Computer simulation of flow and heat transeer

P S Ghoshdastidar Associate Professor Department of Mechanical Engineering lIT, Kanpur The present book covers the fundamentals of what is commonly known as Computational Fluid Oynamics (CFO) The past two decades have witnessed a phenomenal 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 transfer phenomena in a design and predict the system performance before manufacturing The advantages of CFO are numerous, namely, fewer iterations to the final design, shorter time to launch the product, fewer expensive prototypes and so on Furthermore, CPO provides a cost-efficient means of testing new designs and concepts 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 at the Indian Institute of Technology , Kanpur for over a decade It is assumed that the reader has an adequate undergraduate background in Heat Transfer, Fluid Flow, Calculus and Computer Programming in FORTRAN The book is suitable as a text 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 and finite volume are also discussed An emphasis has been laid on the physical understanding of the problems Most of the methods have been illustrated with detailed example problems and the solution procedure Several exercise problems are given at the end of various chapters Readers are encouraged to solve these problems, to get a better understanding of various numerical techniques discussed in the book Chapter gives details of two new numerical methods and their applications Chapter illustrates the application of CFO in solving industrial problems An important subroutine (TOMA) in which tridiagonal matrix algorithm is programmed is listed in Appendix C The softcover version of this book also contains a floppy diskette The diskette contains 21 files comprising 10 programs, I subroutine and 10 output files The programs are given in FORTRAN language and can be run on a PC-AT or Pentium as well as on mainframe computer systems having a FORTRAN 77 compiler 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 grateful 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 P S GHOSHDASTIDAR CONTENTS Preface vii 1.mTRoDucnoN 1.1 1.2 1.3 1.4 What is Computer Simulation? Advantages of Computer Simulation Applications of Fluid Flow and Heat Transfer Why is Computer Simulation Necessary in Fluid Flow and Heat Transfer? 1.5 Basic Approach in Solving a Problem by Numerical Method 1.6 Problem Complexity 1.7 A Comparative Study of Experimental, Analytical and Numerical Methods 1.8 Methods of Discretization 1.9 Justifications for the Choice of the Finite Difference Method Summary References PARTIAL DIFFERENTIAL EQUATIONS 2.1 2.2 2.3 2.4 2.5 2.6 2.7 Classification of PDEs 10 Elliptic, Parabolic and Hyperbolic Equations 11 Initial and Boundary Conditions 13 Initial and Boundary Value Problems 14 Conventions Followed in This Text 15 What About Hyperbolic Problems? 15 How Many Initial and Boundary Conditions You Need for Completely Defining a Problem? 16 Summary 17 References 17 Exercise Problems 17 10 x Contents Contents xi INTRODUCTION TO FINITE DIFFERENCE, 3.1 3.2 3.3 3.4 3.5 3.6 Introduction 21 Central, Forward and Backward Difference Expressions for a Uniform Grid 21 Central Difference Expressions for a Non-Uniform Grid Numerical Errors 28 Accuracy of Solution: Optimum Step Size 29 Method of Choosing Optimum Step Size: Grid Independence Test 29 Summary 30 References 30 Exercise Problems 4.22 False Transient Approach 89 4.23 Three-Dimensional Transient Problems 89 4.24 Problems in Cylindrical Geometry: Handling of Condition at the Centre 90 4.25 Problems in Spherical Geometry 95 4.26 One-Dimensional Transient Conduction in Composite Media 96 4.27 Treatment of Nonlinearities in Heat Conduction 4.28 Irregular Geometry 106 21 NUMERICAL ERRORS AND ACCURACY 25 Summary 110 References 111 Exercise Problems 30 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 4.16 4.17 4.18 4.19 4.20 4.21 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 32 HEAT TRANSFER Applications of Heat Conduction 32 Steady and Unsteady Conduction 32 Dimensionality in Conduction 33 Some Important Examples of Heat Generation in a Body How Does the Classification of a Conduction Problem Help? 33 Basic Approach in Numerical Heat Conduction 33 One-Dimensional Steady State Problem 33 Two-Dimensional Steady State Problem 54 Three-Dimensional Problems 62 Transient One-Dimensional Problem 63 Accuracy of Euler, Crank-Nicholson and Pure Implicit Method 70 Stability: Numerically Induced Oscillations 71 Convection Boundary Condition 76 Stability Limit of the Euler Method from Physical Standpoint 77 Mathematical Representation of All Three Methods by a Single Discretization Equation 78 Physical Representation of All Three Methods 79 Advantages and Disadvantages of Each of the Three Methods 79 How to Choose a Particular Method 80 Consistency 80 Two-Dimensional Transient Problems 81 Example Problem: Two-Dimensional Transient Heat Conduction in a Square Plate 83 112 NUMERICAL METHODS FOR INCOMPRESSmLE FLUID FLOW NUMERICAL METHODS FOR CONDUCTION 4.1 4.2 4.3 4.4 4.5 98 33 5.10 5.11 5.12 5.13 5.14 5.15 5.16 Introduction 119 Governing Equations 120 Difficulties in Solving Navier-Stokes Equations 120 Stream Function-Vorticity Method 121 General Algorithm for Solution by If! - ~ method 123 Creeping Flow (Very Small Reynolds Number) 125 InviscidFlow(Steady) 127 Determination of Pressure for Viscous Flow 128 Is the Transient Approach Used for Solving Steady Flow Problems? 131 If! - ~ Method for 3-D Problems 131 The Primitive- Variables Approach 131 Simple (Semi-Implicit Method for Pressure-Linked Equations) Procedure ofPatankar and Spalding (1972) Computation of Boundary Layer Flow 138 Similarity Solutions of Boundary Layer Equations: Shooting Technique 140 Finite Difference Approach 144 Von Mises Transformation 147 Summary 148 References 149 Exercise Problems ) 119 132 149 NUMERICAL METHODS FOR CONVECTION HEAT TRANSFER 6.1 6.2 6.3 Introduction 153 Convection-Diffusion (Steady, One-Dimensional) 154 Convection-Diffusion (Unsteady, One-Dimensional) 158 153 xii Contents 6.4 6.~ 6.6 6.7 6.8 6.9 Contents xiii Convection-Diffusion (Unsteady, Two-Dimensional) 159 Computation of Thermal Boundary Layer Flows (Part A) 162 Computation of Thermal Boundary Layer Flows (Part B) 163 Transient Free Convection from a Heated Vertical Plate 170 Problem Statement 171 Free Convection in Enclosure 174 Summary 175 References 175 Exercise Problems 176 References SPECIALTOPICS I NEWMEmODS 7.1 7.2 7.3 A.7 Finite Element Method Based on Galerkin's Weighted Residual Approach 268 A.8 Extension to 2-D and 3-D Problems 278 A.9 Accuracy of Solution 279 A.I0 Summary 279 A.ll Control Volume Formulation (also Known as Finite Volume Method) 280 A.12 Closure 287 179 Introduction 179 New Method (I): Application of ADI Method to Solve the Problem of 2-D Transient Heat Transfer from a Straight Composite Fin 179 New Method (II): Alternative to Upwind SchemeApplication of Operator-Splitting Algorithm to Solve Convection- Diffusion Problems 183 C.l C.2 8.5 8.6 190 Introduction 190 Transient Combined Mixed Convection and Radiation from a Vertical Aluminium Fin (Ghoshdastidar and Raju, 1992) 190 Heat Transfer in Rotary Kiln Reactors 205 Modelling of Thermal Transport Processes in Single-Screw Plasticating Extruder with Applications to Polymer and Food Processing 220 Heat Transfer in Metal and Alloy Solidification 237 Cooling of Electronic Equipments 246 Appendix A Other Discretization Methods A.l A.2 A.3 A.4 A.5 A.6 The Essence of the Finite Element Methods 255 Finite Element Method Based on Variational Calculus Convection Boundary Condition 265 Two-Dimensional Steady State Problem 266 Three-Dimensional Problems 267 Summary 268 Appendix D Reference Summary 252 References 252 255 256 291 Subroutine TDMA 291 Demonstration Program Showing Application of Subroutine TDMA 292 Reference Index 288 The Essence of Runge-Kutta Methods 288 Simultaneous Ordinary Differential Equations 289 Solution of Higher Order ODE by R-K Methods 290 References 290 Appendix C Listing of Subroutine IDMA SPECIALTOPICSn 8.3 8.4 Appendix B Runge-Kutta Methods B.l B.2 B.3 Summary 189 References 189 8.1 8.2 287 292 Numerical Method for Radiation in Enclosure with Diffuse-Gray Surfaces: The Absorption Factor Method 293 295 296 INTRODUCTION 1.1 WHAT IS COMPUTER SIMULATION? The simulation of an industrial system on computer involves mathematical representation of the physical processes undergone by the various components of the system, by a set of equations (usually differential equations) transformed to difference equations which are in turn solved as a set of simultaneous algebraic equations At this stage, a reader uninitiated into the numerical methods may ask the question "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 developed when computer was not even invented Now, to return to the original question, the answer is that with the aid of the algorithm of the solution method translated into a programming language like FORTRAN fed into a computer which does 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 simple example will clarify this point One can very easily solve a set of three linear simultaneous algebraic equations by hand through Gaussian Elimination method.Typically in this method, for a system ofn equations the total number of multiplications and divisions is roughly.! n3• Therefore, for n = 3, the number of operations 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 of around (that is, 106 arithmetic operations per sec) will solve the aforesaid problem 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 operations as compared to addition and subtraction on a computer Introduction Computer Simulation of Flow and Heat Transfer • • • • • • • of 0.01 will take about 33 seconds In computer simulation, it is possible to handle 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 • • • • • 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 screw extruders, 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 diffusion made this remark in 1824 "Heat, like gravity, penetrates every substance of the universe; its rays occupy all parts of space The theory of heat will hereafter form one of the most important branches of general physics" Lord Kelvin, in 1864 obtained a rough estimate of the age of earth based on an idea proposed by Fourier in 1820 to be 94 x 106 years (0.094 billion years) by applying the principle of transient heat conduction in a semi-infinite solid Modem dating methods have revealed the age of earth to be approximately 4.7 billion years So Kelvin's result was not really too far off the mark considering the fact that the data for the measured value of the geothermal gradient (rate of increase of temperature of earth with depth), average thermal diffusivity of rock and the initial temperature of molten earth when cooling began available with him at that time were not very ,ccurate The aforesaid example is probably the first known application of heat transfer 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 would find only a handful of analytical (or exact) solutions In actual situations, problems are lot more complex as in those involving non-linear governing equations and/or boundary conditions, and irregular geometry which not allow analytical solutions to be obtained Therefore, it is necessary to use numerical techniques for most problems of practical interest Furthermore, to design and optimize thermal processes and systems, numerical simulation of the relevant transport phenomena is a must, since experimentation is usually too involved and expensive However, necessary experimentation must still be done in checking the accuracy and validity of numerical results Sometimes, numerical model can be refined by input from results of a companion experimental set-up for the same problem Computer Simulation of Flow and Heat Transfer 1.5 BASIC APPROACH IN SOLVING A PROBLEM BY NUMERICAL METIIOD 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 x Grid points (known temperatures) o Grid points (unknown temperatures) Introduction For 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 = (9)3 = 243 PC = 16 x x x 243 = 3.89 x 103 operations Thus, even with a fairly coarse grid, the number of operations is quite large Currently a large number of realistic applications like modelling of supersonic aircraft or weather prediction requires 1012 to 1014 operations per solution (Rajaraman, 1993) If each solution has to be done in about an hour, then the average 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 supercomputers is around 1000 megaflops (Rajaraman, 1993) From the aforesaid example, it is clear why we need supercomputers with speeds in thousands of megaflops range to solve extremely complex problems 1.7 A COMPARATIVE STUDY OF EXPERIMENTAL, ANALYTICAL AND NUMERICAL METIIODS (a) Experimental Method Experimental methods are used to obtain reliable informations about physical processes that are not well understood, e.g combustion and turbulence It may involve full scale, small scale or blown-up scale models The major disadvantages of experimental investigations are high cost, measurement difficulties and probe errors Often, small scale models not always simulate all the features of the full scale set-up The advantage is that it is most realistic Fig 1.1 1.6 x grid for computation of two-dimensional steady-state heat conduction in a square plate PROBLEM COMPLEXI1Y (b) Analytical Method Analytical methods or methods of classical mathematics are used to obtain the solution of a mathematical model consisting of a set of differential equations which represent a physical process within the limit of assumptions made Only a handful of analytical solutions are available in heat transfer and fluid mechanics because analytical methods are inadequate in handling complex boundary and non-linearities in the differential equations and/or boundary conditions Furthermore, the analytical solutions often contain infinite series, -special functions, transcendental equations for eigenvalues, etc the numerical evaluation of which becomes quite cumbersome 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 (c) Numerical Method As explained in Sec 1.5, a numerical prediction works out the consequences of a mathematical_ model but the solution is obtained for variables at discrete grid points in the computational domain in contrast with analytical method which gives closed form solution at all points (theoretically infinite number of points in the solution domain or continuum) The major advantages of numerical solution are its abilities to handle complex geometry and nonlinearities in the governing equation and/or boundary conditions Other advantages of numerical method are briefly described below: Computer Simulation of Flow and Heat Transfer • • • • • Introduction 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 conditions 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 Name of the Method Experimental Advantages • Capable of Being most realistic Analytical • Clean, general information which is usually in formulaform Numerical • No restriction to linearity • Ability to handle irregulargeometry and complicated physics • Low cost and high speed of computation Disadvantages • Equipment required • Scaling problem • Measurement difficulties • Probe errors • High operating costs • Restricted to simple geometry and physics • Usually restricted to linear problems • Cumbersome results-difficult to compute • Truncation and round-off errors • Boundary condition problems - 1.8 METHODS OF DISCRETIZATION There are several methods of discretizing a given differential equation They are briefly described below (a) Finite-Difference Method The ilsual procedure for deriving finite-difference equations consists of approximating derivatives in the differential equation via a truncated Taylor series The method includes the assumption that the variation of the unknown to be computed is somewhat like a polynomial in x, y or z so that higher derivatives are unimportant The great popularity of finite-difference methods is mainly due to their straight-forwardness and relative simplicity by which a newcomer in the field is able to obtain solutions of simple problems As a matter of fact, the subject of Computational Fluid Dynamics (commonly abbreviated as CFD) was born as early as 1933 (remember that world's first computer called ENIAC was built at the University of Pennsylvania, USA, in 1946) with the remarkable work (published in the proceedings of the Royal Society) ofThom (1933) who solved the Navier-Stokes equations for the steady, incompressible viscous flow around a circular cylinder by finite-difference method using hand calculations However, several shortcomings and limitations of finite-difference method came to light when researchers tried to solve problems with increasing degree 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 methods and (ii) spectral methods (b) Finite-Element Method (FEM) Finite-element methods basically seek solutions at discrete spatial regions (called elements) by assuming that the governing differential equations apply to the continuum within each element It is based on integral minimization principle and provides piecewise (or regional) approximations to the governing equations Finite-element methods were already found to be successful in solid mechanics applications Their introduction and ready acceptance in fluid mechanics was due to relative ease by which flow problems with complicated boundary shapes can be modelled, especially when compared with finite-difference methods However, disadvantages of FEM arises from the fact that more complicated matrix operations are required to solve the resulting system of equations Furthermore, meaningful variational formulations are difficult to obtain for high Reynolds number flows Hence, variational principle-based FEM is limited to solutions of creeping flow and heat conduction problems Galerkin's weighted residual method, which is also another finite-element method is a powerful method and circumvents the difficulties faced by variational A.9 ACCURACY OF SOLUTION Comparison of the FEM solution with the exact solution (Eq (A.51)) at the nodal points 1,2, reveals exact matching between the two However, this kind of exact agreement occurs only when a problem is very simple as the present one In other words, it can be said that the use of piecewise linear trial functions has correctly interpolated a quadratic function, indicating the good performance expected in the general case It must be noted, however, that any numerical solution is an approximation, and hence will not agree everywhere with the exact solution The accuracy of the approximate solution depends on the location of evaluation A.tO SUMMARY The finite element method based on Galerkin's weighted residual approach is described in detail for a one-dimensional steady state conduction problem For more details regarding multi-dimensional steady state problems and unsteady transport with fluid motion, readers are referred to Baker and Pepper (1991) 280 A.II A.II.I Computer Simulation of Flow and Heat Transfer CONTROL VOLUMEFORMULATION (ALSO KNOWN AS FINITE VOLUME MEmOD) Basic Concept In the control volume fonnulation, the computational domain is divided into a number of nonoverlapping control volumes such that there is one control volume surrounding each grid point The governing differential equation (GDE) is integrated over each control volume Piecewise profiles expressing the variation of the unknown function (such as temperature, 1) between the grid points are used to evaluate the required integrals This results in the discretization equation consisting of the value of the unknown function for a cluster of grid or nodal points A.II.2 One-Dimensional Steady-State Heat Conduction Problem Let us consider steady state I-D heat conduction in a medium of conductivity 'k' (uniform or non-unifonn) in which heat is generated at the rate of q'" W/m3• The governing differential equation is: A.ll.6 Summary The control volume formulation for the 1-0 steady state conduction problems is described in detail The extension to 2-D and 3-D situations are discussed briefly For transient conduction, flow and convection-diffusion problems, readers are referred to Patankar (1980) A.12 CLOSURE In this chapter three discretization methods other than the finite-difference method, such as variational calculus based FEM, Galerkin's weighted residual approach based FEM and control volume method are discussed in detail with respect to one-dimensional steady state conduction problems Interested readers are encouraged to advance their knowledge in these areas by going through the relevant books given in the references REFERENCES Baker, A J and D W Pepper,Finite Elements 1-2-3, McGraw-Hill Inc., New York, 1991 Galerkin, B G, "Series Occurring in Some Problems of Elastic Stability of Rods and Plates", Engrg Bull., Vol 19, 1915, pp 897-908 Myers, Glen E, Analytical Methods in Conduction Heat Transfer, McGraw-Hill, New York, 1971 Patankar, S V, Numerical Heat Transfer and Fluid Flow, Hemisphere, Washington, DC, 1980 c.t c c c c c c c c c c c SUBROUTINE TDMA SUBROUTINE FOR SOLVING A SYSTEM OF LINEAR SIMULTANEOUS ALGEBRAIC EQUATIONS HAVING A TRIDIAGONAL COEFFICIENT MATRIX THE EQUATIONS ARE NUMBERED FROM I THROUGH N, AND THEIR SUB-DIAGONAL, DIAGONAL AND SUPERDIAGONAL ELEMENTS ARE STORED IN THE ARRAYS A, B, AND C THE RIGHT-HAND COLUMN VECTOR ELEMENTS ARE STORED IN THE ARRAY D THE COMPUTED SOLUTION VECTOR XCI) " X(N) IS STORED IN THE ARRAY X SUBROUTINE TDMA (I, N, A, B, C, D, X) DIMENSION A(1), B(1), C(1), D(1), X(1), BETA (101), GAMMA (101) c *** COMPUTE INTERMEDIATE ARRAYS BET A AND GAMMA ** BETA (I) = B(I) GAMMA (I) = D(I)/BET A(I) 11=1+1 DO J = 11, N BETA (J) = B(J)-A (J)*C(J-1)/BETA(J-1) GAMMA(J) = (D (J) - A(J) * GAMMA (J-1)/BETA (J) c ***** COMPUTE THE SOLUTION VECTOR X * X(N) = GAMMA (N) N1 = N-I DO K = 1, N J = N-K 292 Computer Simulation of Flow and Heat Transfer C.2 X (J) = GAMMA (J) - C(J) RETURN END * X (J + 1) /BETA (J) DEMONSTRATION PROGRAM SHOWING APPliCATION OF SUBROUTINE liMA The application of subroutine TDMA is demonstrated through the solution of Eq (4.40) In the demonstration program A, B, C, D are the coefficient vectors defined in Eq (4.23) Tis the vector of the unknowns In the subroutine TDMA, BET A and GAMMA are vectors of intermediate coefficients Pi and 'ii, respectively defined in Eq (4.28) The level of programming in the demonstration program has been kept intentionally simple to make the readers appreciate how the arrays A, B, C, D are defined The output is given below the main programt c DEMONSTRA nON PROGRAM SHOWING APPLICA nON OF c SUBROUTINE TDMA DIMENSION A(21), B(21), C(21), D(21), T(21) OPEN (UNIT = 76, FILE = 'OUTP') A(2) = - A(3) = - A(4) =-2 B(1) = 2.25 B(2) = 2.25 B(3) = 2.25 B(4) = 2.25 C(l) = - C(2) = - C(3) = - D(1) = D(2) = O D(3) = O D(4) = O CALL TDMA (1, 4, A, B, C, D, T) WRITE (76, 2) (T (1),1 = 1,4) FORMAT (2X, 4F8 3) STOP END 629 415 305 271 REFERENCE Carnahan, Brice, H A Luther, and James,a John Wiley & Sons, New York, 1969 Wilkes, Applied Numerical Methods, The Absorption Factor Method 295 Gebhart also proved a reciprocity relationship between the absorption factors For diffuse radiation and reflection: A-I = C·J B··JI A-J The absorption factors depend only on the geometry, through sh~pe factors and the properties C and P of the surfaces comprising the enclosure They not depend on the surface temperature and the heat input This is particularly advantageous because Bjj values are obtained independent of the surface temperature and the heat input Therefore, in many complicated problems of thermal design of industrial systems, the absorption factor method is more suitable compared to other methods since the numerical solution for Bij is carried out conveniently C·I B·· I} Fig D.I An enclosure of N surfaces Knowing the shape factors and the values of the reflectivity and the emissivity, the values of the Bij's can be determined numerically The follo~ng equations can be written for each of the N surfaces = Flj £j + Fll PI BIj + Fiz Pz BZj + + FIN PN BNj BZj = FZj Cj + FZI PI BIj + Fzz Pz BZj + + Fw PNBNj BNj = FNj Cj + FNl PI Blj + FNZPZ B2j + + FNN PNBNj BIj Gebhart, B., Heat Transfer, 2nd Ed., McGraw-Hill Book Co., New York, 1971 (D.2) = wherecj, Pi' i N represent emissivity and reflectivity oftheith surface F y's are the shape factors These N linear equations can be transposed and rearranged to obtain: =0 F21 PI BIj + (Fzz PZ - l)BZj + , + Fw PN BNj + FZj E.i= FNl PI Blj + FNZPZBZj + (FNN PN-l) BNj + FNj Cj = (Fll PI -l)BIj + FIZ Pz BZj + + FIN PNBNj + FIj Cj (D.3) Equation (D.3) can be solved for the N unknowns Blj' BZj' BNj' Equation (D.3) is valid for any choice of j, that is, 1, 2, N For small N, the direct methods for linear simultaneous equations such as Gaussian Elimination may be employed When N is large, iterative method such as Gauss~Seidel should be used With the values of Bij known, the radiant heat loss or gain qj at each wall element is calculated using Eq (D 1) Some special points of importance should be noted: The distinction between the Bij and F ij is as follows The shape factor represents the fraction of the radiant energy leaving the area Ai and arriving directly at the area Aj while the absorption factor represents the fraction of the radiant energy leaving the areaAj and absorbed by the area Aj following multiple reflections before it finally arrives at the areaAj In the special case of black surfaces: B ij F ij' Since the energy emitted by each surface is absorbed by the collection of N surfaces which form the enclosure, = REFERENCE Index 297 CTCS 71 Cylindrical geometry Absorption factor method 293 Accuracy 21,25,27,28,40,70,279 AD! 82,84,86,89 Advantages of computer simultation Age of earth Algebraic equations 1,4 Algorithm: Operator-S plitting(OS) 183, 184, 189,249 Thomas 40,41,42 Tridiagonal Matrix (TDMA) 40,41, 42 SIMPLE 137 Alloy solidification 239 Alternating Direction Implicit Method 82 Alternative to upwind scheme 183 Analytical method 5,6 Application of ADI to composite fin 179 Application of subroutine TDMA 292 Axisymmetric problems 90 Backward difference 21,24,25 Back substitution 49 Banded coefficient matrix 59 Band width 59 Biharmonic equation 126 Block elimination 170 Boundary conditions 11,13 Boundary conditions, nonlinear Boundary layer 15, 138, 140 Boundary layer flow 138, 144 Boundary value problem 15 BTCS 71 Calculus of variations 256 3,101 Cauchy condition 14 Cement rotary kiln 206 Central difference 21,22,23,24,25,27 CFD Characteristics 15 Checkboard 133 Closed domain 11 Combined modes of heat transfer 190 Compatibility conditions 97,98 Composite fin 179 Composite media 96 Computational fluid dynamics Computer simulation 1, Condition at the centre, handling of 90 Conduction heat transfer 32 Consistency 80 Continuity equation 120 Control volume formulation 8,280 Convection-conduction parameter, CCP 192 Convective (or convection) boundary condition 53, 76,265 Convection heat transfer 153 Convection-diffusion 154, 183, 185, 287 Convection-diffusion (steady, onedimensional) 154 Convection-diffusion (unsteady, onedimensional) 158 Convection-diffusion (unsteady, twodimensional) 159 Cooling of electronic equipments 246 Comer points, handling of 57 Crank-Nicholson method 66, 68, 70 71, 72, 73, 74, 75, 78, 79, 86, 87, 166 Creeping flow 125 Flow over flat plate 140· Flow over a heated flat plate 162 Flow of liquid metals 163 Flow over 2-D wedge 140 Forward difference 21,23,24,25 Fourier, Joseph, J.B Free convection 170 Free convection from a vertical plate 170 Free convection in enclosure 174 FTCS 71 90 Derivatives Diagonal dominance 45, 187 Difference equation Differential equation 1,7,8 Diffuse-gray surfaces 293 Diffusion equation 11 Dirichlet condition 13 Discretization error· 29 Discretization, methods of DufortcFrankel method 81 Eigen-values 74, 75 Eigen-vectors 74 Electronic equipments, cooling of 246 Elliptic PDE 11 ENIAC Equation: Biharmonic 126 Continuity 120 Euler-Lagrange 259 Laplace's 11 Momentum 120 7, 119, 120, 122, 132 Navier-Stokes Poisson's 11, 129, 137 Pressure-correction 137 Vorticity transport 122 Wave 12 Euler~Lagrange equation 259 Euler(or Explicit) method 66, 70, 71, 72,73,74,75,77,78,79,87,88, 100, 145, 163, 165, 172 Exponentialscheme· 158 False diffusion 157 False transient 89 FEM 7,259,265 Fin 33, 179, 190 Finite-difference 1,7 Finite element Finite element basis 271 Finite element method, essence of Finite element method based on variational ca:Iculus 256 Finite volume method 280 Flops I 255 1, 4, 40, 48, 52, Gaussian elimination 59 Gauss-Seidel iterative method 40,42, 44,48,51,52,59 G-E 40 Gebhart, B 293 Grashof number 201 G-S 42,43,45,89 Global diffusion matrix 277 Global load matrix 277 Global matrix 277 Governing differential equation, nonlinear 98 Grid 4,5 Grid independence test 29,87,88 Grid points 11 Heat conduction Higher accuracy 25 Hybrid scheme 158 Hyperbolic PDE 11 Image-point technique 35, 37, 38, 39, 54,93 Implicit 163, 174 Implicit Crank-Nicholson method 166 Implicit Keller-box method 167, 170 Incineration of solid waste 214 Incompressible fluid flow 119 lildustrial problems 190 Initial condition 12,13 Initial value problem 15 Insulation cO!1dition 14 Integral minimization 7,261 Interpolation function 272 Inviscid flow 127 ~,lt Index Index Irregular geometry 3,106 Irrotational flow 122 Jacobi method 43,44 Kronecker delta 275 Lagrange interpolation 272 Laplace's equation 11 Laser cutting 16 Laser surgery 16 L'Hospital's rule 92 Linearity 10 Line-by-Line method 60,.61 LOCA MAC 132 Marker-and~Cell 132 Matrix: banded coefficient 59 global 277 global diffusion 277 global load 277 tridiagonal 39 tridiagon(ll block 170 Mega Magaflop 1;5 Metal and alloy solidification 237 Metal solidification 237 Materials processing 3,16 Method: Absorption factor 293 Alternating direction Implicit(ADI) 82 Control volume 8,280 Crank-Nicholson 66,68, 70, 71, 72, 73,74,75,78,79,86,87,166 Dufort-Frankel' 81 Euier (or Explicit) 66, 70; 71, 72, 73, 74,75,77,78,79,87,88,100, 145, 163, 165, 172 Finite-difference 1,7,21 Finite element 7,256,268 Gaussi.an elimination 1,4,40,48, 52;59 Gauss-Seidel iterative 40, 42, 44, 48,51,52,59 Implicit Crank-Nicholson 166 Implicit Keller-box 167, 170 Line-by-Line 60,61 Newton-Raphson 143 Pure implicit 69 Relaxation 46 Runge-Kutta 288 Shooting 142 Spectral Methods of discretization Mixed convection 153, 190 Mixed convection and radiation 190 Mixed elliptic-parabolic 122 Modelling 190 Momentum equations 120 Natural convection 170 Navier-Stokes equations 7, 119, 120, 122; 132 Neumann condition 14 Newton-Raphson method· 143 Non-axisymmetric problem 93 Non-Fourier conduction 16 NOnlinear boundary conditions 3,101 Nonlinear governing differential equation 3,98 Nonlinearlity 98, 120 Non-Newtonian flow 190 Non-uniform grid 25 Numerical errors 21,28 Numerically induced oscillations 71 Numerical method 5,6 Numerical simulation One-dimensional steady state 33, 259, 268,280,283 Operator-splitting algorithm 189,249 183, 184, Open-ended 11,12 Optimum relaxation factor 47 Optimum step size 29 Order 10 Oscillations, numerically induced Over-relaxation 46 PDE 10 Parabolic PDE 11 Partial differential equations Patankar S.V 132, 135 10 71 Peclet number 157 Phase change 190 Poisson's equation 11 Poisson equation for pressure 129 Poisson equation for pressure-correction 137 Polymer and food.extrusion 126 Polymer and food processing 220 Polynomial Polynomial fitting 37 Prandtl, L 138 Prandtl number 154,201 Pressure-correction equation 137 Pressure-correction equation, Boundary conditions for 137 Pressure-gradient 120 Power-law scheme 158 Primitive-variables approach 131 Problem complexity Pure Implicit Method 69 Radiation boundary condition 103 Radiation in enclosure 190,293 Recursion 41,47 Relaxation factor 47 Relaxation method 46 Reynolds number 7, 154 Robbins condition 14 Rotary kiln, cement 206 Rotary kiln, heat transfer in 205 Rotary kiln incinerator 214 Rotational flow 122 Round-off error 28 Runge- Kutta methods 288 Runge-Kutta methods, essence of 288 Scarborough criterion 44,45· Similarity solution 140 Shooting method 142 Shooting technique 140 SIMPLE' 132, 135, 137 SIMPLE algorithm 137 SIMPLER 138 SIMPLER revised 138 Simultaneous ordinary differential equations 289 Single-screw plasticating extruder Small Reynolds number 125 299 Solution of higher order ODE 290 SOR 47, 89 Source-term linearisation 282 Spalding D.B 132 Spectral method Spherical geometry 95 Stability 68,71,77,78, 173 Stagg~red grid 132, 134 Stream function 122 Stream function-vorticity method 121 Subroutine TDMA 292 Subsonic 16 SUR 47 Supercomputers Supersonic 16 Taylor series 7,21,81,97 TDM 39,59 TDMA 40,41,42,48,49,52,159,187 Thermal boundary layer 162, 163 Thermal radiation 207,293 Thomas algorithm 40,41,42 Three-dimensional problems 62, 267, 286 Three-dimensional transient problems 89 Transient one-dimensional problems 63 Treatment of source term 281 Trial function 270 Tridiagonal block matrix 170 Tridiagonal matrix 39 Tridiagonal matrix algorithm 40,41,42 Truncation error 22,29 Two-dimensional steady state 54, 266, 286 Upwind scheme 156, 157, 159, 161 Upstream differencing 156 Under-relaxation 46 Uniforrii grid 21 220 Variable thermal conductivity 98 Variational principle based FEM 7.256 Vertical fin 190 Vertical plate, Free convection from 170 Von-Mises transformation 147 Vorticity 121 300 Index Vorticity transport equation Wave equation 12 Weak statement 272 122 Weighted residual approach, Galerkin's 7,268 Weight function 270 Wiggles 156 ... 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. .. (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,... subtraction on a computer 2 Introduction Computer Simulation of Flow and Heat Transfer • • • • • • • of 0.01 will take about 33 seconds In computer simulation, it is possible to handle one hundred