Theoretical and Computational Aerodynamics

Aerodynamics has seen many developments due to the growth of scientific computing, which has caused the design cycle time of aerospace vehicles to be heavily reduced. Today computational aerodynamics appears in the preliminary step of a new design, relegating costly, time–consuming wind tunnel testing to the final stages of design.

Theoretical and Computational Aerodynamics is aimed to be a comprehensive textbook, covering classical aerodynamic theories and recent applications made possible by computational aerodynamics. It starts with a discussion on lift and drag from an overall dynamical approach, and after stating the governing Navier–Stokes equation, covers potential flows and panel method. Low aspect ratio and delta wings (including vortex breakdown) are also discussed in detail, and after introducing boundary layer theory, computational aerodynamics is covered for DNS and LES. Other topics covered are on flow transition to analyse NLF airfoils, bypass transition, streamwise and cross–flow instability over swept wings, viscous transonic flow over airfoils, low Reynolds number aerodynamics, high lift devices and flow control.

Key features:

Blends classical theories of incompressible aerodynamics to panel methods Covers lifting surface theories and low aspect ratio wing and wing–body aerodynamics Presents computational aerodynamics from first principles for incompressible and compressible flows Covers unsteady and low Reynolds number aerodynamics Includes an up–to–date account of DNS of airfoil aerodynamics including flow transition for NLF airfoils Contains chapter problems and illustrative examples Accompanied by a website hosting problems and a solution manual

Theoretical and Computational Aerodynamics is an ideal textbook for undergraduate and graduate students, and is also aimed to be a useful resource book on aerodynamics for researchers and practitioners in the research labs and the industry.

Series Preface xv

Preface xvii

Acknowledgements xxi

1 Introduction to Aerodynamics and Atmosphere 1

1.1 Motivation and Scope of Aerodynamics 1

1.2 Conservation Principles 4

1.2.1 Conservation Laws and Reynolds Transport Theorem (RTT)4

1.2.2 Application of RTT: Conservation of Linear Momentum 6

1.3 Origin of Aerodynamic Forces 6

1.3.1 Momentum Integral Theory: Real Fluid Flow 8

1.4 Flow in Accelerating Control Volumes: Application of RTT9

1.5 Atmosphere and Its Role in Aerodynamics 11

1.5.1 Von K´arm´an Line 11

1.5.2 Structure of Atmosphere 11

1.5.3 Armstrong Line or Limit 12

1.5.4 International Standard Atmosphere (ISA) and OtherAtmospheric Details 13

1.5.5 Property Variations in Troposphere and Stratosphere 15

1.6 Static Stability of Atmosphere 17

Bibliography 20

2 Basic Equations of Motion 21

2.1 Introduction 21

2.1.1 Compressibility of Fluid Flow 22

2.2 Conservation Principles 23

2.2.1 Flow Description Method: Eulerian and LagrangianApproaches 23

2.2.2 The Continuity Equation: Mass Conservation 24

2.3 Conservation of Linear Momentum: Integral Form 25

2.4 Conservation of Linear Momentum: Differential Form 26

2.4.1 General Stress System in a Deformable Body 26

2.5 Strain Rate of Fluid Element in Flows 28

2.5.1 Kinematic Interpretation of Strain Tensor 29

2.6 Relation between Stress and Rate of Strain Tensors in FluidFlow 32

2.7 Circulation and Rotationality in Flows 35

2.8 Irrotational Flows and Velocity Potential 36

2.9 Stream Function and Vector Potential 37

2.10 Governing Equation for Irrotational Flows 38

2.11 Kelvin s Theorem and Irrotationality 40

2.12 Bernoulli s Equation: Relation of Pressure andVelocity 41

2.13 Applications of Bernoulli s Equation: Air SpeedIndicator 42

2.13.1 Aircraft Speed Measurement 43

2.13.2 The Pressure Coefficient 44

2.13.3 Compressibility Correction for Air Speed Indicator 44

2.14 Viscous Effects and Boundary Layers 46

2.15 Thermodynamics and Reynolds Transport Theorem 47

2.16 Reynolds Transport Theorem 48

2.17 The Energy Equation 49

2.17.1 The Steady Flow Energy Equation 51

2.18 Energy Conservation Equation 52

2.19 Alternate Forms of Energy Equation 54

2.20 The Energy Equation in Conservation Form 55

2.21 Strong Conservation and Weak Conservation Forms 55

2.22 Second Law of Thermodynamics and Entropy 56

2.23 Propagation of Sound and Mach Number 60

2.24 One–Dimensional Steady Flow 61

2.25 Normal Shock Relation for Steady Flow 62

2.26 Rankine––Hugoniot Relation 64

2.27 Prandtl or Meyer Relation 65

2.28 Oblique ShockWaves 69

2.29 Weak Oblique Shock 71

2.30 Expansion of Supersonic Flows 74

Bibliography 76

3 Theoretical Aerodynamics of Potential Flows 77

3.1 Introduction 77

3.2 Preliminaries of Complex Analysis for 2D IrrotationalFlows:

Cauchy––Riemann Relations 78

3.2.1 Cauchy s Residue Theorem 81

3.2.2 Complex Potential and Complex Velocity 81

3.3 Elementary Singularities in Fluid Flows 81

3.3.1 Superposing Solutions of Irrotational Flows 83

3.4 Blasius Theorem: Forces and Moment for PotentialFlows 90

3.4.1 Force Acting on a Vortex in a Uniform Flow 92

3.4.2 Flow Past a Translating and Rotating Cylinder: LiftGeneration

Mechanism 94

3.4.3 Prandtl s Limit on Maximum Circulation and itsViolation 97

3.4.4 Pressure Distribution on Spinning and Translating Cylinder98

3.5 Method of Images 99

3.6 Conformal Mapping: Use of Cauchy––Riemann Relation 101

3.6.1 Laplacian in the Transformed Plane 102

3.6.2 Relation between Complex Velocity in Two Planes 104

3.6.3 Application of Conformal Transformation 104

3.7 Lift Created by Jukowski Airfoil 111

3.7.1 Kutta Condition and Circulation Generation 113

3.7.2 Lift on Jukowski Airfoil 114

3.7.3 Velocity and Pressure Distribution on Jukowski Airfoil116

3.8 Thin Airfoil Theory 116

3.8.1 Thin Symmetric Flat Plate Airfoil 119

3.8.2 Aerodynamic Centre and Centre of Pressure 122

3.8.3 The Circular Arc Airfoil 124

3.9 General Thin Airfoil Theory 129

3.10 Theodorsen Condition for General Thin Airfoil Theory134

Bibliography 135

4 FiniteWing Theory 137

4.1 Introduction 137

4.2 Fundamental Laws of Vortex Motion 137

4.3 Helmholtz s Theorems of Vortex Motion 138

4.4 The Bound Vortex Element 140

4.5 Starting Vortex Element 140

4.6 Trailing Vortex Element 141

4.7 Horse Shoe Vortex 142

4.8 The Biot–Savart Law 142

4.8.1 Biot–Savart Law for Simplified Cases 144

4.9 Theory for a Finite Wing 146

4.9.1 Relation between Spanwise Loading and Trailing Vortices146

4.10 Consequence of Downwash: Induced Drag 147

4.11 Simple Symmetric Loading: Elliptic Distribution 149

4.11.1 Induced Drag for Elliptic Loading 151

4.11.2 Modified Elliptic Load Distribution 152

4.11.3 The Downwash for Modified Elliptic Loading 153

4.12 General Loading on a Wing 154

4.12.1 Downwash for General Loading 155

4.12.2 Induced Drag on a Finite Wing for General Loading 156

4.12.3 Load Distribution for Minimum Drag 157

4.13 Asymmetric Loading: Rolling and Yawing Moment 157

4.13.1 Rolling Moment () 157

4.13.2 Yawing Moment (N) 159

4.13.3 Effect of Aspect Ratio on Lift Curve Slope 159

4.14 Simplified Horse Shoe Vortex 161

4.15 Applications of Simplified Horse Shoe Vortex System 162

4.15.1 Influence of Downwash on Tailplane 162

4.15.2 Formation–flight of Birds 163

4.15.3 Wing–in–Ground Effect 165

4.16 Prandtl s Lifting Line Equation or the MonoplaneEquation 167

Bibliography 169

5 Panel Methods 171

5.1 Introduction 171

5.2 Line Source Distribution 172

5.2.1 Perturbation Velocity Components due to SourceDistribution 174

5.3 Panel Method due to Hess and Smith 176

5.3.1 Calculation of Influence Coefficients 180

5.4 Some Typical Results 183

Bibliography 188

6 Lifting Surface, Slender Wing and Low Aspect RatioWingTheories 189

6.1 Introduction 189

6.2 Green s Theorems and Their Applications to PotentialFlows 190

6.2.1 Reciprocal Theorem 192

6.3 Irrotational External Flow Field due to a Lifting Surface192

6.3.1 Large Aspect Ratio Wings 197

6.3.2 Wings of Small Aspect Ratio 199

6.4 Slender Wing Theory 201

6.5 Spanwise Loading 205

6.6 Lift on Delta or TriangularWing 206

6.6.1 Low Aspect Ratio Wing Aerodynamics and Vortex Lift 207

6.7 Vortex Breakdown 214

6.7.1 Types of Vortex Breakdown 216

6.8 Slender Body Theory 218

Bibliography 221

7 Boundary Layer Theory 223

7.1 Introduction 223

7.2 Regular and Singular Perturbation Problems in Fluid Flows224

7.3 Boundary Layer Equations 225

7.3.1 Conservation of Mass 226

7.3.2 The –Momentum Equation 226

7.3.3 The –Momentum Equation 227

7.3.4 Use of Boundary Layer Equations 229

7.4 Boundary Layer Thicknesses 230

7.4.1 Boundary Layer Displacement Thickness 231

7.4.2 Boundary Layer Momentum Thickness 232

7.5 Momentum Integral Equation 233

7.6 Validity of Boundary Layer Equation and Separation 235

7.7 Solution of Boundary Layer Equation 237

7.8 Similarity Analysis 238

7.8.1 Zero Pressure Gradient Boundary Layer or Blasius Profile243

7.8.2 Stagnation Point or the Hiemenz Flow 244

7.8.3 Flat Plate Wake at Zero Angle of Attack 245

7.8.4 Two–dimensional Laminar Jet 247

7.8.5 Laminar Mixing Layer 250

7.9 Use of Boundary Layer Equation in Aerodynamics 252

7.9.1 Differential Formulation of Boundary Layer Equation253

7.9.2 Use of Momentum Integral Equation 254

7.9.3 Pohlhausen s Method 254

7.9.4 Thwaite s Method 257

Bibliography 258

8 Computational Aerodynamics 259

8.1 Introduction 259

8.2 A Model Dynamical Equation 260

8.3 Space––Time Resolution of Flows 263

8.3.1 Spatial Scales in Turbulent Flows and Direct NumericalSimulation 264

8.3.2 Computing Unsteady Flows: Dispersion Relation Preserving(DRP)

Methods 265

8.3.3 Spectral or Numerical Amplification Factor 266

8.4 An Improved Orthogonal Grid Generation Method for Aerofoil275

8.5 Orthogonal Grid Generation 279

8.5.1 Grid Generation Algorithm 281

8.6 Orthogonal Grid Generation for an Aerofoil with RoughnessElements 284

8.7 Solution of Navier––Stokes Equation for Flow Past AG24Aerofoil 287

8.7.1 Grid Smoothness vs Deviation from Orthogonality 290

Bibliography 291

9 Instability and Transition in Aerodynamics 295

9.1 Introduction 295

9.2 Temporal and Spatial Instability 298

9.3 Parallel Flow Approximation and Inviscid InstabilityTheorems 299

9.3.1 Inviscid Instability Mechanism 300

9.4 Viscous Instability of Parallel Flows 301

9.4.1 Temporal and Spatial Amplification of Disturbances 303

9.5 Instability Analysis from the Solution of theOrr––Sommerfeld Equation 304

9.5.1 Local and Total Amplification of Disturbances 306

9.5.2 Effects of the Mean Flow Pressure Gradient 308

9.5.3 Transition Prediction Based on Stability Calculation: Method 312

9.5.4 Effects of FST 314

9.5.5 Distinction between Controlled and UncontrolledExcitations 315

9.6 Transition in Three–Dimensional Flows 318

9.7 Infinite Swept Wing Flow 320

9.8 Attachment Line Flow 321

9.9 Boundary Layer Equations in the Transformed Plane 322

9.10 Simplification of Boundary Layer Equations in theTransformed Plane 324

9.11 Instability of Three–Dimensional Flows 325

9.11.1 Effects of Sweep–back and Cross Flow Instability 326

9.12 Linear Viscous Stability Theory for Three–Dimensional Flows328

9.12.1 Temporal Instability of Three–dimensional Flows 329

9.12.2 Spatial Instability of Three–dimensional Flows 330

9.13 Experimental Evidence of Instability on Swept Wings 332

9.14 Infinite Swept Wing Boundary Layer 334

9.15 Stability of the Falkner––Skan––Cooke Profile 337

9.16 StationaryWaves over Swept Geometries 340

9.17 Empirical Transition Prediction Method forThree–Dimensional Flows 340

9.17.1 Streamwise Transition Criterion 341

9.17.2 Cross Flow Transition Criteria 341

9.17.3 Leading Edge Contamination Criterion 343

Bibliography 343

10 Drag Reduction: Analysis and Design of Airfoils347

10.1 Introduction 347

10.2 Laminar Flow Airfoils 350

10.2.1 The Drag Bucket of Six–Digit Series Aerofoils 352

10.2.2 Profiling Modern Laminar Flow Aerofoils 353

10.3 Pressure Recovery of Some Low Drag Airfoils 358

10.4 Flap Operation of Airfoils for NLF 361

10.5 Effects of Roughness and Fixing Transition 362

10.6 Effects of Vortex Generator or Boundary Layer Re–Energizer364

10.7 Section Characteristics of Various Profiles 364

10.8 A High Speed NLF Aerofoil 365

10.9 Direct Simulation of Bypass Transitional Flow Past anAirfoil 369

10.9.1 Governing Equations and Formulation 370

10.9.2 Results and Discussion 371

Bibliography 378

11 Direct Numerical Simulation of 2D Transonic Flows aroundAirfoils 381

11.1 Introduction 381

11.2 Governing Equations and Boundary Conditions 382

11.3 Numerical Procedure 384

11.4 Some Typical Results 387

11.4.1 Validation of Methodologies for Compressible FlowCalculations and Shock Capturing 387

11.4.2 Computing Strong Shock Cases 396

11.4.3 Unsteadiness of Compressible Flows 396

11.4.4 Creation of Rotational Effects 396

11.4.5 Strong Shock and Entropy Gradient 401

11.4.6 Lift and Drag Calculation 404

Bibliography 406

12 Low Reynolds Number Aerodynamics 409

12.1 Introduction 409

12.2 Micro–air Vehicle Aerodynamics 412

12.3 Governing Equations in Inertial and Noninertial Frames413

12.3.1 Pressure Solver 415

12.3.2 Proof of Equation (12.17) 416

12.3.3 Distinction between Low and High Reynolds Number Flows418

12.3.4 Validation Studies of Computations 420

12.4 Flow Past an AG24 Airfoil at Low Reynolds Numbers 425

Bibliography 442

13 High Lift Devices and Flow Control 445

13.1 Introduction 445

13.1.1 High Lift Configuration 446

13.2 Passive Devices: Multi–Element Airfoils with Slats andFlaps 449

13.2.1 Optimization of Flap Placement and Settings 450

13.2.2 Aerodynamic Data of GA(W)–1 Airfoil Fitted with FowlerFlap 453

13.2.3 Physical Explanation of Multi–element Aerofoil Operation455

13.2.4 Vortex Generator 457

13.2.5 Induced Drag and Its Alleviation 461

13.2.6 Theoretical Analysis of Induced Drag 463

13.2.7 Fuselage Drag Reduction 464

13.2.8 Instability of Flow over Nacelle 465

13.3 Flow Control by Plasma Actuation: High Lift Device and DragReduction 465

13.3.1 Control of Bypass Transitional Flow Past an Aerofoil byPlasma Actuation 466

13.4 Governing Equations for Plasma 468

13.4.1 Suzen et al. s Model 470

13.4.2 Orlov s Model 471

13.4.3 Spatio–temporal Lumped–element Circuit Model 472

13.4.4 Algorithm for Calculating Body Force 474

13.4.5 Lemire and Vo s Model 474

13.5 Governing Fluid Dynamic Equations 475

13.6 Results and Discussions 476

Bibliography 484

Index 487

Prof. Sengupta received his basic aeronautical/aerospace education from IIT Kharagpur, IISc Bangalore and Georgia Tech., Atlanta, USA. He has worked in various research organizations and educational institutes at NAL Bangalore, India; Univ. of Cambridge, U.K.; National University of Singapore, Singapore and IIT Kanpur, where he currently holds the PR Dwivedi Chair apart from leading HPCL, IIT Kanpur. His research interests span across fields of scientific and high performance computing; fundamental fluid mechanics and aerodynamics; transition and turbulence. His research teams have refined and expanded areas of scientific computing, HPC, receptivity/instability, transition and turbulence. His interests in fundamental aspects of aerodynamics have resulted in this book containing classical theoretical analyses and newer topics of transonic aerodynamics; natural laminar flow airfoil analysis and design; low Reynolds number aerodynamics; flow control to delay transition and separation. These later topics are outcome of his contributions in direct numerical simulation (DNS) and large eddy simulation (LES).

The book is aimed to be a comprehensive textbook : the classical subject matter, including the transition and stability theory in Chapter 9, would be a useful addition to the literature of any undergraduate or graduate student; the computational sections contain little in terms of fundamentals of numerics but, accepting that useful computational results are the focus, results are presented for several applications that would be of interest to many aerodynamicists.   (The Aeronautical Journal, 3 February 2015)