Theoretical and Computational Aerodynamics

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).

• Series Preface xvPreface xviiAcknowledgements xxi1 Introduction to Aerodynamics and Atmosphere 11.1 Motivation and Scope of Aerodynamics 11.2 Conservation Principles 41.2.1 Conservation Laws and Reynolds Transport Theorem (RTT) 41.2.2 Application of RTT: Conservation of Linear Momentum 61.3 Origin of Aerodynamic Forces 61.3.1 Momentum Integral Theory: Real Fluid Flow 81.4 Flow in Accelerating Control Volumes: Application of RTT 91.5 Atmosphere and Its Role in Aerodynamics 111.5.1 Von Kármán Line 111.5.2 Structure of Atmosphere 111.5.3 Armstrong Line or Limit 121.5.4 International Standard Atmosphere (ISA) and Other Atmospheric Details 131.5.5 Property Variations in Troposphere and Stratosphere 151.6 Static Stability of Atmosphere 17Bibliography 202 Basic Equations of Motion 212.1 Introduction 212.1.1 Compressibility of Fluid Flow 222.2 Conservation Principles 232.2.1 Flow Description Method: Eulerian and Lagrangian Approaches 232.2.2 The Continuity Equation: Mass Conservation 242.3 Conservation of Linear Momentum: Integral Form 252.4 Conservation of Linear Momentum: Differential Form 262.4.1 General Stress System in a Deformable Body 262.5 Strain Rate of Fluid Element in Flows 282.5.1 Kinematic Interpretation of Strain Tensor 292.6 Relation between Stress and Rate of Strain Tensors in Fluid Flow 322.7 Circulation and Rotationality in Flows 352.8 Irrotational Flows and Velocity Potential 362.9 Stream Function and Vector Potential 372.10 Governing Equation for Irrotational Flows 382.11 Kelvin’s Theorem and Irrotationality 402.12 Bernoulli’s Equation: Relation of Pressure and Velocity 412.13 Applications of Bernoulli’s Equation: Air Speed Indicator 422.13.1 Aircraft Speed Measurement 432.13.2 The Pressure Coefficient 442.13.3 Compressibility Correction for Air Speed Indicator 442.14 Viscous Effects and Boundary Layers 462.15 Thermodynamics and Reynolds Transport Theorem 472.16 Reynolds Transport Theorem 482.17 The Energy Equation 492.17.1 The Steady Flow Energy Equation 512.18 Energy Conservation Equation 522.19 Alternate Forms of Energy Equation 542.20 The Energy Equation in Conservation Form 552.21 Strong Conservation and Weak Conservation Forms 552.22 Second Law of Thermodynamics and Entropy 562.23 Propagation of Sound and Mach Number 602.24 One-Dimensional Steady Flow 612.25 Normal Shock Relation for Steady Flow 622.26 Rankine--Hugoniot Relation 642.27 Prandtl or Meyer Relation 652.28 Oblique ShockWaves 692.29 Weak Oblique Shock 712.30 Expansion of Supersonic Flows 74Bibliography 763 Theoretical Aerodynamics of Potential Flows 773.1 Introduction 773.2 Preliminaries of Complex Analysis for 2D Irrotational Flows: Cauchy--Riemann Relations 783.2.1 Cauchy’s Residue Theorem 813.2.2 Complex Potential and Complex Velocity 813.3 Elementary Singularities in Fluid Flows 813.3.1 Superposing Solutions of Irrotational Flows 833.4 Blasius’ Theorem: Forces and Moment for Potential Flows 903.4.1 Force Acting on a Vortex in a Uniform Flow 923.4.2 Flow Past a Translating and Rotating Cylinder: Lift Generation Mechanism 943.4.3 Prandtl’s Limit on Maximum Circulation and its Violation 973.4.4 Pressure Distribution on Spinning and Translating Cylinder 983.5 Method of Images 993.6 Conformal Mapping: Use of Cauchy--Riemann Relation 1013.6.1 Laplacian in the Transformed Plane 1023.6.2 Relation between Complex Velocity in Two Planes 1043.6.3 Application of Conformal Transformation 1043.7 Lift Created by Jukowski Airfoil 1113.7.1 Kutta Condition and Circulation Generation 1133.7.2 Lift on Jukowski Airfoil 1143.7.3 Velocity and Pressure Distribution on Jukowski Airfoil 1163.8 Thin Airfoil Theory 1163.8.1 Thin Symmetric Flat Plate Airfoil 1193.8.2 Aerodynamic Centre and Centre of Pressure 1223.8.3 The Circular Arc Airfoil 1243.9 General Thin Airfoil Theory 1293.10 Theodorsen Condition for General Thin Airfoil Theory 134Bibliography 1354 Finite Wing Theory 1374.1 Introduction 1374.2 Fundamental Laws of Vortex Motion 1374.3 Helmholtz’s Theorems of Vortex Motion 1384.4 The Bound Vortex Element 1404.5 Starting Vortex Element 1404.6 Trailing Vortex Element 1414.7 Horse Shoe Vortex 1424.8 The Biot-Savart Law 1424.8.1 Biot-Savart Law for Simplified Cases 1444.9 Theory for a Finite Wing 1464.9.1 Relation between Spanwise Loading and Trailing Vortices 1464.10 Consequence of Downwash: Induced Drag 1474.11 Simple Symmetric Loading: Elliptic Distribution 1494.11.1 Induced Drag for Elliptic Loading 1514.11.2 Modified Elliptic Load Distribution 1524.11.3 The Downwash for Modified Elliptic Loading 1534.12 General Loading on a Wing 1544.12.1 Downwash for General Loading 1554.12.2 Induced Drag on a Finite Wing for General Loading 1564.12.3 Load Distribution for Minimum Drag 1574.13 Asymmetric Loading: Rolling and Yawing Moment 1574.13.1 Rolling Moment (𝐿𝑅) 1574.13.2 Yawing Moment (N) 1594.13.3 Effect of Aspect Ratio on Lift Curve Slope 1594.14 Simplified Horse Shoe Vortex 1614.15 Applications of Simplified Horse Shoe Vortex System 1624.15.1 Influence of Downwash on Tailplane 1624.15.2 Formation-flight of Birds 1634.15.3 Wing-in-Ground Effect 1654.16 Prandtl’s Lifting Line Equation or the Monoplane Equation 167Bibliography 1695 Panel Methods 1715.1 Introduction 1715.2 Line Source Distribution 1725.2.1 Perturbation Velocity Components due to Source Distribution 1745.3 Panel Method due to Hess and Smith 1765.3.1 Calculation of Influence Coefficients 1805.4 Some Typical Results 183Bibliography 1886 Lifting Surface, Slender Wing and Low Aspect Ratio Wing Theories 1896.1 Introduction 1896.2 Green’s Theorems and Their Applications to Potential Flows 1906.2.1 Reciprocal Theorem 1926.3 Irrotational External Flow Field due to a Lifting Surface 1926.3.1 Large Aspect Ratio Wings 1976.3.2 Wings of Small Aspect Ratio 1996.4 Slender Wing Theory 2016.5 Spanwise Loading 2056.6 Lift on Delta or Triangular Wing 2066.6.1 Low Aspect Ratio Wing Aerodynamics and Vortex Lift 2076.7 Vortex Breakdown 2146.7.1 Types of Vortex Breakdown 2166.8 Slender Body Theory 218Bibliography 2217 Boundary Layer Theory 2237.1 Introduction 2237.2 Regular and Singular Perturbation Problems in Fluid Flows 2247.3 Boundary Layer Equations 2257.3.1 Conservation of Mass 2267.3.2 The 𝑥-Momentum Equation 2267.3.3 The 𝑦-Momentum Equation 2277.3.4 Use of Boundary Layer Equations 2297.4 Boundary Layer Thicknesses 2307.4.1 Boundary Layer Displacement Thickness 2317.4.2 Boundary Layer Momentum Thickness 2327.5 Momentum Integral Equation 2337.6 Validity of Boundary Layer Equation and Separation 2357.7 Solution of Boundary Layer Equation 2377.8 Similarity Analysis 2387.8.1 Zero Pressure Gradient Boundary Layer or Blasius Profile 2437.8.2 Stagnation Point or the Hiemenz Flow 2447.8.3 Flat Plate Wake at Zero Angle of Attack 2457.8.4 Two-dimensional Laminar Jet 2477.8.5 Laminar Mixing Layer 2507.9 Use of Boundary Layer Equation in Aerodynamics 2527.9.1 Differential Formulation of Boundary Layer Equation 2537.9.2 Use of Momentum Integral Equation 2547.9.3 Pohlhausen’s Method 2547.9.4 Thwaite’s Method 257Bibliography 2588 Computational Aerodynamics 2598.1 Introduction 2598.2 A Model Dynamical Equation 2608.3 Space--Time Resolution of Flows 2638.3.1 Spatial Scales in Turbulent Flows and Direct Numerical Simulation 2648.3.2 Computing Unsteady Flows: Dispersion Relation Preserving (DRP) Methods 2658.3.3 Spectral or Numerical Amplification Factor 2668.4 An Improved Orthogonal Grid Generation Method for Aerofoil 2758.5 Orthogonal Grid Generation 2798.5.1 Grid Generation Algorithm 2818.6 Orthogonal Grid Generation for an Aerofoil with Roughness Elements 2848.7 Solution of Navier--Stokes Equation for Flow Past AG24 Aerofoil 2878.7.1 Grid Smoothness vs Deviation from Orthogonality 290Bibliography 2919 Instability and Transition in Aerodynamics 2959.1 Introduction 2959.2 Temporal and Spatial Instability 2989.3 Parallel Flow Approximation and Inviscid Instability Theorems 2999.3.1 Inviscid Instability Mechanism 3009.4 Viscous Instability of Parallel Flows 3019.4.1 Temporal and Spatial Amplification of Disturbances 3039.5 Instability Analysis from the Solution of the Orr--Sommerfeld Equation 3049.5.1 Local and Total Amplification of Disturbances 3069.5.2 Effects of the Mean Flow Pressure Gradient 3089.5.3 Transition Prediction Based on Stability Calculation: 𝑒𝑁 Method 3129.5.4 Effects of FST 3149.5.5 Distinction between Controlled and Uncontrolled Excitations 3159.6 Transition in Three-Dimensional Flows 3189.7 Infinite Swept Wing Flow 3209.8 Attachment Line Flow 3219.9 Boundary Layer Equations in the Transformed Plane 3229.10 Simplification of Boundary Layer Equations in the Transformed Plane 3249.11 Instability of Three-Dimensional Flows 3259.11.1 Effects of Sweep-back and Cross Flow Instability 3269.12 Linear Viscous Stability Theory for Three-Dimensional Flows 3289.12.1 Temporal Instability of Three-dimensional Flows 3299.12.2 Spatial Instability of Three-dimensional Flows 3309.13 Experimental Evidence of Instability on Swept Wings 3329.14 Infinite Swept Wing Boundary Layer 3349.15 Stability of the Falkner--Skan--Cooke Profile 3379.16 Stationary Waves over Swept Geometries 3409.17 Empirical Transition Prediction Method for Three-Dimensional Flows 3409.17.1 Streamwise Transition Criterion 3419.17.2 Cross Flow Transition Criteria 3419.17.3 Leading Edge Contamination Criterion 343Bibliography 34310 Drag Reduction: Analysis and Design of Airfoils 34710.1 Introduction 34710.2 Laminar Flow Airfoils 35010.2.1 The Drag Bucket of Six-Digit Series Aerofoils 35210.2.2 Profiling Modern Laminar Flow Aerofoils 35310.3 Pressure Recovery of Some Low Drag Airfoils 35810.4 Flap Operation of Airfoils for NLF 36110.5 Effects of Roughness and Fixing Transition 36210.6 Effects of Vortex Generator or Boundary Layer Re-Energizer 36410.7 Section Characteristics of Various Profiles 36410.8 A High Speed NLF Aerofoil 36510.9 Direct Simulation of Bypass Transitional Flow Past an Airfoil 36910.9.1 Governing Equations and Formulation 37010.9.2 Results and Discussion 371Bibliography 37811 Direct Numerical Simulation of 2D Transonic Flows around Airfoils 38111.1 Introduction 38111.2 Governing Equations and Boundary Conditions 38211.3 Numerical Procedure 38411.4 Some Typical Results 38711.4.1 Validation of Methodologies for Compressible Flow Calculations and Shock Capturing 38711.4.2 Computing Strong Shock Cases 39611.4.3 Unsteadiness of Compressible Flows 39611.4.4 Creation of Rotational Effects 39611.4.5 Strong Shock and Entropy Gradient 40111.4.6 Lift and Drag Calculation 404Bibliography 40612 Low Reynolds Number Aerodynamics 40912.1 Introduction 40912.2 Micro-air Vehicle Aerodynamics 41212.3 Governing Equations in Inertial and Noninertial Frames 41312.3.1 Pressure Solver 41512.3.2 Proof of Equation (12.17) 41612.3.3 Distinction between Low and High Reynolds Number Flows 41812.3.4 Validation Studies of Computations 42012.4 Flow Past an AG24 Airfoil at Low Reynolds Numbers 425Bibliography 44213 High Lift Devices and Flow Control 44513.1 Introduction 44513.1.1 High Lift Configuration 44613.2 Passive Devices: Multi-Element Airfoils with Slats and Flaps 44913.2.1 Optimization of Flap Placement and Settings 45013.2.2 Aerodynamic Data of GA(W)-1 Airfoil Fitted with Fowler Flap 45313.2.3 Physical Explanation of Multi-element Aerofoil Operation 45513.2.4 Vortex Generator 45713.2.5 Induced Drag and Its Alleviation 46113.2.6 Theoretical Analysis of Induced Drag 46313.2.7 Fuselage Drag Reduction 46413.2.8 Instability of Flow over Nacelle 46513.3 Flow Control by Plasma Actuation: High Lift Device and Drag Reduction 46513.3.1 Control of Bypass Transitional Flow Past an Aerofoil by Plasma Actuation 46613.4 Governing Equations for Plasma 46813.4.1 Suzen et al.’s Model 47013.4.2 Orlov’s Model 47113.4.3 Spatio-temporal Lumped-element Circuit Model 47213.4.4 Algorithm for Calculating Body Force 47413.4.5 Lemire and Vo’s Model 47413.5 Governing Fluid Dynamic Equations 47513.6 Results and Discussions 476Bibliography 484Index 487

“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)