Computer simulation of flow and heat transeer

INTRODUCTION TO FINITE DIFFERENCE, NUMERICAL ERRORS AND ACCURACY 3.1 Introduction 21 3.2 Central, Forward and Backward Difference Expressions for a Uniform Grid 21 3.3 Central Difference

P SGhoshdastidar

Associate Professor

Department of Mechanical Engineering

lIT, Kanpur

The present book covers the fundamentals of what is commonly known asComputational Fluid Oynamics (CFO) The past two decades have witnessed aphenomenal growth in this area due to the developments in the field of computers.CFO has now become an integral part of the engineering design and analysis.Engineers can make use of the CFO tools to simulate fluid flow and heat transferphenomena in a design and predict the system performance before manufacturing.The advantages of CFO are numerous, namely, fewer iterations to the finaldesign, shorter time to launch the product, fewer expensive prototypes and so on.Furthermore, CPO provides a cost-efficient means of testing new designs andconcepts that would otherwise be too expensive and hazardous to investigate.Much of the material in this textbook has been used in a post -graduate course atthe Indian Institute of Technology , Kanpur for over a decade It is assumed that thereader has an adequate undergraduate background in Heat Transfer, Fluid Flow,Calculus and Computer Programming in FORTRAN The book is suitable as atext for a one-semester course at the post-graduate or advanced undergraduate

level It can also be used for self-study by practising engineers.

The book primarily follows finite difference method of discretization.However, in the Appendix A, other important schemes such as finite element andfinite volume are also discussed An emphasis has been laid on the physicalunderstanding of the problems Most of the methods have been illustrated withdetailed example problems and the solution procedure Several exercise problemsare given at the end of various chapters Readers are encouraged to solve theseproblems, to get a better understanding of various numerical techniques discussed

in the book Chapter 7 gives details of two new numerical methods and theirapplications Chapter 8 illustrates the application of CFO in solving industrialproblems An important subroutine (TOMA) in which tridiagonal matrixalgorithm is programmed is listed in Appendix C

The softcover version of this book also contains a floppy diskette The diskettecontains 21 files comprising 10 programs, I subroutine and 10 output files Theprograms are given in FORTRAN language and can be run on a PC-AT orPentium as well as on mainframe computer systems having a FORTRAN 77compiler Basically, the floppy contains programs and solutions to some unsolved

viii Preface

problems in the book and solutions '[programs] to some typical problems

discussed in this book

I would like to acknowledge the interaction with the students, both in and

outside the class, which has greatly contributed towards the shaping of this book

Their suggestions and comments have been useful in writing this text I have also

been benefitted by the lively discussions with some of my colleagues Special

thanks are due to the post-graduate students Suresh Singh, Vipin Kumar,

R Mahesh Kumar and Kali Sanjay who have assisted me in developing some of

the computer programs in the floppy diskette

I also wish to acknowledge the support and encouragement provided by the

editorial and production team of Tata McGraw-Hill, particularly, Ms Vibha

Mahajan and other members of their highly skilled editorial team I am also

grate-ful to the anonymous reviewer whose valuable comments and suggestions for

improvement have gone a long way in the formation of the final version of this

book

The typing was carried out with great care and patience by U S Mishra The

figures were drawn with great competence by B N Srivastava 'J.ne writing of this

book would not have been possible without the generous financial support ofthe

Curriculum Development Cell under the Quality Improvement Programme at the

Indian Institute of Technology, Kanpur

Last but not the least, the greatest contribution to this work has been the

patience and encouragement of my wife Sumita and my daughter Shreya who

often withstood my moody behaviour during the writing cf this book with smiling

faces Gratitude not expressible in words is due to my parents for their blessings

and good wishes

1.5 Basic Approach in Solving a Problem by Numerical Method 41.6 Problem Complexity 4

1.7 A Comparative Study of Experimental, Analyticaland Numerical Methods 5

1.8 Methods of Discretization 71.9 Justifications for the Choice of the Finite DifferenceMethod 8

Summary 8 References 9

2 PARTIAL DIFFERENTIAL EQUATIONS

Summary 17 References 17 Exercise Problems 17

10

x Contents

3. INTRODUCTION TO FINITE DIFFERENCE,

NUMERICAL ERRORS AND ACCURACY

3.1 Introduction 21

3.2 Central, Forward and Backward Difference

Expressions for a Uniform Grid 21

3.3 Central Difference Expressions for a Non-Uniform Grid 25

3.4 Numerical Errors 28

3.5 Accuracy of Solution: Optimum Step Size 29

3.6 Method of Choosing Optimum Step Size:

Grid Independence Test 29

4.1 Applications of Heat Conduction 32

4.2 Steady and Unsteady Conduction 32

4.3 Dimensionality in Conduction 33

4.4 Some Important Examples of Heat Generation in a Body 33

4.5 How Does the Classification of a Conduction

Problem Help? 33

4.6 Basic Approach in Numerical Heat Conduction 33

4.7 One-Dimensional Steady State Problem 33

4.8 Two-Dimensional Steady State Problem 54

4.9 Three-Dimensional Problems 62

4.10 Transient One-Dimensional Problem 63

4.11 Accuracy of Euler, Crank-Nicholson and Pure

Implicit Method 70

4.12 Stability: Numerically Induced Oscillations 71

4.13 Convection Boundary Condition 76

4.14 Stability Limit of the Euler Method from Physical

Standpoint 77

4.15 Mathematical Representation of All Three Methods by

a Single Discretization Equation 78

4.16 Physical Representation of All Three Methods 79

4.17 Advantages and Disadvantages of Each of the

Three Methods 79

4.18 How to Choose a Particular Method 80

4.19 Consistency 80

4.20 Two-Dimensional Transient Problems 81

4.21 Example Problem: Two-Dimensional Transient

Heat Conduction in a Square Plate 83

of Condition at the Centre 90

4.25 Problems in Spherical Geometry 954.26 One-Dimensional Transient Conduction inComposite Media 96

4.27 Treatment of Nonlinearities in Heat Conduction 984.28 Irregular Geometry 106

Summary 110 References 111 Exercise Problems 112

5 NUMERICAL METHODS FOR INCOMPRESSmLE

5.1 Introduction 1195.2 Governing Equations 120

5.3 Difficulties in Solving Navier-Stokes Equations 120

5.4 Stream Function-Vorticity Method 1215.5 General Algorithm for Solution byIf! - ~ method 1235.6 Creeping Flow (Very Small Reynolds Number) 1255.7 InviscidFlow(Steady) 127

5.8 Determination of Pressure for Viscous Flow 1285.9 Is the Transient Approach Used for Solving SteadyFlow Problems? 131

5.10 If! - ~ Method for 3-D Problems 1315.11 The Primitive- Variables Approach 1315.12 Simple (Semi-Implicit Method for Pressure-LinkedEquations) Procedure ofPatankar and Spalding (1972) 1325.13 Computation of Boundary Layer Flow 138

5.14 Similarity Solutions of Boundary Layer Equations:

Shooting Technique 140

5.15 Finite Difference Approach 1445.16 Von Mises Transformation 147

Summary 148 References 149 Exercise Problems 149

6. NUMERICAL METHODS FOR CONVECTION

6.1 Introduction 1536.2 Convection-Diffusion (Steady, One-Dimensional) 1546.3 Convection-Diffusion (Unsteady, One-Dimensional) 158

xii Contents

Contents xiii

8 SPECIALTOPICS n 190

8.1 Introduction 190

8.2 Transient Combined Mixed Convection and

Radiation from a Vertical Aluminium Fin

(Ghoshdastidar and Raju, 1992) 190

8.3 Heat Transfer in Rotary Kiln Reactors 205

8.4 Modelling of Thermal Transport Processes in

Single-Screw Plasticating Extruder with Applications to

Polymer and Food Processing 220

8.5 Heat Transfer in Metal and Alloy Solidification 237

8.6 Cooling of Electronic Equipments 246

Summary 252

References 252

Appendix A Other Discretization Methods 255

A.l The Essence of the Finite Element Methods 255

A.2 Finite Element Method Based on Variational Calculus 256

A.3 Convection Boundary Condition 265.

A.4 Two-Dimensional Steady State Problem 266

A.5 Three-Dimensional Problems 267

A.8 Extension to 2-D and 3-D Problems 278

A.9 Accuracy of Solution 279

Appendix B Runge-Kutta Methods

B.l The Essence of Runge-Kutta Methods 288

B.2 Simultaneous Ordinary Differential Equations 289

B.3 Solution of Higher Order ODE by R-K Methods 290 References 290

Appendix C Listing of Subroutine IDMA

C.l Subroutine TDMA 291

C.2 Demonstration Program Showing Application

of Subroutine TDMA 292

Reference 292

Appendix D Numerical Method for Radiation in

Enclosure with Diffuse-Gray Surfaces:

The Absorption Factor Method 293

Reference 295

179

6.4 Convection-Diffusion (Unsteady, Two-Dimensional) 159

6.~ Computation of Thermal Boundary Layer Flows (Part A) 162

6.6 Computation of Thermal Boundary Layer Flows (Part B) 163

6.7 Transient Free Convection from a Heated Vertical Plate 170

7.2 New Method (I): Application of ADI Method to Solve

the Problem of 2-D Transient Heat Transfer from a

Straight Composite Fin 179

7.3 New Method (II): Alternative to Upwind

Scheme-Application of Operator-Splitting Algorithm to Solve

Convection- Diffusion Problems 183

Summary 189

References 189

7 SPECIALTOPICS I NEWMEmODS

INTRODUCTION

1.1 WHAT IS COMPUTER SIMULATION?

The simulation of an industrial system on computer involves mathematicalrepresentation of the physical processes undergone by the various components ofthe system, by a set of equations (usually differential equations) transformed todifference equations which are in turn solved as a set of simultaneous algebraicequations

At this stage, a reader uninitiated into the numerical methods may ask thequestion "What is the role of computer here?" The aforesaid query is a valid one.Many seem to forget that some of the numerical schemes (e.g Finite-Difference)that are extensively used today for solution of problems on computer were devel-oped when computer was not even invented Now, to return to the original ques-tion, the answer is that with the aid of the algorithm of the solution method trans-lated into a programming language like FORTRAN fed into a computer whichdoes the arithmetic operations at a tremendous speed one can obtain the solution

of mathematical equations in seconds or even in fraction of a second A simpleexample will clarify this point One can very easily solve a set of three linearsimultaneous algebraic equations by hand through Gaussian Eliminationmethod.Typically in this method, for a system ofn equations the total number of

multiplications and divisions is roughly.! n 3• Therefore, for n=3, the number of

3operations is 9, which is clearly manageable by hand calculations However, for

n=10, this number jumps to 333 For n=100, the number skyrockets to 333000

A mainframe computer (e.g VAX 8810) having an average megaflop * rating ofaround 1 (that is, 106arithmetic operations per sec) will solve the aforesaid prob-lem in 0.333 second A personal computer (e.g., IBM PC) with a megaflop rating

* A mega is a million and flops is an abbreviation for floating point operations per

second A floating point operation is an arithmetic operation (addition, subtraction, multiplication and division) on operands which are real numbers with fractional parts NormaUy multiplications and divisions are counted asinajor arithmetic op- erations as compared to addition and subtraction on a computer.

2 Computer Simulation of Flow and Heat Transfer

of 0.01 will take about 33 seconds In computer simulation, it is possible to

han-dle one hundred or more (even greater than thousand) number of equations and

there lies the necessity of using the computer

1.2 ADVANTAGES OF COMPUTER SIMUlATION

Now that the use of computer in simulation is established, let us enumerate some

of the important advantages of computer simulation (also known as numerical

simulation):

• It is possible to see simultaneously the effect of various parameters and

variables on the behaviour of the system since the speed of computing is

very high To study the same in an experimental setup is not only difficult

and time-consuming but in many cases, may be impossible

• It is much cheaper than setting up big experiments or building prototypes

of physical systems

• Numerical modelling is versatile A large variety of problems with

different levels of complexity can be simulated on a computer

• "Numerical experimentation" (another synonym for computer simulation)

allows models and hence physical understanding of the problem to be

improved It is similar to conducting experiments

• In some cases, it is the only feasible substitute for ~xperiments, for

example, modelling loss of coolant accident (LOCA) in nuclear reactors,

numerical simulation of spread of fire in a building and modelling of

incineration of hazardous waste

However, it is to be emphasized that not every problem can be solved by

computer simulation Experiments are still required to get an insight into the

phenomena that are not well understood (and hence cannot be translated into the

language of mathematics) and also to check the validity of the results of computer

simulation of complex problems

1.3 APPUCATIONS OF FLUID FLOW AND HEAT

TRANSFER

Fluid flow and heat transfer playa very important role in nature, living organisms

and in a variety of practical situations More often than not, flow and heat transfer

are coupled and rarely an engineer solves a problem of either pure fluid flow or

pure heat transfer In many applications flow and heat transfer are accompanied

by chemical reaction and/or mass transfer The various applications of fluid flow

and heat transfer are:

• All methods of power production, e.g thermal, nuclear, hydraulic, wind,

and solar power plants

• Heating and air-conditioning of buildings

• Chemical and metallurgical industries, e.g furnaces, heat exchangers,

condensers and reactors

Introduction 3

• Design of IC engines

• Optimization of heat transfer from cooling fins

• Aircraft and spacecraft

• Design of electrical machinery and electronic circuits

• Cooling of computers

• Weather prediction and environmental pollution

• Materials processing such as solidification and melting, metal cutting,welding, rolling, extrusion, plastics and food processing in screwextruders, laser cutting of materials

• Oil exploration

• Production of chemicals such as cement and aluminium oxide

• Drying

• Processing of solid and liquid wastes

• Bio-heat transfer (as in human and animal bodies)

It is no wonder that J.B Joseph Fourier, father of the theory of heat diffusionmade this remark in 1824 "Heat, like gravity, penetrates every substance of theuniverse; its rays occupy all parts of space The theory of heat will hereafter formone of the most important branches of general physics" Lord Kelvin, in 1864obtained a rough estimate of the age of earth based on an idea proposed by Fourier

in 1820 to be 94 x 106years (0.094 billion years) by applying the principle oftransient heat conduction in a semi-infinite solid Modem dating methods haverevealed the age of earth to be approximately 4.7 billion years So Kelvin's resultwas not really too far off the mark considering the fact that the data for the meas-ured value of the geothermal gradient (rate of increase of temperature of earthwith depth), average thermal diffusivity of rock and the initial temperature ofmolten earth when cooling began available with him at that time were not very,ccurate The aforesaid example is probably the first known application of heattransfer simulation

1.4 WHY IS COMPUTER SIMUlATION NECESSARY

IN FLUID FLOW AND HEAT TRANSFER?

Ifope looks at a classical textbook on fluid dynamics and heat transfer, one wouldfind only a handful of analytical (or exact) solutions In actual situations, prob-lems are lot more complex as in those involving non-linear governing equationsand/or boundary conditions, and irregular geometry which do not allow analyti-cal solutions to be obtained Therefore, it is necessary to use numerical tech-niques for most problems of practical interest Furthermore, to design andoptimize thermal processes and systems, numerical simulation of the relevanttransport phenomena is a must, since experimentation is usually too involved andexpensive However, necessary experimentation must still be done in checkingthe accuracy and validity of numerical results Sometimes, numerical model can

be refined by input from results of a companion experimental set-up for the sameproblem

Suppose we wish to obtain the temperature field in the domain as shown in

Fig 1.1 We imagine that the domain is filled by a grid, and seek the values of

temperatures at the grid points Therefore, the energy equation (which is the

governing differential equation for the problem) is valid at all the grid points The

governing differential equation is then transformed into a system of difference

equations resulting in a set of simultaneous algebraic equations which means that

if there are 100 grid points (where variables are not known) there will be 100

equations to solve per variable The simplification inherent in the use of algebraic

equations rather than differential equations is what makes numerical methods so

powerful and widely applicable

4 Computer Simulation of Flow and Heat Transfer

1.5 BASIC APPROACH IN SOLVING A PROBLEM

BY NUMERICAL METIIOD

x Grid points (known temperatures)

o Grid points (unknown temperatures)

steady-state heat conduction in a square plate

1.6 PROBLEM COMPLEXI1Y

In general, the problem complexity is described by the formula

PC= GGx VGxSSxFP

where, PC =problem complexity

GG = geometry of the grid system

VG = variables per grid point

SS=number of steps per simulation for solving problem

FP =number of floating point operations per variable

Introduction 5For the problem shown in Fig 1.1, since there are 16 tiny squares, GG=16 If

a direct method like Gaussian Elimination method is used, then SS= 1,VG= 1,

FP = 1.(9)3=243

3

PC = 16 x 1 x 1 x 243 =3.89 x 103 operationsThus, even with a fairly coarse grid, the number of operations is quite large.Currently a large number of realistic applications like modelling of supersonicaircraft or weather prediction requires 1012 to 1014 operations per solution(Rajaraman, 1993) If each solution has to be done in about an hour, then theaverage speed of computing should be 1014/60 x 60 operations per second which

is equal to 27,700 Megaflops! The peak megaflop rating of modem puters is around 1000 megaflops (Rajaraman, 1993) From the aforesaid exam-ple, it is clear why we need supercomputers with speeds in thousands of megaflopsrange to solve extremely complex problems

supercom-1.7 A COMPARATIVE STUDY OF EXPERIMENTAL, ANALYTICAL AND NUMERICAL METIIODS

(a) Experimental Method Experimental methods are used to obtain reliableinformations about physical processes that are not well understood, e.g combus-tion and turbulence It may involve full scale, small scale or blown-up scalemodels The major disadvantages of experimental investigations are high cost,measurement difficulties and probe errors Often, small scale models do not al-ways simulate all the features of the full scale set-up The advantage is that it ismost realistic

(b) Analytical Method Analytical methods or methods of classical ics are used to obtain the solution of a mathematical model consisting of a set ofdifferential equations which represent a physical process within the limit of as-sumptions made Only a handful of analytical solutions are available in heat trans-fer and fluid mechanics because analytical methods are inadequate in handlingcomplex boundary and non-linearities in the differential equations and/or bound-ary conditions Furthermore, the analytical solutions often contain infinite series,-special functions, transcendental equations for eigenvalues, etc the numericalevaluation of which becomes quite cumbersome

mathemat-(c) Numerical Method As explained in Sec 1.5, a numerical prediction worksout the consequences of a mathematical_ model but the solution is obtained forvariables at discrete grid points in the computational domain in contrast withanalytical method which gives closed form solution at all points (theoreticallyinfinite number of points in the solution domain or continuum) The major advan-tages of numerical solution are its abilities to handle complex geometry and non-linearities in the governing equation and/or boundary conditions Other advan-tages of numerical method are briefly described below:

6 Computer Simulation of Flow and Heat Transfer

• Low Cost While prices of most items are increasing, cost of computing

(mainframe, mini and PC's) is going down every year

• High Speed In the past 20 years there has been a thousand fold increase

in the speed of arithmetic operations of computers

• Complete Information A Computer simulation gives detailed and

complete information of all the variables over the computational domain

• Ability to simulate realistic conditions For a computer program, there

is absolutely no problem in having an area of size 10 km x 10 km or

10-6 m x 10-6 m, 5000 °C or -50°C, hazardous or flammable material

• Ability to simulate ideal conditions There is no problem in idealizations

like two-dimensionality, insulated or isothermal boundary, infinite

reaction rate On the other hand, it is extremely difficult, if not impossible

to set up the same in experiments

The disdvantages of numerical predictions are the associated truncation error

and round-off error and the difficulty in simulating complicated boundary

condi-tions

The aforesaid discussion can now be represented in a capsuled form in

Table 1.1

Table 1.1 Comparison of Experimental, Analytical and

Numerical Methods of Solution

problem

• Measurementdifficulties

• Probe errors

• High operating costs

2 Analytical • Clean, general • Restricted

formula form • Usually

restricted

to linear problems

• Cumbersomeresults-difficult

to compute

to linearity and round-off errors

• Ability to handle • Boundaryirregulargeometry conditionand complicated problemsphysics

• Low cost andhigh speed ofcomputation

finite-differ-a mfinite-differ-atter of ffinite-differ-act, the subject of Computfinite-differ-ationfinite-differ-al Fluid Dynfinite-differ-amics (commonly finite-differ-viated as CFD) was born as early as 1933 (remember that world's first computercalled ENIAC was built at the University of Pennsylvania, USA, in 1946) withthe remarkable work (published in the proceedings of the Royal Society) ofThom(1933) who solved the Navier-Stokes equations for the steady, incompressibleviscous flow around a circular cylinder by finite-difference method using handcalculations However, several shortcomings and limitations of finite-differencemethod came to light when researchers tried to solve problems with increasingdegree of physical complexity such as, for example, flows at higher Reynolds,numbers, flows around arbitrarily shaped bodies, strongly time-dependent flows,etc (Fasel, 1978) This led to a search for and development of superior methods,particularly in the areas where difference methods seemed to have disadvantages.These methods can be divided into two main categories (i) finite-element methodsand (ii) spectral methods

abbre-(b) Finite-Element Method (FEM) Finite-element methods basically seek tions at discrete spatial regions (called elements) by assuming that the governingdifferential equations apply to the continuum within each element It is based onintegral minimization principle and provides piecewise (or regional) approxima-tions to the governing equations

solu-Finite-element methods were already found to be successful in solid ics applications Their introduction and ready acceptance in fluid mechanics wasdue to relative ease by which flow problems with complicated boundary shapescan be modelled, especially when compared with finite-difference methods How-ever, disadvantages of FEM arises from the fact that more complicated matrixoperations are required to solve the resulting system of equations Furthermore,meaningful variational formulations are difficult to obtain for high Reynoldsnumber flows Hence, variational principle-based FEM is limited to solutions ofcreeping flow and heat conduction problems

mechan-Galerkin's weighted residual method, which is also another finite-elementmethod is a powerful method and circumvents the difficulties faced by variational

8 Computer Simulation of Flow and Heat Transfer

calculus based FEM Much current research is in progress in the use of this

method

(c) Spectral Method Spectral methods are generally much more accurate than

simple first or second order finite-difference schemes In spectral methods, in

contrast with the discretization as in finite-difference methods, the approximation

is based on expansions of independent variables into finite (truncated) series of

smooth (mostly orthogonal) functions In addition to its advantage as regards

higher accuracy, it can be easily combined with standard finite-difference

meth-ods A disadvantage of spectral methods is their relative complexity in

compari-son with standard finite-difference methods Also the implementation of complex

boundary conditions appears to be a frequent source of considerable difficulty

(d) Control Volume Formulation In this method, the calculation domain is

divided into a number of non-overlapping control volumes such that there is one

control volume surrounding each grid point The differential equation is integrated

over each control volume Piecewise profiles expressing the variation of the

un-known between the grid points are used to evaluate the required integrals The

result is the discretization equation containing the values of the unknown for a

group of grid points (Patankar, 1980)

The major advantage of this method is its physical soundness The

disadvan-tage is that it is not as straightforward as finite-difference method

1.9 JUSTIFICATIONS FOR TIlE CHOICE OF TIlE

FINITE DIFFERENCE METIIOD

In this book, finite-difference method is chosen as the method of discretization

because (i) for a newcomer in the field of numerical simulation of flow and heat

transfer, this is the best method to begin with simply because of its inherent

straightforwardness and simplicity, and (ii) in recent years tremendous

refine-ments and advances have been made by numerous researchers in the finite

dif~er-ence method, particularly in the areas where it was known to be weak, such as

irregular boundaries or superior accuracy

However, interested readers may look at Appendix' A' containing a brief

dis-cussion on finite element methods and control volume method Spectral method

has been left out in the book for the sake of brevity The readers are however,

encouraged to go through a survey paper by Orszag and Israeli (1974) which

gives a detailed treatment of the spectral method

SUMMARY

This chapter begins with a formal definition of the term Computer Simulation

and its advantages followed by a discussion on the applications and the necessity

of computer simulation of flow and heat ~ansfer Basic approach in the solution

Introduction 9

of a problem by numerical method is then given with the aid of a figure along withthe concept of problem complexity A comparison of advantages and disadvan-tages of numerical methods vis-a-vis analytical and experimental methods is thenprovided in detail Finally, the chapter concludes with a brief description of thevarious methods of discretization and the justifications for the choice of the finitedifference method as the discretization scheme used in this book

REFERENCES

1 Carslaw, H S and J C Jaeger, Conduction of Heat in Solids, 2nd Edition,

Clarendon Press, Oxford, 1959.

2 Fasel, H "Recent Developments in the Numerical Solutions of the Navier-Stokes Equations and Hydrodynamic Stability Problems", Computational Fluid Dynamics, Edited by Wolfgang Kollmann, Hemisphere, Washington, D.C., 1978.

3 Field, George B, Gerrit L Verschuur, and Cyril Ponnamperuma, Cosmic Evolution: An Introduction to Astronomy, Houghton Mifflin Company, Boston,

7 Orszag, S A and M Israeli, "Numerical Simulation of Viscous Incompressible

Flows", Ann Rev Fluid Mech., Vol 6, 1974, p 281.

8 Patankar, S V, Numerical Heat Transfer and Fluid Flow, Hemisphere, Washington, D.C., 1980.

9 Rajaraman, V, Supercomputers, Wiley Eastern, New Delhi, 1993.

10 Ralston, Anthony and Philip Rabinowitz, A First Course in Numerical Analysis.

2nd Edition McGraw-Hill, New York, 1978.

11 Thorn, A "The Flow Past Circular Cylinders at Low Speeds", Proc Roy Soc., A

141, 1933, p 651.

16 Computer Simulation of Flow and Heat Transfer

now-a-days solution of hyperbolic heat conduction problems are receiving tion from researchers because of its application in advanced materials processingtechniques, such as, laser cutting of materials and laser surgery Due to short time

atten-of the pulse in all these processes, non-Fourier conduction effects are introduced

by the finite speed of propagation of heat wave in laser irradiated materials andhence hyperbolic heat conduction equation needs to be solved to accurately repre-sent the physical process Other applications of hyperbolic heat conduction aresituations involving very low temperatures near absolute zero and very high tem-perature gradient

Hyperbolic equations of the first order arise in many gas dynamics problemswhere viscosity is unimportant and the motion is unsteady If the motion is steadythe classification of the systems depends upon the magnitude of the fluid speed,being hyperbolic if it is supersonic and elliptic if subsonic

2.7 HOW MANY INITIAL AND BOUNDARY

COMPLETELY DEFINING A PROBLEM?

The number of initial and boundary conditions in the direction of each ent variable of a problem is equal to the order of the highest derivative of thegoverning differential equation in the same direction

independ-Consider, for example, the conduction equation written in Cartesiancoordinates for a homogeneous isotropic solid moving with velocity

The readers are encouraged to derive Eq (3.10) and Eq (3.11) themselves.

3.2.4 When Do You Use Forward, Backward

and Central Difference Expressions?

• Forward difference expressions are used when data to the left of a point at

which a derivative is desired are not available

• Backward difference expressions are used when data to the right of the

desired point are not available

• Central difference expressions are used when data on both sides of the

desired point are available and are more accurate than either forward or

backward difference expressions

Now, the question is: When does one use a higher order difference scheme?There is no set answer to this It depends on the accuracy requirement of a prob-lem and the analyst will have to use his own judgement

3.3 CENTRAL DIFFERENCE EXPRESSIONS FOR A NON-UNIFORM GRID

Non-uniform grid is used in the region where gradients of the unknown (forexample, near the wall in boundary layer flow and/or heat transfer or heatconduction in a semi-infinite solid) are expected to be high Therefore, by making

a fine grid in the region of large gradients and coarse grid in the region whethergradients are small, one can save on computational memory and execution time.Figure 3.2 shows a typical non-uniform grid, the grid spacings being non-uniform

in the x-direction only

3.4 NUMERICAL ERRORS

-Three most important errors that commonly occur in numerical solution are (a)

round-off error (b) truncation error and (c) discretization error

(a) Round-off Error The round-off error is introduced because of computer's

inability to handle large number of significant digits Typically, in

single-preci-sion, the number of significant figures retained ranges from 7 to 16, although it

may vary from one computer system to another The round-off error arises due to

the fact that a finite number of significant digits or decimal places are retained

and all real numbers are rounded off by the computer The last retained digit is

rounded off if the first discarded digit is equal to or greater than 5 Otherwise, it is

unchanged For an example, if five significant digits are to be kept in place,

5.37527 is rounded oftto 5.3753, and 5.37524 to 5.3752

Introduction to Finite Difference, Numerical Errors and Accuracy 29

(b) Truncation Error The truncation error is due to the replacement of an exactmathematical expression by a numerical approximation This error has alreadybeen discussed earlier in this chapter with respect to finite differenceapproximations Basically, it is the difference between an exact expression andthe corresponding truncated form (for example, truncated Taylor series) used inthe numerical solution

(c) Discretization Error The discretization error is the error in the overall tion that results from the truncation error assuming the round-off error to be neg-ligible Therefore,

solu-Discretization error =exact solution - numerical solution with

However, it is obvious that round-off error increases as total number of metic operations increases Again, total number of arithmetic operations increases

arith-if the step size decreases (that is, when the number of grid points increases).Therefore, round-off error is inversely proportional to the step-size

On the other hand, truncation error decreases as step size decreases (or asnumb,er of grid points increases)

Because of theaf-orementioned opposing effects, an optimum step size is pectedwhich will produce minimum total error in the overall solution

A numerical analyst has to be extremely careful as regards the accuracy of hissolution To get the most accurate solution (that is, the solution with least totalerror), one has to perform a grid independence test The test is carried out byexperimenting with various grid ~izes and watching how the solution changeswith respect to the changes in grid sizes Finally, a stage will come when chang-ing the grid spacings will not affect the solution In other words, the solution hasnow become independent of grid spacing The largest value of grid spacings forwhich the solution is essentially independent of step sizes is chosen so that boththe computational time and effort and the round-off error are minimized A.nex-ample of a grid independence test is given in Chapter 4

30 Computer Simulation of Flow and Heat Transfer

This chapter introduces readers to finite difference approximation of derivatives

of continuous and differentiable functions based on Taylor series expansion Thecentral, forward and backward difference expressions for first and second deriva-tive of a function have been derived Difference expressions of higher order accu-racy are also given Next, application of non-uniform grid and a detailed deriva-tion of schemes for first and second derivatives of function evaluated at a point in

a non-uniform grid follow The concepts of round-off error, truncation error,discretization error and total error have been discussed in detail with reference tosolution accuracy Finally, method of choosing optimum grid size (grid independ-ence test) is briefly mentioned

REFERENCES

1 Carnahan, Brice, H A Luther, and James O Wilkes, Applied Numerical Methods,

John Wiley & Sons, New York 1969.

2 Constantinides, Alkis, Applied Numerical Methods with Personal Computers,

McGraw-Hill Inc., New York, 1987.

3 Jaluria, Yogesh, Computer Methods for Engineering, Allyn and Bacon, Inc., Boston, 1988.

4 James, M L, G M Smith, and J C Wolford, Applied Numerical Methodsfor Digital Computation with FORTRAN, Scraton, International Text Book Co, 1967.

5 Rohsenow, Warren, M and James P Hartnett, Handbook of Heat Transfer,

McGraw-Hill Book Company, New York, 1973.

NUMERICAL METHODS

FOR CONDUCTION

HEAT TRANSFER

Heat conduction has numerous applications in modern technology, in geological

sciences, and in many other evolving areas such as materials processing Some

examples are cooling fins or extended surfaces, solidification and melt1O~ of

metals and alloys in metallurgical industries, welding, metal cuttl~g: quenching,

electrical chemical and nuclear heating, periodic temperature vanatlons of earth

surface, baking oven, furnace walls, heating and cooling of buildings, to name

just a few To analyse thermal stress conditions in a mat~rial, te~perature

distribution must be obtained by solving the heat conduction equation !he

starting point of all such analyses is the differential equation (energy eq.uatlOn)

based on the physical formulations of the phenomena relevant to conduction

Since the numerical treatment of a conduction problem depends on the nature of

the conduction process, all conduction processes are divided broadly into two

categories, namely, steady and unsteady Steady state means that temperature,

density, etc at all points of the conduction region is independent of time

Un-steady state means a change with time, usually only of the te~pera~un~ Unsteady

state problems can be further split into the categories, that IS, penodic an~

tran-sient Daily variation of earth's temperature due to solar eff~cts exam~hfies a

typical periodic heat conduction problem ~nother ex~mple IS the engme wall

temperature variations due to cyclic changes 10combustlon gas temperature !he

immersion of hot steel plate in a cold quenching tank is an example of transIent

conduction Transient periodic heat conduction is also not uncommon

Numerical Methods for Conduction Heat Transfer 33

\~: A CONDUCTION PROBLEM HELP?

I The classification of a problem as mentioned earlier essentially helps the heattransfer analyst to decide on the approach to be used to solve the problem It also tells us the level of difficulty to be encountered in obtaining the solution, that is,.the temperature distribution within the body

HEAT CONDUCTION

.i Many difficult problems arise in conduction, for example, variable thermal

con-I~,

t ductivity, distributed energy sources, radiation boundary conditions for which

! analytical solutions are not available Approximate solution is then obtained bynumerical method The basic approach is to arrive at the relevant governingdifferential equation based on the physics of the particular problem They arethen converted to the required finite difference forms To begin with, the numeri-cal solution procedure for the problem of a simple one-dimensional steady stateheat conduction in a cooling fin is described It is to be noted that simple, closed-form straight forward analytical solution for this problem is available The idea is

to show the use of numerical method and to compare the numerical solution withits analytical counterpart

Consider the one-dimensional steady state heat conduction in an isolated gular horizontal fin as shown in Fig 4.1 The base temperature is maintained at

rectan-T=To and the tip of the fin is insulated The fin is exposed to a convective ronment (neglecting radiation heat transfer from the fin) which is at Too(Too <To).

envi-·

The above algorithm is also known as Thomas Algorithm See Appendix C for

listing of subroutine TDMA

Finally, it is to be noted that Eq (4.7) might also be solved by the Seidel iteration scheme discussed next

Gauss-Gauss-Seidel Iterative Metbod(G-S) For a large number of equations (typically

of the order of several hundred) iterative methods, which initiate the tions with a guessed solution and iterate to the desired solution of the systems ofequations within a specified convergence criterion, using improved guesses in thesecond, third iterations till the final one, are often more efficient In this method,unlike In direct methods like Gaussian elimination the round-off error does not

computa- mulate The round-off error after each iteration simply produces a less

44 Computer Simulation of Flow and Heat Transfer

This process is known as Jacobi iterative method.

Disadvantages of Jacobi Method The main disadvantage is that the computer

storage is needed for the present iteration as well as the previous one This is

because all the values are computed, using previous values before any unknown

is updated

Gauss-Seidel Method: An Improvement over Jacobi Method A significant

improvement in the rate of convergence and in the storage requirements can be

obtained by replacing the values from the previous iteration by new ones as soon

as they are computed Then, only the values of the latest iteration are stored, and

each iterative computation of the unknown employs the most recent values of the

other unknowns This computational scheme is known as point-by-point

Gauss-Seidel method

Numerical Methods for Conduction Heat Transfer 45that is, when the system is diagonally dominant This is also known as

Scarborough Criterion However, convergence may still be possible even if the

above condition is not satisfied Fortunately, it turns out that in fluid flow andheat transfer problems, finite-difference formulation indeed leads to diagonallydominant coefficient matrix which is the reason why for large systems Gauss-Seidel method is so widely used

Application of G-S Iterative Method In order to demonstrate the iterationprocess, the following system of three linear equations is solved by G-S iterative

method using a pocket calculator.

lOx, +x2 +2x3 =44

x, + 2x2 + IOx3 =61

4.8.6 Line-by-Line Method

To alleviate the problem of slow convergence of Gauss-Seidel (G-S) method for

large number of grid points, a line-by-line method which is a convenient

combi-nation of the direct method (TDMA) for one-dimension and the G-S method can

be used Basically, the method makes use ofthe direct nature ofTDMA and low

round-off error of G-S scheme

In the line-by-line method, a grid line (say, in the y-direction) is chosen

assum-ing that the unknowns (say, temperature, 1)along the neighbouring lines (Le., the

x-direction neighbours of the points on the chosen line) are known from their

latest values Now, 1"s along the chosen line (Fig 4.15) is solved by TDMA

This procedure is repeated for alHheJipes inrtb~:Ynd.i.re!Cti.oJl,j,J;lthe first sweep In

Numerical Methods for Conduction Heat Transfer 61

the second sweep, the computation again begins for the first chosen line andsweep continues till the other boundary is reached The same procedure is fol-lowed for third, fourth, till final sweep when there is virtually no change in thetemperature distribution in consecutive sweeps An alternating direction method

is also possible In this method sweeping of the computational domian is donealternately inx (ory) andy(or x) direction

The sweep direction is also important in cases where in one direction one ofthe boundaries has insulation or convection condition while at the other bound-ary, Dirichlet condition is specified For example, in the present problem(Fig 4.15), a right-to-left sweep would transmit the known temperature on theright boundary into the domain; on the other hand, since no te~perature is given

on the left boundary, a left-to-right sweep would bring no such useful tion Based on the same argument, in the y-direction, top-to-bottom sweep is de-sirable

informa-The fast convergence of the line-by-line method is due to the fact that theboundary condition information from the ends of the line is transmitted at once tothe interior of the domain in sharp contrast with point-by-point Gauss-Seidelmethod in which the boundary condition information is transmitted at a rate ofone grid interval per iteration

Fig 4.15 Pictorial representation of the line-by-line method

4.8.7 Check for Accuracy

For the present problem, the accuracy of the numerical results can be checked bycomparing it with the corresponding analytical solution which is available Sub-sequently, a grid independence test must be done to obtain the desired results