An Introductionto ComputationalFluid DynamicsTHE FINITE VOLUME METHOD

Second Edition

H K Versteeg and W Malalasekera

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An Introduction to ComputationalFluid Dynamics

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First published 1995Second edition published 2007

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Contents

Preface xiAcknowledgements xiii

1 Introduction 1

1.1 What is CFD? 11.2 How does a CFD code work? 21.3 Problem solving with CFD 41.4 Scope of this book 6

2 Conservation laws of fluid motion and boundary conditions 9

2.1 Governing equations of uid ow and heat transfer 92.1.1 Mass conservation in three dimensions 102.1.2 Rates of change following a uid particle and for

a uid element 122.1.3 Momentum equation in three dimensions 142.1.4 Energy equation in three dimensions 16

2.2 Equations of state 202.3 NavierStokes equations for a Newtonian uid 212.4 Conservative form of the governing equations of uid ow 242.5 Differential and integral forms of the general transport equations 242.6 Classication of physical behaviours 262.7 The role of characteristics in hyperbolic equations 292.8 Classication method for simple PDEs 322.9 Classication of uid ow equations 332.10 Auxiliary conditions for viscous uid ow equations 352.11 Problems in transonic and supersonic compressible ows 362.12 Summary 38

3 Turbulence and its modelling 40

3.1 What is turbulence? 403.2 Transition from laminar to turbulent ow 443.3 Descriptors of turbulent ow 49

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3.4 Characteristics of simple turbulent ows 523.4.1 Free turbulent ows 533.4.2 Flat plate boundary layer and pipe ow 573.4.3 Summary 61

3.5 The effect of turbulent uctuations on properties of the mean ow 613.6 Turbulent ow calculations 653.7 Reynolds-averaged NavierStokes equations and classical

turbulence models 663.7.1 Mixing length model 693.7.2 The k model 723.7.3 Reynolds stress equation models 803.7.4 Advanced turbulence models 853.7.5 Closing remarks RANS turbulence models 97

3.8 Large eddy simulation 983.8.1 Spacial ltering of unsteady NavierStokes equations 983.8.2 SmagorinksyLilly SGS model 1023.8.3 Higher-order SGS models 1043.8.4 Advanced SGS models 1053.8.5 Initial and boundary conditions for LES 1063.8.6 LES applications in ows with complex geometry 1083.8.7 General comments on performance of LES 109

3.9 Direct numerical simulation 1103.9.1 Numerical issues in DNS 1113.9.2 Some achievements of DNS 113

3.10 Summary 113

4 The finite volume method for diffusion problems 115

4.1 Introduction 1154.2 Finite volume method for one-dimensional steady state diffusion 1154.3 Worked examples: one-dimensional steady state diffusion 1184.4 Finite volume method for two-dimensional diffusion problems 1294.5 Finite volume method for three-dimensional diffusion problems 1314.6 Summary 132

5 The finite volume method for convection---diffusion problems 134

5.1 Introduction 1345.2 Steady one-dimensional convection and diffusion 1355.3 The central differencing scheme 1365.4 Properties of discretisation schemes 141

5.4.1 Conservativeness 1415.4.2 Boundedness 1435.4.3 Transportiveness 143

5.5 Assessment of the central differencing scheme for convectiondiffusion problems 145

5.6 The upwind differencing scheme 1465.6.1 Assessment of the upwind differencing scheme 149

5.7 The hybrid differencing scheme 1515.7.1 Assessment of the hybrid differencing scheme 154

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CONTENTS vii

5.7.2 Hybrid differencing scheme for multi-dimensionalconvectiondiffusion 154

5.8 The power-law scheme 1555.9 Higher-order differencing schemes for convectiondiffusion problems 156

5.9.1 Quadratic upwind differencing scheme: the QUICK scheme 1565.9.2 Assessment of the QUICK scheme 1625.9.3 Stability problems of the QUICK scheme and remedies 1635.9.4 General comments on the QUICK differencing scheme 164

5.10 TVD schemes 1645.10.1 Generalisation of upwind-biased discretisation schemes 1655.10.2 Total variation and TVD schemes 1675.10.3 Criteria for TVD schemes 1685.10.4 Flux limiter functions 1705.10.5 Implementation of TVD schemes 1715.10.6 Evaluation of TVD schemes 175

5.11 Summary 176

6 Solution algorithms for pressure---velocity coupling in steady flows 179

6.1 Introduction 1796.2 The staggered grid 1806.3 The momentum equations 1836.4 The SIMPLE algorithm 1866.5 Assembly of a complete method 1906.6 The SIMPLER algorithm 1916.7 The SIMPLEC algorithm 1936.8 The PISO algorithm 1936.9 General comments on SIMPLE, SIMPLER, SIMPLEC and PISO 1966.10 Worked examples of the SIMPLE algorithm 1976.11 Summary 211

7 Solution of discretised equations 212

7.1 Introduction 2127.2 The TDMA 2137.3 Application of the TDMA to two-dimensional problems 2157.4 Application of the TDMA to three-dimensional problems 2157.5 Examples 216

7.5.1 Closing remarks 2227.6 Point-iterative methods 223

7.6.1 Jacobi iteration method 2247.6.2 GaussSeidel iteration method 2257.6.3 Relaxation methods 226

7.7 Multigrid techniques 2297.7.1 An outline of a multigrid procedure 2317.7.2 An illustrative example 2327.7.3 Multigrid cycles 2397.7.4 Grid generation for the multigrid method 241

7.8 Summary 242

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8 The finite volume method for unsteady flows 243

8.1 Introduction 2438.2 One-dimensional unsteady heat conduction 243

8.2.1 Explicit scheme 2468.2.2 CrankNicolson scheme 2478.2.3 The fully implicit scheme 248

8.3 Illustrative examples 2498.4 Implicit method for two- and three-dimensional problems 2568.5 Discretisation of transient convectiondiffusion equation 2578.6 Worked example of transient convectiondiffusion using QUICK

differencing 2588.7 Solution procedures for unsteady ow calculations 262

8.7.1 Transient SIMPLE 2628.7.2 The transient PISO algorithm 263

8.8 Steady state calculations using the pseudo-transient approach 2658.9 A brief note on other transient schemes 2658.10 Summary 266

9 Implementation of boundary conditions 267

9.1 Introduction 2679.2 Inlet boundary conditions 2689.3 Outlet boundary conditions 2719.4 Wall boundary conditions 2739.5 The constant pressure boundary condition 2799.6 Symmetry boundary condition 2809.7 Periodic or cyclic boundary condition 2819.8 Potential pitfalls and nal remarks 281

10 Errors and uncertainty in CFD modelling 285

10.1 Errors and uncertainty in CFD 28510.2 Numerical errors 28610.3 Input uncertainty 28910.4 Physical model uncertainty 29110.5 Verication and validation 29310.6 Guidelines for best practice in CFD 29810.7 Reporting/documentation of CFD simulation inputs and results 30010.8 Summary 302

11 Methods for dealing with complex geometries 304

11.1 Introduction 30411.2 Body-tted co-ordinate grids for complex geometries 30511.3 Catesian vs. curvilinear grids an example 30611.4 Curvilinear grids difculties 308

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CONTENTS ix

11.5 Block-structured grids 31011.6 Unstructured grids 31111.7 Discretisation in unstructured grids 31211.8 Discretisation of the diffusion term 31611.9 Discretisation of the convective term 32011.10 Treatment of source terms 32411.11 Assembly of discretised equations 32511.12 Example calculations with unstructured grids 32911.13 Pressurevelocity coupling in unstructured meshes 33611.14 Staggered vs. co-located grid arrangements 33711.15 Extension of the face velocity interpolation method to

unstructured meshes 34011.16 Summary 342

12 CFD modelling of combustion 343

12.1 Introduction 34312.2 Application of the rst law of thermodynamics to a combustion system 34412.3 Enthalpy of formation 34512.4 Some important relationships and properties of gaseous mixtures 34612.5 Stoichiometry 34812.6 Equivalence ratio 34812.7 Adiabatic ame temperature 34912.8 Equilibrium and dissociation 35112.9 Mechanisms of combustion and chemical kinetics 35512.10 Overall reactions and intermediate reactions 35512.11 Reaction rate 35612.12 Detailed mechanisms 36112.13 Reduced mechanisms 36112.14 Governing equations for combusting ows 36312.15 The simple chemical reacting system (SCRS) 36712.16 Modelling of a laminar diffusion ame an example 37012.17 CFD calculation of turbulent non-premixed combustion 37612.18 SCRS model for turbulent combustion 38012.19 Probability density function approach 38012.20 Beta pdf 38212.21 The chemical equilibrium model 38412.22 Eddy break-up model of combustion 38512.23 Eddy dissipation concept 38812.24 Laminar amelet model 38812.25 Generation of laminar amelet libraries 39012.26 Statistics of the non-equilibrium parameter 39912.27 Pollutant formation in combustion 40012.28 Modelling of thermal NO formation in combustion 40112.29 Flamelet-based NO modelling 40212.30 An example to illustrate laminar amelet modelling and NO

modelling of a turbulent ame 40312.31 Other models for non-premixed combustion 41512.32 Modelling of premixed combustion 41512.33 Summary 416

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13 Numerical calculation of radiative heat transfer 417

13.1 Introduction 41713.2 Governing equations of radiative heat transfer 42413.3 Solution methods 42613.4 Four popular radiation calculation techniques suitable for CFD 427

13.4.1 The Monte Carlo method 42713.4.2 The discrete transfer method 42913.4.3 Ray tracing 43313.4.4 The discrete ordinates method 43313.4.5 The nite volume method 437

13.5 Illustrative examples 43713.6 Calculation of radiative properties in gaseous mixtures 44213.7 Summary 443

Appendix A Accuracy of a flow simulation 445Appendix B Non-uniform grids 448Appendix C Calculation of source terms 450Appendix D Limiter functions used in Chapter 5 452Appendix E Derivation of one-dimensional governing equations for

steady, incompressible flow through a planar nozzle 456Appendix F Alternative derivation for the term (n . grad Ai) in

Chapter 11 459Appendix G Some examples 462Bibliography 472Index 495

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We were pleasantly surprised by the ready acceptance of the rst edition ofour book by the CFD community and by the amount of positive feedbackreceived over a period of 10 years. To us this has provided justication of our original plan, which was to provide an accessible introduction to thisfast-growing topic to support teaching at senior undergraduate level, post-graduate research and new industrial users of commercial CFD codes. Oursecond edition seeks to enhance and update. The structure and didacticapproach of the rst edition have been retained without change, but aug-mented by a selection of the most important developments in CFD.

In our treatment of the physics of uid ows we have added a summaryof the basic ideas underpinning large-eddy simulation (LES) and directnumerical simulation (DNS). These resource-intensive turbulence predic-tion techniques are likely to have a major impact in the medium term onCFD due to the increased availability of high-end computing capability.

Over the last decade a number of new discretisation techniques and solution approaches have come to the fore in commercial CFD codes. Toreect these developments we have included summaries of TVD techniques,which give stable, higher-order accurate solutions of convectiondiffusionproblems, and of iterative techniques and multi-grid accelerators that arenow commonly used for the solution of systems of discretised equations. Wehave also added examples of the SIMPLE algorithm for pressurevelocitycoupling to illustrate its workings.

At the time of writing our rst edition, CFD was rmly established in theaerospace, automotive and power generation sectors. Subsequently, it hasspread throughout engineering industry. This has gone hand in hand withmajor improvements in the treatment of complex geometries. We havedevoted a new chapter to describing key aspects of unstructured meshing techniques that have made this possible.

Application of CFD results in industrial research and design cruciallyhinges on condence in its outcomes. We have included a new chapter onuncertainty in CFD results. Given the rapid growth in CFD applications itis difcult to cover, within the space of a single introductory volume, even asmall part of the submodelling methodology that is now included in manygeneral-purpose CFD codes. Our selection of advanced application materialcovers combustion and radiation algorithms, which reects our local perspec-tive as mechanical engineers with interest in internal ow and combustion.

Finally, we thank colleagues in UK and overseas universities who haveencouraged us with positive responses and constructive comments on ourrst edition and our proposals for a second edition. We are also grateful toseveral colleagues and postgraduate researchers who have given help in the

Preface

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development of material, particularly Dr Jonathan Henson, Dr MamdudHossain, Dr Naminda Kandamby, Dr Andreas Haselbacher, MurthyRavikanti-Veera and Anand Odedra.

August 2006 H.K. VersteegLoughborough W. Malalasekera

xii PREFACE

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The authors wish to acknowledge the following persons, organisations andpublishers for permission to reproduce from their publications in this book.

Professor H. Nagib for Figure 3.2, from Van Dyke, M. (1982) An Album ofFluid Motion, The Parabolic Press, Stanford; Professor S. Taneda and theJapan Society of Mechanical Engineers for Figure 3.7, from Nakayama, Y.(1988) Visualised Flow, compiled by the Japan Society of MechanicalEngineers, Pergamon Press; Professor W. Fiszdon and the Polish Academyof Sciences for Figure 3.9, from Van Dyke, M. (1982) An Album of FluidMotion, The Parabolic Press, Stanford; Figures 3.11 and 3.14 fromSchlichting, H. (1979) Boundary Layer Theory, 7th edn, reproduced withpermission of The McGraw-Hill Companies; Figure 3.15 reprinted by per-mission of Elsevier Science from Calculation of Turbulent Reacting Flows:A Review by W. P. Jones and J. H. Whitelaw, Combustion and Flame, Vol.48, pp. 126, 1982 by The Combustion Institute; Figure 3.16 fromLeschziner, M. A. (2000) The Computation of Turbulent EngineeringFlows, in R. Peyret and E. Krause (eds) Advanced Turbulent FlowComputations, reproduced with permission from Springer Wien NewYork;Figures 3.17 and 3.18 reprinted from International Journal of Heat and FluidFlow, Vol. 23, Moin, P., Advances in Large Eddy Simulation Methodologyfor Complex Flows, pp. 710712, 2002, with permission from Elsevier;Dr Andreas Haselbacher for Figures 11.2, 11.9 and 11.11, from A Grid-Transparent Numerical Method for Compressible Viscous Flows on MixedUnstructured Grids, thesis, Loughborough University; The CombustionInstitute for Figure 12.8, from Magnussen, B. F. and Hjertager, B. H. (1976)On the Mathematical Modelling of Turbulent Combustion with SpecialEmphasis on Soot Formation and Combustion, Sixteenth Symposium (Int.)on Combustion, and Figure 12.9, from Gosman, A. D., Lockwood, F. C. andSalooja, A. P. (1978) The Prediction of Cylindrical Furnaces GaseousFuelled with Premixed and Diffusion Burners, Seventeenth Symposium(Int.) on Combustion; Gordon and Breach Science Publishers for Figure12.10, from Nikjooy, M., So, R. M. C. and Peck, R. E. (1988) Modelling ofJet- and Swirl-stabilised Reacting Flows in Axisymmetric Combustors,Combust. Sci and Tech. 1988 by Gordon and Breach Science Publishers.

In some instances we have been unable to trace the owners of copyrightmaterial, and we would appreciate any information that would enable us todo so.

Acknowledgements

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Computational uid dynamics or CFD is the analysis of systems involvinguid ow, heat transfer and associated phenomena such as chemical reactionsby means of computer-based simulation. The technique is very powerful andspans a wide range of industrial and non-industrial application areas. Someexamples are:

aerodynamics of aircraft and vehicles: lift and drag hydrodynamics of ships power plant: combustion in internal combustion engines and gas

turbines turbomachinery: ows inside rotating passages, diffusers etc. electrical and electronic engineering: cooling of equipment including

microcircuits chemical process engineering: mixing and separation, polymer moulding external and internal environment of buildings: wind loading and

heating/ventilation marine engineering: loads on off-shore structures environmental engineering: distribution of pollutants and efuents hydrology and oceanography: ows in rivers, estuaries, oceans meteorology: weather prediction biomedical engineering: blood ows through arteries and veins

From the 1960s onwards the aerospace industry has integrated CFD tech-niques into the design, R&D and manufacture of aircraft and jet engines.More recently the methods have been applied to the design of internal combustion engines, combustion chambers of gas turbines and furnaces.Furthermore, motor vehicle manufacturers now routinely predict drag forces,under-bonnet air ows and the in-car environment with CFD. IncreasinglyCFD is becoming a vital component in the design of industrial products andprocesses.

The ultimate aim of developments in the CFD eld is to provide a capability comparable with other CAE (computer-aided engineering) toolssuch as stress analysis codes. The main reason why CFD has lagged behindis the tremendous complexity of the underlying behaviour, which precludesa description of uid ows that is at the same time economical and sufcientlycomplete. The availability of affordable high-performance computing hard-ware and the introduction of user-friendly interfaces have led to a recentupsurge of interest, and CFD has entered into the wider industrial commun-ity since the 1990s.

What is CFD?1.1

Chapter one Introduction

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We estimate the minimum cost of suitable hardware to be between 5,000and 10,000 (plus annual maintenance costs). The perpetual licence fee forcommercial software typically ranges from 10,000 to 50,000 depending onthe number of added extras required. CFD software houses can usually arrangeannual licences as an alternative. Clearly the investment costs of a CFD cap-ability are not small, but the total expense is not normally as great as that of ahigh-quality experimental facility. Moreover, there are several unique advant-ages of CFD over experiment-based approaches to uid systems design:

substantial reduction of lead times and costs of new designs ability to study systems where controlled experiments are difcult or

impossible to perform (e.g. very large systems) ability to study systems under hazardous conditions at and beyond their

normal performance limits (e.g. safety studies and accident scenarios) practically unlimited level of detail of results

The variable cost of an experiment, in terms of facility hire and/or person-hour costs, is proportional to the number of data points and the number of congurations tested. In contrast, CFD codes can produce extremely largevolumes of results at virtually no added expense, and it is very cheap to per-form parametric studies, for instance to optimise equipment performance.

Below we look at the overall structure of a CFD code and discuss the role of the individual building blocks. We also note that, in addition to a substantial investment outlay, an organisation needs qualied people to runthe codes and communicate their results, and briey consider the modellingskills required by CFD users. We complete this otherwise upbeat section bywondering whether the next constraint to the further spread of CFD amongstthe industrial community could be a scarcity of suitably trained personnelinstead of availability and/or cost of hardware and software.

CFD codes are structured around the numerical algorithms that can tackleuid ow problems. In order to provide easy access to their solving power all commercial CFD packages include sophisticated user interfaces to inputproblem parameters and to examine the results. Hence all codes contain threemain elements: (i) a pre-processor, (ii) a solver and (iii) a post-processor. Webriey examine the function of each of these elements within the context ofa CFD code.

Pre-processor

Pre-processing consists of the input of a ow problem to a CFD program bymeans of an operator-friendly interface and the subsequent transformationof this input into a form suitable for use by the solver. The user activities atthe pre-processing stage involve:

Denition of the geometry of the region of interest: the computationaldomain

Grid generation the sub-division of the domain into a number of smaller, non-overlapping sub-domains: a grid (or mesh) of cells(or control volumes or elements)

Selection of the physical and chemical phenomena that need to bemodelled

2 CHAPTER 1 INTRODUCTION

How does a CFD code work?

1.2

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1.2 HOW DOES A CFD CODE WORK? 3

Denition of uid properties Specication of appropriate boundary conditions at cells which coincide

with or touch the domain boundary

The solution to a ow problem (velocity, pressure, temperature etc.) is denedat nodes inside each cell. The accuracy of a CFD solution is governed by thenumber of cells in the grid. In general, the larger the number of cells, the better the solution accuracy. Both the accuracy of a solution and its cost interms of necessary computer hardware and calculation time are dependenton the neness of the grid. Optimal meshes are often non-uniform: ner inareas where large variations occur from point to point and coarser in regionswith relatively little change. Efforts are under way to develop CFD codeswith a (self-)adaptive meshing capability. Ultimately such programs will auto-matically rene the grid in areas of rapid variations. A substantial amount of basic development work still needs to be done before these techniques arerobust enough to be incorporated into commercial CFD codes. At present it is still up to the skills of the CFD user to design a grid that is a suitablecompromise between desired accuracy and solution cost.

Over 50% of the time spent in industry on a CFD project is devoted tothe denition of the domain geometry and grid generation. In order to max-imise productivity of CFD personnel all the major codes now include theirown CAD-style interface and/or facilities to import data from proprietarysurface modellers and mesh generators such as PATRAN and I-DEAS. Up-to-date pre-processors also give the user access to libraries of materialproperties for common uids and a facility to invoke special physical andchemical process models (e.g. turbulence models, radiative heat transfer,combustion models) alongside the main uid ow equations.

Solver

There are three distinct streams of numerical solution techniques: nite difference, nite element and spectral methods. We shall be solely concernedwith the nite volume method, a special nite difference formulation that iscentral to the most well-established CFD codes: CFX/ANSYS, FLUENT,PHOENICS and STAR-CD. In outline the numerical algorithm consists ofthe following steps:

Integration of the governing equations of uid ow over all the (nite)control volumes of the domain

Discretisation conversion of the resulting integral equations into asystem of algebraic equations

Solution of the algebraic equations by an iterative method

The rst step, the control volume integration, distinguishes the nite volumemethod from all other CFD techniques. The resulting statements expressthe (exact) conservation of relevant properties for each nite size cell. Thisclear relationship between the numerical algorithm and the underlying physical conservation principle forms one of the main attractions of the nitevolume method and makes its concepts much more simple to understand byengineers than the nite element and spectral methods. The conservation of a general ow variable , e.g. a velocity component or enthalpy, within a nite control volume can be expressed as a balance between the variousprocesses tending to increase or decrease it. In words we have:

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GRate of change J GNet rate of J GNet rate of J GNet rate of JHof in the K Hincrease of K Hincrease of K Hcreation of KHcontrol volumeK = H due to K + H due to K + H inside theKHwith respect toK Hconvection intoK Hdiffusion intoK Hcontrol KHtime K Hthe control K Hthe control K Hvolume KI L Ivolume L Ivolume L I LCFD codes contain discretisation techniques suitable for the treatment of the key transport phenomena, convection (transport due to uid ow) anddiffusion (transport due to variations of from point to point) as well as forthe source terms (associated with the creation or destruction of ) and therate of change with respect to time. The underlying physical phenomena are complex and non-linear so an iterative solution approach is required. The most popular solution procedures are by the TDMA (tri-diagonalmatrix algorithm) line-by-line solver of the algebraic equations and the SIMPLE algorithm to ensure correct linkage between pressure and velocity.Commercial codes may also give the user a selection of further, more recent, techniques such as GaussSeidel point iterative techniques withmultigrid accelerators and conjugate gradient methods.

Post-processor

As in pre-processing, a huge amount of development work has recently takenplace in the post-processing eld. Due to the increased popularity of engin-eering workstations, many of which have outstanding graphics capabilities,the leading CFD packages are now equipped with versatile data visualisationtools. These include:

Domain geometry and grid display Vector plots Line and shaded contour plots 2D and 3D surface plots Particle tracking View manipulation (translation, rotation, scaling etc.) Colour PostScript output

More recently these facilities may also include animation for dynamic resultdisplay, and in addition to graphics all codes produce trusty alphanumericoutput and have data export facilities for further manipulation external to thecode. As in many other branches of CAE, the graphics output capabilities of CFD codes have revolutionised the communication of ideas to the non-specialist.

In solving uid ow problems we need to be aware that the underlyingphysics is complex and the results generated by a CFD code are at best asgood as the physics (and chemistry) embedded in it and at worst as good asits operator. Elaborating on the latter issue rst, the user of a code must haveskills in a number of areas. Prior to setting up and running a CFD simula-tion there is a stage of identication and formulation of the ow problem interms of the physical and chemical phenomena that need to be considered.Typical decisions that might be needed are whether to model a problem intwo or three dimensions, to exclude the effects of ambient temperature

4 CHAPTER 1 INTRODUCTION

Problem solving with CFD

1.3

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1.3 PROBLEM SOLVING WITH CFD 5

or pressure variations on the density of an air ow, to choose to solve the turbulent ow equations or to neglect the effects of small air bubbles dis-solved in tap water. To make the right choices requires good modellingskills, because in all but the simplest problems we need to make assumptionsto reduce the complexity to a manageable level whilst preserving the salientfeatures of the problem at hand. It is the appropriateness of the simplica-tions introduced at this stage that at least partly governs the quality of theinformation generated by CFD, so the user must continually be aware of allthe assumptions, clear-cut and tacit ones, that have been made.

Performing the computation itself requires operator skills of a differentkind. Specication of the domain geometry and grid design are the maintasks at the input stage and subsequently the user needs to obtain a success-ful simulation result. The two aspects that characterise such a result are convergence and grid independence. The solution algorithm is iterative innature, and in a converged solution the so-called residuals measures of theoverall conservation of the ow properties are very small. Progress towardsa converged solution can be greatly assisted by careful selection of the set-tings of various relaxation factors and acceleration devices. There are nostraightforward guidelines for making these choices since they are problemdependent. Optimisation of the solution speed requires considerable experi-ence with the code itself, which can only be acquired by extensive use. Thereis no formal way of estimating the errors introduced by inadequate griddesign for a general ow. Good initial grid design relies largely on an insightinto the expected properties of the ow. A background in the uid dynamicsof the particular problem certainly helps, and experience with gridding ofsimilar problems is also invaluable. The only way to eliminate errors due to coarseness of a grid is to perform a grid dependence study, which is a procedure of successive renement of an initially coarse grid until certain key results do not change. Then the simulation is grid independent. A sys-tematic search for grid-independent results forms an essential part of allhigh-quality CFD studies.

Every numerical algorithm has its own characteristic error patterns. Well-known CFD euphemisms for the word error are terms such as numerical diffusion, false diffusion or even numerical ow. The likely error patternscan only be guessed on the basis of a thorough knowledge of the algorithms.At the end of a simulation the user must make a judgement whether theresults are good enough. It is impossible to assess the validity of the modelsof physics and chemistry embedded in a program as complex as a CFD codeor the accuracy of its nal results by any means other than comparison withexperimental test work. Anyone wishing to use CFD in a serious way mustrealise that it is no substitute for experimentation, but a very powerful additional problem solving tool. Validation of a CFD code requires highlydetailed information concerning the boundary conditions of a problem, andgenerates a large volume of results. To validate these in a meaningful way itis necessary to produce experimental data of similar scope. This may involvea programme of ow velocity measurements with hot-wire anemometry,laser Doppler anemometry or particle image velocimetry. However, if theenvironment is too hostile for such delicate laboratory equipment or if it issimply not available, static pressure and temperature measurements com-plemented by pitot-static tube traverses can also be useful to validate someaspects of a ow eld.

Sometimes the facilities to perform experimental work may not (yet)exist, in which case the CFD user must rely on (i) previous experience,

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(ii) comparisons with analytical solutions of similar but simpler ows and (iii)comparisons with high-quality data from closely related problems reported inthe literature. Excellent sources of the last type of information can be foundin Transactions of the ASME (in particular the Journal of Fluids Engineering,Journal of Engineering for Gas Turbines and Power and Journal of Heat Transfer),AIAA Journal, Journal of Fluid Mechanics and Proceedings of the IMechE.

CFD computation involves the creation of a set of numbers that (hope-fully) constitutes a realistic approximation of a real-life system. One of theadvantages of CFD is that the user has an almost unlimited choice of thelevel of detail of the results, but in the prescient words of C. Hastings, writtenin the pre-IT days of 1955: The purpose of computing is insight not num-bers. The underlying message is rightly cautionary. We should make surethat the main outcome of any CFD exercise is improved understanding ofthe behaviour of a system, but since there are no cast-iron guarantees withregard to the accuracy of a simulation, we need to validate our results fre-quently and stringently.

It is clear that there are guidelines for good operating practice which canassist the user of a CFD code, and repeated validation plays a key role as thenal quality control mechanism. However, the main ingredients for successin CFD are experience and a thorough understanding of the physics of uidows and the fundamentals of the numerical algorithms. Without these it isvery unlikely that the user will get the best out of a code. It is the intentionof this book to provide all the necessary background material for a goodunderstanding of the internal workings of a CFD code and its successfuloperation.

This book seeks to present all the fundamental material needed for good simulation of uid ows by means of the nite volume method, and is splitinto three parts. The rst part, consisting of Chapters 2 and 3, is concernedwith the fundamentals of uid ows in three dimensions and turbulence.The treatment starts with the derivation of the governing partial differentialequations of uid ows in Cartesian co-ordinates. We stress the commonal-ities in the resulting conservation equations and arrive at the so-called trans-port equation, which is the basic form for the development of the numericalalgorithms that are to follow. Moreover, we look at the auxiliary conditionsrequired to specify a well-posed problem from a general perspective andquote a set of recommended boundary conditions and a number of derivedones that are useful in CFD practice. Chapter 3 represents the developmentof the concepts of turbulence that are necessary for a full appreciation of thener details of CFD in many engineering applications. We look at thephysics of turbulence and the characteristics of some simple turbulent owsand at the consequences of the appearance of random uctuations on the owequations. The resulting equations are not a closed or solvable set unless weintroduce a turbulence model. We discuss the principal turbulence modelsthat are used in industrial CFD, focusing our attention on the k model,which is very popular in general-purpose ow computations. Some of themore recent developments that are likely to have a major impact on CFD inthe near future are also reviewed.

Readers who are already familiar with the derivation of the 3D ow equa-tions can move on to the second part without loss of continuity. Apart from

6 CHAPTER 1 INTRODUCTION

Scope of this book

1.4

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1.4 SCOPE OF THIS BOOK 7

the discussion of the k turbulence model, to which we return later, thematerial in Chapters 2 and 3 is largely self-contained. This allows the use of this book by those wishing to concentrate principally on the numericalalgorithms, but requiring an overview of the uid dynamics and the math-ematics behind it for occasional reference in the same text.

The second part of the book is devoted to the numerical algorithms of the nite volume method and covers Chapters 4 to 9. Discretisation schemesand solution procedures for steady ows are discussed in Chapters 4 to 7.Chapter 4 describes the basic approach and derives the central differencescheme for diffusion phenomena. In Chapter 5 we emphasise the key prop-erties of discretisation schemes, conservativeness, boundedness and trans-portiveness, which are used as a basis for the further development of theupwind, hybrid, QUICK and TVD schemes for the discretisation of con-vective terms. The non-linear nature of the underlying ow phenomena andthe linkage between pressure and velocity in variable density uid owsrequires special treatment, which is the subject of Chapter 6. We introducethe SIMPLE algorithm and some of its more recent derivatives and also discuss the PISO algorithm. In Chapter 7 we describe algorithms for thesolution of the systems of algebraic equations that appear after the discret-isation stage. We focus our attention on the well-known TDMA algorithm,which was the basis of early CFD codes, and point iterative methods withmultigrid accelerators, which are the current solvers of choice.

The theory behind all the numerical methods is developed around a set of worked examples which can be easily programmed on a PC. This pres-entation gives the opportunity for a detailed examination of all aspects of thediscretisation schemes, which form the basic building blocks of practicalCFD codes, including the characteristics of their solutions.

In Chapter 8 we assess the advantages and limitations of various schemesto deal with unsteady ows, and Chapter 9 completes the development of thenumerical algorithms by considering the practical implementation of themost common boundary conditions in the nite volume method.

The book is primarily aimed to support those who have access to a CFDpackage, so that the issues raised in the text can be explored in greater depth.The solution procedures are nevertheless sufciently well documented forthe interested reader to be able to start developing a CFD code from scratch.

The third part of the book consists of a selection of topics relating to theapplication of the nite volume method to complex industrial problems. InChapter 10 we review aspects of accuracy and uncertainty in CFD. It is notpossible to predict the error in a CFD result from rst principles, which creates some problems for the industrial user who wishes to evolve equip-ment design on the basis of insights gleaned from CFD. In order to addressthis issue a systematic process has been developed to assist in the quantica-tion of the uncertainty of CFD output. We discuss methods, the concepts ofverication and validation, and give a summary of rules for best practice thathave been developed by the CFD community to assist users. In Chapter 11we discuss techniques to cope with complex geometry. We review variousapproaches based on structured meshes: Cartesian co-ordinate systems, gen-eralised co-ordinate systems based on transformations, and block-structuredgrids, which enable design of specic meshes tailored to the needs of dif-ferent parts of geometry. We give details of the implementation of the nitevolume method on unstructured meshes. These are not based on a grid oflines to dene nodal positions and can include control volumes that can have

ANIN_C01.qxd 29/12/2006 09:54 AM Page 7

any shape. Consequently, unstructured meshes have the ability to match the boundary shape of CFD solution domains of arbitrary complexity. Thisgreatly facilitates the design and renement of meshes, so that unstructuredmeshes are the most popular method in industrial CFD applications. Theremaining Chapters 12 and 13 are concerned with one of the most signicantengineering applications of CFD: energy technology and combusting systems.In order to provide a self-contained introduction to the most importantaspects of CFD in reacting ows, we introduce in Chapter 12 the basic thermodynamic and chemical concepts of combustion and review the mostimportant models of combustion. Our particular focus is the laminar ameletmodel of non-premixed turbulent combustion, which is the most widelyresearched model with capabilities to predict the main combustion reactionand pollutant species concentrations. In the nal Chapter 13 we discuss CFDtechniques to predict radiative heat transfer, a good understanding of whichis necessary for accurate combustion calculations.

8 CHAPTER 1 INTRODUCTION

ANIN_C01.qxd 29/12/2006 09:54 AM Page 8

In this chapter we develop the mathematical basis for a comprehensive general-purpose model of uid ow and heat transfer from the basic prin-ciples of conservation of mass, momentum and energy. This leads to the governing equations of uid ow and a discussion of the necessary auxiliaryconditions initial and boundary conditions. The main issues covered in thiscontext are:

Derivation of the system of partial differential equations (PDEs) thatgovern ows in Cartesian (x, y, z) co-ordinates

Thermodynamic equations of state Newtonian model of viscous stresses leading to the NavierStokes

equations Commonalities between the governing PDEs and the denition of the

transport equation Integrated forms of the transport equation over a nite time interval and

a nite control volume Classication of physical behaviours into three categories: elliptic,

parabolic and hyperbolic Appropriate boundary conditions for each category Classication of uid ows Auxiliary conditions for viscous uid ows Problems with boundary condition specication in high Reynolds

number and high Mach number ows

The governing equations of uid ow represent mathematical statements ofthe conservation laws of physics:

The mass of a uid is conserved The rate of change of momentum equals the sum of the forces on a uid

particle (Newtons second law) The rate of change of energy is equal to the sum of the rate of heat

addition to and the rate of work done on a uid particle (rst law ofthermodynamics)

The uid will be regarded as a continuum. For the analysis of uid ows at macroscopic length scales (say 1 m and larger) the molecular structure of matter and molecular motions may be ignored. We describe the behaviourof the uid in terms of macroscopic properties, such as velocity, pressure,density and temperature, and their space and time derivatives. These may

Chapter two Conservation laws of fluid motion and boundary conditions

Governing equations of fluid

flow and heat transfer

2.1

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10 CHAPTER 2 CONSERVATION LAWS OF FLUID MOTION

Figure 2.1 Fluid element forconservation laws

be thought of as averages over suitably large numbers of molecules. A uidparticle or point in a uid is then the smallest possible element of uid whosemacroscopic properties are not inuenced by individual molecules.

We consider such a small element of uid with sides x, y and z(Figure 2.1).

The six faces are labelled N, S, E, W, T and B, which stands for North,South, East, West, Top and Bottom. The positive directions along the co-ordinate axes are also given. The centre of the element is located at position(x, y, z). A systematic account of changes in the mass, momentum and energyof the uid element due to uid ow across its boundaries and, where appro-priate, due to the action of sources inside the element, leads to the uid owequations.

All uid properties are functions of space and time so we would strictlyneed to write (x, y, z, t), p(x, y, z, t), T(x, y, z, t) and u(x, y, z, t) for thedensity, pressure, temperature and the velocity vector respectively. To avoidunduly cumbersome notation we will not explicitly state the dependence onspace co-ordinates and time. For instance, the density at the centre (x, y, z)of a uid element at time t is denoted by and the x-derivative of, say, pres-sure p at (x, y, z) and time t by p/x. This practice will also be followed forall other uid properties.

The element under consideration is so small that uid properties at thefaces can be expressed accurately enough by means of the rst two terms of a Taylor series expansion. So, for example, the pressure at the W and Efaces, which are both at a distance of 12 x from the element centre, can beexpressed as

p x and p + x

2.1.1 Mass conservation in three dimensions

The rst step in the derivation of the mass conservation equation is to writedown a mass balance for the uid element:

Rate of increase Net rate of flow of mass in fluid = of mass into element fluid element

1

2

px

1

2

px

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2.1 GOVERNING EQUATIONS OF FLUID FLOW AND HEAT TRANSFER 11

Figure 2.2 Mass ows in andout of uid element

The rate of increase of mass in the uid element is

(xyz) = xyz (2.1)

Next we need to account for the mass ow rate across a face of the element,which is given by the product of density, area and the velocity componentnormal to the face. From Figure 2.2 it can be seen that the net rate of ow ofmass into the element across its boundaries is given by

u x yz u + x yz

+ v y xz v + y xz

+ w z xy w + z xy (2.2)

Flows which are directed into the element produce an increase of mass in theelement and get a positive sign and those ows that are leaving the elementare given a negative sign.

DEF1

2

(w)z

ABCDEF

1

2

(w)z

ABC

DEF1

2

(v)y

ABCDEF

1

2

(v)y

ABC

DEF1

2

(u)x

ABCDEF

1

2

(u)x

ABC

t

t

The rate of increase of mass inside the element (2.1) is now equated to thenet rate of ow of mass into the element across its faces (2.2). All terms of theresulting mass balance are arranged on the left hand side of the equals signand the expression is divided by the element volume xyz. This yields

+ + + = 0 (2.3)

or in more compact vector notation

+ div(u) = 0 (2.4)

Equation (2.4) is the unsteady, three-dimensional mass conservationor continuity equation at a point in a compressible fluid. The rst term

t

(w)z

(v)y

(u)x

t

ANIN_C02.qxd 29/12/2006 09:55 AM Page 11

on the left hand side is the rate of change in time of the density (mass per unitvolume). The second term describes the net ow of mass out of the elementacross its boundaries and is called the convective term.

For an incompressible fluid (i.e. a liquid) the density is constant andequation (2.4) becomes

div u = 0 (2.5)

or in longhand notation

+ + = 0 (2.6)

2.1.2 Rates of change following a fluid particle and for a fluidelement

The momentum and energy conservation laws make statements regardingchanges of properties of a uid particle. This is termed the Lagrangianapproach. Each property of such a particle is a function of the position (x, y, z) of the particle and time t. Let the value of a property per unit massbe denoted by . The total or substantive derivative of with respect to timefollowing a uid particle, written as D/Dt, is

= + + +

A uid particle follows the ow, so dx/dt = u, dy/dt = v and dz/dt = w.Hence the substantive derivative of is given by

= + u + v + w = + u . grad (2.7)

D/Dt denes rate of change of property per unit mass. It is possible to develop numerical methods for uid ow calculations based on theLagrangian approach, i.e. by tracking the motion and computing the rates ofchange of conserved properties for collections of uid particles. However,it is far more common to develop equations for collections of uid elementsmaking up a region xed in space, for example a region dened by a duct, apump, a furnace or similar piece of engineering equipment. This is termedthe Eulerian approach.

As in the case of the mass conservation equation, we are interested indeveloping equations for rates of change per unit volume. The rate of changeof property per unit volume for a uid particle is given by the product ofD/Dt and density , hence

= + u . grad (2.8)

The most useful forms of the conservation laws for uid ow computationare concerned with changes of a ow property for a uid element that is stationary in space. The relationship between the substantive derivative of ,which follows a uid particle, and rate of change of for a uid element isnow developed.

DEFt

ABCDDt

t

z

y

x

t

DDt

dz

dt

z

dy

dt

y

dx

dt

x

t

DDt

wz

vy

ux

12 CHAPTER 2 CONSERVATION LAWS OF FLUID MOTION

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2.1 GOVERNING EQUATIONS OF FLUID FLOW AND HEAT TRANSFER 13

The mass conservation equation contains the mass per unit volume (i.e.the density ) as the conserved quantity. The sum of the rate of change ofdensity in time and the convective term in the mass conservation equation(2.4) for a uid element is

+ div(u)

The generalisation of these terms for an arbitrary conserved property is

+ div(u) (2.9)

Formula (2.9) expresses the rate of change in time of per unit volume plusthe net ow of out of the uid element per unit volume. It is now rewrittento illustrate its relationship with the substantive derivative of :

+ div(u) = + u . grad + + div(u)

= (2.10)

The term [(/t) + div(u)] is equal to zero by virtue of mass conserva-tion (2.4). In words, relationship (2.10) states

Rate of increase Net rate of ow Rate of increase of of uid + of out of = of for a element uid element uid particle

To construct the three components of the momentum equation and theenergy equation the relevant entries for and their rates of change per unitvolume as dened in (2.8) and (2.10) are given below:

x-momentum u + div(uu)

y-momentum v + div(vu)

z-momentum w + div(wu)

energy E + div(Eu)

Both the conservative (or divergence) form and non-conservative form of therate of change can be used as alternatives to express the conservation of aphysical quantity. The non-conservative forms are used in the derivations ofmomentum and energy equations for a uid ow in sections 2.4 and 2.5 forbrevity of notation and to emphasise that the conservation laws are funda-mentally conceived as statements that apply to a particle of uid. In the nal

(E)t

DE

Dt

(w)t

Dw

Dt

(v)t

Dv

Dt

(u)t

Du

Dt

DDt

JKLt

GHIJKL

t

GHI()

t

()t

t

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14 CHAPTER 2 CONSERVATION LAWS OF FLUID MOTION

Figure 2.3 Stress componentson three faces of uid element

section 2.8 we will return to the conservative form that is used in nite vol-ume CFD calculations.

2.1.3 Momentum equation in three dimensions

Newtons second law states that the rate of change of momentum of a uidparticle equals the sum of the forces on the particle:

Rate of increase of Sum of forces momentum of = on uid particle uid particle

The rates of increase of x-, y- and z-momentum per unit volume of auid particle are given by

(2.11)

We distinguish two types of forces on uid particles:

surface forces pressure forces viscous forces gravity force

body forces centrifugal force Coriolis force electromagnetic force

It is common practice to highlight the contributions due to the surface forcesas separate terms in the momentum equation and to include the effects ofbody forces as source terms.

The state of stress of a uid element is dened in terms of the pressureand the nine viscous stress components shown in Figure 2.3. The pressure,a normal stress, is denoted by p. Viscous stresses are denoted by . The usualsufx notation ij is applied to indicate the direction of the viscous stresses.The sufces i and j in ij indicate that the stress component acts in the j-direction on a surface normal to the i-direction.

Dw

Dt

Dv

Dt

Du

Dt

First we consider the x-components of the forces due to pressure p andstress components xx, yx and zx shown in Figure 2.4. The magnitude of a

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2.1 GOVERNING EQUATIONS OF FLUID FLOW AND HEAT TRANSFER 15

force resulting from a surface stress is the product of stress and area. Forcesaligned with the direction of a co-ordinate axis get a positive sign and thosein the opposite direction a negative sign. The net force in the x-direction isthe sum of the force components acting in that direction on the uid element.

Figure 2.4 Stress componentsin the x-direction

On the pair of faces (E, W ) we have

p x xx x yz + p + x

+ xx + x yz = + xyz (2.12a)

The net force in the x-direction on the pair of faces (N, S ) is

yx y xz + yx + y xz = xyz

(2.12b)

Finally the net force in the x-direction on faces T and B is given by

zx z xy + zx + z xy = xyz

(2.12c)

The total force per unit volume on the uid due to these surface stresses isequal to the sum of (2.12a), (2.12b) and (2.12c) divided by the volumexyz:

+ + (2.13)

Without considering the body forces in further detail their overall effect can be included by dening a source SMx of x-momentum per unit volumeper unit time.

The x-component of the momentum equation is found by settingthe rate of change of x-momentum of the uid particle (2.11) equal to the

zxz

yxy

(p + xx)x

zxz

DEF1

2

zxz

ABCDEF

1

2

zxz

ABC

yxy

DEF1

2

yxy

ABCDEF

1

2

yxy

ABC

DEFxxx

px

ABCJKL

DEF1

2

xxx

ABC

DEF1

2

px

ABCGHI

JKL

DEF1

2

xxx

ABCDEF

1

2

px

ABCGHI

ANIN_C02.qxd 29/12/2006 09:55 AM Page 15

16 CHAPTER 2 CONSERVATION LAWS OF FLUID MOTION

total force in the x-direction on the element due to surface stresses (2.13)plus the rate of increase of x-momentum due to sources:

= + + + SMx (2.14a)

It is not too difcult to verify that the y-component of the momentumequation is given by

= + + + SMy (2.14b)

and the z-component of the momentum equation by

= + + + SMz (2.14c)

The sign associated with the pressure is opposite to that associated with thenormal viscous stress, because the usual sign convention takes a tensile stressto be the positive normal stress so that the pressure, which is by denition acompressive normal stress, has a minus sign.

The effects of surface stresses are accounted for explicitly; the sourceterms SMx, SMy and SMz in (2.14ac) include contributions due to body forcesonly. For example, the body force due to gravity would be modelled by SMx = 0, SMy = 0 and SMz = g.

2.1.4 Energy equation in three dimensions

The energy equation is derived from the first law of thermodynamics,which states that the rate of change of energy of a uid particle is equal to therate of heat addition to the uid particle plus the rate of work done on theparticle:

Rate of increase Net rate of Net rate of work of energy of = heat added to + done on uid particle uid particle uid particle

As before, we will be deriving an equation for the rate of increase ofenergy of a uid particle per unit volume, which is given by

(2.15)

Work done by surface forces

The rate of work done on the uid particle in the element by a surfaceforce is equal to the product of the force and velocity component in thedirection of the force. For example, the forces given by (2.12ac) all act inthe x-direction. The work done by these forces is given by

DE

Dt

(p + zz)z

yzy

xzx

Dw

Dt

zyz

(p + yy)y

xyx

Dv

Dt

zxz

yxy

(p + xx)x

Du

Dt

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2.1 GOVERNING EQUATIONS OF FLUID FLOW AND HEAT TRANSFER 17

pu x xxu x

pu + x + xxu + x yz

+ yxu y + yxu + y xz

+ zxu z + zxu + z xy

The net rate of work done by these surface forces acting in the x-direction isgiven by

+ + xyz (2.16a)

Surface stress components in the y- and z-direction also do work on the uidparticle. A repetition of the above process gives the additional rates of workdone on the uid particle due to the work done by these surface forces:

+ + xyz (2.16b)

and

+ + xyz (2.16c)

The total rate of work done per unit volume on the uid particle by all the surface forces is given by the sum of (2.16ac) divided by the volumexyz. The terms containing pressure can be collected together and writtenmore compactly in vector form

= div(pu)

This yields the following total rate of work done on the fluid particle bysurface stresses:

[div(pu)] + + + + +

+ + + + (2.17)

Energy flux due to heat conduction

The heat ux vector q has three components: qx, qy and qz (Figure 2.5).

JKL(wzz)

z(wyz)

y(wxz)

x(vzy)

z

(vyy)y

(vxy)x

(uzx)z

(uyx)y

(uxx)x

GHI

(wp)z

(vp)y

(up)x

JKL(w(p + zz))

z(wyz)

y(wxz)

xGHI

JKL(vzy)

z(v(p + yy))

y(vxy)

xGHI

JKL(uzx)

z(uyx)

y(u(p + xx))

xGHI

JKL

DEF1

2

(zxu)z

ABCDEF

1

2

(zxu)z

ABCGHI

JKL

DEF1

2

(yxu)y

ABCDEF

1

2

(yxu)y

ABCGHI

JKL

DEF1

2

(xxu)x

ABCDEF

1

2

(pu)x

ABC

DEF1

2

(xxu)x

ABCDEF

1

2

(pu)x

ABCGHI

ANIN_C02.qxd 29/12/2006 09:55 AM Page 17

The net rate of heat transfer to the fluid particle due to heat ow inthe x-direction is given by the difference between the rate of heat inputacross face W and the rate of heat loss across face E:

qx x qx + x yz = xyz (2.18a)

Similarly, the net rates of heat transfer to the uid due to heat ows in the y- and z-direction are

xyz and xyz (2.18bc)

The total rate of heat added to the uid particle per unit volume due to heatow across its boundaries is the sum of (2.18ac) divided by the volumexyz:

= div q (2.19)

Fouriers law of heat conduction relates the heat ux to the local temperaturegradient. So

qx = k qy = k qz = k

This can be written in vector form as follows:

q = k grad T (2.20)

Combining (2.19) and (2.20) yields the nal form of the rate of heataddition to the fluid particle due to heat conduction across elementboundaries:

div q = div(k grad T ) (2.21)

Energy equation

Thus far we have not dened the specic energy E of a uid. Often theenergy of a uid is dened as the sum of internal (thermal) energy i, kineticenergy 12 (u

2 + v2 + w2) and gravitational potential energy. This denition

Tz

Ty

Tx

qzz

qyy

qxx

qzz

qyy

qxx

JKL

DEF1

2

qxx

ABCDEF

1

2

qxx

ABCGHI

18 CHAPTER 2 CONSERVATION LAWS OF FLUID MOTION

Figure 2.5 Components of theheat ux vector

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2.1 GOVERNING EQUATIONS OF FLUID FLOW AND HEAT TRANSFER 19

takes the view that the uid element is storing gravitational potential energy.It is also possible to regard the gravitational force as a body force, which doeswork on the uid element as it moves through the gravity eld.

Here we will take the latter view and include the effects of potentialenergy changes as a source term. As before, we dene a source of energy SEper unit volume per unit time. Conservation of energy of the uid particle isensured by equating the rate of change of energy of the uid particle (2.15)to the sum of the net rate of work done on the uid particle (2.17), the netrate of heat addition to the uid (2.21) and the rate of increase of energy dueto sources. The energy equation is

= div(pu) + + + +

+ + + + +

+ div(k grad T ) + SE (2.22)

In equation (2.22) we have E = i + 12 (u2 + v2 + w2).

Although (2.22) is a perfectly adequate energy equation it is commonpractice to extract the changes of the (mechanical) kinetic energy to obtain an equation for internal energy i or temperature T. The part of the energyequation attributable to the kinetic energy can be found by multiplying thex-momentum equation (2.14a) by velocity component u, the y-momentumequation (2.14b) by v and the z-momentum equation (2.14c) by w andadding the results together. It can be shown that this yields the followingconservation equation for the kinetic energy:

= u . grad p + u + +

+ v + +

+ w + + + u . SM (2.23)

Subtracting (2.23) from (2.22) and dening a new source term as Si = SE u . SM yields the internal energy equation

= p div u + div(k grad T ) + xx + yx + zx

+ xy + yy + zy

+ xz + yz + zz + Si (2.24)wz

wy

wx

vz

vy

vx

uz

uy

ux

Di

Dt

DEFzzz

yzy

xzx

ABC

DEFzyz

yyy

xyx

ABC

DEFzxz

yxy

xxx

ABCD[ 12 (u

2 + v2 + w2)]

Dt

JKL(wzz)

z(wyz)

y(wxz)

x(vzy)

z(vyy)

y

(vxy) x

(uzx) z

(uyx) y

(uxx)x

GHIDE

Dt

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20 CHAPTER 2 CONSERVATION LAWS OF FLUID MOTION

For the special case of an incompressible uid we have i = cT, where c is thespecic heat and div u = 0. This allows us to recast (2.24) into a temperatureequation

c = div(k grad T ) + xx + yx + zx + xy

+ yy + zy + xz + yz + zz + Si (2.25)

For compressible ows equation (2.22) is often rearranged to give an equa-tion for the enthalpy. The specic enthalpy h and the specic total enthalpyh0 of a uid are dened as

h = i + p/ and h0 = h + 12 (u2 + v2 + w2)Combining these two denitions with the one for specic energy E we get

h0 = i + p/ + 12 (u2 + v2 + w2) = E + p/ (2.26)Substitution of (2.26) into (2.22) and some rearrangement yields the (total)enthalpy equation

+ div(h0 u) = div(k grad T ) +

+ + +

+ + +

+ + + + Sh (2.27)

It should be stressed that equations (2.24), (2.25) and (2.27) are not new (extra)conservation laws but merely alternative forms of the energy equation (2.22).

The motion of a uid in three dimensions is described by a system of vepartial differential equations: mass conservation (2.4), x-, y- and z-momentumequations (2.14ac) and energy equation (2.22). Among the unknowns arefour thermodynamic variables: , p, i and T. In this brief discussion we pointout the linkage between these four variables. Relationships between the thermodynamic variables can be obtained through the assumption of thermo-dynamic equilibrium. The uid velocities may be large, but they are usually small enough that, even though properties of a uid particle changerapidly from place to place, the uid can thermodynamically adjust itself tonew conditions so quickly that the changes are effectively instantaneous. Thusthe uid always remains in thermodynamic equilibrium. The only exceptionsare certain ows with strong shockwaves, but even some of those are oftenwell enough approximated by equilibrium assumptions.

JKL(wzz)

z(wyz)

y(wxz)

x

(vzy)z

(vyy)y

(vxy)x

(uzx)z

(uyx)y

(uxx)x

GHI

pt

(h0)t

wz

wy

wx

vz

vy

vx

uz

uy

ux

DT

Dt

Equations of state

2.2

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2.3 NAVIER---STOKES EQUATIONS FOR A NEWTONIAN FLUID 21

We can describe the state of a substance in thermodynamic equilibriumby means of just two state variables. Equations of state relate the othervariables to the two state variables. If we use and T as state variables wehave state equations for pressure p and specic internal energy i:

p = p(, T ) and i = i(, T ) (2.28)For a perfect gas the following, well-known, equations of state are useful:

p = RT and i = CvT (2.29)The assumption of thermodynamic equilibrium eliminates all but the twothermodynamic state variables. In the ow of compressible fluids theequations of state provide the linkage between the energy equation on theone hand and mass conservation and momentum equations on the other.This linkage arises through the possibility of density variations as a result ofpressure and temperature variations in the ow eld.

Liquids and gases owing at low speeds behave as incompressiblefluids. Without density variations there is no linkage between the energyequation and the mass conservation and momentum equations. The oweld can often be solved by considering mass conservation and momentumequations only. The energy equation only needs to be solved alongside theothers if the problem involves heat transfer.

The governing equations contain as further unknowns the viscous stresscomponents ij. The most useful forms of the conservation equations foruid ows are obtained by introducing a suitable model for the viscousstresses ij. In many uid ows the viscous stresses can be expressed as func-tions of the local deformation rate or strain rate. In three-dimensional owsthe local rate of deformation is composed of the linear deformation rate andthe volumetric deformation rate.

All gases and many liquids are isotropic. Liquids that contain signic-ant quantities of polymer molecules may exhibit anisotropic or directionalviscous stress properties as a result of the alignment of the chain-like polymermolecules with the ow. Such uids are beyond the scope of this intro-ductory course and we shall continue the development by assuming that theuids are isotropic.

The rate of linear deformation of a uid element has nine components in three dimensions, six of which are independent in isotropic uids(Schlichting, 1979). They are denoted by the symbol sij. The sufx system is identical to that for stress components (see section 2.1.3). There are threelinear elongating deformation components:

sxx = syy = szz = (2.30a)

There are also six shearing linear deformation components:

sxy = syx = + and sxz = szx = +

syz = szy = + (2.30b)DEF

wy

vz

ABC1

2

DEFwx

uz

ABC1

2

DEFvx

uy

ABC1

2

wz

vy

ux

Navier---Stokes equations for aNewtonian fluid

2.3

ANIN_C02.qxd 29/12/2006 09:55 AM Page 21

22 CHAPTER 2 CONSERVATION LAWS OF FLUID MOTION

The volumetric deformation is given by

+ + = div u (2.30c)

In a Newtonian fluid the viscous stresses are proportional to the ratesof deformation. The three-dimensional form of Newtons law of viscosityfor compressible ows involves two constants of proportionality: the rst(dynamic) viscosity, , to relate stresses to linear deformations, and the sec-ond viscosity, , to relate stresses to the volumetric deformation. The nineviscous stress components, of which six are independent, are

xx = 2 + div u yy = 2 + div u zz = 2 + div u

xy = yx = + xz = zx = +

yz = zy = + (2.31)

Not much is known about the second viscosity , because its effect is smallin practice. For gases a good working approximation can be obtained by taking the value = 23 (Schlichting, 1979). Liquids are incompressible sothe mass conservation equation is div u = 0 and the viscous stresses are justtwice the local rate of linear deformation times the dynamic viscosity.

Substitution of the above shear stresses (2.31) into (2.14ac) yields the so-called NavierStokes equations, named after the two nineteenth-centuryscientists who derived them independently:

= + 2 + div u + +

+ + + SMx (2.32a)

= + + + 2 + div u

+ + + SMy (2.32b)

= + + + +

+ 2 + div u + SMz (2.32c)JKL

wz

GHI

z

JKL

DEFwy

vz

ABCGHI

y

JKL

DEFwx

uz

ABCGHI

x

pz

Dw

Dt

JKL

DEFwy

vz

ABCGHI

z

JKL

vy

GHI

y

JKL

DEFvx

uy

ABCGHI

x

py

Dv

Dt

JKL

DEFwx

uz

ABCGHI

z

JKL

DEFvx

uy

ABCGHI

y

JKL

ux

GHI

x

px

Du

Dt

DEFwy

vz

ABC

DEFwx

uz

ABCDEF

vx

uy

ABC

wz

vy

ux

wz

vy

ux

ANIN_C02.qxd 29/12/2006 09:55 AM Page 22

2.3 NAVIER---STOKES EQUATIONS FOR A NEWTONIAN FLUID 23

Often it is useful to rearrange the viscous stress terms as follows:

2 + div u + + + +

= + +

+ + + + ( div u)

= div( grad u) + [sMx]The viscous stresses in the y- and z-component equations can be recast in asimilar manner. We clearly intend to simplify the momentum equations byhiding the bracketed smaller contributions to the viscous stress terms in themomentum source. Dening a new source by

SM = SM + [sM] (2.33)

the NavierStokes equations can be written in the most useful form forthe development of the nite volume method:

= + div( grad u) + SMx (2.34a)

= + div( grad v) + SMy (2.34b)

= + div( grad w) + SMz (2.34c)

If we use the Newtonian model for viscous stresses in the internal energyequation (2.24) we obtain after some rearrangement

= p div u + div(k grad T ) + + Si (2.35)

All the effects due to viscous stresses in this internal energy equation aredescribed by the dissipation function , which, after considerable algebra,can be shown to be equal to

= 22

+

2

+

2

+ +

2

+ +

2

+ +

2

+ (div u)2 (2.36)

54647

DEFwy

vz

ABCDEF

wx

uz

ABCDEF

vx

uy

ABC

JKKLDEF

wz

ABCDEF

vy

ABCDEF

ux

ABCGHHI

14243

Di

Dt

pz

Dw

Dt

py

Dv

Dt

px

Du

Dt

JKL

x

DEFwx

ABCz

DEFvx

ABC

yDEF

ux

ABCx

GHI

DEFuz

ABCz

DEFuy

ABC

yDEF

ux

ABCx

JKL

DEFwx

uz

ABCGHI

z

JKL

DEFvx

uy

ABCGHI

y

JKL

ux

GHI

x

ANIN_C02.qxd 29/12/2006 09:55 AM Page 23

The dissipation function is non-negative since it only contains squared termsand represents a source of internal energy due to deformation work on theuid particle. This work is extracted from the mechanical agency whichcauses the motion and converted into internal energy or heat.

To summarise the ndings thus far, we quote in Table 2.1 the conservativeor divergence form of the system of equations which governs the time-dependent three-dimensional uid ow and heat transfer of a compressibleNewtonian uid.

24 CHAPTER 2 CONSERVATION LAWS OF FLUID MOTION

Table 2.1 Governing equations of the ow of a compressible Newtonian uid

Continuity + div(u) = 0 (2.4)

x-momentum + div(uu) = + div( grad u) + SMx (2.37a)

y-momentum + div(vu) = + div( grad v) + SMy (2.37b)

z-momentum + div(wu) = + div( grad w) + SMz (2.37c)

Energy + div(iu) = p div u + div(k grad T ) + + Si (2.38)

Equations p = p(, T ) and i = i(, T ) (2.28)of state

e.g. perfect gas p = RT and i = CvT (2.29)

(i)t

pz

(w)t

py

(v)t

px

(u)t

t

Momentum source SM and dissipation function are dened by (2.33)and (2.36) respectively.

It is interesting to note that the thermodynamic equilibrium assumption ofsection 2.2 has supplemented the ve ow equations (PDEs) with two furtheralgebraic equations. The further introduction of the Newtonian model, whichexpresses the viscous stresses in terms of gradients of velocity components,has resulted in a system of seven equations with seven unknowns. With anequal number of equations and unknown functions this system is mathemat-ically closed, i.e. it can be solved provided that suitable auxiliary conditions,namely initial and boundary conditions, are supplied.

It is clear from Table 2.1 that there are signicant commonalities betweenthe various equations. If we introduce a general variable the conservativeform of all uid ow equations, including equations for scalar quantities suchas temperature and pollutant concentration etc., can usefully be written inthe following form:

+ div(u) = div( grad ) + S (2.39) ()t

Conservative form of the governing

equations of fluid flow

2.4

Differential and integral forms of the general

transport equations

2.5

ANIN_C02.qxd 29/12/2006 09:55 AM Page 24

2.5 FORMS OF THE GENERAL TRANSPORT EQUATIONS 25

In words,

Rate of increase Net rate of ow Rate of increase Rate of increase of of uid + of out of = of due to + of due to element uid element diffusion sources

Equation (2.39) is the so-called transport equation for property . Itclearly highlights the various transport processes: the rate of change termand the convective term on the left hand side and the diffusive term ( =diffusion coefcient) and the source term respectively on the right handside. In order to bring out the common features we have, of course, had tohide the terms that are not shared between the equations in the source terms.Note that equation (2.39) can be made to work for the internal energy equa-tion by changing i into T or vice versa by means of an equation of state.

Equation (2.39) is used as the starting point for computational proceduresin the nite volume method. By setting equal to 1, u, v, w and i (or T or h0) and selecting appropriate values for diffusion coefcient and sourceterms, we obtain special forms of Table 2.1 for each of the ve PDEs formass, momentum and energy conservation. The key step of the nite volumemethod, which is to be to be developed from Chapter 4 onwards, is the integ-ration of (2.39) over a three-dimensional control volume (CV):

dV + div(u)dV = div( grad )dV + S dV (2.40)

The volume integrals in the second term on the left hand side, the convec-tive term, and in the rst term on the right hand side, the diffusive term, arerewritten as integrals over the entire bounding surface of the control volumeby using Gausss divergence theorem. For a vector a this theorem states

div(a)dV = n . adA (2.41)

The physical interpretation of n.a is the component of vector a in the direction of the vector n normal to surface element dA. Thus the integral of the divergence of a vector a over a volume is equal to the component of ain the direction normal to the surface which bounds the volume summed(integrated) over the entire bounding surface A. Applying Gausss diver-gence theorem, equation (2.40) can be written as follows:

dV + n . (u)dA = n . ( grad )dA + S dV (2.42)

The order of integration and differentiation has been changed in the rstterm on the left hand side of (2.42) to illustrate its physical meaning. Thisterm signies the rate of change of the total amount of fluid property in the control volume. The product n.u expresses the ux com-ponent of property due to uid ow along the outward normal vector n, so the second term on the left hand side of (2.42), the convective term, therefore is the net rate of decrease of fluid property of the fluidelement due to convection.

CV

A

A

DEFCV

ABCt

A

CV

CV

CV

CV

()t

CV

ANIN_C02.qxd 29/12/2006 09:55 AM Page 25

26 CHAPTER 2 CONSERVATION LAWS OF FLUID MOTION

A diffusive ux is positive in the direction of a negative gradient of theuid property , i.e. along direction grad . For instance, heat is conductedin the direction of negative temperature gradients. Thus, the product n . ( grad ) is the component of diffusion ux along the outward normalvector, so out of the uid element. Similarly, the product n . ( grad ),which is also equal to (n . (grad )), can be interpreted as a positive dif-fusion ux in the direction of the inward normal vector n, i.e. into the uidelement. The rst term on the right hand side of (2.42), the diffusive term,is thus associated with a ux into the element and represents the net rate ofincrease of fluid property of the fluid element due to diffusion. Thenal term on the right hand side of this equation gives the rate of increaseof property as a result of sources inside the uid element.

In words, relationship (2.42) can be expressed as follows:

Net rate of decrease Net rate of Net rate of Rate of increase of due to increase of creation of of inside the + convection across = due to diffusion +inside the control volume the control volume across the control control volumeboundaries volume boundaries

This discussion claries that integration of the PDE generates a statement ofthe conservation of a uid property for a nite size (macroscopic) controlvolume.

In steady state problems the rate of change term of (2.42) is equal to zero.This leads to the integrated form of the steady transport equation:

n . (u)dA = n . ( grad )dA + S dV (2.43)

In time-dependent problems it is also necessary to integrate with respect totime t over a small interval t from, say, t until t + t. This yields the mostgeneral integrated form of the transport equation:

dV dt + n . (u)dAdt

= n . ( grad )dAdt + S dVdt (2.44)

Now that we have derived the conservation equations of uid ows the timehas come to turn our attention to the issue of the initial and boundary conditions that are needed in conjunction with the equations to construct a well-posed mathematical model of a uid ow. First we distinguish twoprincipal categories of physical behaviour:

Equilibrium problems Marching problems

CV

t

A

t

A

t

DEFCV

ABCt

t

CV

A

A

Classification of physical behaviours

2.6

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2.6 CLASSIFICATION OF PHYSICAL BEHAVIOURS 27

Equilibrium problems

The problems in the rst category are steady state situations, e.g. the steadystate distribution of temperature in a rod of solid material or the equilibriumstress distribution of a solid object under a given applied load, as well as manysteady uid ows. These and many other steady state problems are governedby elliptic equations. The prototype elliptic equation is Laplaces equa-tion, which describes irrotational ow of an incompressible uid and steadystate conductive heat transfer. In two dimensions we have

+ = 0 (2.45)

A very simple example of an equilibrium problem is the steady state heatconduction (where = T in equation (2.45)) in an insulated rod of metalwhose ends at x = 0 and x = L are kept at constant, but different, tempera-tures T0 and TL (Figure 2.6).

2y2

2x2

Figure 2.6 Steady statetemperature distribution of aninsulated rod

This problem is one-dimensional and governed by the equation kd2T/dx2= 0. Under the given boundary conditions the temperature distribution inthe x-direction will, of course, be a straight line. A unique solution to thisand all elliptic problems can be obtained by specifying conditions on thedependent variable (here the temperature or its normal derivative the heatux) on all the boundaries of the solution domain. Problems requiring dataover the entire boundary are called boundary-value problems.

An important feature of elliptic problems is that a disturbance in the interior of the solution, e.g. a change in temperature due to the suddenappearance of a small local heat source, changes the solution everywhere else.Disturbance signals travel in all directions through the interior solution.Consequently, the solutions to physical problems described by elliptic equa-tions are always smooth even if the boundary conditions are discontinuous,which is a considerable advantage to the designer of numerical methods. Toensure that information propagates in all directions, the numerical techniquesfor elliptic problems must allow events at each point to be inuenced by allits neighbours.

Marching problems

Transient heat transfer, all unsteady ows and wave phenomena are examplesof problems in the second category, the marching or propagation problems.These problems are governed by parabolic or hyperbolic equations.However, not all marching problems are unsteady. We will see further on

ANIN_C02.qxd 29/12/2006 09:55 AM Page 27

that certain steady ows are described by parabolic or hyperbolic equations.In these cases the ow direction acts as a time-like co-ordinate along whichmarching is possible.

Parabolic equations describe time-dependent problems, which involvesignicant amounts of diffusion. Examples are unsteady viscous ows andunsteady heat conduction. The prototype parabolic equation is the diffusionequation

= (2.46)

The transient distribution of temperature (again = T ) in an insulated rodof metal whose ends at x = 0 and x = L are kept at constant and equal tem-perature T0 is governed by the diffusion equation. This problem arises whenthe rod cools down after an initially uniform source is switched off at time t = 0. The temperature distribution at the start is a parabola with a maximumat x = L/2 (Figure 2.7).

2x2

t

28 CHAPTER 2 CONSERVATION LAWS OF FLUID MOTION

The steady state consists of a uniform distribution of temperature T = T0throughout the rod. The solution of the diffusion equation (2.46) yields theexponential decay of the initial quadratic temperature distribution. Initialconditions are needed in the entire rod and conditions on all its boundariesare required for all times t > 0. This type of problem is termed an initialboundary-value problem.

A disturbance at a point in the interior of the solution region (i.e. 0 < x < Land time t1 > 0) can only inuence events at later times t > t1 (unless we allow time travel!). The solutions move forward in time and diffuse in space.The occurrence of diffusive effects ensures that the solutions are alwayssmooth in the interior at times t > 0 even if the initial conditions contain discontinuities. The steady state is reached as time t and is elliptic. Thischange of character can be easily seen by setting /t = 0 in equation (2.46).The governing equation is now equal to the one governing the steady tem-perature distribution in the rod.

Hyperbolic equations dominate the analysis of vibration problems. Ingeneral they appear in time-dependent processes with negligible amounts ofenergy dissipation. The prototype hyperbolic equation is the wave equation

= c2 (2.47)

The above form of the equation governs the transverse displacement ( = y)of a string under tension during small-amplitude vibrations and also acousticoscillations (Figure 2.8). The constant c is the wave speed. It is relatively

2x2

2t2

Figure 2.7 Transientdistribution of temperature in aninsulated rod

ANIN_C02.qxd 29/12/2006 09:55 AM Page 28

2.7 THE ROLE OF CHARACTERISTICS IN HYPERBOLIC EQUATIONS 29

straightforward to compute the fundamental mode of vibration of a string oflength L using (2.47).

Solutions to wave equation (2.47) and other hyperbolic equations can beobtained by specifying two initial conditions on the displacement y of thestring and one condition on all boundaries for times t > 0. Thus hyperbolicproblems are also initialboundary-value problems.

If the initial amplitude is given by a, the solution of this problem is

y(x, t) = a cos sin

The solution shows that the vibration amplitude remains constant, whichdemonstrates the lack of damping in the system. This absence of dampinghas a further important consequence. Consider, for example, initial condi-tions corresponding to a near-triangular initial shape whose apex is a sectionof a circle with very small radius of curvature. This initial shape has a sharpdiscontinuity at the apex, but it can be represented by means of a Fourierseries as a combination of sine waves. The governing equation is linear so each of the individual Fourier components (and also their sum) wouldpersist in time without change of amplitude. The nal result is that the discontinuity remains undiminished due to the absence of a dissipationmechanism to remove the kink in the slope.

Compressible uid ows at speeds close to and above the speed of soundexhibit shockwaves and it turns out that the inviscid ow equations arehyperbolic at these speeds. The shockwave discontinuities are manifestationsof the hyperbolic nature of such ows. Computational algorithms for hyper-bolic problems are shaped by the need to allow for the possible existence ofdiscontinuities in the interior of the solution.

It will be shown that disturbances at a point can only inuence a limitedregion in space. The speed of disturbance propagation through an hyperbolicproblem is nite and equal to the wave speed c. In contrast, parabolic andelliptic models assume innite propagation speeds.

Hyperbolic equations have a special behaviour, which is associated with thenite speed, namely the wave speed, at which information travels throughthe problem. This distinguishes hyperbolic equations from the two othertypes. To develop the ideas about the role of characteristic lines in hyper-bolic problems we consider again a simple hyperbolic problem described by

DEFx

L

ABCDEF

ct

L

ABC

Figure 2.8 Vibrations of a stringunder tension

The role ofcharacteristics in

hyperbolic equations

2.7

ANIN_C02.qxd 29/12/2006 09:55 AM Page 29

wave equation (2.47). It can be shown (The Open University, 1984) that achange of variables to = x ct and = x + ct transforms the wave equationinto the following standard form:

= 0 (2.48)

The transformation requires repeated application of the chain rule for differentiation to express the derivatives of equation (2.47) in terms ofderivatives of the transform variables. Equation (2.48) can be solved veryeasily. The solution is, of course, (, ) = F1( ) + F2(), where F1 and F2can be any function.

A return to the original variables yields the general solution of equation