Theoretical and Computational Aerodynamics
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"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)More details
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Content
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 RTT 9
1.5 Atmosphere and Its Role in Aerodynamics 11
1.5.1 Von Kármán 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 Other Atmospheric 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 Lagrangian Approaches 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 Fluid Flow 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 and Velocity 41
2.13 Applications of Bernoulli's Equation: Air Speed Indicator 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 Irrotational Flows: 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 Potential Flows 90
3.4.1 Force Acting on a Vortex in a Uniform Flow 92
3.4.2 Flow Past a Translating and Rotating Cylinder: Lift Generation Mechanism 94
3.4.3 Prandtl's Limit on Maximum Circulation and its Violation 97
3.4.4 Pressure Distribution on Spinning and Translating Cylinder 98
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 Airfoil 116
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 Theory 134
Bibliography 135
4 Finite Wing 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 Vortices 146
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 Monoplane Equation 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 Source Distribution 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 Ratio Wing Theories 189
6.1 Introduction 189
6.2 Green's Theorems and Their Applications to Potential Flows 190
6.2.1 Reciprocal Theorem 192
6.3 Irrotational External Flow Field due to a Lifting Surface 192
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 Triangular Wing 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 Flows 224
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 Profile 243
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 Equation 253
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 Numerical Simulation 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 Aerofoil 275
8.5 Orthogonal Grid Generation 279
8.5.1 Grid Generation Algorithm 281
8.6 Orthogonal Grid Generation for an Aerofoil with Roughness Elements 284
8.7 Solution of Navier--Stokes Equation for Flow Past AG24 Aerofoil 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 Instability Theorems 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 the Orr--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 Uncontrolled Excitations 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 the Transformed 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 Flows 328
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 Stationary Waves over Swept Geometries 340
9.17 Empirical Transition Prediction Method for Three-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 Airfoils 347
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-Energizer 364
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 an Airfoil 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 around Airfoils 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 Flow Calculations 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 Frames 413
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 Flows 418
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 and Flaps 449
13.2.1 Optimization of Flap Placement and Settings 450
13.2.2 Aerodynamic Data of GA(W)-1 Airfoil Fitted with Fowler Flap 453
13.2.3 Physical Explanation of Multi-element Aerofoil Operation 455
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 Drag Reduction 465
13.3.1 Control of Bypass Transitional Flow Past an Aerofoil by Plasma 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
1
Introduction to Aerodynamics and Atmosphere
1.1 Motivation and Scope of Aerodynamics
Study of aerodynamics involves the ability to predict aerodynamic forces and moments acting on an airborne vehicle. However, it all began with the search for the quintessential shape that will make anything airborne in a sustained manner. Historically, the search for human flight began with lighter than air vehicles, now known variously as aerostats. While airships, blimps and/or dirigibles are still in use, the original search for vehicles heavier than air was the main attraction for human flight. In our quest for flight, we always wanted to emulate the birds, but even today this appears unattainable with present-day technologies. A bird flies in which the flapping wing performs the dual role of propulsive and aerodynamic devices. In man-made devices the propulsive device produces power or thrust for the vehicle to overcome resistance, while the aerodynamic device creates the necessary force to keep the body aloft in a dynamical equilibrium. Any device that imitates the flight of birds is known as an ornithopter and it is to the genius of Sir George Cayley (1773-1857) mankind owes a debt for the conventional aircraft shape and design. In a marked departure, he suggested that such propulsive devices did not exist and that it was more important to understand the analysis and design of aerodynamic aspect of the vehicle first. To do so, he advocated the study of powerless flight of aeronautical shapes of interest in a stable manner. This way of compartmentalizing the different aspects of flight into aerodynamics, propulsion, structures, performance, stability and control is now one essential component in the study of the discipline and was started by the need to understand the basics of flight as pioneered by Cayley. For an historic account of the development of flight and aerodynamics in particular, readers are advised to study it in Anderson (1997).
In this book, the sole motivation would be to study the aerodynamic forces and moments acting on an aircraft. Before we discuss the motivation of studying aerodynamics further, we should familiarize ourselves with different parts of an aircraft and their roles in flight. In Figure 1.1, we show the different main parts of a small aircraft as viewed externally. The external shape of an aircraft is central to the study of aerodynamics, flight stability and controls. As an aircraft is heavier than the ambient displaced air, various surface stresses acting on different parts of the aircraft should be such that there must be a resultant force acting which will sustain the weight of the aircraft, when the aircraft is flying level and steady. This resultant force acting over the aircraft sustaining the weight is called the lift force. In a conventional aircraft, lift force is created by the flow over the wing. This is often modelled by the normal stress or the static pressure acting on the wing surface and is obtained by considering an ideal flow over the wing. When and how this is possible, will be the recurring theme of this book. Such normal stress or pressure distribution acting over the wing also creates a moment acting about a general point along the chord of the wing section and is a concomitant liability of producing the lift force. This is counteracted upon by the smaller amount of lift created on the horizontal stabilizer. One realizes that the main purpose of an aerospace vehicle is to carry payload and this is the reason for having a fuselage. Such a tubular shape of the fuselage, along with different aerodynamic and control surfaces, also creates resistance or drag for the aircraft. The required thrust to overcome drag is provided by the propeller engine located at the nose of the aircraft. The horizontal and vertical stabilizers are required for maintaining the stability of the aircraft for different flight regime.
Figure 1.1 An external view of a small propeller aircraft with tricycle landing gear, identifying different parts of aerodynamic and control surfaces
One of the rudimentary aspects of flight operation is the cruise configuration, in which the aircraft flies level and steady at constant altitude. In Figure 1.2, we show a conventional aircraft flying level and steady in the longitudinal plane. External forces are shown to be in equilibrium, i.e. the weight of the aircraft is balanced by the vertical component of the aerodynamic forces acting on the aircraft called the lift and denoted by L. Most of it is created by the aircraft wing. The horizontal component of the aerodynamic force is termed the drag and denoted by D. Almost half the drag of an aircraft is created by the fuselage and a large portion of it is caused by the creation of lift itself. The total drag is overcome by the thrust (indicated by T), created by the power plant. Thrust is also a form of aerodynamic force created by internal flows through the engine where energy is added by burning fuel and ejecting a well-directed stream of fluid which by reaction creates the thrust. But this is outside the scope of this book and will not be discussed further.
Figure 1.2 A conventional aircraft flying level and steady in the longitudinal plane. Forces and moments acting on the aircraft are shown in equilibrium; the pitching moment is balanced by the lift acting on the tailplane
Note that the drawn free body diagram in Figure 1.2, is an idealization as the forces are not collinear and also the forces and moment created by the stabilizer are not shown in this diagram. At the outset, we want to state the directions of the aerodynamic force as consisting of: the drag force, which is always along the oncoming flow direction, and lift, which is perpendicular to it. For a general motion of the aircraft, additionally another component of aerodynamic force will be experienced, which is perpendicular to both lift and drag and is termed as the side force. The moment experienced by the aircraft in the longitudinal plane has been indicated by the pitching moment. For general motion of the rigid aircraft will give rise to two more components of moment termed as the rolling (about the fuselage axis) and yawing (about the vertical axis of the aircraft) moments. In Figure 1.3, we show a sketch of all three forces and three moments relevant for a rigid aircraft. The reader is made aware of the fact that the aircraft is hardly ever a rigid body and in general the problem constitutes an analysis for very large degrees of freedom. (The static tip deflection of Boeing 747 in cruise from the static position in ground is more than 12 feet!) This constitutes the interesting field of study called aero-elasticity where simultaneous consideration of fluid dynamics and structural dynamics are taken into account. When the design of aircraft is complicated by the fact that the stability and control of the aircraft depends and directly interacts with fluid and structural dynamics of the aircraft, then one is forced to consider aero-servo-elasticity. However creating additional complications by various forces interacting is important for many applications, but the primary goal of this book is to study the aerodynamics of simple lifting surfaces. The moot point is: How have such aerodynamic surfaces evolved to their present state of development?
Figure 1.3 Axes and coordinate system used for any rigid aircraft. Forces and moments acting on the aircraft are shown with conventional usage
1.2 Conservation Principles
We start this discussion with the innocuous question: How does an aerodynamic surface creates lift? Answer to this is related to the simple observation that if a bound vortex induces a circulatory motion in a uniform steady flow by an aerodynamic surface, the resultant superposition of the two yields the flow pattern seen around a lifting airfoil with the flow leaving the trailing edge with a small downward component of velocity. But will any bound vortex do, as in the case of a rotating cylinder in a uniform flow experiencing the Robins-Magnus effect, as explained in White (2008)? It will simply not work due to the fact that, apart from creating lift via rotation, such a body (also referred to as a bluff body) will experience large value of drag. Any aerodynamic surface, to perform efficiently, must provide not only lift but must also yield a very high value of lift to drag ratio. This ratio is also called the aerodynamic efficiency. Instead of getting into technical details of aerofoil aerodynamics, let us first understand what is expected from an aerodynamic surface from first principle of conservation laws in fluid dynamics. For low-speed applications, these would be nothing more than conservation of mass and momentum.
1.2.1 Conservation Laws and Reynolds Transport Theorem (RTT)
All conservation laws used in mechanics are nothing but interaction between system and surrounding, which are separated by real or imaginary boundaries. If the system mass is m, then for a control mass system the conservation of mass implies
(1.1)If the system is moving with a translational velocity, , then the conservation of linear momentum is given by
(1.2)where is the vector sum of all the applied forces acting on the system. In fluid mechanics or aerodynamics, above control mass analysis is often of limited value, as we do not track individual fluid particles. Instead we focus upon a fixed space in the flow domain and this is the rationale for control volume analysis. We will make use of the Reynolds transport theorem...
