Abbott, International Institute for Hydraulic and Environmental Engineering, Delft

,Practical Aspects of

Computational

River Hydraulics/

J.A. ~nge, .E.Mo-Holly;-Jr

Societe Grenobloise d'Etudes at d'Applications Hydrauliques, SOGREAH, Grenoble

A. Verwey

International Institute for Hydraulic and Environmental

Engineering, Delft

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Practical aspects of computational river hYdraulics

(Monographs andIUrVeysin water resources engineering; 3)

Bibliography:p.

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I.Rivers-Mathematical models. 2.Stream measurements-Mathernat ea

models.3.Channels (HYdraulic engineering)-Mathematical models. I.Holly, Fonest M.,joint author. II. Verwey, Adri,jolnt author. 01. TItle. IV. Series.

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Contents

Acknowledgements

List of symbols

1 Introduction 1

2 Mathematical fonnulation of physical processes 7

2.1 Equations of one-dimensional unsteady open channel flow 7 Basic hypotheses 7

Integral relations 9

Differential form of the de 5t Venant equations 13 Supplementary terms and coefficients 18

2.2 Characteristics - boundary and initial conditions 24 Boundary data requirements 29

2.3 Discontinuous solutions - bores 37 2.4 Simplified channel flow equations 44

Effect of neglecting the inertia terms 45

Effect of neglecting the inertia terms and the

ah/ax

term 46 Steady flow equations 48

2.5 Representation of specialflow conditions 48

Localized inapplicability of the channel flow equations 49 Quasi two-dimensional flow 50

3 Solution techniques and their evaluation S3

3.1 Discretization and solution of flow relationships 53 Numerical solution by the method of characteristics 54 Numerical solution by the method of finite differences 59 Some finite difference schemes 62

Discretization of boundary conditions 72

DiscretiZation of the quasi-two-dimensional flow equations 75 3.2 linear analysis of the validity of discretization 77

Convergence and approximation error 77 Numerical stability 80

Amplitude and phase portraits 86

3.3 Discretization of non-linear terms and coefficients 92

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Verwey's variant of the Preissmann scheme 96 Abbott-Ionescu scheme 97

Comments 98

3.4 Algorithmic aspects of modelling systems 103 Iterative matrix methods 105

Double sweep methods 106

3.5 Computational principles of steep front simulation Shock fitting method 122

Pseudoviscosity method 124 Through methods 126

3.6 Representation of topographic and hydraulic data

121

128 4 _{Flow simulation in natural riven 132}

4.1 Introductory remarks 132

4.2 Choice of equations for channel flaw 135 4.3 One.dimensional and two-dimensional modelling

4.4 Topological discretization 143 4.5 Hydraulic discretization 159

4.6 Some computational problems inriver and flood plain flow simulation 175

Smalldepths 175 Weir Oscillations 178 Flooded weir linearization Steady flow calculations 4.7 Concluding remarks 138 180 181 184 5

Model calibration and data needs 185 5.1 Model calibration 185

Steady flow 186 Unsteady flow 193

Example: calibration of the Senegal Valley model 208 Example: calibration of the upper Rhone model 216 Accuracy of calibration 222

5.2 Data needs 225 General remarks 225 Topographic data 228 Hydraulic data 230

6 Modelling of flow regulation inirrigation cana1s and power

"""des

233

6.1 Flow Control in irrigation and water supply canals 233 6.2 Surges in power canals 242

6.3 Flow and energy production control In Power cascades on

canalized rivers 253 ,

6.4 Computational and modelling considerations 260 MOdelling of discontinuous fronts 260

Modelling of transitions and control structures 263 Example of gate simulation 265

7 Movable bed models 271

7.1 The role of movable bed mathematical models in engineering practice 271

7.2 Basic hypotheses and formulation of equations 273 7.3 Boundary conditions in movable bed modelling 278 7.4 Data requirements 280

7.5 Mathematical analysis of the equations 281 Full unsteady flow equations 281

Simplified system of unsteady flow equations 282 7.6 Numerical solutions 287

Full system of three equations 287 Simplified system of two equations 291 7.7 Models of alluvial channel resistance 295

Physical aspects 295

Energy line gradient formulation 298 Solid transport formulation 298 Examples of application 299

7.8 Criteria involved in choosing a modelling method 307 Factors linked to the physical phenomena simulated 308 Factors linked to numerical methods 309

8 Transport of pollutants 312 8.1 Introduction 312

8.2 The dispersion process 313

8.3 One-dlmenslonal dispersion modelling 318

8,4 Evaluation of the longitudinal mixing coefficient 319 8.5 Numerical solution of the one-dimensional convection

equation 320

8.6 Numerical solution of the one-dimensional diffusion equation 331

8.7 Example of one-dimensional dispersion modelling - the Vienne river 333

8.8 Two-dimensional dispersion modelling 336

8.9 Example: simulation of two-dimensional dispersion inthe Missouri river from a continuous source 340

8.1 0 Example: simulation of one-dimensional dispersion in Clinch river 343

8.11 Estimating the transverse distribution of longitudinal velocity 346 8.12 Conclusions 349

9 Specialapplications 350

9.1 Flood forecasting and prediction 350

Strategy for implementation of forecasting models 354 Particular calibration and sensitivity study problems 356

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Simulation nf dam break waves 357 . 357

Physical description of the phenomenon and governing equations Computational problems 360

Unsteady flow modelling in storm drain networks Pressurizedflow 366

Backflow from junctions Backwatereffects 368 Smalldepths 368 looped networks 369 Hydraulicworks 370 9.3 365 368

10 Costs, benefits and qWllity 372

10.1 Cost/benefit and cost/quality ratios 372

10.2 Factors affecting the cost of a mathematical model 374 Algorithm and software development 374

Preliminary study 375

Adaptation of existing software to the project's particular features 376

Construction of the model 377

Supplementary surveysand measurements 378 Modelcalibration 378

Modelexploitation and interpretation of results 379 Transfer of models 380

10.3 Quality of a model 380 Differential equations 382 Finite difference Operator 383

Solution algOrithm and treatment of special features 383

Data

input and output 384

Simulationof observed situations 384 10.4 Modeller-user relationship 384

10.5 Examplesof cost of mathematical models 385 Upper Nile basin 385

MekongRiver Delta 386

II TransferofmathematicaImode1s 391

11.1 Introduction 391

11.2 ModeUer_userrelationship 392

11.3 Computer and operating system problems 396 11.4 Trained personnel problems 397

11.5 Examplesof transfer operations 399

The transfer of the MekongDelta rnathematiclil model from Grenoble to Bangkok 399

The transfer of the SenegalRiver model f~om Grenoble to Dakar 403

Rer_...

407

fndex

417

Acknowledgements

The flrst two authors have acquired most of their experience at the Consulting Engineering firm SOGREAH (Societe Grenobloise d'Etudes et d' Applications Hydrauliques). They wish to express their appreciation to SOGREAH for per-mission to use internal reports and notes, for help in obtaining client's

authorisations to use study material, and for material support in preparing the manuscript, especially the final typing.

The first applications of mathematical modelling techniques to river engineering problems inthe 1960's were made possible by Dr Alexandre Preissmann, member of SOGREAH's Board of Experts. The first two authors consider it a great privilege to be able to work withthe man whose pioneering work launched the present day widespread use of mathematical models. His constant help, guidance and, above all, his criticism have always been a source of encouragement for which we wish to express our profound gratitude.

The third author has greatly appreciated the stimulation of Professorir. L.J. Mostertman, director of the International Institute for Hydraulic and Environ-mental Engineering in Delft, who has alw ays made possible the necessary contacts in this field. The third author is also very much indebted to Dr M. B. Abbott. Working with him at the International Institute in Delft represents not only a professional contact but also a personal and stimulating friendship. The author's accumulation of professional experience was made possible by these two colleagues, starting in 1970 through collaboration with the Versuchsanstalt fur Wasserbau of the Technische Hochschule MOOchen on the development of a mathematical model for a stretch of the river Danube. After completion of this study came the opportunity to develop the hydrodynamic part of the modelling system 'System 11 Siva' at the Computational Hydraulics Centre of the Danish Hydraulic Institute. Here the third author has very much appreciated the useful comments of his colleagues Gaele Rodenhuis, Asger Kej, Peter Hinstrup, and Jens

Aage Bertelsen during the construction of the system and its application to real problems. Last, but not least, thanks are given to the co-authors for the

considerable amount of additional work they did in Grenoble in integrating the third author's contribution into thiswork.

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Additional thanks must be given to Mr Bill Eichert of the U.S. Army Corps of Engineers, Hydrologic Engineering Center, and to Drs Carl Nordin and Robert Baltzer of the U.S. Geological Survey, for having supplied information on Current mathematical modelling activity intheir respective organizations.

Finally, we must acknowledge theinvaluable contribution of Joyce Holly, who typed the early drafts and did considerable proofreading.

List of symbols

All symbols are defined where they first appear inthe text. 'The following list includes only those symbols which retain the same physical or mathematical significance throughout a chapter or the entire book.

(L==length; M==mass; F==force; T==time)

A cross-sectional flow area perpendicular to the flow direction (L2)

As surface area of flood plain cell (L:1)

Ast cross-sectional area available for storage (L2)

a gate opening (L)

b width of cross section at free surface elevation (L)

be channel width (L)

bst storage width (L)

by valley width (L)

C Chezy coefficient inempirical roughness laws (L1!2r-1); concentration (ML -3)

Ca cross-sectional average concentration (ML-3)

0- Courant number

c wave celerity (LT-1)

cp flood peak celerity (LT-1)

dso median bed material diameter (L)

Fe bed friction force acting on control volume (F) Fg gravitational force acting on control volume (F) Fpb FPh FPh Fp2 pressure forces acting on control volume (F)

Fr Frcude number

G sediment volumetric discharge (L311)

---...---,..

g H Hm AH h h*, h. 1m II>/2 i j J I+.J_ i K

K

n ,, _Kz k kStr L M

Me

IW In n n,

acceleration due to gravity (L~)

depth of rectangular Subsection of a composite channel (L);

steady flow depth (L)

depth amplitude of individual component of Fourier series

solution (1)

head I"" (L)

Wale,depth (L)

SUperelevations of wave crests and troughs compared to

undisturbed free surface level (l)

denotes imaginary part of a complex number

cross-sectional moment integrals (L3and L2)

constant "';-1

computational point index

PartiCUlar computational point index

Riemann invariants (Lil)

computational point index; computational cell index

channel conveYance factor (L311)

artificial (numerical) diffusion coefficient (L2TJ)

longitudinal dispersion coefficient (L 2T-t)

Wave number (L -I)

Strickler coefficient in empirical roughness laws (L1/3

r

1)

downstream limit of computational domain,x '"L (L);

distance from source beYond which one dimensional dispersion

occurs (L); differential Operator difference operator

characteristic length in pseudoviSCOsity definition [L];

distance between the cent'e, of two adjacent flood plain cells (L);

turbulent mixing length (L)

number of computational points per Wavelength

net momentum flux into Control volume (FT)

net increase in momentum contained in cont,o! Volume (FT)

index of individual Fourier solution components

. Manning coefficlont in empirical roughness laws (L-'i'T);_{time step index}

p Q q R Re

R,

R, Sf

S"

S.

T Tf

T

u t At U Up u u.

v

v.

v x Ax wetted perimeter(L);

direct inflow into a computational cell (L3

r

1)

volumetric water discharge (L3

r-

I)

continuous lateral inflow per unit lengthibtr-1 );

pseudoviscosity coefficient (L3

r-

2);

discharge per unit width of channel (L2r1)

hydraulic radius AlP (L)

denotes real part of a complex number

amplification factor of numerical solution compared to exact solution, per time step

numerical dispersion factor of numerical solution compared to exact solution, per time step

energy line slope in the x-dlrection (friction slope)

free surface slope

bed slope in the x-direction propagation time (T): wave period (T)

flood forecasting interval (T)

flood forecast updating interval (T) time (T)

time between two computational intervals (T)

steady flow velocity (LT-1);

cross-sectional average velocity (Lr1)

velocity amplitude of individual component of Fourier series solution (Lr1)

flow velocity at flood peak (Lr1)

water velocity in x-dtrection (Lr1)

bed shear (or friction) velocity (Lr1)

volume of water in flood-plain cell (L3)

volume of tracer injected into a river (L3)

discontinuous front propagation velocity. (Lr-1);

velocity in y-direction (Lr1 )

characteristic velocities (Lr1)

velocity in z-direction (Lr-1)

longitudinal space co-ordinate in horizontal plane (L)

distance between two computational points in x-direction (L) xv

Y Yb

Y",

Yw

Z z ~ IJ. 6 e

'm

~

•

8

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p a T, ~ ~ ~

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vertical space co-ordinate above datum (l); water surface elevation (l)

bed elevation (L)

elevation at which overbank flooding begins (L) weir crest elevation

steady flow bed level (L)

transverse space coordinate in horizontal plane (perpendicular to flow direction) (l);

river bed level in Chapter 7 (l)

non-uniform velocity distribution coefficient

initial state error

numerical solution damping factor for one wave period empirical mixing coefficient (L2rl)

molecular diffusivity (L2T"1 )

turbulent diffusivity (L2r-1)

distance of cross section centroid from water surface (L) weighting coefficient infinite difference approximations of functions and their space derivatives

von Karman's constant wavelength (1.) wave frequency (T")

point inspace from which a trajectory originates (L) order of approximation

constant 3.14159 .•• water mass density (ML -3) width of cross section (L)

tangential wall or bed shear stress (FL -2)

combined resistance and bed slope terms (L3T2) discharge coefficient

solution damping factor

we~ting coefficient in finite difference approximations of functions and their time derivatives

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1 Introduction

Mathematical modelling of flow inrivers is rapidly becoming an accepted engineering tool, whose evolution can be compared to that of reduced scale modelling. Scale models came into use as design and verification tools when the complexity and scope of large structures began to present problems which could not be solved using traditional hydraulic methods. but could be accurately and productively modelled at a reduced scale. The use of scale models and inter-pretation of their results provided important feedback into the development of theory (similarity, statistics, turbulence, wave motion, sediment movement) as well as experimental science (measuring equipment, laboratory technique, etc.).

The theoretical foundations of physical scale models were laid down by 19th century precursors such as Froude; what was new in the 1930s and 1940s was the general application of scale models to open channel engineering problems. The early role of models as illustrative examples evolved into a role of providing quantitative, reliable results on which design decisions could be based. But as engineering projects became larger and economy considerations were more and more often integrated into overall planning, scale models reached a natural limit to the scope of their application. It is not sufficient merely to be able to produce a satisfactory design of a given structure or to predict flow conditions in its immediate vicinity; the structure's interaction with the overall development plan of a river basin or region must be considered, from both hydraulic and economic points of view. For example, a sophisticated and well-designed sand-trap structure on an irrigation canal will be useless if the canal system cannot deliver sufficient water and with ahighenough velocity. Even though scale distortion and engineering experience can be used to extend the scope of scale models, there is some point at which new techniques must be used to obtain simulations which are reliable and economic. These new technfques are those of

mathematical modelling.

Mathematical modelling in rivers is the simulation of flow conditions based on the formulation and solution of mathematical relationships expressing known hydraulic principles. The technique finds its origin inthe 19th century work of de St Venant and Boussinesq, who fonnulated the unsteady flow equations, and in the work of Massau, who in 1889 published some early attempts to solve

2 Practlcol Arpecu of Computational River Hydraulics

those equations. Important theoretical concepts were establishe~ ~ the first half f this century but the first engineering applications of these principles to . :atural river c~nditions awaited the development of electronic computers; 10

1952-1953 Isaacson, Stoker and Troesch (1954) constructed and ra~ a mathe-matical model of portions of the Ohio and Mississippi rivers. Following that pioneering effort the use of mathematical modelling inrivers develo~ed at first quite slowly, then began to accelerate; today (1980) we seem to be Inan

expon-ential growth phase, inwhich everyone is trying to build models. E~n when there may appear to be no need for a model inthe planning and design of some projects, contracts invariably call for a mathematical modelling effort.

Mathematical modelling inrivers is much more than the use of computers and computer programs to simulate hydraulics. The engineer who works with data processing techniques develops a particular analytical and experimental attitude, a formalization of intuition and thought, an extension and concretization of the thinking process. Most hydraulic engineers have at least once in their career met a hydraulics expert, an engineer of great experience who applies his

unformalized engineering intuition to solve a problem. Such an expert can say, 'if you do it that way, the dyke will collapse, but if you do it this way, the dyke will hold'. He is almost always right, even though he may not be able to explain, or formalize, his reasoning process, which is thus inaccessible to anyone else. If his reasoning process were formalized and analysed, itcould well serve as the basis of a conceptual mathematical model available to everyone.

This fonnalization of intuitive hydraulic reasoning is an important aspect of current river modelling development. Faced with the need to come up with

answers to design and planning problems as quickly and as economically as possible, engineers have developed a great number of programs and models, formalizing engineering intuition to as great an extent as possible and profiting from modern rapid computing techniques. Generalizations of this experience have led to the development between 1972 and 1976 of what we call modelling systems. Such systems comprise all the programmed procedures (software) necessary to construct and' operate models of a river with its tributaries, inun-dated plains, and existing and future structures inclUding dykes, dams, canals, etc. The engineer using such a system could concern himself only with the physical aspect of the problem,just as does the user of a scale model. But in

order to be able to interpret model results correctly, he must know something of the hypotheses; limitations, and structure of the modelling system _ just as

the scale model user must appreciate the llrititations of similarity laws. In other words, conceptual modelling must be complemented by what is referred to as computational hydraUlics, which is the second important area of mathematical modelling development.

. ~omputational hydraulics was bern out of the conclusion that the new pas. ~bllit~ offered b~ computers dictated a new formulation of hydraulic concepts ID~hich our previous knowledge Would beonly a component. Through inter-action among classical hYdraulics, modelling experience and research basic concep~s have been Ievie~d and. revised so that a n~w body ofknowiedge has

Introduction 3

evolved which provides formal, mathematical support and guidance for the various techniques usedinmodelling systems. It is impossible to develop a modelling system without using computational hydraulics concepts (just as it is impossible to do so without using classical hydraulics, calculus, numerical analysis, programming, data processing concepts, etc.). Thus there has been a constant interaction between modelling system development and computational hydraulics.

The authors are concerned about two attitudes which seem to be developing in mathematical modelling of rivers. The first one is the black-box syndrome, often symbolized by statements inthe press such as '... the data were fed into the computer (the black box) and the results which came out were •• ". The black-box approach, inwhich one seeks to construct a model which responds to input in the same way as a physical system,isthe only possible one when the physical system is poorly understood - incybernetics, for example. But in river hydraulics we do have a good knowledge of the basic physical processes, and models should be built with this knowledge as a foundation. The black-box use of river modelling systems with little or no awareness of their limitations and constraints, is, in our view, irresponsible.

The second attitude which concerns us is the desire to become immersed in computational techniques to the exclusion of practical considerations. The use of esoteric language and the unwillingness to explain practical details and the relationships between theoretical results (often coming from numerical analysis and gas dynamics) and their practical applications, again opens the door to irresponsibility and even charlatanism. Both of these attitudes inhibit technical progress in river mathematical modelling. It is of course much easier to believein

the fairy's magic wand than to be told that the quality of model results is a direct function of the modeller's efforts to understand the problem - the long way to Tipperary never was, and never will be, very popular. The unwillingness to take the long road can have immediate practical consequences, as can already be seen in the attitudes of some serious organizations. Having suffered the abuse of 'models' which produce either false results or no results at all, these

organizations have become understandably discouraged and tend to treat anyone talking about river mathematical modelling as a quack doctor.

We have tried in this book to address ourselves to those who wish to know what are the weak as well as the strong points of mathematical modelling; what can be checked, what can be simulated, and at what price.Asconsultants our-selves, we realize that we are givingour clients a stick to beat us with. We have tried to dismantle various aspects of river modelling in order to point out strengths and weaknesses and clarify the technical language employed. Without doing so, how can we explain the concept of convergence of a finite difference method to a water resource specialist in a language he can understand? And without the notion of convergence, how can we show him that close reproduction of recorded hydrographs is not sufficient proof of the reliability of a model, that another model may be more reliable even though its reproduction of observed events is not quite as good? How can we explain to an applied mathematician or

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4 PracticalAspectt a/Computational Riper Hydraulics

systems analyst the relative importance of physical factors rep:esented .inthe various terms of the flow equations? How can we justify seemingly .arbltrary decisions as to the neglect or not of inertial forces, or the use. of wel~.typeor fluvial-type exchange laws? How can we explain to the technical advisor of a sponsoring organization what he should expect of a mathematical mo~e\ or how one should be chosen, or how the quality of results depends on the pnce: These are the kinds of questions we have considered, and although we may not have been able to provide satisfying answers to all of them, we hope that our efforts will have at least convinced the various parties involved in modelling that only through honest understanding can modelling evolve into a mature engineering tool.

In planning thisbook we were faced with the problem of deciding what kind of reader wewanted to address. One possibility would have been to write for the modelling system developer, staying in the realm of computational hydraulics, numerical methods, solution algorithms, etc. Another possibility would have been to write for the modelling system user,limiting our attention to model schematization, calibration, interpretation of results, examples. But it soon be-came apparent that we would have to try to reach both kinds of readers. Consider first the modelling of flood propagation in large river systems. Although Correct and economical numerical methods must of course be used, the quality of model results depends to a large extent on the skill with which the modeller schematizes thesystem, i.e. on the 'art' he applies to the emplacement of computational points, flood plain storage cells, selection of boundary con. ditions, calibration of bed roughness coefficients, and so on. If on the other hand weconsider flow in artificial canals subject to rapid flow variations due to turbine and gate manoeuvres, we realize that there is very little skill needed in representing the physical system in discrete model form. In this case it is the numerical method which demands the modeller's close attention, since rapidly varying phenomena cannot be correctly modelled with any off-the-shelf, hastily chosen scheme. And to the preoccupations of the two types of potential readers concerned with these modelling problems, i.e. the developer and the user, we must add those of a third party: the buyer,that is to say the organization who pays for model development and use and thus has an interest in appreciating the problems faced in all aspects of modelling.

The structure of this book reflects our attempt to reach these three types of readers. In Chapter 2 we develop the basic integral flow relations from a control volume point of view, then try to show to what extent the normally used differential equations can be considered as eqUivalent to the true momentum conservation expressed by the integral relations. We introduce the method of characteristics to help in explaining boundary and initial condition requirements as well as the development of flow discontinuities. Then we COnsider some of the Slmplifie~ forms of the flow equations and their physical significance, and finally We describe a method of simUlating tWo-dimensional flow on inundated plains. In Chapter 3wedeal with the practical problems involved in nUmerical solution of the flow relations. Our concern iswith the degree to which various methods

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Introduction 5

approximate momentum conservation in a control volume, and with the practical implications and limitations of formal error, convergence, and stability analyses as applied to typical schemes. We try to show that any proposed method must be judged not only on the basis of its behaviour when applied to simplified linear equations, but also on the wayittreats internal and external boundary conditions, non-linear coefficients, and so on. We then outline some solution algorithms which can be used for branched and looped channel and flood plain flow networks.

In Chapters 4 and 5 we discuss the practical problems involved in construct-ing, calibratconstruct-ing, and exploiting models of natural river systems. Our main pre-occupation is the model representation of physical features, and the sensitivity of model quality to this representation. In Chapter 6 we treat the special problems involved in modelling of flow subject to artificial regulation: irrigation systems, derivation canals, and cascade hydroelectric projects.

InChapters 7 and 8 we describe practical problems involved in the modelling of long-term river bed evolution due to sediment transport, and in the simulation of pollutant transport and dispersion in rivers. Although these topics do not directly fall under the heading of unsteady flow modelling, they are becoming more and more often included with, even becoming the object of, river flow modelling. The practical problems arising in the modelling of both sediment and pollution transport are closely related to those of river flow modelling. In

Chapter 9 we briefly consider the modelling problems involved in flood forecast-ing and prediction, dam break wave calculation, and drainage network

simulation. Finally, we throw all caution to the wind and discuss cost/quality and model transferinChapters 10 and 11.

The reader may be surprised to find that we have not described any single modelling systeminits entirety. At the present time (1980) we are stillinthe early stages of development and application of modelling systems, and it would appear useless to describe details which are usually available in other publi-cations and which may become obsolete in a few years' time. We have however tried to describe the general principles of modelling systems, giving particular attention to their industrial development involving the joint efforts of hydraulic engineers, applied mathematicians, and data processing specialists. The authors have been personally involved in the development and application of several current systems: SOGREAH's CARIMA system for the simulation of unsteady flow in multiply connected networks of rivers, canals, and inundated areas; the Danish Hydraulic Institute's SIVA System 21 for unsteady flow in multiply con-nected river and canal networks; SOGREAH's CAREDAS system for unsteady free-surface and pressurized flow in multiply connected drainage networks. We have drawn largely on our own experiences using these systems for examples and illustrations, we hope without displaying any religious fervour for our own methods to the exclusion of others.

We assume that the reader has a basic knowledge of hydraulics and mathematics. We derive certain unsteady flow relationships only when it is necessary to interpret some aspects which are not readily found elsewhere. While

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6 Practical Aspects of Computoti0tll11 River Hydraulics

frequently referring the reader to the literature, we have still tried to maintain the self-contained character of the book, which we hope can be read and under-stood by itself.

2 Mathematical formulation

of physical processes

2.1 EQUATIONS OF ONE-DIMENSIONAL UNSTEADY OPEN CHANNEL FWW

Basic hypotheses

The fundamental notions and hypotheses used inthe mathematical modelling of

rivers are formalized inthe equations of unsteady open channel flow. These

equations are simple models of extremely complex phenomena: they

incorporate only the most important real-life flow influences, discarding those

which are thought to be of secondary importance in view of the purpose of

modelling. When dealing with a set of equations, the modeller must be aware of

the physical phenomena which they do and do not incorporate. Once the equations are established, their subsequent mathematical and numerical treatment (transfonnation, solution, study of properties, etc.) does not change their built-in physical restrictions.

For the sake of clarity. we distinguish between channel flow and flood plain flow. Flood plain flow is usually much more complex and more difficult to des--cribe completely than channel flow; paradoxically, however, its mathematical representation in models is much simpler. Most often the role of flooded plains in flood propagation is to provide storage volume with slow exchange of water from one part of the plain to another. The simulation scale of the whole river is usually such that a global, coarse flood plain schematization may be adopted, a representation which is in the end simpler than that used for the channel, aswe

shall see in Section 2.5. Clearly, this is not the case when the local details of flooded plain flow are to be represented with highaccuracy, but then a physical, reduced scale model is called for.

Throughout this book channel flow is assumed to be strictly one-dimensional, even though truly one-dimensional flow does not exist in nature. One of our major concerns in this chapter is to define what is called a 'one-dimensional situation' and to consider to what extent natural river flow can be modelled as such without violating the basic concepts of the one-dimensional flow equations. We begin with a reminder of the de St Venant hypotheses which we consider to be valid throughout this section and, in general, throughout the book except when

I

,

,

!,

,

Ij

8 1'ractiCilIAspecu ofComputadonal River Hydrtmliu

. h h h areintroduced. The

"me eOrrective facto~ which depart from t e ypot eses h following

de St Venant (1871) equations for unsteady flow are based upon t e series of assumPtions:

'. if m over the cross

(i) The flow is one-dimensionaI Le. the velocity IS

urn

or

section and the water level across the section is horizontal. . Ii lble,

(il)The s"eamline eurvature is small and vertical ac celeratlons are neg gJ

hence the pressure is hydrostatie. d for

(iii) The effects ofboundary friction and turbulence can be accounte

through resistanee laws analogoUs tc those used for steady state flow. gle It

. ha th sine of the an eI

(iv) The average channel bed slope ISsmall so t t e co

makes withthe horizontal may be replaced by Unity.

We note thet the cross section, of the channel cooveylng such a flow are of . arbitnuy shape and may vary along the ebannel axis, althongh the variatinn IS limited by the eonditinn of small streamline eU"rature. . n

One.dbnensional unsteady flnw in ehannels, assuming that the density IS eo • stanr, can be deseribed by two dependent variables; for example the water .st;r.; Y and the disehatge Q at any given riv" eross sectton. These dependent vane

define the state of the flUid mntion along the watereourse and in time, Le ..as a fnnction of two Independent variables (x for space and t for time). Depending on the nature of the Problem One Could define other pairs of dependent Variables, for example, velocity and depth, as We shall show later on. The equations Will of COUrsebe different for different pai~ of variables, but the

phYsical assumptions will be the same.

Since two dependent Variables are sufficieot to describe one-dimensional flow, we need only two equations, each of Which must represent a physical law. However, we can fOoouiate three phYsical laws In such flow: eonservetion of

mass, momentum and energy. Itmay be shown that:

When the flow Variables are not Continuous (hydraulic jump, bore), two representations are pOSSible:conservation of mass and momentum, or conserve. tion of massand energy. The two representation, are not eqUiValent, and only

One of them is correct.

When the flow Variables are continuous, either of the two representations

may beused, and they are eqUiValent,

This

question of choice of conservation laws is extremely important from a theoretical as well as a practical Point of view, and is dealt With In great detail rn the first vUlume of the present series of bOOks (Abbott, 1979) .

. Smc~ the ""s ....momentum COUPleof conservation laws Is applicable to both d""O~tinuous and Continuou, situations While the mas ....energy couple is not, as descnbed by Abbott, we shall base our derivations on mas ....momentum

Mathemlltlctl1 Formulation of Physical Processe, 9

Integral relations

let us consider the control volumeinthe (x.t) plane between cross sections x =Xl andx=x,.and between timest= tl andt= t, shownin Fig. 2.1. In the

same figure is depicted the cartesian co-ordinate system which will be used throughout the book. We shall establish equations of conservation of mass and momentum for the control volume assuming that all the de St Venant

hypotheses are valid. In doing so we shall closely follow liggett (1975), extend-ing his unit-width analysis to a generalized section. The flow is assumed to be nearly horizontal, i.e. the angleabetween the channel bed and the x-axis is sufficiently small so that cosa= I. Based on these hypotheses, the basic equations can be formulated by using the principles of conservation of mass and momentum within a control volume.

The net inflow of mass into the volume is defined by the time integral of the difference between the mass flowrates entering (PuAlx, and leaving (PuA)xz' the control volume:

I

I,

[(PuA)x, - (PuA)x,J dt

I,

(2.1)

This net inflow must be equal to the change of storage in the reach during the time interval:

I

X,

[(PAl"~ - (PA)"J dx

x, (2.2)

wherep= water density;u =u(x, t)= uniform cross-sectional velocity;

y

I.)

Fig. 2.1. Deflnition sketch for derivation of unsteady flow equations: (a) Control volume, section view;

10 PrflCticaI Aspeeu of Computational River HydrauHa - - y y'

•

,

I.)

,

',1

•

_F"pl

11

•

•

F",2

F;g. 2.1. "nl'd (b) ","oss-seetion; (c) pressu,. forces, plan view

MathemotiC/l1 Formulation of PhydC/l1 Proce'lel 11

A = A(x, t)=wetted cross-sectional area. Consequently the mass continuity integral. relation for constant density is

J"

[(A), - (A),

J

dx +

J"

[(Q), - (Q).

J

dt=0

2 I 2"'1

XI '1

(2.3)

whereQ=uA.

The conservation of momentum in the x-direction requires that the change of momentwn in the control volume between times tl and ta be equal to the sum

of the net inflow of momentum into the control volume and the integral. of the external forces acting on it over the same time interval. Momentum is the product of mass and velocity, and momentum flux through the flow section is the product of the mass flow rate and velocity, or

momentum flux =puA x u=pu2A (2.4)

The net momentum flux into the control volume (momentum entering through section x=XI minus the momentum leaving through the section x=x,,) is

and the net momentum inflow between tl and t2is

J

t

Mt= [(pu'Al;, -(pu'Al;,

1

dt

t

,

' ,

(2.5)

The momentum contained in the control volume at any instant is

J"

puA dx

"

and the net increase from 11to t2is

J"

l>M = [(PuA)" -(puA)"J dx

"

(2.6)

We assume that the only important external forces acting upon the control volume in the x-dlreetton arepressure, gravity, and frictional resistance. The pressure force Fl. isthe difference of pressure forces F'pl and

F;l ,

applied at boundaries XI.

x:

of the reach. Atanycross section

x

with free surface elevation y(x) the pressure force is expressed under the hydrostatic distribution

hypothesis by

'VIm

12 hactical AspectlofComputational River Hydrrml;cs

. . . bl aI they ..v;~.h(x r) =water depth;

Where: 11= depth mtegratlOn vena e ong '~ ••

a(x 11)==width ofthe cross section such "thata(x. h)=b(x) ==free s~rf~ce width. Thus the time integral of the net pressure force Fp" when Fp' IS expressed as in Equation (2.7), is

:

:

(2.8)

f

'iX)

where I.

=.

[h(x) -

~J

o(x, ~) d~

For an infinitesimal channel length de, the increase of the pressure force due to

the width variation is represented bytheincrease of the wetted area do. d'1

for constant h=ho times the distance of itscentroid from the free surface

h(x)-~,

This force is to be integrated between ~ = 0 and ~ = h(x) for a given cross

section, and[ramxI tox2 to obtain the total force acting on the control

VOlume. The total integral along the Contour of theControl volume and for the

time interval tl to t2 is

or

(2.9)

WhereI2

=f

h(X} (h-71) [~] d~

o ih' h""h.

Ofcourse Equation (2.9) is not valid if a sndden width change occurs between sections

x,

andx,. In that case supplementary forces will act upon the control volume, and they must be talcen into accOunt. Inany case, in such a situation the cUrvature of streamlines will be nOn-uegbgible, viulating one of the basic

hyPotheses wehave adopted.

c Thefor~e ~g ~ue to 8CSvity,that ia to say the weight component a100g the hann.1aYbaxis,osevaluated by assuming that the channel bottom slope

s.

= -

ax

=

tan •

ia small (Yb being bottom el<vatiao above datum), so that tane ~ sinQ:

c ,".:"~.' ,.;1',,,,- ~"H "i""'~<r"~~~\';:~-{ ~i~iil,.:"·"'l"~'":l,,,,t;:1"<t·:f:I'.~.,"t',,; ji',"" • -~'" '.',

-~---...-~-~-

-" ~-..--~~-~~"--

...

"--"-MathemJItica1 Formulation of Phyric41 ProceneB 13

(2.11)

Frictional resistance force Ffis applied to the control volume through shear along the channel bed and banks. Inorder to generalize the different ways in

which this shear force can be treated, we follow Chow (1959) in expressing the shear force on a unit length of channel aspgASf where Sf is the so-called

friction slope, I.e. the energy gradient needed to overcome frictional resistance in steady flow. The time integral of the resistance force on the control volume is then

(2.12)

The statement of conservation of momentum thus leads to

(2.13)

or, for constant density p,

J

X'

J

t,

[(uA)t -(uA'_{~ Ifl}

J

<Ix

=

[(u'A)x -(u'Alx]_{I 2} dt

X1 ~

J

t,

J't, J

X'

+g [(I1)xl-(ldx,ldt-g pI'1,dxdt

~ ~ ~

(2.14)

Equation (2.14) is the integral form of the momentum conservation equation for unsteady one-dimensional flow in natural channels of arbitrary shape. Equations

(2.3) and (2.14) together arethe integral form of the unsteady flow relations based onthe de St Venant hypothesis.

Differential form of the de St Venant equations

Equations (2.3) and (2.14) are integral relations established without the require-ment that flow variables A. Q.

y, u,

etc., be continuous or differentiable.

Nowhere did we require that the distance X'1,-XI be Infinitely small. As Liggett

(1975) has pointed out, many finite difference schemes are based upon the integral relations and we shall come back to them further 00.The differential

equations of gradually varied unsteady flow may be obtained from integral equations if one assumes that the dependent variables are continuous, differentiable functions. Then, by Taylor series expansions we may write

I

!

I

14 PracticalAqJects ofComputationtl[ River Hydmulics

OA

a'

A /1t'

(A)l, =(A)t/-;-t .6.t +

at"

2

+ ...

u

aQ

a'Q

_J1x' (2.15)

(Q)x, =(Q\,'

a;-

J1x.

ax' 2""

By retaining only first derivatives and assuming that A:r and tJ.t approach zero,

wecan write

fX'f"

OA [(A) -(A),]_r, d>; = .,--dtd>; I tut x, , (2.16)

And the continuity Equation (2.3) becomes

fX'f"

_x,atax

[OA.

aQJ

dtd>;=O

, ,

(2.17)

Ina similarway we may write

a(u'A) a'(u'A) J1x' (u'A) -(u'A'

= _

/1x' _' ... Xz !XI

ax

ax2 2 3Q 32Q I:1r2 (uA), -(uA), =-';-t

11t., _ ' ...

S IU 0/2 iJI1 32[t Ax2 (Il)x -(ldx =-Ax+ __ + 1 lax ih:22

SUbstitution ofthefirst terms of exPansions (2.18) Into Equation (2.14) and

then passage to the

limit

(Ax, t:J.t-+0) leads

to

(2.18)

If the relations (2.17) and (2.19)",e

'0

hold everywhere in the (x, t) plane, then they most hold

for

an Infinitely

sman

volume, and we can Write two differential

equations:

MathematiCilI FOnTWhl.tion of Physictll Procelfe$ IS

momentum equation -+aQ a(u'A) +g-=gA(Soill, -Sr)+gI2

at

ax

ax

(2.21)

Combining thex derivatives in Equation (2.21) and replacing uby Q/A:

(2.22)

Equations (2.20) and (2.22) are written in a special fonn, often called the 'divergent' form of partial differential equations. Ifthe right hand sides of Equations (2.20) and (2.22) are equal to zero, these equations express nil divergence of the mass and momentum vector functions in any closed contour in the (x, t) plane; mass and momentum are conserved. When the right hand side of Equation (2.22) isdifferent from zero, momentum is no longer conserved, the free terms acting as momentum sources or sinks.

Continuing our derivations, still assuming that all dependent variables are differentiable, let us evaluate the derivative of the

st,

term in Equation (2.22):

(2.23)

Applying the Leibniz theorem for differentiation of an integral, and keeping in mind that o(x, h)=b(x)and

r:

odll=A, we obtain

3 3h Ih(X)

ax

(gI,)=g

ax

a(x.~)d~ o

I

h(X)

[3bJ

+g [h(x)-~J - d~ o

ax·

h=const (2.24) (2.25)

Consequently Equation (2.21) may now berewritten

Thus we obtain the 'momentum' equation generally used in engineering practice,

~~ + ~ (uQ)+gA (: -So)+gAS,=O (2.26)

which, as we shall discuss later on, is not in a momentum divergent form. We shall generally refer to the second of the two basic flow relations ~ the

.,--I

j' ,

I '

!

1

!

, : i I'

I'

I

I

I

I

; i ;,

-'77R7EBB

T

.

16 PrtlcticalArpectr ofCompultltioMl Rhy, HYdraulicl

'dynamic' equation, since it is seldom a true statement of momentum

conservation.

Weshallnow consider some of the more often used choices of dependent

variables With which engineers Work, taking the continuity Equation (2.20)

anddynamic Equation (2.26) as a departure point.

(i) Q(x, t), h(x, t)

The

variableA(h) in the continuity equation is eliminated by

While the velocity Uis replaced by Q/Ain the dynamic equation.

The

system of two flow equations becomes

~ +1.

aQ.o

at

b

ax

aQ

a (Q')

ah

at

+

ax

A

+gA

ax

+gA(St-So)' 0

MIere b· b(h),A • A(h). (ij)Q(x. t),y(x, r)

Watet dep'h h·

y - Yb,

\VhereYb(X) is the bottom elevation; hence

ah _

ayah

ay

aYb

ily

at -

at:

ax '" ax

-a;-

==

ax

+80

SUbstitution into Equations (2.27) leads to the system

2.

+.!.

~.Lo

at

b

ax

aQ

a

(0')

ay

at

+

ax

A "+gA

ax

+gASr=o

where b· b(Y),A • A(y).

(Ui)u(x, t), h(x, t)

Let us putinthe COntinuity equation Q==uA:(h), te.

~g

=A ~~ +u ~ =A ~

+u[a4

ah

+

(a4)

]

~

~

ax

ax

aha;-

>::-h

'-14" '" COJlst

MIere the las' ~nns represent the rate of change of A \Vhen 'he depth h is held constant, and

ah •

b. The time and space derivatives of disch"ge are

trans-fonned as follows:

Mathemiltical Formulation of Phyrical Processes 17

aQ

= a(uA) =u aA +A

au

=-u

aQ

+A

au

at

at

at

at

ax

at

axa

(AQ') = 2Q

aQ _

Q' aA A

ax

A2

ax

-2 Q

aQ _

Q' [b

ah

+ (aA) ]

- T ax

A2

ax

ax

h"'const

Substitution of these derivatives into the system of Equation (2.27) yields, after cancelling some terms in the dynamic equation, the following system:

ah

+~

au

+u

ah

+~

(aA) =0

at

b

ax

ax

b

ax

h"'const

au

au

ah

at

+

u

ax

+g

ax

+g(8,-8"0)=0 (2.29) whereA =A(h), b=b(h) (iv) u(x, t),Y(x, t)

The space derivative of discharge Q '" uA(y) is now

aQ=A

au

+U[bay +(aA) ]

ax

ax

ax

ax

y '"const

By substituting this expression and the derivatives

ah

=

ay _ aYb

=

ay

+80'

ah

=

ay

ax

ax

ax

ax

• at

at

into Equation (2.29), we obtain

ay +~au

+u

(ay

+80)+~(aA) =0

at

b

ax

ax

b

ax

y=const

au

au

3y

at

+

u

ax

+g

ax

+g8r=0

(2.30)

The preceding paragraphs may seem to be dedicated to spurious 'algebraic manipulations'. AJJ. a matter of fact, these gymnastics are of some practical importance, since certain numerical techniques may be better adapted to some of the above systems of equations than others. Physical features of a given water-course may also suggest the system which is 'better' (read: easier to integrate numerically without gross errors) than others for that particular case. For example, when the river is steep and its cross-sectional variations are small, the use of hex,t)rather thany(x, r) as a dependent variable is recommended since hand

(ax

aA)h vary slowly from one point to another. On the other

"'const

...

!

I

i i i

f

,

18 PracticlllArpecu ofCo11lPIJtQtioNd Rfrn HJ.drtmlic~

. th I· alee the use of oneo

Moreover, the appearance of surges meso utron may m

the systems compulsolY while rejecting others, as

we

shali see further on: t Equations (2.27H2.3D) are not in dlvergem form, as

we

will have occasion 0

re-emphasize in Section 2.3.

tic,

The systems of Equations (2.3), (2.14) and (2.27H2.3D) are, mathema speaking, eqUiValent if and only if all functions and variables are at least once diffuentiable. If it is impoSSible to consid" the fnnctions and vanables as differentiable, the systems (2.27H2.3D) are not eqnivalent. If the solutions are discontinnons (e.g. mOVinghydraulic jumps appear), differential systems may not be valld at all while integral relations still Will be. Neith" the integral nor differential re!atimo may be Consid"ed as valid when the b~sic de Stven~ rt

hyPothese. are violated, as in the case of an nndular jump with lis tram of 0 Waves which make it impossible to neglect vertical aceel"ations.

Supplementary terms and coefrlCients

The eqnations often used in engineering practice are not always based on the consistent set of assumptions which we used to derive Equations (2.27H2.30). The desire to represent highiy irregular cross sections in natural watercourses has tempted engineers to introduce conective coefficients in ord" to be able to relax de St Venant's assumPtion of uniform velocity in the cross section. The most common example of a situation in which this appears to be necessary is flow in cross sections which include ov"bank areas. Because of its high resistsnce to flow, the overbank area may Contribute only to storage, conveying virtUally no diocharge. In SUchcases the continuity equation is sometimes

Written

M" •

oQ

0

at

~=

where A" io the """;'ectional area aVailabl, for storage, and Isgeneraily not the same area as Used 10 the momentum equation. Often engineers defme a

so-called 'storage_width' b.t, given by

t1A

st _

ay

at- -

bit

at;

bit=b.t(v)

(2.31)

so that the ContinUity equation may be written

oy I oQ

ar+b;"a;=o

~~::",UOU'lateral inflow qper Uuit lengthis added, Equation (2.33)

(2.32)

(2.33)

Mathenuzticlll Formulation of Phydcal Procelsel 19

f As for the dynamic equation, the control volume approach we used in obtaining Equation (2.14) is replaced by a consideration of momentum conservation in a differential element in a cross section in which the velocity is not uniform. Integration over the entire cross section as described by Abbott (1979) assuming

my the water surface and energy slope to be everywhere the same yields another dynamic equation,

aQ+~

(p Q') +gA

ay

+

gAS

=0

at

ax

A

ax

t (2.35)

or, if lateral inflow is considered and uq isits downstream velocity component,

aQ+.~ (pQ')

+gA (ay+Sf)_(U

-~)q=o

at

ax

A

ax

qA (236)

where Q= UA

ii = cross-sectional averagevelocity. and

p

= (2.37)

the subscripts z denoting local values of depth-averaged velocity and depth at position z in the cross section. We shall come back to the meaning of

Ii

later on.

The system of two partial differential equations (2.33), (2.35)/ormally resembles Equations (2.28). The two systems, nevertheless, are founded upon basically different hypothesis. Equations (2.28) were obtained from integral relations which were derived using the assumption ofuni/onn velocity in the whole cross section with a unique definition of discharge Q = Au, A being the cross-sectional area. Equations (2.33) and (2.35) were derived with the discharge defined as Q=Ail, D being anaverage velocity in the cross section and A being a 'live' cross-sectional area, different from the cross-sectional area Ast used in the continuity equation. Before discussing the physical meaning of the different terms of the system (2.33) and (2.35), we would like to stress that, because of these different hypotheses, Equations (2.33) and (2.35) are 1101compatible with the. systems of Equations (2.27}-{2.30). One cannot pass from one to another through formal differentiation. The physical significance of this will be more clear after a discussion of the physical meaning of the resistance tenn (the friction slope Sf)' the coefficient

tJ,

the storage width bst, and lateral inflow. Empirical resistance laws

Inour developments up to this point we have expressed the force per unit channel length due to bed and bank resistance as pgASr in which the 'friction slope' Sf is taken to represent anyone of a number of empirical resistance laws. Most of these empirical laws are based on a relation between discharge and friction losses of the general type

...

. lRive,Hydrtmlia 20 PNetictll Arpects ofComputQtiOJUl

Q=KyS, .~

f the channel and SfIS

where K

=

K(h) is the so-called conveyance factor o. tal Europe the mos

slope of the energy grade line in steady ~ow.1n continen frequently used laws are the Chezy relatIOn

ii=C(lIS,)I; Q=CA(lIS,)1

and the Strickler relation

ii=kSr R'I'v.f,; Q=kS'rAR'I'

v.f,

r • II'/) k the Strickler

where CIsthede Chezy resistance coeffiCient (m s, Sir elation

coefficient (m 1/3/S) andRthe hYdraulicradius, defined by the r

(2

A R=p

. Inthe Anglo Saxon

wherep ~the wetted perimeter of the cross section. .

countries, engineers are more familiar With the Manning relation

(2..

ii= J..R'I'_n

v.f,.

_'nQ= !-AR'I'v.f,

. ". (/ .13) Comparing the

where

n

Is Mannmg'S coeffiCient, m metric units sm. . us res~tsnce taws it will be clea, that the foUOwing relalions link the vane

coefficients:

(2.4

and _(2.43

Jn

the literature the Manning relation isal,o found as Q= ~_n AR'I'v.f,

where Q andRare defined inUsunits of cUhic feet per second and feet, "'PecliveIY.'lhe n'Values in both Equations (2.42) and (2.45) are given

Independently of the SYstem of Units. . nt

Formul" (2.39)--(2.45) impiy a con'tant value of the roughness coefflcle acrossthe section. Jnnat,ue the coefficients kStr, nand C are empirical Panunetors linked to the comPOOitiou of the river bed (forexample, the Strickler coefficient is Inversely pr0Pottional to the I/6th Power of the sand grain diameter) and thu, they generally vary across theWatercourse Width. In c""'Pact chanuels (those Without overbank sections and unifoM bed

rough-(2.44)

MathematiClll Formulation of Physical ProcesGes 21

ness) the roughness coefficient along the wetted perimeter may be nearly constant and the mean velocity can still be computed by one of the formulae (2.39K2.45) without much error (reeChow, 1959). If the cross section has a compound shape such as shown in Fig. 2.2, common practice consists individing the cross section into several distinct subsections (vertical slices) with each sub-section having a different roughness from the others. As we describe in more detail in Chapter 4, it ispossible to define a global conveyance factor for the cross section based upon the hypothesis that the friction slope Seis constant

---I ___ 2 " .40) 1.1 41)

•

I I

I

I

I

1

I

"

1-,

_.,

_/--,

lbl

I

_I

I

---I 2) _./

,

..

,

Fig. 2.2. Consequences of assumptions regarding constant energy level and horizontal water surface in channel flow: I-horizontal water surface implies non-constant energy level, 2-non-constant energy level implies variable water surface. (a) Non-uniform velocity distribution. (b) Flow cross section.

across the watercourse width. Assuming that the resistance equation can be applied separately to each of several vertical slices

t,

we have

0'

(2.46)

(2.47)

r

.,?biF'

fires

22 Practical ArpectJ of ComputatioMl River Hyrbaulic,

Chezy formula, bibeing the width of slice i.

Unfortunately Equation (2.47) is incontradiction with our one.dimen~onal flow hypotheses: if the velocities vary across the width, the energy grade line must vary too inorder to keep the Creesurface level constant, hence . Sr=Sr(z) =f::.const. Itwould thus be more consistent to Write the following equality instead of Equation (2.47):

IK,vWr,=KvWr

,

(2.48)

On the other hand, assuming Sr to be constant but the horizontal velocities to be non-unifonn, thefree water surface can no longer be horizontal. Figure 2.2 depicts this contradiction. Itisonly if the de St Venant hypothesis of a uniform velocity distribution is maintained, that no contradiction arises. In practice itis always assumed that both Sr is constant and the free surface is horizontal; the error thus committed may be considered as a measure of how far the real flow

isfrom an idealized one-dimensional situation.

Following common practice and accepting theerror stemming from the above contradiction, the conveyance of a compound cross section may be com-puted with theaid of Equation (2~47) which yields, for the Strickler fonnula

K= ~_""'" k

s•

_rth·hlt'_{I I} (2.49) i

and for the Chezy formula K= ~

_,

Ctb,hl/'l

(2.50)

Coefficient of non-uniform velocity distribution

The coefficient (Jin Equations (2.35) and (2.36) appears as a result of the assumption of non-unifonn velocity inthe cross section. It reflects the fact that since local momentum flux is proportional to the square of the local velocity, the overall mean channel velocity is not representative of the overall momentum flux unlessit isconected by (lTheoietically (Jcould be evaluated from

~asure.d velOcity distributions UsingEquation (2.37), but this is of course Impractical •

.Sc~onfeld (I951)therefore introduces the follOwing hypothesis: if at every pomt m the cross section the depth.averaged velocity Uz can be computed using th: l~ally applied steady flow reSistance law, these local velocities can then be su ~dtuted mBtoEquation (2.37) and the integration over the cross section

came out. y analogy With .C •

-" dth B· non-unliorm vertIcal velocity proflles (jis often

"4Ue e OUSSlDesqcoefficient. ,

If in fonnula (2 37) the mea

I·, -.

h

• . . n ve OCIyUmt e denominator is replaced by a

resistance equation USin thegl b I

Mathematical Fomw.lation of Physical Processes 23 (2.49) or (2.50), and ifthe friction slope is assumed to be constant with z, i.e. Sr<z)

=

Sf (see once again Fig. 2.2) the integral of Equation (2.37) may be approximated by using vertical slices of finite width bj instead of infinitely narrow tubes db.We obtain

~Clhlbl A ~ Clhlbj

i {

~= = .... _---,--,,---- (2.5 1)

K'

fA

(L

C;b,-hj" ) ,

i

for the Chezy equation, and

~k~rihl/3bi A ~k§trihl/3bi

I i

~ = ---=-;--- =

-=;"---;;:--;-,

J(J/A ( ~ kStrthj/3bi ) 2

{

for the Strickler equation. The coefficients

P

defined according to Equations (2.51) and (2.52) are, for a given cross section, functions of water stagey and as such may be used in the dynamic equation (2.36).

(2.52)

Storagewidthbst

The purpose of introducing the storage width bst, or the storage cross-sectional area Ast

>

A,is to take into account the fact that flooded zones often act only as storage areas. They store water whose velocity is nil in the general flow direction x and, thus, does not contribute to the overall momentum flux in the cross section. If the coefficient

is

is used in the dynamic equation it already takes into account thisphenomenon. Indeed, if for a number of vertical slices the roughness coefficient kStris assumed to be very small, the global

is

coefficient

will increase and the tenn (JJQ'1)/A in Equation (2.36) willincrease accordingly. It would seem superfluous inthis case to admit further that A <Ast. Besides, it is not at all clear how to define the 'live' areaA and width bas distinct from Ast and bst for water stages higher than the river banks as shown in Fig. 2.3. The introduction of a 'live' width b(y)

<

bst opens the door to even more arbitrary judgements as to the model representative of reality.

lAteral inflow

Inflow (or outflow) continuously distributed along the rivercourse is seldom encountered in mathematical modelling of rivers. The most common situations in which such a lateral discharge is to be taken into account arelinked to hydro-logical phenomena: evaporation, rainfall, and infiltration. Usually these

processes can be neglected; intropical areas, however, to neglect them may lead to important errors. A well known example is the Nile River which loses approximately 50% of its discharge between Momgalla and Malaka1 due to