Wave Propagation in Fluids E-Book

Vincent Guinot

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 392

Veröffentlichungsjahr: 2013

Beschreibung

This book presents the physical principles of wave propagation in fluid mechanics and hydraulics. The mathematical techniques that allow the behavior of the waves to be analyzed are presented, along with existing numerical methods for the simulation of wave propagation. Particular attention is paid to discontinuous flows, such as steep fronts and shock waves, and their mathematical treatment. A number of practical examples are taken from various areas fluid mechanics and hydraulics, such as contaminant transport, the motion of immiscible hydrocarbons in aquifers, river flow, pipe transients and gas dynamics. Finite difference methods and finite volume methods are analyzed and applied to practical situations, with particular attention being given to their advantages and disadvantages. Application exercises are given at the end of each chapter, enabling readers to test their understanding of the subject.

Leseprobe

Table of Contents

Introduction

Chapter 1. Scalar Hyperbolic Conservation Laws in One Dimension of Space

1.1. Definitions

1.2. Determination of the solution

1.3. A linear law: the advection equation

1.4. A convex law: the inviscid Burgers equation

1.5. Another convex law: the kinematic wave for free-surface hydraulics

1.6. A non-convex conservation law: the Buckley-Leverett equation

1.7. Advection with adsorption/desorption

1.8. Conclusions

Chapter 2. Hyperbolic Systems of Conservation Laws in One Dimension of Space

2.1. Definitions

2.2. Determination of the solution

2.3. Specific case: compressible flows

2.4. A 2×2 linear system: the water hammer equations

2.5. A nonlinear 2×2 system: the Saint Venant equations

2.6. A nonlinear 3×3 system: the Euler equations

2.7. Summary of Chapter 2

Chapter 3. Weak Solutions and their Properties

3.1. Appearance of discontinuous solutions

3.2. Classification of waves

3.3. Simple waves

3.4. Weak solutions and their properties

3.5. Summary

Chapter 4. The Riemann Problem

4.1. Definitions – solution properties

4.2. Solution for scalar conservation laws

4.3. Solution for hyperbolic systems of conservation laws

4.4. Summary

Chapter 5. Multidimensional Hyperbolic Systems

5.1. Definitions

5.2. Derivation from conservation principles

5.3. Solution properties

5.4. Application to two-dimensional free-surface flow

5.5. Summary

Chapter 6. Finite Difference Methods for Hyperbolic Systems

6.1. Discretization of time and space

6.2. The method of characteristics (MOC)

6.3. Upwind schemes for scalar laws

6.4. The Preissmann scheme

6.5. Centered schemes

6.6. TVD schemes

6.7. The flux splitting technique

6.8. Conservative discretizations: Roe’s matrix

6.9. Multidimensional problems

6.10. Summary

Chapter 7. Finite Volume Methods for Hyperbolic Systems

7.1. Principle

7.2. Godunov’s scheme

7.3. Higher-order Godunov-type schemes

7.4. Summary

Appendix A. Linear Algebra

A.1. Definitions

A.2. Operations on matrices and vectors

A.3. Differential operations using matrices and vectors

A.4. Eigenvalues, eigenvectors

Appendix B. Numerical Analysis

B.1. Consistency

B.2. Stability

B.3. Convergence

Appendix C. Approximate Riemann Solvers

C.1. HLL and HLLC solvers

C.2. Roe’s solver

Appendix D. Summary of the Formulae

References

Index

First published in France by Hermes Science/Lavoisier in 2006 entitled “Ondes en mécanique des fluides” First published in Great Britain and the United States in 2008 by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

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The rights of Vincent Guinot to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Cataloging-in-Publication Data

Guinot, Vincent.

Wave propagation in fluids: models and numerical techniques / Vincent Guinot.

p. cm.

Includes bibliographical references and index.

ISBN 978-1-84821-036-3

1. Fluids--Mathematics. 2. Wave-motion, Theory of. I. Title.

QA927.G85 2008

532′.05930151--dc22

2007043951

British Library Cataloguing-in-Publication Data

A CIP record for this book is available from the British Library

ISBN: 978-1-84821-036-3

Introduction

What is wave propagation?

In a kitchen or in a bathroom, the number of times we turn a tap every day is countless. So is the number of times we watch the liquid stream impacting the sink. The circular flow pattern where the fast and shallow water film diverging from the impact point changes into a deeper, bubbling flow is too familiar to deserve attention. Very few people looking at the circular, bubbling pattern – referred to as a hydraulic jump by the specialists of hydraulics – are aware that they are contemplating a shock wave.

Turning off the tap too quickly may result in a thud sound. This is the audible manifestation of the well-known water hammer phenomenon, a train of pressure waves propagating in the metal pipes as fast as hundreds to thousands of meters per second. The water hammer phenomenon is known to cause considerable damage to hydropower duct systems or water supply networks under the sudden operation of valves, pumps or turbines. The sound is heard because the vibrations of the duct system communicate with the ambient atmosphere, and from there with the operator’s ears.

Everyone has once thrown stones into the water in a pond, watching the concentric ripples propagate on the surface. Less visible and much slower than the ripples is the moving groundwater that displaces a pollutant front in a journey that may last for years.

As ubiquitous and familiar as wave propagation may be, the phenomenon is often poorly understood. The reason why intuition so often fails to grasp the mechanisms of wave propagation may lie in the commonly shared, instinctive perception that waves are made of matter. This, however, is not true. In the example of the hydraulic jump in the sink, the water molecules move across an immobile wave. In the example of the ripples propagating at the free surface of a pond, the waves travel while the water remains immobile.

Waves appear when an object or a system (e.g. the molecules in a fluid, a rigid metallic structure) reacts to a perturbation and transmits it to its neighbors. In many cases, as in the example of the water ripples, the initially perturbed system returns to its initial equilibrium state, while the waves keep propagating. In this respect, waves may be seen as information. The ripples propagating in a pond are a sign that the water molecules “inform” their neighbors that the equilibrium state has been disturbed. A sound is nothing more than information about a perturbation occurring in the atmosphere.

Numerical techniques for wave propagation simulation have been the subject of intensive research over the last 50 years. The advent of fast computers has led to the development of efficient numerical techniques. Engineers and consultants now use simulation software packages for wave propagation on a daily basis. Whether it is for the purpose of acoustics, aerodynamics, flood wave propagation or contaminant transport studies, computer-based simulation tools have become indispensable to almost all domains of engineering. Such tools, however, remain instruments operated by human beings to execute tedious, repetitive operations previously carried out by hand. They cannot, and hopefully never will, replace the expert’s judgment and experience. Human presence remains necessary to the sound assessment of the relevance and accuracy of modeling results. Such an assessment, however, is possible only provided that the very specific type of reasoning required for the correct understanding of wave propagation phenomena has been acquired.

The main purpose of this book is to contribute to a better understanding of wave propagation phenomena and the most commonly used numerical techniques for its simulation. The first three chapters deal with the physics and mathematics of wave propagation. Chapters 4 and 5 provide an insight into more theoretical notions, used in specific numerical techniques. Chapters 6 and 7 are devoted to finite difference and finite volume techniques respectively. Basic notions of linear algebra and numerical methods are presented in Appendices A to C. The various formulae used in the present book are summarized in Appendix D.

What is the intended readership for this book?

This book is intended for the students of professional and research master programs and those engaged in doctoral studies, the curriculum of which contains hydraulics and/or fluid mechanics-related subjects. Engineers and developers in the field of fluid mechanics and hydraulics are also a potential target group. This book was written with the following objectives:

1)To introduce the physics of wave propagation, the governing assumptions and the derivation of the governing equations (in other words, the modeling process) in various domains of fluid mechanics. The application fields are as diverse as contaminant transport, open channel and free surface hydraulics, or aerodynamics.

2)To explain how the behavior of the physical systems can be analyzed using very simple mathematical techniques, thus allowing practical problems to be solved.

3)To introduce the main families of numerical techniques used in most simulation software packages. As today’s practising engineers cannot afford not to master modeling packages, a basic knowledge of the existing numerical techniques appears as an indispensable engineering skill.

How should this book be read?

The chapters are divided into three parts:

– The first part is devoted to the theoretical notions applied in the remainder of the chapter.

– The second part deals with the application of these theoretical notions to various equations of hydraulics and fluid mechanics.

– The third part provides a summary of the key points developed in the chapter, as well as suggestions of application exercises.

The main purpose of the application exercises is to test the reader’s ability to reuse the notions developed in the chapter and apply them to practical problems. The solution principle of the exercises may be accessed from the following URL: http://vincentguinot.free.fr/waves/exercises.htm.

Try to resist the temptation to read the solution immediately. Solving the exercise by yourself should be the primary objective. The solution text is provided only as a help, in case you cannot find a way to start and for you to check the validity of your reasoning after completing the exercise.

Chapter 1

Scalar Hyperbolic Conservation Laws in One Dimension of Space

1.1. Definitions

1.1.1. Hyperbolic scalar conservation laws

A one-dimensional hyperbolic scalar conservation law is a Partial Differential Equation (PDE) that can be written in the form

[1.1]

where t and x are respectively the time- and space-coordinates, U is the so-called conserved variable, F is the flux and S is the source term. Equation [1.1] is said to be the conservation form of the conservation law. The following definitions are used:

– The flux F is the amount of U that passes at the abscissa x per unit time due to the fact that U (also called the transported variable) is being displaced.

– The source term S is the amount of U that appears per unit time and per unit volume, irrespective of the amount transported via the flux F. If U represents the concentration in a given chemical substance, the source term may express degradation phenomena, or radioactive decay. S is positive when the conserved variable appears in the domain, and is negative if U disappears from the domain.

– The conservation law is said to be scalar because it deals with only one dependent variable. When several equations in the form [1.1] are satisfied simultaneously, the term “system of conservation laws” is used. Systems of conservation laws are dealt with in Chapter 2.

Only hyperbolic conservation laws are dealt with in what follows. The conservation law is said to be hyperbolic if the flux F is a function of U (and none of its derivatives) and, possibly, of x and t. Such a dependence is expressed in the form

[1.2]

The function F(U, x, t) is called the “flux function”.

NOTE.– The expression F(U, x, t) in equation [1.2] indicates that F depends on U at the abscissa x at the time t and does not depend on such quantities as derivatives of U with respect to time or space. For instance, the following expression

[1.3]

is a permissible expression [1.2] for F, while the following diffusion flux,

[1.4]

where D is the diffusion coefficient, does not yield a hyperbolic conservation law because the flux F is a function of the first-order derivative of U with respect to space.

In the case of a zero source term, equation [1.1] becomes

[1.5]

In such a case (see section 1.1.2), U is neither created nor destroyed over the domain. The total amount of U over the domain varies only due to the difference between the incoming and outgoing fluxes at the boundaries of the domain.

Depending on the expression of the flux function, the conservation law is said to be convex, concave or non-convex (Figure 1.1):

– The law is convex when the second-order derivative ∂2F/∂U2 of the flux function with respect to U is positive for all U.

– The law is concave when the second-order derivative ∂2F/∂U2 of the flux function with respect to U is negative for all U.

– The law is said to be non-convex when the sign of the second-order derivative ∂2F/∂U2 of the flux function with respect to U changes with U.

Figure 1.1.Typical examples of flux functions: convex (a), concave (b), non-convex (c)

1.1.2. Derivation from general conservation principles

The conservation form [1.1] is derived from a balance over a control volume of unit section defined between x0 and x0 + δx (Figure 1.2). The balance is carried out over the control volume between two times t0 and t0 + δt. The variation in the total amount of U contained in the control volume is then related to the derivatives ∂U/∂t and ∂F/∂x in the limit of vanishing δt and δx.

Figure 1.2.Definition sketch for the balance over a control volume

[1.6]

[1.7]

The variation δS in the amount of U induced by the source term S over the domain between t0 and t0 + δt is given by

[1.8]

The amount δF(x0) of U brought by the flux F across the left-hand side boundary of the control volume between t0 and t0 + δt is given by

[1.9]

A quantity δF(x0 + δx) leaves the domain across the right-hand side boundary

[1.10]

Stating the conservation of U over the control volume [x0, x0 + δx] between t0 and t0 + δt, the following equality is obtained

[1.11]

Substituting equations [1.6–1.10] into equation [1.11] leads to

[1.12]

A first-order Taylor series expansion around (x0, t0) gives

[1.13]

where the quantities O(δt2) and O(δx2) are second- or higher-order polynomials with respect to δt and δx respectively. These polynomials contain the second- and higherorder derivatives of U and F with respect to t and x. When δt and δx tend to zero, the polynomial O(δt2) becomes negligible compared to the quantity δt ∂U/∂t because δt2 decreases faster than δt. The polynomial O(δx2) becomes negligible compared to δx∂F/∂x for the same reason. The relationships [1.13] thus become

[1.14]

A similar reasoning leads to the following equivalence

[1.15]

Substituting equations [1.14] and [1.15] into equation [1.12] leads to

[1.16]

Dividing equation [1.16] by δtδx yields the conservation form [1.1], recalled here

The following comments can be made:

– The PDE [1.1] is a particular case of the more general, integral equation [1.12]. Equation [1.1] is obtained from equation [1.12] using the assumption that δt and δ x tend to zero. Equation [1.12] is the so-called weak form of equation [1.1] (see Chapter 3 for more details).

– The conservation form [1.1] is based on the implicit assumption that F is differentiable with respect to x and U is differentiable with respect to t. Consequently, [1.1] is meaningful only when U is continuous in space and time. In contrast, equation [1.12] is meaningful even when U is discontinuous in space and/or time. This has consequences on the calculation of discontinuous solutions, as shown in Chapter 3.

1.1.3. Non-conservation form

Equation [1.1] can be rewritten in the so-called non-conservation form, which involves only derivatives of U. The non-conservation form of equation [1.1] is

[1.17]

where λ is called the wave celerity, or wave propagation speed, and S′ is a source term that may be identical (although not necessarily) to the source term S in equation [1.1]. Equation [1.17] is obtained from equation [1.1] by rewriting the derivative ∂F/∂x as

[1.18]

[1.19]

i.e.

[1.20]

Comparing equation [1.20] to equation [1.17] leads to the following definitions for λ and S′

[1.21]

Example: assume that the flux function F is defined as in equation [1.3], recalled here

where a is a function of x and t. Equation [1.18] thus becomes

[1.22]

and λ and F′ are given by

[1.23]

1.1.4. Characteristic form – Riemann invariants

Writing a conservation law in non-conservation form leads to the notions of characteristic form and Riemann invariant. Such notions are essential to the understanding of hyperbolic conservation laws. A very convenient way of determining the behavior of the solutions of hyperbolic conservation laws consists of identifying invariant quantities (i.e., quantities that do not change) along certain trajectories, also called “characteristic curves” (or more simply “characteristics”).

The solution is calculated by “following” the invariants along the characteristics, which allows the value of U to be determined at any point. To do this, non-conservation form [1.17] is used

[1.24]

[1.25]

In the particular case of an observer moving at the celerity λ, equation [1.25] becomes

[1.26]

Comparing equations [1.26] and [1.17] leads to

[1.27]

S′ being a function of U, x and t, its value may be calculated at any point (x, t) if the value of U is known. The first-order ODE [1.27] is applicable along the characteristic.

One very important, specific case is where the source term S′ is zero, equation [1.17] becomes

[1.28]

and equation [1.27] becomes

[1.29]

Equation [1.29] can also be written as

[1.30]

Consequently, quantity U is invariant to an observer moving at speed λ. U is called a Riemann invariant.

The physical meaning of the celerity, or wave propagation speed, is the following. The celerity is the speed at which the variations in U (and not U itself) propagate. A perturbation appearing in the profile of U at a given time propagates at speed λ. The celerity can be viewed as the speed at which “information”, or “signals” created by variations in U, propagates in space.

1.2. Determination of the solution

1.2.1. Representation in the phase space

The phase space is a very useful tool in the determination of the behavior of the solutions of hyperbolic conservation laws. The term “phase space” indicates the (x, t) plane formed by space coordinate x and time coordinate t (Figure 1.3).

Figure 1.3.Representation of characteristic curves in the phase space

Figure 1.4.Physically permissible (a) and non-permissible (b) characteristics

The representation in the phase space may be used to determine the behavior of the solutions of conservation law [1.1] using the so-called “characteristics method”. The following simple case is considered:

– Source term S in equation [1.1] is zero.

Characteristic form [1.27] then reduces to equation [1.30], recalled here

[1.31]

From the property of invariance of U along the characteristic, U remains unchanged between A and A′

[1.32]

Extending the reasoning above to any value of x, the following relationship is obtained

[1.33]

where Δt represents the quantity ( t1 − t0) and λis estimated at (x, t).

In general, S and F′ are non-zero. Thus, relationship [1.33] cannot be used because

– U is not invariant along a characteristic line;

– the characteristics are therefore curved lines, the slope of which depends on the local value of x and U.

1.2.2. Initial conditions, boundary conditions

Figure 1.6.Initial and boundary conditions in the phase space

For the sake of clarity, the celerity λ is assumed to be positive over the entire domain (the case where the sign of the celerity changes is examined at the end of the section). Two possibilities arise:

[1.34]

[1.35]

Note that a boundary condition can be prescribed only if the characteristics enter the domain. In the situation illustrated by Figure 1.6, prescribing a boundary condition at the point C′ would be meaningless because the value of U at C′ is entirely determined by the initial condition at C via the characteristic form [1.27] and cannot be prescribed independently of it. Depending on the variations of λ with U, x and t, the number of boundary conditions needed to determine U uniquely over the domain [0, L] may be 0, 1 or 2 (see Figure 1.7).

Figure 1.7.Number of boundary conditions needed depending on the variations of the wave celerity: none (a) one (b) two (c)

1.3. A linear law: the advection equation

1.3.1. Physical context – conservation form

The linear advection equation is the simplest possible hyperbolic conservation law. It is found in many domains of fluid mechanics because it expresses a widespread phenomenon, the transport of a given quantity in a moving fluid. The transported variable may be the temperature of the fluid, the concentration in a given chemical, etc. The expression “advection” is often understood as the advection of a passive scalar, i.e., a quantity that does not influence the behavior of the flow by which it is transported. In a number of cases however, the transported quantity influences the velocity field, a phenomenon known as coupling. This is the case of the inviscid Burgers equation dealt with in section 1.4.

In the present section, a passive scalar is considered. The example of a chemical substance dissolved in water with a concentration variable in space and time is used. The water is assumed to flow in a channel, the transverse dimensions of which are assumed to be negligible compared to the longitudinal dimension. The channel may be an open channel (a river, a canal) or a closed channel (a conduit) with a cross-sectional area variable in space and time. The assumption of negligible transverse dimensions for the channel allows the assumption of a one-dimensional, longitudinal flow and transport process to be used. The channel is represented as a one-dimensional object. The space coordinate is the curvilinear abscissa (Figure 1.8).

Figure 1.8.One-dimensional representation of a channel, the transverse dimension of which can be considered negligible: reality (a) and model (b)

The governing PDE for the one-dimensional transport of a dissolved substance is derived by carrying out a balance as in section 1.1.2. The total quantity M (t) of substance (the “mass” as introduced in section 1.1.2) over a basic slice of channel of length δx (Figure 1.9) is given by

[1.36]

where C is the concentration of the dissolved substance and δV is the volume of the basic channel slice, given by

[1.37]

where A is the cross-sectional area of the channel (Figure 1.9).

Figure 1.9.Perspective view of a basic channel section

Amount δF(x0) of dissolved chemical that passes at x0 during a basic time interval δt is given by

[1.38]

where u is the flow velocity. Using the same reasoning as in equations [1.11–1.16] with a zero source term, the PDE that describes the conservation of mass (also called the continuity equation) is obtained

[1.39]

Equation [1.39] can be written in the form [1.1] by defining the conserved variable U, flux F and source term S as

[1.40]

1.3.2. Characteristic form

Several approaches may be used to rewrite equation [1.39] in characteristic form. A first approach consists of defining the conserved quantity as AC and rewriting equations [1.39–1.40] as

[1.41]

[1.42]

As shown in section 1.1.4 (see equations [1.24–1.27]), equation [1.42] is equivalent to

[1.43]

Equation [1.43] is of limited interest because U does not appear as an invariant quantity along a characteristic line.

In a second approach, equation [1.39] is rewritten with respect to the concentration C by developing the derivatives

[1.44]

equation [1.44] is rewritten as

[1.45]

Noting that the continuity equation for the flow can be written as

[1.46]

substituting equation [1.46] into equation [1.45] yields the following equation

[1.47]

[1.48]

From the developments carried out in section 1.1.4 (see equations [1.24–1.27]), equation [1.48] is known to be equivalent to the following characteristic form

[1.49]

Equation [1.49] is equivalent to

[1.50]

Equation [1.50] is an interesting alternative to equation [1.43] because it allows a Riemann invariant to be derived. The Riemann invariant is the concentration of the dissolved substance. Note that the conserved quantity (the mass AC per unit length of channel) is not identical to the invariant quantity (the concentration).

1.3.3. Example: movement of a contaminant in a river

The difference between a conserved quantity and an invariant quantity is best illustrated by the following example. Consider a river in which a contaminant is transported without degradation or external inflow. The cross-sectional area of the river is variable, with a sudden narrowing at point N. The cross-sectional areas upstream and downstream of the narrowing are denoted by A1 and A2 respectively. Discharge Q is assumed to be constant in both space and time. The numerical parameters are summarized in Table 1.1.

Table 1.1.Parameters of the problem

As shown by the data in Table 1.1 the flow velocity is 0.1 m/s upstream of the narrowing and 0.25 m/s downstream of it. Denoting by tA′ and tB′ the times at which the characteristics passing at A and B reach the narrowing N, the following relationship holds

[1.51]

Figure 1.10.Behavior of the solution in the physical space (top, middle) and the phase space (bottom)

Equation [1.51] can be rewritten as

[1.52]

Consider now a time t1 after the contaminant cloud has left point R. The front and rear of the cloud are denoted by A″ and B″ respectively. Abscissas xA″ and xB″ are related to times tA′ and tB′ as follows

[1.53]

Denoting by L2 the width of the cloud downstream of the narrowing, the following relationship holds

[1.54]

From the parameters in Table 1.1 the cloud is 4 times as large downstream of the narrowing than upstream of it. Note that:

[1.55]

[1.56]

where equation [1.54] is used to relate L2 and L1. Although the source term is zero and no contaminant enters the domain across the boundary, the integral of C with respect to x is not constant. The concentration is an invariant quantity; it is not a conserved quantity,

[1.57]

[1.58]

where equation [1.54] is used again to provide a relationship between L1 and L2. In contrast with the integral of C, the integral of AC (i.e., the total mass of solute contained in the river reach) is conserved. However, AC is equal to A1C0 upstream of N and is equal to A2C0 downstream of N. Consequently, AC is a conserved quantity, not an invariant quantity.

1.3.4. Summary

The conservation form of the linear advection equation for a contaminant in a channel is equation [1.39]. It can be written in the form [1.1] by defining the conserved variable, flux and source terms as in equation [1.40].

The non-conservation form of equation [1.39] is equation [1.48] and its characteristic form is equation [1.49].

The conserved variable is the mass of contaminant AC per unit length of channel. The invariant quantity is the contaminant concentration C.

1.4. A convex law: the inviscid Burgers equation

1.4.1. Physical context – conservation form

The Burgers equation was first introduced [BUR 48] as a simple model to account for nonlinear advection and diffusion. A transformation was proposed independently by Hopf [HOP 50] and Cole [COL 51], whereby the Burgers equation is transformed into a linear PDE. In what follows, only the inviscid form of the equation is considered. If the initial and boundary conditions are not too complex, an analytical solution can be derived for the inviscid Burgers equation. This equation is often used to assess the performance of numerical methods for hyperbolic PDEs.

The inviscid Burgers equation can be viewed as a restriction of the Euler equations of gas dynamics introduced in Chapter 2. It is derived from the momentum equation under the assumption of negligible external forces, pressure gradients and momentum diffusion. If this is the case, the variations in the fluid velocity are due only to the initial distribution of momentum in the fluid. In contrast with the advection equation dealt with in section 1.3, the inviscid Burgers equation describes the advection of an active scalar, in that the value of u influences its own propagation speed.

Consider a slice of fluid, the (infinitesimal) length of which is denoted by δx and the cross-sectional area of which is equal to unity (Figure 1.11). The mass δm and the momentum δq of the slice of fluid are given by

[1.59]

where u and ρ are the fluid velocity and density respectively.

Figure 1.11.Perspective view of a basic volume of width δx and unit cross-sectional area

Both the density and momentum are transported at speed u of the fluid. The mass that crosses the unit section over a basic time interval δt is given by

[1.60]

The momentum that crosses the unit section over time interval δt is given by

[1.61]

Applying reasoning [1.11–1.16] to the conservation of density ρ leads to the so-called continuity equation

[1.62]

[1.63]

1.4.2. Characteristic form

The characteristic form of the inviscid Burgers equation is obtained by developing equation [1.63] into

[1.64]

Substituting equation [1.62] into equation [1.64] yields the following equation

[1.65]

Dividing by the density leads to an equation in u in non-conservation form

[1.66]

As shown in section 1.1.4 (see equations [1.24–1.27]), equation [1.66] is equivalent to

[1.67]

i.e.

[1.68]

[1.69]

Equation [1.69] is the so-called conservation form of the inviscid Burgers equation. It can be written in the form [1.1] by defining U and F as follows

[1.70]

1.4.3. Example: propagation of a perturbation in a fluid

The present example deals with the variations in the fluid density and velocity arising from an initial perturbation in the velocity profile. In what follows, time t is assumed to be small enough for the density and the velocity profiles to remain continuous, i.e., the derivatives of ρ and u with respect to time and space are assumed to take finite values. If this is not the case, the analysis hereafter becomes invalid and a specific treatment must be applied to the discontinuities (also called shocks). Such a treatment is not detailed in this chapter. It is covered in detail in Chapter 3.

Characteristic form [1.67] allows the behavior of the flow velocity to be determined. Points A and C move at speed u0, while point B moves at speed u1. Therefore, point B moves faster than A and C. The profile becomes smoother between A and B and steeper between B and C.

Figure 1.12.Initial profiles for the velocity (top) and the density (bottom)

Table 1.2.Problem parameters

It can easily be shown that the profile, although asymmetric, remains triangular. In fact, differentiating equation [1.66] with respect to x leads to the following equation

[1.71]

Swapping the derivatives in the first term of equation [1.71] and developing the second term yields

[1.72]

which gives the following PDE in ∂u/∂x

[1.73]

The reasoning developed in section 1.1.4 allows equation [1.73] to be rewritten in the form

[1.74]

The solution of equation [1.74] is

[1.75]

Figure 1.13.Determining the variations in ∂u/∂x along a characteristic from the initial velocity profile

Function (x) is given by

[1.76]

This function is piecewise constant and remains such at later times. Applying equation [1.75] along a characteristic line and using equation [1.76] for the initial condition leads to

[1.77]

[1.78]

[1.79]

i.e.

[1.80]

1.4.4. Summary

The conservation form of the inviscid Burgers equation is given by equation [1.69]. It can be written as in equation [1.1] by defining the conserved variable and the flux as in equations [1.70].

The non-conservation and characteristic forms of the equation are given by equation [1.66] and equation [1.67] respectively.

Both the conserved variable and the Riemann invariant are equal to the fluid velocity u.

1.5. Another convex law: the kinematic wave for free-surface hydraulics

1.5.1. Physical context – conservation form

The kinematic wave is a simplified form of the open channel flow equations (Figure 1.15). It is a scalar equation, which makes it easy to solve. A summary of the conditions under which the kinematic wave provides a valid approximation of the open channel flow can be found in [LIG 55]. The underlying hypotheses of the kinematic wave equations are the following:

– hypothesis (H1): the transverse dimensions of the channel are negligible compared to its longitudinal dimensions and the flow can be considered as one-dimensional;

– hypothesis (H2): inertia is negligible and the channel slope is steep enough for the energy slope to be equivalent to the channel bed slope.

Figure 1.15.Flow in an open channel. Cross-sectional view (a), side view (b)

Under such conditions, the energy loss originating from friction against the walls is balanced with the energy gained from the slope. The friction term is assumed to obey Strickler’s law

[1.81]

where A is the cross-sectional area of the channel, KStr is Strickler’s friction coefficient, RH is the hydraulic radius and Sf is the slope of the energy line. The hydraulic radius is defined as the ratio of the cross-sectional area A to the wetted perimeter χ

[1.82]

Since the slope of the energy line is assumed to be equal to the bottom slope S0, equation [1.81] becomes

[1.83]

A mass balance over a control volume defined between abscissa x0 and x0 + δx (Figure 1.15b) leads to the continuity equation (see section 1.2)

[1.84]

where Q is given by equation [1.83]. Note that Q is a known function of A at all points because S0 is known at all points and the law that relates the wetted perimeter χ to A is a known function of the geometry of the channel.

1.5.2. Non-conservation and characteristic forms

Equation [1.84] can be written in a non-conservation form by developing the derivative of Q so as to involve derivatives of A, KStr and S0

[1.85]

Note that the purpose is to rewrite equation [1.85] in the form [1.17], recalled here

To do this, it is sufficient to define U as the cross-sectional area A and λ and S′ as follows

[1.86]

Equation [1.85] thus becomes

[1.87]

The variations in the cross-sectional area A propagate at speed λ, as illustrated by the characteristic form of the equation

[1.88]

Note that the cross-sectional area is a conserved variable (see equation [1.84]) but it is not necessarily an invariant quantity (note the source term in equation [1.88]). If S′ is non-zero, A is not invariant along a characteristic line.

Also note that, Q being a function of A via relationship [1.81], its variations also propagate at speed λ. This can be shown in a more rigorous way by rewriting equation [1.87] as

[1.89]

Dividing by ∂A/∂Q yields

[1.90]

Substituting the first equation [1.86] into equation [1.90] leads to

[1.91]

which leads to the characteristic form for Q

[1.92]

1.5.3. Expression of the celerity

The wave celerity, also called the wave propagation speed, is equal to λQ/∂A. This derivative is difficult to estimate in the general case because the relationship between the wetted perimeter and the cross-sectional area can rarely be determined analytically. This difficulty can be removed by involving the water depth h (i.e., the difference between the elevation of the free surface and the lowest point of the river bed)

[1.93]

The analytical expression of ∂A/∂h is derived very easily by noting that an infinitesimal variation dh in the water depth results in an infinitesimal variation dA in the cross-sectional area (Figure 1.16)

[1.94]

hence the relationship

[1.95]

Figure 1.16.Definition sketch for the derivatives of A and RH with respect to h

Derivative ∂Q/∂h is obtained from equation [1.83]

[1.96]

The expression of ∂χ/∂h is determined by considering the variation dχ caused by an infinitesimal variation dh in the water depth

[1.97]

which leads to

[1.98]

Substituting equations [1.95] and [1.98] into equations [1.96] and [1.93] yields the final expression of λ

[1.99]

Note that the flow velocity u is given by

[1.100]

This allows equation [1.99] to be rewritten as

[1.101]

The following comments can be made:

– The celerity is different from the flow velocity. In the general case, λ is larger than u. In the particular case of a very wide rectangular channel, the ratio of the wave celerity to the flow velocity is exactly 5/3 (see section 1.5.4).

– The steeper the slope, the faster the wave for a given water depth. A large Strickler coefficient, which corresponds to a small friction term, also induces a large celerity. Friction therefore contributes to a reduced celerity.

– For a given value of the hydraulic radius, non-vertical embankments contribute to reduce the wave celerity. Indeed, the milder the embankments, the larger the tangents of angles θ1 and θ2 and, from equation [1.99], the smaller the value of λ. Note that this is true not only when the channel widens from the bottom to the top (i.e. for positive θ1 and θ2), but also when the channel narrows down (i.e. for negative θ1 and θ2). This is because, for a given increase in h, non-vertical embankments induce a larger increase in the wetted perimeter (thus, in the friction) than vertical embankments would. Friction, which is exerted at the walls, increases faster when the embankments are not vertical, which contributes to reduce the wave celerity.

– However, a section that widens from the bottom to the top yields a larger celerity than a section that narrows down from the bottom to the top because the hydraulic radius increases faster with h.

– In the general case, the profile of a perturbation that propagates in the channel is subject to deformation in time. This is because λ is a function of RH and b and therefore of A. Consequently, if A is variable, λ is also variable, which induces a deformation in the profile A(x). This also means that the profile shape of Q is also altered as it travels in the channel. The assumption of a constant shape for the perturbation is valid only if the amplitude of the perturbation is small enough for the celerity to be considered constant.

– A perturbation with a small amplitude may also be subjected to deformation as it travels in the channel. Indeed, if the Strickler coefficient or the bed slope are variable in space, the source term S′ in equation [1.17] is non-zero (see equation [1.86]) and A is not an invariant along a characteristic line.

To summarize, the assumption of zero deformation for a perturbation is valid only if the amplitude of the perturbation is small enough and if the parameters that govern friction (i.e. the Strickler coefficient and the channel slope) are constant in space.

1.5.4. Specific case: flow in a rectangular channel

Consider a rectangular channel, the (constant) width b of which is very large compared to water depth h. The Strickler coefficient and the slope are assumed to be constant. In this case the area and hydraulic radius are given by

[1.102]

If h is very small compared to b, the hydraulic radius is equivalent to the water depth h. This assumption is known as the “wide channel approximation”. The expression for the discharge is thus simplified into

[1.103]

Substituting equation [1.103] into equation [1.84] leads to the conservation form

[1.104]

The non-conservation form of equation [1.104] is

[1.105]

which is equivalent to the non-conservation form [1.87], recalled here

provided that λ and S′ are defined as

[1.106]

From equation [1.103], we obtain

[1.107]

1.5.5. Summary

The conservation form of the kinematic wave equation is given by equation [1.84]. Its non-conservation form is equation [1.87]. Its characteristic form is equation [1.88].

The cross-sectional area A and discharge Q