Patankar Numerical Heat Transfer and Fluid Flow
Trang 2and Physical Processes i Mechanies and
ntgiiongi Afothodks ii Mecldui
ond Pletcher, Computational Pluid Mechanies and leat
sation Methods in Hex Transter ment Computational Fluid Mechanies
“Shiekh, and Litkouhi, leat Conduction Using Green's tical Modeling in Combustion
‘omputatjonal Heat Transfer ‘chan Numerical Heat Trensfor and Fluid Flow
Heinrich, The Finite Element Method: Basic Concepts und plications
Sst, Numerical Heat Transfer PROCEEDINGS
xạ, Ediiar, Finite Elements in Fluids: Volume 8
‘Sheikh, Editor, integral Methods in Science and Engineering-$0 .erical Properties and Methodologies in I¥eat Transfer: af the Second National Symposium
GTITLES
Trang 3
To my wife Rajani ó So ra ete Ten ce SS So cee en NUMERICAL HEAT TRANSFER AND FLUID FLOW
Library of Congress Cataloging in Publication Data Patankar, S¥ ‘Numerical heat unser and ola flow,
(Series in computational methods in mechanice and
shermal sciences) ‘ibtiogsphy: p | Includes index 1 Heat—Transmisson, 2 vid dynamic ; : 3 Numerleal analysis 1 Tile, U, Sess | QCMDPMT 5302 7928386
TRBN 0491164223
“This ook was st in Press Reman by Hemisphere Pubsting Corporation
‘The editors were May A Pips and Edward 1 Milman the progecon | sperior was Rebekah MeKinneysand the types was Susda s
BookGrafter Ine vat pier nd binder
Trang 4= = CONTENTS Preface ụ 1 1 L 1a INTRODUCTION Scope of the Book Methiods of Prediction [24 Experimental Investigation 12-2 Theoretical Caleufation
123 Advantages of a Theoretical Calculation 12-4 Disadvantages of a Theoretical Calculation 12-5 Choice of Prediction Method
Outline of the Book
MATHEMATICAL DESCRIPTION OF PHYSICAL PHENOMENA
2.1 Governing Differential Equations 3-1-1 Meaning of a Differential Equation 21-2 Conservation of a Cheraical Species 21-3 The Energy Equation
2-4 A MomentumEquation
ZS The Time-a (Qed Equations for Turbulent Flow 216 The TurbuleWe-Kinetic-Enerey Equation
21-7 ‘The General Differentiat Equation 2.2 Nature of Coordinates 2.241 Independent Variables
Trang 5Nữ contents contents i x 3 DISCRETIZATION METHODS 25 S25 The tyirid Scheme s 3.1 The Nature of Numerical Methous 2s Be A Gone deed Fetal %
SH on ti 52-8 Consequences of the Various Šchenlei 95
bz ~The Disrette tion Coneert as 5.3 Diseretization Equation for Two Dimensions 96 311-3 The Structure of the Diseretization Equation + FE een ee a sẽ 3.2 Methods of Deriving the Dieretizstion Pquatons BEL Taylor Series Formulation Sad NHA n m 2 si Debietastion Exigilet tor Three Dimers 5.5 A One-Way Space Coordinate 255 the einak Hietatlation Baud 102 tới 39
3⁄23 Method of Weighted Residusls a S.5-1 What Makes a Space Coordinate One-Way 102
3.24 Control-Volume Formulation 2 515-2 The Outflow Soundasy Condition 102 3.3 An filostrative Fxample 31 eee Desk 105
3.4 The Four Basic Rules af 56-1 The Common View of False Diffusion 105 4.3 Closure 2 S.:2 The Proper View of False Diffusion 106
Problems 3 Problems 47 Closue oe 109 4 HEAT CONDUCTION dị
4.1 Objectives of the Chapter ar 6 CALCULATION OF THE FLOW FIELD ne 412 Steady One-dimensional Conduction The Basie Equations e a2 Sứ loa luos Snobl Emeoli= 61-1 The Main Difficulty Hà 3 The Grid Spacing 8 6.12 Vorticity-based Methode ha ‘The Interface Conduetiity ae 6.2 Some Related Difficulties us Nonlinearity Solution of the Linear Algebraic Equations Source-Term Linearization Boundary Conditions so 52 2 4 SATE Monientura Equations 6A 62:1 Representation of the Pressure Gradient Term 6.2-2 Representation of the Continuity Equation Zemedy: The Stapsered Grid nz tế 120 ns 4.3 Unsteady One-sdimensionat Conduction 43:1 The General Discretiration Equation sa Gk Tia presi ond Velueny Codie 123, sa de The Peauecadeciios Eaustion tạ 43:2 Explicit, Crank-Nicolson, and Fully Implicit Schemes 56 65 Ge ahipte aoe 16 43.4 The Fully Implicit Discretization Equation sẽ Bơi .SejlEme of Operiflies
4.4 Two- and Threestimensional Situations 44:1 Diseretization Equation for Two Dimensions 4442 Discretization Equation for Taree Dimensions 89 9 a 61-2 Discussion of the Pressure-Correction Equation 67-3 Boundary Conditions for the Pressure-Correction gusting 129 136 443 Solution of the Algebraic Equations 61 4324 The Relative Nature of Pressure 130 4⁄6 Overrelaxation and Underelsxstion er GAA Revised Algorithm: SIMPLER 131 466 Some Geometric Considerations 4.461 Location of the Conttol-Volume Faces $162 Other Coordinate Systems sẽ s8 i đại Mune 682 ea TAC SIMPLER ALS ‘The Presture Equation 132 133 I 4.7 Closure 2 6.84 Discussion 134
Problems 74 _ 1
: Protlems 135
3 CONVECTION AND DIFFUSION 19
Trang 6
8
T;ã tequlae Geometries
7.31 Orthogonal Curviinear Coordinates 2 Regular Grid with Blocked-of! Regions 3 Conjugate Heat Transfer
Suggestions for ComputerProgrant Preparation and esting
SPECIAL TOPICS
5,1 Two-dimensional Parabolic Flav £2 Threedimensional Parabofie How 8.3 Partlaly Parabolic Flows 84 The Finite-Element Method 84-1 Motivation ` B45 A ControlVol se-based Vinite-Blement Method LLUSTRATIVE APPLICATIONS 9 Developing Flow in a Curved Pipe 2 Combined Convection in a'Ilorizontal Tube
2 Molting around a Vertical Pipe
9.4 Terbulent Flow and Heat Transfer in Internally Finned Tubes 5 4 Deflected Turbulent Jet
oe A Hlypermixing Jet within a Thrust-Augmenting Bjector 47 A Feriodie Fully Developed Duct Flow 913 Thermal Iydraulic Analysis of a Steam Generator 9.9 Closing Remarks tehiture He 146 M47 le 130 Iss l6 157 iss 158 159 160 163 168 6s 169 170 173 178 TH ist 13 185 189 195 ————— PREFACE
ta 1972, 1 taught an informal course on numerical solution of heat transfer ‘und uid flow to a small group of research workers at Imperial College, London, Later the material was expanded and Cormalized for presentation in graduate courses at the University of Waterloo in Canada (in 1974), at the Norwegian Instirute of Technology, Trondheim (in 1977), and at the Uni- versity of Minnesola (ia 19@)1977, and 1979) Dusing the last two years, Ï hhave also presented the sigfe material in a shortcourse format at ASM! rational meetings The enthusiastic response accorded to these courses hi encouraged me to write this book, which can be used as a text fora graduate course as well ss 2 reference book for computational work in heat transfer and fluid flow
Although there is an extensive literature on computational thermofluid analysis, the neweomer to the field has insufficient help available The graduate student, the researcher, and the practicing engineer must struggle through journal articles or be content with elementary presentation in books fon numerical analysis, Often, it is the subtle details that determine the success lure of a computational activity; yet, the practices that are leamed experience by successful computing groups rarely appear in print A ‘consequence is dat many workers either give up the computational approach after many months of frustrating pursuit or struggle through to the end with inefficient computer programs
Being aware of this situation, I have tried to present i this book + self-contained, simple, and practical treatment of the subject The book is introguctory in slyle and is intended for the potential practitioner of numerical heat transfer and fluid llow; itis not designed for the experts in the subject area Ia developing the numerical techniques, 1 emphasize physical significance rather than mathematical manipulation, Indced, most of the
Trang 7
xã PREACH
mathematics used here is limited to simple algebra The result ig that, whereas the book enables the reader (0 travel all the way tn the presentslay frontier ‘of the subject, the journey takes place through delightfully simple and illuminating physical concepts and considerations, In teaching the material with such an approach, [have often heen pleasantly surprised by the fact that the students not only lesm about numerical methods but also develop a better appreciation of the relevant physical processes
‘As a user of numerical techniques, I have come to prefer a certain family ‘of methods and a certain set of practices This reperioire has been collected parly from the liferature and subsequently has been enriched, adapted, and riodified Thus, since 2 considerable amount of sorting and siting of avallahle rethods has already taken place (atbeit with my own bias) Ihave limited the scope of this book to the set of methods that 1 wish to recuniend T do mot attempt to present a comparative stody of ll avatlable methods; otlier methods sre only occasionslly mentioned when they serve to iluninate a specific feature under consideration, In this sense, ahis book represents iny personal view of the subject Although [ am, of course enthusiastic about this Miewpoint, I recognize that my choices have been influenced by my back- ground, personal preferences, and technical objectives Others operating different environments may well come (o preter alternative approaches
To illustrate the application of the material, problems ace given at the end of some chapters Most of the problems can be solved by using a pocket aleulstor, slthough some of them should be programmed for a digital Computer The problems are not meant for testing the student reader, ut are included primarily for extending and enviching the learning process They suggest alternative techniques and present additional material, At times, in my aitempt to give a hint for the problem solution, | alniost uisctose the full answer In such cates, arriving at the correct answer is not the main abjeetive the reader should focus on the message that the problem i designed (9 convey
‘This book carnes the description of the numerical method to a point where the reader could begin (0 write a computer program, Indeed, the reader should be able to construct computer prozrams that generate the Kind of results presented in the final chapter of the book A range of computer programs of varying generality can be designed depending upon the natuse 0Ï the problems to be solved Many readers might have found it helpful if a presentative computer program were included in this book 1 did consider the possibility, However, the task of providing a rexsonably general computer program, its detailed description, and several examples of i's use seemed 50 formidable dat it would have considerably velayed the publication of this book For the time being, { have included 2 section on the preparation and testing of a compuler program (Section 7.4), where many useful procedures and practices gathered through experience are described
‘The completion of this book folfils # Đesre and a drean) that Fave etd emerace xi
Tor a number of years It was in 1971 that Professor D, Dian Spalding and T planned a book oF this kind and wrote a preliminary outline for it Further Progress, however, became difficult because of the geographical distance between us and because of our fovolvement in a variety of demanding aclivities Finally, a joint book seeried impracticable, and 1 proceeded to conver! my feehire Hotes into this textbook The present book has some resemblance to the joint book that we had planned, since | have made liberal uve of Spalding's lectures and writings His direct involvement, however, would ‘ave made this book much better
Tn this undertaking, 1 owe the greatest debt to Professoe Spalding He introduced me to the frscinsling world of computational methods The work that we accomplished togsther represents the most delightful and, creative experience of my professionat Iie The influence of his ideas on my tinking san be seen throughout this book, The concepis of “one-way” and “tworway® coordinates (and the terms themselves) are the product of his linagination, It xay bẻ who organized all the relevant physical processes theough a general differentia! equation of a standard form, Above all, our rapid progress in computational work has resulted from Spalding’s vision and conviction that ‘oe day all practical situations will hecome amenable to computer analysis
1 wish t0 record my sincere thanks {o Professor D Brian Spalding for his creative influence on my professional activities, for continued warm friends ship, and for his direct und indirect centeibutlons 40 this book,
Fofessor Ephraim M Sparngeatas been my most enthusiastic supporter in the activity: of weiting this HOHE interest begin even eadier whisn he attended my graduate course on the subject § have greatly benefited (von lis ‘questions and subsequent discussions, He spent countless hours in reading the mantscript of this book and in suggesting changes and improvements 11 is due to his critical review that [have been able to achieve some measure of clarity and completeness in this book 1 act yery grateful to him for his active interest in this work and for his personal jnterest in me
‘A number of other colleagues and friends have also provided constant Inspiration through their special interest in my work In particular, I wish to lank Professor Richard J Goldstein for his support and encouragement and Professor George D Raithby for many’ stimulating discussions My thanks are also due to the many students fa my graduate courses, who have contributed significantly to this book throvyh thelr questions and discussions and through thoir enthusiasrt and response I am grateful to Mrs, Lucille R Laing, who typed the menuscript so carefully and cheerfully I would like to thank Mr Willam Begell, President of Hemisphere Publishing Corporation, for his Dersonal interest ia publishing this book and the staff at Hemisphere for their competent handling of this project
Trang 9
INTRODUCTION
1.1 SCOPE OF THE BOOK
Importance of heat transfer and fluid flow This book is concerned with heat suid mass transfer, uid flow, chemical resetion, and other elated processes that occut in engineering equipment, in the natural environment, and in living organisms, That these processes play a vital role can be observed in 2 great variety of practical situations Nearly all methods of power production involve fluid flow and heat transfer as essential processes The same processes govern the heating and air conditioning of buildings Major segments of the chertcal and metallurgical industries use components such ss furnaces, heat exchanger, condensers, and reactors, where thermofluid processes are at work, Aircraft and rockets ewe their functioning to fluid Row, heat transfer, and chemical reaction, In the design of electrical machinery and electronic circuits, heat transfer is often the limiting factor The pollution of the natural environment is lagely eaused by heat and mass uansfer, and so are storms, Moods, and fies In the face of changing weather conditions, the human body retorts to heat and snass tvansfer for its temperature control The procesics of heat transfer and fuid flow seem to pervade all aspects of our lif Need for understanding and prediction, Since the processes unde> con- sideration have such an overwhelming impact on human life, we should be able to deal with them effectively This ability can result from an understanding of the nature of the processes and from methodology with which to predict them quantitatively, Armed with this expertise, the designer of an engineesing device can ensure the desired performance-the designer is able to choose the
Trang 104 RUMNERICAL.HEAT.TRANSEEIL AND KLUID FLOW optimum desiga Íiom amtong 4 nuntbar 0F alteinatite possiblities The power af predigtion enables us 1 operate existing, equipment more safely efficiently, Predictions of the selevaot processes help us in foseeasting, even controlling, potential dangers such a5 fluods, tides, and fires tn all these
tions offer economic benefits and cuntribuce fo human well-being wre of prediction, The prediction of behavior in a given physical situation consists of the yalues of the selevant yatta ing the nyavesses of interest, Let us consider a particular example In 3 combustion chamber of a certain deseription, 4 complete prediction should give us the sof velocity, pressure, Temperature, concentrations of the relevant ical gpecies, elz, throughout the domisin of interest; it should also ide the shear stresses, heat Muxes, and mass flow rates at the confining bastion chamber The prediction should stare how any of anuitics would eliange 1a response lo proposed changes ay geometry, ‘hd
books This buck is primarily almed at developing a general fedictiun Joe beat and mass veunster, fluld Mew, and selated among the different methods of predietion, if sonstruet a
prucwais, AS We shell shortly se
2 taPcttea! solution offers great promise In this buok, we 2L Riehud For predicting dhe processes of intezest
lay at possible, our aim will be 10 desiga a numerics! stothod hovlag Jete generality, We shall, therefore, sefrain frum accepting sny fimal 2iaietlanE suekoss twodimensionality, boundary-layer appreximations, and siger-ledsl 1f any restrictions ate temporarily adopted, it will be of presentation and understanding and not because of any intrinsic Hon, We shall begin the subject at a yery elementary level and, from re, rave nearly 10 the frontier of the subject,
This ambitious task cannot, of course, be accomplished in a modest-sized shout leaving out a number of important topics Therefore, the Formulation of the equations that govern the processes of Will be discussed only briefly in this book For the complete iyusien of the tequired equations, the reader must luợi 16 tiandard ogthaoks on che subject The mathematical models (or conyplex procossos like \urbulence, combustion, and radiation wil be assumed to be known ue valable 16 the reader, Even in the aubjeet of numerical solution, we shall not survey all available methods and discuss their merits 2nd demerits Rather, we saa] foeus attention on a particular family of methods What the author bas led, developed, or contributed to, Reference to other methods will be made only when this serves to highlight a certain issue While a general formulation will be attempted, no special attention will be given to supersonic flows, free-surface flows, or two-phase flows
‘An Important characteristic of the qumetical methods to be developed in shis book is that they are strongly based on physical considerations, not just oa mathematical manipulations, Indeed, nothing more sophisticated than ernnpuetion 3 simple algebra and elementary estgulus is used A significant advantage of this Stuatoyy i that the gade, while lcuraiog about the numerical methods, Govclops a ueeper understandfng of, and insight into, the underlying physical wossser THP appreciation for physical significance i very helpful in Inalyzing and ñnerorelng eomphted reall Dut, even if the reader never perfuinis nunisial computations, this study of the numerical methods will provide=it is interesting to note greater fect forthe physical aspects Of heat Teansfer and fluid flow Further, the physical aproach wil equip the reader ‘vith general criteria with: which ti judge ollier existing and future numerical rots 1.2 METHODS OF PREDICTION
Prediction of heat transfer and flufd-flow processes ean be obtained by two yualit methods: experimental investigation and theoretical calculation We shall briefly consider each and then compare the two,
1.21 Experimental Investigation
The most reliable information about a physical process is often given by 2olual ineasurement An experimental investigation involving Fullseale equip- ment can be used to predict how identical copies of the equipment would perform under the same conditions Such fullscale tests are, in most cases, prohibitively expensive and often impossible, The altemative then is to perform experiments on small-scale models The resulting information, howe ver, must be extrapolated to full scale, and general rules for doing this are ofien unavailable, Further, the small-scale models do not always simulate all the features of the fullscale equipment; frequently, important features such as combustion or bolling are omitted from the model tests, This further reduces the usefulness of the test results Finally, it must be remembered that dhere are sctious difficulties of measurement in many situations, and that the measuring instruments ace sot free from errors,
1.2-2 Theoretical Calculation
A theoretical prediction works out the consequences of a mathematical model, rather than those of an actual physleal model For the physical processes of interest here, the mathematical mode! mainly consists of a set of differential equations If the methods of classical mathematics were to be used for solving these equations, there would be little hope of predicting many phenomena of
Trang 11
4 NUMERICAL HEAT TRANSFER AND FLUID FLOW Hesse wall Insulated section Figure 1 Grid leyout for 2 numerical soliton fr the temperature fed
contain innite series, special functions, traneertlental equations For eigen values, ete so that their numerical evaluation may present a formidable task.”
Fortunately, the development of numerical methods and the availability ‘of large digital computers hold the promise that the implications of a mathematical model cant be worked out for almost any practical problem A preliminary idea of the numerical approach to problema solving can be obtained by reference to Fig 1.1 Suppose that we wish to obtain the temperature fleld in the doaiain shown It may be sufficient to know the values of temperature at discrete points of the domain One possible method is to imagine 9 grid thet fills the domain, and 10 seek the values of Temperature at the gid poins We then construct and solve algebraic equations for these unknawn temperatures The simplification inherent in the use of algebraic equations rather than differential equations is what makes umerieal mathods so powerful and widely dpplicable
1
We shall now list the advantages that a theoretical calculation offers over a corresponding experimental Investigation
Advantages of a Theoretical Calculation
“Te sót JAgfied hete that exact anatytealslutisas sce without practical value Indeed, as we shall see Inlet, some fortunes of numerical mesheds are cpartructed by the use of simple analytical safetlons Fuster, there & no beer way ef checking te scnuracy of x numerical methe than hy comparzon with an exact anata) wtution However, there atm tò bê litle dowht tot the methods of elatical malnemalie dst iểet a practical way of solving complex engineering problems
ixTropuerton: 5 Low cost, The most important advantage of a computational prediction is Its low cost In most applications, the cost of a computer run is many orders of magnitude Jower than the cost of a corresponding experimental investiga- tion This factor assumes increasing Importance as the physical situation to be studied becomes larger and more complicated Further, whereas the prices of ‘most Henis are increasing, computing costs are likely to be even lower in the Tu,
Speed A computational investigation can be performed with remarkable speed A designer can stuly the implications of hondreds of different configurations in less than a day and choose the optimum design, On the other hind, a corresponding experimental investigation, It is easy to imagine, would take a very long time
Complete information .\ computer solution of a problem gives detailed and complete information It can provide the values of aif the relevant variables (such as velocity, pressure, temperature, concentration, turbulence intensity) drronghout the domain of interest Unlike the situation in an experiment, there are few [naccessible locations in a computation, and there & no counterpart to the flovt disturbance caused by the probes Obviously no experimental study can be expected to mensure the distributions of all urlables over the entire domain For this reason, even when an experiment i performed, there is great value in obtaining a companion camputer sole supplement the experimental information
Ability 10 simulate realistic conditions In a theorctical calcu! realise conditions can be easily simulated There is no need to resort tu small-scale or cold-flow models For a computer program, there is little difficulty in having very large oc very small dimensions, in treating very low or very high temperatures, in handling toxic or lammable substances, or in Following very fast or very slow processes
Ability) to simulate ideal conditions & prediction method is sometimes used to study a basic phenomenon, rather than a complex engineering application, In the study of a phenomenos, one wants ta focus attention on a few essential parameters and climinate all irrelevant features Thus, many idealizations are desirable—for example, two-dimensionallty, constant density, an adiabatic surface, or infinfte reaction rate, In a computation, such conditions can be easily and exactly set up Ort the other hand, even a very careful experiment can barely approximate the idealization
4.2-4 Disadvantages of a Theoretical Calculation
‘The foregoing adventages we sufficiently impressive to stimulate enthuslasm akout computer analysis, A blind enthusiasm for any cause is, however, Undesirable It is useful 20 be aware of the drawbacks and limitations
Trang 12NUHERICAE HEAT THASSEE AND PLUND EU, Jilly) (SAF, The vabidicy Of the mathematical anodel, therefore, Tots the latest of y computation In tus book, we shall be concerned only with oimputational methods and not with mathematical odes Yet, we mist que | the use af the computer analysis reesives an end product that depends on both the mathematieal model and the numesical method Á pofeEtly satisfactory) mumerical technique can produce worthless results if an na:
uate mathematical model is employe
Far the purpose of discussing the disalvantayes of a theoretival calkuls sion, ibs, therefore, useful to divide all practical problems into (wo groups
Group Az Pusblenis for which an adequate mathematical deaaigtion cạn be suiten (Examples; heat conduction, laminar flows, simple turbulent ‘poundary layers.)
Group #: Problems for which am adequate mathematical description has uot yet been worked out, (Examples: complex tuibulent flows, certain bon-Newtonidn lows, formation of nitric oxides in turbulent combustion, two-phase ows.)
OF courve, the group nta which gives problem falls will be determined by fe prepared 1a consider as an “adequate” deseriptivn,
regi for Group «A, Ue may be stied that, for most problens of fou cả, the theoeéticad caleulaon suffers from a9 disadvantages, The sulahoa thea represeats an alternative that is highly superior to ast nial study Ozsesianally, however, one encounters, some dis: sẻ tesss, TỪ the prediction has a very limited objective (such as finding the ssuie ccop for = spmplleated apparatus),
pensive then an experiment For di
"5¬ ' 1 ‘1 numerical solution may be hard to obtain and would be excessively jpensive if at al possible, Exteemely fast and small-scale pheaomena such as If they ae (a de computed in all their time-dependent deLui by ly Navier-Stokes equations, are still beyoad the practical eormputational methods, Finally, when the miathemticat problem foceascially admits more than one solution, if i aot easy to determine \shethus the computed solution corresponds to reality
in computational methods is aimed st making them mare „ and efficient The disadvantages mentioned here will
arch progresses
‘Disadvantages for Group B The problems of Group B shore ail the sedvantages of Group Aj In addition, there is the uncertainty sbout the ent to which uke computed results would agree with reality In such eases, some experimental backup is highly desirable
Reseatch im mathematical models causss a transfer of problems from foup B inte Group A ThE research consists of proposing a model, working il we INTRODUCTION 7 the results with
‘Out its jinplications by computer analysis, and compari
expevimental dats Thuý, cdmputational methods play a key role in this research A shikivg example of this role can be found in the recent evelopment of turbulence models, The currently populur and widely used woequation turbulence models are primarily based on the work of Kolmogorov (1942) and Prandtl (1945), It was, however, only in the 1970s, when computers and compatational methods became more pawerful, that the turbulenee models were pat te practical use
1.25 Choice of Prediction Method
‘This discussion about the relative merits of computer analysis and experi rental investigation is not aimed at recommending computation to the ‘exclusion of experiment, Am appreciation of the strengths and weaknesses of both approaches is essential to the proper choice of the appropriate technique
‘These ig no doubt dhat experiment is the only method for jnvestigating @ new asic phenomenan Ín this sense, experiment leads and computation follows ft |s in the syathesis of a number of interacting known phenomena that the computation performs more efficiently Even then, sufficient valida- tion of the computed results by comparison with experimental data is fequited On the other hand, for the design of experimental apparatus, picliminaey computations are often helpful, and the amount of experi mentation ean usually be significantly reduced if the investigation is supple- mented by computation,
‘An optimal prediction effort should thus be a judicS@2>ombination of computation and experiment, The proportions of the two ingredients would spend om the nature of the problem, on the objectives of the prediction, and con the economis and other constraints of the situation
1.3 OUTLINE OF THE BOOK
‘This book i desiged ta unfold the subject in a certain sequence, and the reader is urged (o follow the same sequence It will not be profitable to jump to 2 later chapter, as all chapters build upon the material covered in the previous ones The problems at the end of some chapters are intended to sive the reader both dieet experience with and deeper understanding of the pineiples developed in the book
Trang 134 NUMERICAL HEAT TRANSFER AND FLUID FLOW
Before we begin the task of numerical solution, the physical phenomena rust be deseribed via appropriate differential equations This is outlined and discussed in Chapter 2 Of special importance in that chapter is the examina tion of the parabolic or elliptic nature of these equations from a physically meaningful viewpoint
The concept of numerical solution is developed in Chapter 3, where the common procedures of constructing numerical methods are described Among these, the method that lends itself to easy physical interpretation is chosen and illustrated by means of a very simple example This introductory material is used fo formulate general criteria in the form of four basie rules These rules form the guideposts for the development of the sumerieal method in the rest of this book Although the rules are formulated from physical con- siderations alone, they often lead to results that—it js interesting (o observe— are normally derived from purely mathematical analysis Furthermore, these rules guide us to better formulations that may not have been suggested by standard mathematical methods,
The construction of the numerical method begins in Chapter 4 It is carried out in three stages Heat conduction (i the general problem without the convection term) is treated in Chapter 4 Chapter § concentrates on the interaction of convection and conduction, with the flow field regarded as given Finally, the ealculation of the velocity feld itself is dealt with in Chapter 6
Readers who are interested in Muid flow alone, and not in heat transfer, Should note that Chapter 6 is not a solFcontained chapter Tt describes onty the additional features requited for the Nuid-low calculation, the other details hhaving already been given in Chapters 4 and 5 Thus, Chapter 4 does not merely deal with heat conduction: it completes much of the groundwork needed for fluid flow, The treatment of convection in Chapter S is also equally applicable to Mluid-flow calculation, This approach—Hiandling fluid flow through heat transfer—may be unfamiliar to some readers, but if appears 10 be an effective pedagogical technique The eatly focus on heat transfer enables us to conduct all the preliminsry discussion in terms of temperature, which is an casy-to-understand scalar variable It also reinforces the conceptual unity between variables such 25 temperature and momentum, which is useful in understanding and interpreting results
‘Another technique that will be in evidence in these chapters is the use of ‘onedimensional situations ta constrict the basic algorithm which is then ‘quickly generalized to multidimensional cases The one-dimensional problem serves to keep the algebraic complication to a minimum and to focus attention on the significant issues
Chapter 7 ít a compilation of 4 mumiber of elucidatingd@emarks and suggestions that can be properly appreciated after the readfOhas haủ an overview of the” method through familiarity with the first ‘six chapters Chapter 8 deals with calculation procedures that can be considered as special cases of the general method developed in the book The eontrol-volume-based INTRODUCTION 5
Ginite-clement method, which is briefly deseribed in Section 8.4, is, however, an extension rather than a special case of the general method
Trang 14CHAPTER TWO MATHEMATICAL DESCRIPTION OF PHYSICAL PHENOMENA
‘The numerical solution of heat transfer, fluid flow, and other related processes can begla when the laws governing these processes have been expressed in ‘mathematical form, generally in terms of differential equations For a detailed and complete derivation of these equations, the reader should turn to a standard textbook Our purpose here is to develop familiarity with the form and the meaning of these equations It will be shown that all the equations of relevance here possess a common form, the identification of which is the first step toward constructing a general solution procedure We shall also discuss some characteristics of the independent variables used in these equations 2.1 GOVERNING DIFFERENTIAL EQUATIONS
2.1-1 Meaning of a Digs Equation
The individual differential Gquations that we shall encounter express a certain conservation principle Each equation employs a certain physical quantity 2: its dependent variable and implies that there must be a balance among the sarious factors that influence the variable The dependent varables of these diffewental equations are usually specific properties, ie., quantities expressed con a unitmars basis, Examples” are mass frection, velocity (Le., momentum per unit mass), and specific enthalpy
“Temperature, whigh i quite Frequently used as a dependent variable, is not 3 specifi property: it arses from more basic equations cmploying specific internal eneray cor specific enthalpy as the dependent variable
Trang 1512 NUMERICAL HEAT TRANSFER AND FLUID FLOW ‘The terms in a differential equation of this type denote influences on unit-voluume basis An example will make this clear Suppose J denotes a Mux influencing a typical dependent variable ọ Let uẹ consider the conteol volume of dimensions dx, dy, dz shown in Fig 2.1, The flux J, (which is the xedirection component of J) is shown entering one face of area dy dz, while the flux leaving the opposite face is shown as J + (B//@x) dx, Thus, the net efifux is (@q/2x) de dy dz over the sưea of the face, Considering the contributions of the y snd z directions as well and noting that oe dy dz is the volume of the region considered, we have
1%
Net efilux per unit volume per 3
ays G1) ‘This interpretation of div J will be particularly useful to us because, as we shall se¢ later, our numerical method will be constructed by perforining a balance over a control volume
Another example of a term expressed on a unit-wolume basis is the rate-of-change term a(p6)fat If ộ is a specific property and p is the density, then pp denotes the amount of the corresponding extensive @operty con: tained in a unit volume Thus, (og)/2r is the rate of change gl) the relevant
roperty per unit volume,
‘A differential equation is a compilation of such terms, exch representing an influence on s unitwolume basis, and all the terms together implying a Dalanice or conservation We shall now take as examples a few standard
ifferential equations, to find & general form
2.1-2 Conservation of a Chemical Species
Lat my denote the mass fraction” of a chemical species In the presence of 2 velocity feld u, the conservation of my is expressed as
Fp Ped + iv (pum +) = Ry (2) Here a(pm,)/at denotes, as explained eater, the rate of change of the mass of the chemical species per unit volume The quantity pum is the convection Sux of the species, i.e, the flux carried by the general Now fiekd pu The symbol J; stands for the diffusion flux, which is normally caused by the
“The mass fraction my of a chemical specie {is detined as the rato of the mass of the species I (contained in 4 given volume) £9 the totat mass ofthe minture (contained in ine same volume)
MATHEMATICAL DESCRIPTION OF PHYSICAL PHENOMENA 8
pe
Figure 2.1 Fluy balance ver 2 contol volume
gradients of mj The divergence of the two fluxes (convection and diffusion) forms the second term of the differential equation, The quantity Ry on the right-hand side is the rate of generation of the chemical species per unit volume The generation is caused by chemical rezetion Of course, Ry can have 4 positive or negative value depending on whether the reaction actually produces or destrays the chemieal species, and Rj is zero for 2 nonreacting species If the diffusion Oux J; is expressed by the use of Fick’s law of diffusion, wo can write J, == Dy rad my, G3) where Tị the điUson coeffieat The substitution of Eq (2.3) into (2.2) leads te & Comp) + ate (ou = aiv 1 grad m) + Ri G4)
3 The Energy Equation
‘The energy equation in its most general form contains a large number of influences Since we arc primarily interested in the form rather than in ¢ details of the equiation, it wil be suffictent to consider some restricted cases
For a steady low-velocity Slow with negligible viscous dissipation, the ‘nergy equation can be written as
diy (puh) = div (K grad 7) +5, , (2.8) where fis the specific enthalpy, & is the thermal conductivity, T is the temperature, and 5, is the volumetric rate of Hie generation The term div (F rad 7) seprecents the influence of conduction heat transfer within the fluid,
Trang 16
NUMEWICAL HEAT TRAXSEEE AND FUUID FLOW jes] gases and for solids and liquids, we ean write
cand T= gadh, G6) whlere @ Is the constant
yy equation beconies assure specifig heat, With this substitution, the
div (puh) = div é gad 4) tây G7)
ives sanetan, the ~ 7 mlstion simplifies lô Aeer, (2) svhich would fad to iv (pu) = ai Ệ gad 7) G» jn this manner, either the enthalpy or the lemperature em be chosen as the dependent variable ‘The steady heat-conduction situation i obtained by setting the velocity w 0 ern; thus, đi (E grad T) +S, = 0 2.10) 2.14 A Momentum Equation
The differential equation governing the conservation of momentum in a given 4ieston fer a Newtonian fluid can be written slong similar tines; however, the complication ic greater because both shear and normal stresses must be ve# aad bacause the Stokes viscosity law is more complicated than let's law of Feurier’s law With u denoting the x-direction velocity, we write
ouespending momentum equation as
Ầ lạ) + su 080 = gu tia 9 CÃP ty + — GẦN
Hee g Ís thê Vieogity, p is the preesure, đụ i$ the x-lizection body force per unit volume, and V, stands for the viscous terms that are in addition to those
ssed by div (u grad a),
The Time-averaged Equations for Turbulent Flow
‘Turbulent flows are commonly encountered in practical applications It is the time-mean behavior of these flows that is usually of practical interest
MATHEMATICAL, DESCRIPTION OF MUYSICAL PHENOMFNA ts
Therefore, the equations for unsteady laminar flow are converted into the time averaged equations for turbulent flow by an averaging operation in which it is, ‘assumed that there are rapid and random fluctuations about the mean value ‘The “uitional (exis arising from this operation are ihe so-called Reynolds stresses, lurbuleat teat flux, turbulent diffusion Dux, ete To express these Huxes int rerms of the mean properties of the flow is the task of a turbulence mode
Many turbulence models employ the concept of a turbulent viscosity or a turbulent diffusivity to express the turbulent stresses and Aluxes The result is that the time-averaged equations for turbulent flow have the same appearance as the equations for Ia low, but the laminar exchange coefficients such as viscosity, diffusivity, Yu conductivity ate replaced by effective (ie, Jminar plus turbulent)” exchange coefficients From a computation viewpoint, a turbulent flow within this framework is equivalent to a luninar ow with a eather complicated prescription of viscosity (The same idea applicable to non-Newtonian flows, which can be thought of as flows in whi the viscosity depends on the velocity gradient.) 2.1-6 The Turbulence-Kinetic-Energy Equation
The currently popular “two-equation models” of turbulence (Launder and Spilling, 1972, 1274) employ, a6 one of the equations, the equation for tf not energy K of the Muctuating motion, which reads
2 (ok) + div (pub) = Uv (Pe grad +20, (2.12)
where Oy Is tie diffusion coefficient for k, G is the rate of generation of turbulence energy, and ¢ is the kinematic rate of dissipation The quantity G—pe ig the net source term in the equation A similar differential equation governs the variable €;
1-7 The General Differential Equation
‘This wwief journey though some of the relevant differential equations has indicated that all the dependent yarlables of interest here seem to obey a generalized conservation principle, If the dependent variable is denoted by 6, the general dfferentisl equation is
Ponta au)-wirmiats, ean
Trang 17ts NUMERICAL HEAT TRANSFER AND FLUID FLOW
‘The four torms in the general differential equation are the unsteady term, the convection term, the diffusion term, and the source term The dependent variable $ can stand for a variety of different quantities, such as the mass fraction of a chemical species, the enthalpy or the temperature, a velocity component, the turbulence kinetic enerey, of a turbulence length scl Accordingly, for each of these variables, an appropriate meaning yl have to den itis ‘Not all diffusion Muxes are governed by the gradient of Me relevant Fond eae QO
variable The use of div (P grad 4) a the diffusion term dloes not, however, limit the general ¢ equation to gradient-driven diffusion processes, Whatever cannot be fitted into the nominal diffusion term can always be expressed as part of the source term; in fact, the diffusion cocfficient T` can even be set equal to zero if desired A gradientdiffusion term has been explicitly included in the general ở equation because most dependent variables do require a prominent diffusion term of this nature
The density appearing in Eg (2.13) may be slated, via an equation of state, to variables such as mass fraction and temperature, These variables and the velocity components obey the general differential equation Fuether, the flow field should satisfy an additional constraint, namely, the mass conservation of the continuity equation, which is
2 + aiv (ou) = a (ou) = 9 619)
we in Eqs (2.13) and (2.14) in vector form Another useful representation is the Certesian-tensor form of these equations: a a a ag 2 (pe) + © faye = 2 (r ) 4 HO nd) xứ =) 5 G5) a, 2 3P +22 (nụ a 7 (pu) , (2.14)
where the subscript / can take the values J, 2, 3, denoting the three space esordinates When 9 subscript is repeated in a term, a summation of ước terms is implied; for example,
a 4 a 2
Fe (OH am (84) % a, (8B) + (80) (217)
3 ag a ae a ag a ae
2 (p#)a2 (r®)+ 2 (rp) e C0) m 10 )0 1910910 0u us An immediste benefit of the Cartesin-enior ph is thatthe one-dimensional {orm of the equation is obtained by simely dropping the subscript
MATHEMATICAL DESCRIPTION OF PHYSICAL PHENOMENA, „ ‘The procedure for casting any particular differential equation inlo the general farm (2.13) is te manipylate it until, for the choses dependent variable, the unsteady term and the convection and diffusion terms conform to the stundard form, The coefficient of grad ở in the diffusion term is then taken as the expression for P, and the remaining terms on the righthand side ‘ate collectively defined as the source term 5
‘Although we have so far considered all the variables as dimensional ‘quantities, cis at times more convenient to work with dimensionless variables Again, auy particular differential equation written in terms of dimensionless veiables can be regarded as possossing the general form (2.13), with ó standing for the dimensionless dependent yartable, and with I and being the dlimensionless forms of the diffusion coefficient and the source teem, 1 stony’ ceases, the dimensionless value of may siomply be unity, while 8 may t2ke th value of 0 or I
The recognition that all the relevant differential equations for heat and russ transfer, iluid flow, torhulenee, and related phenomena can be thought of as particular cases of the general @ equution is an important time-saving step As a consequence, we need to concern ourselves with the ‘umerical solution of only Eq (2.13) Even in the construction of a computer program, it is sufficient to write 2 generat sequence of instructions for solving Eg, (2.13), which can be repeatedly used for different meanings of ¢ along with appropriate expressions for P and S, and, of course, with appropriate initial and boundary conditions Thus, the concept of the general @ equation cnables us to formulate a general numerical method and to prepare general- purpose computer programs, -
2.2 NATURE OF COORDINATES
So far we have piven attention to the dependent variables Now we shall tum (@ the independent variables and discuss their properties from the computa- ‘ional point of view
2.241 Independent Variables
‘The dependent variable @ would, In general, be a funetion of three space coordinates and time Thus,
#=4G.z 09; (2.19) and 4 ate the independent variables In a numerical solution, we yendent variables at which the values of 3
Were x,ý
stall choose the values of the ind are to be exientated
Fortunately, sot all problems require consierstion of all four
Trang 18
£ NUMERICAL HEAT TRANSFEK AND TH) EEU
at veribles The sioaller We umber af participating idepondeat
‘ite „ tc fewer will be the locations (or grid points) at which thẻ @ ve
cab csoused hosed thst terse ue problems até ot esas
mien) Whon tte iclevant physical quae depend ow sly me apace
ordinates the sulin is elle dieimelgiodi, Dpcyleee aạị Ba ee Sintee lads 10.4 tecrdinontonal sition, aud on tase space irae (0 a tiedincnional uation Whey the proheir code nạ
Zepceauace on le te eal siadp, Othervis, Ie allel sont epcodent Conidering $8 Aepeniseee on face af ie ges
Pelt desalbe a soso an Unley Sredincnsonah ier 4 Gah) The choxe of independent coorotes as expel by (2.19) ot
‘he oly pots, Instead of deeb assay fompestne Uttam on T9, nề mg i6 HERI, (2.20)
shure 2 becomes: the dependent variable that stinds Toe dhe height of an hesmoal sutfae corresponding to T st the location fx, 9) A msthod based cp such a representation has been developéd by Dic and Cizek (1970) ad by
Crenk and co-workers (Crank and Phahle, 1973; Crank and Gupta, 1975; Crank and Crowley, 1978) and is known as the isotherm migration method
The method is, however, limited to temperature fields tht are monotone nelons of the coocdinates; for more general feds, the height = cou have
veral values for given values of 7, x, and y: this makes 2, for computational
vote, unsuitable a a dependent vavable
Proper Choice of Coordinates
tie number of grid points would, in general, be related 10 the number variables, there ig a significant computational saving fo be ey working with fewer independent varlables A judicious choice of inate system can sometimes seduce the ember of independent
we hove used x, j and 2 a8 che space coordinstes, it is not coplied Ut we must ose whe Cartesian coordinate system: any description of che spatial location will do We shall now ilustrate, by a few specific
examples, how the choice of coordinates influences the number of lependent variables
The aw around an aisplane that ig moving with constant velotiey ts v whea viewed from a stationary coordinate system, but steady ‘pect to a moving coordinate system attached to the airplane,
UENOMENA 19
MATITAUATICAL ĐESCMIEFION OF PHYSICA
2, The axisymmetric flow in a circular pipe appears tu be three-dimensional in 1 Cartesian coordinate system but is twe-imensional in cylindrical polar cooninates r, Ủ, 2, since
= 96, z) G3)
with no dependence on 0
3 Transformed coordinates offer further possibilities of fewer independent variables For example:
2 A twodimensional laminar boundary layer on a lat plate gives a similurity behavior such that the velocity 1 depends on 7 alone, where (322)
and where e ¡s a dimensional constant Thus, a two-dimensional problem is reduced to a one-ditensional problem
4 Unsteady heat conduction in a semiinfinite solid has x and £ as the Independent vadiables, However, for some simple boundary conditions, the temperature can be shown ta epend on ‡ alone, wheee
-& =, 2.28
& we Về
tHỦI Đf8NG88008.d609)60nd6 ĐnxGbhdlSSEE0G
da Hưng gỈ Ai dgeulstk valle sus ext tae rates dependant va tes, Phe cunt
fo + By doveeped ek lene, se Wana F detande on the Ai tim con bi 3: nữ Re bon Ai Dục biet Ope
Mắc len e3 et cine vi Lidg KÚP lotpirdo Ta oe tee
8 =00), (224) where
and Ty is the bulk temperature, which varies with x
‘b A plane fee jet is a two-dimensional flow However, we can write
Trang 1930 NUERICALHEAT THANSFER AND FLUID TLOW where
(2.26)
Tere u, represents the centering \elosity, yi the crosssueat coordinate, and 6 is characteristic jor widdh Both vw, and & vary with the streamvise egordinste
‘Although most of the discussion in this book will be conducted in terms of x, ¥, 2, and tas the independent variubles, it should be remembered thị all the’ ideas and practices ate equally applicable to the transformed cr dimensionless variables illustrated hero Indeed, for computational efficiency, numerical methods should always be used with the appropriate choice of coordinates
2.23 One-Way and Two-Way Coordinates
We shell now consider new concepts about the properties uf eon then establish a connection between these and the standard mathematical terminology
‘Definitions qwowa’ cootdinate is such that the conditions at a glen Jocatioa in that coordinate are influenced by changes in conditions on ether side of that location A one-way coordinate is such that the conditions al 4 ven location ia that coordinate are influenced by changes in conditions an only one side of that location,
Examples, One-dimensional steady heat conduction in a rod presides on example of a two-way coordinate, The temperature of any given point in the rod can be influenced by changing the temperature of either end, Normally, space coordinates ae twoway coordinates, Time, on the other hand, is always 2 oneway cootieate During the unsteady cooling of a solid, the temperature at a given instant can fe influenced by chenging only those conditions that prevailed before that instant, Tt is 2 matter of common experience that yesterday's events affect today's Isppenings, tut tomorrow's conditions have no influence on what happens today
‘Space ax a one-way: coordinate What is cvore interesting is that even 3 space coordinate can very pearly became one-way under Hie action uf fd flow, If there is @ strong unidirectional flow in the coordinate direct, then significant influences travel only from upstream to downstream The condi tions at a point are then affected largely by the upstceam conditions, and very Iittle by the dowestream ones, Thé one-way nature of a space coordinate isan approximation, It i vue that convection is 2 one-way process, but diffusion (which is always present) has two-way influences, However, when the flow rate is lage, convection overpowers diffusion and thus fmakes the space
nates and
coordinate nearly one-way
MATHEMATICAL DESCRIPTION O¥ FHIYSICAL FIENOMENA, 2
Parabolic, eliptic hyperbole Ie appears that the mathematical terms parabolic and elipric, which are wsed for the classification of differentia equations, correspond to our computation concepts of one-way and twoeway coordinates The term paraholie indicates a one-way bebavior, while eliptic signifies the (wo-way concent It ould be more meaninefal if situations were described sr being parabolic or elliptic in a given coordinate Thus, the mneteady heat conduction problem, which is normally called parabolic, is actually parabolic ia time and sHitie im the space conmdiuslet, The stealy heat conduction problem is tiliptc in all couedinates A twoslimensional boundary layer is parabolic in the streamuise coordinate and elit fa the erossstteam coordinate, Since such Uescrintions ste nevoventional, a connection with established pyactice can pethans be sehieved by the flowing rule:
A situation is parabolic if theye exists at least one one-way courdinatey otherwise, it is elliptic
A flow sith ene ones) space coordinate i sometimes called a wemmlarglyerlvpe low, while + Row with all twonvay cooedinates is referred to asa recirculating flow [ees the titles of the books by Patankar and Spalding (1970) and Gosmen, Pen, Runchal, Spalting, and Wollthtrin (i969))
Why yout the third category, namely, hyperbolic? It so happens that a
hyperboSeRation Joss not neuUỦy Gt into the computational classification [A hyperbolic problem fas a kind of oneovay hebavior, which is, however, not slong coordinate directions but along specis! lines called characteristics Toere ace numerical methods that make use of the characteristic line, but they are ‘eaticted to hyperbolic problems On the other hind, the numerical method to be developed in this book does not rake advantage of the special nature of
« hyperbolic problem, We shall treat hyperbolfe problems as members of the
cece clas of elite problems (te, ll twoway coordinates) Computational implications The motivation for the foregoing discussion
about cneavay and two-way coordinates is that, if a one-way coordinate can
bo identified sn a piven situation, substents! economy of computer storage
and computer time is possible Let us consider an unsteady two-dimensional
hsat conduction problem, We shall construct a two-dimensional aray of grid points in the ealeulation domain, At any instant of time, there wil be a corresponding twoudimensional temperature field, Such a ld will have to be handed in the computee for cach of the successive instants of ime However, since time is 2 oneway coordinsle, the temperature field at a given tine is sat atfected by the future temperate fstds, Indeed, the entire unsteady roblem can be reduced lo the sequred repetitions of one basie step, namly Ghis: Given the temperature field at time f,fnd the temperature field at time 1# Bt Ths, ninpoterstarnge will be needed only for these two temperature
Trang 20
2 NUMERICAL HEAT TRANSFER AND FLUID FLOW folds; the same storage space can be used, over and over again, for all the
m5 Res
15 this manner, starting with a given dutial temperature field, we are able ro “marek” farwacd to successive instants of time During any time step, only ‘o-diensional array of temperatures forms the unknowas to be treated ously.” They we decoupled from all fiture wlues of temperature, revious values that influence them aro known, Thus, we need! (ọ si simalor set af equations, with 3 consequent saving of computer
ln a shilar manner, a two-dimensional boundary layer is computed by sqarching in the streamwise coordinate, Wil values of Ue dependent variables given along one etosestream line at an upstream stution, the values along cessive crossstzeam lines are obtained, Only one-dimensional computer ge Ít needed for handling the two-dimensional đow, Sinilady, a thiee srsional duct flow that is parabolic in the streamwise direction can be
Tn this book, we shall give only vceasional attention to the oncsway space dinate However, its great potential for saving computer storage and computer time should always be kept in mind, PROBLEMS Be 128 Define an effective pressure by Pap—tudivu,
hoe p i the thermedynamic presure, If the sncosty i constaat but the density pis sant (and hence div U0), show that tac texm Vg in Eq (211) can be ined eh the presure gradient such that - cu ‘ie ss assumed bose đất an implicit method is 19 bE employed This matter is Chapter 4
MATHEMATICAL DESCIIPTION OF FVSICAL PHENOMENA, 2 226 A the continuity equation (2.14) were ta be regarded as a special cise of the general “eduation (2.13), what would be the espresions for Py and SP
27 Conider a mnistore of various chemical species, Define the mixture enchalpy by Jez 2myhp, whore my is the mas fraction of 4 typieal species, and Ay is its specie
`
Trang 21CHAPTER THREE
DISCRETIZATION METHODS
So far we have seen that dete are sigsificant benefits in obtaining a theoretical prediction of physical phenomena, The phenomena of interest here te governed by differential equations, which we have represented by a general equition for the variable ạ Now our main task is to develop the means of solving this equation,
For ease of understanding, we shall assume in this chapter that the variable ¢ is a function of only one independent variable x However, the ideas developed here continue to be applicable when more than one inde pendent variable is active
3.1 THE NATURE OF NUMERICAL METHODS, 3.1-1 The Task
‘A numerical solution of a difforential equation consists of a set of numibers from which the distribution of the dependent vatisble @ can be constructed, In this sense, a numerical method is akin to a laboratory experiment, in which 4 set of instrument readings enables us to establish the distribution of the measured quantity in the domain under investigation, The numerical analyst and the laboratory experimenter both must remain content with only a finite ‘umber of numerical values as the outcome, although this number can, 2t least in principle, be made farge eneugh For practical purposes
Trang 22
36 NUHERICAL HEAY TIAANSEEE ANID PL UND HO Let us suppose that we decide 10 represeit the variation of ý by w polynomial i 2,
' Gu)
BS ay tae tage? +
sé employ a numerical method to find the Finite number of coefficients a5, pq, This will enable us to evaluate at any foration x by Íng the value of x and the values of the a's mo Eq 3.1) This
ig, however, somewhat aeonvenient if our ultimate interest Js to } the taluer of @ al various Tocatons The values af the a's ate, by Fsmuclyes, not particularly meaningful, and the substitution aperation must Wut to arrive at the required values of ¢ This leads us to the vg (ioughs: Why not sinsicuct 2 method that employs dhe valwes of @ athnoee pf given points a the primary unknowns? Indved,, snost lejowtlual methods for solving differential equations do belong ia tus e-egoay, and therefore we shall limit our attention to such methous
ThS, 8 numerical method treats as its basic unknowns the values of de pendent yattable at a finite number of locations (called the grid pers) in the ciloulstion domain, The method includes the tasks of providing a set of quations for these onknowns arid of prescribing at algorithm for i Diseretization Concept
‘scusing sltentiga on the values at the grid points, we have repleced the vnuinuous information contained iq the exact solution of the differential jon with discrete values We have thus discretized the distribution of 9, Ht Js appeopdate to refer to this class of numerical methods as discretize-
vist methods The algebraic equations involving the unknown values of @ at chosen grid itis, which we shall now name the disererization equations, ate derived the vifferential equation governing g In this derivation, we must employ some assumption about how 9 vases befween the grid polnts, Although this profile" of ¢ cauld be chosen such that a single algcbralc expression suffices for the whole ealeuttion domain, it is often more practical 10 vse piecewise roils such that 2 gen sepment describes the variation of @ over only a all region in terms of the @ values at the gr points within and acound that a Thus, iris common ta subdivide the ealeulation domain into a number | subdomains of elements such that a separate profile assumption can be
ted with each subdomain
Ge manner, We encounter the discretization concept in another The continuum calcvlation domain has been discretized It is this tization of space sad of the dependent variables that makes it biseeetizaTion tHUTHGDS 2
ossble fo replace the govcening differential equations with simple algebraic equations, which eaa be solved With elative ase,
3.1-3 The Structure of the Diseretization Equation
A discrettaatinn equation is an algebraic telation connecting the values of ¢ for a group of grid points Such an equation is derived (rom the diffesential equation governing ở and thus expresses the same physical information as the differential equation, That only a few grid points participate in a given liscretizatinn equation is 4 consequence of the piecewise nature of the profiles shosen, The value of ở at a gril point thereby influences the distribution of ở only in its Immedisie neighborhood, As ihe number of grid points becomes very large, the solution of the discretisation cquations is expected to approach de exact’ solution of the corresponding diflerential equation This follows from the consideration that, as the grid points get closer together, the change in @ between neighboring grid points becomes small, and then the actual details of the profile assumption become unimportant
For a given differential equation, the possible discretization equations ate by ma means unique, although sll types of disctetization equations are, in the limit of a very large number of grid points, expected to give the same solition, The different types arise from the differences in the profile asoumptions and in the raethous of derivation Until now we have deliberately refrained fram making reference to Finitediffecence and finite-clement methods, Now it may be stated that these can be thought of as io altemative versions of the discretization method, Which we have described in gencral terms The distinction between the finite-dtference method and the fnite-clement method results from the ways sf choosing the profiles and deriving the discretization equations The method that Js (0 be the main focus of attention in this book has the appearance of x Gnite-dfference method, but iưỂhploys many ideas that are typical of the Finiteslement mnethodoligy TAfeall the present method a finite-difference method might convey an adherence to the conventional finite-difference practice For this reason, we shall refer to it simply as a discretization ‘method, Also, we shall note in Chapter § how 2 method that has the appearance of a finite-element method can be consteucted from the general principles presented ia this book
3.2 METHODS OF DERIVING THE DISCRETIZATION EQUATIONS
Trang 2318 NUMERICAL HEAT TRANSFER AND FLUID FLOW 3.21 TaylorSeries Formutation
‘The usual procedure for deriving finiteifference equations consists of approximating the derivatives in the differential equation via a truncated Taylor series Let us consider the grid points shown in Fig 3.1 For grid point 2, located midway between guid points 1 and 3 such that Ax =x —y = x5 —¥2, the Taylorseries expansion around 2 gives
ha () Las? (2) - (32)
ve cnet ae(H) «Lam (8) «
Traneating the series just after she thied term, and adding and subtracting the two equations, we obtain
Bì-sz
‘The substitution of such expressions into the differential equatfon leads to the finitedifference equation
‘The method includes the assumption that the variation of ¢ Is somewhat like a polynomial in x, so that the higher derivatives are unimportant This assumption, however, leads to an undesirable formulation when, for example, exponential variations are encountered, (We shall refer to this matter gain in Chapter 5.) The Taylorsories formulation is relatively straightforward bì allows less levibility and provides little insight into the physical meanin the terms.” * + Ga) 13.5) af "This is admitedly 2m entirely subjective vow Someune with prope raining may find the Taylorscries method! highly Mumia and meaning a Ea Tavares expansion, Figure 31 Thote euecesiv pid points used for Ds » DISCRETIZATION a nal Formulation 3.22 Variat
Another method of obtaining the diserctization equations is based on the calculus of variations, To understand the method fully, the reader should have sufficient knowledge of this branch of caleulus However, a general spprecia- tion of the main ingredients of the formulation is all that is needed for the present purposes
‘The calculus of variations shows that solving certain differential equations is equivalent to minimizing a related quantity called the functional This equivalence is known as a varlational principle, If the functional is minimized Wwith respect to the grit-point values of the dependent variable, the resulting conditions give the required discretization equations, The variational formulation is very commonly employed in finite-element methods for stress analysis, where it can be linked to the virtual-work principle In addition to its algebraic and conceptual complexity, the min drawback of this formulation is its limited applicability, since a variational principle does not exist for all differential equations of intorest
3.23 Method of Weighted Residuals
‘A powerful method for solving differential equations is the method of weighted residuals, which is described in detail by Finlayson (1972) The basic concept is simple and interesting Let the differential equation be represented
10) 4
Further, let us assume an approximate solution @ that contains ä nuunBet oF
undetermined parameters, for example,
# 19 aye Hage? nhát gu” G7)
the a's being the parameters The substitution of @ into the differential
equation leaves @ residual 2, defined ss
R= LO) G8)
‘We wish to make this residual small in some sense: Let us propose that JWR de #0, G9)
where W ig a weighting function and the integration is performed over the domain of interest By choosing a succession of weighting functions, we cam
Trang 24NUMHHCAI,IEÁT TRANSTER AND FLU FLOW
coverata a8 many equations as are sequired for evaluating (he parameters, Ijiese algebraic equations contuining the paramecers as the unknowns are sued to obtain the approximate solution to the iffereutial equation, Different versions of the method (Known by specific sames) resull frum the choice of different classes of weighting functions
‘The method was very popular in boundary-layer analysis before the Ainke-difference method nearly replaced it However, a conncetion with the Tinitedifference method, or rather with the Uiseretication method, can be tablished if the approximate solution ộ, instead of being a single algebra
ression over the whole domain, is constructed yia piecewise profiles with he grid-paint values of @ as the unknown pacanteters Indecd, mueh of the receat development of the finite-lentent teclinique is also based on piecewise [rolls used in conjunction with a particular weighted.residual practice known s alerkin me
Eiep this, a aumber of equations cua rated by dividing the calculation an ole subdomains o¢ control volumes, and setting the weighting Jonelion) la UE unity over une subdamain at a time aad zeio everywhere else
(QE eke anethod of weighted residuals is called the subdlamaie nà he controtsolume forrmuistion Te impliss that the jnxepiäl uf ee wer cach conteal volume atust Seceme z6v0 Since we shail adopt the proach in this book, a more detatled discussion ss desirable cis aa foltows Control Volume Formulation
demenlan' textbooks on heat transfor desive the DnitediTerenov sạunhòa te the Taylorseries method and then demonstrate thal the resulting squttion f gonsitent with a heat balance over a small regioa susrounding a 2d point, We have also seen that the controlvolume formulation can be garded us a special version of the methed of weighted residuals The basic J22 of “ip cantiobyolume formulation is easy to understand and lends itself Sirest physical intespetation The caleulation domain Is uivided into a 1 of nonoverlapping control volumes such that there is one control lume surrounding each grid point The differential equation is integrated + each contro} volume Piecewise profiles expressing the vevition of & otwoon the grid points are used to evaluate the required integrals The result ® the discretization equation containing the values of 6 fora group of grid points, Te disesetization equation obtained in this manner expresses the con- servation principle for @ for the Gnite control volume, just asthe diferental eguation expresses it for an Inflaltesimal control volume.”
"Indes, deriving the eontiabvelume diereiation squation by intersting the #iiieen(Gl eaustion over a faite conigal volume is 3 rather roendabout process, much
BISCRETIZATION METHODS a ‘The mest atiractive feature of the control-volume formulation is thal (he resulting solulion would imply thal the invegral conservation of quantities such ag mass, miontentun, and encrgy is exactly satisfied over any group of control volumes and, of course, over the whole calculation domain, This chatcteristic exists for any number of grid points—not just in a limiting sense vien the number of grid points becomes large Thus, even the coarse-grid solution exhibits exue¢ jntegral balances,
When the liscretization equations are solved to obtain the grid-point values of the dependent variable, the result cx be viewed in two different ways, la the finite-ctement method and in must weighted-residual methods, the assumed variation of 8= of the grid-point values and the
tron -uie lineoore íx May iets toe Od poate G le ak Be seprotins savior tl ON họledReoro net, Kimuft cờ tì, Pal puin Shien af) 8 combed fo entiale fhe lin, of mỹ Spt lene felelos # x„io keo Ue gil pata Tee het tar pasting! sie Me Be đe teEer of 5 span ie abel te tees the pre Alcs 9 Sa daca ces ohne tly sextet sbuut ibe elses burner dee nats i sur 270080180 1H11
we shall also adopt this view We shall seek the solution in the form of the
Eideeim nlet quy, The miogimdio Tivi: HỆ mơ VÌM ề ciel esas Phúchreiettg Tri dc cun sổ nề HO
Ga ine Cowes Sdetiaon pacar ant sees he potters Hees gat feast, THe sewpoln paren comple Resin taken
Sanit se ye down pile auunon i itamnticn tite
terns ne ctrl equation Tos hs frei ssa HG Cai We Hn UC đá
tanhenluok dlebisui eiaslo lơ Fares a 3.3 AN ILLUSTRATIVE EXAMPLE Lt us consider steady one-dimensional heat conduction governed by afar = ( #) *
‘where & is the thermal conductivity, T is the temperature, and S is the rate of heat generation per unit volume,
(5.10)
Trang 25
2 [NUMERICAL HEAT TRANSFER AND FLUID FLOW Preparation To derive the discretization equation, we shall employ the aridpoint cluster shown in Fig 3.2 We focus attention on the grid point /, ‘which has the grid points £ and W as its neighbors (denotes the cast side, le, the positive x direction, while W stands for west or the negative x direc- tion.) The dashed lines show the faces of the control volume; their exact locations are unimportant for the time being The letters ¢ und w denote these faces For the one-dimensional problem under consideration, we shall assume a unit thickness in the » and 2 ditections, Thus, the volume of the contrat volume shown it AxX1XJ, If we integrate Fq (3.10) over the control volume, we get
Gay
Profile assumption, To make further progress, we need 9 profile assump> tion of an interpolation formula, Two simple profile assumptions are shown in Fig 3.3 The simplest possibility is to assume that the yalue of Tat a grid point prevails over the control volume surrounding it, This =~ stepwise
profile sketched in Fig 3.32 For this protile, the slope a7jidx igigpt detined at the controlvolume faces (it., at t9 or €) A profile that do@Pnot suifer from this difficulty is the piecewiselincar profile (Fig 3.36) Mere, linear interpolation functions are used between the grid points
The discretization equation, It we evaluate the derivatives d7yédx in Fa (B.11) from the piecewise-linear profile, the resulting equation will be
ke(Tp = To) _ k„»— te z 9, 412) :
where Fis the average value of § over the control volume It is usefirl to cast
the discretization equation (3.12) into the following form:
apTp 2 agTe tayTw +b, 3.13) Figure 32 Grid:poin custer for the one-dimensional pobieia, 2 DISCRETIZATION METHODS: a) đi Figure 33 Two simple profile assumptions, () Slepwite p(effe; (5) piêetkftedinear pre: where (2142) / (3.140) (8142) and (3.144) Comments,
Trang 261 NUERICAL (IEAT THÁNSEER AND PLULD HLOW 1¢ subscript nb denotes a neighbor, and the summation is to be et all the neighbors
2 In steriving Ea, (1.13), we have used the simplest profile assumption that bled us t evaluate afd, OF course, many other interpolation functions would have been possible,
Further, it i important to understand hot we need not use the same profile for all quantities For example, 5 need aot be calculated trom 3 linear variation of S between the grid points, wor &¢ from a linear variation of & between Kp and ky
4 Ten far a given variable, the same profile sisuinption need sot be used fr all terms in the equation For example, if Eq (3.10) liad an additional term involving T alone, it would have been permissible tw use stepwise profile for that term, instead of adhering 10 the piecewisodinear prolile used for evaluating đT/đx
ee Oe
vagy oF Stab, Corner ie al WF Ie abe vô Re akg, ul tue pains eg ehean Apis ea a ‘sot Wea ee re so lEmd foidensnf Hi +84 Jae Jaber inca dig ke Hilpnifes We cake rw We soameid olan shoul always axe (1) dhyded sale rarer] (2) cieal bahar
Pave sion expe WR ab ue (ATA il Te seg tham ẤN? má te đc sóc Sri carton shoudl lội lạ tho beendly'emacalar Mhha hơi già ig cold Unrest Approximate, but — Unnslinie
Figure 34 Physically sealise and Unveatisthe behavior
DISCRETIZATION MUA}LODS
by an ambient fluid, the solid cannot acquire a temperature Fower than that of the đhúd, We shall alvays apply such fests t0 our discretization equations ‘The requirement of overall balance implies integral conservation over the ‘wliole ealeulation Unmain, We shall jnsist that the heat Muxes, mass Now rates, and momentum Muxes must correctly give an overall balance with appropriate soutees and sinks—not just ia the limit as the number of grid points becomes very large, but for any number of grid points Our controb-yolume formulation makes this averall balance possible, but vate is needed, as we shall shortly sec, in calculating fluxes at the contrololume interfaces The constraints of physical realism and overall balance wall be used 10 guidết out cholees of profile assumptions and related practices, On the basis of these constraints, we shall develop some basic rules that will enable us to discriminate botweon available formulations and to invent new ones The decisions that are normally governed by mathematical considerations can now be diected by physical reasoning
Treatment of the source term Belore we proceed w devslop the basic ‘ules, we shall give some attention to the source term Sn Eq (3.10) Often, the source term is a function of the dependent variable T itself, and itis then desirible to acknowledge this dependence in constructing the discretization equation We can, however, Formally account for only a linear dependence because, as we shall sce later, the discretization equations will be solved by the tecliniques for linear algebraic equations The procedure for “linearizing” a given S~T relationship will be discussed in the next chapter Here, it is sulffefent lo exptess the average value S as
F=Se4+SeTr, (16)
whore Sq stands for the constant part of 5, while Sp is the coefficient of Tp (Obviously, Sip does not stand for S evaluated xt point P)
‘The appearance of Tp in Eq (3.16) reveals that, in expressing the average value 8, we have presumed that the value 7p prevails over the control volume: in other words, the stepwise profile shown in Fig 3.3a has been used (It should be noted that we are ffee to use the stepwise profile for the source term while using the piecewisedinese profile for the d7de tec.)
Trang 2736 NUMERICAL HIEAT TRANSFER AND FLUID FLOW aw ¬ (3.134) 0% ap sae tay —Sp Ax, (3.186) and b= Sc Ax 6 G18)
“The foregoing introductory discussion provides suffictent background to allow the formulation of the basic rales that our discretization equatious should obey, to ensure physical realism and overall balance These seemingly simple rules have far-reaching implications, and they will guide the develop- ment of methods throughout this book,
3.4 THE FOUR BASIC RULES
Rule 1; Consistency at controlvoleme faces When a face is common to two adjacent control volumes, the flux across it must be represented by the same expression in the discretization cyuativns for the two control volumes
Discussion Obviously, the heat ux that leaves one cantrol vowue through a particular Face must be identical to flux that enters the next control volume though the same face, Otherwieete overall balance wonll ot be satisfied, Although this requirement is easy to understand, subtle siolations must be watched for For the controt volume shown in Fig 3.2, we could have cvsluated the interface heat fluxes k d7idx from a quadratic profile pussing through Ty, Tp, and Tp The use of the same kind of formulation for the next control volume implies thet the gradient dT/dx at the common interface is calculated from different profiles, depending, on which control volume is beiag considered The resulting imconsistency* in Tax (and hence in the heat Mux) is sketched in Fig 3.5 ‘Another practice that could lead to flux inconsistency is to assume that the Muxes at the faces of 3 gen contol volume ace all governed by the center-point conductivity kp, Then the heat flux at the interface ¢ (shown in Fig 3.2) wil be expressed as kp (Tp—Te)MEx)e when the contol volume surrounding the point P is considered, and as kr (Tp —TeWilOx)_ vien the equation with # as the center point is constructed To avoid such incon-
ie so happens that, i the interfaces ate located midwar between the Bed points the type oF quadratic profile shawn in Fig 35 does not che any inconsistency This & Iceause the slope of a parabola a a location midway ferween ro polnts E cxatf eNeal to the slope of the straight line jiaing the tw points But this operty of the pals must be fegarded as fortuitous, and one most, m yencral tesain from shunting the race fix expresion while roing fom one control volume 10 the next DISCRETIZATION METHHOUS 7
stone from ent Slope trom let
Figure 35 Flux inconsistency resulting from quadratic profile,
sistenecies, itis useful to remember thet an interface Mux must he considered in its own right, and not as belonging 10 2 certain control volume,
Rule 2: Positive coefficients Most situations of interest hese will be such lat the value ofa dependent variable at 2 grid paint is influenced by the values at neighboring arid points arly through the processes of convection and aiffusion Then it follows that an increase in the value at one geid point should, with other conditions remaining unchanged, lead to an increase (and mọt a decrease) int the value at the neighboring grid point In Eg (3.13), if an increase in Te must lead to an increase in Tp, it follows that the coefficients ap and ap must have the same sign In other words, for the general equation (3.15), the neighbor coefficients agp and the ‘centerpoint coefficient ap all must be of the sume sign We can, of course, choose to make them all positive or all negative Let us decide to write our discretization equations such that the coefficients are positive: then Rule 2 can be stated 2s follows:
All coefficients (ap afd neighbor coefficients aqy) must always be positive
Comments, The coefficient definitions given in Eys (3.14) show that our HhuatraHwe iseretization equation [Eq, (3.13)] does obey the positive- ‘coefficient rule THawover, as we shall sce later, there are numerous formulae
Trang 28RUMERICAL HHEAT TRANSELK AWD EAD FLOW -0ile saluto Thẻ pfetenee of a usgative neighbor eoelticicut can Wad 1> the sitvation in which an incsouse iy a bouudary teinperstuze causes the Loispermture at the adjacent grid point 19 decrease We sill aecepe oaly these formulations tat guarantee positive coefficients under sil vircunistunces,
4: Negativeslope linearization of the source term If we eunsider the ignt definitions in Eqs (3.18), it appears that, even if the neighbur cacificlents are positive, the eentor-point coefficient ay can became sepasive via the Sp term, OF course, the danger can be completely availed by requiring that Sp will not be positive, Thus, se Foranitate Rule 3 as follows Ro
When the souice term i Hneaszed as 3= Sq + $pT the cnetfcit Sp must always be les than or equal to zero
Kemarits, This rule % not as arbitrary as it suunds Most physical processes do have a aegativeslone rclationship between the source term and the dependent variable, Indeed, if Sp were positive, the physical situation could become unstable A positive Sp implies that, as 7p increases, the source ecm increasgs; if an, effective heat-removal mectanisiy is not available, this Jn turn, lead to an increase in Tp and so on, Computativwaly itis vital ‘eop Sp neyative so that instabilities and physically unrealistic solutions du snot arise, The source-term linearization is further discussed in the next chapter, Tt 1s sufficient to note here that, for computational success, the principle of negative Sp ts essential
Rule 4; Som af the nelghbor coefficients Often the governing differential egicitions contain oly the derivatives of the dependent vaciable Then, if ovents the dependent variable, the functions T and T+ e (where ¢ sa abilrary cunstant) bath sulisiy the differeniial equation, This ry of the differential equation must also be reflected by the otization equatian Thus, Eq, (2.15) should reraain valid even when Tp snd ai) Typ's are theseased by x constant, Krom this requirement, Tt follows that dp must equsl the sum Of the neighbur coefficients, Hence ne statement of Rule 45: We sequite ora Xinh G19) Tor situations where the differential equation continues to cemain satisfied after a constant is added to the dependent variable
Dikeussion, I is easy to see that Eq, (3.13) does satisfy this rule, The rule celes that the centerpoint value Tp isa weighted average of the neighbor
eh ETAT
biScKET ZATION BIULTHODS 3
values Yup Uae Ea, (313), the coefficients iy TH, (3.17) de not obey the rule This is, however, not a violation, tut a case oF inapplicabiity OF the rule When the source term depends on T, both Tand Te do wot satisfy the tiferential equation, Even in such eases, the rule should not be forgotren, but shuld be applied by envisaging a special eae of the equation, If, for example, Sp is set equal to 2eru in Eq, (3.17), the rule becomes applicable and is indeed obeyed When the uitferentisl equation is satisied by both 7 and THe, the dessed temperature field T does not become multivalued or indeterminate The values of T ean be made determinate by appropriate boundary conditions Conformity to Rule 4 ensures that, if, for example, the boundary tempera: tures were increased by a constant, all temperatures would increase by exactly that constant
Another way of looking at Rule 4is this: Whoa Ute source term is absent and the neighbor temperatures Ty, ate all equal, the center temperature Tp mist become equal (0 them Only 4 poor discretization equation would not predict Tp= Typ under these cscunistances,
3.5 CLOSURE
Ia this chapter, we have made certain basic decisions about the type of uiscetiation method io be developed i this book, Through 4 simple iy example, we have been able to formulate four basic rules, which constitute the underlying guiding principles for all further work The discussion has been given in terms of temperature T'as the dependent variable, This wes done simply for conceptual convenience We shall continue with T in Chapter 4, but switch ta the general variable @ from Chapter 5 onward, Of course, the four rules developed in this chapter ute all applicable to the general variabie ¢ ‘The convection term in the general differential equation (2.13) requires special formulation This matter is deferred to Chapter 5 The remaining three temns of Fg, (2.13) are dealt with in Chapter 4 in the framework of heat conduction,
PROBLEMS
3A Using the Taplocseties expansion around point P ih Fig 32, show thar the Tiiteditference approximation for d°T/ds! is given by
er Ge Gh 3 [Tg=T ty — By ~ Teh]
Trang 29é
40 NUMERICAL HEAT TRANSFER AND FLUID TOW saNWenlenes) Let the weighting function W be zero overywhete except beiwcen the Points M and £ In Fig, 3.2 Further, asumte that the weighting funetiun f piecewise linear, with value unity at P and sư at points Wand £ Multiply Eq (3.10) by the ‘weighting function, and integrate ese the teglon from point W to point & Uses Piccewiselines profile for T Compare the resulting Uiseretiation equation ‘with tq 2) (Note that the method outlined hese, which is speciot case of the method of weighted residuals, Is known as the Galerkin metho.)
33 Consider Ea (3:10) and assume that Si constant, but & depends on x Fuither, we 8 uniform grid spacing in Fig 3.2, so thai AT (x)_= (bx) Detve the discretization ‘equation by writing Fa, (3.10) as wT yak ar, ae aa tS% 2d using the approsimations BE Kelis Ty 2 se tàn * ar Te ae "Tar
‘with diffe as a given quantity Noting that dk/dx can be yostive OF negative, find the conditions for which the coefficient 2g or ey would become negative, thes violating Rule 2 (Note Ghat the derivation in Section 3.3, which war busod wn the physical significance of the terms, did not lead to negative coefficients.)
34 Ta an axisymmeical situation œ sesdy onedimendnal conduction problem is
——
ad (yar)
re (« £) a
here rik the radial coordinate Following the procsdute fn Section 3.3, dethe a ‘iscretination equation for this siuation, (fuldply the diferertiah equation hy ry ad then integrate with sepeet 10 r fram ry 10 ¢q) Internect the coefficients ihe CHAPTER FOUR HEAT CONDUCTION 4.1 OBJECTIVES OF THE CHAPTER
In this chapter, we shall begin the task of gaustructing a numerical method for solving the general differential equation which governs the physical Processes of interest here As we have seen, the equation contains four basic terms Here we shall omit the convection term and concentrate on the femaining three terms The construction of the method will be completed in Chapter 5, where the treatment of the conveetion term will be discussed
Omission of the convection term reduces the situation to a conduction- type problem, Hest conduction provides a convenient starting point for our formulation, because the physical processes are easy to understand and the mathematical complication is minimal
‘The objectives of this chapter, however, go far beyond presenting 4 ‘numerical method for hest conduction alone Fist, other physical processes are govemed by very similar mathematical equations Among these are potential Now, mass diffusion, Now through porous media, and some fully deweloped duct flows The numerical techniques described in this chapter are directly applicable to all these processes Electromagitetic field theory, Giffesion models of thermal radiation, and lubrication flows sre further ‘examples of phenomena governed by conduction-type equations Although we shall only occasionally make reference to these related processes, it Is important to remember that the techniques developed in this chapter sre lnmediately available for application tn these different areas
Trang 302 NUNLRICAL HEAT TIANSERILAN FLUID CLOW lụz latsr chápseze, The nroeedure [0r the solttdon of the algebrale squations ts ed hee tag anceméforall manner Later vliap6%5 mIoöly the conten of the algebraic equations, but the sume solutun technique continues to be applicable Thus, oven for the reader who ix exclusively interested in Aukl-ow calculation, an understanding of this chapter is cssential; nuch of the material here (and in the next chapter) is an integral part of the fuid-flow
soltion scheme to be presented in Chapter 6
To be able to soe the similarities between transfer of omentum and ouster of feat and to regard velocity as, in some ways, analogous to re is a gecat conceptual help The use of heat conduction as 2 ‘losk in the Muidflow calculation scheme reinforces this conceptual build unity,
4.2 STEADY ONE-DIMENSIONAL CONDUCTION 4.21 The Basic Equations
© couse of presenting the illustrative example in Section 3.3, which wat ustd as velicle to explain the four basie rules, we have slrerdy uerived the Giseretivation equation for steady conduction in one dimension To review the
cain ingredients, the governing uifferential equation is Lt B)esno uy z8 “This Inads to the discretization equation oeTe tayTy tb 42) (35) ow ge (4.38) ap #02 Fay —Sp Bx (39) b= Sc Ax (434)
‘ho geld points P, E, and Ware shown in Fig 3,2, whe various distances are sated The controbvolume faces ¢ and w are placed between the grid
HEAT CONDUCTION “ point P and it cotresponding acighbors, The exact locations of these faces ca be considered to be arbiteary Many practices for their placement are possible, some of which will be discussed in Section 4.6-1 For the tine being, ‘we shall simply regard the locations of e and w as Auown in relation to the rid points #, 2, and W The quantities Se and Sp aise from the source-term linearization of the form
c + Sen (4)
As to the profile assumptions, the gradient đ7/dv has been evaluated from a plecewise-linear variation of T with x, while for the linearized source term the value Tp is assuyned to prevail throughout the contral volorme, Ít should, of couse, be remembered that other choices of profiles are possible and petmissible, as long as the four basic rules are not violated The policy here is to adopt rather simple profiles within the constraints of these rules and to introduce sophistication only where it js necded,
Many important aspects of the one-dimensional heal-conduction problem still remain to be discussed It is to these topies that we now tuen
4,
forthe ed pol shown in Fig 15, Ì bớt hesesy at the dias 5 ay be sin ded tus of nnamlam aid spa te Se nana cay cannes ee we aha abun an sation only wien he tad 4 sim) No Bt thre no ned to enpey& ne gdh mee wee te See
smn change rath: slowly witht On ie eer hun ae “eel wet the Tr rica ee mconrpion seem to peal ft nosuiorm giất lai t lớn seuac tard nor giác Thec sund tu oven seri The pi ping sod be ly Inked he way Me pentose change ithe eon ý la, te xe mở thoa Ma den The Grid Spa What maximum (or rninimumyfi@tio the adjacent grid Intervals should main- tin
Since the T~.x distribution is not known before the problem is sotved, ow can we design an appropriate nonuniform grid? First, one normally has some qualitative expectations about the solution, from which some guidance can be obtained Second, preliminary cuarsogrid solutions can be used to find the pattern of the 7'~x variation; then, a suitable nonuniform grid can be constructed This is one of the reasons why we insist thet our method should give physically meaningful solutions even for coarse grids An exploratory soarse-grid solution would not be useful if the method gave reasonable solutions only for sufficiently fine grids,
The number of grid points needed for given accuracy and the way they
Trang 31
“ NUMERICAL HEAT TRANSFER AND FLUID FLOW should be distributed in the’ calculation domuin are matters that depend on the nature of the problem to be solved.,Exptaratory calculations using «nly 4 few grid points provide a convenient way of lesming about the solution, After all, this is precisely what is commonly done in a laboratory experiment Proliminsry experiments or tial runs are eonducted, and the resulting information is used to decide the number and loestions of the probes to he installed for the final experiment,
4.23 The Interface Conducti
In Eq, (4.3), the conductivity &- has been used to represent the vilue of k pertaining to the control-volume face ¢; similarly, ky cefers to the interface w, When the conductivity & is a funtion of x, we shall often know the value of # only at the grid points W, P, &, and se on, We then need a prescription for evaluating the interface conductivity, say Re, in terms of thie grid-point values The following discussion is, of course, nat relevant ifqgtuations of
TH g3 Ni Fe 008601906 Cle MEneldreeosỏe d HỆ nogbi Min? Hưng êm, BI Gì Re nà ng Sen mũ My Xà Bế lộ nh HÀ Gia "chẽ ẽ.ẻẽ
Sung ly de ftmaaik BE Hi Vi eee 4 ssilieidir E BiBemh vleteed Cử PA T5 ng cỡ tan Tì He OR ee Perea ee
Se RUE Tes ae eee CR a
ty ace IE WA Le reed or all Rin 88 daily ty ut care ban Gent een eas on ke =fekp + (1 fhe (45) where the interpolation factor fz is a ratio defined in terms of the distances shown in Fig 4.1 Giles tổ); (46) If the interface e were midway between the grid points, f= would be 0.5, and kp would be the arithmetic mean of Kp and ke
We shall shortly show that this simple-minded approach leads to rather incorrect implications in some cases and cannot accurately handle the abrunt changes of conduciivity that may occur in compasite materials Fortunately, a much better alternative of comparable simplicity is available In developing “ NFAT CONDUCTION |-—— ts ———+ a ị so ? if † © é i
Figure 4.1 Distances associated with the interface e,
this alternative, we recognize that itis not the local value of conductivity at the interface © that concerns us primacily Our main ubjective is to obtain a 00d cepresentation for the heat Mux g, at the interface via
Ke(Tp—Te)
(xe (47)
%
which lias, in effect, been used in deriving the discretization eauation (4.2), The desired expression For Xe is the one that fads to z "eorrevf” đc
Let us eomshsr that the gontol wolime surounding the grit point i {ed wth a material of uniform conductivity Rp, and the one around F with a material of conductivity kp For the composite slab helwern points P an
f, steady one-dimensional analysis (without sources) leads t0 ter 0 Chek * Oxhale” (48) Combination of Eqs (4.6}-(4.8) yields fe ( cứ tế) 69) Ñ ‘When the interface es placed midway Between P ad E, ge hoye J2 = 0.5 then 0.5(kg' + kặ) (189 h Ea (4.108) othe
Equations (4.10) show thae ke i= the harmonic mean of ky ane Airy rather
‘than the arithmetic mean which Bq, (4.5) would give when fe = 0
Trang 32UMERICAL JIFAT TRANSEER AND TLUID FLOW ‘The se of Eq (4.9) in the coefficient definitions (L3) leads to the
expression Hoe ag
đán
A similar expression can be written for ay Clearly, aye tepesent thẻ conductance of the material between points # aid E-
The effectiveness of this formulation can be quickly seen in the following sce iiaiting easest
1 Ler kg+0, Then, from By, (4.9),
4e 0 (4.12) “Thus Impliey that the beat ux at the fice of an insulator becomes zero, 2s at shauld, The arithmetiemean formulation, on the other hand, would have siRen 2 nonzero flux in this situation
Let Ep ky Then
(4.13)
ti este Mimo spinon ove ea toda i dor i ote oa, Eaten (O10) Jules tah rel nea Buoy wi aoenieyl ov Bes Ing ig fe cages Sou he Eireonducviny tcc ound pit P vos afer eagle eatace
po corti Win tt mater ound (ihe ante old Rave vetcned the effet of lyon fg) the her implation is ha aca Tres
fe i ot equa to Fay bt rather ff tmes tA ie ection sl shor the givouricons tác bại me weiss a vies ft di vệ
fy (ea he we of Bạ CID) ls
— Helle gu, — Te) vn ‘i
When kp ® kg, the cempetatuce Tp will prevail right up to the interface e, wid whe temperature drop Tp Te wil actustly take place over the istance (Ex}eqs Thus, the correct heat flux will t¢ as gen by Eq: (4.14) in other worts, the factor f in Eq (4.13) can be seen to compensate for the use of the nominal distance (Bx) ia Eq, (4.7),
Cobsidération of these two limiting cases shows that the formutation ean abrupt changes in the conductivity without requiring an excessively
HEAT conUETION “ fine eid in the vienuty of the change This is not only convenient for conduction calculations in composite slabs, hut it has other quile fascinating Implications These ave been described in Patankar (1978) and will be explained in tater chaptors
The recommended interface-conductivity formula (4.9) is based on the Steady, nowource, one-dimensional situation in which the conductivity varies ‘nv stepwise fashion from one contra! volume tù the next, Even in situations swith monzetn sources or cee Vacation of conductivity, i performs
mach better than the ari(@etic-mean formula This is demonstrated in Patankar (1978) for some ses for which exact analytical solutions can be found,
42-4 Nonlinearity
The discretization equation (4.2) is a linear algebraic equation, and we shall solve the set of such equations by the methods for linear algebraic equations We shall, however, frequently encounter nonlinew situations even in heat conduction, The conductivity & may depend on 7, or the source § may be 4 hnontingar function of F Then, the coefficients in the Jiseretization equation ‘sil themselves depend on 7 We shal] handle such situations by iteration This process involves the following steps:
St with § guess of estimate for the vilues of Tat all grid points From these guessed T's, calculate tentative yalues of the coefficients in the discretization equation,
3 Solve the nominally linsar set of algebraic equations to get new values of ii A, With these T's as better guesses, retum to step 2 and cepeat the process ‘until further repetitions (called iterations) cease to produce any significant
changes in the values of T
This Sinal unchaning state is called the convergence of the iterations.* The converged solution is actually the correct solution of the nonlinear equations, although it is artved at by the methods for solving linear equations Wis, however, possible that successive iterations would not ever converge to a solution, The valucs of 7 may steadily deft or oscillate with increasing amplitude, This process, which is the opposite of convergence, is called divergence A good numerical method should minimize the possibilities of divergence As we shall see later, adherence to our four basic rules promotes
‘Sometimes, the term conversenes is used Yor the process by which successive gr teñnenenk brings the mamnerical solution closet to the exset solulion We shall refer ‘his aspect as tho “accuracy” of the numsical solution, and eserve the word convergence for the convergence of iterations
Trang 33“ NUMERICAL HEAT TRANSFER AND FLUID FLOW convergence; we shall also discuss other strategies for avoiding Mergence At this point, it is sufficient to note thet our procedure is not I@ed to linear problems, and that any nonlinearity can, at least in principle, be handled by the iterative technique just outlined,
4.25 Source-Term Linearization
When thie source 5 depends on T, we express the dependence in a linear form gen by Eq (44) This is done because (1) our nominally linear framework ‘would allow only a formally linear dependence, and (2) the incorporation of linear dependence is better than treating S as a constant,
‘When S is a nonlinear function of T, we must linearize it, i., specify the values of Sq and Sp, which may themselves depend on 7 During each iteration cycle, Se and Sp would then be recalculated from the new values of T The linearization of $ should be a good representation of the S~T relationship Further, the basic rule about nonpositive Sp must be obeyed
There are many ways of splitting 2 given exprestion for $ into Sc and SpTp Some of these are ilusteated by the following examples The numbers
appearing in these examples have no particular significance The symbol TP is
used to denote the guess value or the previousiteration value of Tp Example 1 Given: $= §— AT, Some possible linsurizations are
1, Se=$,Sp=—4 This is the most obvious form and is recommended: 2 Se=5=4Tf, Sp=0 This is the approsch of the lay person who to the entire F into Sq and sets Sp equal to zero This approach,
however, is nat impractisable and is perhaps the only choice when the expression for Sis veey complicated,
3 Sc=S+I7$, Sp=—I1, This proposes a steeper $~T relationship than the one actually even, The sult wil he that the consergence of the iterations will slow down Howerer, |f there are other non: linearis in the problem, this slowdown may actully be weleame
Example 2 Given: 5
3 + TT Some possible Linearization are:
1, Se=3, Sp=7 In general this is not acceptable, as it makes Sp positive, If the problem could be solved without iteration, this linearization would give the correct solution, but if iteration is employed for some reason (such as the nonlinearity of other terms}, the presence of a positive Sp may cause divergence
2 Sq=3+7Tf, Sp=0 This is the practice onc should follow when a nepative Sp ig not naturally fortheorning, 3 Sc=3+9Tp, Sp=—2 This is an artificial It will, in goneral, slow down the convergence, ation of a negative Sp MEAT CONDUCTION “ s
Example 3 Give 4 ST* Some possible linearizations are:
1 Sg =4—STP, Sp=0 This is the lazy.person approsch, whieh lạ
to take advantage of the known dependence of ÿ en T
2 S¢=4, Sp=—STH? This Tooks Eke the correct Tinearitation, but the given # ~ 7 eurwe is steeper than this Smplis, 3 Recommended method: ` @ (Tn — Tf) = 4— STƑ" — 15782 (Tp — TH) vĩ, Tins, 4+ 10T”, sp -isng2
This Jineazization represents the tangent to the S~T curve at Tf 4 Se=4+20T7°, Sp >—2STp?_ This linearization, which is steepe=
than the givon $~ T curve, would slaw dawn convergence
ir foue spice lie ation ate-anumi ee 42 lode wil te HU ST be On bua ase apaue ner of pettbe e ‘PO delat hoác Rite 3, Anny fhe cettiesope Ines, le angen tee phon ers nal lc Det sli: Sleeper let reise ut Tepid DUG ed tạ 3n? Ca se: [ai Sie Da ta tớ,
Ma hai resorkcfe dit nh l0 Xi
‘This discussion of the soureesterm linearization is adequate for present purposes Further considerations ate given in Chapter 7
Figure 4.2 The four possible linearizations for Example 3
Trang 341——-—-
su NUMERICAL HEAT TRANSHEX AND FLUID PLOW 4.26 Boundary Conditions
Let us consider that, for the one-dimensional problem, the string of avid is shown in Fig 4,3 is chosen, These is one grid point on each of the two “cundaries The other grid points wll be called the internal points, around Ích is shown a cantrot volume A discretization equation like 1g led as
1 cám be written Tor cách such contiol volume If Ey (4.2) i regal su for Tin we shen have the necessary equations For all the vinknoww at the iuternal grid points, Two ul these equatiuns, however boundary gcid-point cemperaturvs It is througle the teentment of lary teenperatures that the given boundary conditions ace intew these bo
vce into the aumerieal solution schenie
Since ix is not necessary ta discuss the two boundary points separately, \cention will be focused on the left-hand buundary point 22, which is adjacent st internal point / as shown in Fig 4.3 Typleally, three kinds of sre encountered in heat canduction These are:
soumdary’ condi
|, LZXen boundary temperature Given boundary hese lux
5 Boundary heat Mux specified wa a wat transfer cveitivient ant the temperature of the surrounding Quid
If the boundary temperature is given (Le, if the salue of Tp is known), co particular difficulty srises, and oo additional equations are roquired When the boundary temperature is nor given, we need to construct an additional ccuation for Tg This is done by integrating the differential equation ver the “Sulf? control volume shown adjacent (© the boundary in Fig, 4.3 (This control volume extends only on one side of the grid point B This Is why we to it a8 the half control volume.) An enlarged view of this control velume is given in Fig 4.4 Integrating Ey, (4.1) ower this control volume and noting thet the heat flux q stands for —É d7/Úx, we get gaa + Sot SpTa) Ax = 0, 415) Š tà hán) UA gure 4.3 Control volomes for he itersal wad boundary pass st EAT CONDUCTION Figure 44 Half control volume near the boundary
where the source term hus been linearized in the usual fashion The iaterfaee heat Mux đụ cam be weitten along the lines of Faq, (4.7), The result is
„ /Ứn — Tị) tuy * Set SeTa) Ax
* (4.16)
Further implementation of this equation depends on what is given about the bounduy heat Aux gq If the value of gp itself is given, the required equation for Tp becomes ate ees fan at * _ 180 b=Se dx tan (4.180) Og =a; — Sp Ax (4.186)
Trang 3532 NUMERICAL HEAT TRANSFER AND FLUID FLOW b= Se dx +t, , (4218) tp = a) —Sp Ax th (4212
In this manner we are able to construct (le requitel mumber of equations for the unknown temperatures We shall now describe the method for solving them,
4.2-7 Solution of the Linear Algebraic Equations
The solution of the discretization equations for the one-dimensional situation can be obtained by the standard Goussian-elimination method Because of the particularly simple form of the equations, the elimination process tums into a clightfully convenient algorithm This’ is sometimes called the Thomas algorithm or the TDMA (TiiDiagonal-Matrix Algorithm) The designation TDMA refers to the fact that when the matrix of the coefficients of these equations is written, all the nonzero coefficients align themselves slong three isgonals of the matrix, For convenience in presenting the algorithm, it is necessary to use somewhat different nomenclature, Suppose the giả points in Fig 4,3 were numbered 1, 2, 3, , À, with points 1 and VY denoting the boundary points ‘The discretization equations can be written as
OTe = dT + oT + oy (4.22) for #=1, 2, 3, ., MV Thus, the temperature jis related to the neighboring Temperatures Ti+1 snd T;—1 To account far the special form of the boundary-point equations, let us set
=O and by =0 4.23) so that the temperatures Ty and Tis will not have any meaningful rote to play (When the boundary temperatures are given, these boundary-point equations take a rather trivial form For example, if Tị is given, we have a, =I, by =0, cị =0, and đị = the given value of Ty.) for #=2 isa relation between 7ì, Tz, and 75 But, since T, can be expressed These conditions imply that T, is known in terms of Ty The equation in terms of Tz, this relation reduces 20 a relation between T and Ty In other words, Tz can be expressed in terms of 7 This process of substitution gen be continued unti Ty is formally expressed in terms of Tye But, Because Tyy¢1 has no meaningful existence, we actually obtsin the numerical value of Ty at this stage This enables us to begin the “hack-substitution”” process in which Ty is obtained from Ty Ty—a ftom Ty1, - Ts from Ts, and 7; from T, This is the essence of the TDMA rar conpuction a Suppose in the forward-substitution process, we seek a relation TPT + Q (4.24) alter we have just obtained % V (425) Subsdtution of Ea, (4.25) into Ky, (4.22) leads to Te = Bi Tees + CPT + ei) + dy (4.26) Which can be rearranged 10 took like Eq, (4.24) In other words, the soefficlemts P; and Q, then stand for (4270) (4278) and tis
‘These are recurtence relations, since they sive By and Q; in teams of P; Q.-1 To stait the recurrence process, we note that Eq (4.22) for 7 almost of the form (4,24), Thus, the values nf Py and Qr are given by
4
(428)
[Nt is interesting t0 note that dese expressions do follow from Eq (4.27) after the substitution e+ = 0.)
‘At the other end of the F, Q; sequence, we note that by = 0 This leads to Py =0, and hence from Eq (4.24) we obtain
Tx = On (429) Now we ave in a position to start the back substitution via Eq (4.24), Summary of the algorichm 1 Calculate Py and Q, from Eq (428),
Trang 36
$e NUMERICAL HEAT TRANSEER AND FLUID FLOW ‘The tridiagonal-matrix algorithm is a very powerful and convenient caualion solver whenever the algebraic equations can be tepreseatel im thự orm of Eq, (4.22) Unlike general matrix methods, the TOMA requires jouer storege and computer time proportional only tw N, rather than tò
4.3 UNSTEADY ONE-DIMENSIONAL CONDUCTION
3-1 The General Diseretization Equation
Widt reference to the general differentiat equation for ộ, we have now seen at ‘cast in the one-dimensional context, how to handle the diffusion term snd = source term, Here, we turn to the unsteady ferm and temporarily drop the source term, since nothing new needs to be said about it Thus, we seek to zelve the unsteady one-dimensional heat-conduction equation
~-¿ (9) mà
ner, for convenfence, we shall assume pe to be constant (la Chapter 2, c ‘as shown how the heat conduction equation could be modified to take secount of the variable specific heat e See Problem 2.2.)
Since time is 2 one-way coordinate, we obtain the solution by marching in time from a given initial disteibution of temperature Thus, in a typical “ime step” the task is this: Given the arid-point values of T at time ¢, find the valoes of T at time ¢+Ar The “old” (given) values of T at the grid
points will be denoted by Tz, TE, Thy, and the “new" (unknown) values at net + At by TẢ, TẾ, T
‘The discretization equation is now derived by integrating Eq, (4.30) over ‘the control volume shown in Fig, 3.2 and over the time interval from £ to t+ Ap Thus,
ar = Bee ihe
nf [ga [PED ae an
where the order of the integrations is chosen according to the nature of the For the representation of the term A7/@¢, we shall assume that the evid-point value of T prevails thoughout the control valume, Then, «ff F atae peas (7h — 12) 432) near CONDUCTION 35 Following our stessly-state practice for k A7/Ax, we obtain pe Ax (TA — TB) -ƒ [:
‘nis at this poiot that we need sn assumption about how Tp, Tis, and Ty wary with time Irom ¢ to t+ At, Many assumptions are possible, and some of them ean be generalized by proposing
— =] a3
[ Tự dt = VTE +(L—/ĐTỆ] at, đáo
Where fis a weighting factor between 0 and 1 Using similar formulas for the integrals of Ty and 7y, we derive from Eq (4.33)
ax — 78) =z |#kữ? = Tế: oo SE Th = [beth )
ap [eR ae) _ &„Œ? - Tỳ;
toon PAD — tah TB sao
Trang 3756 NUMERICAL EAT TRANSFER AND FLUID FLOW
4.3-2_ Explicit, Crank-Nicolson, and Fully Implicit Schemes
For certain specific values of the weighting factor f, the disctetizaion equation reduces to one of the well-known schemes for parabolic differential equations In particular, f= 0 leads to the explicit scheme, f=0.5 10 the Crank-Nicolson scheme, and to the fully implicit selieme We shall briefly discuss these schemes and finally indicate the fully Implicit scheme as our preference
The different values of f can be interpreted in terms of the 7p~/ variations shown in Fig 4.5 The explicit scheme essentially assumes that the
old value Tp prevails throughout the entise time step except at time ¢+Ar
The fully implicit scheme postulates that, at time 1 Tp suddenly drops fiom
Tp to Tp and then stays at Tp over the whole of the time step; thus the
temperature during the time step is characterized by Tp, the new value The
Crank-Nicolson scheme assunies linear variation if Tp At first sight, the linear variation would appear more sensible than the two other alternatives Why thea would we prefer the fully implicit scheme? The answer will emerge very shortly For the explicit scheme (f= 0), Eq, (4.36) becomes apTp = agTE + ayTw + (ah —ap —ay) TS (4.38)
This means that Tp is not related t0 other unknowns syggs Te ot Ty, but is explicitly obtainable in terms of the known temperaiulSOEY, 72, 19) Th Js-why the schema is called explicit, Any scheme with #0 sauld be init Espler Đrank.Nebbem uly imalict 124.5 Variation of temperature with tn for thee different schemas A ear conpucTiON 27 that is, Zp would be linked to the unknowns Ti and Ty, and the sclation of 2 set of simoltancous equations would be necessary The conyenience of the explicit scheme in this regard is, however, offset by a sorious limitation 1f we remember the basic role about positive coefficients (Rule 2) and examine Eq, ), we note that the coefficient of 72 can become negative, (We consider TẾ as 4 neighbor of Tp in the time diection,) Indeed, for this cosficient to be positive, the time step A/ would have to be small enovgh so that a3 txeeeds ag-Fay For uniform conductivity and Ax =(8x)- =(8x)u, this condition can be expressed 25 pe{dx)? Ar< (439)
1 this condition is violated, physically unrealistic results could emerge, because the negative coefficient implies that a higher 72 results in a tower Ty, Equation (4.39) is the well-known stability criterion for the explicit scheme, It js interesting to note that we ltave been able to derive this from physical arguments based on one of our basic rules, The troublesome featute about condition (4.39) is that, as we reduce Ax to improve the spatial accuracy, we are Forced to use a rovcl smaller Az
‘The Crank-Nicolson scheme is usually described as unconditionally stable ‘Aa inexperienced user often interprets this to imply that a physically realistic solution will result no matter how farge the time step, and such a user is, therefore, surprised to encounter oscillatory solutions ‘The “stability” in a mathematical sense simply ensures that these oscillations will eventually dle ‘out, but it does not guarantee physically plausible solutions Some examples ‘of unrealistic solutions siven by the Crank-Nieolhon scheme cay) be found in Patankar and Baliga (1978)
In our fraiework, this behavior b easy, t0 explain, For f=035, the coetticient of 72 in Eq (4.36) becomes uB~(ag +ay)/2 Far uniform conductivity and uniform grid spacing, this coefficent ean be seen to bạ pc Ax/At—k/Sx, Again, whenever the time step is not sufficiently small, this cnefficient could hecame negative with its potential for physically unseatistic results The seemingly reasonable near peofile in Fig 4.5 is a good representation of the temperaturesime relationship for only small time intervals Over a larger interval, the intrinsically exponential decay of temp: tute is akin to a steep drop in the beginning, followed by a flat tail, The assumptions made in the fully implicit scheme are thus closer to reality than the linear profile used in the Crank-Nicolson scheme, especially for iarge time steps
Trang 38
sẽ MUMENICAL HEAT] RANSFER AND FLUID FLOW! behavior 14 is for this reason that we shall adape the fully implicit scheme iu this book
In must bo admitted that for small time steps the Sully Hapliie sclieme is feat as aeeuzate as the Crank-Nicolson schieme Again, the reason can be seem fom Fig 4.5; the temperature-time curve i nearly linear for small time Iniervals It is tempting to seek a scheme that combines the advantages of oth schemes and shares the disadvantages of neither Indeed, this Irs heen jone, and the resuit, called the exponential scheme, bas boon deseribed by sakar and Baliga (3978), That scheme, kowever, i somewhat complicated, Es fschiston am this book, ia which many” other hems are bo be aed, would have made the tveatmeat quite intricate
43:3 The Fully Implicit Discretization Equation
it form of Eq (4.36) tn doing s0, we shall termi, which wo fad temporarily droped The ine fully: imp! w linearized sours agTe + ayTw +b (440) te aie (ala) & iy eg Kw (send) pee (sate) baSe Ox tabTh (ata ap = an baw tap —Sp Ax (4416) fon reduces to our steady-state can bo seen that, as Ar—+ee, this enuat AGiscietization equation
The main principle of the fully implicit scheme is that the new value Tp ails over the entie time step Thus, if the conductivity kp depended on temporatue, it should be iteratively secslevated from Tp, exactly 25 in out ate procedure, Other aspects of the steady-state procedure, such as »auadary conditions, souree-term linearization, and the TOMA, ‘are do
plicable to the unsteady situation ‘Dur detailed consideration of the one-dimensional problem has aow set HHEAT CONDUEHOS % thế slage fy extension Lo two and three dia The extension is surprisingly easy
4-4 TWO- AND THREE-DIMENSIONAL SITUATIONS 44-1 Diseretization Equation for Two Dimensions
A portion of « twoslimensional grid is shown in Fig 4.6, For the ged point P, points # and W ar ils xeirection neighbors, while N and S (denoting north and south) are the ylieetion neighbors The control volume around P is shown by dashed lines Its thickness in the 2 direction is asused to be unity The aomeielatute introduced in Fig 3.2 for distanecs A, (6x)e, ete isto be extended 10 two dimensions here, The question of the actual Iveation of the control volume faces in relation 10 the grid points is still left open, Locating thor exactly midwar between the neighboring grid points is an obvious possibility, Dut other praclices can also be employed, some of which wil be diseuseed in Section 4.6-1 Here we shall derive discretization equations that willbe applicable to any such practice, We lave seen how to caleulate the heat Mux ge at the controbvolume face vetween P and £, We shall assume that ge, thus obtained, prevails over the entire face of area Ay X 1, Heat flow rates through the other faces can be ohtaiaed in a simaar fashion In this manner, the differential equation cất 2 (5P Pe 3E «em q
Trang 3960 NUMERICAL HEAT TRANSFER AND FLUID SLOW can be instantly tumed into the discretization equation O97 = apTe + ayTy + ayTy + o5Ts +, (43) where 444) (aaa) (444) (644) Am — A ¢ (4446 } Se Ax Ay +0878 (444) đp Ca † aw + ay +a + ah Sp Ax Ay (444g) ‘The product Ax Ap is the volume of the control volume
4.42 Discretization Equation for Three Dimensions
Fy, ee wo estos Tan 2 up a nn = direction to compete the tyeedimensionl confgvaton The điẾP ;uuon
equation can easly be seen to be
pT p = pT + Oy Ty + ayTy + a5Ts tayTr +apTg +h, (445) aye Tiên Ẵ (4.466) quế Đâu, ca HEAT CONDUCTION 6 (469) A6) (446) # (4460) b= Sc dx dy At + TP, (4.46) 1g bay tay tas tap tay tah — Sp ax dy Az, (4/460) sp
At this point, tis interesting to examine the physizalsignifeaRce of the various coefficients in the discretization equation The neighbor coefficients Oy dnp Gy, , 8g tepfesent the conductance between the point P and the corresponding neighbor, The term a&72 is the internat energy (divided by Af) contained in the control volume at time 1 The constant term ð consists of this internal energy and the rate of heat generation in the control volume resulting fiom Se The centerpoint coefficient ap is the @rof all neighbor coefficients (including @8, which is the coefficient of the “Nt neighbor” 7) and contains a contribution from the linearized source term
4.4-3 Solution of the Algebraic Equations
Ir should be noted that, while constructing he discretization equations, we cast them into a linear form but did not assume that a particular method ‘would be used for ther solution, Therefore, any suitable solution method can be employed at this stage, It is useful to consider the derivation of the equations und their solution as two distinet operations, and there is no need for the choices in one to influence the other Ina computer progrem, the two operations can be conveniently performed in separate sections, and either seetion can be independently rnodified when desiced
So far, we have obtained the multidimensional discretization equations by 4 straightforward extension of the one-dimensional situation, One procedure that cannot so easily be extended to multiple dimensions is the triểjagonal- matrix algorithm (TDMA) Direct methods Gie., those requiring no iteration) for solving the algebraic equations sriing in twe- of three-dimensional problems are much more complicated and require rather large amounts of computer storage and time For a linest problem, which requires the solution of the algebraic equations only once, a direct method may be acceptable; but
——~
Trang 40
¿ IV RICAL NHẤT TRANSEEE AND TEUID ELONE
+ đonlinear problems, since the equations have to be solved repeatedly with ae coefficients, the use of a dizeet method ie usually not economical We shail, therefore, exclude direct methods from further consideration, exeept to say that 2 computer program for the direct solution of discretization equations in two dimensions has been published by King (1976)
The alternative, then, is iterative methods for the solution Uf algebraic equations These start from a guessed ffeld of T (the dependent variable) and the algebraic equations In some manner to obtain an improved old Successive tepetitions of the algorithm fiually lead to a solution that is safficiently close to the correct solution of the algebraic equations lerative ‘methods usually require very soul alditional storage in the computer, and they are especislly attractive for handling nonlinearities In 4 nonlinear oblem, if is not aecessary of wise lo take the solution of the algebraic seljeys {0 Ainsi convergence for x fixed set of evefficieat values, With a sven set of tese values, a few iterations of the equation-solving algorithm ase nt before tke updating of the coefficients is performed, It scems that, r goreral there should be a certain balance between the effort required to late the ccefficients and that spent un solving the equations Once the Ticients ae ealculsted, we must perform sufficient iterations of the «tion solver fo extisct substantial benefit from the coeifvients, bul it wise 10 spend an excessive amount of effort on solving equations that ace ed en only tentative coefficients
There are many iterative methods for solving algebsaie equations We shall ssiotibe oaly two methods the first will sot the background, recummended for uss and the seeond Fhe Gans
idel point-by-point method The simplest of all iterative methous + the Gauss-Seidel methad in which the values of the variable are calculated ty visiting each grid point in a certain order Only one set of T°s is held in sompater storage ln the beginning, these represent the initial guess or values ‘mi the previous iteration, As each grid point is visited, the correspon ic of T in the computer storage is altered as follows: If the discretization esaation is written as app = Z aanTay +b, (440 viene the subscript nb denotes 2 neighbor point, then Tp al the visited grid point is caleblated fram
Tp = 2 ameTan th ap (4.48) whore Tih stands for the neighbor-point value pfesent ia the computer corage For neighbors that have already been visited during the current
HEAL CONDUCTION 63
iteration, Tay & the freshly valeulated value; for yekto-beidled neighbors, Thy ts the Value frou the previous iteration, fu any case, Ti is the latest salable value for the mgt temperature When all grid points have been visited im this imanner, ABP iteration of the Gauss-Seidel method is complete To illustrate the mrcthod, we shall consider two very simple examples, Fquations: ĐẠT; +02, (4490) Ty=T, +0 (4.499) Solution’ Wonton 0 1 5 0 02 D68 0833 0349 0986 «Lo 2 3 8 3 = 3 0 L2 bes LẾU 1949 1880 29 Faquations: 1 (4.502) (4,500) Solution: Itevationne, 0 r 0 T ụ