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Introduction to

Computational Fluid Dynamics

Development Application and Analysis

Dr Atul Sharma

Professor Deptt of Mech EngineeringIIT Bombay

Introduction to



Introduction to Computational Fluid DynamicsDevelopment Application and Analysis

Dr Atul Sharma

copy Author 2017

This Edition Published by

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Library Congress Cataloging-in-Publication DataA catalogue record for this book is available from the British Library

Cover Image The image for C F and D in the cover page is obtained from a Computational Fluid Dynamics simulations as follows

C Curvilinear-Grid and Resultant-Velocity Contours for Fluid-Flow in a C-shaped Bend of a Pipe

F Temperature-Contours in a Heat Conduction Problem

D Vertical-Velocity Contours for a vertical-wall (moving upward like a conveyor belt) driven flow in a D shaped cavity

DisclaimerThe contents sources and the information provided by the authors in different chapters are the sole responsibility of the authors themselves including the copyright issues Editors and Publishers have no liabilities whatsoever towards these in any manner

This book is dedicated

to

Suhas V Patankar

Dr Suhas V Patankar1 (born February 22 1941) is an Indian

mechanical engineer He is a pioneer in the eld of computational

uid dynamics (CFD) and nite volume method He received his

BE from the University of Pune in 1962 MTech from IIT Bombay

in 1964 and PhD from Imperial College London in 1967

Dr Patankar is currently a Professor Emeritus in the Mechanical

Engineering Department at the University of Minnesota where he

worked for 25 years (from 1975-2000) Earlier he held teaching and

research positions at IIT Kanpur Imperial College and University

of Waterloo He is also the President of Innovative Research Inc He

has authored or co-authored four books published over 150 papers

advised 35 completed PhD theses and lectured extensively in the

USA and abroad His 1980 book Numerical heat Transfer and Fluid

Flow is considered to be a groundbreaking contribution to CFD He

is one of the most cited authors in science and engineering

For excellence in teaching Dr Patankar received the 1983 George

Taylor Distinguished Teaching Award and the 1989-90 Morse-Alumni

1Source (for the Biography) Personal Communication

v

vi

Award for Outstanding Contributions to Undergraduate Education

For his research contributions to computational heat transfer he was

given the 1991 ASME Heat Transfer Memorial Award and the 1997

Classic Paper Award He was awarded the 2008 Max Jakob Award

which is considered to be the highest international honor in the eld of

heat transfer In 2015 an International Conference on Computational

Heat Transfer (CHT-15) held at Rutgers University in New Jersey

was dedicated to Professor Patankar

Dr Patankars widespread inuence on research and engineering

education has been recognized in many ways In 2007 the Editors of

the International Journal of Heat and Mass Transfer wrote There is

no person who has made a more profound and enduring impact on the

theory and practice of numerical simulation in mechanical engineering

than Professor Patankar

FOREWORD

by Prof K Muralidhar

Indian Institute of Technology Kanpur INDIA

The subject of computational uid dynamics (CFD) emerged in the

academic context but in the past two decades it has developed to a

point where it is extensively employed as a design and analysis tool in

the industry The impact of the subject is deeply felt in diverse disci-

plines ranging from the aerospace spreading all the way to the chem-

ical industry New compact and ecient designs are enabled and it

has become that much more convenient to locate optimum conditions

for the operation of engineering systems As it stands CFD is com-

pulsory learning in several branches of engineering Many universities

now opt to develop theoretical courses around it The great utility of

this tool has inuenced adjacent disciplines and the process of solv-

ing complex dierential equations by the process of discretization has

become a staple for addressing engineering challenges

For an emerging discipline it is necessary that new books fre-

quently appear on the horizon Students can choose from the se-

lection and will stand to gain from it Such books should have a

perspective of the past and the future and it is all the more signi-

cant when the author has exclusive training in the subject Dr Atul

Sharma represents a combination of perspective training and passion

that combines well and delivers a strong text

I have personal knowledge of the fact that Dr Atul Sharma has

developed a whole suite of computer programs on CFD from scratch

He has wondered and pondered over why approximations work and

conditions under which they yield meaningful solutions His spread

vii

viii Foreword

of experience over fteen and odd years includes two and three di-

mensional Navier-Stokes equations heat transfer interfacial dynam-

ics and turbulent ow He has championed the nite volume method

which is now the industry standard The programming experience

is important because it brings in the precision needed for assembling

the jigsaw puzzle The sprinkling of computer programs in the book

is no coincidence

Atul knows the conventional method of discretizing dierential

equations but has never been satised with it He has felt an ele-

ment of discomfort going away from a physical reality into a world

of numbers connected by arithmetic operations As a result he has

developed a principle that physical laws that characterize the dier-

ential equations should be reected at every stage of discretization

and every stage of approximation indeed at every scale whether a

single tetrahedron or the sub-domain or the region as a whole This

idea permeates the book where it is shown that discretized versions

must provide reasonable answers because they continue to represent

the laws of nature

This new CFD book is comprehensive and has a stamp of origi-

nality of the author It will bring students closer to the subject and

enable them to contribute to it

Kanpur

August 2016 K Muralidhar

PREFACE

As a Post-Graduate (PG) student in 1997 I got introduced to a

course on CFD at Indian Institute of Science (IISc) Bangalore Dur-

ing the introductory course on CFD although I struggled hard

in the physical understanding and computer-programming it didnt

stop my interest and enthusiasm to do research teaching and writ-

ing a text-book on a introductory course in CFD Since then I did

my Masters project work on CFD (1997-98) introduced and taught

this course to Under-Graduate (UG) students at National Institute

of Technology Hamirpur HP (1999-2000) did my Doctoral work on

this subject at Indian Institute of Technology (IIT) Kanpur (2000-

2004) and co-authored a Chapter on nite volume method in a re-

vised edition of an edited book on computational uid ow and heat

transfer during my PhD in 2003 After joining as a faculty in 2004

I taught this course to UG as well as PG students of dierent depart-

ments at IIT Bombay as well as to the students of dierent collages in

India through CDEEP (Center for Distance Engineering Programme)

IIT Bombay from 2007-2010 Furthermore I gave series of lecture

on CFD at various collages and industries in India More recently

in 2012 I delivered a ve-days lecture and lab-session on CFD (to

around 1400 collage teachers from dierent parts of India) as a part

of a project funded by ministry of human resource development under

NMEICT (National Mission on Education through information and

communication technology)

The increasing need for the development of the customized CFD

software (app) and wide spread CFD application as well as analysis

for the design optimization and innovation of various types of engi-

neering systems always motivated me to write this book Due to the

increasing importance of CFD another motivation is to present CFD

ix

x Preface

simple enough to introduce the rst-course at early under-graduate

curriculum rather than the post-graduate curriculum The moti-

vation got strengthened by my own as well as others (students

teachers researchers and practitioners) struggle on physical under-

standing of the mathematics involved during the Partial Dierential

Equation (PDE) based algebraic-formulation (nite volume method)

along with the continuous frustrating pursuit to eectively convert

the theory of CFD into the computer-program

This led me to come up with an alternate physical-law (PDE-

free) based nite volume method as well as solution methodol-

ogy which are much easier to comprehend Mostly the success of

the programming eort is decided by the perfect execution of the

implementations-details which rarely appears in-print (existing books

and journal articles) The physical law based CFD development

approach and the implementations-details are the novelties in the

present book Furthermore using an open-source software for nu-

merical computations (Scilab) computer programs are given in this

book on the basic modules of conduction advection and convection

Indeed the reader can generalize and extend these codes for the de-

velopment of Navier-Stokes solver and generate the results presented

in the chapters towards the end of this book

I have limited the scope of this book to the numerical-techniques

which I wish to recommend for the book although there are numerous

other advanced and better methods in the published literature I do

not claim that the programming-practice presented here are the most

ecient minus they are presented here more for the ease in programming

(understand as your program) than for the computational eciency

Furthermore although I am enthusiastic about my presentation of

the physical law based CFD approach it may not be more ecient

than the alternate mathematics based CFD approach The emphasis

in this book is on the ease of understanding of the formulations and

programming-practice for the introductory course on CFD

I would hereby acknowledge that I owe my greatest debt to Prof

J Srinivasan (my ME Project supervisor at IISc Bangalore) who

did a hand-holding and wonderful job in patiently introducing me to

the fascinating world of research in uid-dynamics and heat trans-

fer I would also acknowledge the excellent training from Prof V

Introduction to CFD xi

Eswaran (my PhD supervisor at IIT Kanpur) from whom I learned

systematic way of CFD development application and analysis His

inspiring comments and suggestions on my writing-style and elabora-

tion of the rst chapter in this book is also gratefully acknowledged

I also want to record my sincere thanks to Prof Pradip Datta IISc

Bangalore and Prof Gautam Biswas IIT Kanpur minus my teachers for

CFD course Special thanks to Prof K Muralidhar IIT Kanpur for

his personal encouragement academic support and motivation (for

the book writing) He also cleared my dilemma in the proper presen-

tation of the novelty in this book I thank him also for writing the

foreword of this book

I am grateful to Prof Kannan M Moudgalya and Prof Deepak B

Phatak from IIT Bombay for giving me opportunity to teach CFD

in a distance education mode I am also grateful to Prof Amit

Agrawal (my colleague at IIT Bombay) who diligently read through

all the chapters of this book and suggested some really remarkable

changes in the book I also thank him for the research interactions

over the last decade which had substantially improved my under-

standing of thermal and uid science I would also like to thank my

PhD student Namshad T who did a great job in implementing

my ideas on generating the CFD simulation based CFD image for

the cover-page and couple of gures in the rst chapter (Fig 12

and 13) Special mention that the Scilab codes developed (for the

CFD course under NMEICT) by Vishesh Aggarwal greatly helped

me to develop the codes presented in this book his contributions is

gratefully acknowledged I thank Malhar Malushte for testing some

of the formulations presented in this book

I thank my research students (Sachin B Paramane D Datta C

M Sewatkar Vinesh H Gada Kaushal Prasad Mukul Shrivastava

Absar M Lakdawala Mausam Sarkar and Javed Shaikh) for the

diligence and sincerity they have shown in unraveling some of our

ideas on CFD Apart from these several other research students and

the students teachers and CFD practitioners who took my lectures

on CFD in the past decade have contributed to my better under-

standing of CFD and inspired me to write the book I thank all of

them for their attention and enthusiastic response during the CFD

courses The inuence of the interactions with all of them can be seen

xii Preface

throughout this book I am grateful to IIT Bombay for giving me

one year sabbatical leave to write this book The ambiance of aca-

demic freedom considerate support by the faculties and the friendly

atmosphere inside the campus of IIT Bombay has greatly contributed

in the solo eort of book writing I thanks all my colleagues at IIT

Bombay for shaping my perceptions on CFD

During this long endeavor the patient support by Mr Sunil

Saxena Director Athena Academic Ltd UK is gratefully acknowl-

edged I also thank John Wiley for coming forward to take this

book to international markets Lastly I would acknowledge that I

am greatly inspired by the writing style of J D Anderson and the

book on CFD by S V Patankar Of course you will nd that I

have heavily cross-referred the CFD book by the two most acknowl-

edged pioneers on CFD I dedicate this book to Dr S V Patankar

whose book nicely introduced me to the computational as well as

ow physics in CFD and greatly helped me to introduce a physical

approach of CFD development in this book

The culmination of the book writing process fullls a long cher-

ished dream It would not have been possible without the excel-

lent environment continuous support and complete understanding

of my wife Anubha and daughter Anshita Their kindness has greatly

helped to fulll my dream now I plan to spend more time with my

wife and daughter

Mumbai

August 2016 Atul Sharma

Contents

I Introduction and Essentials

1

1 Introduction 3

11 CFD What is it 3

111 CFD as a Scientic and Engineering Analysis

Tool 5

112 Analogy with a Video-Camera 13

12 CFD Why to study 15

13 Novelty Scope and Purpose of this Book 16

2 Introduction to CFD Development Application and

Analysis 23

21 CFD Development 23

211 Grid Generation Pre-Processor 24

212 Discretization Method Algebraic Formulation 26

213 Solution Methodology Solver 29

214 Computation of Engineering-Parameters

Post-Processor 34

215 Testing 35

22 CFD Application 35

23 CFD Analysis 38

24 Closure 39

xiii

xiv Contents

3 Essentials of Fluid-Dynamics and Heat-Transfer for

CFD 41

31 Physical Laws 42

311 FundamentalConservation Laws 43

312 Subsidiary Laws 45

32 Momentum and Energy Transport Mechanisms 46

33 Physical Law based Dierential Formulation 48

331 Continuity Equation 49

332 Transport Equation 51

34 Generalized Volumetric and Flux Terms and their Dif-

ferential Formulation 57

341 Volumetric Term 58

342 Flux-Term 58

343 Discussion 62

35 Mathematical Formulation 63

351 Dimensional Study 63

352 Non-Dimensional Study 67

36 Closure 71

4 Essentials of Numerical-Methods for CFD 73

41 Finite Dierence Method A Dierential to Algebraic

Formulation for Governing PDE and BCs 75

411 Grid Generation 75

412 Finite Dierence Method 78

413 Applications to CFD 92

42 Iterative Solution of System of LAEs for a Flow Property 93

421 Iterative Methods 94


43 Numerical Dierentiation for Local Engineering-

Parameters 105

431 Dierentiation Formulas 106


44 Numerical Integration for the Total value of

Engineering-Parameters 110

441 Integration Rules 111


45 Closure 116

Problems 116

Introduction to CFD xv

II CFD for a Cartesian-Geometry

119

5 Computational Heat Conduction 12151 Physical Law based Finite Volume Method 122

511 Energy Conservation Law for a Control Volume 123512 Algebraic Formulation 124513 Approximations 127514 Approximated Algebraic-Formulation 130515 Discussion 133

52 Finite Dierence Method for Boundary Conditions 13853 Flux based Solution Methodology on a Uniform Grid

Explicit-Method 139531 One-Dimensional Conduction 139532 Two-Dimensional Conduction 147

54 Coecients of LAEs based Solution Methodology on a

Non-Uniform Grid Explicit and Implicit Method 154541 One-Dimensional Conduction 154542 Two-Dimensional Conduction 166

Problems 176

6 Computational Heat Advection 17961 Physical Law based Finite Volume Method 180

611 Energy Conservation Law for a Control Volume 181612 Algebraic Formulation 181613 Approximations 183614 Approximated Algebraic Formulation 191615 Discussion 191

62 Flux based Solution Methodology on a Uniform Grid

Explicit-Method 194621 Explicit-Method 197622 Implementation Details 198623 Solution Algorithm 199


Non-Uniform Grid Explicit and Implicit Method 207631 Advection Scheme on a Non-Uniform Grid 207632 Explicit and Implicit Method 210633 Implementation Details 216634 Solution Algorithm 220

Problems 228

xvi Contents

7 Computational Heat Convection 22971 Physical Law based Finite Volume Method 229

711 Energy Conservation Law for a Control Volume 229712 Algebraic Formulation 231713 Approximated Algebraic Formulation 232




Non-Uniform Grid Explicit and Implicit Method 242Problems 248

8 Computational Fluid Dynamics Physical Law based

Finite Volume Method 25181 Generalized Variables for the Combined Heat and

Fluid Flow 25282 Conservation Laws for a Control Volume 25583 Algebraic Formulation 25984 Approximations 26085 Approximated Algebraic Formulation 263

851 Mass Conservation 263852 MomentumEnergy Conservation 264

86 Closure 269

9 Computational Fluid Dynamics on a Staggered Grid 27191 Challenges in the CFD Development 273

911 Non-Linearity 273912 Equation for Pressure 273913 Pressure-Velocity Decoupling 273

92 A Staggered Grid to avoid Pressure-Velocity Decoupling27593 Physical Law based FVM for a Staggered Grid 27794 Flux based Solution Methodology on a Uniform Grid

Semi-Explicit Method 281941 Philosophy of Pressure-Correction Method 283942 Semi-Explicit Method 286943 Implementation Details 292944 Solution Algorithm 298

95 Initial and Boundary Conditions 300951 Initial Condition 300952 Boundary Condition 301

Problems 306

Introduction to CFD xvii

10 Computational Fluid Dynamics on a Co-located Grid309

101 Momentum-Interpolation Method Strategy to avoid

the Pressure-Velocity Decoupling on a Co-located Grid 310

102 Coecients of LAEs based Solution Methodology on

a Non-Uniform Grid Semi-Explicit and Semi-Implicit

Method 314

1021 Predictor Step 316

1022 Corrector Step 318

1023 Solution Algorithm 323

Problems 329

III CFD for a Complex-Geometry

331

11 Computational Heat Conduction on a Curvilinear

Grid 333

111 Curvilinear Grid Generation 333

1111 Algebraic Grid Generation 334

1112 Elliptic Grid Generation 336

112 Physical Law based Finite Volume Method 343

1121 Unsteady and Source Term 343

1122 Diusion Term 344

1123 All Terms 349

113 Computation of Geometrical Properties 349

114 Flux based Solution Methodology 352

1141 Explicit Method 354

1142 Implementation Details 354

Problems 359

12 Computational Fluid Dynamics on a Curvilinear Grid361


1211 Mass Conservation 363

1212 Momentum Conservation 363

122 Solution Methodology Semi-Explicit Method 372



Problems 380

References 383

Index 389

Part I

Introduction and Essentials

The book on an introductory course in CFD starts with the two intro-

ductory chapters on CFD followed by one chapter each on essentials

of two prerequisite courses minus uid-dynamics and heat transfer and

numerical-methods The rst chapter on introduction presents an

eventful thoughtful and intuitive discussion on What is CFD and

Why to study CFD whereas How CFD works which involves lots

of details is presented separately in the second chapter The third-

chapter on essentials of uid-dynamics and heat transfer presents

physical-laws transport-mechanisms physical-law based dierential-

formulation volumetric and ux terms and mathematical formula-

tion The fourth-chapter on essentials of numerical-methods presents

nite dierence method iterative solution of the system of linear alge-

braic equation numerical dierentiation and numerical integration

The presentation for the essential of the two prerequisite courses is

customized to CFD development application and analysis

1

Introduction to Computational Fluid Dynamics Development Application and Analysis First Edition Atul Sharma copy Author 2017 Published by Athena Academic Ltd and John Wiley amp Sons Ltd

1

Introduction

The introduction of any course consists of the three basic questions

What is it Why to study How it works An eventful thoughtful

and intuitive answer to the rst two questions are presented in this

chapter The third question which involves lots of working details

on the three aspects of the Computational Fluid Dynamics (CFD)

(development application and analysis) is presented separately in

the next chapter The novelty scope and purpose of this book is

also presented at the end of this chapter

11 CFD What is it

CFD is a theoretical-method of scientic and engineering investiga-

tion concerned with the development and application of a video-

camera like tool (a software) which is used for a unied cause-and-

eect based analysis of a uid-dynamics as well as heat and mass

transfer problem

CFD is a subject where you rst learn how to develop a product

minus a software which acts like a virtual video-camera Then you learn

how to apply or use this product to generate uid-dynamics movies

Finally you learn how to analyze the detailed spatial and temporal

uid-dynamics information in the movies The analysis is done to

come up with a scientically-exciting as well as engineering-relevant

story (unied cause-and-eect study) of a uid-dynamics situation in

nature as well as industrial-applications With a continuous develop-

ment and a wider application of CFD the word uid-dynamics in

3


4 1 Introduction

the CFD has become more generic as it also corresponds to heat and

mass transfer as well as chemical reaction

Fluid-dynamics information are of two types scientic and en-

gineering Scientic-information corresponds to a structure of the

heat and uid ow due to a physical phenomenon in uid-dynamics

such as boundary-layer ow-separation wake-formation and vortex-

shedding The ow structures are obtained from a temporal variation

of ow-properties (such as velocity pressure temperature and many

other variables which characterizes a ow) called as scientically-

exciting movies Engineering-information corresponds to certain pa-

rameters which are important in an engineering application called

as engineering-parameters for example lift force drag force wall

shear stress pressure drop and rate of heat-transfer They are ob-

tained as the temporal variation of engineering-parameters called as

engineering-relevant movies The engineering-parameters are the ef-

fect which are caused by the ow structures minus their correlation leads

to the unied cause-and-eect study A uid-dynamics movie consist

of a time-wise varying series of pictures which give uid-dynamics

information Each ow-property and engineering-parameter results

in one scientically-exciting and engineering-relevant movie respec-

tively When the large number of both the types of movies are

played synchronized in-time the detailed spatial as well as tempo-

ral uid-dynamic information greatly helps in the unied cause-and-

eect study of a uid-dynamics problem

Alternatively CFD is a subject concerned with the development of

computer-programs for the computer-simulation and study of a natu-

ral or an engineering uid-dynamics system The development starts

with the virtual-development of the system involving a geometri-

cal information based computational development of a solid-model

for the system the system may be already-existing or yet-to-be-

developed physically Thereafter the development involves numerical

solution of the governing algebraic equations for CFD formulated

from the physical laws (conservation of mass momentum and en-

ergy) and initial as well as boundary conditions corresponding to the

system CFD study involves the analysis of the uid-dynamics and

heat-transfer results obtained from the computer-simulations of the

virtual-system which is subjected to certain governing-parameters

Introduction to CFD 5

111 CFD as a Scientic and Engineering Analysis Tool

For a better understanding of the denition of the CFD and its ap-

plication as a scientic and engineering analysis tool two example

problems are presented rst conduction heat transfer in a plate

and second uid ow across a circular pillar of a bridge The rst-

problem correspond to an engineering-application for electronic cool-

ing and the second-problem is studied for an engineering-design of

the structure subjected to uid-dynamics forces and are shown in

Fig 11 Both the problems are considered as two-dimensional and

unsteady However the results asymptotes from the unsteady to a

steady state in the rst-problem and to a periodic (dynamic-steady)

state in the second-problem

Figure 11 Schematic and the associated parameters for the two example prob-lems (a) thermal conduction module of a heat sink and (b) theclassical ow across a circular cylinder

6 1 Introduction

The rst-problem is taken from the classic book on heat-transfer

by Incropera and Dewitt (1996) shown in Fig 11(a) The g-

ure shows a periodic module of the plate where a constant heat-ux

qW = 10Wcm2 (dissipated from certain electronic devices) is rst con-

ducted to an aluminum plate and then convected to the water ow-

ing through the rectangular channel grooved in the plate The forced

convection results in a convection coecient of h = 5000Wm2K in-

side the channel The unsteady heat-transfer results in an interesting

time-wise varying conduction heat ow in the plate and instantaneous

convection heat transfer rate Qprimeconv (per unit width of the plate per-

pendicular to the plane shown in Fig 11(a)) into the water

The second-problem is encountered if you are standing over a

bridge and look below to the water passing over a pillar (of circu-

lar cross-section) of the bridge shown in Fig 11(b) as a general case

of 2D ow across a cylinder The uid-dynamics results in a beau-

tiful time-wise varying structure of the uid ow just downstream

of the pillar and certain engineering-parameters The parameters

correspond to the uid-dynamic forces acting in the horizontal (or

streamwise) and vertical (or transverse) direction called as the drag-

force FD and lift-force FL respectively The respective force is pre-

sented below in a non-dimensional form called as the drag-coecient

CD and lift-coecient CL equations shown in Fig 11(b)

If you use your video camera to capture a movie for the two

problems you will get series of pictures for the plate in the rst-

problem and the owing uid in the second-problem Since these

movies result in a pictorial information but not the ow-property

based uid-dynamic information such movies are not relevant for

the uid-dynamics and heat-transfer study Instead if you use a

CFD software for computer simulation you can capture various types

of movies The various ow-properties based scientically-exciting

movies are presented here as the pictures at certain time-instant in

Fig 12(a)-(d) and Fig 13(a1)-(d2) The former gure shows gray-

colored ooded contour for the temperature whereas the latter gure

shows gray-colored ooded contour for the pressure and vorticity and

vector-plot for the velocity For the gray-colored ooded contours a

color bar can be seen in the gures for the relative magnitude of the

variable These gures also show line contours of stream-function


and heat-function as streamlines and heatlines respectively They

are the visualization techniques for a 2D steady heat and uid ow

The heatlines are analogous to streamlines and represents the direc-

tion of heat ow under steady state condition proposed by Kimura

and Bejan (1983) later presented in a heat-transfer book by Bejan

(1984) The steady-state heatlines can be seen in Fig 12(d) where

the direction for the conduction-ux is tangential at any point on the

heatlines analogous to that for the mass-ux in a streamline Thus

for a better presentation of the structure of heat and uid ow the

scientically-exciting movies are not only limited to ow-properties

but also correspond to certain ow visualization techniquesAn engineering-parameter based engineering-relevant uid dy-

namic movie is presented in Fig 12(e) for the convection heat trans-

fer rate per unit width at the various walls on the channel Qprimew at the

west-wall Qprimen at the north-wall Q

primee at the east-wall and their cumu-

lative value Qprimeconv (refer Fig 11(a)) Another engineering-relevant

movie is presented in Fig 13(e) for the lift and drag coecient Note

that the movie here corresponds to the motion of a lled symbol over

a line for the temporal variation of an engineering-parameter Thus

the engineering-relevant movies are one-dimensional lower than the

scientically-exciting movies here for the 2D problems the former

movie is presented as 1D and the latter as 2D results

1111 Heat-Conduction

For the computational set-up of the heat conduction problem shown

in Fig 11(a) the results obtained from a 2D CFD simulation of

the conduction heat transfer are shown in Fig 12 For a uni-

form initial temperature of the plate as Tinfin = 15oC Fig 12(a)-(d)

shows the isotherms and heatlines in the plate as the pictures of the

scientically-exciting movies at certain discrete time instants t = 25

5 10 and 20 sec For the same discrete time instants the temporal

variation of the position of the symbols over a line plot is presented in

Fig 12(e) The gure is shown for each of the heat transfer rate and

is obtained from a engineering-relevant 1D movie There is a tempo-

ral variation of picture in the scientically-exciting movie and that

of position of symbol (along the line plot) in the engineering-relevant

movieAccording to caloric theory of the famous French scientist An-

tonie Lavoisier heat is considered as an invisible tasteless odorless

8 1 Introduction

t(sec)

Qrsquo (W

m)

0 5 10 15 20 250

200

400

600

800

1000Qrsquo

conv

Qrsquow

Qrsquon

Qrsquoe

Qrsquoin

(e)

(b)

(c)(d)

(a)

(a) t=25 sec31

29

27

25

23

21

19

17

15

T(oC)

(c) t=10 sec

(b) t=5 sec

(d) t=20 sec

Figure 12 Thermal conduction module of a heat sink temporal variation of(a)-(d) isotherms and heatlines and (e) the convection heat transferrate per unit width at the various walls of the channels and theircumulative total value Q

primeconv For the discrete time-instant corre-

sponding to heat ow patterns in (a)-(d) temporal variation of thevarious heat transfer rates are shown as symbols in (e) (a)-(c) arethe transient and (d) represents a steady state result

weightless uid which he called caloric uid Presently the caloric

theory is overtaken by mechanical theory of heat minus heat is not a sub-

stance but a dynamical form of mechanical eect (Thomson 1851)

Nevertheless there are certain problems involving heat ow for which

Lavoisiers approach is rather useful such as the discussion on heat-

lines here

The role of heatlines for heat ow is analogous to that of stream-

lines for uid ow (Paramane and Sharma 2009) For the uid-ow

the dierence of stream-function values represents the rate of uid

ow the function remains constant on a solid wall the tangent to

a streamline represents the direction of the uid-ow (with no ow

in the normal direction) and the streamline originate or emerge at


a mass source Analogously for the heat-ow the dierence of heat-

function values represents the rate of heat ow the function remains

constant on an adiabatic wall the tangent to a heatline represents the

direction of the heat-ow (with no ow in the normal direction) and

the heatline originate or emerge at the heat-source For the present

problem in Fig 11(a) the heat-source is at the inlet (or left) bound-

ary and the heat-sink is at the surface of the channel Thus the

steady-state heatlines in Fig 12(d) shows the heat ow emerging

from the inletsource and ending on the channel-wallssink Further-

more it can be seen in the gure that the heatlines which are close

to the adiabatic walls are parallel to the walls The source sink and

adiabatic-walls can be seen in Fig 11(a)

Figure 12(e) shows an asymptotic time-wise increase in the con-

vection heat transfer per unit width at the various walls of the channel

and its cumulative value Qprimeconv (= Q

primew+Q

primen+Q

primee) It can be also be seen

that the heat-transfer per unit width Qprimew at the west-wall and Q

primen at

the north-wall are larger than Qprimee at the east-wall of the channel As

time progresses the gure shows that Qprimeconv increases monotonically

and becomes equal to the inlet heat transfer rate Qprimein = 1000W un-

der steady state condition refer Fig 11(a) The dierence between

the Qprimein and Q

primeconv under the transient condition is utilized for the

sensible heating and results in the increase in the temperature of the

plate Under the steady state condition note from Fig 12(d) that

the maximum temperature is close to 32oC minus well within the per-

missible limit for the reliable and ecient operation of the electronic

device

1112 Fluid-Dynamics

For the free-stream ow across a circular cylinderpillar at a Reynolds

number of 100 the non-dimensional results obtained from a 2D CFD

simulation are shown in Fig 13 In contrast to the previous prob-

lem which reaches steady state the present problem reaches to an un-

steady periodic-ow minus both ow-patterns and engineering-parametersstarts repeating after certain time-period Thereafter for four dier-

ent increasing time instants within one time-period of the periodic-

ow the results are presented in Fig 13(a)-(d) The gure shows

the close-up view of the pictures of the scientically-exciting movies

for ow-properties (velocity pressure and vorticity) and streamlines

10 1 Introduction

Figure 13 Free-stream ow across a cylinder at a Reynolds number of100 temporal variation of (a1)-(d1) velocity-vector and vorticity-contour and (a2)-(d2) streamlines and pressure contour for avortex-shedding ow For the discrete time-instant correspondingto the uid ow patterns in (a)-(d) temporal variation of the liftand drag coecient are shown as symbols in (e) Two consecutivegures in (a)-(d) and (e)-(h) are separated by a time-interval ofone-fourth of the time-period of the periodic-ow

Introduction to



Dr Atul Sharma


Introduction to




Dr Atul Sharma

copy Author 2017




Under licence from


ISBN 978-11-1900-299-4









to

Suhas V Patankar



















v

vi















FOREWORD



























vii

viii Foreword







is no coincidence











the laws of nature




Kanpur


PREFACE





























ix

x Preface












































































xii Preface























wife and daughter

Mumbai


Contents


1

1 Introduction 3

11 CFD What is it 3


Tool 5





Analysis 23






Post-Processor 34

215 Testing 35


23 CFD Analysis 38

24 Closure 39

xiii

xiv Contents


CFD 41

31 Physical Laws 42










342 Flux-Term 58

343 Discussion 62




36 Closure 71











Parameters 105







45 Closure 116

Problems 116



119







Problems 176







Problems 228

xvi Contents











86 Closure 269






Problems 306







Method 314




Problems 329


331


Grid 333







1123 All Terms 349





Problems 359








Problems 380

References 383

Index 389

Part I

















1


1

Introduction









11 CFD What is it





transfer problem










3


4 1 Introduction




















































6 1 Introduction












































































8 1 Introduction

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the Qprimein and Q






device

1112 Fluid-Dynamics










10 1 Introduction


Dr Atul Sharma


Introduction to




Dr Atul Sharma

copy Author 2017




Under licence from


ISBN 978-11-1900-299-4









to

Suhas V Patankar



















v

vi















FOREWORD



























vii

viii Foreword







is no coincidence











the laws of nature




Kanpur


PREFACE





























ix

x Preface












































































xii Preface























wife and daughter

Mumbai


Contents


1

1 Introduction 3

11 CFD What is it 3


Tool 5





Analysis 23






Post-Processor 34

215 Testing 35


23 CFD Analysis 38

24 Closure 39

xiii

xiv Contents


CFD 41

31 Physical Laws 42










342 Flux-Term 58

343 Discussion 62




36 Closure 71











Parameters 105







45 Closure 116

Problems 116



119







Problems 176







Problems 228

xvi Contents











86 Closure 269






Problems 306







Method 314




Problems 329


331


Grid 333







1123 All Terms 349





Problems 359








Problems 380

References 383

Index 389

Part I

















1


1

Introduction









11 CFD What is it





transfer problem










3


4 1 Introduction




















































6 1 Introduction












































































8 1 Introduction

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primew+Q

primen+Q



primen at





the Qprimein and Q






device

1112 Fluid-Dynamics










10 1 Introduction



Dr Atul Sharma

copy Author 2017




Under licence from


ISBN 978-11-1900-299-4









to

Suhas V Patankar



















v

vi















FOREWORD



























vii

viii Foreword







is no coincidence











the laws of nature




Kanpur


PREFACE





























ix

x Preface












































































xii Preface























wife and daughter

Mumbai


Contents


1

1 Introduction 3

11 CFD What is it 3


Tool 5





Analysis 23






Post-Processor 34

215 Testing 35


23 CFD Analysis 38

24 Closure 39

xiii

xiv Contents


CFD 41

31 Physical Laws 42










342 Flux-Term 58

343 Discussion 62




36 Closure 71











Parameters 105







45 Closure 116

Problems 116



119







Problems 176







Problems 228

xvi Contents











86 Closure 269






Problems 306







Method 314




Problems 329


331


Grid 333







1123 All Terms 349





Problems 359








Problems 380

References 383

Index 389

Part I

















1


1

Introduction









11 CFD What is it





transfer problem










3


4 1 Introduction




















































6 1 Introduction












































































8 1 Introduction

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primen at





the Qprimein and Q






device

1112 Fluid-Dynamics










10 1 Introduction



to

Suhas V Patankar



















v

vi















FOREWORD



























vii

viii Foreword







is no coincidence











the laws of nature




Kanpur


PREFACE





























ix

x Preface












































































xii Preface























wife and daughter

Mumbai


Contents


1

1 Introduction 3

11 CFD What is it 3


Tool 5





Analysis 23






Post-Processor 34

215 Testing 35


23 CFD Analysis 38

24 Closure 39

xiii

xiv Contents


CFD 41

31 Physical Laws 42










342 Flux-Term 58

343 Discussion 62




36 Closure 71











Parameters 105







45 Closure 116

Problems 116



119







Problems 176







Problems 228

xvi Contents











86 Closure 269






Problems 306







Method 314




Problems 329


331


Grid 333







1123 All Terms 349





Problems 359








Problems 380

References 383

Index 389

Part I

















1


1

Introduction









11 CFD What is it





transfer problem










3


4 1 Introduction




















































6 1 Introduction












































































8 1 Introduction

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(b) t=5 sec

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primen+Q



primen at





the Qprimein and Q






device

1112 Fluid-Dynamics










10 1 Introduction


vi















FOREWORD



























vii

viii Foreword







is no coincidence











the laws of nature




Kanpur


PREFACE





























ix

x Preface












































































xii Preface























wife and daughter

Mumbai


Contents


1

1 Introduction 3

11 CFD What is it 3


Tool 5





Analysis 23






Post-Processor 34

215 Testing 35


23 CFD Analysis 38

24 Closure 39

xiii

xiv Contents


CFD 41

31 Physical Laws 42










342 Flux-Term 58

343 Discussion 62




36 Closure 71











Parameters 105







45 Closure 116

Problems 116



119







Problems 176







Problems 228

xvi Contents











86 Closure 269






Problems 306







Method 314




Problems 329


331


Grid 333







1123 All Terms 349





Problems 359








Problems 380

References 383

Index 389

Part I

















1


1

Introduction









11 CFD What is it





transfer problem










3


4 1 Introduction




















































6 1 Introduction












































































8 1 Introduction

t(sec)

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(b) t=5 sec

(d) t=20 sec









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primew+Q

primen+Q



primen at





the Qprimein and Q






device

1112 Fluid-Dynamics










10 1 Introduction


FOREWORD



























vii

viii Foreword







is no coincidence











the laws of nature




Kanpur


PREFACE





























ix

x Preface












































































xii Preface























wife and daughter

Mumbai


Contents


1

1 Introduction 3

11 CFD What is it 3


Tool 5





Analysis 23






Post-Processor 34

215 Testing 35


23 CFD Analysis 38

24 Closure 39

xiii

xiv Contents


CFD 41

31 Physical Laws 42










342 Flux-Term 58

343 Discussion 62




36 Closure 71











Parameters 105







45 Closure 116

Problems 116



119







Problems 176







Problems 228

xvi Contents











86 Closure 269






Problems 306







Method 314




Problems 329


331


Grid 333







1123 All Terms 349





Problems 359








Problems 380

References 383

Index 389

Part I

















1


1

Introduction









11 CFD What is it





transfer problem










3


4 1 Introduction




















































6 1 Introduction












































































8 1 Introduction

t(sec)

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15

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(c) t=10 sec

(b) t=5 sec

(d) t=20 sec









lines here























primew+Q

primen+Q



primen at





the Qprimein and Q






device

1112 Fluid-Dynamics










10 1 Introduction


viii Foreword







is no coincidence











the laws of nature




Kanpur


PREFACE





























ix

x Preface












































































xii Preface























wife and daughter

Mumbai


Contents


1

1 Introduction 3

11 CFD What is it 3


Tool 5





Analysis 23






Post-Processor 34

215 Testing 35


23 CFD Analysis 38

24 Closure 39

xiii

xiv Contents


CFD 41

31 Physical Laws 42










342 Flux-Term 58

343 Discussion 62




36 Closure 71











Parameters 105







45 Closure 116

Problems 116



119







Problems 176







Problems 228

xvi Contents











86 Closure 269






Problems 306







Method 314




Problems 329


331


Grid 333







1123 All Terms 349





Problems 359








Problems 380

References 383

Index 389

Part I

















1


1

Introduction









11 CFD What is it





transfer problem










3


4 1 Introduction




















































6 1 Introduction












































































8 1 Introduction

t(sec)

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25

23

21

19

17

15

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(c) t=10 sec

(b) t=5 sec

(d) t=20 sec









lines here























primew+Q

primen+Q



primen at





the Qprimein and Q






device

1112 Fluid-Dynamics










10 1 Introduction


PREFACE





























ix

x Preface












































































xii Preface























wife and daughter

Mumbai


Contents


1

1 Introduction 3

11 CFD What is it 3


Tool 5





Analysis 23






Post-Processor 34

215 Testing 35


23 CFD Analysis 38

24 Closure 39

xiii

xiv Contents


CFD 41

31 Physical Laws 42










342 Flux-Term 58

343 Discussion 62




36 Closure 71











Parameters 105







45 Closure 116

Problems 116



119







Problems 176







Problems 228

xvi Contents











86 Closure 269






Problems 306







Method 314




Problems 329


331


Grid 333







1123 All Terms 349





Problems 359








Problems 380

References 383

Index 389

Part I

















1


1

Introduction









11 CFD What is it





transfer problem










3


4 1 Introduction




















































6 1 Introduction












































































8 1 Introduction

t(sec)

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21

19

17

15

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(c) t=10 sec

(b) t=5 sec

(d) t=20 sec









lines here























primew+Q

primen+Q



primen at





the Qprimein and Q






device

1112 Fluid-Dynamics










10 1 Introduction


x Preface












































































xii Preface























wife and daughter

Mumbai


Contents


1

1 Introduction 3

11 CFD What is it 3


Tool 5





Analysis 23






Post-Processor 34

215 Testing 35


23 CFD Analysis 38

24 Closure 39

xiii

xiv Contents


CFD 41

31 Physical Laws 42










342 Flux-Term 58

343 Discussion 62




36 Closure 71











Parameters 105







45 Closure 116

Problems 116



119







Problems 176







Problems 228

xvi Contents











86 Closure 269






Problems 306







Method 314




Problems 329


331


Grid 333







1123 All Terms 349





Problems 359








Problems 380

References 383

Index 389

Part I

















1


1

Introduction









11 CFD What is it





transfer problem










3


4 1 Introduction




















































6 1 Introduction












































































8 1 Introduction

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(b) t=5 sec

(d) t=20 sec









lines here























primew+Q

primen+Q



primen at





the Qprimein and Q






device

1112 Fluid-Dynamics










10 1 Introduction








































xii Preface























wife and daughter

Mumbai


Contents


1

1 Introduction 3

11 CFD What is it 3


Tool 5





Analysis 23






Post-Processor 34

215 Testing 35


23 CFD Analysis 38

24 Closure 39

xiii

xiv Contents


CFD 41

31 Physical Laws 42










342 Flux-Term 58

343 Discussion 62




36 Closure 71











Parameters 105







45 Closure 116

Problems 116



119







Problems 176







Problems 228

xvi Contents











86 Closure 269






Problems 306







Method 314




Problems 329


331


Grid 333







1123 All Terms 349





Problems 359








Problems 380

References 383

Index 389

Part I

















1


1

Introduction









11 CFD What is it





transfer problem










3


4 1 Introduction




















































6 1 Introduction












































































8 1 Introduction

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17

15

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(c) t=10 sec

(b) t=5 sec

(d) t=20 sec









lines here























primew+Q

primen+Q



primen at





the Qprimein and Q






device

1112 Fluid-Dynamics










10 1 Introduction


Contents


1

1 Introduction 3

11 CFD What is it 3


Tool 5





Analysis 23






Post-Processor 34

215 Testing 35


23 CFD Analysis 38

24 Closure 39

xiii

xiv Contents


CFD 41

31 Physical Laws 42










342 Flux-Term 58

343 Discussion 62




36 Closure 71











Parameters 105







45 Closure 116

Problems 116



119







Problems 176







Problems 228

xvi Contents











86 Closure 269






Problems 306







Method 314




Problems 329


331


Grid 333







1123 All Terms 349





Problems 359








Problems 380

References 383

Index 389

Part I

















1


1

Introduction









11 CFD What is it





transfer problem










3


4 1 Introduction




















































6 1 Introduction












































































8 1 Introduction

t(sec)

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15

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(c) t=10 sec

(b) t=5 sec

(d) t=20 sec









lines here























primew+Q

primen+Q



primen at





the Qprimein and Q






device

1112 Fluid-Dynamics










10 1 Introduction


xiv Contents


CFD 41

31 Physical Laws 42










342 Flux-Term 58

343 Discussion 62




36 Closure 71











Parameters 105







45 Closure 116

Problems 116



119







Problems 176







Problems 228

xvi Contents











86 Closure 269






Problems 306







Method 314




Problems 329


331


Grid 333







1123 All Terms 349





Problems 359








Problems 380

References 383

Index 389

Part I

















1


1

Introduction









11 CFD What is it





transfer problem










3


4 1 Introduction




















































6 1 Introduction












































































8 1 Introduction

t(sec)

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(c) t=10 sec

(b) t=5 sec

(d) t=20 sec









lines here























primew+Q

primen+Q



primen at





the Qprimein and Q






device

1112 Fluid-Dynamics










10 1 Introduction




119







Problems 176







Problems 228

xvi Contents











86 Closure 269






Problems 306







Method 314




Problems 329


331


Grid 333







1123 All Terms 349





Problems 359








Problems 380

References 383

Index 389

Part I

















1


1

Introduction









11 CFD What is it





transfer problem










3


4 1 Introduction




















































6 1 Introduction












































































8 1 Introduction

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(b) t=5 sec

(d) t=20 sec









lines here























primew+Q

primen+Q



primen at





the Qprimein and Q






device

1112 Fluid-Dynamics










10 1 Introduction


xvi Contents











86 Closure 269






Problems 306







Method 314




Problems 329


331


Grid 333







1123 All Terms 349





Problems 359








Problems 380

References 383

Index 389

Part I

















1


1

Introduction









11 CFD What is it





transfer problem










3


4 1 Introduction




















































6 1 Introduction












































































8 1 Introduction

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(b) t=5 sec

(d) t=20 sec









lines here























primew+Q

primen+Q



primen at





the Qprimein and Q






device

1112 Fluid-Dynamics










10 1 Introduction








Method 314




Problems 329


331


Grid 333







1123 All Terms 349





Problems 359








Problems 380

References 383

Index 389

Part I

















1


1

Introduction









11 CFD What is it





transfer problem










3


4 1 Introduction




















































6 1 Introduction












































































8 1 Introduction

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(c) t=10 sec

(b) t=5 sec

(d) t=20 sec









lines here























primew+Q

primen+Q



primen at





the Qprimein and Q






device

1112 Fluid-Dynamics










10 1 Introduction









Problems 380

References 383

Index 389

Part I

















1


1

Introduction









11 CFD What is it





transfer problem










3


4 1 Introduction




















































6 1 Introduction












































































8 1 Introduction

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(d) t=20 sec









lines here























primew+Q

primen+Q



primen at





the Qprimein and Q






device

1112 Fluid-Dynamics










10 1 Introduction


Part I

















1


1

Introduction









11 CFD What is it





transfer problem










3


4 1 Introduction




















































6 1 Introduction












































































8 1 Introduction

t(sec)

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1112 Fluid-Dynamics










10 1 Introduction


1

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11 CFD What is it





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3


4 1 Introduction




















































6 1 Introduction












































































8 1 Introduction

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1112 Fluid-Dynamics










10 1 Introduction


4 1 Introduction




















































6 1 Introduction












































































8 1 Introduction

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1112 Fluid-Dynamics










10 1 Introduction
















6 1 Introduction












































































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1112 Fluid-Dynamics










10 1 Introduction








































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the Qprimein and Q






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1112 Fluid-Dynamics










10 1 Introduction


















primew+Q

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primen at





the Qprimein and Q






device

1112 Fluid-Dynamics










10 1 Introduction


10 1 Introduction


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