computational hydraulics

WL | delft hydraulics

Computational HydraulicsVersion 20061

© 2005 Adri Verwey

Lecture notes

Computational HydraulicsVersion 20061

Copyright © 2005, Adri Verwey

The right of Adri Verwey to be identified as the author of this work has been asserted in accordancewith the Copyright, Designs and Patents Act 1988

Lecture notes

Computational Hydraulics © 2005 Adri Verwey UNESCO IHEVersion 20061

WL | Delft Hydraulics 3

Contents

1 Introduction...............................................................................................................7

1.1 Introductory remarks......................................................................................7

1.2 Unsteady flow hydraulics ..............................................................................8

1.3 Numerical methods......................................................................................11

1.4 Mathematical modelling model ...................................................................11

2 Introduction to numerical methods.......................................................................17

2.1 Introduction..................................................................................................17

2.2 Introduction to finite difference approximations .........................................18

2.3 The Euler scheme ........................................................................................18

2.4 The improved Euler scheme ........................................................................21

2.5 The implicit scheme.....................................................................................22

2.6 The NewtonRaphson scheme .....................................................................22

2.7 A more formal analysis of accuracy ............................................................23

2.8 Stability........................................................................................................26

2.9 Consistency and convergence ......................................................................27

2.10 The choice of a time step .............................................................................27

2.11 Reservoir routing .........................................................................................29

2.12 Routing of a decayable pollutant through a reservoir ..................................33

2.13 Nonuniform steady flow in channels..........................................................34

2.14 An inverse scheme for the backwater curve computation ...........................38

2.15 Nonuniform flow in nonuniform channels................................................40

2.16 What have we learnt?...................................................................................41

2.17 Questions and small assignments ................................................................41



3 Basic Unsteady Channel Flow Equations .............................................................43

3.1 Introduction..................................................................................................43

3.2 Continuity equation .....................................................................................43

3.3 Momentum equation....................................................................................44

3.4 Transformation to the characteristic form....................................................45

3.5 The significance of the characteristics.........................................................48

3.6 The method of characteristics ......................................................................51

3.7 More complex boundary conditions ............................................................55

3.8 The formation of positive and negative hydraulic jumps ............................57

3.9 The limited practical importance of the method of characteristics..............59

3.10 Questions and assignments ..........................................................................60

4 Introducing Numerical Solutions for Partial Differential Equations.................61

4.1 Introduction..................................................................................................61

4.2 The advection equation................................................................................63

4.3 The characteristic solution ...........................................................................64

4.4 Finite difference schemes ............................................................................65

4.5 Characteristic solutions on a fixed grid .......................................................70

4.6 Introducing diffusion ...................................................................................72

4.7 An explicit finite difference scheme............................................................73

4.8 Explicit schemes for the combination of advection and diffusion...............76

4.9 Implicit schemes for the diffusion equation.................................................77

5 De Saint Venant equations and their solutions.....................................................81

5.1 Introduction..................................................................................................81

5.2 The continuity equation ...............................................................................82

5.3 The momentum equation .............................................................................84

5.4 Numerical solutions .....................................................................................88



5.5 Description of hydraulic structures..............................................................92

5.6 Topological model schematisation...............................................................94

5.7 Hydraulic model schematisation..................................................................95

5.8 Boundary and initial conditions..................................................................97

5.9 Model calibration and validation .................................................................99

6 Mathematical modelling of floods .......................................................................101

6.1 Introduction................................................................................................101

6.2 Flood model requirements .........................................................................101

6.3 The role of new data collection technologies ............................................103

6.4 The nature of flood wave propagation.......................................................104

6.5 Deformation of flood waves ......................................................................105

6.5.1 The role of varying celerities ........................................................105

6.5.2 The role of the diffusion term.......................................................106

6.5.3 The role of the lateral flow terms..................................................107

6.6 Link to hydrologic flood routing models ...................................................107

6.7 Twodimensional modelling of floods .......................................................108

6.8 Integrated 1D/2D modelling ......................................................................113

6.9 Exercise......................................................................................................116

7 Water Hammer .....................................................................................................117

7.1 Introduction................................................................................................117

7.2 Water Hammer Equations..........................................................................119

7.3 The Method of Characteristics for Water Hammer....................................125

7.4 Exercise......................................................................................................128

8 References..............................................................................................................129



1 Introduction

1.1 Introductory remarks

The topic of computational hydraulics is dealing with the question how waterresources engineers and planners can be assisted in dealing with complex hydraulicproblems. When these problems are very local, they can generally be addressed byusing empirical relationships and a well trained engineer usually is equipped withmethods and tools dealing with it. In these cases, the use of laws and relationships inhydraulics is often limited to steady flow approximations.

For larger scale problems, however, the unsteady nature of flows becomes moredominant and methods used will become more complex. Whereas until variousdecades ago, the focus has been on developing approximations, partly empirical, ofthe full hydrodynamic behaviour of the water system, the use of computers has madeit possible to describe hydraulic systems quite accurately. Over the past decadesenormous progress has been made in developing simulators or mathematical modelsfor all kinds of hydraulic systems, such as rivers, drainage networks, irrigationnetworks, water distribution networks etc.

Currently, the level of accuracy of such simulators is primarily limited by the qualityof data available to construct and calibrate the models. A variety of good softwarepackages is available to construct such models. However, good schematisations ofhydraulic systems in models requires some insight in the laws and techniques behindthese modelling systems in order to use them correctly in building one’s own model.

In this series of lectures we address this need by providing insight into the nature ofonedimensional unsteady flow. After the introduction of the unsteady flowequations, the link between the equations and the physical system is shown by thecharacteristic celerities of disturbances propagating along channels. This providesthe basis for the numerical schemes developed to solve the equations. Moreover, itshows clearly the effect of boundary variations and, in particular, the effects ofcontrol of hydraulic systems.

With this understanding as a basis, numerical method is introduced. First, only thesocalled ordinary differential equations are treated, enabling us to do simplebackwater computations, water quality simulation in well mixed reservoirs etc.However, at the same time numerical concepts and their evaluation are introducedrelating to the accuracy, stability, robustness and efficiency of numerical operations.This will serve as a necessary basis for those who want to develop their own models,as well as for those applying existing modelling systems.



Subsequently, numerical descriptions of more complex problems described bypartial differential equations are discussed, primarily to provide enough insight as abasis for good mathematical modelling practice. Also here, the role and choice ofnumerical parameters in mathematical model simulation is emphasised.

Finally, the overall construction of good mathematical models is discussed. Basedupon a clear idea about the objective of model use, rules for the most economicaland correct schematisation are developed, starting with a topological schematisationof the system. In addition, the derivation of correct physical parameters is discussed,such as those describing channel conveyance, storage and lateral flows. Moreover,the possibilities and limitations of such models is discussed, with special attentiongiven to the use of models under extrapolated conditions, such as often the case inthe modelling of floods.

1.2 Unsteady flow hydraulics

In teaching, the topic of hydraulics is generally first introduced from the steady flowconcept. In principle, it would be more logical to start with the unsteady flowconcept and than introduce steady flow as a special case of unsteady flow. In thiscase a clear indication has to be made of the underlying assumptions and thesimplification of the problem. In each application these assumptions have to beverified.

It should be realised that a real steady flow does not exist in nature. There willalways be some small variations in the flow distribution, even if there are noobservable water level variations. The steady flow is a concept of our engineeringmind, which sometimes can simplify engineering without unacceptable differencesfrom reality. However, the danger exists that engineers turn too easily towards theconcept of steady flow, even in cases where this is a quite wrong schematization ofreality and where this approach may lead to quite wrong conclusions.

For this reason, it is important to familiarize oneself with problems which typicallyshow more significant variations both in time and in space. Hydraulic problems inchannels are governed by two important concepts: storage and conveyance. Forincompressible flow, the first concept deals with water volume conservation. Thesecond concept deals with balance of forces acting upon the water mass and theireffect on the momentum balance. Of particular importance in this description is themagnitude of flow momentum losses due to channel friction relative to the gain ofmomentum due to gravity or other forces.

Before we can discuss more in detail the difference between steady and unsteadyflow, the concept of boundary conditions has to be introduced. In general, one isonly interested in a specific part of a hydraulic system. To illustrate this idea, it isuseful to consider the hydrological cycle. Water evaporates from the sea surface,precipitates partly above land and flows via the land surface, or via infiltrationthrough the subsurface, to the rivers and, in most cases, back to the sea. Let us nowconsider the river part of this cycle, or even a small part of this river system. Thelink to the upstream part of this river subsystem is specified in the form of aboundary condition and, more specifically, as the upstream boundary condition.



The link to the lower part of the river system, or to the sea, is specified as thedownstream boundary condition. In the sequel, we will discuss these boundaryconditions in more detail, including the question of the real need for such conditionsin our computations.

The question of the flow type is very much linked to the nature of the changes at theboundaries of the area described (or modelled). When the part of the river systemstudied adapts its state only slowly to the changes at the boundaries, the system isconsidered to be unsteady. However, when the river system adapts its state nearlyinstantaneously to the changes at the boundaries, the system is said to be quasisteady and some steady flow concepts might be used. This is usually the case whenlocal or nearfield problems are studied, such as local structures with their typicalbackwater effects.

One of the important parameters influencing this ease to adaptation is the storage inthe system. If this storage capacity is large compared to the difference betweeninflow and outflow the time scale of adaptation is also large. It is very likely, then,that we will observe a strongly unsteady flow phenomenon. If, however, there islittle storage capacity available between the boundaries, the adaptation of the state ofthe system to the new boundary conditions may be fast and the flow may passthrough a sequence of nearlysteady states. This adaptation is also dependent on thefacility of the flow to accelerate or decelerate. If the adaptation of flow particlevelocities to changing boundary conditions is fast, the system will pass through aseries of nearlysteady states. Where the adaptation is slow compared to the changesat the boundaries, the result will be clearly an unsteady flow.

These concepts are best illustrated by giving some practical examples.

Flood waves in rivers

Floods in a river are the result of the surface and subsurface runoff generated duringperiods of intense rainfall. This response usually leads to a typical hydrograph shapedischarge and water level variation in the river, which is more pronounced when therain is uniformly distributed over the period of the rain event. However, as rainfall isgenerally not uniformly distributed in time, the discharge distribution from thecatchments into the river is usually less irregular. While the flood wave propagatesdown the river it undergoes further deformations due to varying storage andconveyance characteristics of the river channel and due to additional lateral inflowsfrom other catchments. These processes together form a typical unsteady flowphenomenon, which can only be studied by simulations on the basis of unsteadyflow equations.

Flow in an urban drainage or combined sewer system

Drainage from urban catchments is to be seen as a special case of flood routing.Differences with a rural catchment are the faster response to the rainfall, the smalleramount of storage usually available and the more important role of channelconveyance, compared to channel or reservoir storage. In systems with drainagepipes, the open channel flow may temporarily change into a pressurized flow. Theunsteadiness of the phenomenon, however, is very pronounced.


WL | Delft Hydraulics 1 0

Despite these differences, it will be shown in the sequel that these problems requirethe same modelling tools as the ones used for the study of flood routing in rivers.

Flow in a desert wadi

In wadis, the unsteady nature of the flow is even more pronounced than in manyother rivers, due to the infiltration of the water into the river and flood plain bed.This leads to the steepening up of the flood wave front. Also, if often leads to thecomplete disappearance of the flood wave after some time.

Flow in irrigation systems

Flow in irrigation systems is usually controlled for the optimum use of the waterresources. This control introduces variations in time. A fast control may even lead tothe formation of hydraulic jumps travelling along the channel. Although the designof the irrigation canals is often based on steady flow concepts, the performanceunder operation usually requires checks on the basis of unsteady flow computations.

Flow over or through a hydraulic structure

The description of flow through a hydraulic structure within a river branch is usuallybased on an assumption of steady flow. The discharge at each moment in time isdirectly dependent on the water level boundary conditions. Although these waterlevels may vary rather fast, the discharge generated will respond more or lessinstantaneously. The immediate adaptation of the flow to changing boundaryconditions is the result of the lack of storage between the upstream and downstreamsection and the presence of a relatively small water mass to be accelerated ordecelerated. As will be shown in the sequel, the possibility to link a steady flowchannel element to channel elements where the flow has a distinctly unsteady nature,depends on the relative importance of the various terms of the equations describingthe flow problem.

Flow in pipe networks

Flow in pipe networks for water distribution in a town is often computed as steadyflow. For given constant water demands at various places in town and constant waterlevels in reservoirs, the flow and pressure distribution over the complete networkcan indeed be computed. These computed pressures can be checked againstminimum pressure requirements. However, the assumption of constant demands isan oversimplification of reality. Water demands usually have daily and weeklycycles. Reservoirs may be filled during the night at cheaper electricity rates andemptied during the day, when demands are higher. Fire fighting may suddenlychange the water demands over the network. These varying demands and storagelead to unsteady flow phenomena in pipe networks, although these are still oftencomputed as series of quasisteady states. Complete unsteady flow may occur as aresult of sudden changes caused by failures or misoperation of the distributionnetwork. This may cause unacceptable water hammer and cavitation effects. In thiscase computations are based on the use of the full unsteady flow equations for pipeflow, including storage of water resulting from water compressibility and pipediameter expansion.



1.3 Numerical methods

Although numerical methods have been developed already for centuries, thepossibilities offered by computers have given a strong impulse to the furtherdevelopment of these methods. With numerical methods one tries to solve sets ofdifferential equations, such as the De Saint Venant equations for unsteady flow inriver and channel systems. In the derivation of the differential equations one startswith finite control volumes for the definition of balance equations and than assumesthat these volumes reduce to an infinitesimal small size. Under this assumption,these equations provide a correct and valid description of continuous flows.

With numerical methods this process is reversed and balance equations are derivedover finite control volumes, starting from the differential equations. For this reason,one cannot expect that this procedure leads to the same results as those which mighthave been obtained by solving the partial differential equations directly. However, inmost cases such solutions do not exist, particularly when the equations are nonlinear. This, unfortunately, is usually the case when solving practical problems inhydraulics.

Currently, for nearly all applications in hydraulics, numerical methods have beendeveloped that work well and have the potential of limiting the differences betweenexact solutions and the approximate solutions. In these lectures we are discussing thedifferences by dealing with concept such as consistency, stability, robustness andeconomy of numerical operations. In particular, it will be shown how partialdifferential equations can be transformed into linear finite difference equations, towhich extent these linearization’s require iterations and what sort of algorithms existto solve the systems of equations in an economical way.

It will also be shown, wherever applicable, that numerical behaviour is related to thephysical behaviour described. Most obvious is the relation between boundary datarequirements and the way changes at boundaries are affected by the hydraulicsystem. However, also the performance of iteration techniques are influenced by thephysical behaviour of the system.

The objective of dealing with numerical methods in this lecture series is to provideenough background information to serve as a basis for the correct development ofmodels and for the best choice of modelling systems offered for use in a project.

1.4 Mathematical modelling

Modelling has become a frequently used tool for studies in hydraulic andenvironmental engineering. Whereas in the past many engineers turned to physicallybased models or simplified descriptions for the support of engineering studies, theincreasing availability of personal computers and the powerful developments incomputer graphics, data bases and online control, software has brought computersupport to the desk of consulting engineers. In line with these developments we alsosee a strongly increased availability and use of mathematical modelling softwaretools.



For a better understanding of what a model represents, let us look at one of the manypossible ways of defining such a tool:

a model is a physical or mathematical description of a physical system,including the interaction with its outside world, which can be used to simulatethe effect of changes in the system itself or the effect of changes in the conditionsimposed upon it.

In the development of a mathematical model one may distinguish the followingmain elements:

· definition of objectives· schematisation· equations and conditions· solution algorithm· software choice or development· data collection· model calibration· model verification· simulations

Definition of objectives

Definition of objectives is a very important and often underestimated element in thedecision process which leads to the use of a model. The first question to be posed iswhether a model can add important information to what is already understood abouta system. It also involves the estimation of the possibility to save project costs byusing a model and the economic value that may be attached to the development anduse of the model. This process will also lead to the choice of the level of complexityof model description. The use of models is generally associated to their use forsimulations. However, another interesting field of application is in the analysis ofpossible hypothesis about empirical relations. Finally, models define relationsbetween the various variables describing a state of a physical system andconsequently models may be used for data consistency checking. As typicalexamples of model objectives in the fields of environmental, hydraulic andhydrological engineering one may mention:

· effects of hydraulic works;· simulation of the impacts of floods;· online flood forecasting;· flood prediction;· design of urban drainage systems;· simulation of the impacts of dam breaks;· study of field irrigation water supply;· control of salt intrusion in estuaries;· BODDO computations along rivers;· ecological effects of heat loads from power plants;· estimation of sedimentation in reservoirs;



· soil remediation studies;· effects of sewerage overflows upon the receiving waters;· consistency checking of water quality data;· consistency checking of hydrological data.

Model schematization

The schematisation of the physical system follows from the complexity of theprocesses and the economical interest in studying these processes in all their details.The use of simplified models, based on a simplified description of the physicalprocesses is only justified if the results of the model can still be used reliably in thedesign process. In other words, if the results still fall within the reliability range ofdata used by the designer. All processes in nature are of a threedimensional andunsteady nature. The choice in the model schematisation is primarily:

· choice of the number of spatial dimensions;· choice of time variability.

From the spatial dimensions and the time, the modeller selects one or moreindependent variables x,y,z or t, or other independent variables if certaintransformations are applied, e.g. r and in a polar coordinate system. Such a choiceis strongly linked to the model objectives. For example, a reservoir can beschematized into a single point, if one wants to study the water level variations andthe reservoir outflow as a function of time. The same reservoir, however, willrequire a threedimensional schematization in space in a study of windinducedcirculations or velocity patterns following from density stratification.

Equations and boundary conditions

A mathematical model is based primarily on the choice of equations describing thestate of the physical system. Water levels, discharges, velocities, temperatures,salinities etc. are socalled state variables or dependent variables. One equation isneeded for each of these variables describing the state of the physical system. Mostof these equations are balance equations of mass, or, simplified, of volumes. Otherequations are based on balances of other physical quantities, such as momentum,energy or turbulence. Often, also, simplified forms of these equations are used forthe description of processes within our computational domain in space and time. Inspace this domain might represent the axis of a river flowing from town A to town Bfurther downstream and in time the duration of a typical flood wave, including anantecedent period. Additional conditions have to be provided at the modelboundaries, to specify the interaction of the outside world with the domain describedby our model. These conditions will follow from the nature of the physicalprocesses, translated into mathematical conditions through the equations describingthe system.



Solution algorithms

The balance equations in space and time provide us, generally, with partialdifferential equations. These equations are transformed from a continuous form intoa discrete form by writing them out as relations between variables at points in thecomputational domain. Such formulations are based on finite differences, finiteelements or boundary integrals and provide systems of linear equations. Completedwith a linearized formulation of boundary conditions and special elements, such ashydraulic structures, the total system of linear equations may be solved with avariety of algorithms, ranging from simple Gaussian elimination as a direct matrixsolver, to iterative techniques such as the conjugate gradient method. The solutionalgorithm usually has to follow a specific sequence of operations, consistent with thephysical links in the system.

Software

For most environmental studies standard software tools are available. For simpleproblems engineers usually turn to spread sheet packages, whereas for problemsrelated to open channel flow in networks, reservoirs, flow in pipe networks, sewersystems, water hammer, coastal management, short waves computations etc. varioussoftware products are available, developed at specialized hydraulic researchinstitutes. The use of these packages assures a flexible user environment and areliable solution of all sorts of numerical problems.

Data collection

Over the past years more and more effort has been spent on the collection of all sortsof data and the processing and storage of these data in data bases. However, formodel development, data available in data bases is not always sufficient, as thecalibration of models often requires data measured over short periods in time,available at various locations simultaneously. For this reason, the models are usuallyset up with whatever data available in the standard data base, completed with datacollected during some specially organised campaigns.

Model calibration and verification

Some data required for a model can be collected directly in the field. Examples aresalinities, channel crosssections, discharges, water levels and concentrations ofdissolved substances. Some data can only be collected with a certain degree ofuncertainty, such as the details of the topography in the flood plains. Other types ofdata cannot be measured at all, such as Manning numbers and diffusion coefficients.Such data can only be estimated on the basis of a sound engineering judgement,based on the interpretation of recorded values of other variables and parameters. Themore uncertainty we have in the model parameters, the more we are dependent on agood set of calibration data. The fit between measurements and computations andthe knowledge of the processes enables the adjustment of the parameters until anacceptable fit has been obtained. For model calibration one will usually select anumber of events, which are complementary to each other in terms of the calibrationparameters.



A calibration of a river model, for example, will typically start with low flowcalibration and finish up with the calibration of some typical flood events. Thisprocedure allows for the calibration of channel roughness parameters, prior to thecalibration of flood plain conveyance and storage parameters. After completion of amodel calibration the model should be verified on a set of data not yet used forparameter estimation. However, in practise it is not easy to reserve such a set andeven if such verification runs are made, the differences between model andprototype performance may lead to lengthy arguments about the quality of the modeldata. In other words, why should one trust the results of verification more than theresults of a model calibration?

Simulations

Once the model has been accepted it can be used for the typical simulationsfollowing from the definition of the model objectives. It should be kept in mind thatthe use of the model with modified parameters may, in turn, modify otherparameters. As an example, the construction of river embankments may also changethe bed roughness. The use of such models, therefore, should always beaccompanied with sound engineering judgement based on a thorough knowledge ofthe physical processes.



2 Introduction to numerical methods

2.1 Introduction

This chapter deals with the solution of problems described on the basis of one singleindependent variable, such as x or t. A typical example is the simulation of flowthrough a reservoir, where water level variations in the horizontal plan are neglectedand the point water level fluctuates in time as a result of timevariations in inflowand outflow. The water volume balance leads to an ordinary differential equation ofthe first order which can conveniently be solved by a finite differenceapproximation.

Another typical example connected to this problem is the description of the variationin time of the concentration of a chemical substance in the reservoir water, assumingthat this reservoir is wellmixed. The balance equation for this substance also leadsto an ordinary differential equation with time as the independent variable.

In a further extension one may consider the interaction of various substances,leading to coupled systems of firstorder ordinary differential equations. A wellknown example is given by the StreeterPhelps equations, describing the decay oforganic pollutants and the effects on the oxygen concentration in a wellmixed pond.The same set of equations is found by considering the BOD and DO concentrationsin a particle of water moving with the stream velocity in a river and neglecting theinfluence of exchange by diffusion between various water particles. Although thisset of equations may be described as a function of time, it might also be convenientto apply a transformation which takes the channel axis x as the independent variable.Such transformation facilitates the inclusion of typical influences along the river,such as point effluent loads, weirs providing additional reaeration and the effects ofvariations in the channel topography on stream velocity and reaeration.

Although the simplest problems of this category can be studied by exact solutions ofthe differential equations, the more flexible formulation of model equations andparameters requires solutions formulated by numerical schemes, such as finitedifference schemes or finite element schemes. In the sequel, a range of finitedifference approximation are introduced and discussed in terms of accuracy, stabilityand convergence of solutions.

Figure 2.1 Sketch of a cylindrical groundwater reservoir



2.2 Introduction to finite difference approximations

Let us consider one of the simplest ordinary differential equations expressing thevolume balance of groundwater in a cylindrical reservoir as shown in Figure (2.1).Assuming a constant reservoir surface area A, a constant porosity n and an outflowdefined as a linear function of the hydraulic head h, the water level is described bythe continuity equation and Darcy law, respectively, as

dhnA Qdt

= - ; Q kAh= -

or, by elimination of Q,

Article 2. dh hdt

a= - (2.1)

where is a linear reservoir coefficient.

This equation has the exact solution

0th h e a-= (2.2)

where h0 is the initial water level in the reservoir.

This exact solution is given here primarily with the purpose of comparing variousnumerical schemes, solved with a variety of numerical parameters for theseschemes. The numerical schemes that will be introduced successively are the

· Euler scheme;· Improved Euler scheme;· Implicit scheme;· NewtonRaphson scheme.

The numerical parameter in this example is the time step of numerical integration t.

2.3 The Euler scheme

For the definition of a finite difference approximation we will first return to theconcept of derivatives and the differential equations that may be constructed fromthese. The meaning of Equation (2.1) is that at any point along t, the derivative dh/dtto the function h(t) is equal to the value of the right hand side of that equation.



Figure 2.2 Situation sketch for the definition of a derivative (tangent) to a function h(t)

Keeping in mind (Figure 2.2) that this derivative is defined as

0

( ) ( )imt

dh h t t h tLdt tD ®

+ D -=

D(2.3)

one may use the inverse of this definition to construct a finite difference scheme ofEquation (23) on the basis of the approximation

1n nndh h h h h

dt t ta

+D -@ = @ -

D Dor

( )1 1n nh t ha+ @ - D (2.4)

where it should be noted that n is introduced as a counter for the time step andwritten as a superscript to the variable h or to any other quantity defined at a gridpoint at time t=n t. This notation should not be confused with an exponent. Thenumerical scheme introduced this way is called an Euler scheme. The right handside of Equation (2.6) is taken at time t=n t. Assuming that the value of h is knownat that point, we call such a scheme a forward difference scheme as we construct asolution forward in time proceeding from a point where the solution is alreadyknown.

Figure 2.3 Sketch of a finite difference approximation as the inverse of the definition of aderivative at a point A at t=n t



Figure (2.3) also shows that for any nonlinear solution the finite difference schemeintroduces at each time step an error , defined as the difference between the exactsolution of the equation and the solution obtained with the scheme. From the figureit may also be concluded that this error usually increases with an increasing timestep. It will also be clear that the error depends on the curvature of the function,expressed by the influence of the higherorder derivatives. In other words, forsolutions deviating slowly from straight lines one may take larger time steps than forsolutions which deviate fast from straight lines, if one wants to obtain the samerelative accuracy of the solution.

Let us demonstrate the effect of the choice of time step by solving Equations (2.1)and with the Euler scheme of Equation (2.4) for the following data:

h0 = 10 mm = 0.1 day1

Table 2.1 shows results at T=4 days for the exact solution of the equations,compared with numerical solutions obtained with Equation (2.4) for time steps t =0.5 days, 1.0 day and 2.0 days respectively. Differences between the exact solutionand the numerical solution are 1 %, 2 % and 4.5 %, respectively. It is also observedthat the errors are larger than those found at T = 2 days, demonstrating theaccumulative effect of the errors during the integration of the differential equationalong the time axis.

Table 2.1 Influence of the numerical scheme and the time step on the accuracy of results

time Exact Euler Improved Euler Implicit(days) t=½

(days)t=1

(days)t=2

(days)t=1

(days)t=2

(days)t=1

(days)t=2

(days)0.0 10.00 10.00 10.00 10.00 10.00 10.00 10.00 10.000.5 9.50 9.501.0 9.05 9.03 9.00 9.05 9.00 9.051.5 8.57 8.602.0 8.19 8.15 8.10 8.00 8.19 8.20 8.19 8.182.5 7.74 7.783.0 7.41 7.35 7.29 7.41 7.38 7.413.5 6.98 7.044.0 6.70 6.63 6.56 6.40 6.71 6.72 6.70 6.69



Figure 2.4 Various derivatives used in the finite difference approximation: (a) derivative used inthe Euler scheme; (b) ideal choice of derivative bringing the solution from point A topoint B; (c) approximate derivative used in the centred schemes

2.4 The improved Euler scheme

From Figure (2.4) it is observed that one possible way of reducing the errors is byselecting a better location for the derivative along the time axis. A good locationwould be a place C, approximately halfway the time levels n t and (n+1) t.However, the exact location of the point is not known, due to the influence of thehigherorder derivatives on the shape of the solution function. Moreover, even if thisexact location would be known, the solution of the function is not known at thispoint and, hence the value of the derivative. The principle of a better positioning ofthe derivative leads us to the improved Euler scheme based on an additional iterationof the solution. As a first step in the improved Euler method the value of hn+½,halfway the time step, is approximated by

( )½ 1 ½n nh t ha+ @ - D (2.5)Substitution of this approximate value in the expression of the derivative gives thefinite difference approximation over the total time step from grid point n to (n+1) as

1½

n nnh h h

ta

++-

@ -D

or1 ½n n nh h t ha+ +@ - D (2.6)

Results of this scheme, also given in Table (2.1), show considerable improvementsin accuracy, with errors of 0.15 % and 0.3 % for time steps of 1.0 and 2.0 days,respectively. Even considering that the amount of computational work done with theimproved Euler scheme is approximately twice the amount of work done with thenormal Euler scheme, the improvement in terms of efficiency is still remarkable.

For an equivalent computational effort the improved Euler scheme produces only 15% of the error of the normal Euler scheme. Although similar improvements inaccuracy are not always obtained for all problems, the example demonstrates thepotential of efficiency improvements by using higheraccurate numerical schemes.



2.5 The implicit scheme

The numerical schemes introduced so far are based on the concept that the derivativeof the function is known at the initial point in the computation over a time step anddoes not change during that time step or, at best, is adapted in its value and locationduring an additional iteration. A next logical step for improvement, then, is toinclude the variables composing the derivative in the expression for the yet unknownvariable h at time level (n + 1) t. For Equation (2.1) this leads to a centred implicitfinite difference scheme of the form

( )1

1½ ½n n

n nh h h ht

a a+

+-@ - +

D

or1 1 ½

1 ½n nth h

taa

+ - D=

+ D(2.7)

For the given value of and a time step of 1 day this leads to the simple relation

1 0.9048n nh h+ = .

Results of this computation are shown in Table (2.1). The errors are 0.017% and0.13% for time steps of 1.0 and 2.0 days, respectively. In terms of accuracy, thisapproach gives another considerable improvement over the earlier introduced socalled explicit schemes. Although this conclusion may not be generalized to othernumerical schemes without exceptions, the implicit schemes, in general, have thepotential of providing more accurate results, at lower computations cost, due to thebetter centring of the finite difference equations. However, for problems involvingmore unknown variables, the implicit schemes lead to systems of linear equations,which have to be solved simultaneously through matrix operations.

In most cases the overall solution algorithm leads to more numerical operations pertime step. However, this is generally compensated by the much larger time steps thatcan be taken. As a consequence, currently most numerical algorithms are based uponimplicit schemes. Apart from the higher accuracy, another important advantage ofimplicit schemes is their improved stability or robustness behaviour, as discussed in§ 2.8.

2.6 The NewtonRaphson scheme

One special form of implicit schemes is the NewtonRaphson scheme. Although theNewton Raphson approach is generally presented as a method for solving nonlinearequations, we introduce it here as an approach to the formulation of linear finitedifference schemes and notably the linearization of individual terms and coefficientsin the scheme.



Assume, for example, that the coefficient in Equation (2.1) is given as a linearfunction of h by the relation

0 1a a ha = +

where a0 and a1 are constants. Substitution into Equation (2.1) gives

20 1

dh a h a h Fdt

= - - = (2.8)

where F is a general function of h. An implicit finite difference scheme, centredhalfway the time step, could be written as

½nh F Ft

D= + D

D

As F is a function of h, the NewtonRaphson approach gives

( )0 12dFF h a a h hdh

D = D = - + D

leading to the finite difference scheme

( )0 11 ½ 2

n

n

F tht a a h

DD =

+ D +(2.9)

The advantage of this approach is that the change in the value of the coefficient isalready partly taken into account during the integration over the time step. FromEquation (2.13) one may conclude that the process is not yet ideal, as the value of hin the right hand side of the expression is taken at grid point n, whereas thesubstitution of a value at grid point (n+½) would be more precise. Referring, again,to the implicit scheme of § 2.5, the value of would be taken at grid point n,whereas an improved centring would require an additional iteration, similar to theapproach followed in the improved Euler method. The Newton Raphson implicitscheme is generally used without such additional iteration as in most cases this extrastep is hardly cost effective.

2.7 A more formal analysis of accuracy

Although the concept of accuracy was discussed on the basis of a common senseapproach and such a reasoning should always accompany the use of numerical andphysical concepts, it is also useful to introduce more fundamental analysistechniques. One of the most frequently used tools for the analysis of the accuracy ofnumerical schemes is the Taylor's series expansion.



Referring to Equation (2.2) and assuming a continuous behaviour of the function hand all its first and higherorder derivatives in time, the value of h at grid point n+1can be expanded from its value at grid point n through the infinite Taylor's series

1

1 !

nk kn n

kk

t d hh hk dt

¥+

=

æ öD= + ç ÷

è øå (2.10)

with all derivatives taken at grid point n. In Equation (2.10) k! should be read as “kfactorial”, defined as the product 1*2*3*… .*k.

It will be useful to discuss the meaning of the term under the summation sign in apragmatic way. The term tk refers to a step in time raised to the power k andcancels, at least in magnitude, against the contribution dtk. The notation dkh has themeaning of expressing differences in function values at points in the vicinity of thepoint where the Taylor's series is expanded upon and has a value comparable inorder of magnitude to the average function value at these points. As the value of k!increases rapidly with increasing k, it may be expected then that the contribution ofthe higherorder derivative terms in this expanded series decreases rapidly withincreasing k.

Referring to Equation (2.10), one may divide all terms by t to give

1 2 2 3

2 3 . . .2 6

n nnn nh h dh t d h t d h h o tt dt dt dt

+ æ ö æ ö- D Dæ ö= + + +ç ÷ ç ÷ç ÷D è ø è ø è ø(2.11)

where h.o.t. refers to all higherorder terms in this series expansion. Substitution ofthe Euler scheme of Equation (2.4) into Equation (2.11) and dropping the superscriptn provides

2 2 3

2 3 . . .2 6

dh t d h t d hh h o tdt dt dt

TE

aD D

= - - - +

-14444244443

(2.12)

Comparison of Equation (2.12)with Equation (2.1) leads to a difference TE betweenthe differential equation and the Euler scheme defined with the intention to solvethis equation.

This difference TE is called the truncation error of the finite difference scheme. Forthe improved Euler scheme, Figure (2.3) visualizes this truncation error by themagnitude . For the special case of Equation (2.1), the magnitude of all higherorder derivatives can be expressed in terms of the value of h at those points, asshown for constant by successive differentiation of the equation with respect to t.



At all points in time this gives the truncation error in the form

2 32 . . .

2 6h hTE t t h o ta a

= D - D + (2.13)

From Equation (2.13) it is readily seen that by decreasing t in the integration ofEquation (2.1), the total error in h decreases linearly with t with respect to the firstterm in the right hand side of Equation (2.13) and decreases even faster than linearlywith respect to the remaining terms in the truncation error. A truncation error of thistype is said to be of first order in t or simply O( t), while the numerical scheme issaid to be first order accurate.

As the numerical integration proceeds in time, the errors of each individual time stepare accumulated. It should be remarked here that the accumulative error for any t issubject to a decay by virtue of the meaning of Equation (2.1) and for any realistictime step t the accumulated error will tend to zero for t à . Another interestingobservation regarding the truncation error in the numerical integration of Equation(2.1) is the nearlylinear decrease of the error when reducing the time step t from2.0 days to 0.5 days, as shown in Table 2. 1. This behaviour points at a rapidlydecreasing influence of the higherorder derivatives in the truncation error, for thisapplication.

The much smaller truncation error in the implicit scheme of Equation (2.7) can alsoconveniently be demonstrated by a Taylor's series expansion. This derivationfollows a more common introduction of Taylor's series in numerical schemes. Afterthe selection of the appropriate centre point of the scheme all values of thedependent variables introduced at neighbouring grid points are expanded from thatcentre point. For the implicit scheme, centred at point n+½, the Taylor's seriesexpansion gives

( ) ( ) ( )

( ) ( ) ( ) ( )

2 32 3

2 3

2 32 3

2 3

½ ½1 ½ ½ . . .

2 6

½ ½1 ½ ½ . . .

2 6

t tdh d h d ht h t h o tdt dt dt

t tdh d h d ht h t h o tdt dt dt

a

a

æ öD D+ D + D + + + =ç ÷

ç ÷è øæ öD D

- D + - D + + +ç ÷ç ÷è ø

where all values of h and the derivatives with respect to time are taken at grid pointn+½. Expanding these expressions further leads to the equation

2 32 2

2 3

1 . . .8 24

TE

dh d h d hh t t h o tdt dt dt

aa

-

= - - D - D +144444424444443

(2.14)

which is the equation in reality solved by the implicit scheme.



Considering, again, that for this application all third and higherorder derivativesare small, the remaining part of the truncation error is indeed much smaller than thatof Equation (2.12) for any realistic choice of t. A realistic time step in this contextis a choice which relates t to the value of as discussed in § 2.10.

Figure 2.5 Oscillations and instabilities generated for large time steps: (a) exact solution; (b) stableoscillatory solution for t=18 days; (c) unstable solution for t=22 days

2.8 Stability

Despite the truncation error in the computation presented in Table (2.1) most resultsare quite acceptable for practical purposes. However, it is interesting to observewhat kind of results would have been produced if the time step had been taken muchlarger. As an example, let us consider a time step of 25 days in an application of theEuler scheme for the same equation and data as used in Table (2.1). For successivetime steps the sequence of results would be 10, 15, 22.5, 33.75, 50.63, 75.94 etc.,leading to infinity or an exponent overflow message on a digital computer after asufficiently large number of time steps. In any hand computation the sequence ofoperations would have been interrupted after one or a few time steps as the resultswould appear to be unrealistic for any practical interpretation. A computation of thiskind is called an instability. In an unstable computation results will always exceed alimit which has been set by the engineer as a realistic maximum or minimum value,for which exceedance is not to be expected (Figure 2.5). A frequently used analysisfor the definition of stability criteria is based on the notion of amplification factorsbetween results at successive time steps. If the absolute value of the amplificationexceeds unity at each and every step in time, one has sufficient proof of the unstablenature of the computation. The application of this analysis to the Euler scheme ofEquation (2.4) gives

1

| | 1 1n

n

hA th

a+

= = - D £ (2.15)

as a stability condition for the scheme. Since t is definite positive the stabilitycondition for the Euler scheme is derived as t 2.



Applying the same reasoning to the implicit scheme of Equation (2.7) leads to

1 ½ 11 ½

tt

aa

- D£

+ D(2.16)

and to the conclusion that the implicit scheme is unconditionally stable. However,the stability limit of t=20 days obtained for this application of the Euler scheme isfar beyond any time step that would be set as a maximum from the accuracy point ofview. Even for the implicit scheme the results obtained with this time step are veryinaccurate, as h drops to zero over the first time step and remains zero over allsuccessive time steps, whereas in the exact solution the value of h decreasesexponentially from the given initial value and only approaches the value of zero fortà .

It may be included that the unconditional stability of the implicit scheme does nothave special advantages in this application to the solution of ordinary differentialequations. When moving to applications on partial differential equations, however,the increased stability of the implicit schemes will turn out to be of such greatimportance that currently, nearly all finite difference schemes are based uponimplicit formulations.

2.9 Consistency and convergence

The Taylor's series expansion of the numerical scheme visualizes the differencebetween the differential equation as a continuous description of the physical systemand the finite difference equation as a discrete description on a set of grid points. As

t decreases, the difference between both equations reduces to the extent that theybecome equivalent as tà0. In such case the difference equation is said to beconsistent With the differential equation. In the case of ordinary differentialequations this generally implies that the results also converge towards the exactsolution as tà0. An additional condition for such convergence is that the resultsare computed under stable conditions.

2.10 The choice of a time step

The choice of a time step in the numerical integration of the differential equation is abalance between the maximum tolerable error and the economy of the numericaloperations. In any model application, errors are introduced from the followingsources :

· accuracy of basic data;· choice of additional parameter values;· model schematization;· choice of simulation data;· numerical errors.



Of all these sources of errors the numerical error is easiest controlled and a generalrequirement in model simulations is that the numerical error does not add to theuncertainties in the results introduced by the other sources. To satisfy this condition,the numerical error should be of an order of magnitude smaller than the overallexpected error. So, if the overall error is expected to be some centimetres in level,the admissible numerical error should not exceed a few millimetres. Whereas it isalready difficult to estimate the magnitude of the overall error, it is even moredifficult to estimate the numerical error generated during one single time step andeven more so the accumulated effect over various steps.

As the truncation error contains higherorder derivatives, a first estimate of the timestep is based on an idea about the curvature of the solution function. Strongly curvedsolution functions require smaller computational steps than solutions with moregentle variations.

Figure 2.6 Process of diminishing errors by reducing the time step t

A better estimate of the time step is based on a sensitivity analysis. By takingsuccessively smaller grid steps, solutions are compared and if the differences appearto be sufficiently small, similar computations can be made with that acceptable gridstep. It should be noted that the difference obtained by successively halving thetime step is less than half the total numerical error , as shown in Figure (2.6), wherethis total numerical error at a given point in time is plotted as a function of thenumber of computational time steps N over a given period of integration. This leadsto the conclusion that it is more efficient to refine coarse grids than refining furtheralready fine grids.

In a pragmatic approach one might also limit the allowable change in the functionderivative over a single time step. For the simple Equation (2.4), this condition isequivalent to setting a maximum to the change in the value of h from one time stepto the other. Setting, for example, as a criterion that over a single time step a changeof 5% is allowed, this criterion leads to t 0.05, or t 0.5 days.

Even in this simple case, however, it remains difficult to estimate the accumulatedeffect of this error over various time steps. In an attempt to do so and keeping inmind the decaying nature of the error, the accumulated error E at time step n=N, isapproximated as

( )2

1½ 1

NN nn

nE t h ta a -

=

= D - Då (2.17)



where in this notation Nn represents an exponent. In principle, it is possible tofollow the magnitude of E during the computational process, assuming that theeffect of third and higherorder derivative contributions to the truncation error arenegligible. Such an approach, however, is not practical for more complex problemsand it is better to turn to sensitivity analysis for determining an acceptable time step.

2.11 Reservoir routing

Consider a reservoir with a leveldependent storage area A, an inflow Qi given as ahydrograph and a leveldependent spillway outflow Q0, as shown in Figure (2.7).

Figure 2.7 Sketch of a reservoir with spillway flow

Reservoir volume balance and spillway flow are given by the following set ofequations

( )i odhA Q t Qdt

= - (2.18)

( )1.50 1.71 crQ m L h h= - (2.19)

whereA level dependent surface area of the reservoir;h reservoir water level above a general reference (e.g. mean sea level

MSL);hcr level of the spillway crest;L length of the crest;m discharge coefficient.

In principle, Equation (2.19) may be substituted into Equation (2.18) to give onesingle equation with h as the only dependent variable. However, in general it ispreferred to do such substitutions at the level of the numerical formulation afterlinearization of the equation and/or the finite difference formulation.

In a simple Euler approach the equation reads as follows:

( )1.5

0 1.71n ncrQ m L h h= - (2.20)



( ){ }1n n n ni on

th h Q t QA

+ D= + - (2.21)

where n, again, has the meaning of a superscript indicating the grid point along thetime axis. To demonstrate the computational algorithm, a small example is workedout with the following data:

hinitial= 24.70 mt(hours)

Qi(t)(m3/s)

h(m)

A(h)m2

hcr= 24.00 m 0 50 24 0.4*106

L = 30 m 1 150 25 0.8*106

t = 1 hour 2 360 26 1.0*106

m = 1.1 3 340 27 1.1*106

The results of the computation are given in Table (2.2).

Table 2.2 Results of the reservoir routing simulation with the Euler method

time(hrs)

gridpoint

hn

(m)Qo

n

(m3/s)Qi

n

(m3/s)An

(m2)h

(m)0.0 0 24.700 33.05 50 6.80*105 0.0901.0 1 24.790 39.62 150 7.16*105 0.5552.0 2 25.345 88.02 360 8.69*105 1.1273.0 3 26.472 219.32 340

With reference to Figure (2.8) it is readily seen that the time step of 1 hour is toolarge, as the error introduced by using a reservoir surface area at time n t issignificant. Moreover, the use of the discharge at time n t contributes to the error inthe time derivative, although it does not affect the volume balance in a direct way.

Figure 2.8 Volume error at successive time steps in reservoir routine using the Euler scheme

A much improved formulation is based on the NewtonRaphson approach. Apartfrom the better centring of the derivative of Equation (2.23) the variation in thesurface area is included implicitly. Although this variation is only introduced as alinear function, it is a great improvement over including A as a constant over acomputational step, defined at time level n t.



For the NewtonRaphson formulation the equations are rewritten in the form

( )( ) ( )1 , ,= - =i o i odh Q t Q F h Q Qdt A

(2.22)

and discretized, jointly with Equation (2.20), as

½nh F Ft

D= + D

D(2.23)

( )n

n n oo o o

dQQ Q Q h hdh

æ ö+ D = + Dç ÷è ø

(2.24)

where

( )( )

( )( )21 1

i oi o

nn n

i o i on n

F F FF Q Q hQ Q h

dAQ Q Q t Q hA dhA

¶ ¶ ¶D = D + D + D

¶ ¶ ¶

æ ö= D - D - - Dç ÷è ø

(2.25)

and

( )0.51.5*1.71*ocr

dQ m L h hdh

= - (2.26)

Special attention is drawn to the use of Q0(hn) in Equation (2.24) instead ofsubstituting the already known value Q0

n. As shown in Figure (29) the value Q0n

was obtained from a linearized Qh relation at time level (n1) t (line a at point A).The solution should proceed from point B along the line b during the next time step.If in the right hand side of Equation (2.24) Qo

n had been used instead of Qo(hn), thesolution would proceed from point C along the dotted line c and the accumulation oferrors over various time steps would bring us further and further away from thecorrect Qh relation. Note that for consistency reasons this correction is not includedin the volume balance equation.

Figure 2.9 Linearization of the Q0h relation



The coefficients in the two linear equations can be collected to the form

11 12 1

21 22 2

o

o

a h a Q ba h a Q b

D + D =D + D =

(2.27)

or

A x b- -

= (2.28)

with matrix A and vectors x-

and b-

given as

11 12 1

021 22 2

; ;ha a b

A x bQa a b

- -Dé ùé ù é ù= = =ê úê ú ê úDë û ë ûë û

(2.29)

where

( )( )

( )( )( )

11

12

021

22

1

2

2 1

1

1

2 ½

nnn n

i on

n

n ni i o

n no o

A dAa Q t Qt A dh

a

dQadh

a

b Q t Q Q

b Q Q h

æ ö= + - ç ÷D è ø=

æ ö= ç ÷è ø

= -

= + D -

= -

(2.30)

Elimination of Q leads to

1 2

11 21

b bha a

+D =

+(2.31)

1 11Q b a hD = - D (2.32)

The computation over the same time steps as taken for the demonstration of theEuler method gives results as shown in Table (2.3). The results, indeed, are morerealistic than those of Table 2.2. However, the differences between Q0

n and Q0(hn)indicate that it is advisable to use a smaller time step of integration.



Table 2.3 Results of the reservoir routing simulation with the implicit NewtonRaphson method

time (hrs) gridpoint

hn (m) Q0n (m3/s) Q0(hn) (m3/s) Qi(t)

(m3/s)An*105

(m2)dA/dh*105 (m)

dQ0/dh(m2/s)

0.0 0 24.700 33.05 33.05 50 6.80 4.00 70.821.0 1 24.992 53.73 55.75 150 7.97 4.00 84.312.0 2 25.688 114.45 123.78 360 9.38 2.00 109.983.0 3 26.379 199.71 207.02 340

time (hrs) gridpoint

a11 a21 b1 b2 h (m) Q0 (m3/s)

0.0 0 388 70.8 133.9 0.00 0.292 20.681.0 1 491 84.3 402.5 2.03 0.696 60.722.0 2 573 110.0 481.1 9.33 0.690 85.273.0 3

2.12 Routing of a decayable pollutant through a reservoir

Let us consider next a similar reservoir which is polluted by a decayable substancewith concentration c. Assuming that the reservoir volume can be schematized aswell mixed and introducing a first order decay, with reaction coefficient k, gives

( ) i i od cV c Q cQ k cVdt

= - - (2.33)

or

( ) i i odc dhV c cA c Q cQ k cVdt dt

+ = - - (2.34)

Substitution of Equation (2.18) into Equation (2.34) and division of all terms by thereservoir volume V leads to

( )ii

Qdc c c kcdt V

= - - (2.35)

where the volume V, as a function of the reservoir level, follows from the integrationof the surface area A along h. Such integration is best based upon a simpletrapezium rule as the surface area, obtained from planimetering a topography from amap, is never accurate enough to justify higheraccuracy integrations of the Ahrelation, such as the Simpson's rule or even integrations based on cubic splinefunctions. Moreover, linear Ah relations are used in the NewtonRaphson methodand the use of the trapezium rule for the integration of the Vh relation is consistentwith this approach. Although in a quick analysis the dilution coefficient

iQDV

= (2.36)

is often set as a constant, giving for Equation (2.35) the exact solution

( )( ) ( )0 1k D t k D t

iDc c e c e

k D- + - += - -

+(2.37)



We will focus here on the general case where D is a function of time and the exactsolution of Equation (2.35) does not exist. The simplest numerical solution ofEquation (2.35) is, again, based on the Euler method, giving

( )1 nn n nii

Qc c t c c kc tV

+ æ ö= + D - - Dç ÷è ø

(2.38)

Numerical solutions have the advantage over analytical solutions that all parameterscan easily be made a function of the dependent variables c and h and of theindependent variable t.

Examples of such further generalization of relations both in the water quantity andquality part are:

· outflow partly given as a user demand, possibly affected by reservoir operationlevels;

· inclusion of the effects of precipitation to and evapotranspiration from thereservoir surface in a longer term water balance simulation;

· stage dependent width of the spillway crest;· controlled spillway crest level as a function of h or t;· the use of m, calibrated as a function of the reservoir level;· various sources of pollutant inflows into the reservoir;· decay described as a function of the timevarying water temperature;· inclusion of additional substances, such as dissolved oxygen with dependence on

windgenerated reaeration, photosynthesis, bottom sediment processes etc.

Again, in this example more accurate results are obtained for a given time step withthe improved Euler method and implicit methods, including the NewtonRaphsonformulation. As the water balance is not affected by the pollutant concentration, anyimplicit technique leads to a system of linear equations with the unknownconcentration at the new time level decoupled from the unknown level and outflowing discharge. It is also rather common to simulate the water quantity part first,write results to a data base and subsequently retrieve the necessary information in aseparate water quantity computation. We will return to this approach whendiscussing water quality studies in rivers, coastal areas and in reservoirs where theassumption of a well mixed estuary is not correct.

2.13 Nonuniform steady flow in channels

As an example of computations which describe steady processes with variations inthe spatial xdirection let us consider the backwater computation in a uniformchannel. A typical channel crosssection is shown in Figure (2.10), including theconveyance K plotted as a function of the water level, where

2 /3

1

1jj

j jj j

K A Rn=

= å (2.39)



with

nj = Manning roughness coefficient for the sub section j;Aj = sub area of the crosssection j;Rj = local hydraulic radius of the sub area j;jj = number of vertical slices used in the integration across the section.

Figure 2.10 Crosssection and crosssectional parameters used in the nonuniform flow computation

Assuming the absence of lateral flow, the steady state De Saint Venant equationsreduce to the form

tancons tQ Q= (2.40)2 2

0 2

1 0d Q dh QIgA dx A dx K

æ ö+ + + =ç ÷

è ø(2.41)

Further differentiation of the first term of Equation (2.41) gives, for constant Q,

2 2

03 21 0Q dA dh QIgA dh dx K

æ ö- + + =ç ÷

è ø(2.42)

or2

02 2

11

dh QIdx Fr K

æ ö= +ç ÷- è ø

(2.43)

with the dimensionless Froude number Fr defined as a function of the flow velocityu, the crosssectional area A and the storage width b, as

2 22

3

s

u Q dAFr A gA dhgb

= = (2.44)

For a given discharge, the right hand side of Equation (2.43) is a function of thewater level above the channel bottom, giving

( )dh F hdx

= (2.45)



Table (2.4) gives an example of such function F(h), where it is readily seen at whichdepth the equilibrium slope is reached. This depth, for which F(h) = 0, is called thenormal depth.

The solution can easily be found in a NewtonRaphson approach, where, followingthe usual notation, a subscript i has been introduced as a grid point counter along thexaxis. The implicit finite difference scheme reads as

½ii i

i

h dFF hx dh

D æ ö= + Dç ÷D è ø(2.46)

or

1 ½

ii

i

F xhdF xdh

DD =

æ ö- Dç ÷è ø

(2.47)

For the crosssectional data given in Figure (2.10), a discharge Q=3.40 m3/s, a bedslope I0=104, the uniform flow depth is found for dh/dx=0, or Q2=K2I0, givingK=340 m3/s and h=1.00 m from the conveyance function shown in Table (2.4). Atthe downstream end of the channel the flow is controlled by a weir with a freeoverflow given by Equation (2.19) with parameter values:

m = 1.1 (weir coefficient);L = 11.0 m (weir crest length);hcr = 2.00 m (weir crest level),

giving a water level above the bottom just upstream of the weir of h0=2.30 m. Thevalue serves as the downstream boundary condition for the nonuniform flowcomputation. In the situation sketch of Figure (2.11) attention is drawn to the factthat the numerical integration of Equation (2.43) proceeds in the negative xdirection, requiring a negative value for the distance step x.

A suitable choice for this distance step follows from the consideration that in thefirst integration step the change in h should be limited to a reasonably small fractionof the difference between the downstream boundary water depth and the uniformflow depth. Considering that the NewtonRaphson approach gives fairly accurateresults, a first step h = 0.10 m, covering approximately 10 % of the total difference,should be acceptable. Linear interpolation in Table 2.3 gives a value K = 1498 m3/sat the downstream boundary. For a friction slope Q2/K2 = 0.052 * 104, giving adifference of 0.95 * 104 with the bed slope, a choice x = 1000 m satisfies thecriterion selected. Completing the table with the water level gradient function F ofEquation (2.43) for the given discharge (Table 2.4), the upstream water depths canbe computed by applying Equation (2.47). Results are shown in Table (2.5) over adistance of 12 km upstream of the weir.



Figure 2.11 Positive backwater upstream of a hydraulic structure

Table 2.4 Crosssectional data given as a tabulated function of the depth, including dischargedependent functions for Q = 3.4 m3/s (see also Figure 2.10)

waterdepth h(m)

bs (m) A (m2) K (m3/s) Q2/K2 *104 Fr F *104 (dF/dh)*104

(m1)1.0 10.0 9.00 340 1.000 0.126 0.000

1.5121.5 11.0 14.25 688 0.244 0.066 0.756

0.3122.0 12.0 20.00 1143 0.088 0.042 0.912

0.1002.5 20.0 28.13 1734 0.038 0.032 0.962

Table 2.5 Nonuniform flow computation based on the NewtonRaphson implicit approach (a)and compared with results of the Euler method (b)

(a) Newton Raphson (b) Euler

x(m)

gridpoint i

hi(m)

Fi*104

(dF/dh)i*104

(m1)

h(m)

hi(m)

Fi*104

h(m)

0 0 2.300 0.942 0.100 0.369 2.300 0.942 0.0941000 1 2.206 0.933 0.0932000 2 2.113 0.923 0.0923000 3 2.021 0.914 0.0914000 4 1.930 0.890 0.312 0.335 1.930 0.890 0.0895000 5 1.841 0.862 0.0866000 6 1.755 0.836 0.0847000 7 1.671 0.809 0.0818000 8 1.595 0.786 0.312 0.296 1.590 0.784 0.0789000 9 1.512 0.760 0.07610000 10 1.436 0.659 0.06611000 11 1.370 0.559 0.05612000 12 1.327 1.314

The results are also computed by using the Euler scheme which differs from theNewton Raphson scheme by the second term in the denominator of Equation (2.47).Assuming that the NewtonRaphson results are rather accurate, for this distance step,the Euler scheme produces results which might be considered acceptable forpractical use. Although the difference of 1.3 cm at x=12000 m will still increasefurther upstream, the crosssection schematization, the constant bed slope and thechoice of roughness coefficients will give rise to significantly larger uncertaintiesabout the correct water levels.



For another observation we refer to Table (2.4). The number of levels for which thecross sectional parameters are given is fairly small and produce unacceptable errorsin the linear interpolation of these values. The errors introduced are most likelylarger than those caused by using a lower accuracy numerical integration scheme.Although this might be acceptable in a quick analysis by hand computation, anycomputer code would be programmed to set up the table with much smaller steps.This table step size should not exceed substantially the step h in the numericalintegration. For our problem it would have been advisable to present the functionsbs, A, K, Fr and F at intervals of 10 cm.

The advantage of the NewtonRaphson scheme over the Euler scheme is the higheraccuracy obtained at the cost of the simple additional computation of thedenominator of Equation (2.47). For a comparison of its efficiency with that of thenormal Euler scheme let us recompute the solution with a four times larger distancestep, as shown in Table (2.6).

Table 2.6 Recomputation of the backwater curve with x=4,000 m

(a) Newton Raphson (b) Euler

x (m) gridpoint i

hi(m)

Fi*104

(dF/dh)i*104

h(m)

hi(m)

Fi*104

h(m)

0 0 2.300 0.942 0.100 0.369 2.300 0.942 0.3774,000 1 1.931 0.890 0.312 0.335 1.923 0.888 0.3558,000 2 1.596 0.786 0.312 0.296 1.568 0.777 0.31112,000 3 1.300 1.257

The NewtonRaphson approach still maintains its high accuracy during the first twosteps. During the third step the effect of the large table step in felt in the correctionthrough the term dF/dh. The difference of 1.7 cm with the computation taking x =1,000 m, would certainly have been much smaller if Table (2.4) had been set up withsmaller steps h. The result obtained with the Euler method for this case differs 7.0cm from the most accurate result and is quite unacceptable.

The advantage of the NewtonRaphson scheme becomes clearer when negativebackwaters with much stronger water profile curvatures are computed. Suchcomputations start from downstream boundary conditions with depths below thenormal depth h=1.00m. There is no doubt that the NewtonRaphson scheme is farmore efficient in its application than the Euler scheme.

2.14 An inverse scheme for the backwater curvecomputation

It has been demonstrated that once the table of the function F of Equation (2.45) hasbeen constructed for a uniform crosssection, the numerical operations for the nonuniform flow computation are fairly simple. For uniform channels with a constantbed slope we will now introduce a scheme which combines a high accuracy with anextreme simplicity. The computation of depths with the NewtonRaphson methodintroduced in the previous section is an asymptotic process, requiring continuouscomputations with ever decreasing changes in the depth.



Of course, one could have decided to increase the distance step as the depth comescloser and closer to the normal depth.

However, returning to Equation (2.45), one could also introduce a numerical schemetaking equal steps h in the dependent variable h, a socalled inverse scheme. Theadvantage of this approach is that the right hand side of the equation, being only afunction of h, can be centred without requiring the iteration of the improved Eulerscheme. The simple inverse scheme looks as follows:

1/ 2i

i

hxF +

DD = (2.48)

and results are shown in Table (2.7) for steps h = 0.10 m. These results are basedon the same interval for presenting topographic data as given in Table (2.4).Interpolation at x = 8,000 m in Table (2.7) gives a depth of 1.596 m, which onlydiffers 0.1 cm from the result obtained with the NewtonRaphson scheme. At x = 12,000 m, however, the difference is 0.5 cm, which difference must mainly beattributed to the NewtonRaphson integration based on too large table intervals (gridpoint 910).

The normal depth h=1.00 m is found at x = 33 km, whereas theoretically this shouldbe at x = . However, this deviation is of no practical importance at all. Althoughthe inverse scheme is very handy for uniform channel computations, it looses itsadvantage or even does not work at all in applications where the channel parametersvary along the channel axis.

Table 2.7 Results of the inverse scheme for h = 0.10 m

i xi (m) hi (m) Fi+½ * 104 xi (m)0 0 2.300

2.25 0.9370 1,0671 1,067 2.200

2.15 0.9270 1,0792 2,146 2.100

2.05 0.9170 1,0913 3,237 2.000

1.95 0.8964 1,1164 4,353 1.900

1.85 0.8652 1,1565 5,509 1.800

1.75 0.8340 1,1996 6,708 1.700

1.65 0.8028 1,2467 7,954 1.600

1.55 0.7716 1,2968 9,250 1.500

1.45 0.6804 1,4709 10,720 1.400

1.35 0.5292 1,89010 12,610 1.300

1.25 0.3780 2,64611 15,256 1.200

1.15 0.2268 4,409



12 19,665 1.1001.05 0.0756 13,228

13 32,893 1.000

2.15 Nonuniform flow in nonuniform channels

Let us now consider the somewhat more complex case of steady flow in nonuniform channels. For such channels the crosssectional area is a function of both hand x, giving the variation in the xt space

A AdA dx dhx h

¶ ¶= +

¶ ¶(2.49)

ordA A A dhdx x h dx

¶ ¶= +

¶ ¶(2.50)

Equation (2.43) should now read as

2 2

02 2 3

11

dh Q Q AIdx Fr K gA x

æ ö¶= + -ç ÷- ¶è ø

(2.51)

For the solution of this equation it is most practical to process the function F (x, h)as a tabulated function at equal depth steps h, starting from a reference point nearthe channel bottom. It is advisable to choose a h that has the same value at allcrosssections given along the river. For each river section, between two successivecrosssection points, a set of parallel lines is found along which F (x, h) reflects thechange in F along x, while keeping h constant. This approach will require at eachcrosssection the processing of one table reflecting the values dA/dx applicable tothe left hand channel section and another table reflecting their values applicable tothe right hand section.

The NewtonRaphson integration now proceeds with the scheme

½ ½ii i

i i

h F FF x hx x h

D ¶ ¶æ ö æ ö= + D + Dç ÷ ç ÷D ¶ ¶è ø è ø(2.52)

or2½

1 ½

ii

i

i

FF x xxh

F xh

¶æ öD + Dç ÷¶è øD =¶æ ö- Dç ÷¶è ø

(2.53)

in an integration performed over an integer number of distance steps along eachsuccessive channel section marked by its end crosssections and its local referenceslope. Again here, the procedure is most efficient if only one NewtonRaphsoniteration is made per time step and care is taken that the distance steps are not toolarge.



Some final remarks:

1. Both methods applied to uniform and to nonuniform channels can easily beextended to include the effects of lateral flows.

2. Both methods apply to fully irregular crosssection geometry and allow for avariation of roughness coefficients across the channel.

3. The methods are applicable for any formulation of the friction laws entered intothe integration of the channel conveyance K.

4. The efficient inverse scheme of Equation (2.48) can, in general, not be applied tononuniform channel flow computations.

5. Both methods can be applied to sub critical flow (Fr<1) and to supercritical flow(Fr>1), if the appropriate direction of integration (algorithmic structure) isfollowed.

2.16 What have we learnt?

In this Chapter we introduced various numerical methods for the solution of (sets of)ordinary differential equations. Many of these problems can simply be programmedin Excel. This is very straightforward in cases where equation parameters areconstants. Cases with nonlinear parameters can still be programmed easily by usingmacros for the function descriptions.

When programming solutions for incidental application it is recommended to use thesimple Euler method and to make sure that a sufficiently small grid step is used.Higher accuracy methods only justify the additional effort of programming andtesting of the code when repetitive use of this code is made and a single computationis not finalized instantaneously. With the current computer speed this is no longerthe case for the relatively simple problems programmed in Excel.

The higher accuracy integration methods have been introduced primarily, as theprinciples demonstrated are commonly applied in the numerical methods for solvingpartial differential equations. Applications of iterative methods as applied in theimproved Euler method, implicit schemes and Newton Raphson principles, will bediscussed in Chapters 4, 5 and 6.

2.17 Questions and small assignments

1. Extend the computation of all cases shown in Table (2.1) up to T = 6.0 days.

2. Explain in your own words why the improved Euler method is more attractive inuse than the standard Euler method.

3. Determine the type of numerical scheme used in the wellknown reservoirrouting equation

( )1 2 1 212

S I I O Ot

D= + - -

D



where S = reservoir storage ; I = reservoir inflow ;

O = reservoir outflow,

with subscripts 1 and 2 referring to time levels n t and (n+l) t respectively.

4. Explain why the resulting functions in Table (2.1), both obtained with the Eulermethod and the implicit scheme, is below the exact solution. Also explain whythe resulting function obtained with the improved Euler method lies above theexact solution.

5. Find with a computation carrying more decimal positions than shown in Table(2.1) the ratio between the errors obtained with the Euler scheme and the implicitscheme for a time step of 1 day. Discuss this ratio in the light of the truncationerrors for both schemes.

6. Verify Equation (2.14).

7. Show with a more precise computation of the results of the implicit schemecomputation in Table (2.1) that at T = 4 days the error obtained with t = 2 daysis four time larger than the error obtained with t = 1 day. Explain thisdifference from the truncation error of the scheme as shown in Equation (2.14).

8. Give the fourth order derivative term of the truncation error of Equation (2.14).

9. Define an Euler scheme for Equation (2.1) based on a numerical integration inthe negative time direction and show that the scheme is unconditionally unstable.

10. Extend the computation shown in Table (2.2) with one additional time step.

11. Extend the computation shown in Table (2.3) with one additional distance step.

12. Extend the computation shown in Table (2.5) with one additional distance step.

13. Explain why the depths calculated with the Euler scheme in Table (2.5) aresmaller than those obtained with the NewtonRaphson scheme.

14. Explain why in Table (2.6) the depth obtained in a computation with a distancestep of 4 km is smaller than the depth obtained with a 1 km step.

15. Recompute the solution of Table (2.7) with h = 0.05 m over the range 2.1 h 2.3 m. Comment on the results.

16. Explain what sort of problems may result from the application of Equation (2.48)for the computation of nonuniform channel flow.



3 Basic Unsteady Channel Flow Equations

3.1 Introduction

Unsteady channel flow will be first derived for a simple uniform channel of unitwidth, a horizontal bottom with friction along the walls and the bottom neglected.Referring to Figure (3.1), the water depth is symbolized by h and the average flowvelocity by u. These variables u and h are usually called the dependent variables,defined as functions of the independent variables x (space coordinate) and t (timecoordinate). The definition of two dependent variables requires the formulation oftwo equations for their solution. In channel flow these equations are usually basedupon volume and momentum conservation. The volume conservation approachassumes incompressible flow and constant water density. This is a quite realisticassumption for free surface fresh water channel systems. The equation followingfrom this volume conservation principle is called the continuity equation.

Figure 3.1 Control volume along a channel axis

3.2 Continuity equation

The derivation of the equation is based on the concept that over a given time thedifference between inflow to and outflow from a control volume along the channelbalances the change in the storage in this control volume over this same time.Referring, again, to Figure (3.1), it is readily seen that the inflow per unit time intothe control volume, or the flux of water volume through the upstream crosssection,is given by

inflow = uh

Assuming that both u and h are continuous functions of x, the outflow at thedownstream section of the control volume can then be defined as

outflow = uh + (uh) dxx

¶¶



The storage in the control section per unit time is expressed in terms of the change inwater level as

storage = (h dx)t

¶¶

based on the concept that for continuous water level changes with initial depth h0,the depth h1 after a step in time dt equals

1 0h = + dth h t

¶¶

Balancing terms gives the continuity equation in the form

h + (uh) = 0t x

¶ ¶¶ ¶

(3.1)

Differentiating out the second term leads to another, often used, form

h h u + u + h = 0t x x

¶ ¶ ¶¶ ¶ ¶

(3.2)

3.3 Momentum equation

It should be recalled that momentum is defined as the product of velocity and mass.The total momentum M of the fluid contained in our control volume of unit width,therefore, is defined as

M = u h dxr

where M and u are both vectors acting along the channel axis. Over a certain step dtin time, the change of M is balanced by the net amount of momentum carried by thefluid velocity through the boundaries of the control volume and the impulsegenerated by the forces acting upon it. Both quantities are to be taken with theircomponent acting in the xdirection. Following the same principles as used in thederivation of the continuity equation, the net amount of momentum carried with theflow velocity is defined by

a = u uh [u uh + (u uh) dx ]M xr r r¶

¶

where the first term refers to the inflow of momentum and the term within braces tothe outflow of momentum per unit time.

We will furthermore recall Newton's second law

F dt = d(mu)



where F is a force acting on the fluid with mass m and velocity u. The product of theforce and the time period over which it is acting is called an impulse. Newton'ssecond law states that this impulse generates a change in momentum.

Figure 3.2 Forces acting on a horizontal uniform channel section

Under the simplifications introduced above, the only forces acting on the controlvolume are the hydrostatic forces at the upstream and downstream ends of thecontrol section. Integrating the hydrostatic pressures over the verticals (Figure 3.2)gives the net force

½ ½ ½2 2 2dF = g [ g + ( g ) dx ]h h hxr r r¶

¶Balancing all terms over a unit time step leads, after division of all terms by theconstant dx, to the equation

( ) ( ) ( )2 ½r r r¶ ¶ ¶¶ ¶ ¶

2uh + u h + g = 0ht x x

Dividing, furthermore, all terms by the constant , leads to

( )2(¶ ¶ ¶¶ ¶ ¶

h uh) + u h + gh = 0t x x

(3.3)

Differentiating out the product terms in the derivatives, substituting the continuityequation and dividing all the remaining terms by h, leads to

u u h + u + g = 0t x x

¶ ¶ ¶¶ ¶ ¶

(3.4)

3.4 Transformation to the characteristic form

Although the form in which the set of continuity and momentum Equations (3.1) and(3.3) is shown is very convenient for the formulation of solutions on the basis offinite differences, it is useful to introduce a transformation which enables us tounderstand better the meaning of these equations. For this purpose we will introducethe auxiliary variable



c = gh (3.5)

Multiplication of Equation (3.1), substitution of Equation (3.3) it and division of allterms by c gives

c c u2 + 2u + c = 0t x x

¶ ¶ ¶¶ ¶ ¶

(3.6)

Similarly, the substitution of Equation (3.4)into Equation (3.3)s

u u c + u + 2c = 0t x x

¶ ¶ ¶¶ ¶ ¶

(3.7)

The set of Equations (3.5), (3.6) represents the characteristic form of the flowequations. The special significance of this form will become clear when respectivelyEquation (3.5) is added to and subtracted from Equation (3.6), to give the set ofequations in the form

(u+2c) + (u+c) (u+2c) = 0t x

¶ ¶¶ ¶

(3.8)

and

(u 2c) + (u c) (u 2c) = 0t x

¶ ¶¶ ¶

(3.9)

Before drawing any conclusions from these relations, we will introduce the generalmathematical formulation expressing the change of the value of a function f whenmoving from a given point P to a neighbouring point Q.

Figure 3.3 Relation between function values at neighbouring points

As shown in Figure (3.3), this change results from the partial derivatives of thefunction in tdirection and xdirection respectively as

f fdf = dt + dxt x

¶ ¶¶ ¶

(3.10)

or, after division of both sides by dt



df f dx f +dt t dt x

¶ ¶=

¶ ¶(3.11)

Taking for f the function u+2c, readily shows by comparison of Equations (3.7) and(3.10) that along the line with direction

dx = u+cdt

(3.12)

the expressiond (u+2c) = 0dt

(3.13)

holds, while the comparison of Equations (3.8) and (3.10) shows that along the linewith direction

dx = u cdt

(3.14)

the expression

d (u 2c) = 0dt

(3.15)

holds (Figure 3.4).

Figure 3.4 Characteristic directions and their Riemannn invariants

The line with the direction expressed by Equation (3.11) is called the c+characteristic, whereas the line with the direction expressed by Equation (3.13) iscalled the c characteristic.

The integration of Equation (3.12) gives

+u+2c = = constantJ (3.16)

whereas the integration of Equation (3.14) yields the expression



u 2c = = constantJ (3.17)

In this formulation J+ and J are the socalled Riemann invariants. For the simplifiedchannel flow equations these invariants have constant values along the socalled c+and c characteristics respectively. When introducing additional terms in theequation, such as lateral flow, bottom slope and bottom friction, these Riemanninvariants will contain correction terms and will no longer be "invariant".

3.5 The significance of the characteristics

The conditions derived along the characteristics enable us to compute solutions fromknown conditions at an earlier point in time. Referring to Figure (3.5), it is seen thatthe state of the fluid at point P can be computed from given states at points A and B.Given, for example, that at point A, u=1.0 m/s and h=5 m, while at point B, u=1.2m/s and h=4.8 m, the Riemann invariants are computed as J+ = u+2Ögh = 15 m/s atthe points A and P, and as J = u2Ögh = 12.52 m/s at the points B and P. Solvingthese two equations representing the Riemann invariants at point P gives u=1.24 m/sand h=4.83 m.

Figure 3.5 Characterisitics leaving from points A and B, defining the solution of the flow state at apoint P

The significance of the characteristics, however, goes far beyond the possibility tomake such computations. The fact that along the characteristics a special condition isvalid, implies that they pass on some information on the state of the fluid. In otherwords, the characteristics are to be seen as lines along which information on the stateof the fluid propagates. The uniqueness of the equivalence between Equation (3.10)on the one hand and Equations (3.7) and (3.8) on the other hand, implies that theselines along which information propagates are unique lines and that information onlytravels along just these characteristics.

The auxiliary variable c introduced in Equation (3.4) appears to get a specialmeaning as part of the direction of the characteristics. For the special case whereu=0, both characteristic directions have the same magnitude, though their signs areopposite (Figure 3.6a). As c is a function of the water depth, this means that theeffect of any disturbance or control at a point along the river, propagates in bothchannel directions with the same speed, which is a function of the square root of thewater depth. The variable c gets here the meaning of a celerity with whichdisturbances propagate in stagnant water.



Figure 3.6 Characteristic directions for: (a) and (b) subcritical flow; (c) critical flow and (d)supercritical flow

For velocities u>0, the characteristic directions no longer have the same magnitudeand, at a particular point, information on the fluid state travels faster in thedownstream then in the upstream direction (Figure 3.6b). For increasing watervelocities this leads to the particular case where the celerity c equals the watervelocity u (Figure 3.6c) and at this point information is no longer able to travel inthe upstream direction. Introducing the Froude number, defined as

| |uFr =g h

(3.18)

it is readily seen that for the situation of Figure (3.6c), where u=c, the Froudenumber equals unity. This special case is called critical flow, whereas the cases ofFigures (3.6a, 3.6b), where the Froude numbers are less than unity, represent subcritical flow. For water velocities exceeding the celerity c, the Froude numbersexceed unity and the flow is called super critical (Figure 3.6d).

From Figure (3.6) it is readily seen that for the case of sub critical flow, the state ofthe fluid at any point of space and at any point in time is controlled by both theupstream and the downstream conditions. In the case of super critical flow, however,the state of the fluid is only controlled by the upstream conditions. So, if at any pointA at a channel, we have supercritical flow, downstream control of the conditions atA is impossible, as long as the super critical state at A is maintained. It should benoted, however, that this supercritical state may change into a subcritical state whendownstream control, by a weir for example, generates a region of subcritical flowupstream of the control structure to the extent that this subcritical flow regionreaches point A.

From Figure (3.6) it may also be concluded that it takes a certain time before theeffect of the opening of the gate at the upstream end of an irrigation channel reachesthe point where the water is required. Taking, for example, a channel with stagnantwater at 1 meter depth, giving a celerity of 3.13 m/s, it lasts nearly 1 hour beforeadditional water reaches a demand point after the further opening of a gate 10 kmupstream.



Figure 3.7 Regions of determinacy of the state at a point P for various flow conditions

To make these principles even more clear, Figure (3.7) shows various cases ofsubcritical and supercritical flow, where the shaded areas show the region ofdeterminacy of the state at point P. Any disturbance or control within the shadedregion will have some effect on the water velocity and depth at point P, while anydisturbance or control outside the shaded region will not be able to affect conditionsat point P.The propagation of information along the characteristics will also determine theinitial and boundary condition requirements for the solution inside a computationaldomain, bounded in the xt plane by the lines t=0 and t=T and the lines x=0 andx=L. Referring first to Figure (3.8a), point A at time t=0 receives two characteristicscoming from outside the computational domain. If point A would have been locatedinside the computational domain, the solution would have followed from theRiemann invariants along the two characteristics passing through it, much similar tothe situation sketched in Figure (3.5). With A located at the boundary of thecomputational domain, it is can be concluded that the information on the Riemanninvariants has to be replaced by initial conditions, as we have no information on thestate of the fluid outside our computational domain and the values of the Riemanninvariants associated with it.

Figure 3.8 Relation between characterisitic directions and the boundary and initial datarequirements

Similarly, at point B in Figure (3.8a), one characteristic arrives from inside thecomputational domain and is supposed to carry information from another pointwithin the computational domain. The other characteristic arrives from outside thedomain, where the computation is not made and the information, which wouldnormally be passed on via the Riemann invariant, has to be replaced by a boundarycondition. Such boundary condition is usually provided in the form of a givenvelocity, water level, discharge or stagedischarge relationship.



The same applies to point C, where one characteristic carries information frominside the computational domain, while the other characteristic comes from outsideand requires a boundary condition to replace the information which would otherwisebe provided by its Riemann invariant.

Figure (3.8b) shows the case of supercritical flow with positive u. The problem iscompletely upstream controlled and two upstream boundary conditions are neededfor a solution. We define this need as a twopoint boundary condition against a onepoint boundary condition for the case of subcritical flow, given in Figure (3.8a).Initial data requirements are the same as for the subcritical flow case and twopointinitial data are required for all types of flow. Usually, values for u and h are to begiven.

Figure (3.8c), finally, shows the case of a supercritical flow with velocity u in thenegative xdirection. For the same reasons as pointed out earlier, this systemrequires twopoint initial data and twopoint boundary data at the upstream end,which in this case is located at the point x=L.

From the discussion above the general conclusion can be derived:

one initial or boundary condition is required for every characteristicentering the computational domain,

where entering is defined by following the characteristic forward in time.

3.6 The method of characteristics

In § 3.3 we have introduced the Riemann invariants and the computation of the statevariables u and h with these Riemann invariants. Where necessary, these Riemanninvariants are complemented with the initial and boundary conditions. We may nowproceed to the description of a complete solution algorithm, defined as the methodof characteristics.

This method is based on the construction of a network of points where the statevariables are computed. As shown in Figure (3.5), from a given set of two points onthe network a new point may be constructed by drawing a characteristic of onefamily from one point, which intersects with a characteristic of the other familydrawn from the other point. In general, these directions are not known initially, asthey depend on the solution of the state variables at the intersecting point. However,the solution of these state variables can be found through the Riemann invariants,even if the exact location of the point is not known yet. The nearly exact direction ofthese characteristics can then be drawn based on the average characteristic directionat the connecting points.

The procedure is best explained by considering a practical example. Figure (3.9)shows the application of the method on the problem of closing a gate in an irrigationcanal. Initially, the water flows at steady state with a velocity of 1.0 m/s at a depth of1.60 meter. The gate is closed over a period of 100 seconds, controlled to give alinearly decreasing water velocity immediately downstream of the gate.



This assumption serves our exercise, as most gate control operations are based uponwater level sensoring. For a further simplification in this example it is also assumedthat the downstream channel is very long and that the wave, generated by the gateclosure, is not reflected at a downstream side.

Figure 3.9 Hodograph and physical planes for a graphical solution of the computations alongcharacteristics

For ease of computation the acceleration due to gravity is taken as g=10 m2/s. Forthe initial steady state, the following data can be derived:

c+ = u+ Ögh = 5.00 m/sc = u Ögh = 3.00 m/sJ+ = u+2Ögh = 9.00 m/sJ = u2Ögh = 7.00 m/s

The disturbance generated by the gate operation will travel in the downstreamdirection with a celerity c+=5.00 m/s. This implies, for example, that at a point Lmeter downstream of the gate, the flow remains undisturbed over a period of L/c+seconds.



As a result, one may expect a steady state region downstream of the gate, shown asregion A in Figure (3.9). This is the region enclosed by the lines (t=0; x 0) and(x=5.00 t; x 0).

The solution will first be defined at points along the taxis, immediately downstreamof the gate.As a first step a set of points will be selected and numbered from 1 onward atintervals of 25 seconds. Subscripts u and d will be used for the locations of thesepoints upstream and downstream of the gate respectively.

Let us first consider the solution of the problem downstream of the gate. Thesolution at point 1d of Figure (3.9) is found by applying the boundary conditionu=0.75 m/s at t=25 s (Figure 4.9b), and the condition that the Riemann invariant J=7.00 m/s at point 1d, as it keeps its constant value with which it arrives along thenegative characteristic from the steady state region A. The solution of this set of twoequations gives h=1.50 m.

Similarly, the solution at points 2d,3d and 4d is found by applying the velocityboundary condition combined with the Riemann invariant carried along the negativecharacteristic arriving from the steady state region A as:

point 2d: u=0.50 m/s & J=7.00 m/s > h=1.41 m;point 3d: u=0.25 m/s & J=7.00 m/s > h=1.31 m;point 4d: u=0.00 m/s & J=7.00 m/s > h=1.22 m.

A summary of these data is shown in Table 3.1.

Table 3.1 Data at various characteristic net points of the physical plane

Point u(m/s)

h(m)

q(m2/s)

gh(m/s)

c+(m/s)

c(m/s)

2 gh (m/s)

J+(m/s)

J(m/s)

0 1.0 1.60 1.600 4.00 5.00 3.00 8.00 9.00 7.00

1d2d3d4d

0.750.500.250.00

1.501.411.311.22

1.1250.7050.3280.000

3.883.753.623.50

4.634.253.883.50

3.133.253.373.50

7.757.507.257.00

8.508.007.507.00

7.007.007.007.00

1u2u3u4u

0.640.380.170.00

1.761.861.932.03

1.1250.7050.3280.000

4.204.314.394.50

4.844.694.564.50

3.563.934.224.50

8.408.628.799.00

9.009.009.009.00

7.768.248.629.00

Let us now turn to points x 0 away from the boundary and draw the positivecharacteristic leaving from point 1d. Conditions at all points along this line are givenby the same Riemann invariants: J+ = u+2Ögh = 9.00 m/s and J = u2Ögh = 7.00m/s. Consequently, the same solution is found at all points along this line, givingu=0.25 m/s and h=1.50 m. This characteristic, then, is a straight line defined by acelerity c+ = u1+Ögh1 = 4.63 m/s.



Similarly, the positive characteristics leaving from points 2,3 and 4 must be straightlines with celerities of 4.25, 3.88 and 3.50 m/s respectively.The region between the positive characteristics leaving from points 1d and 4d isdefined as a simple wave region, characterised by straight (positive) characteristicsof one family and curved characteristics of the other family. However, it is notnecessary here to draw the curved c characteristics, as the visualisation of theselines does not add any essential information on the state of the flow in such a simplewave region.

For the same reason, only the characteristics bordering a steady state region aredrawn and none of the characteristics inside the steady state region.

Having defined a simple wave region one might define a complex wave region asone where both families of characteristics are curved lines, indicating that velocitiesand depths vary from point to point in both directions. In such a complex waveregion a network of both families of characteristics is drawn. This situation wouldoccur if the simple wave generated from points 1d to 4d would be reflected against adownstream boundary as could easily be verified by introducing, for example, aconstant water level condition at a point, say, 400 m downstream of the gate. As willbe explained later, however, we do not want to carry the presentation of the methodof characteristics too far and refer the interested reader to standard text books on thetopic (e.g. Abbott, 1979).

However, as an example of a curved characteristic and for demonstration purposeonly, consider the negative characteristic between point 1d and the steady stateregion A shown in Figure (3.9). The direction of this characteristic is varying frompoint to point, as follows from the varying J+values, while moving along thischaracteristic from point 1d to region A. As a reasonable approximation the averagedirection is taken as

1d A1d A

+ 3.13 3.00c c 3.07 m/sc 2 2® = = =

and the approximate characteristic is drawn backward in time, leaving from point 1until its arrival at the steady state region.

From this procedure it may be concluded that, although the method of characteristicsis generally considered a quite accurate solution of the simplified partial differentialequations describing channel flow, it does not represent an exact solution of thesedifferential equations. The approximation is in the location of the points of thenetwork. The solution of the simplified equations at the network points, however, isexact. The more generally applied finite difference methods, to the contrary, haveexact locations of the network or grid points, while the solution of the equations atthese points, as a rule, are approximations of the true solutions.



3.7 More complex boundary conditions

Let us now consider the upstream part of the channel, where the water velocity isreduced and the flux of momentum from upstream is likely to lead to a piling up ofthe water against the control structure. The effect of any control will be travelling inupstream direction with the celerity c = 3.00 m/s.As also in this case some time will elapse before the disturbance reaches anupstream point along the canal, we will also find here a steady state region shown asD in Figure (3.9).The solution upstream of the gate requires gate boundary conditions. These are notthe same as given for the downstream part, as the velocities given are related tocrosssections and water levels just downstream of the gate. The solution at thisdownstream section, however, enables the combination of velocity and depth to givethe specific discharges, which are the same just upstream and downstream of thestructure.The solution of the state at point 1u then follows from the computed specificdischarge q=udhd=1.125 m2/s and the positive Riemann invariant arriving fromsteady state region D, J+ = u+2Ögh = 9.00 m/s. Substitution of one equation into theother leads to a third degree polynomial in h, which requires a NewtonRaphsoniteration algorithm for their solution.

For noncomputerized solutions it is therefore easier to turn to a graphical method.Graphically, the solution of any set of two equations is found at the intersection ofthe lines representing these equations in an appropriate coordinate system. For thegiven problem, a suitable choice of the coordinates is a plot of values 2Ögh linearlyalong the vertical axis and values u along the horizontal axis. The plane formed bythese two variables is called the hodograph plane. The choice of the vertical scaleleads to the representation of the Riemann invariants as lines under 45o (Figure3.9c). By plotting the 2Öghaxis vertically down, for subcritical flow the sign of thedirections coincides with that of the characteristic directions in the physical plane.This convention is quite convenient in its practical use.

Table 3.2 Data for drawing qrelations in the hodograph plane

qm2/s

um/s

hm

ghm/s

2 ghm/s

1.125 .60.70

1.871.61

4.324.01

8.658.02

0.705 .35.40

2.011.76

4.484.20

8.978.39

0.328 .18.16

1.822.05

4.274.53

8.539.06



Figure (3.10) also shows various boundary conditions, which may all easily bepresented in graphical form, such as u given (Figure 4.10c), h given (Figure 3.10d),a given specific discharge q (Figure 3.10e) and a critical flow condition (Figure3.10f). This last relation is a special form of a rating curve, or Qh relation, which,for the simplified channel flow equations, would normally be presented as a uhrelation. The plot of a given specific discharge, as applied at point 2u of Figure(3.9c), is easily made by selecting a number of uvalues and calculating theassociated h and 2Öghvalues. It should be realised that q should always be plottedwith the appropriate sign.Also, the line is curved and requires a certain number of computed points in theexpected neighbourhood of intersection for a sufficient accuracy.

Figure 3.10 Various boundary conditions represented graphically in the hodograph plane

We return again to Figure (3.9c), which shows the hodograph plane belonging to thephysical plane of Figure (3.9a). The boundary conditions q=1.125, 0.705 and 0.328m2/s at points 1, 2 and 3 respectively, are presented by three different lines in thehodograph plane, whereas the discharge q=0 is represented by the line u=0. Thesolutions at points 1u,2u,3u and 4u are shown in the hodograph plane as theintersections of the lines representing the specific discharge at those points and thecondition J+=9.00 m/s. The characteristics leaving the taxis over the period of theclosure, again, define a simple wave region, with characteristics now convergingtowards each other. Further upstream this will lead to the intersection of these ccharacteristics. Such intersection of characteristics of the same family leads to twovalued solutions for both u and h at such intersection, or, in other words, to theformation of a hydraulic jump. It should be realised that the intersection ofcharacteristics is found here on the basis of the equations which do not contain bedslope and bed friction. Where these terms are playing a significant role thecomputations become more complex and are the subject of another chapter.



Although not frequent in nature or in manmade systems, such travelling hydraulicjumps are known phenomena, for example, as tidal bores in estuaries or as undularhydraulic jumps in tail race channels of peak power generating hydropower systems.

3.8 The formation of positive and negative hydraulicjumps

The discussion on the formation of hydraulic jumps is simplest presented on thebasis of an extreme operation mode of the gate. Figure (3.11) shows the case of aninstantaneous closure of otherwise the same problem as discussed in Figure (3.9).

Considering first the upstream side of the gate, the positive Riemann invariantcarried from the steady state region D has to be combined with the condition u=0 atx=0, for t>0. As shown in the hodograph plane the solution of these two conditionsis u=0 m/s; h=2.03 m. The negative characteristic celerity is c=4.50 m/s, which isgreater than c=3.00 m/s found over the steady state region D.This difference in celerities leads to the immediate intersection of the negativecharacteristic directions just upstream of the gate at t=0 and to the formation of apositive hydraulic jump.



Figure 3.11 Method of characteristics for the case of a sudden gate closure

We have assumed that despite this intersection of characteristics, the positiveRiemann invariant remains constant across the jump.We will, furthermore, assume that the celerity of the jump is approximated by theaverage of the two celerities of the intersecting characteristics forming this jump.Both hypotheses have been investigated and appear to be approximately true forsmall jumps (Abbott 1979). Based on these assumptions the celerity of the jump iscomputed at c=3.75 m/s.

The line representing this propagation in the physical plane separates the steady stateregion D from another steady state region C bordered by the line x=0, for t>0 andthe path of the hydraulic jump.



A similar construction at the downstream side of the gate gives a steady state regionB, where h=1.22 m. Steady state regions A and B give characteristic celeritiesc+=5.00 m/s and c+=3.50 m/s respectively, which show diverging directionsoriginating from point (x=0, t=0). Only at the very origin, a discontinuity is present,which rapidly spreads along x while travelling downstream. We call such specialform of a discontinuity a negative hydraulic jump and the subsequent simple waveregion a centred simple wave. In this simple wave region the characteristics drawncan be selected freely and Figure (3.11) shows the choice of some characteristicsrepresenting steps of 10 cm in h.

The results of the computation along x, at T=125 s are shown in Figure (3.11c). Thediscontinuity formed at the positive jump and the spreading negative hydraulic jumpin the downstream part are clearly recognised.

3.9 The limited practical importance of the method ofcharacteristics

The method of characteristics has the principal advantage of visualising the way inwhich flow disturbances or the effects of flow control travel through the hydraulicsystem. As will be shown in the Chapters dealing with mathematical modelling, thestructure of the net of characteristics is also very important in understanding thenumerical procedures required for the practical solution of hydraulic phenomena. Inother words, a good understanding of the physics of a hydraulic phenomena guidesus in the choice of appropriate numerical solution techniques and their numericalparameters. As already discussed in § 3.4, the characteristics show us the initial andboundary data requirements for any hydraulic engineering problem simulation.

However, where the concept of characteristics is very important to us, the method ofcharacteristics provides us only for simple practical cases with an algorithm for thecomputation of unsteady flow systems. For this reason we do not expand here on theinclusion of energy generating or dissipating terms, such as gravity terms andbottom friction, especially not for nonuniform channels with irregular topographies.In this text we also do not show how to include lateral flows. Inclusion of such termsis much easier and flexibly handled in programmable solution algorithms based onimplicit finite difference schemes, as discussed in Chapter 5.

We will return, however, to the practical use of characteristics in the case of waterhammer and flood propagation simulations.



3.10 Questions and assignments

3.10.1 Derive Equations (3.1) and (3.3) again for the case where a channelwidth b is introduced.

3.10.2 The simplified model is extended with a lateral flow term, byintroducing the lateral flow per unit width and length of the channel q(m/s).a. Derive the new form of Equations (3.1) and (3.3).b. Give the transformation to the characteristic form.c. Does this term give any change in the characteristicdirections?d. Derive the new expressions for the Riemann invariants.

3.10.3 Calculate u and h at point P of Figure (3.5) for the following data:Point A: u=0.5 m/s; h=3.00 m;Point B: u=0.6 m/s; h=2.80 m.

3.10.4 A rectangular channel of 10 meter width carries a discharge of 6 m3/s.For a water velocity u=2m/s, calculate the state of the flow. Whatinitial and boundary conditions do we need for modelling a certainreach of the channel?

3.10.5 At the downstream end of a channel the flow is critical. A positivecharacteristic arrives at this boundary carrying a Riemann invariantJ+=7 m/s. Calculate the outflow velocity and depth.

3.10.6. A gate in an irrigation canal of unit width is closed over a period of 4minutes. The initial velocity of the water in the canal is 1.0 m/s at adepth of 0.9 m. Assume a linear decrease of the discharge during theclosure.

a. Sketch the structure of the characteristics upstream anddownstream of the gate. Take a total time of 8 minutes and achannel length of 2000 meter upstream and downstream of thegate. Take one characteristic every 2 minutes (only for the purposeof this exercise, as for practical solutions a somewhat denser netwill be needed).

b. Draw the hodograph plane and solve the depths and velocities atthe intersection points of the characteristics.

c. Draw a more precise network of characteristics.d. Draw the water level and velocity functions along the canal at 8

minutes after the initiation of the closure.e. What is going to happen at a later moment in time?

3.10.7. Repeat the same exercise for an instantaneous closure of the gate.



4 Introducing Numerical Solutions for PartialDifferential Equations

4.1 Introduction

Partial differential equations are introduced when the dependent variables ofdifferential equations are a function of more than one independent variable. In manypractical applications these independent variables are x for space and t for time. Inthis case, derivatives of the dependent variables have fractions both in space and intime. For this reason we speak about partial derivatives, when dealing with aderivative along one of these two or more independent variables. Consequently, wespeak about partial differential equations when rates of changes of these dependentvariables are related along the various dependent variables.

Depending of the nature of the physical processes, these partial differentialequations can be classified as belonging to one of the following types:

· hyperbolic equations;· parabolic equations; and· elliptic equations.

The classification of the Equation (s) for a certain problem leads to distinctdifferences in which these equations are solved. Typical application areas for thesevarious classes are:

· hyperbolic equations: unsteady surface water flow; unsteady flow in theunsaturated soil zone; flood wave propagation; transport of pollutants, channelmorphology;

· parabolic type: dispersion of pollutants; flood wave peak dampening; dispersionof heat or energy;

· elliptic equations: steady surface water flow; steady groundwater flow.

The difference in type lies in the way in which disturbances propagate from onepoint of a given (physical) system to other points. In the case of hyperbolic problemsit takes a definite time before a change of the state of the system at one pointinfluences the state at another point. The effect of these changes propagates alongsocalled characteristics, or lines along which disturbances propagate. An obviousexample is the propagation of flood waves along rivers. Generally, it takes a fewhours or a number of days before a rain event in the upper catchments leads tofloods at locations downstream. In parabolic problems, the effect of a changesomewhere in the system, immediately affects the state everywhere else. In this case,the characteristics travel at an infinite speed.



In a step further, elliptic problems even have imaginary characteristics only andthese lead to an even more pronounced effect of the influence of the change of thestate at one single point on the state of the total system. Examples are statedescriptions of steady surface water or groundwater flow. It should be realised thatthese are rather artificial descriptions, as true steady states do not exist in nature. Forthis reason, most parabolic or elliptic problems are solved as if one is dealing with ahyperbolic problem.

The simplest of the equations of hyperbolic type is the socalled advection equation(see Equation (4.3)). Advection describes how some quantity, such as mass,momentum, energy, heat, a pollutant etc. is transported by a carrier, such as water orair. When dealing with the water sector, there is a variety of problems which can bedescribed by equations where advection processes play an important role. Examplesare:

· convective momentum term in the unsteady flow equations. Although often notof much importance in most gradually varied flow situations, the advection term(or convective momentum term) plays a dominant role in the formation ofhydraulic jumps, including the special appearances of moving hydraulic jumps,such as tidal bores;

· flood wave propagation, as a special case of the application of unsteady flowequations. Unsteady river or channel flow is described by the De Saint Venantequations. These form a set of second order partial differential equations ofhyperbolic type with two characteristic directions. In the most common case ofsub critical flow, these characteristics have opposite directions. In the specialcase of flood wave propagation along rivers, gravity and friction play a dominantrole. This results in the emergence of an underlying simplified advectiondiffusion equation, of which the advection part is only first order hyperbolic. Thecharacteristic direction of this part is shown to us by the propagation celerity ofthe flood wave peak;

· in hydrology also the movement of groundwater in the unsaturated zone isbased upon a combination of advection and diffusion processes, generallydescribed by the Richards equation. The advection part of this equation is drivenby gravity and by soil suction forces (matric forces). Due to the complex relationbetween soil permeability and soil moisture, the process is extremely nonlinear,leading to the propagation of soil moisture shock fronts, much similar to thepropagation of hydraulic jumps in free surface flow;

· transport and dispersion of substances in water. A good description of theseprocesses is essential for the assessment of water quality processes, such as thespreading of pollutants and heat, growth of algae, sedimentation processes etc.An important element of this description, again, is the advection part of theadvectiondispersion equation.



4.2 The advection equation

As an example, consider the advection of a conservative matter (e.g. a chemicalpollutant) in a channel. The derivation of the equation describing this processproceeds via the introduction of a control element along the channel axis as shownin Figure (4.1).

Figure 4.1 Control element for the derivation of the advection equation.

As a first step, the equation of mass conservation for water is derived. referring toFigure (4.1), the mass balance for a control element with length dx along the xaxisis set up by defining the change in storage to be equal to the difference betweeninflow and outflow, or over a time period dt,

( ) ( )V dt Q Q Q dx dtt x

r r r r¶ ì ¶ üé ù= - +í ýê ú¶ ¶ë ûî þ

Replacing the volume V by the product of crosssectional area A and distance stepdx it is found, for constant density , that

0A Qt x

¶ ¶+ =

¶ ¶(4.1)

In a similar way, the mass balance for a conservative matter in the water withconcentration c can be derived as

( ) ( ) 0Ac Qct x

¶ ¶+ =

¶ ¶(4.2)

or, after differentiating out the terms of this equation and substituting Equation (4.1),

0c cut x

¶ ¶+ =

¶ ¶(4.3)

where the velocity u is defined as Q/A.

In this derivation it is assumed that the concentration c is uniformly distributed overthe crosssection. For the time being, it is also assumed that the crosssectional area



for flow is equal to the crosssectional area representing storage. Equation (4.3) isvalid at every point of the xt plane and shows the relation at the solution surfacebetween the tangent in xdirection (to the function which shows the distribution of calong the river at a fixed time) and the tangent in tdirection (to the function showinghow c varies in time at a fixed point along the channel).

4.3 The characteristic solution

To get more insight into the meaning of Equation (4.3) let us consider the totalderivative

c cdc dt dxt x

¶ ¶= +

¶ ¶(4.4)

This equation simply states how the variation of c in any direction in the xt plane iscomposed of variations of c along the t and xaxis, respectively. Division by dtshows that, by equivalence of the Equations (4.3) and (4.4) along the line

dx udt

= (4.5)

the condition holds that

0dcdt

= (4.6)

or, after integration of Equation (4.6), that c remains constant along this line. Theline defined by Equation (4.5) is called the characteristic of Equation (4.3). InFigure (4.2) it is shown how a solution of Equation (4.3) is obtained, for the simplecase where u=constant and where, consequently, the characteristics are parallel linesin the xt plane.

Figure 4.2 Solution of the advection equation with the method of characteristics.It is also seen that a solution is only obtained when in the computational domaininitial data are defined along the line t=0 and, where an upstream boundary is



present, also boundary data along the line x=0. As will be shown later, thesecharacteristics play an important role in defining where initial and boundary dataare required, as well as the number and the type of these conditions. For practicalapplications the method of characteristics is not often applied in this form, mainlybecause it has the disadvantage that it is not based upon a fixed grid in space.This makes it more complicated to include in the solution such processes as sourcesand sinks, decay and diffusion, which will be discussed later.

4.4 Finite difference schemes

On the grid, shown in Figure (4.3), the derivative c/ x can be written in severalways in terms of finite differences and, at grid point (i,n) more in particular as

1n ni ic cc

x x--¶

@¶ D

backward difference approximation;

1 1

2

n ni ic cc

x x+ --¶

@¶ D

centred difference approximation;

1n ni ic cc

x x+ -¶

@¶ D

forward difference approximation.

Figure 4.3 Grid for defining finite difference schemes of the advection equation.

In a similar way finite difference approximations for c/ t can be defined. For theadvection scheme these approximations can be combined to nine possible finitedifference schemes, already. This number even increases when combinations of 1),2) and 3) are made by introducing, for example, other time levels than n t. Not allthese schemes will give satisfactory results and it is needed, therefore, to definecriteria for judging which scheme can be selected as the best out of the large numberof possible schemes.

Let us consider, in the first instance, a scheme approximated by a backwarddifference in the xdirection and a forward difference in the tdirection. Equation(4.3) than reads as



11 0

n n n ni i i ic c c cu

t x

+-- -

+ =D D

or, expressing cin+1 in terms of the other variables and defining

u trx

D=

D(4.7)

11 (1 )n n n

i i ic r c r c+-= + - (4.8)

This relation can be represented graphically by the operator shown in Figure (4.4).

Figure 4.4 Operator for the scheme of Equation Error! Reference source not found..

A solution using this scheme is shown in Figure (4.5) for the following data:

u = 0.0185 m/s;x = 100 m;t = 0.75 hours.

Upstream boundary data are set as c=0 and the initial data are given by the Gaussiandistribution of the form

2½ ( )e

x

cm

s-

-= (4.9)

with mean value =500 m and standard deviation =100 m.



Figure 4.5 Numerical solution of the advection equation using the scheme given by Equation (4.8)with r=½ and various values for x.

Figure (4.5) shows the effect of varying numerical parameters on the solution. Theexact solution is defined by a shift of the initial Gaussian distribution over 400 malong the xaxis. Numerically, the same solution is found for r=l. For r=½ the waveis strongly dampened. This effect is called numerical diffusion of the wave. Thefigure also shows that a smaller value for x, while maintaining the same value r=½,reduces the numerical diffusion. For a value r=2 the solution will not look realistic.Negative values for c are found as well as values greater than 1. These values cannotpossibly be correct as the method of characteristics shows that in the absence ofsource and sink terms extreme values for c are found along the boundaries wherecharacteristics enter the computational domain. The problems faced here are relatedto the stability of the scheme.

It is clear that not only the choice of a specific scheme is important for obtaininggood results in a computation, but also the choice of the numerical parameters, suchas x and t, for that scheme. The accuracy aspects of a particular scheme can bejudged by writing each term in a Taylor's series expansion from the selected centrepoint of the scheme. At the centre point (i,n) we obtain for the scheme given byEquation (4.8):

2 2 3 31

2 3 . . .1! 2! 3!

n nnn ni i

i i i

t c t c t cc c h o tt t t

+ æ ö æ öD ¶ D ¶ D ¶æ ö= + + + +ç ÷ ç ÷ç ÷¶ ¶ ¶è ø è ø è ø

and( )32 2 3

1 2 3 . . .1! 2! 3!

n nnn ni i

i i i

xx c x c cc c h o tx x x-

-Dæ ö æ ö-D ¶ D ¶ ¶æ ö= + + + +ç ÷ ç ÷ç ÷¶ ¶ ¶è ø è ø è ø



where h.o.t. stands for higher order terms. Collecting terms in Equation (4.8) anddividing by t gives at point (i,n):

2 2 2 2 3 3 3

2 2 3 3 . . .2 2 6 6

c c t c r x c t c r x cu h o tt x t t x t t x

¶ ¶ -D ¶ D ¶ D ¶ D ¶+ = + - - +

¶ ¶ ¶ D ¶ ¶ D ¶ (4.10)

The right hand side of this equation is called the truncation error of the finitedifference scheme. It can be concluded that by using the scheme of Equation (4.8),another equation, containing higher order derivatives, is solved than the differentialequation (4.3). For looking more closely at the lowest order terms of the truncationerror, Equation (4.3) is differentiated with respect to t to give, for constant u,

2 2 2 22

2 2

c c c cu u ut x t t x x

¶ ¶ ¶ ¶= - = - =

¶ ¶ ¶ ¶ ¶ ¶

Substitution of this relation into Equation (4.10) gives

( )2

2½ 1 . . .c c cu u x r h o tt x x

¶ ¶ ¶+ = D - +

¶ ¶ ¶(4.11)

Referring already to Equation (4.17), it will be shown in Chapter 4 that this equationadds diffusion to the advection equation. For the example of Figure (4.5), with r = ½and x = 100 m, the numerical diffusion coefficient is 0.463 m2/s. Comparison ofFigure (4.5) with Figure (4.9) of § 4.7 confirms the magnitude of the diffusive effectof the truncation error.

Equation (4.11) shows, furthermore, that the lowest order term of the truncationerror goes to zero for r=1. The computed results for r = l indicate that a furthersubstitution will cancel out all the higher order terms in the truncation error for thisspecific value of r, to give the exact solution of the differential equation. For values r= ½ the numerical diffusion can be reduced by taking smaller and smaller values for

x. In the limit, for x > 0, the truncation error will go to zero (see Equation(4.11)) so that the numerical solution converges towards the true solution ofEquation (4.3). This convergence will not be found for the solutions with r=2. It canbe shown that for this rvalue subsequent reduction of x will give a solutiondeviating more and more from the true solution. The results become more and moreunstable. There are several ways in which this stability problem can be investigated:

1. by interpreting the dominant effects of the truncation error of the scheme;2. by showing that the values in the computational domain will always remain

between the extreme values specified at the boundaries;3. by an analysis based on the role of characteristics on the solution (Courant

FriedrichLewy or CFLcondition);4. by a Fourier series analysis of the solution function and its transformation

from one time level to another.

Ad 1) For positive values of the numerical diffusion coefficient in Equation (4.10),the solution will be dampened within the computational domain.



Negative diffusion coefficients are physically unrealistic and would lead toamplification of the solution. This would affect most strongly the shorter wavelengths as those have the largest values for 2c/ x2. In this way, small errors willrapidly be blown up to large values. This leads to the conclusion that only positivediffusion coefficients will give stable results and thus, from Equation (4.11), that rshould not exceed unity.

Ad 2) The exact solution of Equation (4.3) shows that extremes in the computationaldomain should always remain between minimum and maximum values given at theboundaries of this domain. It can be shown that the scheme, indeed, gives valueswhich are limited by the extremes at the boundaries for r 1 (Abbott, 1979).

Ad 3) Referring to Figure (4.7) and following the characteristics through point(i,n+l), the scheme will be stable when the characteristic intersects the line t=n tbetween the grid points (i1,n) and (i,n), as in this case ci

n+1 is found from aninterpolation of earlier computed results. When the intersection of the characteristicand the line t=n t is outside the line element between the points (il,n) and (i,n), thevalue ci

n+1 follows from an extrapolation of data and the scheme will becomeunstable. The time step can be compared with the time tc required to travel alongthe characteristic over a distance x. The ratio t/ tc is defined as the Courantnumber Cr. For Cr 1 the scheme will be stable and it will be unstable for Cr>1. Thiscase is shown in Figure (4.7) when, for the characteristic leaving the point (i1,n+1)backward in time, the numerical scheme of Equation (4.8) is applied at the grid point(i1,n).

Ad 4) A solution at time level n t is decomposed into the Fourier seriesjx ikikn n n jjl

j k kk k

c e epp

x x= =å å (4.12)

whereim = indicator for imaginary numbers;k = wave component number;

kn = wave amplitude at time level n t;

L = length of the computational domain along x;ii = highest grid point number over domain L.

Stability of the solution is guaranteed if, in the transformation of the solution fromtime level n t to time level (n+l) t by the scheme, none of the wave amplitudes k

n

are amplified. Substitution of Equation (4.12) into Equation (4.8) gives

11 (1 )m m m

i i ii k i k i kn nii ii iik ke re r e

p p px x

-+ ì ü

= + -í ýî þ

or1

1 mkn ik ii

k nk

A r epx

x

+

= = - + (4.13)



For a specific wave component the amplification |Ak| is defined as the length of thevector composed of the real and imaginary parts of Equation (4.13). From Figure(4.6) it can be concluded that |Ak| 1 for r 1.

Figure 4.6 Argand wave amplification diagram for the scheme of Equation (4.8)

Table 4.1 Example of unstable solution of Equation (4.8) for r=2

i => 0 1 2 3 4 5 6 7 8 9 10x (m) => 0 100 200 300 400 500 600 700 800 900 1000

n t (s)0 0 0 1 0 0 0 0 0 0 0 0 01 200 0 1 2 0 0 0 0 0 0 0 02 400 0 1 4 4 0 0 0 0 0 0 03 600 0 1 6 12 8 0 0 0 0 0 04 800 0 1 8 24 32 16 0 0 0 0 05 1000 0 1 10 40 80 80 32 0 0 0 06 1200 0 1 12 60 160 240 192 64 0 0 07 1400 0 1 14 84 280 560 672 448 128 0 08 1600 0 1 16 112 448 1120 1792 1792 1024 256 09 1800 0 1 18 144 672 2016 4032 5376 4608 2304 512

10 2000 0 1 20 180 960 3360 8064 #### #### #### 512011 2200 0 1 22 220 1320 5280 #### #### #### #### ####12 2400 0 1 24 264 1760 7920 #### #### #### #### ####13 2600 0 1 26 312 2288 #### #### #### #### #### ####14 2800 0 1 28 364 2912 #### #### #### #### #### ####15 3000 0 1 30 420 3640 #### #### #### #### #### ####

4.5 Characteristic solutions on a fixed grid

The accuracy of the scheme given by Equation (4.8) is rather low for many practicalapplications. It may be possible to choose rvalues equal or close to unity on someparts of the grid, but varying velocities may make it necessary to work with lowervalues elsewhere. Reduction of x increases rapidly the number of operations andmay demand much computer time.



Higher accuracy schemes introduce the need to define relations between more gridpoints. Programming wise this may lead to a rather complicated organisation. For arelatively clear and accurate procedure we will return now to the method ofcharacteristics. For the fixed grid, as shown in Figure (4.7), one may follow thecharacteristic line from grid point (i,n+l) back to the time level n t.

Figure 4.7. Characteristic solution on a fixed grid.

As the value for c remains constant along this characteristic, the accuracy of thesolution technique depends on the accurate positioning of the intersection point attime level n t and on an accurate interpolation. For a constant velocity u and alinear interpolation between cvalues at points (il,n) and (i,n) it is found that

11

n n ni i i

u t x u tc c cx x

+-

D D - D= +

D D

This relation is equivalent to the finite difference scheme given by Equation (4.8).The advantage of this method is that it can be extended to Courant numbers greaterthan unity (r>1; see Figure (4.7). An improvement of the accuracy is obtained byusing a cubic spline function interpolation (Preissmann and Holly, 1977). In thisinterpolation not only the values for c are used at the grid points (il,n) and (i,n), butalso their first derivatives in x. The interpolation is than given by

11 1 2 3 4

1

n nn n ni i i

i i

c cc c cx x

a a a a+-

-

¶ ¶æ ö æ ö= + + +ç ÷ ç ÷¶ ¶è ø è ø(4.14)

with2

1

2 1

23

24

(3 2 )

1

(1 )

(1 )

r r

r r x

r r x

a

a a

a

a

= -

= -

= - D

= - - D

(4.15)

Preissmann and Holly describe the computation of c/ x values via an advectionscheme. As under simplified conditions the advection does not alter the shape of thetranslated function, it must be possible to find an equation for the advection of boththe first and the higher order derivatives. For the fractioned step approachesdescribed later, this procedure, however, cannot be followed. A simpler, but also lessaccurate procedure, is found by approximating the derivatives from earlier computedcvalues as



1 1

2

n ni ic cc

x x+ --¶

@¶ D

(4.16)

The results obtained with this method of characteristics, using cubic spline functionsfor interpolation, show a rather low numerical diffusion. However, some parasiticwaves still appear in the tail of the function.

4.6 Introducing diffusion

The derivation of the equation for advectiondiffusion processes proceeds verysimilar to the one for advection alone. Referring again to Figure (4.1), the advectionthrough the element boundary has to be increased with the contribution of molecularand turbulent exchange processes that can be described by Fick's law as

dispersioncT DAx

¶= -

¶

where D is a diffusion coefficient. The balance equation now reads

( ) ( ) cAc Qc DAt x x x

¶ ¶ ¶ ¶æ ö+ = - -ç ÷¶ ¶ ¶ ¶è ø

or, assuming that the effects of spatial variation in crosssection and diffusioncoefficients can be neglected,

2

2

c c cu Dt x x

¶ ¶ ¶+ =

¶ ¶ ¶(4.17)

The contribution of molecular diffusion to D is negligible in practical applications.Furthermore, in a schematization where u and t are averaged over the crosssection,the value of D is much larger than following from turbulent diffusion alone. Much ofthe spreading of the matter along the channel axis is caused by differentialadvection. Differences in advection are caused by velocity variations in the verticalprofile, in the direction perpendicular to the flow lines, as a result of spiral flow inbends, due to storage elements along the channel banks etc. The combined effect ofturbulent diffusion and differential advection is called dispersion, so that D inEquation (4.17) generally stands for a dispersion coefficient.

Equation (4.17) has an exact solution defined as

( )

( )2

0 4½( , )

2

x utDtMc x t e

A Dtp

--

= (4.18)

where M0 is the total mass of a tracer which is injected instantaneously anduniformly distributed over the complete crosssection. The initial distribution is



defined as a Dirac delta distribution, which tends to be dispersed along the channelaxis as a Gaussian distribution with mean value =x+ut and with a standarddeviation = 4Dt .

For an initial Gaussian distribution the value of 0 can be transformed into a value,defined as the time which it would have taken to come from a delta distribution tothe given Gaussian form. This leads to

20

0 4t

Ds

= (4.19)

and the exact solution defined as

( ){ }

( ){ }( )

20

040½

0

( , )2

x x utD t tMc x t e

A D t tp

- +-

+=+

(4.20)

where M0, in this case, is the total mass of the tracer integrated along the channelsection at time t=0.

4.7 An explicit finite difference scheme

Exact solutions are defined only for very particular cases. The following limitationsof the applicability can be mentioned:

· the solution is defined only for single point instantaneous sources ;· the channel characteristics are often not constant along x.

More general, again, practical solutions are found with finite differenceapproximations, which will be demonstrate here, in the first instance, on a schemefor the diffusion part of the equation only, given by

2

2

c cDt x

¶ ¶=

¶ ¶(4.21)

Consider a grid, as shown in Figure (3.8).

Figure 4.8 Operator for an explicit scheme for the diffusion equation.



The time derivative can be given as a forward finite difference approximation in theform

1n ni ic cc

t t

+ -¶@

¶ D

The second derivative in x is approximated as

2½ ½

2

1 1

1 12

2

n n

i i

n n n ni i i i

n n ni i i

c cc c x x

x x x x

c c c cx x c c c

x x

+ -

+ -

- +

¶ ¶æ ö æ ö-ç ÷ ç ÷¶ ¶ ¶ ¶ ¶æ ö è ø è ø= @ @ç ÷¶ ¶ ¶ Dè øæ ö æ ö- -

-ç ÷ ç ÷D D - +è ø è ø =D D

Substitution of these finite difference approximations into Equation (4.21) gives thescheme, with cj

n+1 expressed in terms of variables at time level n t,

11 1(1 2 )n n n n

i i i ic rc r c rc+- += + - + (4.22)

where

2

tr Dx

D=

D(4.23)

As the value at grid point (i,n+l) is computed explicitly from known values at timelevel n t, it is customary to call this an explicit scheme. At a later stage we willintroduce a somewhat refined definition for the types of finite difference schemes. ATaylor's series development shows that, in reality, we are solving with the finitedifference scheme (4.22) the equation

2 2 42

2 2 4

1 + h.o.t.2 12

c c t c cD D xt x t x

¶ ¶ -D ¶ ¶- = + D

¶ ¶ ¶ ¶

An analysis similar to the one for the advection equation leads to the cancellation ofthe lowest order terms in the truncation error for r=1/6. However, for this same valueof r, cancellation of the higher order terms does not occur. A complete analysis willlead to the conclusion that, for this scheme, it is impossible to find any rvalue whichwill produce the exact solution.

As an example, consider a channel section of length L, where initially a conservativepollutant with concentration c is present, given as a Gaussian distribution withstandard deviation = x. Selecting a distance step x=100 m and a dispersioncoefficient D=0.463 m2/s, a time step t=l hour will give an rvalue 1/6.



Figure 4.9 Results with the numerical scheme of Equation (4.22) for various values of r.

Consistent with the theory of partial differential equations, the solution requires onepoint initial data at t=0 and one point boundary data at x=0 and x=L. In the case ofthis example, constant values c=0 have been set at both boundaries. For the secondorder space derivative of this parabolic partial differential equation, both boundarydata could also have been given as first order derivatives of c along x.

In Figure (4.9), a comparison is shown between the exact solution for this problemand numerical solutions obtained for different grid steps. With respect to the choiceof the grid steps it is interesting to note that, unlike for the advection scheme, thehighest accuracy is not found at the limit of stability. For practical applications it isbest to choose a grid which gives rvalues around 1/6. However, Figure (4.9) alsoshows that the differences are minor. This is certainly true when compared with thesensitivity of numerical parameters in advection schemes. In all cases, it is importantto make sure that the stability limit r=½ is never exceeded. The existance of suchlimit can be investigated again with a Fourier series analysis, to give anamplification factor

21 4 sin2

km xA r D= - (4.24)

so that, since the sin2term is definite positive, the stability condition |A| 1 impliesthat

½r £ (4.25) or

2

2xtD

DD £ (4.26)



4.8 Explicit schemes for the combination of advection anddiffusion

For a first discussion, let us return to the advection scheme given by Equation (4.8).Comparing the Taylor's series expansion of this scheme shown by Equation (4.11),to the advectiondiffusion Equation (4.17) it is seen that physical and numericaldiffusion are equal when

( )½ 1D u x r= D - (4.27)

In principle, advectiondiffusion processes could be computed with a scheme for theadvection equation alone, by selecting numerical parameters which just give thecorrect amount of diffusion by satisfying Equation (4.27). In practice this cannoteasily be realised as u varies in general with x and t and the grid steps cannot beadjusted from one to the other grid point. Furthermore, the numerical diffusioncoefficient has been derived from the idealized case where u and D are constant. Itis, therefore, more practical to combine a diffusion scheme with a scheme foradvection, which has a numerical diffusion which is small as compared to thephysical diffusion.A convenient approach is the fractioned step method where the contributions to thevariations in c by advection and dispersion are solved sequentially. As a first step,the equation

0advection

c cut x

¶ ¶+ =

¶ ¶(4.28)

is solved, so as to give the cvalues at (n+l) t resulting from transport alone, whileas a second step the equation

2

2dispersion advection

c c c cDt t t x

¶ ¶ ¶ ¶æ ö æ ö= - =ç ÷ ç ÷¶ ¶ ¶ ¶è ø è ø(4.29)

is solved, giving the complete solution for advection and dispersion together.Equation (4.28) can, again, be solved with the fixed grid characteristic method usingthe spline function interpolation (see Equation (4.14)), giving intermediate values cj*defined by advection alone. Subsequently, the dispersion step can be given in thefinite difference form as

1 * * * *1 1

2

2ni i i i ic c c c cD

t x

+- +- - +

@D D

or

1 * * *1 1(1 2 )n

i i i ic rc r c rc+- += + - + (4.30)



The advectiondiffusion equation can easily be extended to the twodimensionalcase, giving

2 2

2 2x yc c c c cu v D Dt x y x y

¶ ¶ ¶ ¶ ¶+ + = +

¶ ¶ ¶ ¶ ¶(4.31)

where y is the direction in the horizontal plane orthogonal to x, v is the watervelocity in ydirection and Dx and Dy are dispersion coefficients in x and ydirection, respectively. This equation can be solved, following the same principles,in the sequence of fractioned steps:

1) advection along x;2) advection along y;3) dispersion along x;4) dispersion along y.

4.9 Implicit schemes for the diffusion equation

The stability criterion for the diffusion scheme of Equation (4.22) limits the timestep of the solution method and this is not always desirable. For this reason implicitschemes can be introduced, which have less stringent stability conditions. As shownin Figure (4.10), in implicit schemes the unknown variables at the new time level(n+1) t are connected by the partial definition of the space derivative of Equation(4.21) at the new time level as

( )( ) ( ){ }1

1 1 11 1 1 12 1 2 2

n nn n n n n ni ii i i i i i

c c D c c c c c ct x

q q+

+ + +- + - +

-@ - - + + - +

D D (4.32)

where defines a weighting of the space derivative between the time levels n t and(n+1) t. This equation can be rewritten to the form

1 1 11 1

n n ni i i i i i ic c ca b g d+ + +

- -+ + = (4.33)

where

( ) ( )( ) ( )1 1

1 2

1 1 2 1 1

i

i

i

n n ni i i i

rr

r

r c r c r c

a qb qg q

d q q q- +

= -

= += -

= - + - - + -

(4.34)

As an alternative, the implicit finite difference scheme may be written in terms ofchanges c from time level n t to (n+1) t as

( ) ( ){ }1 1 1 12 2 2n n nii i i i i i

c D c c c c c ct x

q- + - +

D@ - + + D - D + D

D D(4.35)

which can be written in a form similar to that of Equation (4.33).



Figure 4.10 Grid for the implicit scheme of the diffusion equation

The system of Equations (4.33) can be solved with the double sweep algorithm, aspecial form of the Gauss elimination, based upon tridiagonal materices. In thisalgorithm, in a first sweep the coefficients in the lower diagonal are eliminated,starting with the left hand boundary condition. In a second sweep values for theunknows are computed, by subsequently eliminating the upper diagonal coefficients.

Figure 4.11 Schematic presentation of the tridiagonal double sweep matrix solution algorithm

Schematically, the algorithm is presented in Figure (4.10). Assuming the left handand right hand boundary conditions, respectively, to be of the form

0 0 0 1 0c cb g d+ = (4.36)

1ii ii ii ii iic ca b d- + = (4.37)

Gaussian elimination of the lower diagonal leads to the set of equations

1 * 1 *1

n ni i i ic cg d+ +

-+ = (4.38)

with the new coefficients in the matrix (now marked with a superscript *), computedfrom grid point 1 until ii1 by the relations

** * 1

* *1 1

; i ii i

i i

d adgg db ag b ag

-

- -

-= =

- -(4.39)



For the more general case of nonlinear equations, with coefficients derived as afunction of x, also the coeficients , and in Equation (4.33) will get a subscript i.These subscripts will than also appear in the relations given by Equation (4.39).

The elimination is started by assigning the starting values for the new coefficients atgrid point 0 by substitution of the boundary relation given by Equation (4.36) intoEquation (4.33) applied at gid point 1, giving

* *0 00 0

0 0

;g dg d

b b= = (4.40)

The same result would also have been obtained simply by setting the value of 0 tozero in Equation (4.39). In this case the assignment of new values for i

* and i*

could have started at grid point 0 instead of 1.

After completion of the first sweep, the right hand boundary value is obtained bysubstitution of the boundary relation given by Equation (4.37) into Equation (4.39)at point ii1. Successive back substitution (back sweep) of the values for c leads tothe overall solution of the system of equations.



5 De Saint Venant equations and theirsolutions

5.1 Introduction

In Chapter 3 the unsteady channel flow equations were introduced for canals withsimple rectangular crosssections. These equations will now be extended to the moregeneral case of irregular crosssections. In this approach, the flow in rivers and othertype of channels is assumed to be correctly defined by the state variables ordependent variables discharge, Q and water level, as a function of the independentvariables t for time and x for space.

Basic assumptions and/or limiting conditions are:

· the discharge is sufficiently well defined as the integral of the velocities througha crosssection, perpendicular to the xaxis and perpendicular to the flowvelocity vectors in the flood plain;

· the water level is constant along the crosssection. This implies that at any timethe water level at all points along a given crosssection should rise or fall at thesame rate. This assumption is generally justified when the widths of river andflood plain are of the same scale and free of obstacles such as natural levees orembankments;

· the water level slope or gradient in the xdirection is constant along the crosssection. However, it should be noted that in case this condition is not satisfied,correction factors may be applied to reduce significantly errors in the modelparameters, as discussed in this contribution.

Quantitative analysis of the two state variables requires two independent equations.Usually, the following equations are used:

· the continuity equation, based upon volume conservation in a control volumedefined between two successive points along the channel axis;

· the momentum equation, based upon the conservation of momentum, includingthe effect of impulses generated by forces acting upon the water contained in thecontrol volume.

Making, furthermore, the following assumptions:

· the pressure distribution in the vertical is hydrostatic;· the resistance relationship for steady flow is also applicable for unsteady flow;

and· the bed slope is moderately steep, so that the cosine of the slope can be replaced

by unity,



De Saint Venant (1871) derived the following equations (presented in a slightlyadapted form here):

sl

A Q + = qt x

¶ ¶¶ ¶

(5.1)

{ }2

2

1 0Q QQ Q+ ( ) +

gA t x A x Kz¶ ¶ ¶

+ =¶ ¶ ¶

(5.2)

where As is the crosssectional area representative for storage over a control volume(m2), t the time (s), Q the discharge (m3/s), x the position along the channel axis (m),ql the lateral discharge per unit length of channel (m2/s), A the flow conveying crosssectional area (m2), the stage or water level above a selected horizontal referenceplane (m) and K the channel conveyance (m3/s). The meaning of the parameters inthese equations becomes clear when we follow the derivation of these equations.

5.2 The continuity equation

The continuity equation for channels with arbitrary crosssection can be derived onthe basis of the same principles as applied to the case of the simplified rectangularprofile, as described in Chapter 3. It should be realised that in channels and riverswith arbitrary crosssections even the topography between two successive crosssections is irregular. For this reason, we will base the derivation of the continuityequation upon a storage volume V( ), as a function of stage .

Figure 5.1 Control volume for the derivation of the continuity equation

From Figure (5.1) it is readily seen that the following volume conservation equationholds:

lV Q + dx = q dxt x

¶ ¶¶ ¶

As for each control volume between two successive points along x, the volume is afunction of the water level, the following substitution can be made



ssurfV dV= = dx= A bt d t t t

V V VV

¶ ¶ ¶ ¶¶ ¶ ¶ ¶

(5.3)

where Asurf is the storage area in the horizontal plane defined between twosuccessive control sections. It is common practice, however, to define the storage asa crosssection property. For this reason, we usually transform the parameter Asurf toa parameter linked to a crosssection, by introducing a storage width bs of thechannel. This level dependent channel crosssection storage parameter bs is definedas Asurf (for a given level) divided by the distance between two successive controlsections (Equation (5.3)). This definition leads to the continuity equation of thecommonly used form

s lQ + qb t x

V¶ ¶=

¶ ¶(5.4)

Integration of bs over the height of the crosssection leads to the definition of thestorage crosssection

( )b

s sz

A b dzV

V = ò (5.5)

as introduced earlier in Equation (5.1). It should be realised that the storage crosssection parameters bs and As are often different from the parameters B and A used inthe momentum equation (see § 5.3). These differences will be discussed in moredetail in § 5.7.

The second important parameter in the continuity equation is the lateral flow.Examples of lateral flow contributions, either positive or negative, are:

· catchment runoff hydrographs, either given as point sources or as distributedsources. These consists both of surface water and groundwater components;

· drainage releases, for example, via pumps;· irrigation water extraction;· flow over side weirs;· water exchange at the river bed, possible linked to a groundwater storage

description;· rainfall and evapotranspiration directly linked to the water surface.



5.3 The momentum equation

For the derivation of the momentum equation we will limit ourselves here to apragmatic derivation. Recalling § 3.3, with the derivation of the momentum equationfor a uniform rectangular crosssection, one of the intermediate forms has been

2 h (uh) + ( h) + gh = 0ut x x¶ ¶ ¶¶ ¶ ¶

(5.6)

Referring to Figure (5.2), the momentum equation of the overall crosssection isobtained by integrating each term of Equation over the width.

Figure 5.2 Crosssection for the integration of the momentum equation

Integration of the first component leads to the acceleration term

j j j jo o

Q ( )dy dyu h u ht t t

B B¶ ¶ ¶= =

¶ ¶ ¶ò ò

Integration of the second component leads to the convective momentum term

22 2i i j j

o o

Q ( ) dy ( ) dy ( u Q) ( )u h u hx x x x Ab b

B B¶ ¶ ¶ ¶= = =

¶ ¶ ¶ ¶ò ò

where is the Boussinesq coefficient of velocity distribution. It corrects for the factthat we have used the average flow velocity u in the integration of the momentumcarried by the flow through the crosssection, instead of the local velocities.Consequently, the value of is greater than unity. It should be noted that theimportance of this convective momentum term is generally limited and in mostpractical applications the value of ß is simply set to unity. In some practicalapplications the value of ß is even set to zero, which means that the complete term isneglected (see diffusive wave approximation of Chapter 6).



Figure 5.3 Definition of the gravity component

Integration of the second component leads to the combined hydrostatic pressure andgravity term

j jO O

g dy g dy gAh hx x xV V VB B¶ ¶ ¶

= =¶ ¶ ¶ò ò

In the integration it has been assumed that the term (zb+h)/ x, or / x, is constantacross the section. In order to separate the effects of hydrostatic pressure gradientsand gravity, the term can be split up again on the basis of a kind of representativedepth hr above a representative bottom level zbr as follows

0br r r

rz h hgA gA gA I

x x x xV ¶ ¶ ¶¶ æ ö æ ö= + = +ç ÷ç ÷¶ ¶ ¶ ¶è øè ø

(5.7)

It can be concluded from Equation (5.7) that the way in which we define this bottomlevel does not affect the results of computations. This definition is a product of ourmind and not that of the river itself. The river feels for its flow only the effect of thewater level slope (combined with crosssectional parameters) and not that of a bedslope, as such.

Finally, a friction term has to be added. This friction description is rather empiricaland it is common practice to follow the same definition as applying to steady flow.

In steady flow, the bottom friction for sub section j of the crosssection is supposedto balance the gravity term, or

) rt obottom jj( = g h I

Furthermore, t =t (u2) for turbulent flow, leading to the Chezy relation

1/ 2oj jj = Cu h I

or to the Manning relation



2 /30j j

j

1 = h Iun

Proceeding with the Manning relationship and taking into account the side slope ofthe bottom profile, the local depth is generally replaced by a local hydraulic radiusRj, giving

2 /30j j

j

1 = R Iun

For unsteady flow, the slope expressed by this shear force relation deviates from thebottom slope and for each sub section a friction slope may be introduced defined as

2 2

2

2/31

2j j j

f 2j

j jj

Qu AI

KA R

n

= =æ öç ÷ç ÷è ø

where Kj is the channel conveyance for subsection j.

As per definition of a onedimensional schematisation If is constant over the crosssection, integration of the friction term over this crosssection leads to a contributionto the momentum equation is given as

2

f 2

QIK

= (5.8)

with the total channel conveyance K defined as

2 /3

=

= åjj

j jjj 1

1K A Rn

(5.9)

where jj is the number of sub sections in the channel crosssection.

The concept of the total conveyance of a crosssection is based on the integration orsummation of the contributions Kj for individual vertical slices of the crosssection.For each individual slice (Figure 5.2) the concept of the hydraulic radius is usedwhere

jj

j

A =RP

(5.10)

with the wet perimeter Pj representing the contact length of water and channel bed.



It should be remarked that the concept of a wet perimeter is based on the assumptionthat the shear force is equal at all points along this perimeter. This is approximatelytrue for a small subsection or slice of the channel crosssection. However, it is nottrue for the complete crosssection, as the local shear force is a linear function of thelocal depth. It is, therefore, strongly advised not to use the concept of the hydraulicradius for rather irregular crosssections. For such sections one should always workwith the concept of the conveyance, or discharge capacity K, obtained through theintegration of the local conveyance across the section.

In the special cases where the hydraulic radius concept can be used for the oveallcrosssection, the Manning and Chezy coefficients are related through

2 /31Q K I AR In

= = (5.11)

and

1/ 2Q K I CAR I= = (5.12)

giving the relation between the Manning and Chezy coefficients as

1/ 6 1/ 61 1C R ; or n Rn C

= = (5.13)

It is quite well accepted that the Manning coefficient is more constant with depththan the Chezy coefficient. Especially at relatively low flow depth, such as occurringon flood plains, it is recommended to use the Manning coefficient for the frictiondescription.

The terms of the momentum equation can now be collected to give the form

0

2r

r 2

hQ Q |Q |Q + ( ) + gA + + gA 0gAIt x A x K

b ¶¶ ¶=

¶ ¶ ¶(5.14)

Division of all terms by gA gives the equation in dimensionless form as

{ }2

r0r 2

kinematic wave

diffusive wave

dynamic wave

Q Qh1 Q Q+ ( ) + I 0gA t x A x K

¶¶ ¶+ + =

¶ ¶ ¶ 14243

144424443

1444444442444444443

(5.15)

which is equivalent to Equation (5.2). Equation (5.12) also shows how themomentum term can be simplified when some parts of this equation are of lesserimportance. These kinematic wave and diffusive wave approximations will bediscussed in § 6.4.



5.4 Numerical solutions

Hydraulic modelling techniques for 1D are currently based upon the numericalsolution of the de Saint Venant equations, including the full convective momentumterm. These numerical solutions are nearly exclusively based upon implicit finitedifference methods. These offer the advantage of unconditional numerical stability,while the various robustness problems of the past, relating to nonlinear effects andflooding and drying of channels and flood plains, have been solved satisfactorily.The application of finite elements and finite volume techniques does not providespecific advantages as, in 1D modelling, most of these techniques lead to equivalentforms derived through finite difference formulations.

Numerical methods for the de Saint Venant equations may be based upon socalledstaggered and nonstaggered schemes. The first category represents formulationswhere the dependent variables Q and are defined alternatingly at successive gridpoints along the xaxis. For nonstaggered schemes, however, the variables Q and are defined at the same grid points. At first sight this last definition offersadvantages through the availability of the state variables discharge and water level atthe same points along the channel axis. It has been shown, however, that thestaggered grid approach offers distinct advantages over nonstaggered grids byguaranteeing the convergence of numerical solutions and the better ability to handleflooding and drying of grid sections, as shown by Stelling et al. (1998).

For the numerical solution of the de Saint Venant Equations (5.1) and (5.2), we willconsider their Eulerian form per unit width of channel by first neglecting the lateralflow and further simplifying the equations to

( ) 0uht xV¶ ¶

+ =¶ ¶

(5.16)

and

0f

u uu uu g ct x x h

V¶ ¶ ¶+ + + =

¶ ¶ ¶(5.17)

where is the water level defined as = h+zb with h defined as the local water depth(m) and zb as the local bottom level (m), u the flow velocity (m/s) and cf thedimensionless bottom friction coefficient.

Figure 5.4 Staggered grid for unsteady channel flow



Referring to Figure 1 and to Stelling and Duinmeijer (2003) for further details, thestaggered grid approach requires that alternatingly at and upoints, Equations(5.16) and (5.17) are transformed into finite difference form (see also Abbott (1979),Cunge et al. (1986), Hirsch (1990) and Toro (1999)). Taking a point as the onlyfeasible choice for the transformation of the continuity equation, the finitedifference form relates three successive unknowns 1

1/ 2niu +- , 1n

iV + and 11/ 2

niu ++ defined at

the time level (n+1) t to known values at time level n t as

1 * *1/ 2 1/ 2 1/ 2 1/ 2 0

n n n n n ni i i i i ih u h u

t x

q qV V+ + ++ + - -- -

+ =D D

(5.18)

where( ) 11n n nu u uq q q+ += - + (5.19)

and

t = the time step along the taxis (s);x = the space step along the xaxis (m);

n = a superscript denoting the time step number along t;i = a subscript denoting the space step number along x;

= the time step weighting coefficient.

The symbol * in Equation (5.18) indicates that the value of h at this grid point has tobe approximated by its nearest available upstream value along the grid. For apositive flow direction, for example, this means that hi+½ is approximated by thevalue of hi.

Equation (5.18) can be reformulated as

1 1 11/ 2 1/ 21 1 1 1n n n

i i i i i i iu ua b V g d+ + +- ++ + = (5.20)

where 1i, 1i, 1i and 1i are the coefficients of the linearized implicit finitedifference scheme.

Equation (5.18) can also be rewritten to provide a choice of time step that guaranteesthe computation of positive water depths, as follows

11/ 2 1/ 2 11n n n n n

i i i i it th u h u hx x

q q+ + ++ - -

D Dæ ö= - +ç ÷D Dè ø(5.21)

For

1/ 2 1/ 20 0n ni iu and uq q+ +- +³ ³ (5.22)

Equation (5.21) shows that for these positive water velocities and for positive waterdepths, the velocity Courant number condition



1/ 21/ 2

1; orni n

i

t xu tx u

qq

++ +

+

D D£ D £

D(5.23)

can be applied to provide a sufficient condition for obtaining positive water depths atthe new time level. This implies that for the time step limitation of Equation (5.23)newly computed water levels can never fall below the bottom of the channel. As aconsequence, no artificial bottom slots or other arrangements are required to avoidnumerical robustness problems for small water depths. A similar condition can bederived for negative flow directions.

Following the same procedure as applied to the continuity equation, the momentumEquation (5.17) can now be defined at a velocity point, by the finite difference form

( )11

1/ 2 1/ 21/ 2 1/ 2 1*

1/ 2

, 0n nn n n ni in ni i i i

f ni

u uu u a u u g ct x h

q qV V++ + +

+ ++ + +

+

- -+ + + =

D D(5.24)

where the symbol * has the same meaning as in Equation (5.18) and a(un,un) is ageneralization of the discretization of the convective momentum term.

Appropriate formulations for a(un,un) follow from the physical conditions, asdetailed by Stelling and Duinmeijer (2003). In the paper it is shown that the correctformulation of the convective momentum term depends on the way in which theconvective speed of momentum is interpolated on the grid. As a rule, thediscretization is based upon a transformation of the convective momentum term to

1 ( )u uq qu ux h x x

¶ ¶ ¶ì ü= -í ý¶ ¶ ¶î þ(5.25)

and its discretization

* *1 1 1

1/ 21/ 2

1 i ii i i ii

i

u q u q q quu ux x xh

+ + ++

+

æ ö- -¶-ç ÷ç ÷¶ D Dè ø

; (5.26)

where

*1 1/ 2 1/ 21/ 2 1/ 2 1/ 2 1/ 2;

2 2i i i i

i i i iih h q qh q and q h u+ - +

+ + + +

+ += = =

Also here the value of *h has the same meaning as in Equation (5.18) The values of*ui are missing at points on the grid and are approximated by first order unwindingas

* *1/ 2 1/ 2 1/ 2 1/ 21/ 2 1/ 2, 0 , 0

2 2i i i i

i i i iq q q qu u if and u u if+ +

+

+ += ³ = < (5.27)

For positive flow direction this yields the simple expression



_

1/ 2 1/ 2_

1/ 2

i i i

i

q u uuux xh

+ -

+

-¶ æ öç ÷¶ Dè ø

; (5.28)

This momentum conservative approximation is applied in cases of gradually variedflow or in flow expansions. Referring to Stelling and Duinmeijer (2003) again, thisformulation should not be used in strong flow contractions, as this will breed energy.Instead, the energy head conservation formulation should be used, given by

2

2u uu g gx x x g

V Væ ö¶ ¶ ¶

+ = +ç ÷¶ ¶ ¶ è ø(5.29)

and, for positive values of u, the energy head conservative upwind discretization

( )2 2

1/ 2 1/ 2 1/ 2 1/ 21/ 2 1/ 2

12 2

i i i ii i

u u u uuu u ux x x

+ - + -- +

- -¶ æ ö= + ç ÷¶ D Dè ø; (5.30)

For negative flow velocity, this complete expression is shifted one grid point in thepositive xdirection. Comparison of Equations (5.28) and (5.30) leads to theconclusion that the difference in obtaining momentum or energy conservation lies inthe way in which the convective velocity of momentum is interpolated from the flowfield.

Terms in Equation (5.24) including (5.26) or (5.30) can be collected to give thegeneralized relation

1 1 11/ 2 12 2 2 2n n n

i i i i i i iua V b g V d+ + ++ ++ + = (5.31)

By successive elimination of all equations at upoints, the remaining set is given by

1 1 11 1

n n ni i i i i i ia V b V g V d+ + +

- ++ + = (5.32)

and is solved by applying elimination or conjugate gradient techniques, as discussedfurther down in this contribution.

The numerical discretization is second order accurate for all terms of the de SaintVenant equations, except for the convective momentum term, which is first orderaccurate. For many practical applications the convective momentum term is of lesserimportance and this lower discretization accuracy is quite acceptable then. However,nearly second order accuracy can be achieved by up winded second orderextrapolation of uvalues, combined with slope limiters, as described by Stelling andDuinmeijer (2003), again.

In rapidly varied flow the convective momentum term becomes locally dominant.Modelling these types of flow requires an appropriate implementation of theconvective momentum term. By doing so, Delft Hydraulics’ SOBEK package has



been enabled the highly accurate and robust modelling of phenomena such assupercritical flow in steep channels and moving hydraulic jumps.

The numerical scheme given by Equations (5.20) and (5.31) only provides linearequations at internal grid points. At channel boundaries, additional equations arerequired. As the de Saint Venant equations are of second order hyperbolic type, oneboundary condition is required for each of its characteristic lines entering thecomputational domain. Boundary condition requirements for 1D river models werediscussed extensively by Cunge, Holly and Verwey (1986). Here we will limitourselves by stating that, in most practical applications, inflowing discharges arespecified at the upstream ends of channels entering the flood model domain andwater levels or rating curves at channels leaving the model domain. At internalboundaries, such as channel junctions, usually a modified continuity equation isapplied, jointly with water level compatibility at all channel boundaries at thatjunction.

5.5 Description of hydraulic structures

Nearly all flood simulation models are dealing with hydraulic structures, such asdams, weirs, bridges etc. The length of the hydraulic structure along the xaxis isusually supposed to be negligible on the scale of the river or channel and thereforethe structure can be seen as one single point along x, where both water and energylevels are discontinuous and the de Saint Venant equations do not apply. At thispoint a relation is established between the upstream water level, the dischargethrough the structure and the downstream water level. In this relation it is assumedthat the upstream water level is taken at the nearest point just upstream of thestructure, where vertical accelerations can still be neglected. Similarly, thedownstream water level is taken at the nearest location just downstream of thestructure, where the flow can be seen as nearly horizontal and the structure flow nolonger contributes to the energy dissipation. Usually, it is also assumed that storageof water around the structure is negligible, so that the relationship is applicable atany moment in time. On this basis, there is a variety of ways in which the flowthrough hydraulic structures can be formulated:

· in most cases hydraulic structure descriptions are based upon empirical lawsrelating the discharge through the structure to the upstream and downstreamwater level. The relation has the general form

( ),crest up downQ Q V V= (5.33)

where up is the water level upstream of the structure (m), Qcrest the dischargethrough the structure (m3/s) and down the water level downstream of thestructure (m). This formulation includes the possibility of using energy levelsinstead of water levels. Equation (5.33) may be linearized to

up crest downQa V b g V d+ + = (5.34)



where , , and are the local values of nonlinear coefficients at a given stateof the flow. A wide variety of structure descriptions is available from literature,e.g. the classical books of Chow (1959), Henderson (1966) and, more recently,Chanson (1999);

· as an alternative, the state of the flow in the section just upstream of thestructure, where the flow is contracted and vertical accelerations occur, can bedescribed by an energy conservation principle. Similarly, the state of the flowjust downstream of the structure, where the flow expands with energy lossesassociated to it, can be described by a momentum and impulse balancerelationship. By internal elimination of unknowns in these relationships, thewater level at the structure crest or at its most contracted section can beeliminated and a relation similar to Equation (5.34) is obtained;

· in cases where it is difficult to define the state of the flow in terms of equations,laboratory experiments may be set up to define a matrix relating upstream anddownstream water levels to the structure discharge. As an alternative, thestructure may be modelled in detail by a 3D numerical code, leading to a similarset of matrix coefficients. Conditions are that a fine grid is used, and that thecode is based upon an appropriate numerical description of convectivemomentum terms and the effect of turbulence.

In all cases shown above, the structure equations could be based upon energy levels,instead of water levels. Through suitable transformations, these relations can bereworked to the form of Equation (5.34) Similar descriptions can be made for localenergy loss descriptions along a channel.

Appropriate linearization of Equation (5.33) or of the other structure flowdescriptions, leads to a numerical scheme of the form of Equation (5.31). In the totalset of equations for channel flow this structure relationship replaces the momentumequation that is generally applied between successive points.

In the case of a closed structure or a discharge specified by a pump, the coefficients and of Equation (5.31) are both zero and the structure presents itself as an

internal boundary condition with a discharge given. In the case of free flow in thepositive xdirection, only the coefficient equals zero, showing also in thenumerical relationship that the structure discharge is only dependent on the upstreamwater level.

In literature an abundant number of structure equations can be found (Chow, 1959,Henderson, 1966 and, more recently, Chanson, 1999).Implementing these in a numerical code requires attention for dischargecompatibility between the various flow states, especially at the transition betweenfree and submerged flow. Incompatibility in the definition of the discharge may leadto oscillations in the computed flow, in particular at the moment of flow reversal.Problems may be suppressed by defining some inertia to the structure flow. This isimplemented by adding a small term to the coefficients and . The numericalstructure behaviour is generally better when applying an energy relationship in theaccelerating flow section upstream of the structure combined with a momentumimpulse relationship for the section downstream of the structure.



Where appropriate, energy losses may be added to the description of flow in thecontracted section. Internal elimination of local variables leads, again, to arelationship as given by Equation (5.31) . In all cases, the different flow states onlyaffect the computation of the coefficients , , and and have no bearing on thesolution algorithm of the resulting system of equations.

5.6 Topological model schematisation

Mathematical models for the simulation of flow in river and channel networks arebased upon the De Saint Venant equations, hydraulic structure equations and initialand boundary conditions. The modeller has to simplify the complete system to achoice of schematization objects which will give a satisfactory representation of thephysical behaviour of the hydraulic system. Two processes are of primaryimportance: storage and conveyance.

The actual model construction will start with the selection of the network branches:the topological schematisation. In this process it will be decided up to which level ofdetail network branches are to be included in the schematization. If too many minorchannels are included, the amount of work in data preparation may be unnecessaryhigh. In addition, model simulations may become too slow. The practice of goodmodel construction requires a good insight into the relative importance of thesecontributions.

Starting with conveyance, let us discuss the following issues:

· what is the relative contribution of a channel to the overall conveyance of thesystem? For a good judgement, the Manning equation shows that if two channelsgive flow into the same direction, their relative importance is found bycomparing their conveyance. Applying the Manning concept, Equation (5.11)shows, for example, that for similar roughness, a channel with half the width anddepth of a reference channel, has only 16 % of the discharge capacity of thatreference channel;

· can a minor channel form a short cut? Not only the conveyance of a channel isimportant. Certain channels may provide a short cut to the flow from one part ofthe network to the other. Even for small conveyance values the discharges maybecome substantial due to a steep water level slope. Special attention has to begiven to a possible over bank flow at higher stages;

· is the water level always the same all over the channel crosssections? If this isnot the case it may be decided to split the channel up in two or more parallelbranches and describe cross flow between the parallel channels via crosschannels or hydraulic structure descriptions. In general, such schematisationslead to significantly more complex model networks.

This last point is also important for the correct representation of storage in themodel. Natural levees along rivers may delay the storage in the flood plain andhence, affect the celerity of flood wave propagation along the river and thedampening of flood wave peaks.



Other questions addressed are:

· is a storage area well connected to the network? This means, for example, that itcan be fed with the correct conveyances by flow from all relevant directions;

· are there any barriers in the flood plain? If part of the flood plain is schematizedas a storage element, it may contain various embankments or levees. Examplesare levees raised around paddy fields. During floods, part of the describedstorage area may not get flooded at all.

Finally, lateral flows may have their impact on the topological schematisation asfollows:

· should we extend the schematisation to include part of a tributary? For example,how far does the backwater influence the tributary and does this effect justify theinclusion of part of the tributary in the topological schematisation? If ameasuring station is available to provide the lateral flow time series, it may bequite far from the main river. In this case the schematisation may be extended toinclude this location;

· should we couple other models to our hydrodynamic model? For example, theexchange of surface water with the sub soil may be substantial or it may at leastbe important in relation to the objective of the model development. Rainfallrunoff models may become an integrated part of the total description. In floodwave propagation the role of infiltration of surface water into the sub soil mayhave to be included in the model.

5.7 Hydraulic model schematisation

Once decisions have been made about the channel network schematisation and itsvarious model elements, choices have to be made about equations and theirassociated data. In most cases, channel flow will be based upon the description ofthe full De Saint Venant equations, although also mixed models may be constructed.It could make sense, for example, to describe the flow in upstream, steep riverbranches with hydrologic models or artificial neural networks (ANNs), whereas inthe principal rivers the full De Saint Venant equations are applied. In all cases,however, the models must include representative values for channel storage andconveyance, whether these are specified directly or hidden in the parameters of ahydrologic model. In principle, the model must compensate for conveyance orstorage which has been neglected in the topological schematisation.

One important decision is the choice of distance step x, which should be basedupon a good representation of the hydraulic processes. The choice is based upon:

· The wave length modelled. A rule of thumb is to model waves with at least 50 to100 grid point along important wave components;

· The representation of the local variations in hydraulic parameters, such assudden contractions and expansions;

· Placement of grid points closely around hydraulic structures or otherdiscontinuites in the system.



Channel conveyance is described at crosssections. Under all circumstances thesecrosssections must be lines perpendicular to the direction of mean velocities. In thecase of meandering rivers these crosssection alignments have to be defined on thebasis of sound engineering judgement. A complicating factor is the change ofvelocity vector directions with changing stage.

For manmade channels the specification of crosssection parameters is ratherstraight forward. As the dimensions are usually quite accurately known, the mostsensitive input parameter is the roughness coefficient. For simple channelgeometries, the use of the hydraulic radius as an approximation of a representativedepth is quite acceptable. The hydraulic radius is defined as the surface of theconveying crosssection divided by the wet perimeter (Equation (5.11) or (5.12)) andassumes a uniform distribution of the shear force along this wet perimeter. Forcrosssections with varying depths this assumption is incorrect. The local shear forcein the crosssection is a linear function of the local depth. For this reason, theconveyance of compound channels has to be based upon a summation of theconveyances of individual subsections where for each of them this shear force ismore or less uniform (Equation (5.9)).

An advantage of such integration is that each sub section can be given its ownroughness value. Another advantage is that by keeping track of the individualcontribution of the sub sections to the overall conveyance of the channel, computeddischarges can be redistributed across the section to provide water velocities foreach sub section. This ability may be important in simulating water quality ormorphological processes in flood plains where the use of local velocities is required.In the integration, contributions of sub sections with small conveyance are withoutany loss of accuracy added to those of the other sub sections. There is no need toneglect the conveyance of shallow and highly resistant parts of the crosssections asthese may still give substantial contributions during floods or may be important inthe overall assessment of the morphologic or water quality behaviour.

One of the basic assumptions in a onedimensional schematisation is the use of aconstant water level slope all along the crosssection. In meandering channels,however, the slope may be quite different for different sub sections. One way toinclude this effect in the derivation of the conveyance is by giving each sub section aweight in the integration, as a function of the distance to its equivalent part in thenext crosssection. Equation (5.11) will then be modified as follows

( )( )

( )( )

½ ½2/3

½ ½

jj

j jjj 1 j

1Q K A Rnx x

V V

=

D D= =

D Då

or

( )2 /3

½/

jjj j

j 1 j j

A RK

x xn=

=D D

å (5.35)



where x is the distance for which the crosssection parameters are representativealong the xaxis and xj a similar value for sub section j.

Storage parameters usually are given as a function of stage and linked to crosssections. For uniform channels, as often designed in irrigation and drainage systems,the storage width is equivalent to the flow width. Storage width data are simplyextracted from crosssection information. In natural rivers, however, meandering andirregular flood plain topography requires more complex procedures. § 5.2 gives anextensive discussion of the correct definition of the storage width parameter bs. Itfollows that storage parameters have to be extracted from information provided bytopographic maps, currently mostly available in the form of digital elevation models(DEMs) in GIS. For successive compartments along the river axis, approximate waterlevel slopes have to be assumed to extract from these maps storage area as a function ofstage. Division of these areas by the length of the compartment along the xaxis,provides the parameter values for bs. For models of lesser economic value, proceduresmay be simplified, if needed.

Storage connected to channels which have been neglected in the topologicalschematisation must be added to the model as additional storage. In the crosssectionsthe compensation can be made in the form of additional storage width. It can also beintroduced in the schematisation in the form of additional storage areas at nodes. It isrecommended to keep track of the changes that this correction phase has given inparameter values derived during the primary phase of schematisation. It is alwaysadvised to keep good records of all the steps taken in the development of a model andthe way the model parameters have been defined. Without such records it is practicallyimpossible to introduce future improvements or modifications correctly. This is evenmore so when various persons are involved in the model development and/or thedevelopment takes place over a considerable length of time.

Hydraulic structures may be described by their empirical relationships, by theapplication of an energy conservation principle upstream and a momentumconservation principle downstream or by specifying the discharge water levelrelationships in matrix form. In case of empirical relationships, care must be taken thatthe parameters describing free flow and submerged flow, successively, give aconsistent computed discharge at the moment of transition from one flow state to theother. If applicable, additional entrance energy losses must be specified. Compositecrosssections of the structure require an adaptation of the topological schematisationby defining parallel structures. Similar to the definition of conveyance, the totaldischarge through the composite structure is defined by adding up the contributions ofthe individual structure components.

5.8 Boundary and initial conditions

In relation to boundary conditions two issues are important: the location of a modelboundary and the condition applied at this point. The location depends on the modelobjective. For models meant to study hydraulic problems, a strict rule for a boundarylocation is that the boundary data given at this point should not affect the outcomecomputations of various scenarios.



In other words, the model changes introduced with these scenarios and the resultingchanges of the hydraulic state at the boundaries of the models should not reflectback on the study area. Exceptions to this rule are rare, but may be justified in somecases of water quality modelling, for example.

There are various ways in which this can be achieved. Water levels at a seaboundary of a river model are generally not affected by the river discharge. Usually,it is safe to use a sea boundary as a boundary location in a river model and providewater levels as boundary condition.

In other cases it must be guaranteed that the downstream river boundary is located ata sufficient distance from the area studied. How far, depends on how an error, or achange in the downstream boundary condition, affects the area under study. In thefirst place, this depends on the magnitude of the error or change. In the second placethis depends on the hydraulic characteristics of the river. A good measure for therequired distance is the length of a backwater effect. In case of doubt it isrecommended to do a sensitivity analysis on the effect of an error in the downstreamboundary condition.

At upstream boundaries the same reasoning applies. In this case the scenario appliedshould not reflect back on the data or condition applied at the upstream boundary.Also here, the required distance follows from an analysis of backwater effects.

The type of boundary condition follows from the theory of characteristics, statingthat one boundary condition has to be given for each characteristic entering thecomputational domain. In practice, however, this always means that one boundarycondition is given at each boundary of the model. For sub critical flow this is usuallya discharge hydrograph at an upstream end and water levels or a rating curve at adownstream end.

A rating curve at the upstream end will usually lead to model instabilities as suchcondition will be nearly equivalent to the information contained in the channeltopography. In this way, this boundary condition does not provide additionalinformation that is not already available and the state of the flow remainsundetermined. A rating curve conflicting with the topographic information leadsmathematically to indeterminacy of the solution and hence, to instabilities.

At downstream boundaries, usually water levels or rating curves are given. In riverapplications one could define this as a weak boundary condition, as it will be shownin Chapter 6 that the wave propagating in a river is usually an advection typephenomenon, only requiring an upstream boundary condition. This implies that forthe use of the full hydrodynamic equations the downstream boundary conditionbecomes redundant.

For similar reasons, it is sufficient in practice to give only one upstream boundarycondition for super critical flow. The other condition is already implied in therelation between topographic data and the water level slopes following from this.



At the downstream boundary, any condition given, will at worst lead to incorrectresults over a small stretch of the downstream region. As a rule, the downstreamboundary condition given will turn the outflow locally into a state of sub criticalflow.Initial data are simplest supplied by starting up flow simulations on a dry bed or alow initial water depth. In the case of tidal models this specification is replaced by aconstant mean sea level initially and zero discharges throughout. In tidal flow,usually two tidal cycles are sufficient to arrive at a consistent set of initial data, atleast in terms of water levels. Most modelling systems provide the option ofgenerating a set of consistent initial data and writing this state to a file to be used asa socalled hotstart file for further simulations.

5.9 Model calibration and validation

Calibration of a model is the process of removing, or at least reducing, theuncertainties in the choice of model parameter values. In principle, nearly all modeldata can be collected from the field, except for the roughness values. Theseparameter values can only be obtained indirectly by comparison of measured andcomputed variables, such as water levels and discharges. In principle, a modelcalibration could focus only on the optimization of roughness coefficient values.

However, in practice there are uncertainties also in relation to crosssection data,flood plain topography, lateral flow data and inflowing hydrographs. This meansthat in a calibration also the correctness of storage parameters and measuredhydrographs will have to be checked and possibly corrected. Part of the uncertaintyis also caused by the schematization of the real system into the model. Neglectedstorage and conveyance and an inappropriate compensation for these missing effectsalso influence the calibration results.

It is possible, then, that corrections are applied to the wrong parameters and still leadto reasonable calibration results. As will be shown in Chapter 6, flood wavepropagation celerity and dampening both depend on storage and conveyance. Anerror in the storage could, therefore, be corrected through the channel resistance andstill improve calibration results. It is, however, not certain that the sameimprovement is maintained under extrapolated conditions.

For river channels, calibration of roughness data should proceed from lowerdischarges to higher discharges. Only after the roughness values in the main channelhave been calibrated, those of the flood plain should be determined.

Model validation is meant to provide confidence in the quality of a calibration. Inprinciple, model validation should be applied on a case representing an extrapolationof the calibration events. Many models are developed for use beyond the range ofdata for which they could be calibrated. Flood models, for example, often are usedfor studies on flood frequencies of several hundreds of years and it cannot beexpected that data of such events is available for model calibration.

Moreover, one of the principal problems with model calibration is that results thathave been measured during low frequency events, are not very reliable.


WL | Delft Hydraulics 1 0 0

Rating curves for these low frequencies are rather unreliable as measurementconditions are difficult and the staff of catchment authorities generally have otherworries than getting the most accurate discharge measurements during an extremestorm event.

Only water levels have a reasonable chance of being reliable, also during extremeconditions. A possibility, therefore, is the extrapolation of rating curves on the basisof the use of local 2D models, based upon an accurately measured topography andcalibration of roughness values during the higher frequency events.

However, this approach is not common practice yet. On the other hand, PublicWorks in The Netherlands uses detailed 2D models of the flood plains of theprincipal rivers for their calibration of 1D models. These 1D models can than beused with more confidence in the range of extreme events. It should be realised thatthis procedure is feasible only if these 2D models are based upon very small sizegrid cells.



6 Mathematical modelling of floods

6.1 Introduction

All over history, floods have challenged scientists and engineers to master thisphenomenon, both by developing mathematical descritions and by taking floodprotection measures. Over the past decades, these efforts even have increased. In thefirst place, because a rapidly increasing world population has shown a strong driveto settle in flood prone coastal zones and river flood plains. In addition, there is anincreasing awareness about climate changes which appear to bring more and moreprecipitation to river catchments. There also are numerous intellectual challenges, asscientific and technological developments in a wide range of fields open up manynew ways of supporting flood related studies. In particular, important progress hasbeen made in computer speed and in new measuring technologies.

Although numerical methods for solving unsteady flow equations were developedalready half a century ago, e.g. Arakawa (1966), it was only recently that numericaltechniques behind flood simulation models have reached an acceptable level ofperfection. Hydraulic engineers and hydrologists are now able to model flow inchannels and over flood plains, irrespective of their bathymetric or topographiccomplexity; irrespective of the number and location of embankments for floodprotection, roads and railways and irrespective of the number and complexity ofhydraulic structures and the way we control these.

6.2 Flood model requirements

Flood simulation models may have different requirements, depending on theirobjective. Criteria for the selection of the appropriate tool are often based on:engineering staff time needed for model development, overall consultancy time forproduct delivery, speed of computation, completion time for a simulation, accuracylevel of results, data requirements, numerical robustness, userfriendliness of thesoftware and possibly others, depending on the objective of the model. Theseobjectives may be related to flood risk analysis, flood forecasting, flood control andbe based upon a variety of causes, such as storms, dam or dike breaks, hurricanes,typhoons or similar low atmospheric pressure phenomena. Recently (2004) attentionhas also been drawn once again to the devastating effects of tsunamis. All theseapplication areas of numerical models have their own requirements, as will bediscussed briefly in the sequel.

In many countries, insurance companies are using flood risk maps, sometimes basedupon relatively simple and quick estimates obtained via simple rules in GIS. In thesecases it is assumed that the value of insured property does not justify the morecomplex laws defining the detailed flow of water.



If the economic interest is greater, a first improvement is found by applying 1D(onedimensional) steady flow models with GIS post processing to developtopography based flood frequency contour lines (e.g. FEMA procedures:www.floodmaps.fema.gov/fhm/). However, there is a tendency now to base floodrisk analysis on more detailed combined 1D and 2D unsteady flow models for floodprone areas with valuable assets and a complex infrastructure. Federal and localgovernments have also become more aware of the potential of using such models forevacuation planning. In The Netherlands, for example, more than 60 % of thecountry is subject to flood risk and for most of these areas integrated 1D and 2Dhydrodynamic models have been developed to study the effects of potential dikebreaks and to provide guide lines to authorities in setting up evacuation plans.Besides producing flood depths, these models have to be capable of providingaccurate estimates of flood wave propagation celerities over dry beds.

Flood forecasting sets quite different requirements. Speed of producing a forecast isone of the most important criteria, especially in areas where flash floods occur. Forthis reason, numerical models behind a river catchment flood forecasting system areusually 1D hydrodynamic models, gradually replacing the simpler hydrologicalrouting techniques. There is a tendency to include partly 2D hydrodynamic models,which is already common practice in flood forecasting systems for coastal areas andseas. Numerical models for flood forecasting are usually embedded in a floodforecasting platform, such as the Delft FEWS system (Werner et al., 2004), whichhas recently been installed in the UK to provide flood forecasts for nearly all riverbasins in the country.

Important criteria for numerical models supporting flood control are accuracy,flexible schematization options, numerical robustness and consultancy time formodel development and use. Currently, stateoftheart for flood control is the use ofcombined 1D and 2D models (e.g. Hesselink, 2003). The former use of flood cellshas been replaced by complete 2D flow descriptions, whereas subgrid channel flowis still better described in 1D. Flood control models should be based upon reliablephysical descriptions and schematizations, as part of their use is in extrapolation ofcalibrated models to extreme situations which have never occurred. One of thereasons to build models for flood control is the study of downstream impacts,especially cross border effects. Downstream impacts of flood control are changedflood wave celerity and changed flood peak attenuation. Higher flood wavecelerities result from deepening of the river and the construction of embankments.This, in turn, leads to increased peak floods downstream. The construction of floodretention areas has opposite impacts and may be used to compensate the negativeimpacts. Model selection criteria then follow from the detail in which potentialeconomic, environmental and social impacts have to be studied.

The analysis of floods caused by dam and dike breaks requires extremely robustnumerical methods, especially for the description of flooding of dry areas and thecorrect propagation of the wave front. Moreover, model accuracy, partly based uponthe ability to describe the full hydrodynamic equations, is important, as will bediscussed in the section on software and model validation. As dam and dike breaksimulations are nearly always made for the prediction of their potential effects, datafor model calibration is rarely available.



The quality of the model fully depends on its descriptive capabilities of the physicalsystem in terms of topographic and roughness data, the representativeness of theequations and the numerical methods applied. However, it has to be kept in mindthat the overall model accuracy also follows from the quality of the description ofthe dam failure mechanism and the assumptions made here.

Floods generated by the passage of low atmospheric pressure zones such ashurricanes, typhoons and the geologically induced tsunamis require the modelling of2D flow in coastal zones, seas and oceans and may set requirements such as thedescription of Coriolis forces, the use of spherical coordinates and curvilinear grids,the specification of moving atmospheric pressure fields, special ways of handlinginitial data etc. A possible integrated use of 1D, 2D and 3D models may provideadvantages here.

6.3 The role of new data collection technologies

Models need good quality data if one wants to draw reliable conclusion from theiruse. In addition to developments in numerical methods, new technologies for thecollection of data have recently led to complete changes in the selection of the typeof numerical models. In particular, the development of GPS and DGPS technologyhas led to far cheaper methods of collecting bathymetric and topographic data(Moglen and Maidment, hsa025). In turn, this has led to the gradual replacement ofthe hydrograph based hydrologic models by the bathymetric and topographic basedhydrodynamic or hydraulic models. Similarly, the collection of detailed digitalterrain data in river and coastal flood plains has led to the replacement of 1D modelsby 2D models.

Let us first consider the impact of LIDAR. The use of this laser technology,scanning the earth surface with laser beams from airplanes or helicopters, hasprovided the means to generate highly accurate digital elevation models at relativelow cost. As an example, the whole area of The Netherlands has been remappedover the past years, with an accuracy of approximately 10 cm in the vertical at adensity of 1 point per 16 m2. Total cost of this project was approximately 10 millioneuro, or approximately 250 euro per km2 (Verwey, 2001). This new topographicinformation has been essential for the flood risk analyses and the evacuationplanning studies mentioned earlier.

Also river bathymetries are obtained at relative low cost now by boats equipped withmultibeam echo sounders. Also here, the position of the boat is recorded via DPGS,while the spatial sound signals record bottom depths relative to the boat at anaccuracy of approximately 10 cm in the vertical. At a typical boat speed of 4 m/s anda swat width of the order of magnitude of the river depth, the bathymetry of quitelarge river beds can be obtained in a few days. Experiments are also being madewith laser beams in the green range, as these are able to pass through clean waterand enable the application of LIDAR technology also for river bathymetries.

In similar ways other technologies are advancing rapidly enabling large amounts ofdata to be collected at relatively low cost.



Worth mentioning are the ADCP (Acoustic Doppler Current Profiler) technology fortidal discharge measurements and the increased precision of spatially distributedprecipitation measurements with radar.

6.4 The nature of flood wave propagation

The set of Equations (5.1) and (5.11) forms the socalled kinematic waveapproximation for flood propagation, if the slope I is taken as the bed slope. Aftersubstitution of Equation (5.11), or the kinematic wave part of Equation (5.15), intoEquation (5.1) and neglecting the lateral flow term, a further simplication isachieved of the form

0Q Qct x

¶ ¶+ =

¶ ¶(6.1)

with a flood wave celerity c (m/s) expressed as

1

s

dQcb dh

= (6.2)

It is important to realise that, as an essential assumption in the derivation of thiskinematic wave form, the channel bed slope has been used in the conveyancerelationship. Equation (6.1) has the form of an advection equation which expressesthat flood waves propagate with a celerity c which is inversely proportional to theavailable channel storage width bs (m) and a linear function of the derivative of thelocal flow rating curve. The characteristic celerity of this kinematic wave is lowerthan the celerity of the dynamic wave characteristic in the same direction. Asdiscussed by Abbott (1979) it is this mechanism that leads to roll waves at floodwave fronts, limiting their propagation speed (see also Stoker, 1957).

The first order partial differential equation (6.1) also expresses that along itscharacteristic celerity c the discharge remains constant, and so does the peak of theflood wave. In other words: there is no dampening effect of the flood peak. Thoughthis is approximately true for rivers with steep slopes, the stretches with milderslopes require a lesser simplification of the De Saint Venant equations. Following,for example, Chaudhry (1993) and defining I of Equation (5.11) as the water levelslope / z, substitution of the last three terms of Equation (5.15) into Equation (5.1)gives the socalled diffusive wave approximation

2

2

Q Q Qc Dt x x

¶ ¶ ¶+ =

¶ ¶ ¶(6.3)

with a flood wave diffusion coefficient D (m2/s) derived as

2 s

KDb I

= (6.4)



Including the lateral flow term the diffusive wave approximation reads

2

2

Q Q Qc D cqt x x

¶ ¶ ¶+ = +

¶ ¶ ¶(6.5)

Returning to Equation (5.15) again, the variable I represents the bed slope in thecase of the kinematic wave approximation and the water level slope in the case ofthe diffusive wave approximation. The description based upon the full set ofEquations (5.1) and (5.15) is defined as the full dynamic wave description.

6.5 Deformation of flood waves

6.5.1 The role of varying celerities

Returning to Equation (6.2), it is readily seen that for constant celerity at all waterlevels, there is no deformation of the flood waves. The waves are just translatedalong the river axis. However, in practice the wave celerity c generally increaseswith increasing river discharge, especially when the river banks are steep (Figure6.1a). Moreover, when the flood waves arrive at bank level, the storage width oftenincreases more rapidly than the increase in dQ/dh, resulting in a temporary dip inthis celerity function (Figure 6.1b). When the water level rises further, the celeritiesstart increasing again.

Figure 6.1 Flood wave celerity for various crosssection shapes

The higher celerities for increasing discharges cause a steepening of the wave at thefront and a stretching of the falling limb of the flood hydrograph (Figure 6.2). This isthe most common form of flood wave deformation in natural river systems.

Figure 6.2 Flood wave deformations resulting from celerity varying with stage



6.5.2 The role of the diffusion term

Equation (6.3) shows that along the characteristic line c=dx/dt the flood peaks aredampened by the integral of the diffusion term. As the diffusive wave approximationof the De Saint Venant equations may include other effects as well (e.g. the form ofEquation (6.5)), extending the number of terms of the right hand side, we willdiscuss these various contributions in a fractioned step approach.

Returning now to the right hand side terms of Equation (6.3), the influence of thediffusion fraction can be written as

2

1 2

dQ Q( = D)dt x

¶¶

(6.6)

which gives the equation for flood peak attenuation as

2

1 2peak

Q = D dtdQx

æ ö¶ç ÷¶è ø

(6.7)

The diffusion mechanism is commonly explained by the fact that the water levelslope during rising water level is steeper than the bed slope. Consequently, thedischarge at the wave front is higher than might be expected from the rating curveand the water balance in that section forces a lowering of the wave peak. This isfurther enforced by the lower discharges at the rear of the wave as compared to thedischarge which would follow from the stagedischarge relationship. Here, the waterlevel slopes are smaller than the bed slope (Figure 6.3).

Figure 6.3 Illustration of the effect of varying water level slopes on flood wave deformation

Returning to Equation (6.7), it is readily seen that this fraction gives a dampening ofthe wave peak, as the term ¶2Q/¶x2 is negative, while it leads to an increase of thedischarge at the front and at the rear of the wave, where ¶2Q/¶x2 is positive.

Observation of the diffusion coefficient given by Equation (6.4) leads to aninteresting paradox. It is a well known phenomenon that an increase in storage alsoincreases the dampening of flood waves. This, however, does not followimmediately from Equation (6.4), which shows a decrease in the wave diffusioncoefficient with increasing storage width of the river. It should be realised that anincrease in storage width also decreases the flood wave celerity, leading to aproportional decrease in the flood wave length. Consequently, the contribution of the



term ¶2Q/¶x2 shows a quadratic increase in the wave dampening. Furthermore, thedecrease in the wave celerity increases the travel time, or the time over whichEquation (6.7) is integrated, over a given stretch of the river, compensating for thedecrease in the diffusion coefficient. Taking all these effects into account, anincrease in the storage width by a factor two, gives a total increase in waveattenuation of a factor four.

6.5.3 The role of the lateral flow terms

Including also the effect of lateral flow, Equation (6.5) shows its influence.Applying, again, the fractioned step approach, the contribution of the lateral flow tothe change in peak discharge is given by

2dQ( = c q)dt

(6.8)

or

2 peak peakdQ cq dt q dx= = (6.9)

Equation (6.9) simply tells us that the peak value of the flood wave is increased bythe total lateral flow added to the flood as its peak travels along the river.

Interesting conclusions can now be drawn on the case where flood water infiltratesinto the soil as the flood passes the flood plain. As the infiltration is a loss of water,the lateral flow is negative and leads to a reduction of discharges along the pathsgiven by the characteristics. A typical infiltration function along the flood wavewould show the highest water losses during the passage of the front of the wave,which shows up as a pronounced effect when the bed material is highly permeable.A high porosity of the bed material, gravel for example, causes a rapid increase inthe infiltration when water is made available by the wave. During the further rise ofthe water level, the stream width increases and increases the storage volumeavailable for infiltration. At the passage of the peak, the infiltration is more or lesscompleted and the function drops down to zero.

A typical response of this infiltration is a pronounced steepening up of the floodwave front. This is why such floods may have a very short lead time for warning thepopulation and may be quite destructive. A typical example is given by floods inwadis. However, also in more moderate climates this effect is known. Note that it isnecessary in such cases to include in a model the effect of exchange with thegroundwater, as the typical wave deformation found in this case cannot becalibrated by modifying friction parameter values.

6.6 Link to hydrologic flood routing models

Equations (6.1) and (6.3) are rarely used in discretized form anymore. However,they are useful for the flood modeler as to provide insight into the physical nature offlood wave propagation. Moreover, they link hydrologic and hydraulic flood routing



techniques. By applying a Taylor’s series expansion to the Muskingum equation,Cunge (1969) has demonstrated that the wellknown hydrological Muskingum floodrouting technique emulates the solution of the advectiondiffusion Equation (6.3).This insight provides guidelines for a suitable choice of the Muskingum parameterson the basis of the expressions for c and D (Equations (6.2) and (6.4), respectively).

The numerical Muskingum method belongs to the class of hydrological routingtechniques. These methods are based upon the notion that flood wave propagationcharacteristics can be derived from measured flood hydrographs along the river,rather than from detailed topographic information, as discussed in the WMO reporton flood forecasting models by Serban et al. (2005). The basis of these methods isthat flood wave propagation and diffusion behavior can be represented by a limitednumber of parameters, which can be calibrated from observed hydrographs. Usuallythere are only two parameters, as in the case of the Muskingum routing method. Thelimitation of this assumption is clearly shown by analyzing the stage dependentexpressions for celerity and diffusion given by Equations (6.2) and (6.4). Althoughthe current methods for flood routing are far more precise, there are still manysituations where the use of hydrological flood routing techniques is still justified. Inmany river catchments, and especially in tributaries, the collection of detailedinformation on river bathymetry and flood plain topography is economically notalways justified, despite the emergence of relatively cheap new technologies.

An alternative to hydrological forecasting methods is provided by newly developedartificial neural network (ANN) concepts (Minns & Hall, hsa018). Also thistechnology relies on measured hydrographs to derive relationships betweeninflowing and out flowing hydrographs, though in practice ANN’s are mostlyapplied to the development of relationships between rainfall and river catchmentrunoff.

Both hydrological routing and ANN techniques provide limited reliability for therange of events outside those used for calibration. However, this is exactly the rangeone is interested in when dealing with extreme flood events, which rarely occur andfor which measurements are even more rarely available. So even though ANNs andother datadriven modelling techniques may prove quite valuable in e.g. determiningrainfallrunoff relations (Solomatine, hsa021), physically based descriptions, such asprovided by hydraulic routing techniques and based upon the full use of Equations(5.1) and (5.2), offer better extrapolation possibilities than hydrological routingmethods or ANN based models. For this reason, we will focus on hydraulicmodelling techniques in the sequel. However, it should be realized that simplifiedequations, such as Equation (6.5), remain useful, as these offer us a good insight intothe physical nature of flood propagation, in particular the concepts of flood peakarrival time and the attenuation of peak discharges.

6.7 Twodimensional modelling of floods

In the modelling of floods, flows often take short cuts through flood plains wherethe 1D description may become quite inaccurate. This is even more the case fordam or embankment failures, where the flow may leave the flood plain completelyand inundate natural terrains. For this reason the twodimensional (2D) shallow



water equations will be introduced. Following the same principles as for 1D flow,these equations read

( ) ( ) 0h uh vht x y

¶ ¶ ¶+ + =

¶ ¶ ¶(6.10)

2 2( ) 0bf

u u vh zu u uu v g ct x y x h

+¶ +¶ ¶ ¶+ + + + =

¶ ¶ ¶ ¶(6.11)

2 2( ) 0bf

v u vh zv v vu v g ct x y y h

+¶ +¶ ¶ ¶+ + + + =

¶ ¶ ¶ ¶(6.12)

where we now also introduce the yaxis, orthogonal to the xaxis, with its flowvelocity v (m/s) associated to it. The friction term, with the dimensionless frictioncoefficient cf, has in both momentum equations a shear force component derivedfrom the quadratic head loss description along a stream line of the 2D flow. Basicassumptions are similar to those given for the 1D equations, as far as applicable inthis form of schematization.

Figure 6.4 Staggered grid for 2D flow simulations

Referring to the 2D grid shown in Figure 2, a volume conservative finite differenceform of the continuity equation is given by

1 * *, , 1/ 2, 1/ 2, 1/ 2, 1/ 2,

* *, 1/ 2 , 1/ 2 , 1/ 2 , 1/ 2 0

n n n n n ni j i j i j i j i j i j

n n n ni j i j i j i j

h u h ut x

h v h vy

q q

q q

V V+ + ++ + - -

+ ++ + - -

- -+ +

D D-

=D

(6.13)



( ) ( )

( )

11/ 2, 1/ 2, 1, ,

11 12

2 21

1/ 2, 1/ 2, 1/ 2,

*1/ 2,

, ,

0

n n n ni j i j i j i jn n n n

n n ni j i j i j

f ni j

u ua u u a v u g

t x

u u vc

h

q qV V+ + ++ + +

++ + +

+

- -+ + + +

D D

æ ö+ç ÷

è ø =

(6.14)

( ) ( )

( )

1, 1/ 2 , 1/ 2 , 1 ,

21 22

221

, 1/ 2 , 1/ 2 , 1/ 2

*, 1/ 2

, ,

0

n n n ni j i j i j i jn n n n

n n ni j i j i j

f ni j

v va u v a v v g

t y

v u vc

h

q qV V+ + ++ + +

++ + +

+

- -+ + + +

D D

æ ö+ç ÷

è ø =

(6.15)

where the symbol * has the same meaning as in Equation (5.18) again and a11(un,un),a12(vn,un), a21(un,vn) and a22(vn,vn) are generalizations of the discretization of theconvective momentum term.The long double bar over the velocity in the friction term means that this velocity isobtained by averaging over values at four surrounding grid points. The friction termrequires special treatment in case of flooding of dry terrain. At the wave front thewater velocity rapidly accelerates from zero. Overshoot of velocities can beprevented by a predictor corrector approach.

The convective momentum terms are subject to the same principles as discussed forthe 1D approximations. For example, for positive flow velocities the momentumconservative discretization of the term a12(un,vn) is given by

1 1/ 2, 1/ 2 1/ 2, 1/ 2, 112

1/ 2,

( , )x

v j i j i jn nxi j

q u ua v u

yh+ - + + -

+

-æ öç ÷Dè ø

; (6.16)

whereas it is given by1/ 2, 1/ 2, 1

1/ 2, 1/ 212 ( , )x i j i jn ni j

u ua v u v

y+ + -

+ --æ ö

ç ÷Dè ø; (6.17)

for the energy conservative discretization. In the first expressionx

v q means thespecific discharge in ydirection, averaged over two surrounding points along thelocal xaxis. In the last expression

xv has the same meaning in relation to the

velocity v.

The treatment of the convective momentum terms shown above is numerically veryrobust and allows for the correct description of the effects of sudden expansions andcontractions and similar changes in the topography, such as steps in the bed level.Moreover, it allows for the 2D simulation of supercritical flows and the propagationof hydraulic jumps.



Figure 6.5 Layout of the instantaneous dam break experiment

Figure 6.6 Top view of results of the numerical scheme of Equations (6.13), (6.14) and (6.15) (lower half),compared with video monitored measurements in a physical model (upper half)

As described by Stelling and Duinmeijer (2003), the correct modelling of thesephenomena has been demonstrated in a software validation study, where results ofthe numerical scheme of Equations (6.13), (6.14) and (6.15) were compared withvideo monitored measurements in a physical model (Figure 6.6). The setup consistsof two reservoirs with different water levels, separated by a wall. The wall containsa gate which can be lifted. The width of both reservoirs is 8.30 m, the length of the



upper reservoir 2 m and the length of the lower reservoir 29 m. The gate has a widthof 0.4 m and has been placed at the middle of the wall.

The numerical experiment was made on a grid of 0.1*0.1 m, giving a total ofapproximately 25000 grid cells. The time step was set to 0.005 s. The gate was liftedat a speed of 0.16 m/s, to produce a flow spreading out into a 2dimensional plain.Initial data were set at a depth of 0.60 m for the upstream reservoir and a depth of0.05 m for the downstream reservoir.

Figure 6.7 Comparison of measured and computed wave front position at various times (see case Figure 4)

Figure 4 shows the results of this simulation. The upper half of this figure presents avideo recorded view from above. The lower part of the figure presents the computedresults. It is clearly seen that the front propagation, the propagation of the hydraulicjump and the side spreading of the wave are represented reasonably well. Figure 5shows a comparison of the measured and computed position of the wave front atvarious times.

Simulations were made with various Manning roughness coefficients and both for aninitially wet and a dry downstream reservoir. For the propagation of the flood on thedry bed the Manning roughness turned out to be a sensitive parameter. If for dambreak models a reliable topography is available, the roughness parameter remainsthe only parameter to be estimated. Currently, researchers are focusing on betterdescriptions of roughness parameters by deriving depth dependent relationships on



the basis of vegetation characteristics, e.g. Uittenbogaard (2003), Rodriguez (2004)and Baptist (2005).

6.8 Integrated 1D/2D modelling

In flood modelling, there are numerous practical examples where flows are bestdescribed by combinations of 1D and 2D schematizations. An obvious example isthe flooding of deltaic areas, often characterized by a flat topography with complexnetworks of natural levees, polder dikes, drainage channels, elevated roads andrailways and a large variety of hydraulic structures. Flow over the terrain is bestdescribed by the 2D equations, whereas channel flow and the role of hydraulicstructures are satisfactorily described in 1D. Flow over higher elevated lineelements, such as roads and embankments can be reasonably modeled in 2D byraising the bottom of computational cells to embankment level. However, for ahigher accuracy of the numerical description adapted formulations have to beapplied, such as energy conservation upstream of overtopped embankments.

Another example is the flood propagation in a meandering river, with shortcuts viathe flood plain when over bank flow occurs. In large scale models, the flow betweenthe river banks is satisfactorily described by the de Saint Venant equations solvedwith 1D grid steps several times the width of the channel. An equivalent accuracy ofdescription in 2D would require a large number of grid cells, with step sizes being afraction of the channel width. However, flow in the flood plain may be betterdescribed in 2D and may allow for 2D grid steps often exceeding the width of theriver.

For this reason, hybrid 1D and 2D schematizations are often used. Basically thereare two approaches: one with interfaces defined between 1D and 2D along verticalplanes and the other approach with schematization interfaces in almost horizontalplanes.

Coupling along vertical planes, gives a full separation in the horizontal space of the1D and 2D modeled domains. In the 1D domain the flow is modeled with the deSaint Venant equations applied over the full water depth. The direction of flow inthe 1D domain is assumed to follow the channel xaxis and in the model it carries itsmomentum in this direction, also above bank level. Without special provisions, thereis no momentum transfer accounting applied between the 1D and 2D domains.Momentum and volume entering or leaving the 2D domain at these interfaces, aregenerated by the compatibility condition applied. As a result, the coupling cannot beexpected to be momentum conservative. Depending on the numerical solutionapplied, the linkage may either be on water level or on discharge compatibility.Particular care has to be taken in applying this form of schematization if waterquality processes are to be included in the model.In a model coupled along an almost horizontal plane, 2D grid cells are placed abovethe 1D domain, as shown in Figure 6. In this schematization, the de Saint Venantequations are applied only up to bank level. Above this level, the flow description inthe 2D cell takes over. For relatively small channel widths compared to the 2D cellsize, errors in neglecting the effect of momentum transfer at the interface are minor.For wider channels it is recommended to modify each 2D cell depth used in the



momentum equation by adding a layer defined by the local hydraulic radius for thatpart of the 1D crosssection which underlies a 2D cell.Further refinements are possible, including terms describing the momentum transferbetween the 1D and 2D domains.

Numerical solutions are obtained by discretizing separately the 1D and 2D domains.Assuming that for both domains implicit numerical schemes are applied, theinterface compatibility conditions can be modeled either as an explicit or an implicitlink. Applying explicit links, first the solutions for the 1D and 2D domains aregenerated sequentially. Subsequently, exchange flows are computed and added aslateral flows at the next time step. Implicit links are based upon water levelcompatibility. These equations are then added to the complete sets of equationsgenerated separately for the 1D and 2D domains. There are many approaches tosolving the complete set of equations. With the current state of the art, it is no longernecessary to apply for the 1D domain different solvers for socalled simply ormultiply connected channel networks. Similarly, in 2D there is no real need anymorefor alternating direction algorithms, as the efficiency of the conjugate gradientsolvers has increased significantly over the past years.

Figure 6.8 Coupling of 1D and 2D domains in SOBEK

As an example, Delft Hydraulics has developed its combined 1D2D packageSOBEK for the modelling of integrated fresh water systems (www.sobek.nl;www.wldelft.nl). The 1D part of hybrid models is based upon the numerical schemeof Equation (5.20) and (5.31). The 2D part is described by the single step 2D schemegiven by Equations (6.13), (6.14) and (6.15). For efficiency reasons, the continuityequations for the 1D and 2D domains are combined into one single equation atpoints where 1D grid sections underlie a 2D cell. As a first step in reducing the totalnumber of equations, SOBEK eliminates all equations at velocity grid points. Thesecond step in the solution algorithm is the elimination of a large number ofunknowns by applying a minimum connection search between unknown waterlevels. As a rule, this leads to an efficient elimination of nearly all unknowns of the1D domain and a substantial number of unknowns in the 2D domain.



This direct solver carries its elimination on, until nearly every second equation in the2D domain has been eliminated. Beyond this point, it is more economical to applythe conjugate gradient solver to solve the remaining set of equations.

Apart from its efficiency, an additional advantage of eliminating nearly everysecond 2D equation is the improved conditioning of the resulting matrix. Thisfollows from the fact that elimination of an unknown water level at a 2D grid pointhas the effect of increasing the spatial distance between the remaining adjacentpoints, where water levels are still unknown. This, in turn, reduces Courant numbersand as a consequence, it leads to changed coefficients at the main diagonal of thematrix which is now more dominant in relation to the other diagonals, e.g. Verwey(1994).

Figure 6.9 Flood modelling of the Vallei and Eem area, The Netherlands

An example of a combined 1D2D model is shown in Figure 7. It represents theschematization of a model of the Eem Valley area in The Netherlands applied in astudy of the potential effects of a River Rhine dike breach.This model has been used to provide information on warning lead times and flooddepths for evacuation planning. The Rhine branch upstream of the breach has beenmodeled in 1D. At its upstream end a design hydrograph was specified, whereas thedownstream boundary condition of this relatively short branch is given by a ratingcurve. Such a short downstream reach is permissible as the rating curveautomatically corrects for most of the effects of flow deviated through the breach atthis boundary. The breach itself has been described in 1D as a structure with avelocity dependent breach growth.



North of the dike the 1D link discharges into the 2D domain, given by a 100 * 100m grid with bottom levels derived from a digital elevation model and resistancecoefficients derived from land use maps. Elevated roads and railways are presentedas flow barriers by raising the underlying cell bottom levels up to the levels of theseembankments. The resulting flood depths presented in Figure 7 clearly show theeffect of the 1D channel in the schematization. Due to their greater depth, floodwaves propagate faster in these channels than over land. Further downstream, thisleads to first signs of the progressing flood wave already one or two days before themain flood arrives.

6.9 Exercise

At a discharge measuring station a crosssection is measured as shown below.

a) Compute the conveyance of the channel for levels h=2.00 m and h=3.00 m,respectively. The Manning numbers vary along the crosssection and theirvalues have been indicated in Figure 6.8..

b) Compute discharges for both levels given an approximate bed slope I0=103.c) Compute approximate flood wave celerities for the given water levels. Explain

the difference in the celerities.d) During the passage of a flood wave, the water level rises from 2.00 m to 3.00

m above the reference level over a period of 10 minutes. Compute theapproximate deviation between the water level slope and the bed slope. (Note:transform the time to a length via the wave celerity).

e) Compute the approximate difference in percentage between the expectedmeasured discharges and the discharges based on a rating curve.

Figure 6.10 Crosssection for the computation of flood wave propagation celerities



7 Water Hammer

7.1 Introduction

Water hammer is the phenomenon of highamplitude pressure waves travelling inpipes, caused by the rapid changes in the velocity of the transported fluid. Thesevariations usually result from the operation of flow and pressure control devices,such as valves, pumps and turbines.

D

d slot

Figure 7.1 Sketch of pipe with artificial slot at the top to represent storage effects of pipe expansionand water compressibility

In principle, the equations used for the computation of these pressure waves are thesame as those derived for open channel flow. As an analogy one may consider a pipewhich has an artificial slot at the top, in which a free surface water level may riseafter complete filling of the pipe (Figure 7.1).

The wetted area of the slot represents the storage of water resulting from changes inthe water pressure. The storage capacity consists of three principal contributions:

1. the elasticity of the pipe wall which leads to a pipe crosssection expansion atincreasing water pressure. It is evident that this type of storage depends on thecrosssection shape and on the wall material properties and its composition.

2. compressibility of the water, which usually is neglected in free surface flowcomputations. In the case of water hammer, however, the compressibility plays adominant role. As the free surface flow equations are based upon a volumebalance, the compression of the water represents a virtual water storage;

3. compressibility of air bubbles or vacuum bubbles contained in the water;

Water hammer is important in engineering for the following reasons:

1. high pressures may build up in a pipe line to the extent that it may lead tobursting of the pipe;



2. vacuum bubbles may be generated when the pressure drops to values belowapproximately 30% of the atmospheric pressure. The resulting cavitation ishighly aggressive to the pipe material and may lead to considerable damage;

3. in other cases, air may be entrained in the fluid at moments and locations wherethe pressures are low.

For these reasons, any design related to transport of fluid in pipes should be checkedagainst the risk of water hammer damage. There are various reasons why waterhammer may occur. For details, reference is made to text books, e.g. Chaudhry(1987). Typical reasons are:

· instantaneous or very fast valve closure. Many of us know this phenomenonwhen closing a tab at home. A wrong design of the valve may cause a suddendeceleration of the water flow and a high pressure builds up due to thetransformation of momentum of the water into an impulse;

· sudden energy demand changes in a hydropower station or turbine failure;· pump failure, for example, by a power cut. At the upstream end of the pump the

pressure will increase, while at the downstream end a negative wave is generatedwhich may cause cavitation or implosion of the pipe;

· burst of a pipe line, which in turn generates pressure surges along the pipe.

In many cases it is the way of operating the system that causes the water hammerproblems. In other cases there may be accidental causes. In the design of thehydraulic system such operation problems may be foreseen and water hammer maybe prevented by designing antiwater hammer arrangements. Examples are:

· strict control of valve manipulations. As will be shown by the computationalprocedure, slow closure of valves reduce the over pressures. Slow has to be seenin relation to the length of the pipe and the celerity with which pressure surgesare travelling along the pipe;

· design and construction of a surge chamber as closely as possible upstream ofthe turbine or pump. This surge chamber serves as an escape for the pressurewave and reduces effectively the dangerous operation time of closure;

· closed and vented air vessels, which have a function similar to surge chambers.However, in this case the surface is formed by a pressurized air chamber. Thesedevices are used, for example, at the downstream end of sewer pumps in order toprevent excessive under pressures;

· fly wheels, which prevent the sudden change of the pump rotational speed;· bypasses with a checkvalve placed in parallel to a pump. It opens during pump

rundown and supplies water to the pipe downstream of the pump;· pressure release valves, which open when the water pressure exceeds a

maximum value admitted.

For a more extensive list of options reference is made again to the literature in thisfield. As water hammer computations may be complex, use is often made ofstandard software packages, such as the WANDA system of Delft Hydraulics(www.wldelft.nl). However, simple problems may also be investigated by usingExcel. The next paragraphs provide the basis for setting up such computations.



7.2 Water Hammer Equations

For a discussion on the water hammer equations reference is made to the open channelflow equations (Chapter 5), given here as

sH Q + = 0b t x

¶ ¶¶ ¶

(7.1)

2Q H fQ + + gA + Q | Q | = 0t x A x 2DA

æ ö¶ ¶ ¶ç ÷¶ ¶ ¶è ø

(7.2)

where

Q = pipe discharge (m3/s); H = piezometric head above any horizontal reference level (m); bs = storage width due to fluid compressibility and pipe expansion (m); A = crosssectional area of the pipe (m2); f = DarcyWeisbach friction factor; D = pipe diameter (m).

In the momentum equation the friction term has been replaced by a term based on theDarcyWeisbach concept. In this form, the coefficient expresses the energy head lossDH as the fraction of the velocity head which is lost over a pipe length equivalent toits diameter D. For a length L of the pipe the head loss is then defined as

2L uH = fD 2g

D (7.3)

In water hammer, the role of the convective momentum is small and usuallyneglected. Introducing, furthermore, a characteristic celerity a as

s

Aa = gb

(7.4)

the water hammer equations are best known in the form

2

gA H Q + = 0t xa

¶ ¶¶ ¶

(7.5)

Q H f + gA + Q | Q | = 0t x 2DA

¶ ¶¶ ¶

(7.6)

The physical behaviour of water hammer is easily understood by transforming theseequations into the form



gA Q H + a = 0t a x

¶ ¶æ öç ÷¶ ¶è ø

(7.7)

Q gA + a H = Et x a

¶ ¶ æ öç ÷¶ ¶ è ø

(7.8)

where E is to be seen as a sink term representing the effect of friction losses.

fE Q|Q|2DA

= - (7.9)

Successive addition and subtraction of Equations (7.8) and (7.7) gives

gA gA Q + H + a Q + H = Et a x a

¶ ¶æ ö æ öç ÷ ç ÷¶ ¶è ø è ø

(7.10)

gA gA Q H a Q H = Et a x a

¶ ¶æ ö æ ö- - -ç ÷ ç ÷¶ ¶è ø è ø(7.11)

Referring to the concept of total derivatives, as introduced in Chapter 3, Equations(7.10, 7.11) express the integration of the equation

d gA Q + H = Edt a

æ öç ÷è ø

(7.12)

along the characteristic line, or simply characteristic, defined by

dx = adt

(7.13)

and the integration of

d gA Q H = Edt a

æ ö-ç ÷è ø

(7.14)

along the characteristic line

dx = adt

- (7.15)

Neglecting the influences of friction, Equation (7.12) represents the condition

gAQ + H = constanta

(7.16)



along the positive characteristic given by Equation (7.13), while Equation (7.14) leadsto

gAQ H = constanta

- (7.17)

along the negative characteristic, given by Equation (7.15).

Before discussing the role of these equations in the algorithm leading to the solutionof water hammer problems, let us consider again the meaning of the characteristiccelerity a in relation to water storage. In analogy with Hooke's law expressing theelastic deformation of a string under the influence of the axial stress

= Es e (7.18)

wheres = axial stress (N/m2);e = strain per unit length of the string;

E = Young's modulus of elasticity.

The compression of water under the influence of changing pressures is seen as anelastic process described by

dVdp = KV

- (7.19)

where

dp = change in fluid pressure (N/m2); K = bulk modulus of water elasticity (N/m2); V = water volume; dV = change of this water volume under the influence of pressure change dp.

For a control volume of length dx the compression of fluid in the pipe can be madeequivalent to a virtual storage of an incompressible fluid in a virtual pipe slot by therelation

sVdV = dp = dH dxbK

- - (7.20)

where the change in pressure level dH is related to the change in water pressure dp, by

dp = g dHr (7.21)

assuming that the contribution of the change in fluid density in this relation can beneglected.



Substitution of (7.21) into (7.20) and replacing dV by Adx, gives

s g A=b K

r (7.22)

For the derivation of the effect of pipe deformation it will be assumed that the pipewall is elastic and that the pipe has expansion joints all along its axis. These jointswill prevent the development of axial stresses under the influence of changing waterpressure in the pipe. The general stressstrain relation for elastic pipe material underthe influence of stresses in axial and tangential direction reads as

1 2= = Em es s (7.23)

wheres1 = hoop stress (tangential direction) (N/m2);s2 = axial stress (N/m2);m = Poisson's ratio.

For the case where the axial stress cannot develop the equation reads, for a change intangential stress

1d = Ees (7.24)

The expression of e in terms of the change in the circular pipe circumference withradius r

1d(2 r) d= =2 r E

p sep

(7.25)

now leads to the relation

1dr d=r E

s (7.26)

or

1dA 2d=A E

s (7.27)

Referring to Figure (7.2) the change in pipe material stress is related to the change inpressure dp as

2dF = Ddp (7.28)

or



12ed = Ddps (7.29)

D

P

tt

e

Figure 7.2 Forces on semicylinder under the influence of water hammer induced pressures

Substitution of Equations (7.29), (7.21) into Equation (7.27) gives

DdA = gA dHeE

r (7.30)

Relating this compression volume to the water storage in the fictive pipe slot gives

sDdV = dA dx = g A dH dx = dH dxbe

r

or

sD= g Ab eE

r (7.31)

The combined effects of water compressibility and pipe expansion leads to the virtualpipe slot width

s g A DK= 1 +bK eE

r æ öç ÷è ø

(7.32)

Substitution of this relation into Equation (7.41) gives the expression for the pressurewave celerity

K

a = DK1 +eE

r (7.33)

The effect of axial stresses and composite pipe walls requires a further generalisationof this equation to the form



K

a = K1 +E

r

y(7.34)

with

D=e

y (7.35)

for the case of homogeneous elastic pipe material and the presence of expansion jointsall along the pipe.

Referring to Chaudhry (1987), the following relations can be derived for Y:

2D= (1 )e

y m (7.36)

for the case of thinwalled elastic conduits anchored against axial movementthroughout their length;

D= (1 0.5 )e

y m (7.37)

for the case of thinwalled elastic conduits anchored against axial movement at theupper end;

= 1 ; E = Gy (7.38)

for the case of an unlined tunnel with a modulus of rock elasticity G, and

DE=GD + Ee

y (7.39)

for a steellined tunnel through rock.

For values of the various material properties reference is made to literature and toTable 7.1 and Table 7.2 for some very common ones.



Table 7.1 Pipe material properties

Material Modulus of elasticity E

(Gpa)

Poisson’s ratio

Mild steel 200 212 0.27

Polyethylene 0.8 0.46

Concrete 14 – 30 0.10 – 0.15

Cast iron 80 – 170 0.25

Table 7.2 Fluid properties

Fluid Temperature

(oC)

Density

(kg/m3)

Bulk modulus of elasticity K

(GPa)

Fresh water 20 999 2.19

Sea water 15 1025 2.27

Oil 15 900 1.5

7.3 The Method of Characteristics for Water Hammer

The method of characteristics is by far the most commonly used equation solver inwater hammer analysis. In the discussion of Chapter 3 it was concluded that themethod of characteristics is a rather unpractical approach to solving free surfaceflow problems. Most of the reasons given there, however, do not hold for waterhammer analysis. As shown by Equations (7.13, 7.15 and 7.33), the characteristicdirections for the water hammer equations usually are straight lines, which allow forthe construction of an equidistant network, as shown by the following example.

Consider the hypothetical case of a pipe with frictionless flow supplied at theupstream end from a large reservoir and controlled at the downstream end by avalve, as shown in Figure (7.3). Data for this problem are given as follows:

Hres = 50 mQ = 1 m3/sA = 1 m2

L = 1000 ma = 1000 m/sg = 10 m/s2

The valve is closed over a period of 5 seconds, controlled to provide a linearlydecreasing discharge over this time.



The characteristic grid for this problem is shown in Figure 7.4. Initially alldischarges and pressure levels are equal to 1 m3/s and 50 m respectively.

Referring, again, to Equations (7.16 and 7.17), the solution at point A1 is found bysolving the boundary condition

1AH = 50 m

and by Equation (7.17) applied along the negative characteristic from point B0 to pointA1

1 1 0 0

3A A B BQ gA/a H = Q gA/a H = 1 0.01*50 = 0.50 m /s

giving

1 1

3A AH = 50 m; Q = 1.0 m /s

Note that this solution is still equivalent to the initial steady state condition along thepipeline between A and B. This is explained by the fact that the effect of the valveclosure starting at point B0 cannot influence conditions at the upstream boundarybefore the negative characteristic through point B0 has arrived at point A1. Thetriangle formed by the points A0, B0 and A1, therefore, is a steady state region.

The solution at point B1 is found in a similar way by combining the downstreamboundary condition at t=1 s, given as

1BQ = 0.8 m3/s

with the condition along the positive characteristic passing from A0 to B1, given as

1 1 0 0

3B B A AQ + gA/a H = Q + gA/a H = 1 + 0.01*50 = 1.50 m /s

resulting in

1 1

3B BH = 70 m; Q = 0.8 m /s

Similarly, solutions at subsequent points can be computed, with results shown inTable 7.3. The maximum pressure found at point B is 90 m water column.

The gradual valve closure gives a considerable operation improvement over the caseof a sudden valve closure. Closing the valve instantaneously at time t=0 gives apressure H=150 m at point B1, as can be verified easily by applying Equation (7.16)with QB1=0 m3/s. Further improvements are found by slowing the closure operationfurther down. This example, therefore, demonstrates clearly the origin of waterhammer problems and the essential approach to water hammer prevention.



H

L

reservoir

supply pipe

Figure 7.3 Situation sketch of valve closure problem

10

9

8

7

6

5

4

3

2

1

0

10

9

8

7

6

5

4

3

2

1

0

t (s)

t(s)

Q (m3 /s)x (m)

0 100 0 1.0

A B

0.2

0.4

0.6

0.8

1.0

Figure 7.4 Characteristic lines for the water hammer problem

Table 7.3 Pressures and discharges computed for the problem of valve closure

Point 0 1 2 3 4 5 6 7 8 9 10 variable

A 1.0 1.0 0.6 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 Q (m3/s)50 50 50 50 50 50 50 50 50 50 50 H (m)

B 1.0 0.8 0.6 0.4 0.2 0.0 0.0 0.0 0.0 0.0 0.0 Q (m3/s)50 70 90 70 50 70 70 30 30 70 70 H (m)



7.4 Exercise

Water flows from a reservoir through a horizontal pipe with a length L=10 km. Thereservoir water level HR is 50 m +MSL. The pipe has a diameter D of 1.00 meter. Atthe end of the pipe the outflow is controlled by a valve. The steady state water velocityu=2 m/s. The DarcyWeisbach friction coefficient f is 0.01. Use g=10 m/s2 in yourcomputations.

Questions

1. Compute the pressure distribution along the pipe line for the steady state flow,assuming that the energy loss at the valve can be neglected.

2. For the computation of Questions 37 we will assume that the complete energyloss along the pipe is now concentrated at the valve. Recompute the pressuredistribution.

3. Compute the characteristic pressure wave celerity for the data given below.4. Compute, with the rigid water column theory, the pressure increase at the pipe

end if the valve is closed over a period of 1 second. Is this pressure increaserealistic?

5. Compute the pressure increase with the water hammer theory for fast closure.6. Compute the pressure as a function of time at the location of the valve by

using the method of characteristics. Use a valve closure function which leadsto a linear decrease in out flowing discharge over a period of 1 minute. Neglectfriction.

7. While keeping the other parameters constant, what will be the effect on themaximum pressure increase if:a) the closure time is increased;b) the steel pipe is replaced by a PVC pipe;c) the pipe wall thickness is increased;d) the pipe diameter is increased;e) the pipe length is decreased;f) air is entrained into the flow.

8. Repeat the computations for the case where pipe wall friction is included andthe energy loss at the valve is neglected. Produce a graph of the results.

Additional data:

e = 0.01 m;K = 2 GPa;E = 200 GPa.

The pipe has frequent expansion joints.



8 References

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Stelling, G.S., Kernkamp, H.W.J. and Laguzzi, M.M. (1998). Delft Flooding System: apowerful tool for inundation assessment based upon a positive flow simulation,Hydroinformatics ‘98, Babovi & Larsen (Eds.), Balkema, Rotterdam, 449456.

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