i | F o r m o r e i n f o . r e f e r : w w w . g a t e a e r o s p a c e f o r u m . c o m

Aerodynamics Third Edition

Ashish Garg (M.E. IISc, Bangalore)

GATE Aerospace Forum Educational Services

ii | F o r m o r e i n f o . r e f e r : w w w . g a t e a e r o s p a c e f o r u m . c o m

Aerodynamics Ashish Garg All right reserved. No part of this publication may be reproduced or transmitted in any form or by any means electronic or mechanical, including photocopy, recording or any information storage and retrieval system- without permission in writing from GATE Aerospace Forum Educational Services. © 2015 by Author This book is manufactured in India and is authorized for sale only by GATE Aerospace Forum

Edutional Services.

First Edition : 2013

Second Edition : 2014

Third Edition : 2015

iii | F o r m o r e i n f o . r e f e r : w w w . g a t e a e r o s p a c e f o r u m . c o m

Dedicated to

My Grandparents

(whose blessings and love have always been with me)

My Teachers and Parents

(who have made me capable of what I am today)

Scientists

(who work throughout their lives in contributing theories to mankind worth applying,

feeling, reading and writing again and again)

My Friends and Colleagues

(whose support enabled me to complete it)

My Students

(who have been my teachers and suggested me to start the project)

iv | F o r m o r e i n f o . r e f e r : w w w . g a t e a e r o s p a c e f o r u m . c o m

Preface

This book is an attempt to explain fundamental laws of fluids in a simple language as well as

to create a compendium of the concepts of Aerodynamics. It has been principally designed

and documented keeping in mind the course structure and understanding level of students

majoring Aeronautical or Aerospace Engineering in India. This book will definitely

complement their specialist reading of Aerodynamics.

Salient features of the book are as follows :

Includes all important definitions and detailed mathematical explanations.

Derivations are followed by various numerical examples for better understanding.

Covers GATE (AE) Aerodynamics Syllabus.

Covers previous year¡¦s questions of GATE (AE) Aerodynamics along with their

solutions.

This 3rd

edition also includes Simitude, Dynamics of Sport balls, Birds and Fish, more

solved Examples and extended Exercises Questions compare to last edition.

I would like to express my deepest gratitude for my Professor Dr. Joseph Mathew for his

invaluable guidance, continuous support and motivation and Professor Dr. Praveen

Chandrashekar for his precious teachings including computational techniques which can be

beautifully applied to frontier problems in fluid studies. I would like to thank TIFR-CAM and

IISc, Bangalore for providing me an excellent working environment and my seniors who

have been a constant source of inspiration and support for me.

I express sincere and heartfelt thanks to my student Sonakshi Shyam Arora for helping me in

preparing this book.

I would express my deepest gratitude for my parents and other family members for their

whole-hearted affection, inspiration, constant encouragement and support.

Last but not the least, I am very much thankful to some of the greatest prodigies in the field

of Aerospace, Shri J.R.D. Tata (Founder - IISc), Dr. Satish Dawan (Father of Experimental

Fluid Dynamics), Dr. A.P.J. Abdul Kalam (Former President of India), Kalpana Chawla

(Indian Astronaut - NASA), Prof. Roddam Narasimha (Director of - JNCASR) for being an

unmatched inspiration for thousands of scholars like me in Aerospace Engineering.

v | F o r m o r e i n f o . r e f e r : w w w . g a t e a e r o s p a c e f o r u m . c o m

Table of Contents

Chapter 1: Basics of Fluids .......................................................................................... 1

Basic Flow Analysis Techniques ........................................................................................................... 1

Computational ............................................................................................................................. 1

Experimental ................................................................................................................................ 1

Continuity Equation (Mass Conservation) ................................................................................... 1

Types of Fluid flow .........................................................................................................................12

Substantial Derivative ....................................................................................................................14

Streamline, Streakline, Pathline ....................................................................................................15

Streamline ..................................................................................................................................15

Pathline .......................................................................................................................................15

Streakline ....................................................................................................................................15

Newton’s Law of viscosity..............................................................................................................19

Shear Stresses in Non-Newtonian Fluids ......................................................................................20

Time Independent Fluids ...........................................................................................................20

Time dependent Fluids...............................................................................................................21

Visco Elastic Fluids .....................................................................................................................21

Conservation of Momentum Equation(Differential form) ..........................................................21

Circulation .......................................................................................................................................25

Kelvin Theorem ...............................................................................................................................27

Bernoulli’s Equation .......................................................................................................................30

Case 1: Steady irrotational flow...............................................................................................30

Case 2: Steady but rotational flow:- .........................................................................................31

Helmholtz Theorem ........................................................................................................................36

Biot – Savart’s Law .........................................................................................................................41

Energy Equation .............................................................................................................................49

Simitude ..........................................................................................................................................52

Laws of Similarities ........................................................................................................................52

Geometric Similarity ..................................................................................................................52

Kinematic Similarity ...................................................................................................................52

Dynamic Similarity .....................................................................................................................53

Force ratios – Dimensionless Numbers.........................................................................................54

The inertial Force ratio ..............................................................................................................54

vi | F o r m o r e i n f o . r e f e r : w w w . g a t e a e r o s p a c e f o r u m . c o m

The inertia - Viscous Force ratio – Reynold’s Number ............................................................54

The Inertia - Gravity Force ratio – Froude Number .................................................................55

The Inertia – Pressure Force ratio – Euler Number .................................................................55

The Inertia – Elasticity Force ratio – Mach Number ................................................................55

The Inertia – Surface Tension Force ratio – Weber Number ..................................................56

Concept of Dynamic Similarity ......................................................................................................56

Reynolds Model Law ..................................................................................................................56

Froude Model Law .....................................................................................................................57

Euler Model Law ........................................................................................................................57

Mach Model Law .......................................................................................................................57

Weber Model Law ......................................................................................................................57

Exercise Questions .........................................................................................................................60

Chapter 2: Airfoil Nomenclature ......................................................................................65

NACA Airfoils Nomenclature .........................................................................................................66

Four-digit series..........................................................................................................................66

Five-digit series ..........................................................................................................................66

Modified 4 Digit and 5 Digit Series ...........................................................................................67

1-series ........................................................................................................................................67

6-series ........................................................................................................................................67

7-series ........................................................................................................................................68

8-series ........................................................................................................................................68

Objective Questions ...................................................................................................................69

Chapter 3: Potential Function .......................................................................................78

Stream function(𝜓)........................................................................................................................78

Potential Function𝜙 .......................................................................................................................79

Types of flows .................................................................................................................................81

Uniform Flow ..............................................................................................................................81

Uniform Flow inclined at an angle 𝛼 ........................................................................................81

Point Source(+) or Sink(-) ...........................................................................................................82

Vortex Flow.................................................................................................................................84

Combination of Elementary flows.................................................................................................85

Half Rankine Oval ......................................................................................................................85

Full Rankine Oval.......................................................................................................................89

Source – Sink Doublet ................................................................................................................90

vii | F o r m o r e i n f o . r e f e r : w w w . g a t e a e r o s p a c e f o r u m . c o m

Vortex Doublet ..........................................................................................................................92

Stationary or Non-rotating Cylinder (Uniform flow + Source-sink Doublet ...........................94

Rotating Cylinder (Uniform flow + Source-sink Doublet + Vortex) .........................................99

Corner Flows ............................................................................................................................ 110

Spiral Flows.............................................................................................................................. 113

Von Karman Vortex Street ..................................................................................................... 114

Blasius Theorem .......................................................................................................................... 115

Lift and Drag on Stationary cylinder .......................................................................................... 118

Lift and Drag on Rotating cylinder ............................................................................................. 119

Relation between Acceleration and Velocity potential ............................................................ 123

Method of Images ....................................................................................................................... 124

For Plane Boundaries .............................................................................................................. 124

For Spherical and Curvilinear Boundaries ............................................................................ 129

Transformation and Mapping .................................................................................................... 132

Kutta – Zhukovsky Transformation ............................................................................................ 133

To transform a circle into a straight line ............................................................................... 134

To transform a circle into an elliptical airfoil ........................................................................ 134

To transform a circle into a symmetrical airfoil profile........................................................ 136

To transform a circle into a cambered airfoil profile ........................................................... 137

To transform a circle into a circular arc ................................................................................ 139

Modified Kutta – Zhukovsky Transformation........................................................................... 141

Exercise Questions ...................................................................................................................... 144

Chapter 4 : Laminar Viscous Flows .................................................................................... 157

Internal Viscous Flows ............................................................................................................... 159

General 2d flow , between two infinite parallel plates ............................................................ 159

Couette’s Flow ............................................................................................................................. 162

Plane - Poiseuille’s Flow ............................................................................................................. 167

Different Boundary condition on plates.................................................................................... 169

Axisymmetric - Poisueilli’s Flow ................................................................................................. 171

Velocity profile of fluid between two drums moving in tangential direction ......................... 175

Exercise Questions ...................................................................................................................... 184

Chapter 5 : Boundary Layer .................................................................................... 192

Boundary layer thickness ............................................................................................................ 194

Displacement Thickness .............................................................................................................. 194

viii | F o r m o r e i n f o . r e f e r : w w w . g a t e a e r o s p a c e f o r u m . c o m

Momentum Thickness ................................................................................................................. 196

Von Karmann’s Momentum Integral Equation ......................................................................... 199

For Boundary layer over a flat plate ...................................................................................... 202

Turbulent Boundary Layer .......................................................................................................... 209

Commonly used boundary layer control method to prevent separation ................................ 212

Cricket Ball Dynamics .................................................................................................................. 212

Tennis Ball Dynamics .................................................................................................................. 214

Baseball Dynamics ...................................................................................................................... 215

Exercise Questions ...................................................................................................................... 216

Chapter 6: Gas Dynamics ................................................................................. 226

Laws of Thermodynamics ........................................................................................................... 227

0th Law ......................................................................................................................... 227

1st Law ......................................................................................................................... 227

2nd Law ........................................................................................................................ 227

3rd Law ......................................................................................................................... 228

Compressibility(𝜏) ....................................................................................................................... 230

Speed of Sound ............................................................................................................................ 230

Energy Conservation ................................................................................................................... 231

Shocks........................................................................................................................................... 235

Static Normal Shock .................................................................................................................... 236

Moving Normal Shocks ............................................................................................................... 243

Oblique Shocks ............................................................................................................................ 245

Multiple Shock Waves ................................................................................................................. 253

The Detached Shock Wave ......................................................................................................... 256

Expansion Waves ........................................................................................................................ 256

Nozzles and Nozzle Flows ........................................................................................................... 258

Chocked Flow .......................................................................................................................... 260

Nozzle performance Characteristics .......................................................................................... 266

Nozzle Coefficients .................................................................................................................. 266

Velocity Coefficient ................................................................................................................. 267

Angularity Coefficient ............................................................................................................. 267

Diffusers ....................................................................................................................................... 268

Exercise Questions ...................................................................................................................... 275

Chapter – 7: ..................................................................................................................................... 279

ix | F o r m o r e i n f o . r e f e r : w w w . g a t e a e r o s p a c e f o r u m . c o m

Thin Airfoil & Finite Wing Theory ................................................................................................... 279

Starting Vortex ............................................................................................................................ 279

Kutta’s Condition ......................................................................................................................... 280

Thin Airfoil Theory ....................................................................................................................... 281

Thin Cambered Airfoil ................................................................................................................. 286

Prandtl’s Finite Wing Theory ..................................................................................................... 296

Special Case : Elliptical Lift distribution over finite wing .......................................................... 298

Effect of Aspect Ratio .................................................................................................................. 303

High Lift Devices .......................................................................................................................... 309

Flight of Birds ............................................................................................................................... 313

Locomotion of Fish ...................................................................................................................... 313

Exercise Questions ...................................................................................................................... 314

Chapter – 8: Linearized Theory..................................................................................... 317

Linearized Velocity Potential Equation ...................................................................................... 320

Linearized pressure coefficient ................................................................................................... 321

Prandtl – Glauert Compressibility Correction............................................................................ 323

Subsonic flow over a Wavy Wall ................................................................................................ 326

Linearized Supersonic Pressure Coefficient ............................................................................... 328

Supersonic flow over Diamond Airfoil ....................................................................................... 333

Case 1 (𝛼 = 0 ) ....................................................................................................................... 333

Case 2 (𝛼 < 𝛿) ....................................................................................................................... 334

Case 3 (𝛼 = 𝛿) ....................................................................................................................... 336

Case 4 (𝛼 > 𝛿) ....................................................................................................................... 338

Linearized Pressure Coefficient from oblique – shock relation ................................................ 339

Exercise Questions ...................................................................................................................... 341

Chapter 9: Wind Tunnel Testing ................................................................................... 343

Pressure Measuement ................................................................................................................ 343

Pitot Tube ................................................................................................................................ 345

Flow Rate Measurement ............................................................................................................ 347

Venturi meter .......................................................................................................................... 348

Orifice meter ........................................................................................................................... 349

Temperature Measurement ....................................................................................................... 351

Velocity Measurement ................................................................................................................ 354

Dot Matrix. .............................................................................................................................. 355

x | F o r m o r e i n f o . r e f e r : w w w . g a t e a e r o s p a c e f o r u m . c o m

Hot Wire Anemometry ........................................................................................................... 355

Laser Doppler Velocimetry ..................................................................................................... 357

Particle Image Velocimetry .................................................................................................... 358

Density Measurement................................................................................................................. 359

Schlieren Method .................................................................................................................... 363

Shadowgraph Technique ........................................................................................................ 365

The Interferometer.................................................................................................................. 367

Exercise Questions ...................................................................................................................... 370

Questions for Practice .................................................................................................................... 380

Bibliography ..................................................................................................................................... 392

Appendix A: Glauert Integral Derivation ....................................................................................... 394

Appendix B: Prandtl Meyer Function ............................................................................................ 397

Appendix C: Airfoil Integrals ........................................................................................................... 400